A few example designs and data sets for this module are available in the R package apts.doe, which can be installed from GitHub

library(devtools)
install_github("statsdavew/apts.doe", quiet = T)
library(apts.doe)

References will be provided throughout but some good general purpose texts are

• Atkinson, Donev and Tobias (2007). Optimum Experimental Design, with SAS. OUP
• Wu and Hamada (2009). Experiments: Planning, Analysis, and Parameter Design Optimization (2nd ed.). Wiley.
• Morris (2011). Design of Experiments: An Introduction based on Linear Models. Chapman and Hall/CRC Press.
• Santner, Williams and Notz (2019). The Design and Analysis of Computer Experiments (2nd ed.). Springer.

These notes and other resources can be found at https://statsdavew.github.io/apts.doe/

• Observational studies
• Sample surveys
• Designed experiments

Definition: An experiment is a procedure whereby controllable factors, or features, of a system or process are deliberately varied in order to understand the impact of these changes on one or more measurable responses.

See Luca and Bazerman (2020) for further history, annecdotes and examples, especially from psychology and technology.

Why do we experiment?

• key to the scientific method
  (hypothesis – experiment – observe – infer – conclude)

• potential to establish causality …

• … and to understand/improve complex systems depending on many factors

• comparison of treatments, factor screening, prediction, optimisation, …

Design of experiments: a statistical approach to the arrangement of the operational details of the experiment (eg sample size, specific experimental conditions investigated, …) so that the quality of the answers to be derived from the data is as high as possible.

1. Multi-factor experiment in pharmaceutical development.

Key to developing new medicines is the identification of optimal and robust process conditions (e.g. settings of temperature, pressure etc.) at which the active pharmaceutical ingredient should be synthesized.

[Somewhat confusinging, the FDA refer to this as identification of a “design space”.]

An important step in is this methodology is a robustness experiment to assess the sensitivity of identified conditions to changes in all (or at least very many) controllable factors.

While developing a new melanoma drug, GlaxoSmithKline performed an experiment to investigate sensitivity to 20 factors. Their experimental budget allowed only 10 individual experiments (runs) to be performed.

2. Computer experiments to optimise ride performance in luxury cars

Suspension settings can be used to improve the ride performance in cars. Optimising settings across many different car models would take many hundreds of hours of testing, so computer simulations are used.

Jaguar-Land Rover wanted to find suspension settings robust across different car models using a computer experiment (KTN workshop).

3. Optimal design to calibrate a physical model.

Physical (mechanistic, mathematical, …) models are used in many scientific fields. Typically, they are derived from fundamental understanding of the physics, chemistry, biology …

Most commonly, these models are solutions to differential equations. The models usually contain unknown parameters that should be estimated from experimental data.

Biologists at Southampton were studying the transfer of amino acids between mother and baby through the placenta. They could control the times at which observations were taken and the initial concentrations of amino acids (see Overstall, Woods, and Parker 2019).

Consider an experiment to compare two treatments (eg drugs, diets, fertilisers, \(\ldots\)).

We have \(n\) subjects (eg people, mice, plots of land, \(\ldots\)), each of which can be assigned to one of the two treatments.

A response (eg protein measurement, weight, yield, \(\ldots\)) is then measured from each subject.

Question: How should the two treatments be assigned to the subjects to gain the most precise inference about the difference in expected response from the two treatments.

Assume a linear model for the response \[ y_i = \beta_0 + \beta_1 x_i + \varepsilon_i\,,\qquad i=1,\ldots,n\,, \] with \(\varepsilon_i\sim N(0, \sigma^2)\) independently, \(\beta_0,\beta_1\) unknown parameters and \[ x_i = \left\{ \begin{array}{cc} -1 & \mbox{if treatment 1 is applied to subject $i$}\,, \\ +1 & \mbox{if treatment 2 is applied to subject $i$} \end{array} \right. \] The difference in expected response between treatment 1 and 2 is \[ E(y_i\,|\, x_i = +1) - E(y_i\,|\, x_i = -1) = \beta_0 + \beta_1 - \beta_0 + \beta_1 = 2\beta_1 \] So we need the most precise possible estimator of \(\beta_1\)

Both \(\beta_0\) and \(\beta_1\) can be estimated using least squares (or equivalently maximum likelihood).

Writing \[ \boldsymbol{y}= X\boldsymbol{\beta}+ \boldsymbol{\varepsilon}\,, \] we obtain estimators \[ \hat{\boldsymbol{\beta}} = \left(X^\mathrm{T}X\right)^{-1}X^\mathrm{T}\boldsymbol{y} \] with \[ \mbox{Var}(\hat{\boldsymbol{\beta}}) = \left(X^\mathrm{T}X\right)^{-1}\sigma^2 \] In this simple example, we are interesting in estimating \(\beta_1\), and we have \[ \begin{split} \mbox{Var}(\hat{\beta_1}) & = \frac{n\sigma^2}{n\sum x_i^2 - \left(\sum x_i\right)^2}\\ & = \frac{n\sigma^2}{n^2 - \left(\sum x_i\right)^2} \end{split} \]

Hence, we need to pick \(x_1,\ldots,x_n\) to minimise \(\left(\sum x_i\right)^2 = (n_1 - n_2)^2\)

• denote as \(n_1\) the number of subjects assigned to treatment 1, and \(n_2\) the number assigned to treatment 2, with \(n_1+n_2 = n\)
• it is obvious that \(\sum x_i = 0\) if and only if \(n_1 = n_2\)

Assuming \(n\) is even, the “optimal design” has \(n_1 = n_2 = n/2\)

For \(n\) odd, let \(n_1 = \frac{n+1}{2}\) and \(n_2 = \frac{n-1}{2}\)

We can assess a designs, labelled \(\xi\), via its efficiency relative to the optimal design \(\xi^\star\): \[ \mbox{Eff($\xi$)} = \frac{\mbox{Var}(\hat{\beta_1}\,|\,\xi^\star)}{\mbox{Var}(\hat{\beta_1}\,|\,\xi)} \]

n <- 50
eff <- function(n1) 1 - ((2 * n1 - n) / n)^2
curve(eff, from = 0, to = n, ylab = "Eff", xlab = expression(n[1]))

• Treatment – entities of scientific interest to be studied in the experiment
  eg varieties of crop, doses of a drug, combinations of temperature and pressure

• Unit – smallest subdivision of the experimental material such that two units may receive different treatments
  eg plots of land, subjects in a clinical trial, samples of reagent

• Run – application of a treatment to a unit

An initial step in fabricating integrated circuits is the growth of an epitaxial layer on polished silicon wafers via chemical deposition (see Wu and Hamada 2009, p155).

\[ y_{ij} = \mu + \tau_i + \varepsilon_{ij}\,,\qquad i=1,\ldots,t;\,j=1,\ldots,n_i\,, \] where

• \(y_{ij}\) : measured response from the \(j\)th unit to which treatment \(i\) has been applied

• \(\mu\) : overall mean response (often labelled \(\beta_0\))

• \(\tau_i\) : treatment effect (\(\tau_i\) is the expected difference in response from the overall mean after application of the \(i\)th treatment)

• \(\varepsilon_{ij}\) : random deviation from the expected response [typically \(\sim N(0,\sigma^2)\)]

The aims of the experiment are achieved by estimating comparisons between the treatment effects, \(\tau_k - \tau_l\).

Experimental precision and accuracy are largely obtained through control and comparison.

Three key model assumptions are:

• additivity (response = treatment effect + unit effect)
• constancy of treatment effects (treatment effect does not depend on the unit to which it is applied)
• no interference between units (the effect of a treatment applied to unit \(j\) does not depend on the treatment applied to any other unit)

See Dasgupta, Pillai, and Rubin (2015) for discussion of these assumptions for factorial experiments

Stratification (blocking)

• account for systematic differences between batches of experimental units by arranging them in homogeneous sets (blocks)
  • if the same treatment was applied to all units, within-block variation in the response would be much less than between-block
  • compare treatments within the same block and hence eliminate block effects

Replication

• the application of each treatment to multiple experimental units
  • provides an estimate of experimental error against which to judge treatment differences
  • reduces the variance of the estimators of treatment differences

Randomisation

• we randomise features such as the allocation of units to treatments, the order in which treatments are applied, …
  • protects against lurking (uncontrolled) variables (model-robust) and subjectively in the allocation of treatments to units

Randomisation is perhaps the key principle in the design of experiments

• it protects against model misspecification (bias), and hence allows causality to be established
  • a clear difference between treatments can only be an accident of the randomisation or a consequence of the treatments
• unbiased estimation of \(\tau\) and \(\sigma^2\), even if the errors are not normally distributed
• exact tests for differences between treatment effects are available (Basu 1980)

Without randomisation, unobserved confounders (\(U\)) can induce a dependency between
the response (\(Y\)) and treatment (\(T\)) cf Cox and Reid (2000), p.35

With randomisation, unobserved confounders (\(U\)) are independent of the treatment (\(T\)). Marginalisation over \(U\) does not induce an edge between \(T\) and \(Y\) cf Cox and Reid (2000), p.35

Fabrication of integrated circuits (Wu and Hamada 2009, p155)

Treatment

• combination of settings of the factors
  • A : rotation method (\(x_1\))
  • B : nozzle position (\(x_2\))
  • C : deposition temperature (\(x_3\))
  • D : deposition time (\(x_4\))

Assume each factor has two-levels, coded -1 and +1

Each factor has two levels \(x_k = \pm 1,\, k=1,\ldots,4\)

A treatment is then defined as a combination of four values of \(-1, +1\)

• eg \(x_1 = -1, x_2 = -1, x_3 = +1, x_4 = -1\)
• specifies a setting of the process

Assume each treatment effect is determined by a regression model in the four factors, eg \[ \tau(\boldsymbol{x}) = \sum_{i=1}^4\beta_ix_i + \sum_{j=1}^4\sum_{i>j}^4\beta_{ij}x_ix_j \]

with(cirfab, cirfab[order(x1, x2, x3, x4), ])

##    x1 x2 x3 x4     ybar
## 2  -1 -1 -1 -1 13.58983
## 1  -1 -1 -1  1 14.59000
## 4  -1 -1  1 -1 14.04983
## 3  -1 -1  1  1 14.24000
## 6  -1  1 -1 -1 13.94000
## 5  -1  1 -1  1 14.65000
## 8  -1  1  1 -1 14.14017
## 7  -1  1  1  1 14.40000
## 10  1 -1 -1 -1 13.72000
## 9   1 -1 -1  1 14.67000
## 12  1 -1  1 -1 13.90000
## 11  1 -1  1  1 13.84017
## 14  1  1 -1 -1 13.87983
## 13  1  1 -1  1 14.56000
## 16  1  1  1 -1 14.11017
## 15  1  1  1  1 14.30000

• treatments in standard order

• \(\bar{y}\) - average response from the six wafers

\[ \boldsymbol{Y} = X\boldsymbol{\beta} + \boldsymbol{\varepsilon}\,,\qquad \boldsymbol{\varepsilon}\sim N(\boldsymbol{0}, \sigma^2I)\,,\qquad \hat{\boldsymbol{\beta}} = \left(X^\mathrm{T}X\right)^{-1}X^\mathrm{T}\boldsymbol{Y} \]

• model matrix \(X\) has columns corresponding to intercept, linear and cross-product terms

• information matrix \(X^\mathrm{T}X = nI\)

• regression coefficients are estimated by independent contrasts in the data

cirfab.lm <- lm(ybar ~ (.) ^ 2, data = cirfab)
coef(cirfab.lm)

##  (Intercept)           x1           x2           x3           x4        x1:x2 
## 14.161250000 -0.038729167  0.086270833 -0.038708333  0.245020833  0.003708333 
##        x1:x3        x1:x4        x2:x3        x2:x4        x3:x4 
## -0.046229167 -0.025000000  0.028770833 -0.015041667 -0.172520833

Main effect of \(x_k\): \[ [\text{Avg. response when $x_k = 1$}]\, -\, [\text{Avg. response when $x_k = -1$}] \]

Interaction between \(x_j\) and \(x_k\): \[ [\text{Avg. response when $x_jx_k= 1$}]\, -\, [\text{Avg. response when $x_jx_k = -1$}] \]

Higher-order interactions defined similarly

Assuming -1,+1 coding, there is a straightforward relationship between factorial effects and regression coefficients

• main effect of \(x_k\) is equal to \(2\beta_k\)
• interaction between \(x_j\) and \(x_k\) is equal to \(2\beta_{jk}\)

Using the effects package:

library(effects)
plot(Effect("x1", cirfab.lm), main = "", rug = F, ylim = c(13.5, 14.5), aspect = 1)
plot(Effect("x2", cirfab.lm), main = "", rug = F, ylim = c(13.5, 14.5), aspect = 1)
plot(Effect("x3", cirfab.lm), main = "", rug = F, ylim = c(13.5, 14.5), aspect = 1)
plot(Effect("x4", cirfab.lm), main = "", rug = F, ylim = c(13.5, 14.5), aspect = 1)

plot(Effect(c("x3", "x4"), cirfab.lm, xlevels = 2), main = "", rug = F, ylim = c(13.5, 15), 
     x.var = "x4")

\(X^\mathrm{T}X = nI \Rightarrow \hat{\boldsymbol{\beta}}\) are independently normally distributed with equal variance

Hence, we can treat the identification of important effects (ie large \(\beta\)) as an outlier identification problem

• plot (absolute) ordered factorial effects against (absolute) quantiles from a standard normal
• outlying effects are identified as important

Cuthbert (1959)

Using the FrF2 package

library(FrF2)
par(pty = "s", mar = c(8, 4, 1, 2))
DanielPlot(cirfab.lm, main = "", datax = F, half = T)

An unreplicated factorial design provides no model-independent estimate of \(\sigma^2\) (Gilmour and Trinca 2012)

• any unsaturated model does provide an estimate, but it may be biased by ignored (significant) model terms
• this is one reason why graphical (or associated) analysis methods are popular

Replication also increases the power of the design

• common to replicate a centre point
• allows a portmanteau test of curvature

Refit a linear model to the fabricated circuits experiment excluding interactions.

cirfab2.lm <- lm(ybar ~ (.), data = cirfab)
coef(cirfab.lm)

##  (Intercept)           x1           x2           x3           x4        x1:x2 
## 14.161250000 -0.038729167  0.086270833 -0.038708333  0.245020833  0.003708333 
##        x1:x3        x1:x4        x2:x3        x2:x4        x3:x4 
## -0.046229167 -0.025000000  0.028770833 -0.015041667 -0.172520833

coef(cirfab2.lm)

## (Intercept)          x1          x2          x3          x4 
## 14.16125000 -0.03872917  0.08627083 -0.03870833  0.24502083

cbind(sigma1 = sigma(cirfab.lm), df1 = df.residual(cirfab.lm),  
      sigma2 = sigma(cirfab2.lm), df2 = df.residual(cirfab2.lm))

##        sigma1 df1    sigma2 df2
## [1,] 0.137152   5 0.2396108  11

Effect sparsity

• the number of important effects in a factorial experiment is small relative to the total number of effects investigated (cf Box and Meyer 1986)

Effect hierarchy

• lower-order effects are more likely to be important than higher-order effects
• effects of the same order are equally likely to be important

Effect heredity

• interactions where at least one parent main effect is important are more likely to be important themselves

Wu and Hamada (2009), pp.172–172

Factorial designs can require a large number of runs for only a moderate number of factors (\(2^5 = 32\))

Resource constraints (eg cost) may mean not all \(2^m\) combinations can be run

Lots of degrees of freedom are devoted to estimating higher-order interactions

• eg in a \(2^5\) experiment, 16 degrees of freedom are used to estimate three-factor and higher-order interactions
• principles of effect hierarchy and sparsity suggest this may be wasteful

Need to trade-off what you need to estimate against the number of runs you can afford

Production of bacteriocin, a targetted antibacterial used as a natural food preservative (Morris 2011, p231)

Assume each factor has two-levels, coded -1 and +1

Find an \(n=8\) run design using FrF2

bact.design <- FrF2(8, 5, factor.names = paste0("x", 1:5), 
     generators = list(c(1, 3), c(2, 3)), randomize = F, alias.info = 3)
bact.design

##   x1 x2 x3 x4 x5
## 1 -1 -1 -1  1  1
## 2  1 -1 -1 -1  1
## 3 -1  1 -1  1 -1
## 4  1  1 -1 -1 -1
## 5 -1 -1  1 -1 -1
## 6  1 -1  1  1 -1
## 7 -1  1  1 -1  1
## 8  1  1  1  1  1
## class=design, type= FrF2.generators

• \(8\) = \(32/4\) = \(2^5/2^2\) = \(2^{5-2}\)
• we need a principled way of choosing one-quarter of the runs from the factorial design that leads to clarity in the analysis

Assuming the number of runs is a power of two, \(n = 2^{k-q}\), we can construct \(2^{k-q} -1\) orthogonal vectors (with inner product zero), spanned by \(k-q = \log_2(n)\) vectors

• construct the full factorial design for \(k-q\) factors
• assign the remaining \(q\) factors to interaction columns

model.matrix(~ (x1 + x2 + x3) ^ 3, bact.design[, 1:3])[, -1]

##   x11 x21 x31 x11:x21 x11:x31 x21:x31 x11:x21:x31
## 1  -1  -1  -1       1       1       1          -1
## 2   1  -1  -1      -1      -1       1           1
## 3  -1   1  -1      -1       1      -1           1
## 4   1   1  -1       1      -1      -1          -1
## 5  -1  -1   1       1      -1      -1           1
## 6   1  -1   1      -1       1      -1          -1
## 7  -1   1   1      -1      -1       1          -1
## 8   1   1   1       1       1       1           1

The design has been deliberately chosen so that

• \(x_4 = x_1x_3\)
• \(x_5 = x_2x_3\)

[\(x_1x_2\) is shorthand for the Hadamard (Schur or entry wise) product of two vectors, \(x_1\circ x_2\)]

What other consequences are there?

• \(x_4x_5 = x_1x_3x_2x_3 = x_1x_2x_3^2\)
• the product of any column with itself is the constant column (the identity)
• hence, \(x_4x_5 = x_1x_2\)

Now we can obtain the defining relation \(\ldots\)

• \(I = x_1x_3x_4 = x_2x_3x_5 = x_1x_2x_4x_5\)

\(\ldots\) and the complete aliasing scheme

• \(x_1 = x_3x_4 = x_1x_2x_3x_5 = x_2x_4x_5\)
• \(x_2 = x_1x_2x_3x_4 = x_3x_5 = x_1x_4x_5\)
• \(x_3 = x_1x_4 = x_2x_5 = x_1x_2x_3x_4x_5\)
• \(x_4 = x_1x_3 = x_2x_3x_4x_5 = x_1x_2x_5\)
• \(x_5 = x_1x_3x_4x_5 = x_2x_3 = x_1x_2x_4\)
• \(x_1x_2 = x_2x_3x_4 = x_1x_3x_5 = x_4x_5\)
• \(x_1x_5 = x_3x_4x_5 = x_1x_2x_3 = x_2x_4\)

FrF2 will summarise the aliasing amongst main effects and two- and three-factor interactions.

design.info(bact.design)$aliased 

## $legend
## [1] "A=x1" "B=x2" "C=x3" "D=x4" "E=x5"
## 
## $main
## [1] "A=CD=BDE" "B=CE=ADE" "C=AD=BE"  "D=AC=ABE" "E=BC=ABD"
## 
## $fi2
## [1] "AB=DE=ACE=BCD" "AE=BD=ABC=CDE"
## 
## $fi3
## [1] "ACD=BCE"

What is the consequence of this aliasing?

If more than one effect in each alias string is non-zero, the least squares estimators will be biased

• assumed model \(\boldsymbol{Y}= X_1\boldsymbol{\beta}_1 + \boldsymbol{\varepsilon}\)
• true model \(\boldsymbol{Y}= X_1\boldsymbol{\beta}_1 + X_2\boldsymbol{\beta}_2 + \boldsymbol{\varepsilon}\)

\[ \begin{split} E\left(\hat{\boldsymbol{\beta}}_1\right) & = \left(X_1^\mathrm{T}X_1\right)^{-1}X^\mathrm{T}_1E(\boldsymbol{Y}) \\ & = \left(X^\mathrm{T}_1X_1\right)^{-1}X_1^\mathrm{T}\left(X_1\boldsymbol{\beta}_1 + X_2\boldsymbol{\beta}_2\right) \\ & = \beta_1 + \left(X_1^\mathrm{T}X_1\right)^{-1}X_1^\mathrm{T}X_2\boldsymbol{\beta}_2 \\ & = \boldsymbol{\beta}_1 + A\boldsymbol{\beta}_2\\ \end{split} \]

\(A\) is the alias matrix

• if the columns of \(X_1\) and \(X_2\) are not orthogonal, \(\hat{\boldsymbol{\beta}}_1\) is biased

For the \(2^{5-2}\) example considering “main effects” and “two-factor interactions”:

• \(X_1\) is an \(8\times 8\) matrix with columns for the intercept, five linear and two product terms
• \(X_2\) is an \(8\times 8\) matrix with columns for the 8 remaining product terms

The transpose of the alias matrix is provided by the alias function.

ff.alias <- alias(y ~ (.)^2, data = data.frame(bact.design, y = vector(length = 8)))
t(ff.alias$Complete)[, order(rownames(ff.alias$Complete))]

##             x11:x31 x11:x41 x21:x31 x21:x41 x21:x51 x31:x41 x31:x51 x41:x51
## (Intercept) 0       0       0       0       0       0       0       0      
## x11         0       0       0       0       0       1       0       0      
## x21         0       0       0       0       0       0       1       0      
## x31         0       1       0       0       1       0       0       0      
## x41         1       0       0       0       0       0       0       0      
## x51         0       0       1       0       0       0       0       0      
## x11:x21     0       0       0       0       0       0       0       1      
## x11:x51     0       0       0       1       0       0       0       0

For a regular design, the matrix \(A\) will only have entries 0, \(\pm 1\) (no aliasing or complete aliasing)

These linear dependencies can be seen if we attempt to fit a linear model

bact.lm <- lm(yB ~ (x1 + x2 + x3 + x4 + x5)^2, data = bact)
summary(bact.lm)

## 
## Call:
## lm.default(formula = yB ~ (x1 + x2 + x3 + x4 + x5)^2, data = bact)
## 
## Residuals:
## ALL 8 residuals are 0: no residual degrees of freedom!
## 
## Coefficients: (8 not defined because of singularities)
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept)  4.42625        NaN     NaN      NaN
## x1           0.26625        NaN     NaN      NaN
## x2           0.24625        NaN     NaN      NaN
## x3          -0.22875        NaN     NaN      NaN
## x4          -1.11875        NaN     NaN      NaN
## x5          -0.66375        NaN     NaN      NaN
## x1:x2       -0.05375        NaN     NaN      NaN
## x1:x3             NA         NA      NA       NA
## x1:x4             NA         NA      NA       NA
## x1:x5       -0.13375        NaN     NaN      NaN
## x2:x3             NA         NA      NA       NA
## x2:x4             NA         NA      NA       NA
## x2:x5             NA         NA      NA       NA
## x3:x4             NA         NA      NA       NA
## x3:x5             NA         NA      NA       NA
## x4:x5             NA         NA      NA       NA
## 
## Residual standard error: NaN on 0 degrees of freedom
## Multiple R-squared:      1,  Adjusted R-squared:    NaN 
## F-statistic:   NaN on 7 and 0 DF,  p-value: NA

Typically, in a first experiment, fractional factorial designs are used in screening

• investigate which of many factors have a substantive effect on the response
• main effects and two-factor interactions
• centre points to check for curvature

At second and later stages, augment the design

• to resolve ambiguities due to the aliasing of factorial effects (“break the alias strings”)
• to allow estimation of curvature and prediction from a more complex model

Regular fractional factorial designs have the number of runs equal to a power of the number of levels

• eg \(2^{5-2}\), \(3^{3-1}\times 2\)
• this inflexibility in run sizes can be a problem in practical experiments

Non-regular designs can have any number of runs (usually with \(n>p\), the number of parameters to be estimated)

Often the clarity provided by a regular design is lost

• no defining relation or straightforward aliasing scheme
• partial aliasing and fractional entries in \(A\)

One approach to finding non-regular designs is via a design optimality criterion

Notation: let \(\xi = [\boldsymbol{x}_1,\ldots,\boldsymbol{x}_n]\) denote a design (choice of treatments and their replications)

Assuming the model \(\boldsymbol{Y}= X\boldsymbol{\beta}+ \boldsymbol{\varepsilon}\), with \(\boldsymbol{\varepsilon}\sim N(0, \sigma^2I_n)\), a \(D\)-optimal design maximises \[ \phi(\xi) = \mathrm{det}\left(X^\mathrm{T}X\right) \]

That is, a \(D\)-optimal design maximises the determinant of the (expected) Fisher information matrix

• equivalent to minimising the volume of the joint confidence ellipsoid for \(\boldsymbol{\beta}\)

Also useful to define a Bayesian version, with \(R\) a prior precision matrix \[ \phi_B(\xi) = \mathrm{det}\left(X^\mathrm{T}X + R\right) \] (See later)

\(D\)-optimal designs are model dependent

• if the model (ie the columns of \(X\)) changes, the optimal design may change
• model-robust design is an active area of research

\(D\)-optimality promotes orthogonality in the \(X\) matrix

• if there are sufficient runs, the \(D\)-optimal design will usually be orthogonal
• for particular models and choices of \(n\), regular fractional factorial designs are \(D\)-optimal

There are many other optimality criteria, tailored to other experimental goals

• prediction, model discrimination, space-filling, …

\(k=11\) factors in \(n=12\) runs, first-order (main effects) model (Plackett and Burman 1946)

A particular \(D\)-optimal design is the following orthogonal array

Using the pb function in the FrF2 package:

pb.design <- pb(12, factor.names = paste0("x", 1:11))
pb.design

##    x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11
## 1   1 -1 -1 -1  1 -1  1  1 -1   1   1
## 2   1  1  1 -1 -1 -1  1 -1  1   1  -1
## 3  -1 -1  1 -1  1  1 -1  1  1   1  -1
## 4  -1  1  1  1 -1 -1 -1  1 -1   1   1
## 5   1  1 -1  1  1  1 -1 -1 -1   1  -1
## 6   1  1 -1 -1 -1  1 -1  1  1  -1   1
## 7  -1 -1 -1  1 -1  1  1 -1  1   1   1
## 8   1 -1  1  1  1 -1 -1 -1  1  -1   1
## 9  -1 -1 -1 -1 -1 -1 -1 -1 -1  -1  -1
## 10 -1  1  1 -1  1  1  1 -1 -1  -1   1
## 11  1 -1  1  1 -1  1  1  1 -1  -1  -1
## 12 -1  1 -1  1  1 -1  1  1  1  -1  -1
## class=design, type= pb

This 12-run PB design is probably the most studied non-regular design

• orthogonal columns
• complex aliasing between main effects and two-factor interactions

pb.alias <- alias(y ~ (.)^2, data = data.frame(pb.design, y = vector(length = 12)))
t(pb.alias$Complete)[, 1:8]

##             x11:x21 x11:x31 x11:x41 x11:x51 x11:x61 x11:x71 x11:x81 x11:x91
## (Intercept)    0       0       0       0       0       0       0       0   
## x11            0       0       0       0       0       0       0       0   
## x21            0    -1/3    -1/3    -1/3     1/3    -1/3    -1/3     1/3   
## x31         -1/3       0     1/3    -1/3    -1/3     1/3    -1/3     1/3   
## x41         -1/3     1/3       0     1/3     1/3    -1/3    -1/3    -1/3   
## x51         -1/3    -1/3     1/3       0    -1/3    -1/3    -1/3    -1/3   
## x61          1/3    -1/3     1/3    -1/3       0    -1/3     1/3    -1/3   
## x71         -1/3     1/3    -1/3    -1/3    -1/3       0     1/3    -1/3   
## x81         -1/3    -1/3    -1/3    -1/3     1/3     1/3       0    -1/3   
## x91          1/3     1/3    -1/3    -1/3    -1/3    -1/3    -1/3       0   
## x101         1/3    -1/3    -1/3     1/3    -1/3     1/3    -1/3    -1/3   
## x111        -1/3    -1/3    -1/3     1/3    -1/3    -1/3     1/3     1/3

Screening designs with fewer runs than factors (see Woods and Lewis 2017)

• can’t use ordinary least squares/maximum likelihood as \(X\) does not have full column rank
• Bayesian \(D\)-optimality with \(R = [0\,|\, \tau I_m]\)

Supersaturated experiment used by GlaxoSmithKline in the development of a new oncology drug

• \(k=16\) factors: e.g. temperature, solvent amount, reaction time
• \(n=10\) runs
• Bayesian \(D\)-optimal design with \(\tau = 0.2\)

ssd

##    x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16
## 1   1  1 -1  1  1 -1 -1 -1 -1  -1  -1   1   1   1   1   1
## 2   1  1  1 -1 -1 -1 -1 -1  1   1  -1  -1   1  -1  -1   1
## 3  -1 -1  1 -1  1 -1  1 -1 -1   1  -1  -1  -1   1   1  -1
## 4  -1  1  1  1  1 -1 -1  1 -1  -1   1  -1  -1  -1  -1  -1
## 5  -1 -1 -1 -1 -1  1 -1  1  1  -1  -1  -1  -1  -1   1   1
## 6   1  1  1 -1  1  1 -1  1  1   1   1   1   1   1   1  -1
## 7  -1 -1  1  1 -1 -1  1 -1  1  -1   1   1   1  -1   1  -1
## 8   1 -1 -1  1  1  1  1  1 -1   1  -1   1   1  -1  -1  -1
## 9  -1  1 -1  1 -1  1  1 -1  1   1   1  -1  -1   1  -1   1
## 10  1 -1  1 -1 -1 -1  1  1 -1  -1   1   1  -1   1  -1   1

Partial aliasing between main effects

Heatmap of column correlations:

library(fields)
par(mar=c(8,2,0,0))
image.plot(1:16,1:16, cor(ssd), zlim = c(-1, 1), xlab = "Factors", 
           ylab = "", asp = 1, axes = F)
axis(1, at = seq(2, 16, by = 2), line = .5)
axis(2, at = seq(2, 16, by = 2), line = -5)

Analysis via regularised (shrinkage) methods (eg lasso, Dantzig selector; see APTS High Dimensional Statistics)

• small coefficients shrunk to zero

Many physical and social processes can be approximated by computer codes which encapsulate mathematical models

• eg partial differential equations solved using finite element methods
• eg reaction kinetics modelling in computational biology, in-silico chemistry

- computer code: numerical implementation of the mathematical model

Key feature: the model does not have a closed-form; it can only be evaluated numerically, and this is typically (relatively) expensive

We will focus on deterministic computer models

Assumption: \(g(\boldsymbol{x})\) can only be evaluated numerically; i.e. \(g(\boldsymbol{x})\) can be computed for a given \(\boldsymbol{x}\) but the general form is unknown

How do we learn about the function \(g(\boldsymbol{x})\)?

In an analogy to a physical system, we experiment on \(g(\boldsymbol{x})\), i.e.

• choose a design \(\xi = (\boldsymbol{x}_1,\ldots, \boldsymbol{x}_n)\)
• evaluate \(g(\boldsymbol{x}_i)\) (run the computer code)

Use the “data” \(\left\{\boldsymbol{x}_i, g(\boldsymbol{x}_i)\right\}\) to build statistical models linking \(\boldsymbol{x}\) and \(g(\boldsymbol{x})\)

• called emulators; typically use a Gaussian process

See Santner, Williams, and Notz (2019)

Climate modelling involves the solution of many intractable equations, leading to mathematical models evaluated via computationally expensive computer codes

• lots of applications of computer experiments

We will illustrate methods on a very simple example: a time-stepping advective/diffusive surface layer meridional EBM (energy balance model)

• 2D earth with no land
• each surface object has a percentage of ice cover
• different albedo (fraction of solar energy reflected) for ice vs non-ice surfaces
• ocean circulation is explicitly modelled (cf Atlantic gulf stream)
• two variables: \(x_1\) - solar constant; \(x_2\) - non-ice albedo
• output is mean temperature

See https://wiki.aston.ac.uk/foswiki/bin/view/MUCM/SurfebmModel (temporarily unavailable - hopefully back soon!)

## design and data are in 'ebm'
library(akima)
fld <- interp(x = ebm$x1, y = ebm$x2, z = ebm$y, extrap = TRUE, linear = FALSE)
filled.contour(x = fld$x, y = fld$y, z = fld$z, ylim = c(-1, 1), xlim = c(-1, 1), asp = 1, 
               plot.title = title(xlab = "solar constant", ylab = "non-ice albedo"), 
               plot.axes = {axis(1, seq(-1, 1, l = 5))
                 axis(2, seq(-1, 1, l = 5))})

As we will see later, emulators are usually constructed using nonparametric statistical models

This choice leads naturally to using space-filling designs

• such designs do not rely on the functional form of the relationship between the code inputs and the response
• good coverage is important for prediction (we will predict “better” near points we have already run the computer model)

Common designs are chosen to optimise some space-filling metric, or formed from (stratified) random sampling

Space-filling designs do not have replication, so ideal for deterministic computer models

Many designs proposed for computer experiments are related to ideas underpinning quadrature, and the approximation of an expectation.

Let \(\bar{g} = \frac{1}{n}\sum_{i=1}^n g(\boldsymbol{x}_i)\), the sample mean of \(g(\cdot)\) for \(\xi\). Then

\[ |E_\boldsymbol{x}[g(\boldsymbol{x})] - \bar{g}| \le \mbox{constant}\times D(\xi) \] where \(D(\xi)\) is the star discrepancy of the design

• \(D(\xi)\) is a measure of the uniformity of the design points

This relationship leads to the criterion of design selection via minimising discrepancy

• \(D(\xi)\) is difficult to compute for moderate to high numbers of dimensions
• therefore, it is more common to minimise the related centred \(L_2\)-discrepancy

Fang, Li, and Sudjianto (2006), Ch.3

Two sensible criteria for the selection of a space-filling design are

• make sure no two points in the design are too close together
• make sure no point in the design region is too far from a design point

(Johnson, Moore, and Ylvisaker 1990)

The Euclidean distance between points \(\boldsymbol{x}\) and \(\boldsymbol{x}^\prime\) is given by

\[ \delta(\boldsymbol{x}, \boldsymbol{x}^\prime) = \sqrt{\sum_{i=1}^k \left(x_{j} - x^\prime_j\right)^2} \]

Using Euclidean distance, we can define

• maximin (Mm) criterion: maximise

\[ \min_{\boldsymbol{x}_i, \boldsymbol{x}_j\in\xi}\delta(\boldsymbol{x}_i, \boldsymbol{x}_j) \]

• minimax (mM) criterion: minimise

\[ \max_{\boldsymbol{x}}\delta(\boldsymbol{x}, \xi) \] where the distance between a point \(\boldsymbol{x}\) and a design \(\xi\) is defined as

\[ \delta(\boldsymbol{x}, \xi) = \min_{\boldsymbol{x}_j\in\xi}\delta(\boldsymbol{x}, \boldsymbol{x}_i) \]

Roughly speaking, an Mm design spreads out the design points, and an mM design covers the design region

Intuitively, covering the design region seems more desirable (eg for prediction), but optimising the mM objective function is computationally challenging. Hence, Mm designs are more commonly used

For high-dimensional problems, space-filling is difficult

• many points are required to adequate space-fill a high-dimensional space (curse of dimensionality)

Latin hypercube designs (LHDs) are randomly chosen sets of points with the restriction of uniform one-dimensional projections (McKay, Beckman, and Conover 1979)

• each variable has no overlapping points, and good coverage (compare with a factorial design, which has hidden replication)
• can be easily constructed using permutations of integers

An LHD only guarantees space-filling properties in each one-dimensional projection, not overall. So we normally combine the Latin hypercube principle with a space-filling criteria, eg to find a Mm LHD

LH <- function(n = 3, d = 2) {
    D <- NULL
    for(i in 1:d) D <- cbind(D, sample(1:n, n))
    D 
}
set.seed(4)
par(mar=c(5,6,2,4)+0.1, pty = "s")
plot((LH() - .5) / 3, xlim = c(0, 1), ylim = c(0, 1), pty = "s", xlab = expression(x[1]), 
     ylab = expression(x[2]), pch = 16, cex.lab = 1.5, cex.axis = 1, cex = 2)
abline(v = c(0, 1/3, 2/3, 1), lty = 2)
abline(h = c(0, 1/3, 2/3, 1), lty = 2)

The DiceDesign package has functions to generate various LHDs

library(DiceDesign)
lhs.d <- lhsDesign(9, 2)
plot(lhs.d$design, xlim = c(0, 1), ylim = c(0, 1), pty = "s", xlab = expression(x[1]), 
     ylab = expression(x[2]), pch = 16, cex.lab = 2, cex.axis = 2, cex = 2, 
     main = "random", cex.main = 2)
abline(v = seq(0, 9) / 9, lty = 2)
abline(h = seq(0, 9) / 9, lty = 2)

discrep.d <- discrepSA_LHS(lhs.d$design, criterion = "C2")
plot(discrep.d$design, xlim = c(0, 1), ylim = c(0, 1), pty = "s", xlab = expression(x[1]), 
     ylab = expression(x[2]), pch = 16, cex.lab = 2, cex.axis = 2, cex = 2, 
     main = "discrepancy", cex.main = 2)
abline(v = seq(0, 9) / 9, lty = 2)
abline(h = seq(0, 9) / 9, lty = 2)

maximin.d <- maximinSA_LHS(discrep.d$design)
plot(maximin.d$design, xlim = c(0, 1), ylim = c(0, 1), pty = "s", xlab = expression(x[1]), 
     ylab = expression(x[2]), pch = 16, cex.lab = 2, cex.axis = 2, cex = 2, 
     main = "maximin", cex.main = 2)
abline(v = seq(0, 9) / 9, lty = 2)
abline(h = seq(0, 9) / 9, lty = 2)

The design for the EBM example is a Mm LHD

par(mar=c(5,6,2,4)+0.1, pty = "s")
plot(ebm[, 2:3], xlim = c(-1, 1), ylim = c(-1, 1), pty = "s", xlab = expression(x[1]), 
     ylab = expression(x[2]), pch = 16, asp = 1)
abline(v = 2 * seq(0:20) / 20 - 1, lty = 2)
abline(h = 2 * seq(0:20) / 20 - 1, lty = 2)

The most common statistical model used to emulate computer models is the Gaussian process (GP)

• flexible, nonparametric regression model (few assumptions made about \(g(\boldsymbol{x})\))
• naturally allows for uncertainty quantification (eg prediction intervals)
• interpolates observed responses

An intuitive way to think about a GP is as a prior for the unknown function \(g(\boldsymbol{x})\) within a Bayesian framework

We say that

\[ g(\boldsymbol{x})\sim \text{GP}\left(\mathbf{f}(\boldsymbol{x})^\mathrm{T}\boldsymbol{\beta}, \sigma^2\kappa(\boldsymbol{x},\boldsymbol{x}^\prime;\,\boldsymbol{\theta})\right)\,, \] where \(\mathbf{f}(\boldsymbol{x})^\mathrm{T}\boldsymbol{\beta}\) is the mean, \(\kappa(\boldsymbol{x},\boldsymbol{x}^\prime;\,\boldsymbol{\phi})\) is the correlation function, \(\boldsymbol{\theta}\) is the vector of correlation parameters and \(\sigma^2\) is the constant variance, if:

• any vector \(\boldsymbol{g}= \left(g(\boldsymbol{x}_1), \dots , g(\boldsymbol{x}_n)\right)^{\mathrm{T}}\) satisfies \[\boldsymbol{g}\sim N\left(F\boldsymbol{\beta}, \sigma^2 K(\boldsymbol{\theta})\right)\,,\] with \(F\) a model matrix and \(K\) the \(m\times m\) covariance matrix defined by \(K(\boldsymbol{\theta})_{ij} = \kappa(\boldsymbol{x}_i,\boldsymbol{x}_j;\boldsymbol{\theta})\).

See Rasmussen and Williams (2006)

Typically, very simple mean functions are chosen for the GP, eg

• constant: \(\mathbf{f}(\boldsymbol{x})^\mathrm{T}\boldsymbol{\beta}= \beta_0\) (sometimes called ordinary kriging)
• linear: \(\mathbf{f}(\boldsymbol{x})^\mathrm{T}\boldsymbol{\beta}= \beta_0 + \sum_{j=1}^k\beta_jx_j\) (universal kriging)

The most commonly used correlation functions are separable and stationary

• squared exponential:

\[ \kappa(\boldsymbol{x}, \boldsymbol{x}^\prime;\,\boldsymbol{\theta})=\exp\left[-\sum_j\left(\frac{|x_{j} - x^\prime_{j} |}{\theta_j }\right)^2\right] \]

• Matérn \(\nu = 5/2\)

\[ \kappa(\boldsymbol{x}, \boldsymbol{x}^\prime; \,\boldsymbol{\theta}) = \prod_{j}\left(1 + \sqrt{5}\frac{|x_j - x_j^\prime|}{\theta_j} + \frac{5}{3}\left(\frac{|x_j - x_j^\prime|}{\theta_j}\right)^2\right)\exp\left(-\sqrt{5}\frac{|x_j - x_j^\prime|}{\theta_j}\right) \] The Matérn function can be defined for other values of \(\nu\); for \(\nu\rightarrow\infty\), the squared exponential function is obtained

Given model evaluations \(\boldsymbol{g}= \left[g(\boldsymbol{x}_1), \ldots, g(\boldsymbol{x}_n)\right]\), a posterior GP can be obtained:

\[ g(\boldsymbol{x})\,|\, \boldsymbol{g},\boldsymbol{\beta},\boldsymbol{\theta},\sigma^2 \sim \text{GP}\left(m(\boldsymbol{x}), s^2(\boldsymbol{x})\right) \]

• \(m(\boldsymbol{x}) = \mathbf{f}(\boldsymbol{x})^\mathrm{T}\boldsymbol{\beta}+ \boldsymbol{\kappa}_n^\mathrm{T}K^{-1}(\boldsymbol{g}- F\boldsymbol{\beta})\)
• \(s^2(\boldsymbol{x}) = \sigma^2\left(1 - \boldsymbol{\kappa}_n^\mathrm{T}K^{-1}\boldsymbol{\kappa}_n\right)\)

where \(\boldsymbol{\kappa}_n = [\kappa(\boldsymbol{x},\boldsymbol{x}_i\,;\,\boldsymbol{\theta})]_{i=1}^n\) is a vector of correlations between \(g(\boldsymbol{x})\) and \(g(\boldsymbol{x}_1),\ldots,g(\boldsymbol{x}_n)\)

The updating of the prior mean and variance depends on the “distance” between \(\boldsymbol{x}\) and the points in \(\xi\)

• the posterior mean will be adjusted more for points closer to the design
• predictions at these points will have smaller posterior variance

If \(\boldsymbol{x}= \boldsymbol{x}_i\) (so we are predicting at a design point), \(K^{-1}\boldsymbol{\kappa}_n = \boldsymbol{e}_i\), the \(i\)th unit vector

• \(m(\boldsymbol{x}_i) = \mathbf{f}(\boldsymbol{x}_i)^\mathrm{T}\boldsymbol{\beta}+ \boldsymbol{e}_i^\mathrm{T}(\boldsymbol{g}- F\boldsymbol{\beta}) = g(\boldsymbol{x}_i)\)
• \(s^2(\boldsymbol{x}_i) = \sigma^2\left(1 - \boldsymbol{\kappa}_n^\mathrm{T}\boldsymbol{e}_i\right) = \sigma^2\left(1 - \kappa(\boldsymbol{x}_i,\boldsymbol{x}_i\,;\,\boldsymbol{\theta})\right) = 0\)

The posterior GP interpolates - exactly what you want for a deterministic computer code

Inference unconditional on all the hyperparameters requires numerical approximation (eg Markov chain Monte Carlo)

• it is common to estimate the parameters, eg using maximum likelihood, to “plug-in” to the posterior predictive distribution

A simple example: \(g(x) = \sin(2\pi x)\) using the DiceKriging package

library(DiceDesign)
library(DiceKriging)
xi <- lhsDesign(6, 1)$design
y <- sin(2 * pi * xi)
gp <- km(design = xi, response = y, control = list(trace = F))
xs <- sort(c(seq(-.1, 1.1, length = 100), xi))
gpp <- predict(gp, newdata = xs, type = "SK")

plot(xs, gpp$mean, ylim = c(-2, 2), type = "l", col = "red", lwd = 3, ylab = "", xlab = "x")
points(xi, y, pch = 4,lwd = 4, col = "blue")
lines(xs, sin(2 * pi * xs), lty = 1, col = "blue")
lines(xs, gpp$upper95, lty = 2, lwd = 3)
lines(xs, gpp$lower95, lty = 2, lwd = 3)
legend(x = "topright", legend = c("posterior mean of g", "posterior quantiles for g", 
                                  expression(paste("observed data ", g(x[i]))), 
                                    "function g(.)"), lty = c(1, 2, NA, 1), 
       pch = c(NA, NA, 4, NA), lwd = c(2, 2, 2, 2), col = c("red", "black", "blue", "blue"))

Return to the EBM example

gpebm <- km(formula = ~., design = ebm[, 2:3], response = ebm[, 1], control = list(trace = F))
gpebm

## 
## Call:
## km(formula = ~., design = ebm[, 2:3], response = ebm[, 1], control = list(trace = F))
## 
## Trend  coeff.:
##                Estimate
##  (Intercept)    16.3262
##           x1     2.4078
##           x2   -28.9973
## 
## Covar. type  : matern5_2 
## Covar. coeff.:
##                Estimate
##    theta(x1)     2.8829
##    theta(x2)     0.2722
## 
## Variance estimate: 2.215046

xs1 <- sort(c(seq(-1, 1, length = 10), ebm[, 2]))
xs2 <- sort(c(seq(-1, 1, length = 10), ebm[, 3]))
xs <- expand.grid(x1 = xs1, x2 = xs2)
gppebm <- predict(gpebm, newdata = xs, type = "UK")
filled.contour(x = xs1, y = xs2, z = matrix(gppebm$mean, nrow = length(xs1)))
filled.contour(x = xs1, y = xs2, z = matrix(gppebm$sd, nrow = length(xs1)),
               plot.axis = {axis(1); axis(2); points(ebm[, 2:3])})

A common task is optimisation of \(g(\boldsymbol{x})\)

When \(g(\boldsymbol{x})\) is computationally expensive to evaluate, computer experiments and emulators can be used to facilitate the optimisation.

The field of Bayesian optimisation uses sequentially collected evaluations of \(g(\boldsymbol{x})\)

• place a prior distribution (eg GP) on \(g(\boldsymbol{x})\)
• collect function evaluations at points chosen sequentially via an acquisition function
• update the prior to a posterior distribution, and infer the maximum/minimum of \(g(\boldsymbol{x})\)

Uncertainty in the posterior (i.e. for \(g(\boldsymbol{x})\) at unobserved \(\boldsymbol{x}\)) leads to exploration/exploitation trade-off

The most common acquisition function is expected improvement (EI)

See Jones, Schonlau, and Welch (1998)

For a deterministic computer model and a minimisation problem, the improvement from performing one more run is given by: \[ \max(g_\min - g(\boldsymbol{x}), 0) \] where \(g_\min\) is the minimum across the model runs performed to date

This quantity is a random variable - we are uncertain about \(g(\boldsymbol{x})\) at a point we have not observed.

EI chooses \(\boldsymbol{x}\) to maximise \[ E_g\left[\max(g_\min - g(\boldsymbol{x}), 0)\,;\, \boldsymbol{g}\right] = \left[g_\min - m(\boldsymbol{x})\right]\Phi\left(\frac{g_\min - m(\boldsymbol{x})}{s(\boldsymbol{x})}\right) + s(\boldsymbol{x})\phi\left(\frac{g_\min - m(\boldsymbol{x})}{s(\boldsymbol{x})}\right) \] where \(\phi\) and \(\Phi\) are the standard normal pdf and cdf, respectively

EI is an decreasing function of \(m(\boldsymbol{x})\) and an increasing function of \(s^2(\boldsymbol{x})\), so it leads to choosing design points that either minimise the posterior mean or maximise the posterior variance

• experiment either where our uncertainty is high or near where we predict the minimum to be (explore or exploit)

A simple example: \(g(\boldsymbol{x}) = \sin(2\pi x)\) but with a different starting design using DiceOptim

xi <- matrix(c(0.1, 0.8, 0.9), ncol = 1)
fn <- function(x) sin(2 * pi * x)
y <- fn(xi)
gp <- km(design = xi, response = y, control = list(trace = F))
xs <- sort(c(seq(0, 1, length = 100), xi))
gpp <- predict(gp, newdata = xs, type = "SK")

plot(xs, gpp$mean, ylim = c(-2, 2), type = "l", col = "red", lwd = 3, ylab = "", xlab = "x")
points(xi, y, pch = 4,lwd = 4, col = "blue")
lines(xs, fn(xs), lty = 1, col = "blue")
lines(xs, gpp$upper95, lty = 2, lwd = 3)
lines(xs, gpp$lower95, lty = 2, lwd = 3)

library(DiceOptim)
xin <- max_EI(model = gp, lower = 0, upper = 1)$par

## 
## 
## Mon Jul 25 08:22:01 2022
## Domains:
##  0.000000e+00   <=  X1   <=    1.000000e+00 
## 
## NOTE: The total number of operators greater than population size
## NOTE: I'm increasing the population size to 10 (operators+1).
## 
## Data Type: Floating Point
## Operators (code number, name, population) 
##  (1) Cloning...........................  0
##  (2) Uniform Mutation..................  1
##  (3) Boundary Mutation.................  1
##  (4) Non-Uniform Mutation..............  1
##  (5) Polytope Crossover................  1
##  (6) Simple Crossover..................  2
##  (7) Whole Non-Uniform Mutation........  1
##  (8) Heuristic Crossover...............  2
##  (9) Local-Minimum Crossover...........  0
## 
## HARD Maximum Number of Generations: 12
## Maximum Nonchanging Generations: 2
## Population size       : 10
## Convergence Tolerance: 1.000000e-21
## 
## Using the BFGS Derivative Based Optimizer on the Best Individual Each Generation.
## Not Checking Gradients before Stopping.
## Not Using Out of Bounds Individuals and Not Allowing Trespassing.
## 
## Maximization Problem.
## 
## 
## Generation#      Solution Value
## 
##       0  1.174912e-01
##       1  1.196487e-01
##       2  1.208143e-01
##       3  1.211283e-01
##       4  1.211283e-01
##       6  1.211283e-01
## 
## 'wait.generations' limit reached.
## No significant improvement in 2 generations.
## 
## Solution Fitness Value: 1.211283e-01
## 
## Parameters at the Solution (parameter, gradient):
## 
##  X[ 1] : 6.719061e-01    G[ 1] : 6.611444e-09
## 
## Solution Found Generation 6
## Number of Generations Run 9
## 
## Mon Jul 25 08:22:02 2022
## Total run time : 0 hours 0 minutes and 1 seconds

EI(xin, gp)

## [1] 0.1211283

plot(xs, gpp$mean, ylim = c(-2, 2), type = "l", col = "red", lwd = 3, ylab = "", xlab = "x", cex.lab = 2)
points(xi, y, pch = 4,lwd = 4, col = "blue")
lines(xs, gpp$upper95, lty = 2, lwd = 3)
lines(xs, gpp$lower95, lty = 2, lwd = 3)
abline(v = xin)
plot(xs, sapply(xs, EI, model = gp), type = "l", lwd = 3, ylab = "", xlab = "x", cex.lab = 2)
abline(v = xin)

xi <- rbind(xi, xin)
y <- c(y, fn(xin))
gp2 <- km(design = xi, response = y, control = list(trace = F))
xin <- max_EI(model = gp2, lower = 0, upper = 1, control = list(trace = F))$par

## 
## 
## Mon Jul 25 08:22:02 2022
## Domains:
##  0.000000e+00   <=  X1   <=    1.000000e+00 
## 
## NOTE: The total number of operators greater than population size
## NOTE: I'm increasing the population size to 10 (operators+1).
## 
## Data Type: Floating Point
## Operators (code number, name, population) 
##  (1) Cloning...........................  0
##  (2) Uniform Mutation..................  1
##  (3) Boundary Mutation.................  1
##  (4) Non-Uniform Mutation..............  1
##  (5) Polytope Crossover................  1
##  (6) Simple Crossover..................  2
##  (7) Whole Non-Uniform Mutation........  1
##  (8) Heuristic Crossover...............  2
##  (9) Local-Minimum Crossover...........  0
## 
## HARD Maximum Number of Generations: 12
## Maximum Nonchanging Generations: 2
## Population size       : 10
## Convergence Tolerance: 1.000000e-21
## 
## Using the BFGS Derivative Based Optimizer on the Best Individual Each Generation.
## Not Checking Gradients before Stopping.
## Not Using Out of Bounds Individuals and Not Allowing Trespassing.
## 
## Maximization Problem.
## 
## 
## Generation#      Solution Value
## 
##       0  9.527998e-03
##       2  9.546776e-03
##       3  9.561379e-03
##       4  9.561379e-03
##       5  9.561379e-03
## 
## 'wait.generations' limit reached.
## No significant improvement in 2 generations.
## 
## Solution Fitness Value: 9.561379e-03
## 
## Parameters at the Solution (parameter, gradient):
## 
##  X[ 1] : 4.891087e-01    G[ 1] : 1.035347e-10
## 
## Solution Found Generation 5
## Number of Generations Run 8
## 
## Mon Jul 25 08:22:02 2022
## Total run time : 0 hours 0 minutes and 0 seconds

EI(xin, gp2)

## [1] 0.009561379

xs <- sort(c(seq(0, 1, length = 100), xi))
gpp <- predict(gp2, newdata = xs, type = "SK")
plot(xs, gpp$mean, ylim = c(-2, 2), type = "l", col = "red", lwd = 3, ylab = "", xlab = "x", cex.lab = 2)
points(xi, y, pch = 4,lwd = 4, col = "blue")
lines(xs, gpp$upper95, lty = 2, lwd = 3)
lines(xs, gpp$lower95, lty = 2, lwd = 3)
abline(v = xin)
plot(xs, sapply(xs, EI, model = gp2), type = "l", lwd = 3, ylab = "", xlab = "x", cex.lab = 2)
abline(v = xin)

xi <- rbind(xi, xin)
y <- c(y, fn(xin))
gp3 <- km(design = xi, response = y, control = list(trace = F))
xin <- max_EI(model = gp3, lower = 0, upper = 1, control = list(trace = F))$par

## 
## 
## Mon Jul 25 08:22:02 2022
## Domains:
##  0.000000e+00   <=  X1   <=    1.000000e+00 
## 
## NOTE: The total number of operators greater than population size
## NOTE: I'm increasing the population size to 10 (operators+1).
## 
## Data Type: Floating Point
## Operators (code number, name, population) 
##  (1) Cloning...........................  0
##  (2) Uniform Mutation..................  1
##  (3) Boundary Mutation.................  1
##  (4) Non-Uniform Mutation..............  1
##  (5) Polytope Crossover................  1
##  (6) Simple Crossover..................  2
##  (7) Whole Non-Uniform Mutation........  1
##  (8) Heuristic Crossover...............  2
##  (9) Local-Minimum Crossover...........  0
## 
## HARD Maximum Number of Generations: 12
## Maximum Nonchanging Generations: 2
## Population size       : 10
## Convergence Tolerance: 1.000000e-21
## 
## Using the BFGS Derivative Based Optimizer on the Best Individual Each Generation.
## Not Checking Gradients before Stopping.
## Not Using Out of Bounds Individuals and Not Allowing Trespassing.
## 
## Maximization Problem.
## 
## 
## Generation#      Solution Value
## 
##       0  1.741815e-05
##       2  6.544479e-02
##       3  6.717128e-02
##       4  6.717128e-02
##       5  6.717128e-02
## 
## 'wait.generations' limit reached.
## No significant improvement in 2 generations.
## 
## Solution Fitness Value: 6.717128e-02
## 
## Parameters at the Solution (parameter, gradient):
## 
##  X[ 1] : 7.459677e-01    G[ 1] : -3.642919e-15
## 
## Solution Found Generation 5
## Number of Generations Run 8
## 
## Mon Jul 25 08:22:02 2022
## Total run time : 0 hours 0 minutes and 0 seconds

EI(xin, gp3)

## [1] 0.06717128

xs <- sort(c(seq(0, 1, length = 100), xi))
gpp <- predict(gp3, newdata = xs, type = "SK")
plot(xs, gpp$mean, ylim = c(-2, 2), type = "l", col = "red", lwd = 3, ylab = "", xlab = "x", cex.lab = 2)
points(xi, y, pch = 4,lwd = 4, col = "blue")
lines(xs, gpp$upper95, lty = 2, lwd = 3)
lines(xs, gpp$lower95, lty = 2, lwd = 3)
abline(v = xin)
plot(xs, sapply(xs, EI, model = gp3), type = "l", lwd = 3, ylab = "", xlab = "x", cex.lab = 2)
abline(v = xin)

Computer experiments are an important statistical contribution to the field of uncertainty quantification (UQ)

• interdisciplinary topic on the interface of Statistics and Applied Maths
• methodologies for taking account of uncertainties when mathematical and computer models are used to describe real-world phenomena

Space-filling designs and (GP) emulators are very general, and can be applied to a variety of black box computer models

• typically require a lot less knowledge about the model than alternative methods from numerical analysis (although at some loss of efficiency)

GP emulators can be used as priors for Bayesian calibration of computer models (Kennedy and O’Hagan 2001)

• eg learning tuning parameters (cf parameter estimation, albeit with various important nuances around interpretation and physical understanding)
• data fusion: combining computer model runs and data from real experiments

Now consider a more general class of models (cf preliminary material).

Let \(\boldsymbol{y}= (y_1,\ldots,y_n)^\mathrm{T}\) be iid observations from a distribution with density/mass function \(\pi(y_i\,;\,\boldsymbol{\theta},\boldsymbol{x}_i)\)

• \(\boldsymbol{\theta}\) is a \(q-\)vector of unknown parameters
• \(\boldsymbol{x}_i =(x_{1i},\ldots,x_{ki})^\mathrm{T}\) is a vector of values of \(k\) controllable variables.

The (expected) information matrix \[ M(\boldsymbol{\theta}) = E_y\left[-\frac{\partial^2l(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}\partial\boldsymbol{\theta}^\mathrm{T}}\right] \] is an important quantity for design, where \(l(\boldsymbol{\theta}) = \sum_{i=1}^n\log\pi(y_i;\,\boldsymbol{\theta},\boldsymbol{x}_i)\) (the log-likelihood).

• \(M(\boldsymbol{\theta})\) is the (asymptotic) precision for the maximum likelihood estimators \(\hat{\boldsymbol{\theta}}\).
• \(M(\boldsymbol{\theta})\) is also an asymptotic approximation to the posterior precision for \(\boldsymbol{\theta}\) in a Bayesian analysis.

Example 1: Compartmental model \[ y_i \sim N\left(c(\boldsymbol{\theta})\mu(\boldsymbol{\theta};\,x_i), \sigma^2\nu(\boldsymbol{\theta};\,x_i)\right)\,,\quad x_i\in[0,24]\,, \] with \[ \mu(\boldsymbol{\theta};\,x) = \exp(-\theta_1x)-exp(-\theta_2x)\,,\quad c(\boldsymbol{\theta}) = \frac{400\theta_2}{\theta_3(\theta_2-\theta_1)}\,,\quad \nu(\boldsymbol{\theta};\,x) = 1 + \frac{\tau^2}{\sigma^2}c(\boldsymbol{\theta})^2\mu(\boldsymbol{\theta};\,x)\,, \] for \(\theta_1, \theta_2, \theta_3, \tau^2, \sigma^2>0\).

Prior distributions (for later use):

• \(\log\theta_i\sim N(m_i, 0.05)\), with \(m_1 = \log 0.1, m_2 = 0, m_3 = \log 20\)

Ryan et al. (2014)

comp <- function(x, theta, D = 400) {
    mu <- exp(-theta[1] * x) - exp(-theta[2] * x)
    c <- (D / theta[3]) * (theta[2]) / (theta[2] - theta[1])
    c * mu }
theta <- c(.1, 1, 20)
M <- 100
par(mar = c(6, 4, 0, 1) + .1)
lapply(1:M, function(l) {
  thetat <- rlnorm(3,log(theta),rep(0.05,3))
  curve(comp(x, theta = thetat), from = 0, to = 24, ylab = "Expected concentration", 
        xlab = "Time", ylim = c(0, 20), xlim = c(0, 24), add = l!=1) })

## [[1]]
## [[1]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[1]]$y
##   [1]  0.000000  4.745565  8.246307 10.802291 12.641815 13.938560 14.824644
##   [8] 15.400557 15.742719 15.909234 15.944267 15.881380 15.746063 15.557665
##  [15] 15.330866 15.076789 14.803855 14.518427 14.225304 13.928098 13.629513
##  [22] 13.331570 13.035765 12.743199 12.454672 12.170757 11.891852 11.618228
##  [29] 11.350054 11.087427 10.830388 10.578935 10.333038 10.092640  9.857671
##  [36]  9.628047  9.403678  9.184464  8.970306  8.761100  8.556741  8.357124
##  [43]  8.162146  7.971703  7.785693  7.604015  7.426570  7.253261  7.083993
##  [50]  6.918672  6.757208  6.599510  6.445491  6.295066  6.148150  6.004663
##  [57]  5.864524  5.727656  5.593981  5.463427  5.335918  5.211386  5.089760
##  [64]  4.970973  4.854958  4.741650  4.630987  4.522906  4.417348  4.314254
##  [71]  4.213565  4.115227  4.019183  3.925381  3.833768  3.744294  3.656907
##  [78]  3.571560  3.488205  3.406796  3.327286  3.249632  3.173790  3.099718
##  [85]  3.027375  2.956721  2.887715  2.820320  2.754498  2.690212  2.627426
##  [92]  2.566106  2.506217  2.447725  2.390599  2.334806  2.280315  2.227095
##  [99]  2.175118  2.124354  2.074775
## 
## 
## [[2]]
## [[2]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[2]]$y
##   [1]  0.000000  4.417998  7.699465 10.112354 11.861971 13.105604 13.963749
##   [8] 14.528698 14.871133 15.045173 15.092245 15.044049 14.924833 14.753134
##  [15] 14.543111 14.305573 14.048758 13.778939 13.500880 13.218192 12.933603
##  [22] 12.649165 12.366411 12.086481 11.810211 11.538207 11.270898 11.008580
##  [29] 10.751447 10.499614 10.253139 10.012036  9.776286  9.545843  9.320646
##  [36]  9.100619  8.885677  8.675727  8.470674  8.270418  8.074858  7.883894
##  [43]  7.697423  7.515347  7.337563  7.163976  6.994488  6.829003  6.667429
##  [50]  6.509675  6.355651  6.205269  6.058443  5.915091  5.775129  5.638479
##  [57]  5.505061  5.374800  5.247621  5.123451  5.002219  4.883855  4.768292
##  [64]  4.655463  4.545304  4.437752  4.332745  4.230222  4.130125  4.032397
##  [71]  3.936981  3.843823  3.752869  3.664067  3.577367  3.492718  3.410072
##  [78]  3.329382  3.250601  3.173684  3.098588  3.025268  2.953683  2.883792
##  [85]  2.815555  2.748932  2.683886  2.620379  2.558375  2.497838  2.438733
##  [92]  2.381027  2.324686  2.269679  2.215973  2.163538  2.112343  2.062360
##  [99]  2.013560  1.965915  1.919397
## 
## 
## [[3]]
## [[3]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[3]]$y
##   [1]  0.000000  4.485092  7.904503 10.488364 12.417610 13.834485 14.850802
##   [8] 15.554446 16.014479 16.285163 16.409122 16.419824 16.343543 16.200892
##  [15] 16.008032 15.777628 15.519592 15.241676 14.949932 14.649074 14.342770
##  [22] 14.033863 13.724546 13.416507 13.111033 12.809101 12.511442 12.218592
##  [29] 11.930942 11.648763 11.372235 11.101467 10.836515 10.577389 10.324068
##  [36] 10.076507  9.834641  9.598390  9.367665  9.142369  8.922399  8.707648
##  [43]  8.498009  8.293373  8.093629  7.898667  7.708381  7.522661  7.341402
##  [50]  7.164500  6.991852  6.823359  6.658920  6.498440  6.341825  6.188982
##  [57]  6.039820  5.894252  5.752190  5.613552  5.478255  5.346218  5.217362
##  [64]  5.091612  4.968893  4.849131  4.732256  4.618197  4.506888  4.398261
##  [71]  4.292252  4.188798  4.087838  3.989311  3.893158  3.799324  3.707751
##  [78]  3.618385  3.531172  3.446062  3.363004  3.281947  3.202844  3.125647
##  [85]  3.050311  2.976791  2.905043  2.835024  2.766693  2.700009  2.634932
##  [92]  2.571424  2.509446  2.448962  2.389936  2.332332  2.276117  2.221257
##  [99]  2.167719  2.115472  2.064484
## 
## 
## [[4]]
## [[4]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[4]]$y
##   [1]  0.000000  3.954780  6.994173  9.309142 11.051292 12.340967 13.273685
##   [8] 13.925248 14.355800 14.613052 14.734840 14.751157 14.685773 14.557512
##  [15] 14.381273 14.168841 13.929523 13.670668 13.398062 13.116258 12.828826
##  [22] 12.538559 12.247634 11.957737 11.670170 11.385926 11.105757 10.830222
##  [29] 10.559732 10.294574 10.034947  9.780971  9.532712  9.290189  9.053388
##  [36]  8.822267  8.596764  8.376800  8.162286  7.953124  7.749209  7.550433
##  [43]  7.356685  7.167853  6.983822  6.804481  6.629718  6.459420  6.293479
##  [50]  6.131787  5.974238  5.820728  5.671155  5.525421  5.383426  5.245078
##  [57]  5.110282  4.978948  4.850987  4.726314  4.604843  4.486494  4.371186
##  [64]  4.258840  4.149382  4.042736  3.938832  3.837597  3.738964  3.642866
##  [71]  3.549238  3.458016  3.369139  3.282546  3.198179  3.115979  3.035893
##  [78]  2.957865  2.881842  2.807773  2.735608  2.665298  2.596795  2.530052
##  [85]  2.465025  2.401669  2.339942  2.279801  2.221205  2.164116  2.108494
##  [92]  2.054302  2.001503  1.950060  1.899940  1.851108  1.803531  1.757177
##  [99]  1.712014  1.668012  1.625141
## 
## 
## [[5]]
## [[5]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[5]]$y
##   [1]  0.000000  4.091836  7.215135  9.577902 11.343936 12.642215 13.574286
##   [8] 14.220089 14.642542 14.891160 15.004896 15.014385 14.943711 14.811801
##  [15] 14.633515 14.420516 14.181946 13.924964 13.655166 13.376922 13.093630
##  [22] 12.807931 12.521865 12.237003 11.954545 11.675402 11.400255 11.129608
##  [29] 10.863825 10.603160 10.347782 10.097792  9.853242  9.614141  9.380471
##  [36]  9.152187  8.929227  8.711517  8.498973  8.291501  8.089004  7.891384
##  [43]  7.698536  7.510357  7.326744  7.147593  6.972801  6.802267  6.635891
##  [50]  6.473573  6.315218  6.160730  6.010016  5.862985  5.719548  5.579618
##  [57]  5.443109  5.309939  5.180025  5.053289  4.929653  4.809041  4.691379
##  [64]  4.576597  4.464622  4.355386  4.248823  4.144868  4.043455  3.944524
##  [71]  3.848013  3.753864  3.662018  3.572419  3.485012  3.399744  3.316563
##  [78]  3.235416  3.156255  3.079030  3.003695  2.930204  2.858510  2.788571
##  [85]  2.720343  2.653784  2.588853  2.525512  2.463720  2.403440  2.344634
##  [92]  2.287268  2.231305  2.176712  2.123454  2.071499  2.020816  1.971372
##  [99]  1.923138  1.876085  1.830182
## 
## 
## [[6]]
## [[6]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[6]]$y
##   [1]  0.000000  4.421152  7.699804 10.105569 11.845010 13.076371 13.920865
##   [8] 14.471328 14.798846 14.957843 14.989967 14.927083 14.793558 14.608009
##  [15] 14.384656 14.134342 13.865330 13.583903 13.294828 13.001713 12.707278
##  [22] 12.413562 12.122087 11.833977 11.550051 11.270898 10.996930 10.728427
##  [29] 10.465563 10.208438  9.957093  9.711524  9.471696  9.237549  9.009004
##  [36]  8.785972  8.568353  8.356041  8.148925  7.946894  7.749834  7.557632
##  [43]  7.370175  7.187351  7.009048  6.835159  6.665577  6.500196  6.338914
##  [50]  6.181630  6.028246  5.878666  5.732796  5.590545  5.451822  5.316541
##  [57]  5.184616  5.055964  4.930504  4.808158  4.688847  4.572496  4.459033
##  [64]  4.348385  4.240483  4.135258  4.032644  3.932576  3.834992  3.739829
##  [71]  3.647027  3.556529  3.468275  3.382212  3.298285  3.216440  3.136626
##  [78]  3.058792  2.982890  2.908871  2.836689  2.766299  2.697655  2.630714
##  [85]  2.565434  2.501775  2.439694  2.379155  2.320118  2.262545  2.206402
##  [92]  2.151651  2.098259  2.046192  1.995417  1.945902  1.897615  1.850527
##  [99]  1.804607  1.759827  1.716158
## 
## 
## [[7]]
## [[7]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[7]]$y
##   [1]  0.000000  4.199058  7.363451  9.725719 11.466624 12.726667 13.615044
##   [8] 14.216612 14.597302 14.808328 14.889465 14.871592 14.778672 14.629294
##  [15] 14.437864 14.215541 13.970958 13.710787 13.440175 13.163081 12.882547
##  [22] 12.600900 12.319910 12.040919 11.764933 11.492701 11.224769 10.961532
##  [29] 10.703263 10.450141 10.202276  9.959724  9.722499  9.490581  9.263930
##  [36]  9.042484  8.826172  8.614908  8.408604  8.207165  8.010493  7.818489
##  [43]  7.631051  7.448078  7.269472  7.095132  6.924960  6.758859  6.596734
##  [50]  6.438493  6.284042  6.133293  5.986157  5.842549  5.702384  5.565580
##  [57]  5.432058  5.301738  5.174543  5.050400  4.929235  4.810976  4.695555
##  [64]  4.582902  4.472952  4.365639  4.260901  4.158676  4.058903  3.961524
##  [71]  3.866481  3.773719  3.683181  3.594816  3.508571  3.424395  3.342239
##  [78]  3.262054  3.183792  3.107408  3.032857  2.960094  2.889077  2.819763
##  [85]  2.752113  2.686085  2.621642  2.558745  2.497357  2.437441  2.378964
##  [92]  2.321889  2.266183  2.211814  2.158749  2.106957  2.056408  2.007072
##  [99]  1.958919  1.911922  1.866052
## 
## 
## [[8]]
## [[8]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[8]]$y
##   [1]  0.000000  4.033760  7.158026  9.559288 11.386175 12.757136 13.766555
##   [8] 14.489637 14.986295 15.304250 15.481508 15.548328 15.528794 15.442069
##  [15] 15.303393 15.124877 14.916144 14.684828 14.436983 14.177400 13.909868
##  [22] 13.637373 13.362267 13.086393 12.811191 12.537780 12.267023 11.999581
##  [29] 11.735953 11.476511 11.221525 10.971185 10.725617 10.484898 10.249066
##  [36] 10.018127  9.792064  9.570841  9.354408  9.142703  8.935657  8.733194
##  [43]  8.535235  8.341695  8.152491  7.967536  7.786743  7.610025  7.437296
##  [50]  7.268470  7.103463  6.942191  6.784572  6.630525  6.479969  6.332828
##  [57]  6.189024  6.048483  5.911132  5.776897  5.645709  5.517500  5.392201
##  [64]  5.269747  5.150073  5.033116  4.918815  4.807109  4.697941  4.591251
##  [71]  4.486983  4.385084  4.285499  4.188175  4.093061  4.000108  3.909265
##  [78]  3.820485  3.733722  3.648929  3.566061  3.485076  3.405929  3.328580
##  [85]  3.252988  3.179112  3.106914  3.036356  2.967400  2.900010  2.834150
##  [92]  2.769787  2.706884  2.645411  2.585333  2.526620  2.469240  2.413164
##  [99]  2.358360  2.304802  2.252459
## 
## 
## [[9]]
## [[9]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[9]]$y
##   [1]  0.000000  4.651546  8.178899 10.829868 12.798145 14.235110 15.259062
##   [8] 15.962441 16.417464 16.680543 16.795729 16.797415 16.712441 16.561741
##  [15] 16.361637 16.124842 15.861252 15.578559 15.282734 14.978401 14.669136
##  [22] 14.357695 14.046190 13.736238 13.429062 13.125583 12.826485 12.532268
##  [29] 12.243289 11.959791 11.681936 11.409814 11.143467 10.882897 10.628075
##  [36] 10.378951 10.135455  9.897508  9.665018  9.437888  9.216018  8.999302
##  [43]  8.787634  8.580906  8.379013  8.181847  7.989302  7.801275  7.617662
##  [50]  7.438361  7.263275  7.092304  6.925354  6.762330  6.603142  6.447699
##  [57]  6.295913  6.147700  6.002975  5.861656  5.723663  5.588919  5.457346
##  [64]  5.328870  5.203419  5.080921  4.961307  4.844508  4.730459  4.619095
##  [71]  4.510353  4.404170  4.300488  4.199246  4.100387  4.003856  3.909598
##  [78]  3.817558  3.727685  3.639928  3.554237  3.470564  3.388860  3.309079
##  [85]  3.231177  3.155109  3.080831  3.008302  2.937481  2.868327  2.800801
##  [92]  2.734865  2.670480  2.607612  2.546224  2.486281  2.427749  2.370595
##  [99]  2.314786  2.260292  2.207080
## 
## 
## [[10]]
## [[10]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[10]]$y
##   [1]  0.000000  4.673728  8.186011 10.798711 12.715265 14.093710 15.056792
##   [8] 15.699838 16.096864 16.305325 16.369806 16.324894 16.197404 16.008114
##  [15] 15.773112 15.504842 15.212916 14.904749 14.586050 14.261199 13.933551
##  [22] 13.605661 13.279465 12.956420 12.637611 12.323835 12.015667 11.713510
##  [29] 11.417634 11.128210 10.845329 10.569021 10.299273 10.036037  9.779239
##  [36]  9.528788  9.284576  9.046487  8.814397  8.588180  8.367705  8.152841
##  [43]  7.943455  7.739417  7.540596  7.346865  7.158097  6.974169  6.794958
##  [50]  6.620345  6.450215  6.284452  6.122947  5.965589  5.812274  5.662897
##  [57]  5.517359  5.375559  5.237404  5.102798  4.971652  4.843876  4.719383
##  [64]  4.598090  4.479915  4.364776  4.252596  4.143300  4.036812  3.933062
##  [71]  3.831977  3.733491  3.637536  3.544047  3.452961  3.364216  3.277752
##  [78]  3.193509  3.111432  3.031465  2.953553  2.877643  2.803684  2.731626
##  [85]  2.661420  2.593018  2.526375  2.461444  2.398182  2.336546  2.276494
##  [92]  2.217985  2.160980  2.105441  2.051328  1.998607  1.947240  1.897194
##  [99]  1.848434  1.800927  1.754641
## 
## 
## [[11]]
## [[11]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[11]]$y
##   [1]  0.000000  4.062568  7.170692  9.529080 11.298934 12.607187 13.553780
##   [8] 14.217392 14.659957 14.930216 15.066524 15.099051 15.051525 14.942596
##  [15] 14.786919 14.595999 14.378866 14.142594 13.892725 13.633587 13.368560
##  [22] 13.100273 12.830764 12.561611 12.294026 12.028933 11.767033 11.508850
##  [29] 11.254770 11.005069 10.759940 10.519508 10.283849 10.052997  9.826954
##  [36]  9.605701  9.389199  9.177394  8.970224  8.767618  8.569499  8.375788
##  [43]  8.186399  8.001250  7.820254  7.643325  7.470378  7.301328  7.136090
##  [50]  6.974581  6.816720  6.662425  6.511617  6.364219  6.220154  6.079348
##  [57]  5.941728  5.807221  5.675758  5.547270  5.421690  5.298952  5.178993
##  [64]  5.061749  4.947158  4.835162  4.725701  4.618718  4.514156  4.411962
##  [71]  4.312081  4.214461  4.119051  4.025802  3.934663  3.845587  3.758528
##  [78]  3.673440  3.590278  3.508999  3.429559  3.351919  3.276035  3.201870
##  [85]  3.129384  3.058539  2.989297  2.921624  2.855482  2.790837  2.727656
##  [92]  2.665906  2.605553  2.546567  2.488916  2.432570  2.377500  2.323676
##  [99]  2.271071  2.219657  2.169407
## 
## 
## [[12]]
## [[12]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[12]]$y
##   [1]  0.000000  4.520807  7.949447 10.527554 12.443754 13.845284 14.847063
##   [8] 15.538779 15.990414 16.256569 16.379834 16.393422 16.323222 16.189411
##  [15] 16.007700 15.790320 15.546781 15.284470 15.009120 14.725170 14.436051
##  [22] 14.144408 13.852271 13.561192 13.272353 12.986642 12.704723 12.427085
##  [29] 12.154079 11.885950 11.622862 11.364915 11.112160 10.864611 10.622251
##  [36] 10.385045 10.152937  9.925863  9.703747  9.486507  9.274057  9.066307
##  [43]  8.863166  8.664542  8.470342  8.280473  8.094843  7.913362  7.735939
##  [50]  7.562487  7.392917  7.227145  7.065085  6.906657  6.751780  6.600373
##  [57]  6.452360  6.307666  6.166215  6.027936  5.892757  5.760609  5.631425
##  [64]  5.505137  5.381681  5.260993  5.143012  5.027676  4.914927  4.804707
##  [71]  4.696958  4.591625  4.488655  4.387993  4.289589  4.193392  4.099352
##  [78]  4.007421  3.917552  3.829698  3.743814  3.659856  3.577781  3.497547
##  [85]  3.419112  3.342436  3.267479  3.194204  3.122571  3.052545  2.984090
##  [92]  2.917169  2.851750  2.787797  2.725279  2.664162  2.604416  2.546010
##  [99]  2.488914  2.433098  2.378534
## 
## 
## [[13]]
## [[13]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[13]]$y
##   [1]  0.000000  4.539722  8.016916 10.656198 12.635244 14.094589 15.145376
##   [8] 15.875490 16.354414 16.637069 16.766850 16.778041 16.697709 16.547215
##  [15] 16.343408 16.099563 15.826131 15.531332 15.221614 14.902034 14.576542
##  [22] 14.248216 13.919445 13.592074 13.267519 12.946857 12.630897 12.320239
##  [29] 12.015317 11.716437 11.423800 11.137531 10.857691 10.584292 10.317311
##  [36] 10.056696  9.802373  9.554250  9.312224  9.076186  8.846015  8.621591
##  [43]  8.402790  8.189484  7.981548  7.778857  7.581284  7.388708  7.201005
##  [50]  7.018058  6.839747  6.665957  6.496577  6.331495  6.170604  6.013797
##  [57]  5.860973  5.712030  5.566870  5.425399  5.287521  5.153146  5.022186
##  [64]  4.894553  4.770163  4.648935  4.530787  4.415641  4.303422  4.194054
##  [71]  4.087466  3.983587  3.882347  3.783680  3.687521  3.593806  3.502472
##  [78]  3.413460  3.326709  3.242164  3.159766  3.079463  3.001201  2.924928
##  [85]  2.850593  2.778148  2.707543  2.638733  2.571672  2.506315  2.442619
##  [92]  2.380541  2.320042  2.261080  2.203616  2.147613  2.093033  2.039840
##  [99]  1.987999  1.937476  1.888236
## 
## 
## [[14]]
## [[14]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[14]]$y
##   [1]  0.000000  4.250857  7.501751  9.965605 11.810501 13.169112 14.146150
##   [8] 14.824245 15.268593 15.530622 15.650888 15.661364 15.587247 15.448380
##  [15] 15.260384 15.035544 14.783515 14.511872 14.226552 13.932198 13.632431
##  [22] 13.330070 13.027298 12.725799 12.426861 12.131465 11.840346 11.554048
##  [29] 11.272965 10.997372 10.727453 10.463319 10.205026  9.952586  9.705978
##  [36]  9.465153  9.230047  9.000577  8.776652  8.558171  8.345028  8.137115
##  [43]  7.934319  7.736528  7.543629  7.355509  7.172056  6.993159  6.818709
##  [50]  6.648599  6.482724  6.320979  6.163265  6.009480  5.859529  5.713317
##  [57]  5.570751  5.431741  5.296198  5.164036  5.035172  4.909522  4.787007
##  [64]  4.667549  4.551072  4.437501  4.326765  4.218791  4.113512  4.010859
##  [71]  3.910769  3.813176  3.718018  3.625236  3.534768  3.446558  3.360549
##  [78]  3.276687  3.194917  3.115188  3.037449  2.961649  2.887741  2.815678
##  [85]  2.745413  2.676901  2.610099  2.544964  2.481454  2.419530  2.359150
##  [92]  2.300278  2.242874  2.186904  2.132329  2.079117  2.027233  1.976643
##  [99]  1.927316  1.879220  1.832324
## 
## 
## [[15]]
## [[15]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[15]]$y
##   [1]  0.000000  3.889432  6.836466  9.049658 10.691826 11.890069 12.743597
##   [8] 13.329838 13.709213 13.928863 14.025559 14.027976 13.958466 13.834443
##  [15] 13.669469 13.474098 13.256534 13.023148 12.778880 12.527555 12.272124
##  [22] 12.014861 11.757508 11.501397 11.247535 10.996681 10.749396 10.506092
##  [29] 10.267062 10.032509  9.802563  9.577301  9.356757  9.140935  8.929812
##  [36]  8.723348  8.521487  8.324164  8.131307  7.942837  7.758672  7.578728
##  [43]  7.402918  7.231157  7.063358  6.899434  6.739300  6.582871  6.430065
##  [50]  6.280799  6.134993  5.992567  5.853444  5.717549  5.584807  5.455145
##  [57]  5.328492  5.204779  5.083937  4.965900  4.850604  4.737984  4.627978
##  [64]  4.520526  4.415569  4.313049  4.212909  4.115094  4.019550  3.926224
##  [71]  3.835065  3.746023  3.659048  3.574092  3.491109  3.410052  3.330877
##  [78]  3.253541  3.178000  3.104213  3.032140  2.961739  2.892974  2.825805
##  [85]  2.760195  2.696109  2.633510  2.572366  2.512640  2.454302  2.397318
##  [92]  2.341657  2.287288  2.234182  2.182309  2.131640  2.082147  2.033804
##  [99]  1.986583  1.940458  1.895405
## 
## 
## [[16]]
## [[16]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[16]]$y
##   [1]  0.000000  4.369574  7.658068 10.110786 11.917823 13.226485 14.150908
##   [8] 14.779517 15.180810 15.407838 15.501681 15.494138 15.409817 15.267752
##  [15] 15.082658 14.865900 14.626252 14.370477 14.103781 13.830163 13.552688
##  [22] 13.273700 12.994980 12.717877 12.443406 12.172320 11.905175 11.642372
##  [29] 11.384190 11.130821 10.882383 10.638941 10.400521 10.167113  9.938689
##  [36]  9.715199  9.496581  9.282764  9.073669  8.869212  8.669307  8.473864
##  [43]  8.282795  8.096008  7.913413  7.734921  7.560443  7.389891  7.223180
##  [50]  7.060224  6.900939  6.745246  6.593062  6.444309  6.298911  6.156793
##  [57]  6.017880  5.882100  5.749384  5.619662  5.492866  5.368930  5.247791
##  [64]  5.129385  5.013651  4.900527  4.789956  4.681880  4.576242  4.472988
##  [71]  4.372064  4.273416  4.176995  4.082749  3.990629  3.900588  3.812578
##  [78]  3.726555  3.642472  3.560286  3.479955  3.401436  3.324689  3.249674
##  [85]  3.176351  3.104683  3.034631  2.966161  2.899235  2.833819  2.769879
##  [92]  2.707382  2.646295  2.586586  2.528225  2.471180  2.415422  2.360923
##  [99]  2.307653  2.255585  2.204692
## 
## 
## [[17]]
## [[17]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[17]]$y
##   [1]  0.000000  4.155454  7.324417  9.720047 11.509894 12.825644 13.770788
##   [8] 14.426647 14.857114 15.112376 15.231848 15.246475 15.180540 15.053097
##  [15] 14.879079 14.670190 14.435587 14.182433 13.916318 13.641601 13.361671
##  [22] 13.079155 12.796084 12.514021 12.234157 11.957399 11.684425 11.415739
##  [29] 11.151703 10.892573 10.638521 10.389654 10.146025  9.907651  9.674517
##  [36]  9.446587  9.223803  9.006099  8.793395  8.585607  8.382644  8.184412
##  [43]  7.990815  7.801757  7.617138  7.436863  7.260835  7.088957  6.921136
##  [50]  6.757277  6.597291  6.441086  6.288575  6.139671  5.994291  5.852350
##  [57]  5.713769  5.578467  5.446369  5.317398  5.191480  5.068543  4.948517
##  [64]  4.831333  4.716924  4.605224  4.496169  4.389696  4.285744  4.184255
##  [71]  4.085168  3.988428  3.893978  3.801766  3.711736  3.623839  3.538024
##  [78]  3.454240  3.372441  3.292578  3.214607  3.138482  3.064160  2.991598
##  [85]  2.920754  2.851588  2.784060  2.718131  2.653763  2.590920  2.529565
##  [92]  2.469662  2.411178  2.354080  2.298333  2.243906  2.190769  2.138889
##  [99]  2.088238  2.038787  1.990507
## 
## 
## [[18]]
## [[18]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[18]]$y
##   [1]  0.000000  3.969482  7.037032  9.386964 11.166398 12.492753 13.459733
##   [8] 14.142104 14.599511 14.879525 15.020076 15.051398 14.997580 14.877804
##  [15] 14.707336 14.498317 14.260391 14.001213 13.726847 13.442091 13.150731
##  [22] 12.855745 12.559473 12.263737 11.969956 11.679222 11.392368 11.110023
##  [29] 10.832652 10.560592 10.294077 10.033261  9.778233  9.529032  9.285660
##  [36]  9.048086  8.816257  8.590103  8.369537  8.154465  7.944785  7.740388
##  [43]  7.541164  7.346998  7.157776  6.973384  6.793707  6.618631  6.448045
##  [50]  6.281838  6.119900  5.962126  5.808410  5.658650  5.512745  5.370598
##  [57]  5.232113  5.097195  4.965754  4.837701  4.712948  4.591411  4.473008
##  [64]  4.357657  4.245280  4.135801  4.029144  3.925238  3.824011  3.725395
##  [71]  3.629321  3.535725  3.444543  3.355712  3.269172  3.184863  3.102729
##  [78]  3.022713  2.944760  2.868818  2.794834  2.722758  2.652541  2.584134
##  [85]  2.517492  2.452569  2.389319  2.327701  2.267672  2.209191  2.152218
##  [92]  2.096715  2.042642  1.989965  1.938646  1.888650  1.839943  1.792493
##  [99]  1.746266  1.701232  1.657359
## 
## 
## [[19]]
## [[19]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[19]]$y
##   [1]  0.000000  4.550325  7.999872 10.591003 12.513237 13.914764 14.911439
##   [8] 15.593832 16.032729 16.283446 16.389198 16.383736 16.293408 16.138772
##  [15] 15.935860 15.697164 15.432407 15.149150 14.853260 14.549283 14.240731
##  [22] 13.930309 13.620092 13.311664 13.006223 12.704668 12.407665 12.115699
##  [29] 11.829114 11.548143 11.272935 11.003573 10.740091 10.482483 10.230713
##  [36]  9.984723  9.744441  9.509777  9.280637  9.056919  8.838515  8.625318
##  [43]  8.417215  8.214096  8.015849  7.822364  7.633532  7.449243  7.269394
##  [50]  7.093877  6.922592  6.755437  6.592315  6.433128  6.277782  6.126186
##  [57]  5.978249  5.833883  5.693003  5.555524  5.421364  5.290444  5.162685
##  [64]  5.038010  4.916347  4.797621  4.681762  4.568702  4.458371  4.350705
##  [71]  4.245638  4.143109  4.043056  3.945419  3.850140  3.757162  3.666429
##  [78]  3.577887  3.491484  3.407167  3.324886  3.244593  3.166238  3.089776
##  [85]  3.015160  2.942346  2.871290  2.801950  2.734285  2.668254  2.603818
##  [92]  2.540937  2.479575  2.419695  2.361261  2.304238  2.248593  2.194291
##  [99]  2.141300  2.089589  2.039127
## 
## 
## [[20]]
## [[20]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[20]]$y
##   [1]  0.000000  3.879558  6.904368  9.245165 11.038939 12.395602 13.403327
##   [8] 14.132843 14.640877 14.972920 15.165444 15.247683 15.243061 15.170344
##  [15] 15.044554 14.877716 14.679442 14.457415 14.217767 13.965387 13.704166
##  [22] 13.437195 13.166927 12.895296 12.623829 12.353721 12.085903 11.821094
##  [29] 11.559846 11.302574 11.049586 10.801106 10.557285 10.318225 10.083981
##  [36]  9.854576  9.630006  9.410245  9.195251  8.984971  8.779340  8.578287
##  [43]  8.381736  8.189605  8.001813  7.818274  7.638902  7.463612  7.292316
##  [50]  7.124930  6.961368  6.801546  6.645383  6.492796  6.343705  6.198031
##  [57]  6.055699  5.916631  5.780753  5.647994  5.518281  5.391546  5.267721
##  [64]  5.146738  5.028533  4.913042  4.800203  4.689955  4.582239  4.476997
##  [71]  4.374172  4.273708  4.175551  4.079649  3.985949  3.894401  3.804956
##  [78]  3.717565  3.632181  3.548759  3.467252  3.387617  3.309811  3.233793
##  [85]  3.159520  3.086953  3.016053  2.946781  2.879100  2.812974  2.748366
##  [92]  2.685243  2.623569  2.563311  2.504438  2.446917  2.390717  2.335807
##  [99]  2.282159  2.229743  2.178531
## 
## 
## [[21]]
## [[21]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[21]]$y
##   [1]  0.000000  3.998131  7.065139  9.398176 11.153067 12.452963 13.395182
##   [8] 14.056618 14.498022 14.767379 14.902581 14.933544 14.883880 14.772212
##  [15] 14.613226 14.418491 14.197116 13.956265 13.701564 13.437425 13.167301
##  [22] 12.893887 12.619280 12.345104 12.072613 11.802765 11.536286 11.273721
##  [29] 11.015471 10.761826 10.512985 10.269078 10.030183  9.796333  9.567532
##  [36]  9.343754  9.124958  8.911086  8.702068  8.497828  8.298283  8.103346
##  [43]  7.912926  7.726932  7.545272  7.367852  7.194579  7.025362  6.860111
##  [50]  6.698734  6.541145  6.387255  6.236979  6.090235  5.946940  5.807013
##  [57]  5.670376  5.536953  5.406667  5.279446  5.155218  5.033912  4.915460
##  [64]  4.799795  4.686851  4.576565  4.468873  4.363716  4.261032  4.160765
##  [71]  4.062858  3.967254  3.873899  3.782742  3.693729  3.606811  3.521938
##  [78]  3.439062  3.358137  3.279116  3.201954  3.126608  3.053035  2.981193
##  [85]  2.911042  2.842541  2.775653  2.710338  2.646560  2.584283  2.523472
##  [92]  2.464091  2.406108  2.349489  2.294203  2.240217  2.187502  2.136027
##  [99]  2.085764  2.036683  1.988757
## 
## 
## [[22]]
## [[22]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[22]]$y
##   [1]  0.000000  5.110801  8.880134 11.632108 13.613109 15.010426 15.966412
##   [8] 16.589251 16.961150 17.144564 17.186928 17.124260 16.983897 16.786575
##  [15] 16.548014 16.280118 15.991894 15.690145 15.380000 15.065315 14.748982
##  [22] 14.433156 14.119437 13.809002 13.502708 13.201167 12.904811 12.613930
##  [29] 12.328710 12.049259 11.775625 11.507812 11.245790 10.989508 10.738892
##  [36] 10.493861 10.254321 10.020173  9.791313  9.567637  9.349037  9.135406
##  [43]  8.926637  8.722625  8.523264  8.328451  8.138085  7.952065  7.770293
##  [50]  7.592673  7.419111  7.249515  7.083795  6.921862  6.763630  6.609015
##  [57]  6.457934  6.310306  6.166053  6.025097  5.887363  5.752777  5.621269
##  [64]  5.492766  5.367201  5.244507  5.124617  5.007467  4.892996  4.781142
##  [71]  4.671845  4.565046  4.460688  4.358717  4.259076  4.161713  4.066576
##  [78]  3.973614  3.882776  3.794016  3.707284  3.622535  3.539724  3.458805
##  [85]  3.379737  3.302476  3.226981  3.153212  3.081129  3.010694  2.941869
##  [92]  2.874618  2.808904  2.744692  2.681948  2.620638  2.560730  2.502192
##  [99]  2.444991  2.389099  2.334484
## 
## 
## [[23]]
## [[23]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[23]]$y
##   [1]  0.000000  4.128537  7.295869  9.704361 11.514253 12.852415 13.819284
##   [8] 14.494361 14.940564 15.207678 15.335082 15.353918 15.288802 15.159181
##  [15] 14.980412 14.764609 14.521323 14.258072 13.980767 13.694044 13.401537
##  [22] 13.106078 12.809874 12.514630 12.221660 11.931967 11.646307 11.365245
##  [29] 11.089192 10.818440 10.553188 10.293560 10.039623  9.791398  9.548875
##  [36]  9.312012  9.080750  8.855015  8.634720  8.419769  8.210061  8.005491
##  [43]  7.805951  7.611330  7.421520  7.236409  7.055889  6.879851  6.708188
##  [50]  6.540796  6.377569  6.218408  6.063213  5.911885  5.764330  5.620455
##  [57]  5.480168  5.343381  5.210006  5.079959  4.953158  4.829520  4.708968
##  [64]  4.591425  4.476816  4.365067  4.256107  4.149866  4.046278  3.945275
##  [71]  3.846793  3.750770  3.657143  3.565854  3.476843  3.390054  3.305432
##  [78]  3.222921  3.142471  3.064028  2.987544  2.912969  2.840255  2.769357
##  [85]  2.700228  2.632825  2.567104  2.503024  2.440543  2.379622  2.320222
##  [92]  2.262305  2.205833  2.150771  2.097083  2.044736  1.993695  1.943928
##  [99]  1.895404  1.848091  1.801959
## 
## 
## [[24]]
## [[24]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[24]]$y
##   [1]  0.000000  4.144861  7.323521  9.741172 11.559848 12.907488 13.885102
##   [8] 14.572433 15.032422 15.314741 15.458585 15.494867 15.447967 15.337098
##  [15] 15.177400 14.980791 14.756647 14.512337 14.253645 13.985106 13.710264
##  [22] 13.431888 13.152129 12.872656 12.594754 12.319408 12.047366 11.779188
##  [29] 11.515289 11.255968 11.001432 10.751821 10.507216 10.267658 10.033151
##  [36]  9.803677  9.579197  9.359654  9.144984  8.935111  8.729955  8.529432
##  [43]  8.333452  8.141927  7.954764  7.771873  7.593164  7.418544  7.247925
##  [50]  7.081218  6.918336  6.759194  6.603706  6.451790  6.303366  6.158353
##  [57]  6.016674  5.878252  5.743014  5.610886  5.481797  5.355677  5.232459
##  [64]  5.112074  4.994459  4.879550  4.767285  4.657602  4.550442  4.445748
##  [71]  4.343463  4.243531  4.145898  4.050511  3.957318  3.866270  3.777317
##  [78]  3.690410  3.605503  3.522549  3.441504  3.362323  3.284965  3.209386
##  [85]  3.135545  3.063404  2.992923  2.924063  2.856787  2.791060  2.726844
##  [92]  2.664106  2.602812  2.542927  2.484421  2.427260  2.371415  2.316854
##  [99]  2.263549  2.211471  2.160590
## 
## 
## [[25]]
## [[25]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[25]]$y
##   [1]  0.000000  5.159759  9.023260 11.888497 13.985547 15.492039 16.545094
##   [8] 17.250552 17.690092 17.926738 18.009105 17.974685 17.852374 17.664439
##  [15] 17.428020 17.156306 16.859431 16.545173 16.219489 15.886936 15.550984
##  [22] 15.214270 14.878787 14.546030 14.217112 13.892853 13.573846 13.260509
##  [29] 12.953127 12.651886 12.356888 12.068182 11.785767 11.509610 11.239654
##  [36] 10.975820 10.718018 10.466147 10.220099  9.979760  9.745016  9.515750
##  [43]  9.291843  9.073178  8.859638  8.651109  8.447475  8.248625  8.054449
##  [50]  7.864838  7.679687  7.498891  7.322349  7.149961  6.981629  6.817260
##  [57]  6.656760  6.500037  6.347004  6.197573  6.051660  5.909182  5.770059
##  [64]  5.634210  5.501560  5.372033  5.245556  5.122056  5.001464  4.883710
##  [71]  4.768730  4.656456  4.546825  4.439776  4.335247  4.233179  4.133514
##  [78]  4.036196  3.941169  3.848379  3.757773  3.669301  3.582912  3.498557
##  [85]  3.416188  3.335758  3.257221  3.180534  3.105653  3.032534  2.961137
##  [92]  2.891420  2.823345  2.756873  2.691966  2.628587  2.566700  2.506270
##  [99]  2.447263  2.389646  2.333384
## 
## 
## [[26]]
## [[26]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[26]]$y
##   [1]  0.000000  3.985361  7.047008  9.380493 11.140339 12.448645 13.401878
##   [8] 14.076248 14.531952 14.816536 14.967539 15.014599 14.981105 14.885507
##  [15] 14.742357 14.563120 14.356827 14.130582 13.889971 13.639375 13.382228
##  [22] 13.121215 12.858428 12.595494 12.333668 12.073918 11.816981 11.563414
##  [29] 11.313630 11.067934 10.826539 10.589593 10.357187 10.129372  9.906165
##  [36]  9.687558  9.473523  9.264019  9.058990  8.858375  8.662104  8.470105
##  [43]  8.282299  8.098610  7.918956  7.743257  7.571433  7.403402  7.239086
##  [50]  7.078404  6.921280  6.767636  6.617397  6.470489  6.326839  6.186374
##  [57]  6.049026  5.914726  5.783406  5.655000  5.529444  5.406676  5.286632
##  [64]  5.169254  5.054481  4.942256  4.832523  4.725226  4.620311  4.517726
##  [71]  4.417418  4.319337  4.223434  4.129660  4.037968  3.948312  3.860647
##  [78]  3.774928  3.691113  3.609158  3.529023  3.450667  3.374051  3.299136
##  [85]  3.225885  3.154260  3.084225  3.015745  2.948786  2.883313  2.819294
##  [92]  2.756697  2.695489  2.635640  2.577121  2.519900  2.463950  2.409242
##  [99]  2.355749  2.303444  2.252300
## 
## 
## [[27]]
## [[27]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[27]]$y
##   [1]  0.000000  4.116698  7.274704  9.677045 11.484179 12.822910 13.793438
##   [8] 14.474931 14.929929 15.207828 15.347635 15.380143 15.329652 15.215332
##  [15] 15.052294 14.852447 14.625163 14.377814 14.116190 13.844831 13.567290
##  [22] 13.286340 13.004139 12.722359 12.442288 12.164915 11.890986 11.621063
##  [29] 11.355560 11.094773 10.838910 10.588105 10.342438 10.101944  9.866627
##  [36]  9.636463  9.411409  9.191405  8.976381  8.766260  8.560956  8.360379
##  [43]  8.164439  7.973041  7.786090  7.603491  7.425150  7.250972  7.080865
##  [50]  6.914736  6.752495  6.594053  6.439323  6.288218  6.140656  5.996554
##  [57]  5.855830  5.718408  5.584209  5.453158  5.325181  5.200208  5.078166
##  [64]  4.958989  4.842608  4.728958  4.617975  4.509596  4.403761  4.300409
##  [71]  4.199483  4.100926  4.004681  3.910696  3.818916  3.729289  3.641767
##  [78]  3.556298  3.472835  3.391331  3.311740  3.234017  3.158118  3.084000
##  [85]  3.011621  2.940942  2.871921  2.804520  2.738700  2.674426  2.611659
##  [92]  2.550366  2.490512  2.432062  2.374984  2.319245  2.264815  2.211662
##  [99]  2.159756  2.109069  2.059571
## 
## 
## [[28]]
## [[28]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[28]]$y
##   [1]  0.000000  4.810790  8.420171 11.103795 13.074565 14.496905 15.497788
##   [8] 16.175259 16.605012 16.845477 16.941749 16.928618 16.832918 16.675336
##  [15] 16.471810 16.234613 15.973184 15.694778 15.404958 15.107986 14.807117
##  [22] 14.504828 14.202999 13.903046 13.606031 13.312740 13.023748 12.739467
##  [29] 12.460186 12.186095 11.917315 11.653908 11.395894 11.143261 10.895973
##  [36] 10.653974 10.417198 10.185566  9.958993  9.737389  9.520663  9.308718
##  [43]  9.101459  8.898789  8.700613  8.506836  8.317362  8.132100  7.950958
##  [50]  7.773845  7.600673  7.431356  7.265808  7.103946  6.945688  6.790955
##  [57]  6.639668  6.491751  6.347129  6.205728  6.067476  5.932305  5.800145
##  [64]  5.670928  5.544591  5.421068  5.300296  5.182216  5.066765  4.953887
##  [71]  4.843524  4.735619  4.630118  4.526967  4.426115  4.327509  4.231100
##  [78]  4.136838  4.044677  3.954569  3.866468  3.780331  3.696112  3.613769
##  [85]  3.533261  3.454546  3.377585  3.302339  3.228768  3.156837  3.086509
##  [92]  3.017747  2.950517  2.884785  2.820517  2.757681  2.696245  2.636177
##  [99]  2.577448  2.520027  2.463886
## 
## 
## [[29]]
## [[29]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[29]]$y
##   [1]  0.000000  4.544311  8.002598 10.611628 12.557028 13.984320 15.007572
##   [8] 15.716191 16.180241 16.454619 16.582328 16.597044 16.525130 16.387217
##  [15] 16.199438 15.974400 15.721950 15.449767 15.163831 14.868795 14.568265
##  [22] 14.265033 13.961252 13.658572 13.358253 13.061248 12.768269 12.479845
##  [29] 12.196353 11.918062 11.645149 11.377723 11.115842 10.859520 10.608742
##  [36] 10.363467 10.123637  9.889180  9.660014  9.436050  9.217193  9.003346
##  [43]  8.794407  8.590276  8.390851  8.196030  8.005713  7.819799  7.638191
##  [50]  7.460791  7.287503  7.118235  6.952893  6.791388  6.633632  6.479539
##  [57]  6.329023  6.182002  6.038395  5.898123  5.761109  5.627278  5.496555
##  [64]  5.368868  5.244148  5.122324  5.003330  4.887101  4.773571  4.662679
##  [71]  4.554363  4.448563  4.345220  4.244279  4.145682  4.049376  3.955306
##  [78]  3.863423  3.773673  3.686009  3.600381  3.516742  3.435046  3.355248
##  [85]  3.277304  3.201171  3.126806  3.054169  2.983219  2.913917  2.846225
##  [92]  2.780106  2.715522  2.652439  2.590822  2.530636  2.471848  2.414425
##  [99]  2.358337  2.303551  2.250039
## 
## 
## [[30]]
## [[30]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[30]]$y
##   [1]  0.000000  4.013045  7.113445  9.489147 11.289815 12.634615 13.618424
##   [8] 14.316769 14.789761 15.085241 15.241265 15.288106 15.249832 15.145571
##  [15] 14.990516 14.796729 14.573775 14.329234 14.069105 13.798128 13.520040
##  [22] 13.237783 12.953663 12.669484 12.386645 12.106232 11.829074 11.555801
##  [29] 11.286884 11.022667 10.763395 10.509234 10.260288 10.016614  9.778229
##  [36]  9.545119  9.317251  9.094572  8.877013  8.664499  8.456946  8.254263
##  [43]  8.056357  7.863131  7.674489  7.490332  7.310561  7.135079  6.963788
##  [50]  6.796593  6.633400  6.474114  6.318645  6.166903  6.018799  5.874248
##  [57]  5.733166  5.595469  5.461077  5.329911  5.201895  5.076952  4.955009
##  [64]  4.835994  4.719838  4.606471  4.495827  4.387840  4.282446  4.179584
##  [71]  4.079193  3.981212  3.885585  3.792255  3.701167  3.612266  3.525501
##  [78]  3.440820  3.358173  3.277511  3.198786  3.121953  3.046964  2.973777
##  [85]  2.902348  2.832635  2.764596  2.698191  2.633382  2.570129  2.508395
##  [92]  2.448145  2.389341  2.331950  2.275937  2.221270  2.167916  2.115843
##  [99]  2.065021  2.015420  1.967011
## 
## 
## [[31]]
## [[31]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[31]]$y
##   [1]  0.000000  4.069420  7.142316  9.440785 11.137909 12.368559 13.237818
##   [8] 13.827551 14.201525 14.409396 14.489825 14.472898 14.382024 14.235402
##  [15] 14.047173 13.828316 13.587346 13.330858 13.063950 12.790556 12.513703
##  [22] 12.235711 11.958352 11.682969 11.410573 11.141916 10.877551 10.617874
##  [29] 10.363163 10.113601  9.869300  9.630317  9.396665  9.168328  8.945263
##  [36]  8.727410  8.514695  8.307033  8.104334  7.906502  7.713436  7.525037
##  [43]  7.341202  7.161828  6.986814  6.816060  6.649464  6.486930  6.328360
##  [50]  6.173659  6.022735  5.875497  5.731855  5.591722  5.455014  5.321646
##  [57]  5.191538  5.064610  4.940784  4.819985  4.702140  4.587175  4.475021
##  [64]  4.365609  4.258871  4.154744  4.053162  3.954063  3.857388  3.763076
##  [71]  3.671070  3.581313  3.493751  3.408330  3.324997  3.243702  3.164394
##  [78]  3.087026  3.011549  2.937917  2.866086  2.796011  2.727649  2.660959
##  [85]  2.595899  2.532430  2.470513  2.410110  2.351183  2.293697  2.237617
##  [92]  2.182908  2.129536  2.077470  2.026676  1.977124  1.928784  1.881626
##  [99]  1.835621  1.790740  1.746957
## 
## 
## [[32]]
## [[32]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[32]]$y
##   [1]  0.000000  4.236143  7.460551  9.891396 11.700345 13.022463 13.964001
##   [8] 14.608532 15.021774 15.255397 15.350011 15.337522 15.242986 15.086067
##  [15] 14.882188 14.643435 14.379265 14.097072 13.802624 13.500410 13.193916
##  [22] 12.885840 12.578255 12.272753 11.970538 11.672519 11.379367 11.091571
##  [29] 10.809475 10.533312 10.263228  9.999301  9.741557  9.489980  9.244526
##  [36]  9.005125  8.771693  8.544128  8.322324  8.106165  7.895531  7.690300
##  [43]  7.490348  7.295552  7.105788  6.920933  6.740865  6.565466  6.394618
##  [50]  6.228205  6.066115  5.908237  5.754462  5.604686  5.458806  5.316719
##  [57]  5.178330  5.043540  4.912259  4.784393  4.659855  4.538558  4.420418
##  [64]  4.305353  4.193283  4.084130  3.977818  3.874273  3.773424  3.675199
##  [71]  3.579531  3.486354  3.395602  3.307212  3.221123  3.137275  3.055610
##  [78]  2.976070  2.898601  2.823148  2.749660  2.678084  2.608372  2.540474
##  [85]  2.474344  2.409935  2.347203  2.286103  2.226594  2.168635  2.112184
##  [92]  2.057202  2.003652  1.951495  1.900696  1.851220  1.803031  1.756097
##  [99]  1.710385  1.665862  1.622499
## 
## 
## [[33]]
## [[33]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[33]]$y
##   [1]  0.000000  4.504068  7.952679 10.570831 12.536031 13.988309 15.038116
##   [8] 15.772565 16.260351 16.555633 16.701103 16.730404 16.670037 16.540869
##  [15] 16.359322 16.138309 15.887981 15.616301 15.329514 15.032508 14.729099
##  [22] 14.422260 14.114298 13.806999 13.501734 13.199550 12.901238 12.607389
##  [29] 12.318436 12.034689 11.756361 11.483587 11.216448 10.954974 10.699163
##  [36] 10.448984 10.204385  9.965300  9.731648  9.503343  9.280289  9.062388
##  [43]  8.849538  8.641636  8.438577  8.240258  8.046575  7.857424  7.672704
##  [50]  7.492314  7.316155  7.144131  6.976145  6.812104  6.651917  6.495494
##  [57]  6.342747  6.193589  6.047938  5.905711  5.766828  5.631210  5.498781
##  [64]  5.369466  5.243191  5.119886  4.999480  4.881906  4.767097  4.654988
##  [71]  4.545515  4.438616  4.334232  4.232302  4.132769  4.035577  3.940671
##  [78]  3.847996  3.757502  3.669135  3.582846  3.498587  3.416309  3.335966
##  [85]  3.257513  3.180905  3.106098  3.033051  2.961721  2.892069  2.824055
##  [92]  2.757641  2.692788  2.629461  2.567622  2.507239  2.448275  2.390698
##  [99]  2.334475  2.279574  2.225964
## 
## 
## [[34]]
## [[34]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[34]]$y
##   [1]  0.000000  3.949141  7.002563  9.344738 11.122537 12.452886 13.428858
##   [8] 14.124535 14.598875 14.898786 15.061588 15.116959 15.088495 14.994950
##  [15] 14.851219 14.669128 14.458058 14.225443 13.977169 13.717888 13.451272
##  [22] 13.180211 12.906975 12.633341 12.360691 12.090098 11.822385 11.558179
##  [29] 11.297951 11.042051 10.790727 10.544153 10.302441 10.065654  9.833820
##  [36]  9.606936  9.384976  9.167896  8.955641  8.748144  8.545329  8.347117
##  [43]  8.153423  7.964160  7.779241  7.598576  7.422074  7.249647  7.081205
##  [50]  6.916661  6.755928  6.598919  6.445552  6.295742  6.149409  6.006473
##  [57]  5.866857  5.730483  5.597277  5.467165  5.340077  5.215942  5.094692
##  [64]  4.976260  4.860580  4.747589  4.637225  4.529425  4.424132  4.321286
##  [71]  4.220830  4.122710  4.026871  3.933260  3.841824  3.752514  3.665281
##  [78]  3.580075  3.496850  3.415559  3.336159  3.258604  3.182852  3.108861
##  [85]  3.036590  2.965999  2.897049  2.829702  2.763921  2.699669  2.636910
##  [92]  2.575610  2.515736  2.457253  2.400130  2.344335  2.289836  2.236605
##  [99]  2.184611  2.133826  2.084221
## 
## 
## [[35]]
## [[35]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[35]]$y
##   [1]  0.000000  4.334218  7.598123 10.032835 11.825654 13.122066 14.035064
##   [8] 14.652393 15.042182 15.257318 15.338845 15.318606 15.221296 15.066055
##  [15] 14.867709 14.637733 14.385001 14.116360 13.837094 13.551262 13.261980
##  [22] 12.971629 12.682022 12.394528 12.110175 11.829725 11.553736 11.282607
##  [29] 11.016614 10.755940 10.500694 10.250931 10.006663  9.767868  9.534502
##  [36]  9.306503  9.083792  8.866286  8.653889  8.446504  8.244029  8.046362
##  [43]  7.853399  7.665035  7.481168  7.301694  7.126513  6.955525  6.788632
##  [50]  6.625737  6.466745  6.311566  6.160107  6.012280  5.868000  5.727180
##  [57]  5.589739  5.455595  5.324669  5.196886  5.072168  4.950443  4.831639
##  [64]  4.715687  4.602516  4.492062  4.384258  4.279041  4.176350  4.076122
##  [71]  3.978300  3.882826  3.789643  3.698696  3.609932  3.523298  3.438743
##  [78]  3.356217  3.275672  3.197060  3.120334  3.045450  2.972363  2.901030
##  [85]  2.831408  2.763458  2.697138  2.632410  2.569236  2.507577  2.447398
##  [92]  2.388663  2.331338  2.275389  2.220782  2.167486  2.115469  2.064700
##  [99]  2.015150  1.966789  1.919588
## 
## 
## [[36]]
## [[36]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[36]]$y
##   [1]  0.000000  3.857475  6.903486  9.292154 11.148666 12.574700 13.652832
##   [8] 14.450117 15.021001 15.409683 15.652038 15.777176 15.808714 15.765803
##  [15] 15.663969 15.515792 15.331458 15.119211 14.885713 14.636349 14.375459
##  [22] 14.106539 13.832400 13.555297 13.277030 12.999036 12.722453 12.448180
##  [29] 12.176918 11.909215 11.645487 11.386048 11.131130 10.880897 10.635458
##  [36] 10.394882 10.159200  9.928417  9.702515  9.481459  9.265201  9.053682
##  [43]  8.846834  8.644584  8.446854  8.253562  8.064625  7.879957  7.699472
##  [50]  7.523084  7.350707  7.182256  7.017645  6.856791  6.699611  6.546023
##  [57]  6.395948  6.249306  6.106021  5.966016  5.829218  5.695554  5.564952
##  [64]  5.437342  5.312657  5.190831  5.071796  4.955491  4.841852  4.730818
##  [71]  4.622330  4.516329  4.412759  4.311564  4.212689  4.116082  4.021690
##  [78]  3.929462  3.839350  3.751303  3.665276  3.581222  3.499095  3.418852
##  [85]  3.340448  3.263843  3.188994  3.115862  3.044407  2.974591  2.906375
##  [92]  2.839725  2.774602  2.710973  2.648803  2.588059  2.528708  2.470718
##  [99]  2.414058  2.358697  2.304606
## 
## 
## [[37]]
## [[37]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[37]]$y
##   [1]  0.000000  3.385821  6.050967  8.135083  9.751017 10.989958 11.925587
##   [8] 12.617430 13.113579 13.452878 13.666696 13.780365 13.814333 13.785103
##  [15] 13.705989 13.587726 13.438970 13.266691 13.076501 12.872910 12.659545
##  [22] 12.439312 12.214540 11.987090 11.758444 11.529780 11.302032 11.075932
##  [29] 10.852056 10.630848 10.412649 10.197717  9.986244  9.778365  9.574175
##  [36]  9.373733  9.177070  8.984197  8.795104  8.609772  8.428166  8.250248
##  [43]  8.075969  7.905278  7.738118  7.574432  7.414159  7.257236  7.103603
##  [50]  6.953196  6.805953  6.661810  6.520706  6.382580  6.247371  6.115018
##  [57]  5.985464  5.858650  5.734518  5.613014  5.494081  5.377667  5.263717
##  [64]  5.152181  5.043007  4.936146  4.831548  4.729166  4.628953  4.530864
##  [71]  4.434852  4.340875  4.248890  4.158853  4.070724  3.984462  3.900029
##  [78]  3.817384  3.736491  3.657312  3.579810  3.503951  3.429700  3.357022
##  [85]  3.285884  3.216253  3.148098  3.081387  3.016090  2.952177  2.889618
##  [92]  2.828384  2.768448  2.709783  2.652360  2.596154  2.541140  2.487291
##  [99]  2.434583  2.382992  2.332495
## 
## 
## [[38]]
## [[38]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[38]]$y
##   [1]  0.000000  4.528467  7.993808 10.623978 12.598495 14.058723 15.115967
##   [8] 15.857861 16.353387 16.656842 16.810950 16.849323 16.798395 16.678949
##  [15] 16.507313 16.296310 16.056002 15.794276 15.517309 15.229928 14.935901
##  [22] 14.638161 14.338984 14.040130 13.742951 13.448481 13.157505 12.870607
##  [29] 12.588221 12.310658 12.038133 11.770791 11.508715 11.251946 11.000491
##  [36] 10.754327 10.513415 10.277696 10.047102  9.821555  9.600973  9.385266
##  [43]  9.174345  8.968115  8.766482  8.569353  8.376633  8.188228  8.004047
##  [50]  7.823996  7.647987  7.475930  7.307738  7.143325  6.982608  6.825504
##  [57]  6.671933  6.521816  6.375074  6.231633  6.091419  5.954359  5.820383
##  [64]  5.689420  5.561405  5.436269  5.313948  5.194380  5.077502  4.963254
##  [71]  4.851577  4.742412  4.635703  4.531396  4.429435  4.329769  4.232345
##  [78]  4.137113  4.044024  3.953030  3.864083  3.777138  3.692149  3.609072
##  [85]  3.527864  3.448484  3.370890  3.295042  3.220900  3.148427  3.077584
##  [92]  3.008336  2.940645  2.874478  2.809800  2.746576  2.684776  2.624366
##  [99]  2.565315  2.507593  2.451170
## 
## 
## [[39]]
## [[39]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[39]]$y
##   [1]  0.000000  4.148044  7.286483  9.638592 11.378772 12.643229 13.538332
##   [8] 14.147133 14.534472 14.750955 14.836072 14.820623 14.728625 14.578793
##  [15] 14.385702 14.160692 13.912576 13.648197 13.372856 13.090651 12.804745
##  [22] 12.517566 12.230971 11.946372 11.664835 11.387156 11.113918 10.845546
##  [29] 10.582332 10.324475 10.072095  9.825256  9.583974  9.348234  9.117992
##  [36]  8.893187  8.673741  8.459567  8.250570  8.046648  7.847699  7.653615
##  [43]  7.464290  7.279615  7.099484  6.923791  6.752429  6.585297  6.422292
##  [50]  6.263315  6.108267  5.957053  5.809578  5.665752  5.525484  5.388688
##  [57]  5.255276  5.125167  4.998278  4.874530  4.753845  4.636148  4.521364
##  [64]  4.409422  4.300251  4.193784  4.089952  3.988690  3.889936  3.793626
##  [71]  3.699702  3.608102  3.518770  3.431650  3.346687  3.263828  3.183020
##  [78]  3.104213  3.027357  2.952403  2.879306  2.808018  2.738495  2.670694
##  [85]  2.604571  2.540085  2.477196  2.415864  2.356051  2.297718  2.240830
##  [92]  2.185350  2.131243  2.078477  2.027016  1.976830  1.927887  1.880155
##  [99]  1.833605  1.788207  1.743933
## 
## 
## [[40]]
## [[40]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[40]]$y
##   [1]  0.000000  3.624961  6.435547  8.597626 10.243642 11.479342 12.389146
##   [8] 13.040446 13.487027 13.771811 13.929042 13.986034 13.964564 13.881991
##  [15] 13.752144 13.586033 13.392415 13.178255 12.949079 12.709271 12.462299
##  [22] 12.210904 11.957242 11.703009 11.449528 11.197829 10.948706 10.702767
##  [29] 10.460471 10.222159  9.988077  9.758400  9.533240  9.312666  9.096708
##  [36]  8.885370  8.678631  8.476454  8.278790  8.085576  7.896746  7.712224
##  [43]  7.531934  7.355794  7.183722  7.015634  6.851446  6.691075  6.534436
##  [50]  6.381448  6.232028  6.086096  5.943573  5.804380  5.668442  5.535683
##  [57]  5.406030  5.279411  5.155755  5.034994  4.917060  4.801887  4.689411
##  [64]  4.579569  4.472299  4.367542  4.265238  4.165329  4.067761  3.972478
##  [71]  3.879427  3.788555  3.699812  3.613148  3.528513  3.445861  3.365145
##  [78]  3.286319  3.209340  3.134165  3.060749  2.989054  2.919038  2.850662
##  [85]  2.783888  2.718678  2.654995  2.592804  2.532070  2.472759  2.414836
##  [92]  2.358271  2.303031  2.249084  2.196401  2.144952  2.094709  2.045642
##  [99]  1.997725  1.950930  1.905231
## 
## 
## [[41]]
## [[41]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[41]]$y
##   [1]  0.000000  4.037579  7.160020  9.556495 11.377439 12.742466 13.746684
##   [8] 14.465703 14.959626 15.276223 15.453454 15.521476 15.504240 15.420764
##  [15] 15.286139 15.112335 14.908841 14.683172 14.441273 14.187844 13.926590
##  [22] 13.660431 13.391658 13.122068 12.853060 12.585721 12.320888 12.059202
##  [29] 11.801145 11.547078 11.297263 11.051885 10.811066 10.574883 10.343373
##  [36] 10.116544  9.894381  9.676853  9.463912  9.255502  9.051558  8.852011
##  [43]  8.656784  8.465801  8.278983  8.096247  7.917514  7.742702  7.571730
##  [50]  7.404517  7.240985  7.081054  6.924648  6.771690  6.622106  6.475822
##  [57]  6.332767  6.192869  6.056060  5.922271  5.791437  5.663492  5.538373
##  [64]  5.416017  5.296364  5.179354  5.064929  4.953031  4.843606  4.736597
##  [71]  4.631953  4.529620  4.429549  4.331688  4.235988  4.142404  4.050886
##  [78]  3.961391  3.873873  3.788288  3.704594  3.622749  3.542712  3.464444
##  [85]  3.387904  3.313056  3.239861  3.168283  3.098287  3.029837  2.962899
##  [92]  2.897440  2.833428  2.770829  2.709614  2.649751  2.591210  2.533963
##  [99]  2.477980  2.423235  2.369699
## 
## 
## [[42]]
## [[42]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[42]]$y
##   [1]  0.000000  4.233738  7.455433  9.884975 11.694959 13.020842 13.968926
##   [8] 14.622623 15.047370 15.294494 15.404245 15.408174 15.331006 15.192106
##  [15] 15.006633 14.786443 14.540798 14.276931 14.000474 13.715811 13.426342
##  [22] 13.134696 12.842901 12.552509 12.264705 11.980379 11.700199 11.424652
##  [29] 11.154090 10.888754 10.628804 10.374331 10.125381  9.881956  9.644032
##  [36]  9.411560  9.184475  8.962699  8.746144  8.534717  8.328319  8.126847
##  [43]  7.930198  7.738268  7.550952  7.368145  7.189745  7.015649  6.845757
##  [50]  6.679970  6.518190  6.360323  6.206274  6.055954  5.909271  5.766139
##  [57]  5.626472  5.490186  5.357201  5.227436  5.100814  4.977258  4.856695
##  [64]  4.739052  4.624258  4.512245  4.402945  4.296292  4.192223  4.090674
##  [71]  3.991585  3.894897  3.800550  3.708489  3.618658  3.531003  3.445471
##  [78]  3.362011  3.280572  3.201106  3.123566  3.047903  2.974073  2.902032
##  [85]  2.831735  2.763142  2.696210  2.630899  2.567171  2.504986  2.444307
##  [92]  2.385098  2.327324  2.270948  2.215939  2.162262  2.109885  2.058777
##  [99]  2.008907  1.960245  1.912762
## 
## 
## [[43]]
## [[43]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[43]]$y
##   [1]  0.000000  4.080472  7.174687  9.500211 11.227054 12.488058 13.387006
##   [8] 14.004978 14.405300 14.637426 14.739967 14.743061 14.670220 14.539783
##  [15] 14.366043 14.160130 13.930708 13.684509 13.426758 13.161504 12.891874
##  [22] 12.620279 12.348570 12.078159 11.810119 11.545255 11.284168 11.027296
##  [29] 10.774951 10.527350 10.284633 10.046883  9.814137  9.586398  9.363644
##  [36]  9.145831  8.932902  8.724786  8.521407  8.322681  8.128521  7.938837
##  [43]  7.753538  7.572531  7.395724  7.223025  7.054344  6.889589  6.728673
##  [50]  6.571508  6.418008  6.268089  6.121668  5.978665  5.839000  5.702597
##  [57]  5.569378  5.439270  5.312202  5.188101  5.066898  4.948527  4.832921
##  [64]  4.720015  4.609747  4.502055  4.396878  4.294159  4.193839  4.095863
##  [71]  4.000176  3.906724  3.815455  3.726319  3.639265  3.554244  3.471210
##  [78]  3.390116  3.310916  3.233566  3.158024  3.084246  3.012192  2.941821
##  [85]  2.873095  2.805973  2.740420  2.676399  2.613873  2.552807  2.493169
##  [92]  2.434924  2.378039  2.322483  2.268225  2.215235  2.163483  2.112940
##  [99]  2.063577  2.015368  1.968285
## 
## 
## [[44]]
## [[44]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[44]]$y
##   [1]  0.000000  4.235513  7.504204 10.006290 11.900958 13.314765 14.348315
##   [8] 15.081574 15.578091 15.888361 16.052497 16.102358 16.063234 15.955197
##  [15] 15.794166 15.592761 15.360976 15.106719 14.836240 14.554472 14.265298
##  [22] 13.971772 13.676285 13.380704 13.086478 12.794728 12.506309 12.221873
##  [29] 11.941905 11.666759 11.396689 11.131866 10.872397 10.618341 10.369717
##  [36] 10.126513  9.888694  9.656206  9.428981  9.206943  8.990007  8.778080
##  [43]  8.571068  8.368874  8.171398  7.978542  7.790205  7.606288  7.426693
##  [50]  7.251322  7.080079  6.912870  6.749601  6.590182  6.434524  6.282538
##  [57]  6.134138  5.989241  5.847765  5.709629  5.574755  5.443066  5.314487
##  [64]  5.188944  5.066367  4.946685  4.829830  4.715735  4.604335  4.495566
##  [71]  4.389367  4.285676  4.184435  4.085586  3.989071  3.894837  3.802828
##  [78]  3.712993  3.625280  3.539639  3.456022  3.374379  3.294666  3.216835
##  [85]  3.140843  3.066646  2.994202  2.923469  2.854408  2.786977  2.721140
##  [92]  2.656858  2.594094  2.532813  2.472980  2.414560  2.357521  2.301828
##  [99]  2.247452  2.194360  2.142522
## 
## 
## [[45]]
## [[45]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[45]]$y
##   [1]  0.000000  4.094413  7.203346  9.542120 11.279494 12.547721 13.450440
##   [8] 14.068848 14.466552 14.693370 14.788308 14.781895 14.698018 14.555357
##  [15] 14.368508 14.148868 13.905329 13.644817 13.372716 13.093210 12.809533
##  [22] 12.524183 12.239078 11.955683 11.675111 11.398195 11.125555 10.857642
##  [29] 10.594773 10.337166 10.084959  9.838229  9.597005  9.361280  9.131018
##  [36]  8.906165  8.686649  8.472386  8.263285  8.059248  7.860174  7.665957
##  [43]  7.476493  7.291675  7.111398  6.935555  6.764042  6.596757  6.433599
##  [50]  6.274467  6.119265  5.967897  5.820269  5.676289  5.535869  5.398920
##  [57]  5.265358  5.135099  5.008061  4.884166  4.763334  4.645492  4.530565
##  [64]  4.418481  4.309169  4.202561  4.098591  3.997193  3.898303  3.801860
##  [71]  3.707803  3.616073  3.526612  3.439364  3.354275  3.271291  3.190359
##  [78]  3.111430  3.034454  2.959382  2.886167  2.814764  2.745127  2.677213
##  [85]  2.610979  2.546384  2.483387  2.421948  2.362030  2.303593  2.246603
##  [92]  2.191022  2.136817  2.083952  2.032396  1.982114  1.933077  1.885253
##  [99]  1.838612  1.793125  1.748764
## 
## 
## [[46]]
## [[46]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[46]]$y
##   [1]  0.000000  4.042759  7.174670  9.581929 11.413069 12.786564 13.796909
##   [8] 14.519460 15.014305 15.329347 15.502769 15.564995 15.540263 15.447871
##  [15] 15.303181 15.118409 14.903267 14.665467 14.411126 14.145090 13.871190
##  [22] 13.592450 13.311248 13.029451 12.748516 12.469575 12.193500 11.920957
##  [29] 11.652451 11.388353 11.128934 10.874382 10.624821 10.380324 10.140925
##  [36]  9.906625  9.677402  9.453214  9.234005  9.019709  8.810249  8.605545
##  [43]  8.405510  8.210055  8.019091  7.832523  7.650262  7.472213  7.298286
##  [50]  7.128389  6.962433  6.800329  6.641990  6.487331  6.336267  6.188717
##  [57]  6.044598  5.903833  5.766344  5.632054  5.500891  5.372781  5.247653
##  [64]  5.125439  5.006070  4.889482  4.775608  4.664385  4.555753  4.449651
##  [71]  4.346019  4.244801  4.145941  4.049382  3.955073  3.862959  3.772992
##  [78]  3.685119  3.599293  3.515466  3.433591  3.353623  3.275517  3.199230
##  [85]  3.124721  3.051946  2.980866  2.911442  2.843635  2.777407  2.712721
##  [92]  2.649542  2.587834  2.527564  2.468697  2.411201  2.355045  2.300196
##  [99]  2.246624  2.194301  2.143195
## 
## 
## [[47]]
## [[47]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[47]]$y
##   [1]  0.000000  4.167559  7.370307  9.811659 11.652558 13.020328 14.015692
##   [8] 14.718319 15.191215 15.484203 15.636670 15.679743 15.638020 15.530925
##  [15] 15.373791 15.178715 14.955233 14.710854 14.451486 14.181768 13.905337
##  [22] 13.625038 13.343090 13.061215 12.780745 12.502705 12.227873 11.956838
##  [29] 11.690035 11.427783 11.170303 10.917747 10.670206 10.427728 10.190325
##  [36]  9.957983  9.730667  9.508324  9.290891  9.078295  8.870457  8.667292
##  [43]  8.468715  8.274633  8.084958  7.899597  7.718460  7.541456  7.368494
##  [50]  7.199486  7.034344  6.872983  6.715316  6.561261  6.410736  6.263661
##  [57]  6.119958  5.979550  5.842361  5.708319  5.577351  5.449387  5.324359
##  [64]  5.202198  5.082840  4.966220  4.852276  4.740946  4.632170  4.525889
##  [71]  4.422047  4.320588  4.221456  4.124599  4.029964  3.937500  3.847158
##  [78]  3.758889  3.672645  3.588379  3.506047  3.425604  3.347007  3.270213
##  [85]  3.195181  3.121870  3.050242  2.980257  2.911878  2.845067  2.779790
##  [92]  2.716010  2.653694  2.592807  2.533318  2.475193  2.418402  2.362914
##  [99]  2.308699  2.255728  2.203972
## 
## 
## [[48]]
## [[48]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[48]]$y
##   [1]  0.000000  4.378578  7.640459 10.045430 11.793398 13.038186 13.898135
##   [8] 14.464284 14.806648 14.979068 15.022929 14.970034 14.844807 14.666000
##  [15] 14.447991 14.201800 13.935861 13.656615 13.368976 13.076680 12.782558
##  [22] 12.488744 12.196842 11.908041 11.623218 11.343008 11.067863 10.798091
##  [29] 10.533893 10.275390 10.022638  9.775647  9.534391  9.298819  9.068860
##  [36]  8.844428  8.625427  8.411754  8.203300  7.999957  7.801611  7.608149
##  [43]  7.419459  7.235429  7.055948  6.880908  6.710201  6.543722  6.381368
##  [50]  6.223039  6.068634  5.918058  5.771216  5.628017  5.488369  5.352186
##  [57]  5.219381  5.089871  4.963574  4.840411  4.720303  4.603176  4.488955
##  [64]  4.377567  4.268944  4.163016  4.059717  3.958980  3.860744  3.764945
##  [71]  3.671523  3.580419  3.491575  3.404937  3.320448  3.238055  3.157707
##  [78]  3.079353  3.002943  2.928429  2.855764  2.784902  2.715798  2.648409
##  [85]  2.582693  2.518606  2.456111  2.395165  2.335733  2.277775  2.221255
##  [92]  2.166137  2.112387  2.059971  2.008856  1.959009  1.910398  1.862994
##  [99]  1.816767  1.771686  1.727724
## 
## 
## [[49]]
## [[49]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[49]]$y
##   [1]  0.000000  4.412967  7.764154 10.287957 12.167449 13.545602 14.534068
##   [8] 15.220025 15.671540 15.941745 16.072109 16.094987 16.035618 15.913683
##  [15] 15.744518 15.540070 15.309635 15.060442 14.798106 14.526977 14.250425
##  [22] 13.971048 13.690847 13.411354 13.133737 12.858879 12.587441 12.319913
##  [29] 12.056651 11.797905 11.543846 11.294582 11.050172 10.810638 10.575974
##  [36] 10.346152 10.121128  9.900844  9.685235  9.474228  9.267744  9.065704
##  [43]  8.868023  8.674618  8.485403  8.300295  8.119208  7.942058  7.768764
##  [50]  7.599243  7.433415  7.271201  7.112523  6.957305  6.805471  6.656950
##  [57]  6.511668  6.369556  6.230545  6.094566  5.961555  5.831447  5.704177
##  [64]  5.579685  5.457910  5.338792  5.222274  5.108299  4.996811  4.887757
##  [71]  4.781082  4.676736  4.574667  4.474825  4.377163  4.281632  4.188185
##  [78]  4.096779  4.007367  3.919907  3.834355  3.750671  3.668813  3.588741
##  [85]  3.510418  3.433803  3.358861  3.285554  3.213847  3.143705  3.075094
##  [92]  3.007981  2.942332  2.878116  2.815301  2.753858  2.693755  2.634964
##  [99]  2.577456  2.521204  2.466179
## 
## 
## [[50]]
## [[50]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[50]]$y
##   [1]  0.000000  4.397644  7.818188 10.457475 12.472559 13.989372 15.108884
##   [8] 15.912039 16.463716 16.815909 17.010273 17.080175 17.052328 16.948114
##  [15] 16.784634 16.575561 16.331815 16.062110 15.773392 15.471189 15.159893
##  [22] 14.842985 14.523216 14.202754 13.883299 13.566175 13.252408 12.942786
##  [29] 12.637904 12.338206 12.044013 11.755551 11.472968 11.196352 10.925743
##  [36] 10.661144 10.402528 10.149843  9.903022  9.661984  9.426636  9.196879
##  [43]  8.972609  8.753717  8.540092  8.331621  8.128193  7.929693  7.736012
##  [50]  7.547036  7.362658  7.182768  7.007261  6.836032  6.668980  6.506003
##  [57]  6.347004  6.191886  6.040556  5.892922  5.748894  5.608384  5.471308
##  [64]  5.337580  5.207120  5.079849  4.955687  4.834559  4.716392  4.601113
##  [71]  4.488651  4.378938  4.271907  4.167491  4.065627  3.966254  3.869309
##  [78]  3.774733  3.682469  3.592461  3.504652  3.418989  3.335421  3.253895
##  [85]  3.174361  3.096772  3.021079  2.947236  2.875198  2.804921  2.736362
##  [92]  2.669478  2.604229  2.540575  2.478477  2.417897  2.358798  2.301143
##  [99]  2.244897  2.190026  2.136496
## 
## 
## [[51]]
## [[51]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[51]]$y
##   [1]  0.000000  4.109387  7.284909  9.719052 11.565043 12.944851 13.955550
##   [8] 14.674394 15.162853 15.469830 15.634220 15.686954 15.652614 15.550737
##  [15] 15.396833 15.203217 14.979651 14.733873 14.472002 14.198878 13.918314
##  [22] 13.633314 13.346233 13.058913 12.772786 12.488963 12.208292 11.931422
##  [29] 11.658836 11.390891 11.127842 10.869861 10.617062 10.369506 10.127214
##  [36]  9.890181  9.658374  9.431745  9.210230  8.993755  8.782239  8.575594
##  [43]  8.373730  8.176553  7.983966  7.795875  7.612182  7.432790  7.257606
##  [50]  7.086534  6.919481  6.756355  6.597067  6.441527  6.289650  6.141349
##  [57]  5.996541  5.855145  5.717081  5.582271  5.450638  5.322108  5.196609
##  [64]  5.074068  4.954416  4.837585  4.723508  4.612122  4.503362  4.397166
##  [71]  4.293475  4.192229  4.093370  3.996842  3.902590  3.810561  3.720703
##  [78]  3.632963  3.547292  3.463641  3.381963  3.302211  3.224340  3.148305
##  [85]  3.074063  3.001572  2.930790  2.861678  2.794195  2.728304  2.663966
##  [92]  2.601145  2.539806  2.479914  2.421434  2.364332  2.308578  2.254138
##  [99]  2.200982  2.149079  2.098401
## 
## 
## [[52]]
## [[52]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[52]]$y
##   [1]  0.000000  4.396520  7.756610 10.301939 12.207263 13.610337 14.619744
##   [8] 15.321054 15.781696 16.054790 16.182176 16.196794 16.124573 15.985907
##  [15] 15.796833 15.569949 15.315142 15.040164 14.751084 14.452642 14.148535
##  [22] 13.841638 13.534174 13.227859 12.924007 12.623616 12.327435 12.036021
##  [29] 11.749775 11.468982 11.193830 10.924438 10.660865 10.403128 10.151210
##  [36]  9.905066  9.664634  9.429837  9.200584  8.976778  8.758318  8.545096
##  [43]  8.337003  8.133928  7.935763  7.742394  7.553714  7.369613  7.189984
##  [50]  7.014721  6.843722  6.676884  6.514107  6.355294  6.200349  6.049180
##  [57]  5.901693  5.757801  5.617416  5.480452  5.346827  5.216459  5.089270
##  [64]  4.965181  4.844118  4.726006  4.610773  4.498351  4.388669  4.281662
##  [71]  4.177263  4.075410  3.976040  3.879094  3.784511  3.692234  3.602207
##  [78]  3.514375  3.428685  3.345084  3.263522  3.183948  3.106315  3.030574
##  [85]  2.956681  2.884589  2.814254  2.745635  2.678689  2.613375  2.549654
##  [92]  2.487486  2.426834  2.367661  2.309931  2.253609  2.198659  2.145050
##  [99]  2.092748  2.041721  1.991938
## 
## 
## [[53]]
## [[53]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[53]]$y
##   [1]  0.000000  4.415174  7.739344 10.219091 12.045751 13.367782 14.300360
##   [8] 14.932830 15.334490 15.559078 15.648256 15.634314 15.542268 15.391490
##  [15] 15.196974 14.970313 14.720463 14.454336 14.177256 13.893317 13.605657
##  [22] 13.316676 13.028198 12.741607 12.457938 12.177964 11.902249 11.631200
##  [29] 11.365102 11.104143 10.848440 10.598054 10.353004 10.113276  9.878833
##  [36]  9.649617  9.425559  9.206579  8.992590  8.783499  8.579211  8.379630
##  [43]  8.184656  7.994192  7.808138  7.626399  7.448877  7.275477  7.106106
##  [50]  6.940672  6.779084  6.621255  6.467098  6.316528  6.169462  6.025818
##  [57]  5.885518  5.748484  5.614640  5.483911  5.356227  5.231514  5.109706
##  [64]  4.990733  4.874530  4.761033  4.650178  4.541905  4.436152  4.332862
##  [71]  4.231977  4.133440  4.037198  3.943197  3.851384  3.761709  3.674122
##  [78]  3.588575  3.505019  3.423409  3.343699  3.265845  3.189804  3.115533
##  [85]  3.042992  2.972139  2.902937  2.835345  2.769328  2.704847  2.641868
##  [92]  2.580355  2.520275  2.461593  2.404278  2.348297  2.293620  2.240216
##  [99]  2.188055  2.137109  2.087349
## 
## 
## [[54]]
## [[54]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[54]]$y
##   [1]  0.000000  4.512862  7.915776 10.456201 12.327013 13.678531 14.627891
##   [8] 15.266357 15.665022 15.879254 15.952159 15.917284 15.800725 15.622769
##  [15] 15.399178 15.142182 14.861263 14.563758 14.255336 13.940365 13.622203
##  [22] 13.303414 12.985955 12.671301 12.360560 12.054548 11.753862 11.458922
##  [29] 11.170017 10.887332 10.610971 10.340978 10.077351  9.820052  9.569015
##  [36]  9.324158  9.085380  8.852571  8.625615  8.404389  8.188768  7.978626
##  [43]  7.773834  7.574266  7.379796  7.190300  7.005653  6.825737  6.650431
##  [50]  6.479621  6.313191  6.151033  5.993035  5.839094  5.689105  5.542966
##  [57]  5.400581  5.261852  5.126686  4.994991  4.866679  4.741663  4.619857
##  [64]  4.501181  4.385553  4.272895  4.163131  4.056186  3.951989  3.850469
##  [71]  3.751556  3.655184  3.561288  3.469803  3.380669  3.293825  3.209211
##  [78]  3.126771  3.046449  2.968190  2.891942  2.817652  2.745271  2.674749
##  [85]  2.606039  2.539093  2.473868  2.410318  2.348400  2.288073  2.229296
##  [92]  2.172028  2.116232  2.061869  2.008903  1.957297  1.907017  1.858029
##  [99]  1.810298  1.763795  1.718485
## 
## 
## [[55]]
## [[55]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[55]]$y
##   [1]  0.000000  4.004889  7.085128  9.433965 11.204706 12.518955 13.473161
##   [8] 14.143811 14.591551 14.864459 15.000638 15.030278 14.977288 14.860596
##  [15] 14.695177 14.492872 14.263031 14.013035 13.748699 13.474595 13.194316
##  [22] 12.910670 12.625853 12.341567 12.059128 11.779547 11.503590 11.231833
##  [29] 10.964701 10.702499 10.445438 10.193657  9.947234  9.706204  9.470569
##  [36]  9.240299  9.015346  8.795647  8.581124  8.371693  8.167262  7.967734
##  [43]  7.773011  7.582991  7.397572  7.216651  7.040127  6.867900  6.699868
##  [50]  6.535933  6.375998  6.219968  6.067750  5.919251  5.774382  5.633055
##  [57]  5.495184  5.360686  5.229478  5.101480  4.976613  4.854802  4.735972
##  [64]  4.620050  4.506965  4.396648  4.289030  4.184047  4.081633  3.981725
##  [71]  3.884263  3.789187  3.696438  3.605959  3.517694  3.431590  3.347594
##  [78]  3.265653  3.185719  3.107741  3.031671  2.957464  2.885072  2.814453
##  [85]  2.745563  2.678358  2.612799  2.548845  2.486455  2.425593  2.366221
##  [92]  2.308302  2.251801  2.196682  2.142913  2.090460  2.039291  1.989375
##  [99]  1.940680  1.893177  1.846837
## 
## 
## [[56]]
## [[56]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[56]]$y
##   [1]  0.000000  4.608002  8.087927 10.691886 12.616157 14.013542 15.002984
##   [8] 15.677055 16.107798 16.351261 16.451045 16.441052 16.347637 16.191273
##  [15] 15.987858 15.749728 15.486442 15.205402 14.912328 14.611634 14.306714
##  [22] 14.000173 13.693999 13.389703 13.088422 12.791007 12.498084 12.210106
##  [29] 11.927392 11.650155 11.378530 11.112592 10.852365 10.597839 10.348977
##  [36] 10.105721  9.867998  9.635723  9.408802  9.187138  8.970629  8.759168
##  [43]  8.552651  8.350971  8.154021  7.961697  7.773894  7.590509  7.411441
##  [50]  7.236590  7.065859  6.899151  6.736373  6.577433  6.422241  6.270709
##  [57]  6.122752  5.978284  5.837224  5.699493  5.565010  5.433700  5.305489
##  [64]  5.180302  5.058069  4.938720  4.822187  4.708404  4.597306  4.488829
##  [71]  4.382911  4.279493  4.178515  4.079919  3.983650  3.889652  3.797873
##  [78]  3.708259  3.620759  3.535325  3.451906  3.370455  3.290926  3.213274
##  [85]  3.137454  3.063423  2.991139  2.920561  2.851648  2.784361  2.718661
##  [92]  2.654512  2.591877  2.530719  2.471005  2.412699  2.355770  2.300183
##  [99]  2.245909  2.192915  2.141171
## 
## 
## [[57]]
## [[57]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[57]]$y
##   [1]  0.000000  4.441538  7.780105 10.268500 12.101984 13.431367 14.373122
##   [8] 15.017193 15.433028 15.674246 15.782234 15.788930 15.718975 15.591373
##  [15] 15.420771 15.218453 14.993107 14.751412 14.498499 14.238302 13.973832
##  [22] 13.707386 13.440711 13.175130 12.911639 12.650980 12.393702 12.140205
##  [29] 11.890771 11.645596 11.404807 11.168481 10.936653 10.709329 10.486492
##  [36] 10.268108 10.054129  9.844500  9.639158  9.438032  9.241053  9.048146
##  [43]  8.859235  8.674246  8.493101  8.315725  8.142043  7.971980  7.805462
##  [50]  7.642418  7.482776  7.326466  7.173418  7.023566  6.876843  6.733184
##  [57]  6.592525  6.454804  6.319960  6.187932  6.058662  5.932092  5.808166
##  [64]  5.686829  5.568027  5.451707  5.337816  5.226305  5.117123  5.010222
##  [71]  4.905554  4.803073  4.702733  4.604489  4.508298  4.414115  4.321901
##  [78]  4.231613  4.143211  4.056656  3.971909  3.888932  3.807689  3.728143
##  [85]  3.650259  3.574002  3.499338  3.426234  3.354657  3.284576  3.215958
##  [92]  3.148774  3.082994  3.018587  2.955526  2.893783  2.833329  2.774139
##  [99]  2.716185  2.659441  2.603883
## 
## 
## [[58]]
## [[58]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[58]]$y
##   [1]  0.000000  4.846473  8.472146 11.159823 13.127299 14.542287 15.533926
##   [8] 16.201642 16.621973 16.853832 16.942550 16.923002 16.822008 16.660182
##  [15] 16.453363 16.213710 15.950547 15.671018 15.380587 15.083426 14.782710
##  [22] 14.480850 14.179666 13.880528 13.584455 13.292201 13.004312 12.721179
##  [29] 12.443071 12.170167 11.902573 11.640343 11.383489 11.131995 10.885819
##  [36] 10.644904 10.409180 10.178568  9.952981  9.732329  9.516520  9.305458
##  [43]  9.099047  8.897192  8.699798  8.506769  8.318014  8.133438  7.952952
##  [50]  7.776466  7.603893  7.435147  7.270144  7.108801  6.951037  6.796773
##  [57]  6.645933  6.498439  6.354218  6.213197  6.075306  5.940475  5.808636
##  [64]  5.679723  5.553671  5.430416  5.309897  5.192052  5.076823  4.964151
##  [71]  4.853980  4.746253  4.640918  4.537920  4.437208  4.338731  4.242440
##  [78]  4.148286  4.056221  3.966200  3.878176  3.792106  3.707946  3.625654
##  [85]  3.545189  3.466509  3.389575  3.314349  3.240792  3.168868  3.098540
##  [92]  3.029773  2.962532  2.896783  2.832493  2.769631  2.708163  2.648060
##  [99]  2.589290  2.531825  2.475635
## 
## 
## [[59]]
## [[59]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[59]]$y
##   [1]  0.000000  4.659324  8.212652 10.899578 12.908270 14.386493 15.450278
##   [8] 16.190723 16.679343 16.972270 17.113550 17.137738 17.071932 16.937376
##  [15] 16.750714 16.524975 16.270356 15.994822 15.704591 15.404510 15.098344
##  [22] 14.789016 14.478783 14.169384 13.862147 13.558082 13.257949 12.962312
##  [29] 12.671581 12.386047 12.105907 11.831288 11.562256 11.298839 11.041028
##  [36] 10.788789 10.542069 10.300799 10.064900  9.834285  9.608862  9.388533
##  [43]  9.173198  8.962758  8.757109  8.556152  8.359784  8.167906  7.980418
##  [50]  7.797224  7.618226  7.443331  7.272447  7.105481  6.942345  6.782953
##  [57]  6.627218  6.475057  6.326388  6.181132  6.039211  5.900547  5.765067
##  [64]  5.632697  5.503366  5.377005  5.253544  5.132918  5.015062  4.899912
##  [71]  4.787406  4.677482  4.570083  4.465150  4.362626  4.262456  4.164586
##  [78]  4.068963  3.975536  3.884254  3.795068  3.707930  3.622792  3.539609
##  [85]  3.458337  3.378930  3.301346  3.225544  3.151483  3.079122  3.008422
##  [92]  2.939346  2.871856  2.805915  2.741489  2.678542  2.617040  2.556950
##  [99]  2.498240  2.440878  2.384833
## 
## 
## [[60]]
## [[60]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[60]]$y
##   [1]  0.000000  4.731780  8.247792 10.833224 12.706975 14.037046 14.952361
##   [8] 15.551846 15.911389 16.089195 16.129894 16.067691 15.928793 15.733261
##  [15] 15.496447 15.230079 14.943113 14.642371 14.333041 14.019059 13.703396
##  [22] 13.388289 13.075408 12.765992 12.460949 12.160932 11.866403 11.577677
##  [29] 11.294953 11.018349 10.747917 10.483661 10.225547  9.973515  9.727487
##  [36]  9.487367  9.253051  9.024427  8.801380  8.583790  8.371536  8.164498
##  [43]  7.962555  7.765587  7.573477  7.386108  7.203366  7.025138  6.851315
##  [50]  6.681789  6.516454  6.355209  6.197951  6.044583  5.895010  5.749136
##  [57]  5.606872  5.468128  5.332816  5.200853  5.072155  4.946641  4.824233
##  [64]  4.704854  4.588430  4.474886  4.364151  4.256157  4.150836  4.048120
##  [71]  3.947947  3.850252  3.754975  3.662055  3.571435  3.483057  3.396866
##  [78]  3.312808  3.230831  3.150881  3.072910  2.996869  2.922709  2.850385
##  [85]  2.779850  2.711060  2.643973  2.578546  2.514738  2.452509  2.391820
##  [92]  2.332633  2.274910  2.218616  2.163714  2.110172  2.057954  2.007028
##  [99]  1.957363  1.908926  1.861689
## 
## 
## [[61]]
## [[61]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[61]]$y
##   [1]  0.000000  4.078864  7.196812  9.558935 11.327025 12.628699 13.564612
##   [8] 14.214138 14.639862 14.891119 15.006789 15.017503 14.947383 14.815416
##  [15] 14.636539 14.422492 14.182493 13.923774 13.651996 13.371587 13.085998
##  [22] 12.797912 12.509409 12.222089 11.937180 11.655612 11.378086 11.105118
##  [29] 10.837083 10.574243 10.316773 10.064780  9.818317  9.577395  9.341996
##  [36]  9.112076  8.887572  8.668408  8.454498  8.245748  8.042058  7.843326
##  [43]  7.649446  7.460312  7.275818  7.095858  6.920327  6.749120  6.582134
##  [50]  6.419269  6.260424  6.105504  5.954411  5.807054  5.663339  5.523179
##  [57]  5.386485  5.253173  5.123159  4.996361  4.872701  4.752101  4.634485
##  [64]  4.519780  4.407914  4.298816  4.192418  4.088653  3.987456  3.888764
##  [71]  3.792515  3.698648  3.607104  3.517825  3.430757  3.345843  3.263031
##  [78]  3.182269  3.103505  3.026691  2.951779  2.878720  2.807470  2.737983
##  [85]  2.670216  2.604126  2.539672  2.476813  2.415510  2.355725  2.297419
##  [92]  2.240556  2.185101  2.131018  2.078273  2.026835  1.976669  1.927745
##  [99]  1.880032  1.833500  1.788119
## 
## 
## [[62]]
## [[62]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[62]]$y
##   [1]  0.000000  4.069747  7.168812  9.510118 11.260239 12.549484 13.479779
##   [8] 14.130849 14.565041 14.831107 14.967166 15.003008 14.961915 14.862074
##  [15] 14.717684 14.539827 14.337144 14.116371 13.882747 13.640346 13.392330
##  [22] 13.141144 12.888679 12.636387 12.385381 12.136506 11.890400 11.647542
##  [29] 11.408280 11.172866 10.941476 10.714225 10.491185 10.272388 10.057841
##  [36]  9.847532  9.641428  9.439489  9.241662  9.047890  8.858109  8.672252
##  [43]  8.490251  8.312035  8.137534  7.966674  7.799386  7.635598  7.475239
##  [50]  7.318240  7.164532  7.014048  6.866721  6.722485  6.581277  6.443033
##  [57]  6.307692  6.175192  6.045475  5.918482  5.794156  5.672442  5.553284
##  [64]  5.436628  5.322423  5.210617  5.101160  4.994002  4.889094  4.786391
##  [71]  4.685844  4.587410  4.491044  4.396702  4.304342  4.213922  4.125401
##  [78]  4.038740  3.953899  3.870841  3.789527  3.709921  3.631988  3.555692
##  [85]  3.480998  3.407874  3.336286  3.266201  3.197589  3.130418  3.064658
##  [92]  3.000280  2.937254  2.875552  2.815146  2.756009  2.698114  2.641436
##  [99]  2.585948  2.531625  2.478444
## 
## 
## [[63]]
## [[63]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[63]]$y
##   [1]  0.000000  4.283629  7.539796  9.992923 11.818885 13.155501 14.110756
##   [8] 14.769252 15.197256 15.446669 15.558127 15.563436 15.487485 15.349738
##  [15] 15.165406 14.946372 14.701907 14.439237 14.163986 13.880521 13.592228
##  [22] 13.301722 13.011016 12.721652 12.434803 12.151352 11.871960 11.597109
##  [29] 11.327148 11.062317 10.802773 10.548609 10.299869 10.056560  9.818656
##  [36]  9.586113  9.358868  9.136846  8.919962  8.708125  8.501240  8.299208
##  [43]  8.101928  7.909300  7.721222  7.537593  7.358312  7.183282  7.012403
##  [50]  6.845581  6.682720  6.523728  6.368515  6.216991  6.069070  5.924666
##  [57]  5.783697  5.646080  5.511737  5.380590  5.252562  5.127581  5.005572
##  [64]  4.886467  4.770196  4.656691  4.545886  4.437718  4.332124  4.229043
##  [71]  4.128414  4.030179  3.934282  3.840667  3.749279  3.660066  3.572976
##  [78]  3.487957  3.404962  3.323942  3.244850  3.167639  3.092266  3.018686
##  [85]  2.946857  2.876737  2.808286  2.741464  2.676231  2.612551  2.550386
##  [92]  2.489700  2.430458  2.372626  2.316170  2.261057  2.207256  2.154734
##  [99]  2.103463  2.053412  2.004551
## 
## 
## [[64]]
## [[64]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[64]]$y
##   [1]  0.000000  4.156909  7.346326  9.771605 11.593848 12.940676 13.913169
##   [8] 14.591382 15.038700 15.305302 15.430900 15.446911 15.378182 15.244351
##  [15] 15.060932 14.840167 14.591713 14.323172 14.040525 13.748463 13.450663
##  [22] 13.149992 12.848684 12.548464 12.250661 11.956289 11.666113 11.380702
##  [29] 11.100469 10.825710 10.556619 10.293322 10.035882  9.784319  9.538615
##  [36]  9.298729  9.064596  8.836136  8.613259  8.395864  8.183846  7.977095
##  [43]  7.775497  7.578940  7.387307  7.200486  7.018362  6.840822  6.667757
##  [50]  6.499056  6.334613  6.174322  6.018080  5.865787  5.717344  5.572653
##  [57]  5.431622  5.294158  5.160171  5.029573  4.902280  4.778208  4.657275
##  [64]  4.539402  4.424513  4.312530  4.203382  4.096996  3.993302  3.892233
##  [71]  3.793722  3.697704  3.604116  3.512896  3.423986  3.337325  3.252858
##  [78]  3.170529  3.090283  3.012069  2.935834  2.861528  2.789104  2.718512
##  [85]  2.649707  2.582643  2.517277  2.453565  2.391466  2.330938  2.271942
##  [92]  2.214440  2.158393  2.103764  2.050518  1.998620  1.948035  1.898731
##  [99]  1.850674  1.803834  1.758179
## 
## 
## [[65]]
## [[65]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[65]]$y
##   [1]  0.000000  4.009767  7.050668  9.335236 11.029864 12.264807 13.142023
##   [8] 13.741304 14.125088 14.342222 14.430907 14.421012 14.335881 14.193745
##  [15] 14.008836 13.792253 13.552646 13.296741 13.029767 12.755773 12.477893
##  [22] 12.198539 11.919563 11.642375 11.368045 11.097372 10.830949 10.569206
##  [29] 10.312443 10.060867  9.814605  9.573725  9.338254  9.108178  8.883462
##  [36]  8.664048  8.449864  8.240826  8.036842  7.837817  7.643650  7.454236
##  [43]  7.269473  7.089254  6.913477  6.742037  6.574832  6.411761  6.252725
##  [50]  6.097625  5.946366  5.798855  5.654999  5.514709  5.377897  5.244477
##  [57]  5.114365  4.987481  4.863744  4.743075  4.625400  4.510644  4.398735
##  [64]  4.289602  4.183177  4.079391  3.978181  3.879481  3.783231  3.689368
##  [71]  3.597833  3.508570  3.421522  3.336633  3.253850  3.173121  3.094395
##  [78]  3.017622  2.942754  2.869743  2.798544  2.729111  2.661401  2.595371
##  [85]  2.530979  2.468185  2.406948  2.347231  2.288995  2.232205  2.176823
##  [92]  2.122815  2.070148  2.018787  1.968700  1.919856  1.872224  1.825773
##  [99]  1.780475  1.736301  1.693223
## 
## 
## [[66]]
## [[66]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[66]]$y
##   [1]  0.000000  4.340751  7.634527 10.110803 11.949266 13.290605 14.244961
##   [8] 14.898551 15.318850 15.558656 15.659272 15.652996 15.565076 15.415238
##  [15] 15.218883 14.988026 14.732032 14.458190 14.172164 13.878347 13.580138
##  [22] 13.280155 12.980411 12.682440 12.387408 12.096186 11.809424 11.527592
##  [29] 11.251024 10.979946 10.714503 10.454776 10.200796  9.952555  9.710019
##  [36]  9.473128  9.241811  9.015980  8.795541  8.580392  8.370428  8.165541
##  [43]  7.965621  7.770559  7.580244  7.394567  7.213421  7.036698  6.864293
##  [50]  6.696104  6.532030  6.371970  6.215828  6.063509  5.914920  5.769970
##  [57]  5.628571  5.490636  5.356080  5.224821  5.096778  4.971872  4.850027
##  [64]  4.731168  4.615222  4.502117  4.391783  4.284153  4.179161  4.076742
##  [71]  3.976833  3.879373  3.784300  3.691558  3.601089  3.512836  3.426747
##  [78]  3.342767  3.260845  3.180931  3.102976  3.026931  2.952749  2.880386
##  [85]  2.809796  2.740936  2.673763  2.608237  2.544317  2.481963  2.421137
##  [92]  2.361802  2.303921  2.247458  2.192379  2.138650  2.086238  2.035111
##  [99]  1.985236  1.936583  1.889123
## 
## 
## [[67]]
## [[67]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[67]]$y
##   [1]  0.000000  4.408922  7.757669 10.280535 12.160438 13.540177 14.531223
##   [8] 15.220582 15.676161 15.950947 16.086282 16.114412 16.060482 15.944093
##  [15] 15.780516 15.581643 15.356727 15.112960 14.855930 14.589965 14.318418
##  [22] 14.043876 13.768330 13.493306 13.219969 12.949200 12.681662 12.417844
##  [29] 12.158105 11.902698 11.651798 11.405517 11.163917 10.927027 10.694844
##  [36] 10.467346 10.244492 10.026231  9.812502  9.603236  9.398361  9.197800
##  [43]  9.001475  8.809306  8.621213  8.437115  8.256931  8.080583  7.907992
##  [50]  7.739078  7.573767  7.411982  7.253649  7.098696  6.947051  6.798643
##  [57]  6.653404  6.511267  6.372165  6.236035  6.102812  5.972435  5.844843
##  [64]  5.719976  5.597777  5.478189  5.361155  5.246621  5.134534  5.024841
##  [71]  4.917492  4.812437  4.709625  4.609010  4.510545  4.414183  4.319880
##  [78]  4.227591  4.137274  4.048887  3.962387  3.877736  3.794893  3.713820
##  [85]  3.634479  3.556833  3.480846  3.406482  3.333707  3.262487  3.192788
##  [92]  3.124578  3.057826  2.992499  2.928568  2.866003  2.804775  2.744855
##  [99]  2.686214  2.628827  2.572665
## 
## 
## [[68]]
## [[68]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[68]]$y
##   [1]  0.000000  3.901357  6.882073  9.142022 10.838013 12.093063 13.003670
##   [8] 13.645526 14.077994 14.347627 14.490923 14.536495 14.506766 14.419302
##  [15] 14.287859 14.123206 13.933764 13.726118 13.505406 13.275637 13.039931
##  [22] 12.800713 12.559861 12.318826 12.078725 11.840411 11.604531 11.371573
##  [29] 11.141898 10.915769 10.693372 10.474833 10.260233 10.049616  9.842998
##  [36]  9.640373  9.441720  9.247003  9.056179  8.869196  8.685998  8.506525
##  [43]  8.330713  8.158498  7.989815  7.824597  7.662778  7.504291  7.349071
##  [50]  7.197054  7.048174  6.902369  6.759576  6.619733  6.482782  6.348661
##  [57]  6.217314  6.088683  5.962712  5.839347  5.718533  5.600219  5.484352
##  [64]  5.370882  5.259760  5.150936  5.044364  4.939997  4.837789  4.737695
##  [71]  4.639673  4.543678  4.449670  4.357606  4.267448  4.179155  4.092688
##  [78]  4.008010  3.925085  3.843875  3.764345  3.686461  3.610189  3.535494
##  [85]  3.462345  3.390709  3.320555  3.251853  3.184572  3.118684  3.054158
##  [92]  2.990968  2.929085  2.868482  2.809133  2.751012  2.694094  2.638353
##  [99]  2.583766  2.530308  2.477956
## 
## 
## [[69]]
## [[69]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[69]]$y
##   [1]  0.000000  4.212168  7.386284  9.756857 11.505853 12.774459 13.672216
##   [8] 14.284096 14.676010 14.899070 14.992907 14.988236 14.908861 14.773214
##  [15] 14.595565 14.386952 14.155904 13.909005 13.651330 13.386785 13.118367
##  [22] 12.848367 12.578535 12.310194 12.044343 11.781725 11.522890 11.268232
##  [29] 11.018033 10.772482 10.531700 10.295753 10.064668  9.838441  9.617043
##  [36]  9.400430  9.188542  8.981311  8.778661  8.580511  8.386779  8.197377
##  [43]  8.012218  7.831217  7.654283  7.481332  7.312276  7.147031  6.985513
##  [50]  6.827639  6.673329  6.522503  6.375083  6.230994  6.090159  5.952506
##  [57]  5.817963  5.686461  5.557931  5.432305  5.309518  5.189507  5.072207
##  [64]  4.957559  4.845503  4.735979  4.628930  4.524302  4.422038  4.322085
##  [71]  4.224392  4.128907  4.035580  3.944363  3.855207  3.768067  3.682896
##  [78]  3.599650  3.518286  3.438762  3.361034  3.285064  3.210811  3.138236
##  [85]  3.067301  2.997970  2.930206  2.863974  2.799239  2.735966  2.674125
##  [92]  2.613681  2.554603  2.496860  2.440423  2.385261  2.331347  2.278651
##  [99]  2.227146  2.176805  2.127602
## 
## 
## [[70]]
## [[70]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[70]]$y
##   [1]  0.000000  5.069992  8.812447 11.547083 13.517206 14.907958 15.860206
##   [8] 16.481120 16.852219 17.035501 17.078103 17.015852 16.875970 16.679125
##  [15] 16.441003 16.173494 15.885603 15.584138 15.274236 14.959765 14.643626
##  [22] 14.327985 14.014450 13.704206 13.398117 13.096799 12.800688 12.510077
##  [29] 12.225156 11.946032 11.672754 11.405328 11.143723 10.887887 10.637748
##  [36] 10.393221 10.154213  9.920623  9.692347  9.469279  9.251311  9.038334
##  [43]  8.830240  8.626922  8.428273  8.234190  8.044570  7.859311  7.678315
##  [50]  7.501484  7.328723  7.159940  6.995042  6.833941  6.676549  6.522781
##  [57]  6.372555  6.225788  6.082401  5.942317  5.805458  5.671751  5.541124
##  [64]  5.413505  5.288825  5.167017  5.048014  4.931752  4.818168  4.707199
##  [71]  4.598787  4.492871  4.389394  4.288301  4.189536  4.093046  3.998778
##  [78]  3.906681  3.816705  3.728801  3.642922  3.559021  3.477052  3.396971
##  [85]  3.318735  3.242300  3.167626  3.094672  3.023397  2.953765  2.885736
##  [92]  2.819274  2.754342  2.690906  2.628931  2.568384  2.509230  2.451440
##  [99]  2.394980  2.339820  2.285931
## 
## 
## [[71]]
## [[71]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[71]]$y
##   [1]  0.000000  4.288802  7.561974 10.037412 11.886768 13.245252 14.219355
##   [8] 14.892939 15.332034 15.588618 15.703594 15.709146 15.630578 15.487782
##  [15] 15.296383 15.068642 14.814176 14.540513 14.253542 13.957857 13.657035
##  [22] 13.353852 13.050452 12.748486 12.449214 12.153591 11.862330 11.575957
##  [29] 11.294851 11.019273 10.749398 10.485328 10.227112  9.974755  9.728231
##  [36]  9.487490  9.252463  9.023065  8.799203  8.580776  8.367676  8.159795
##  [43]  7.957019  7.759237  7.566334  7.378199  7.194719  7.015784  6.841285
##  [50]  6.671115  6.505170  6.343345  6.185541  6.031658  5.881600  5.735272
##  [57]  5.592583  5.453443  5.317762  5.185457  5.056442  4.930637  4.807961
##  [64]  4.688337  4.571689  4.457943  4.347027  4.238870  4.133405  4.030563
##  [71]  3.930280  3.832492  3.737137  3.644154  3.553485  3.465072  3.378858
##  [78]  3.294790  3.212813  3.132876  3.054928  2.978919  2.904801  2.832528
##  [85]  2.762052  2.693330  2.626318  2.560973  2.497255  2.435121  2.374533
##  [92]  2.315453  2.257843  2.201666  2.146887  2.093471  2.041384  1.990593
##  [99]  1.941065  1.892770  1.845677
## 
## 
## [[72]]
## [[72]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[72]]$y
##   [1]  0.000000  4.146025  7.295441  9.667432 11.433396 12.727351 13.654076
##   [8] 14.295487 14.715612 14.964491 15.081222 15.096348 15.033719 14.911950
##  [15] 14.745566 14.545889 14.321740 14.079983 13.825952 13.563785 13.296687
##  [22] 13.027131 12.757022 12.487816 12.220625 11.956286 11.695428 11.438513
##  [29] 11.185874 10.937748 10.694291 10.455599 10.221724  9.992682  9.768459
##  [36]  9.549024  9.334327  9.124307  8.918897  8.718019  8.521594  8.329538
##  [43]  8.141768  7.958196  7.778736  7.603303  7.431809  7.264171  7.100304
##  [50]  6.940126  6.783555  6.630512  6.480918  6.334696  6.191770  6.052068
##  [57]  5.915516  5.782044  5.651583  5.524065  5.399423  5.277594  5.158513
##  [64]  5.042118  4.928350  4.817148  4.708456  4.602216  4.498373  4.396873
##  [71]  4.297663  4.200692  4.105909  4.013264  3.922710  3.834199  3.747685
##  [78]  3.663123  3.580469  3.499680  3.420714  3.343530  3.268088  3.194347
##  [85]  3.122271  3.051821  2.982960  2.915653  2.849865  2.785562  2.722709
##  [92]  2.661274  2.601226  2.542533  2.485164  2.429089  2.374280  2.320707
##  [99]  2.268343  2.217161  2.167133
## 
## 
## [[73]]
## [[73]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[73]]$y
##   [1]  0.000000  4.063096  7.177007  9.543672 11.322483 12.639235 13.593204
##   [8] 14.262722 14.709591 14.982563 15.120085 15.152467 15.103603 14.992315
##  [15] 14.833423 14.638591 14.416989 14.175820 13.920735 13.656161 13.385557
##  [22] 13.111621 12.836450 12.561668 12.288525 12.017977 11.750748 11.487382
##  [29] 11.228279 10.973726 10.723922 10.479000 10.239035 10.004064  9.774088
##  [36]  9.549087  9.329018  9.113826  8.903444  8.697796  8.496803  8.300379
##  [43]  8.108436  7.920884  7.737634  7.558593  7.383672  7.212780  7.045830
##  [50]  6.882732  6.723401  6.567750  6.415698  6.267162  6.122060  5.980316
##  [57]  5.841851  5.706591  5.574460  5.445388  5.319304  5.196139  5.075824
##  [64]  4.958296  4.843488  4.731338  4.621785  4.514768  4.410230  4.308111
##  [71]  4.208357  4.110913  4.015725  3.922742  3.831911  3.743183  3.656510
##  [78]  3.571844  3.489138  3.408347  3.329427  3.252334  3.177026  3.103462
##  [85]  3.031602  2.961405  2.892834  2.825850  2.760418  2.696501  2.634063
##  [92]  2.573072  2.513492  2.455292  2.398440  2.342904  2.288654  2.235661
##  [99]  2.183894  2.133326  2.083929
## 
## 
## [[74]]
## [[74]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[74]]$y
##   [1]  0.000000  4.060316  7.161364  9.509735 11.267947 12.563837 13.497951
##   [8] 14.149366 14.580259 14.839514 14.965556 14.988576 14.932292 14.815326
##  [15] 14.652292 14.454652 14.231385 13.989523 13.734559 13.470784 13.201537
##  [22] 12.929413 12.656421 12.384108 12.113662 11.845986 11.581760 11.321490
##  [29] 11.065546 10.814190 10.567602 10.325896 10.089136  9.857345  9.630519
##  [36]  9.408629  9.191628  8.979456  8.772043  8.569313  8.371182  8.177565
##  [43]  7.988373  7.803517  7.622905  7.446448  7.274055  7.105638  6.941107
##  [50]  6.780376  6.623359  6.469972  6.320132  6.173759  6.030773  5.891096
##  [57]  5.754652  5.621367  5.491168  5.363984  5.239744  5.118382  4.999830
##  [64]  4.884024  4.770900  4.660396  4.552451  4.447006  4.344003  4.243387
##  [71]  4.145100  4.049090  3.955304  3.863690  3.774198  3.686779  3.601385
##  [78]  3.517969  3.436484  3.356887  3.279134  3.203182  3.128989  3.056514
##  [85]  2.985718  2.916562  2.849008  2.783018  2.718557  2.655589  2.594079
##  [92]  2.533994  2.475301  2.417968  2.361962  2.307253  2.253812  2.201609
##  [99]  2.150614  2.100801  2.052142
## 
## 
## [[75]]
## [[75]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[75]]$y
##   [1]  0.000000  4.319291  7.583954 10.028992 11.837523 13.152224 14.084248
##   [8] 14.720185 15.127473 15.358627 15.454533 15.447008 15.360808 15.215182
##  [15] 15.025088 14.802143 14.555359 14.291717 14.016619 13.734236 13.447780
##  [22] 13.159717 12.871930 12.585854 12.302570 12.022887 11.747401 11.476545
##  [29] 11.210622 10.949841 10.694330 10.444162 10.199362  9.959924  9.725814
##  [36]  9.496978  9.273348  9.054845  8.841383  8.632868  8.429205  8.230296
##  [43]  8.036040  7.846339  7.661091  7.480198  7.303561  7.131084  6.962671
##  [50]  6.798229  6.637664  6.480888  6.327812  6.178348  6.032413  5.889923
##  [57]  5.750798  5.614958  5.482326  5.352827  5.226386  5.102931  4.982392
##  [64]  4.864701  4.749789  4.637591  4.528044  4.421084  4.316651  4.214684
##  [71]  4.115126  4.017920  3.923010  3.830342  3.739863  3.651521  3.565266
##  [78]  3.481048  3.398820  3.318534  3.240145  3.163607  3.088877  3.015913
##  [85]  2.944672  2.875114  2.807199  2.740888  2.676143  2.612928  2.551206
##  [92]  2.490943  2.432102  2.374652  2.318559  2.263790  2.210316  2.158104
##  [99]  2.107126  2.057352  2.008754
## 
## 
## [[76]]
## [[76]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[76]]$y
##   [1]  0.000000  4.491773  7.857471 10.355322 12.184856 13.500228 14.420533
##   [8] 15.037763 15.422964 15.630999 15.704217 15.675302 15.569466 15.406147
##  [15] 15.200324 14.963528 14.704626 14.430430 14.146162 13.855816 13.562439
##  [22] 13.268348 12.975294 12.684593 12.397227 12.113916 11.835185 11.561404
##  [29] 11.292826 11.029614 10.771862 10.519612 10.272867 10.031597  9.795754
##  [36]  9.565269  9.340064  9.120050  8.905133  8.695214  8.490192  8.289965
##  [43]  8.094429  7.903482  7.717021  7.534944  7.357153  7.183548  7.014033
##  [50]  6.848514  6.686896  6.529090  6.375005  6.224555  6.077654  5.934219
##  [57]  5.794168  5.657422  5.523902  5.393534  5.266242  5.141954  5.020599
##  [64]  4.902107  4.786413  4.673449  4.563150  4.455455  4.350302  4.247630
##  [71]  4.147382  4.049499  3.953926  3.860610  3.769495  3.680531  3.593666
##  [78]  3.508852  3.426039  3.345181  3.266231  3.189145  3.113877  3.040386
##  [85]  2.968630  2.898567  2.830158  2.763363  2.698145  2.634466  2.572289
##  [92]  2.511581  2.452304  2.394427  2.337916  2.282739  2.228864  2.176260
##  [99]  2.124898  2.074748  2.025782
## 
## 
## [[77]]
## [[77]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[77]]$y
##   [1]  0.000000  3.808277  6.737471  8.971437 10.655991 11.906772 12.815475
##   [8] 13.454802 13.882383 14.143897 14.275536 14.305970 14.257903 14.149303
##  [15] 13.994383 13.804379 13.588166 13.352745 13.103633 12.845168 12.580757
##  [22] 12.313066 12.044177 11.775706 11.508904 11.244728 10.983910 10.726996
##  [29] 10.474391 10.226387  9.983186  9.744922  9.511672  9.283473  9.060325
##  [36]  8.842207  8.629073  8.420866  8.217515  8.018942  7.825062  7.635786
##  [43]  7.451022  7.270676  7.094654  6.922860  6.755199  6.591579  6.431905
##  [50]  6.276085  6.124030  5.975651  5.830861  5.689573  5.551705  5.417174
##  [57]  5.285901  5.157807  5.032815  4.910851  4.791842  4.675716  4.562403
##  [64]  4.451836  4.343949  4.238675  4.135952  4.035719  3.937915  3.842480
##  [71]  3.749359  3.658494  3.569831  3.483317  3.398899  3.316528  3.236152
##  [78]  3.157724  3.081197  3.006525  2.933662  2.862566  2.793192  2.725499
##  [85]  2.659447  2.594996  2.532106  2.470741  2.410863  2.352436  2.295425
##  [92]  2.239796  2.185515  2.132549  2.080867  2.030437  1.981230  1.933215
##  [99]  1.886364  1.840648  1.796040
## 
## 
## [[78]]
## [[78]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[78]]$y
##   [1]  0.000000  3.975311  7.032370  9.362817 11.118751 12.420880 13.364987
##   [8] 14.027067 14.467403 14.733808 14.864191 14.888599 14.830840 14.709769
##  [15] 14.540306 14.334253 14.100935 13.847710 13.580378 13.303501 13.020661
##  [22] 12.734659 12.447684 12.161431 11.877212 11.596031 11.318649 11.045637
##  [29] 10.777413 10.514276 10.256433 10.004016  9.757098  9.515710  9.279845
##  [36]  9.049469  8.824531  8.604959  8.390672  8.181580  7.977586  7.778589
##  [43]  7.584485  7.395169  7.210533  7.030472  6.854879  6.683650  6.516679
##  [50]  6.353866  6.195110  6.040311  5.889374  5.742202  5.598704  5.458788
##  [57]  5.322366  5.189351  5.059658  4.933206  4.809913  4.689700  4.572491
##  [64]  4.458210  4.346786  4.238146  4.132221  4.028943  3.928247  3.830067
##  [71]  3.734340  3.641006  3.550005  3.461278  3.374769  3.290422  3.208182
##  [78]  3.127999  3.049819  2.973593  2.899273  2.826810  2.756158  2.687272
##  [85]  2.620108  2.554622  2.490773  2.428520  2.367822  2.308642  2.250941
##  [92]  2.194682  2.139829  2.086347  2.034202  1.983360  1.933789  1.885457
##  [99]  1.838333  1.792386  1.747588
## 
## 
## [[79]]
## [[79]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[79]]$y
##   [1]  0.000000  3.879100  6.844271  9.090396 10.771256 12.008192 12.896936
##   [8] 13.513005 13.915957 14.152756 14.260426 14.268141 14.198885 14.070750
##  [15] 13.897973 13.691749 13.460867 13.212225 12.951222 12.682082 12.408098
##  [22] 12.131829 11.855257 11.579910 11.306956 11.037284 10.771558 10.510269
##  [29] 10.253774 10.002318  9.756068  9.515121  9.279527  9.049293  8.824398
##  [36]  8.604798  8.390431  8.181221  7.977082  7.777923  7.583646  7.394151
##  [43]  7.209334  7.029093  6.853323  6.681920  6.514782  6.351808  6.192897
##  [50]  6.037952  5.886874  5.739570  5.595946  5.455913  5.319380  5.186261
##  [57]  5.056472  4.929929  4.806551  4.686261  4.568980  4.454633  4.343148
##  [64]  4.234452  4.128477  4.025153  3.924415  3.826198  3.730439  3.637077
##  [71]  3.546051  3.457303  3.370776  3.286415  3.204165  3.123974  3.045789
##  [78]  2.969561  2.895241  2.822781  2.752134  2.683256  2.616101  2.550627
##  [85]  2.486792  2.424554  2.363874  2.304713  2.247032  2.190795  2.135965
##  [92]  2.082508  2.030388  1.979573  1.930030  1.881726  1.834632  1.788716
##  [99]  1.743949  1.700303  1.657749
## 
## 
## [[80]]
## [[80]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[80]]$y
##   [1]  0.000000  3.925841  6.926053  9.198623 10.899634 12.152122 13.053063
##   [8] 13.678881 14.089797 14.333250 14.446605 14.459279 14.394427 14.270262
##  [15] 14.101104 13.898201 13.670382 13.424570 13.166181 12.899450 12.627677
##  [22] 12.353426 12.078685 11.804983 11.533493 11.265107 11.000492 10.740144
##  [29] 10.484424 10.233582  9.987788  9.747146  9.511710  9.281493  9.056480
##  [36]  8.836632  8.621892  8.412190  8.207446  8.007573  7.812478  7.622068
##  [43]  7.436243  7.254904  7.077954  6.905293  6.736822  6.572445  6.412065
##  [50]  6.255589  6.102922  5.953975  5.808658  5.666884  5.528567  5.393624
##  [57]  5.261972  5.133532  5.008226  4.885977  4.766712  4.650358  4.536843
##  [64]  4.426099  4.318058  4.212653  4.109822  4.009500  3.911628  3.816144
##  [71]  3.722991  3.632112  3.543451  3.456954  3.372569  3.290243  3.209927
##  [78]  3.131572  3.055129  2.980553  2.907796  2.836816  2.767569  2.700011
##  [85]  2.634103  2.569804  2.507074  2.445876  2.386171  2.327924  2.271098
##  [92]  2.215660  2.161575  2.108810  2.057334  2.007113  1.958119  1.910321
##  [99]  1.863689  1.818196  1.773813
## 
## 
## [[81]]
## [[81]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[81]]$y
##   [1]  0.000000  4.450820  7.814665 10.333087 12.194482 13.545780 14.501556
##   [8] 15.151145 15.564189 15.794967 15.885774 15.869553 15.771954 15.612938
##  [15] 15.408030 15.169291 14.906083 14.625666 14.333656 14.034390 13.731210
##  [22] 13.426678 13.122755 12.820926 12.522314 12.227753 11.937860 11.653076
##  [29] 11.373711 11.099971 10.831983 10.569814 10.313481 10.062968  9.818231
##  [36]  9.579204  9.345809  9.117952  8.895537  8.678458  8.466606  8.259871
##  [43]  8.058142  7.861306  7.669253  7.481872  7.299053  7.120688  6.946673
##  [50]  6.776903  6.611277  6.449693  6.292055  6.138268  5.988236  5.841871
##  [57]  5.699081  5.559781  5.423884  5.291309  5.161974  5.035799  4.912709
##  [64]  4.792627  4.675480  4.561196  4.449706  4.340940  4.234834  4.131320
##  [71]  4.030337  3.931823  3.835716  3.741958  3.650492  3.561262  3.474213
##  [78]  3.389292  3.306446  3.225626  3.146781  3.069863  2.994825  2.921622
##  [85]  2.850208  2.780539  2.712573  2.646269  2.581585  2.518483  2.456923
##  [92]  2.396867  2.338280  2.281125  2.225366  2.170971  2.117905  2.066137
##  [99]  2.015633  1.966364  1.918300
## 
## 
## [[82]]
## [[82]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[82]]$y
##   [1]  0.000000  4.419211  7.823516 10.424686 12.390748 13.855015 14.923247
##   [8] 15.679335 16.189813 16.507441 16.674047 16.722783 16.679914 16.566244
##  [15] 16.398239 16.188926 15.948601 15.685397 15.405726 15.114640 14.816108
##  [22] 14.513246 14.208486 13.903726 13.600434 13.299745 13.002521 12.709419
##  [29] 12.420924 12.137392 11.859075 11.586141 11.318698 11.056800 10.800464
##  [36] 10.549675 10.304396 10.064571  9.830129  9.600992  9.377071  9.158274
##  [43]  8.944504  8.735661  8.531645  8.332355  8.137689  7.947545  7.761826
##  [50]  7.580430  7.403262  7.230224  7.061223  6.896166  6.734963  6.577523
##  [57]  6.423762  6.273592  6.126930  5.983696  5.843809  5.707192  5.573767
##  [64]  5.443461  5.316201  5.191916  5.070536  4.951993  4.836222  4.723157
##  [71]  4.612735  4.504895  4.399576  4.296719  4.196266  4.098162  4.002352
##  [78]  3.908782  3.817399  3.728152  3.640992  3.555870  3.472737  3.391549
##  [85]  3.312258  3.234821  3.159194  3.085336  3.013204  2.942759  2.873960
##  [92]  2.806770  2.741151  2.677066  2.614479  2.553355  2.493661  2.435362
##  [99]  2.378426  2.322821  2.268516
## 
## 
## [[83]]
## [[83]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[83]]$y
##   [1]  0.000000  5.026844  8.833085 11.689524 13.807415 15.351554 16.450479
##   [8] 17.204436 17.691596 17.972905 18.095868 18.097503 18.006643 17.845740
##  [15] 17.632264 17.379799 17.098899 16.797754 16.482709 16.158671 15.829428
##  [22] 15.497893 15.166298 14.836344 14.509322 14.186198 13.867690 13.554322
##  [29] 13.246467 12.944380 12.648229 12.358109 12.074064 11.796094 11.524169
##  [36] 11.258238 10.998228 10.744055 10.495626 10.252838 10.015587  9.783763
##  [43]  9.557257  9.335957  9.119752  8.908530  8.702182  8.500601  8.303677
##  [50]  8.111307  7.923387  7.739815  7.560493  7.385322  7.214207  7.047054
##  [57]  6.883773  6.724275  6.568470  6.416276  6.267607  6.122382  5.980522
##  [64]  5.841949  5.706586  5.574360  5.445197  5.319027  5.195781  5.075390
##  [71]  4.957788  4.842912  4.730697  4.621083  4.514008  4.409414  4.307244
##  [78]  4.207441  4.109951  4.014719  3.921694  3.830825  3.742061  3.655354
##  [85]  3.570656  3.487921  3.407102  3.328156  3.251040  3.175710  3.102126
##  [92]  3.030247  2.960033  2.891446  2.824449  2.759004  2.695075  2.632628
##  [99]  2.571627  2.512040  2.453834
## 
## 
## [[84]]
## [[84]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[84]]$y
##   [1]  0.000000  4.153120  7.334882  9.751155 11.564647 12.903938 13.870629
##   [8] 14.544992 14.990438 15.257045 15.384355 15.403579 15.339345 15.211072
##  [15] 15.034069 14.820391 14.579524 14.318925 14.044444 13.760666 13.471175
##  [22] 13.178762 12.885598 12.593359 12.303333 12.016502 11.733608 11.455203
##  [29] 11.181688 10.913349 10.650378 10.392899 10.140975  9.894630  9.653851
##  [36]  9.418601  9.188823  8.964444  8.745381  8.531542  8.322831  8.119144
##  [43]  7.920379  7.726429  7.537189  7.352552  7.172413  6.996669  6.825214
##  [50]  6.657949  6.494774  6.335590  6.180301  6.028814  5.881036  5.736877
##  [57]  5.596250  5.459068  5.325248  5.194706  5.067364  4.943142  4.821966
##  [64]  4.703759  4.588450  4.475967  4.366241  4.259205  4.154793  4.052940
##  [71]  3.953584  3.856664  3.762119  3.669893  3.579927  3.492166  3.406557
##  [78]  3.323047  3.241584  3.162117  3.084599  3.008981  2.935217  2.863262
##  [85]  2.793070  2.724599  2.657806  2.592651  2.529093  2.467094  2.406614
##  [92]  2.347616  2.290066  2.233925  2.179162  2.125740  2.073629  2.022794
##  [99]  1.973206  1.924834  1.877647
## 
## 
## [[85]]
## [[85]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[85]]$y
##   [1]  0.000000  4.445120  7.808199 10.328874 12.194234 13.550326 14.511148
##   [8] 15.165673 15.583323 15.818263 15.912736 15.899679 15.804763 15.647984
##  [15] 15.444907 15.207641 14.945594 14.666064 14.374710 14.075902 13.773012
##  [22] 13.468630 13.164739 12.862845 12.564087 12.269314 11.979153 11.694056
##  [29] 11.414341 11.140223 10.871832 10.609241 10.352471 10.101508  9.856312
##  [36]  9.616820  9.382953  9.154622  8.931730  8.714172  8.501842  8.294629
##  [43]  8.092422  7.895110  7.702583  7.514730  7.331441  7.152611  6.978132
##  [50]  6.807902  6.641819  6.479782  6.321695  6.167462  6.016989  5.870186
##  [57]  5.726964  5.587234  5.450913  5.317917  5.188166  5.061580  4.938083
##  [64]  4.817598  4.700053  4.585376  4.473496  4.364347  4.257860  4.153972
##  [71]  4.052618  3.953737  3.857269  3.763155  3.671336  3.581759  3.494366
##  [78]  3.409106  3.325927  3.244776  3.165606  3.088368  3.013014  2.939499
##  [85]  2.867777  2.797805  2.729541  2.662942  2.597968  2.534580  2.472738
##  [92]  2.412405  2.353544  2.296119  2.240096  2.185439  2.132116  2.080094
##  [99]  2.029341  1.979826  1.931520
## 
## 
## [[86]]
## [[86]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[86]]$y
##   [1]  0.000000  4.338103  7.636400 10.122084 11.973174 13.329151 14.299285
##   [8] 14.969169 15.405844 15.661803 15.778146 15.787035 15.713636 15.577628
##  [15] 15.394390 15.175933 14.931631 14.668788 14.393090 14.108952 13.819799
##  [22] 13.528276 13.236419 12.945791 12.657579 12.372681 12.091767 11.815328
##  [29] 11.543720 11.277189 11.015897 10.759944 10.509377 10.264205 10.024408
##  [36]  9.789943  9.560750  9.336758  9.117883  8.904037  8.695127  8.491056
##  [43]  8.291725  8.097035  7.906886  7.721179  7.539816  7.362698  7.189729
##  [50]  7.020814  6.855861  6.694779  6.537476  6.383867  6.233864  6.087383
##  [57]  5.944343  5.804663  5.668264  5.535069  5.405004  5.277994  5.153969
##  [64]  5.032858  4.914592  4.799105  4.686332  4.576209  4.468674  4.363665
##  [71]  4.261124  4.160993  4.063214  3.967733  3.874496  3.783450  3.694543
##  [78]  3.607726  3.522948  3.440163  3.359323  3.280383  3.203297  3.128023
##  [85]  3.054518  2.982741  2.912650  2.844206  2.777370  2.712105  2.648373
##  [92]  2.586140  2.525368  2.466025  2.408076  2.351489  2.296232  2.242273
##  [99]  2.189582  2.138129  2.087886
## 
## 
## [[87]]
## [[87]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[87]]$y
##   [1]  0.000000  4.399362  7.714708 10.190623 12.016973 13.341138 14.277509
##   [8] 14.914860 15.322077 15.552603 15.647891 15.640084 15.554096 15.409232
##  [15] 15.220439 14.999283 14.754705 14.493608 14.221315 13.941920 13.658568
##  [22] 13.373664 13.089040 12.806087 12.525850 12.249108 11.976434 11.708242
##  [29] 11.444825 11.186378 10.933025 10.684834 10.441831 10.204007  9.971330
##  [36]  9.743749  9.521201  9.303609  9.090892  8.882962  8.679729  8.481100
##  [43]  8.286981  8.097278  7.911895  7.730740  7.553720  7.380744  7.211721
##  [50]  7.046562  6.885181  6.727492  6.573412  6.422859  6.275752  6.132013
##  [57]  5.991565  5.854334  5.720245  5.589226  5.461208  5.336122  5.213901
##  [64]  5.094479  4.977792  4.863778  4.752375  4.643523  4.537165  4.433243
##  [71]  4.331701  4.232485  4.135542  4.040818  3.948265  3.857831  3.769469
##  [78]  3.683130  3.598769  3.516341  3.435800  3.357104  3.280211  3.205079
##  [85]  3.131668  3.059938  2.989851  2.921369  2.854456  2.789076  2.725193
##  [92]  2.662773  2.601783  2.542190  2.483962  2.427068  2.371477  2.317159
##  [99]  2.264085  2.212227  2.161557
## 
## 
## [[88]]
## [[88]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[88]]$y
##   [1]  0.000000  4.021914  7.102875  9.443535 11.202162 12.503582 13.446239
##   [8] 14.107762 14.549354 14.819266 14.955521 14.988082 14.940549 14.831502
##  [15] 14.675563 14.484232 14.266544 14.029593 13.778940 13.518939 13.252993
##  [22] 12.983752 12.713275 12.443157 12.174625 11.908616 11.645841 11.386834
##  [29] 11.131985 10.881578 10.635808 10.394804 10.158641  9.927355  9.700950
##  [36]  9.479405  9.262679  9.050719  8.843460  8.640830  8.442749  8.249137
##  [43]  8.059906  7.874970  7.694242  7.517633  7.345056  7.176422  7.011646
##  [50]  6.850643  6.693328  6.539618  6.389433  6.242693  6.099320  5.959236
##  [57]  5.822368  5.688642  5.557986  5.430329  5.305604  5.183743  5.064681
##  [64]  4.948352  4.834696  4.723649  4.615153  4.509149  4.405579  4.304388
##  [71]  4.205522  4.108926  4.014549  3.922339  3.832248  3.744225  3.658225
##  [78]  3.574200  3.492104  3.411895  3.333527  3.256960  3.182151  3.109061
##  [85]  3.037649  2.967878  2.899709  2.833106  2.768033  2.704454  2.642336
##  [92]  2.581645  2.522347  2.464412  2.407807  2.352502  2.298468  2.245675
##  [99]  2.194094  2.143698  2.094460
## 
## 
## [[89]]
## [[89]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[89]]$y
##   [1]  0.000000  4.612623  8.115180 10.751840 12.713541 14.149582 15.176696
##   [8] 15.886141 16.349250 16.621771 16.747265 16.759756 16.685816 16.546181
##  [15] 16.357031 16.130976 15.877842 15.605270 15.319200 15.024238 14.723948
##  [22] 14.421083 14.117758 13.815592 13.515818 13.219364 12.926926 12.639014
##  [29] 12.355995 12.078126 11.805577 11.538452 11.276804 11.020646 10.769958
##  [36] 10.524702 10.284818 10.050236  9.820874  9.596645  9.377457  9.163212
##  [43]  8.953813  8.749162  8.549158  8.353704  8.162699  7.976048  7.793653
##  [50]  7.615421  7.441258  7.271073  7.104775  6.942278  6.783495  6.628341
##  [57]  6.476735  6.328595  6.183843  6.042400  5.904192  5.769145  5.637187
##  [64]  5.508247  5.382255  5.259145  5.138852  5.021309  4.906455  4.794228
##  [71]  4.684568  4.577416  4.472715  4.370409  4.270443  4.172764  4.077319
##  [78]  3.984057  3.892928  3.803884  3.716876  3.631858  3.548785  3.467613
##  [85]  3.388297  3.310795  3.235066  3.161069  3.088765  3.018114  2.949080
##  [92]  2.881624  2.815712  2.751307  2.688375  2.626883  2.566797  2.508086
##  [99]  2.450717  2.394661  2.339887
## 
## 
## [[90]]
## [[90]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[90]]$y
##   [1]  0.000000  4.320847  7.579883 10.014180 11.808417 13.106441 14.020287
##   [8] 14.637194 15.025089 15.236841 15.313596 15.287358 15.183013 15.019900
##  [15] 14.813037 14.574075 14.312046 14.033937 13.745146 13.449834 13.151198
##  [22] 12.851686 12.553161 12.257035 11.964367 11.675941 11.392332 11.113949
##  [29] 10.841072 10.573887 10.312500 10.056963  9.807281  9.563426  9.325346
##  [36]  9.092966  8.866202  8.644956  8.429124  8.218597  8.013264  7.813011
##  [43]  7.617722  7.427284  7.241583  7.060506  6.883943  6.711784  6.543921
##  [50]  6.380250  6.220667  6.065072  5.913365  5.765450  5.621233  5.480622
##  [57]  5.343527  5.209861  5.079537  4.952473  4.828587  4.707799  4.590033
##  [64]  4.475213  4.363264  4.254116  4.147698  4.043943  3.942782  3.844152
##  [71]  3.747990  3.654232  3.562821  3.473695  3.386800  3.302078  3.219475
##  [78]  3.138939  3.060417  2.983860  2.909218  2.836442  2.765488  2.696308
##  [85]  2.628859  2.563097  2.498981  2.436468  2.375519  2.316094  2.258156
##  [92]  2.201668  2.146592  2.092894  2.040540  1.989495  1.939727  1.891204
##  [99]  1.843895  1.797769  1.752798
## 
## 
## [[91]]
## [[91]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[91]]$y
##   [1]  0.000000  4.312748  7.619455 10.133841 12.024661 13.425154 14.440509
##   [8] 15.153762 15.630449 15.922288 16.070081 16.106013 16.055458 15.938418
##  [15] 15.770646 15.564546 15.329875 15.074298 14.803833 14.523195 14.236072
##  [22] 13.945340 13.653238 13.361496 13.071452 12.784127 12.500296 12.220541
##  [29] 11.945290 11.674853 11.409444 11.149205 10.894220 10.644527 10.400133
##  [36] 10.161013  9.927127  9.698414  9.474806  9.256222  9.042578  8.833783
##  [43]  8.629746  8.430370  8.235560  8.045220  7.859255  7.677568  7.500065
##  [50]  7.326654  7.157243  6.991742  6.830061  6.672114  6.517816  6.367083
##  [57]  6.219834  6.075988  5.935468  5.798196  5.664098  5.533101  5.405133
##  [64]  5.280123  5.158005  5.038711  4.922176  4.808335  4.697128  4.588492
##  [71]  4.482369  4.378700  4.277429  4.178500  4.081859  3.987453  3.895230
##  [78]  3.805141  3.717135  3.631164  3.547182  3.465142  3.384999  3.306710
##  [85]  3.230232  3.155523  3.082541  3.011247  2.941603  2.873569  2.807108
##  [92]  2.742185  2.678763  2.616808  2.556286  2.497164  2.439409  2.382990
##  [99]  2.327875  2.274036  2.221441
## 
## 
## [[92]]
## [[92]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[92]]$y
##   [1]  0.000000  4.198493  7.379128  9.768796 11.544198 12.842943 13.772184
##   [8] 14.415355 14.837411 15.088912 15.209210 15.228922 15.171865 15.056556
##  [15] 14.897387 14.705538 14.489685 14.256555 14.011359 13.758125 13.499962
##  [22] 13.239265 12.977869 12.717178 12.458258 12.201914 11.948747 11.699200
##  [29] 11.453593 11.212154 10.975033 10.742327 10.514089 10.290335 10.071060
##  [36]  9.856236  9.645823  9.439768  9.238011  9.040484  8.847118  8.657838
##  [43]  8.472569  8.291235  8.113759  7.940063  7.770072  7.603709  7.440900
##  [50]  7.281569  7.125646  6.973057  6.823732  6.677603  6.534601  6.394660
##  [57]  6.257715  6.123702  5.992558  5.864222  5.738634  5.615735  5.495468
##  [64]  5.377776  5.262605  5.149900  5.039609  4.931680  4.826062  4.722706
##  [71]  4.621563  4.522587  4.425730  4.330947  4.238195  4.147428  4.058606
##  [78]  3.971686  3.886627  3.803390  3.721936  3.642226  3.564223  3.487891
##  [85]  3.413193  3.340095  3.268563  3.198562  3.130061  3.063027  2.997428
##  [92]  2.933235  2.870416  2.808942  2.748785  2.689916  2.632308  2.575934
##  [99]  2.520767  2.466782  2.413953
## 
## 
## [[93]]
## [[93]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[93]]$y
##   [1]  0.000000  4.123289  7.272211  9.655793 11.438680 12.750552 13.693535
##   [8] 14.348049 14.777405 15.031442 15.149379 15.162069 15.093777 14.963573
##  [15] 14.786439 14.574136 14.335886 14.078914 13.808869 13.530165 13.246237
##  [22] 12.959755 12.672783 12.386912 12.103360 11.823049 11.546674 11.274748
##  [29] 11.007641 10.745615 10.488844 10.237435  9.991443  9.750882  9.515734
##  [36]  9.285960  9.061499  8.842279  8.628216  8.419222  8.215199  8.016048
##  [43]  7.821670  7.631961  7.446818  7.266139  7.089822  6.917766  6.749873
##  [50]  6.586043  6.426182  6.270194  6.117988  5.969472  5.824558  5.683160
##  [57]  5.545192  5.410572  5.279219  5.151054  5.026000  4.903981  4.784924
##  [64]  4.668757  4.555409  4.444814  4.336903  4.231612  4.128877  4.028637
##  [71]  3.930829  3.835397  3.742281  3.651426  3.562776  3.476279  3.391882
##  [78]  3.309534  3.229185  3.150786  3.074291  2.999653  2.926828  2.855770
##  [85]  2.786437  2.718788  2.652781  2.588377  2.525536  2.464221  2.404394
##  [92]  2.346020  2.289064  2.233490  2.179265  2.126356  2.074733  2.024362
##  [99]  1.975215  1.927260  1.880470
## 
## 
## [[94]]
## [[94]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[94]]$y
##   [1]  0.000000  4.307001  7.646770 10.215763 12.170979 13.637875 14.716691
##   [8] 15.487510 16.014300 16.348149 16.529846 16.591952 16.560449 16.456058
##  [15] 16.295300 16.091338 15.854647 15.593561 15.314699 15.023312 14.723556
##  [22] 14.418714 14.111374 13.803566 13.496875 13.192535 12.891496 12.594482
##  [29] 12.302042 12.014578 11.732384 11.455660 11.184539 10.919095 10.659361
##  [36] 10.405334 10.156983  9.914258  9.677093  9.445408  9.219117  8.998123
##  [43]  8.782329  8.571632  8.365926  8.165107  7.969068  7.777704  7.590909
##  [50]  7.408580  7.230614  7.056910  6.887369  6.721892  6.560384  6.402752
##  [57]  6.248902  6.098747  5.952196  5.809165  5.669569  5.533327  5.400357
##  [64]  5.270581  5.143924  5.020310  4.899665  4.781920  4.667004  4.554849
##  [71]  4.445390  4.338560  4.234298  4.132542  4.033230  3.936305  3.841710
##  [78]  3.749387  3.659284  3.571345  3.485520  3.401757  3.320008  3.240222
##  [85]  3.162355  3.086358  3.012188  2.939800  2.869152  2.800201  2.732908
##  [92]  2.667232  2.603134  2.540576  2.479522  2.419935  2.361780  2.305023
##  [99]  2.249629  2.195567  2.142804
## 
## 
## [[95]]
## [[95]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[95]]$y
##   [1]  0.000000  4.020660  7.158306  9.587887 11.450097 12.858073 13.902783
##   [8] 14.657362 15.180596 15.519731 15.712730 15.790085 15.776284 15.690980
##  [15] 15.549943 15.365815 15.148725 14.906781 14.646467 14.372960 14.090384
##  [22] 13.802024 13.510483 13.217820 12.925659 12.635271 12.347645 12.063545
##  [29] 11.783555 11.508114 11.237543 10.972073 10.711862 10.457006 10.207560
##  [36]  9.963536  9.724923  9.491683  9.263763  9.041096  8.823603  8.611199
##  [43]  8.403794  8.201291  8.003594  7.810603  7.622218  7.438337  7.258861
##  [50]  7.083691  6.912728  6.745874  6.583036  6.424117  6.269026  6.117673
##  [57]  5.969968  5.825825  5.685159  5.547887  5.413926  5.283199  5.155626
##  [64]  5.031133  4.909645  4.791090  4.675397  4.562498  4.452324  4.344810
##  [71]  4.239892  4.137508  4.037596  3.940096  3.844950  3.752102  3.661497
##  [78]  3.573079  3.486796  3.402596  3.320430  3.240248  3.162002  3.085646
##  [85]  3.011133  2.938420  2.867463  2.798219  2.730647  2.664707  2.600360
##  [92]  2.537566  2.476289  2.416491  2.358137  2.301193  2.245623  2.191395
##  [99]  2.138477  2.086837  2.036444
## 
## 
## [[96]]
## [[96]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[96]]$y
##   [1]  0.000000  3.952865  6.960896  9.230080 10.921914 12.163006 13.052596
##   [8] 13.668471 14.071588 14.309709 14.420247 14.432502 14.369411 14.248923
##  [15] 14.085073 13.888831 13.668763 13.431548 13.182392 12.925338 12.663527
##  [22] 12.399386 12.134787 11.871166 11.609617 11.350970 11.095844 10.844697
##  [29] 10.597858 10.355560 10.117956  9.885141  9.657164  9.434037  9.215746
##  [36]  9.002256  8.793515  8.589461  8.390020  8.195116  8.004666  7.818583
##  [43]  7.636780  7.459169  7.285661  7.116166  6.950598  6.788868  6.630890
##  [50]  6.476581  6.325856  6.178633  6.034833  5.894376  5.757186  5.623188
##  [57]  5.492306  5.364470  5.239608  5.117651  4.998533  4.882187  4.768548
##  [64]  4.657555  4.549144  4.443257  4.339835  4.238819  4.140155  4.043787
##  [71]  3.949662  3.857728  3.767934  3.680230  3.594568  3.510899  3.429178
##  [78]  3.349359  3.271398  3.195252  3.120878  3.048235  2.977283  2.907982
##  [85]  2.840295  2.774183  2.709610  2.646540  2.584938  2.524770  2.466002
##  [92]  2.408602  2.352539  2.297780  2.244296  2.192056  2.141033  2.091198
##  [99]  2.042522  1.994979  1.948543
## 
## 
## [[97]]
## [[97]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[97]]$y
##   [1]  0.000000  4.401813  7.767668 10.320149 12.234453 13.648463 14.670692
##   [8] 15.386527 15.863146 16.153398 16.298844 16.332165 16.279048 16.159676
##  [15] 15.989896 15.782150 15.546189 15.289657 15.018533 14.737488 14.450164
##  [22] 14.159394 13.867375 13.575803 13.285981 12.998903 12.715320 12.435795
##  [29] 12.160740 11.890451 11.625133 11.364919 11.109887 10.860073 10.615478
##  [36] 10.376079 10.141832  9.912679  9.688551  9.469370  9.255053  9.045512
##  [43]  8.840657  8.640395  8.444633  8.253277  8.066235  7.883414  7.704723
##  [50]  7.530071  7.359370  7.192531  7.029469  6.870100  6.714341  6.562110
##  [57]  6.413329  6.267919  6.125805  5.986912  5.851168  5.718501  5.588841
##  [64]  5.462121  5.338274  5.217234  5.098939  4.983326  4.870334  4.759904
##  [71]  4.651978  4.546499  4.443411  4.342661  4.244196  4.147962  4.053911
##  [78]  3.961993  3.872158  3.784361  3.698554  3.614693  3.532733  3.452631
##  [85]  3.374346  3.297836  3.223061  3.149981  3.078558  3.008754  2.940534
##  [92]  2.873860  2.808698  2.745013  2.682772  2.621943  2.562493  2.504391
##  [99]  2.447606  2.392109  2.337870
## 
## 
## [[98]]
## [[98]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[98]]$y
##   [1]  0.000000  3.998589  7.081571  9.439082 11.222187 12.550907 13.520580
##   [8] 14.206934 14.670110 14.957861 15.108094 15.150894 15.110127 15.004720
##  [15] 14.849673 14.656867 14.435707 14.193625 13.936492 13.668937 13.394601
##  [22] 13.116344 12.836403 12.556522 12.278055 12.002043 11.729285 11.460383
##  [29] 11.195786 10.935821 10.680719 10.430636 10.185669  9.945866  9.711242
##  [36]  9.481782  9.257448  9.038187  8.823933  8.614610  8.410134  8.210419
##  [43]  8.015372  7.824899  7.638906  7.457296  7.279974  7.106846  6.937815
##  [50]  6.772790  6.611679  6.454390  6.300836  6.150929  6.004584  5.861717
##  [57]  5.722247  5.586092  5.453175  5.323420  5.196750  5.073094  4.952380
##  [64]  4.834537  4.719498  4.607196  4.497565  4.390544  4.286068  4.184079
##  [71]  4.084516  3.987323  3.892442  3.799819  3.709400  3.621132  3.534965
##  [78]  3.450848  3.368732  3.288571  3.210317  3.133926  3.059352  2.986552
##  [85]  2.915485  2.846109  2.778384  2.712270  2.647730  2.584725  2.523220
##  [92]  2.463178  2.404565  2.347347  2.291490  2.236962  2.183732  2.131769
##  [99]  2.081042  2.031522  1.983180
## 
## 
## [[99]]
## [[99]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[99]]$y
##   [1]  0.000000  3.815924  6.749425  8.984208 10.666202 11.911320 12.811615
##   [8] 13.440186 13.855066 14.102319 14.218494 14.232584 14.167577 14.041689
##  [15] 13.869344 13.661957 13.428551 13.176249 12.910665 12.636217 12.356371
##  [22] 12.073839 11.790734 11.508697 11.228991 10.952580 10.680195 10.412379
##  [29] 10.149528  9.891923  9.639752  9.393134  9.152128  8.916753  8.686991
##  [36]  8.462801  8.244119  8.030868  7.822957  7.620290  7.422763  7.230268
##  [43]  7.042695  6.859933  6.681870  6.508393  6.339392  6.174758  6.014382
##  [50]  5.858157  5.705978  5.557745  5.413355  5.272711  5.135716  5.002277
##  [57]  4.872303  4.745703  4.622391  4.502282  4.385293  4.271343  4.160353
##  [64]  4.052246  3.946948  3.844386  3.744489  3.647187  3.552414  3.460103
##  [71]  3.370191  3.282615  3.197315  3.114231  3.033307  2.954485  2.877711
##  [78]  2.802932  2.730096  2.659153  2.590054  2.522750  2.457195  2.393344
##  [85]  2.331151  2.270575  2.211573  2.154104  2.098128  2.043607  1.990503
##  [92]  1.938779  1.888399  1.839328  1.791532  1.744978  1.699634  1.655468
##  [99]  1.612449  1.570549  1.529737
## 
## 
## [[100]]
## [[100]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[100]]$y
##   [1]  0.000000  4.851309  8.420542 11.017197 12.876732 14.178256 15.057997
##   [8] 15.619556 15.941704 16.084317 16.092895 16.001997 15.837861 15.620391
##  [15] 15.364675 15.082133 14.781400 14.468992 14.149812 13.827537 13.504914
##  [22] 13.183983 12.866246 12.552800 12.244430 11.941689 11.644954 11.354466
##  [29] 11.070369 10.792729 10.521556 10.256820  9.998457  9.746385  9.500501
##  [36]  9.260694  9.026844  8.798826  8.576513  8.359773  8.148479  7.942501
##  [43]  7.741710  7.545982  7.355191  7.169216  6.987937  6.811237  6.639002
##  [50]  6.471119  6.307479  6.147976  5.992505  5.840965  5.693256  5.549282
##  [57]  5.408948  5.272163  5.138837  5.008882  4.882214  4.758749  4.638406
##  [64]  4.521106  4.406773  4.295330  4.186707  4.080830  3.977630  3.877040
##  [71]  3.778995  3.683428  3.590278  3.499484  3.410986  3.324727  3.240648
##  [78]  3.158696  3.078816  3.000956  2.925065  2.851094  2.778993  2.708715
##  [85]  2.640215  2.573447  2.508368  2.444934  2.383104  2.322838  2.264096
##  [92]  2.206840  2.151031  2.096634  2.043613  1.991932  1.941558  1.892459
##  [99]  1.844601  1.797953  1.752485

Example 2: multi-factor experiments to build (hierarchical) logistic regression models for pharmaceutical salt formation

Four controllable variables:

• rate of agitation during mixing (\(x_1\))
• volume of composition (\(x_2\))
• temperature (\(x_3\))
• evaporation rate (\(x_4\))

For the \(j\)th observation in the \(i\)th group \((i=1,\ldots,g;\, j=1,\ldots,n_g)\): \[ y_{ij} \sim \mbox{Bernoulli}\left(\rho(\boldsymbol{x}_{ij})\right) \] with \[ \log\left(\frac{\rho(\boldsymbol{x}_{ij})}{1-\rho(\boldsymbol{x}_{ij})}\right) = \left(\beta_0 + \omega_{i0}\right) + \sum_{r=1}^k\left(\beta_r + \omega_{ir}\right)x_{ijr}\,, \] where \(x_{ijr}\) is the value taken by the \(r\)th variable.

• \(\boldsymbol{\beta}= (\beta_0,\beta_1,\ldots,\beta_{q-1})^\mathrm{T}\) are unknown parameters of interest
• \(\boldsymbol{\omega}_i = (\omega_{i0}, \omega_{i1}, \ldots, \omega_{iq-1})^\mathrm{T}\) are group specific parameters for the \(i\)th group

Prior distributions (for later use):

• \(\beta_0 \sim U(-3,3)\), \(\beta_1 \sim U(4, 10)\), \(\beta_2 \sim U(5, 11)\), \(\beta_3 \sim U(-6, 0)\), \(\beta_4 \sim U(-2.5, 3.5)\)

1. standard logistic regression - \(\omega_{ir} = 0\)
2. hierarchical logistic regression - \(\omega_{ir} \sim U(-s_r, +s_r)\). with \(s_{r}>0\) following a triangular distribution

Many Frequentist criteria for finding optimal designs for both linear and nonlinear models optimise a function of the information matrix; see Atkinson, Donev, and Tobias (2007), ch.10

• we have already seen \(D\)-optimality

Let \(\xi = (\boldsymbol{x}_1,\ldots,\boldsymbol{x}_n)^\mathrm{T}\) denote a design, and set \(M(\xi;\,\boldsymbol{\theta}) = M(\boldsymbol{\theta})\) to explicitly acknowledge the dependence of the information matrix on the design

• \(D\)-optimality: maximise \(\phi_D(\xi) = \mbox{det}\, M(\xi;\,\boldsymbol{\theta})\)
• \(A\)-optimality: minimise \(\phi_A(\xi) = \mbox{trace}\, M(\xi;\,\boldsymbol{\theta})^{-1}\)
• \(G\)-optimality: minimise \(\phi_G(\xi) = \max_\boldsymbol{x}\mbox{Var}(\hat{y}(\boldsymbol{x}))\)
  • where \(\hat{y}(x)\) is the predicted response at \(\boldsymbol{x}\) and the (asymptotic) prediction variance is a function of \(M(\xi;\,\boldsymbol{\theta})\)
• \(V\)- (or \(I\)-) optimality - minimise \(\phi_V(\xi) = \int_\mathcal{X} \mbox{Var}\left(\hat{y}(\boldsymbol{x})\right)\,\mathrm{d}\boldsymbol{x}\)

For most nonlinear models, \(M(\xi;\,\boldsymbol{\theta})\) will be a function of the unknown parameters \(\boldsymbol{\theta}\) (unlike for the linear model, where \(M(\xi;\,\boldsymbol{\beta}) = X^\mathrm{T}X / \sigma^2\))

This leads to a “chicken and egg” situation

• if you can tell me the values of the unknown parameters, I can give you an optimal design
• but if you knew the value of \(\boldsymbol{\theta}\), you probably wouldn’t need to perform the experiment!

For some models/experiments, the quality of a design may change a lot with the value of \(\boldsymbol{\theta}\)

rho <- function(x, beta0 = 0, beta1 = 1) {
  eta <- beta0 + beta1 * x
  1 / (1 + exp(-eta))
}
par(mar = c(8, 4, 1, 2) + 0.1)
curve(rho, from = -5, to = 5, ylab = expression(rho), xlab = expression(italic(x)), cex.lab = 1.5, 
      cex.axis = 1.5, ylim = c(0, 1), lwd = 2)

For simple logistic regression, the information matrix has the form \[ M(\xi;\,\boldsymbol{\beta}) = X^\mathrm{T}W X\,, \] with \(X\) the \(n\times 2\) model matrix and \(W = \mbox{diag}\left\{\rho(x_i)[1-\rho(x_i)]\right\}\)

For example with \(n=2\), \(\xi = (-1, 1)\), \(\beta_0=0\) and \(\beta_1 = 1\) \[ M(\xi;\,\boldsymbol{\beta}) = \left( \begin{array}{cc} 1 & 1 \\ -1 & 1 \end{array} \right) \left( \begin{array}{cc} 0.2 & 0 \\ 0 & 0.2 \end{array} \right) \left( \begin{array}{cc} 1 & -1 \\ 1 & 1 \end{array} \right) \]

Minfo <- function(xi, beta0 = 0, beta1 = 1) {
  X <- cbind(c(1, 1), xi)
  v <- function(x) rho(x, beta0, beta1) * (1 - rho(x, beta0, beta1))
  W <- diag(c(v(xi[1]), v(xi[2])))
  t(X) %*% W %*% X
}
Dcrit <- function(xi, beta0 = 0, beta1 = 1) {
  d <- det(Minfo(xi, beta0, beta1))
  ifelse(is.nan(d), -Inf, d)
}

Locally \(D\)-optimal designs

\(\beta_0 = 0, \beta_1 = 1\)

dopt <- optim(par = c(-1, 1), Dcrit, control = list(fnscale = -1))
xi.opt1 <- dopt$par
xi.opt1

## [1] -1.543421  1.543530

Locally \(D\)-optimal designs

\(\beta_0 = 0, \beta_1 = 2\)

dopt <- optim(par = c(-1, 1), Dcrit, control = list(fnscale = -1), beta1 = 2)
xi.opt2 <- dopt$par
xi.opt2

## [1] -0.7717705  0.7717418

Locally \(D\)-optimal designs

\(\beta_0 = 0, \beta_1 = 0.5\)

dopt <- optim(par = c(-1, 1), Dcrit, control = list(fnscale = -1), beta1 = .5)
xi.opt3 <- dopt$par
xi.opt3

## [1] -3.086913  3.086981

Getting \(\beta_1\) wrong: design for \(\beta_1 = .5\) when actually \(\beta_1 = 2\)

\(D\)-efficiency

(Dcrit(xi.opt3, beta1 = 2) / Dcrit(xi.opt2, beta1 = 2)) ^ (1 / 2)

## [1] 0.05720905

Use of the “wrong” design can lead to uninformative experiments (with “small” information matrices)

For the logistic regression example, the drop in efficiency is closely related to the phenomenon of separation (see Firth 1993)

Motivates the need for designs which are robust to the values of the model parameters

• maximin designs (focus on worst case performance)
• Bayesian designs

Decision-theoretic design starts with a utility function \(u(\xi,\boldsymbol{y},\boldsymbol{\theta})\) that defines the usefulness of a design for a particular purpose, given data \(\boldsymbol{y}\) and parameters \(\boldsymbol{\theta}\)

Common choices of utility function include

• negative squared error loss \[u(\xi, \boldsymbol{y}, \boldsymbol{\theta}) = -\left[\boldsymbol{\theta}- E(\boldsymbol{\theta}\,|\,\boldsymbol{y})\right]^2\]
  • negative squared difference between \(\boldsymbol{\theta}\) and the posterior mean
• surprisal or self information \[ \begin{split} u(\xi, \boldsymbol{y}, \boldsymbol{\theta}) & = \log \pi(\boldsymbol{\theta}\,|\,\boldsymbol{y},\xi) - \log \pi(\boldsymbol{\theta}) \\ & = \log \pi(\boldsymbol{y}\,|\,\boldsymbol{\theta},\xi) - \log \pi(\boldsymbol{y}\,|\,\xi) \end{split} \]
  • difference between log posterior and log prior densities, or between the log-likelihood and the log-evidence

A priori (before the experiment), we do not know \(\boldsymbol{y}\) or \(\boldsymbol{\theta}\) (we will never know \(\boldsymbol{\theta}\))

So, we take the expectation of the utility function with respect to the joint distribution of \(\boldsymbol{y},\boldsymbol{\theta}\)

\[ \begin{split} U(\xi) & = E_{\boldsymbol{y},\boldsymbol{\theta}\,|\,\xi}\left[u(\xi,\boldsymbol{y},\boldsymbol{\theta})\right]\\ & = \int u(\xi,\boldsymbol{y},\boldsymbol{\theta})\pi(\boldsymbol{y},\boldsymbol{\theta}\,|\,\xi)\,\mathrm{d}\boldsymbol{\theta}\,\mathrm{d}\boldsymbol{y}\\ & = \int u(\xi, \boldsymbol{y}, \boldsymbol{\theta})\pi(\boldsymbol{\theta}\,|\,\boldsymbol{y},\xi)\pi(\boldsymbol{y}\,|\,\xi)\,\mathrm{d}\boldsymbol{\theta}\,\mathrm{d}\boldsymbol{y}\\ & = \int u(\xi, \boldsymbol{y}, \boldsymbol{\theta})\pi(\boldsymbol{y}\,|\,\boldsymbol{\theta},\xi)\pi(\boldsymbol{\theta}\,|\,\xi)\,\mathrm{d}\boldsymbol{\theta}\,\mathrm{d}\boldsymbol{y} \end{split} \] The equivalence of the third and fourth equations follows from Bayes theorem

• the third equation more clearly shows the dependence on the posterior distribution
• the fourth equation is often more useful for calculations and computation

See Chaloner and Verdinelli (1995)

Surprisal \[ \begin{split} U(\xi) & = \int \log \frac{\pi(\boldsymbol{\theta}\,|\,\boldsymbol{y},\xi)}{\pi(\boldsymbol{\theta})}\pi(\boldsymbol{y},\boldsymbol{\theta}\,|\,\xi)\,\mathrm{d}\boldsymbol{\theta}\,\mathrm{d}\boldsymbol{y}\\ & = \int \log \frac{\pi(\boldsymbol{y}\,|\,\boldsymbol{\theta},\xi)}{\pi(\boldsymbol{y}\,|\,\xi)}\pi(\boldsymbol{y},\boldsymbol{\theta}\,|\,\xi)\,\mathrm{d}\boldsymbol{\theta}\,\mathrm{d}\boldsymbol{y} \end{split} \] - the expected Shannon information gain (SIG) or expected Kullback-Liebler divergence between prior and posterior densities

Negative squared error loss \[ \begin{split} U(\xi) & = - \int \left[\boldsymbol{\theta}- E(\boldsymbol{\theta}\,|\,\boldsymbol{y})\right]^2\pi(\boldsymbol{y},\boldsymbol{\theta}\,|\,\xi)\,\mathrm{d}\boldsymbol{\theta}\,\mathrm{d}\boldsymbol{y}\\ & = - \int \mbox{tr}\left\{\mbox{Var}(\boldsymbol{\theta}\,|\,\boldsymbol{y},\xi)\pi(\boldsymbol{y}\,|\,\xi)\right\}\,\mathrm{d}\boldsymbol{y} \end{split} \] - the expected negative squared error loss (NSEL)

In general, Bayesian design is easy in principle but hard in practice

1. For most nonlinear models, the expected utility will be intractable and involves high-dimensional integrals with respect to \(\boldsymbol{y}\) - often, obtaining the utility function itself requires the solution of intractable integrals (cf both ESIG and NSEL) - numerical or analytical approximation is required (eg Ryan et al. 2016)

2. A high-dimensional optimisation problem results for multi-factor experiments with many design points

For large \(n\), the inverse information matrix \(M(\xi;\,\boldsymbol{\theta})\) is an asymptotic approximation to the posterior variance-covariance matrix

Using this approximation, we can define Bayesian analogues of classical optimality criteria

\(D\)-optimality: maximise \[ U_D(\xi) = \int \log\mbox{det} M(\xi;\,\boldsymbol{\theta})\pi(\boldsymbol{\theta})\,\mathrm{d}\boldsymbol{\theta} \]

• approximation to ESIG

\(A\)-optimality: maximise \[ U_A(\xi) = - \int \mbox{tr} M^{-1}(\xi;\,\boldsymbol{\theta})\pi(\boldsymbol{\theta})\,\mathrm{d}\boldsymbol{\theta} \]

• approximation to NSEL

These integrals, with respect to \(\boldsymbol{\theta}\), are lower dimensional and more amenable to deterministic (quadrature) approximation, eg Gotwalt, Jones, and Steinberg (2009)

The acebayes package provides functions for constructing approximations to expected utilities

• default is to use quadrature to approximate the Bayesian \(D\)-optimality objective function

library(acebayes)
prior <- list(support = matrix(c(0, 0, .5, 2), nrow = 2))
logreg.util <- utilityglm(formula = ~ x, family = binomial, prior = prior)$utility
BDcrit <- function(xi) logreg.util(data.frame(x = xi))
bdopt <- optim(par = c(-1, 1), BDcrit, control = list(fnscale = -1))
bdopt$par

## [1] -1.202960  1.203132

As an alternative to analytical approximations, Monte Carlo approximation to the expected utility is simple to implement and intuitively appealing

\[ \tilde{U}(\xi) = \frac{1}{B}\sum_{i=1}^B\tilde{u}(\xi, \boldsymbol{y}_i, \boldsymbol{\theta}_i) \] where

• \(\left\{\boldsymbol{\theta}_h, \boldsymbol{y}_h\right\}_{h=1}^B\) is a random sample from \(\pi(\boldsymbol{\theta},\boldsymbol{y}\,|\,\xi)\)
• \(\tilde{u}(\xi,\boldsymbol{y},\boldsymbol{\theta})\) is, where necessary, an approximation to the utility function (often, nested Monte Carlo is required)

How to construct the approximation \(\tilde{u}(\xi,\boldsymbol{y},\boldsymbol{\theta})\) is an active area of research, eg Overstall, McGree, and Drovandi (2018), Beck et al. (2018)

Find an optimal design using Monte Carlo:

priorMC <- function(B) cbind(rep(0, B), runif(n = B, min = .5, max = 2))
logreg.utilSIG <- utilityglm(formula = ~ x, family = binomial, prior = priorMC, criterion = "SIG")$utility
BDcritSIG <- function(xi, B = 1000) mean(logreg.utilSIG(data.frame(x = xi), B))
bdoptSIG <- optim(par = c(-1, 1), BDcritSIG, control = list(fnscale = -1))
bdoptSIG$par

## [1] -1.001514  0.999707

bdopt$par

## [1] -1.202960  1.203132

Larger Monte Carlo sample sizes will produce results more similar to the design found using quadrature (in this example)

In general, direct optimisation of the Monte Carlo approximation requires large \(B\) to generate suitable smooth objective function and/or expensive stochastic algorithms (eg genetic algorithms)

Hamada et al. (2001)

Alternatively, the optimisation can be embedded within a simulation scheme and samples generated from the joint artificial distribution of \(\xi,\boldsymbol{y},\boldsymbol{\theta}\)

• take \(\xi^*\), the optimal design, to be the posterior mode of the marginal distribution
• most effective for small experiments (both numbers of variables and runs)

Müller (1999), Müller, Sansó, and De Iorio (2004)

Instead of directly minimising a Monte Carlo approximation to the expected utility, find designs via curve fitting (Müller and Parmigiani 1996)

1. Evaluate the Monte Carlo approximation \(\tilde{U}(\xi)\) for a small number of designs, \(\xi_1,\ldots,\xi_Q\)
2. Smooth the “data” \(\left\{\xi_i, \tilde{U}(\xi_i)\right\}\), i.e. fit a statistical model, to obtain a surrogate \(\hat{U}(\xi)\)
3. Find \(\xi\) that maximises \(\hat{U}(\xi)\)

Return to Example 1, compartmental model

• find a design with \(n=2\) runs, with fixed \(x_1 = 5\)
• use Monte Carlo approximation to SIG for 10 values of \(x_2\)

library(DiceKriging)
library(DiceDesign)
n <- 10; x1<- -0.583; x2 <- 2 * maximinSA_LHS(lhsDesign(n, 1)$design)$design- 1
u <- NULL; for(i in 1:n) u[i] <- mean(utilcomp15sig(c(x1, x2[i]), B = 1000))
par(mar = c(4, 4, 2, 2) + 0.1)
plot(12 * (x2 + 1), u, xlab = expression(x[2]), ylab = "Approx. expected SIG", xlim = c(0, 24), 
     ylim = c(0, 2), pch = 16, cex = 1.5); abline(v = 12 * (x1 + 1), lwd = 2)
usmooth <- km(design = 12 * (x2 + 1), response = u, nugget = 1e-3, control = list(trace = F))
xgrid <- matrix(seq(0, 24, l = 1000), ncol = 1); pred <- predict(usmooth, xgrid, type = "SK")$mean
lines(seq(0, 24, l = 1000), pred, col = "blue", lwd = 2); abline(v = xgrid[which.max(pred), ], lty = 2)

Coordinate exchange, a version of cyclic ascent, is a popular algorithm for finding optimal designs (Meyer and Nachtsheim 1995)

• optimisation of \(\xi = (\boldsymbol{x}_1,\ldots,\boldsymbol{x}_n)\) proceeds coordinate-wise, i.e. just one of the \(x_{ij}\) is varied at a time

Approximate coordinate exchange (ACE) combines coordinate exchange with smoothing to find high-dimensional designs under computationally expensive approximate expected utilities

• a nonparametric regression model (a Gaussian process) is used to smooth the Monte Carlo approximations of \(U(\xi)\) as a function of one coordinate
  • reduces the computational burden
  • facilitates optimisation of a noisy function

Overstall and Woods (2017)

Return to the multifactor logistic regression (crystallography) example

## set up prior
priorMFL <- function(B) {
  b0 <- runif(B, -3, 3)
  b1 <- runif(B, 4, 10)
  b2 <- runif(B, 5, 11)
  b3 <- runif(B, -6, 0)
  b4 <- runif(B, -2.5, 2.5)
  cbind(b0, b1, b2, b3, b4)
}
## define the utility function
MFL.utilSIG <- utilityglm(formula = ~ x1 + x2 + x3 + x4, family = binomial, prior = priorMFL, 
                          criterion = "SIG")$utility
## starting design with n=18 runs, on [-1, 1]
d <- 2 * randomLHS(18, 4) - 1
colnames(d) <- paste0("x", 1:4)

## not run - quite computationally expensive
MLF.ace <- ace(utility = MFL.utilSIG, start.d = d, progress = T)

For this logistic regression example, acebayes has some designs precomputed

pairs(optdeslrsig(18), pch = 16, 
      labels=c(expression(x[1]), expression(x[2]), expression(x[3]), expression(x[4])), cex = 2)

Hierachical logistic regression with \(g=3\) groups (blocks, eg wellplates)

pairs(optdeshlrsig(18), pch = 16, 
      labels=c(expression(x[1]), expression(x[2]), expression(x[3]), expression(x[4])),
      col = c("black", "red", "blue")[rep(1:3, rep(6, 3))], cex = 2)

Reasonably recent overviews of the topics discussed here, and many more, are given in the Handbook of Design and Analysis of Experiments (2015, eds Dean, Morris, Stufken, Bingham; CRC Press).

Some nice examples of recent experiments in technology, economics and social science are described by Luca and Bazerman (2020, The Power of Experiments; MIT Press).

Good online resources include

• Meier’s short intro to ANOVA
• Taback’s design of experiments and observational studies
• Krennrich’s experimental design and process optimisation in R