exponential Rosenbrock-type methods

(1)

Exp. Rosenbrock methods ReLPM Numerical experiments

Implementation of

exponential Rosenbrock-type methods

M. Caliari

^∗

A. Ostermann

^∗

3rd Austrian Numerical Analysis Day, TU Wien April 26–27, 2007

∗Universit¨at Innsbruck, Institut f¨ur Mathematik

M. Caliari, A. Ostermann Implementation of exponential Rosenbrock-type methods

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Exponential Rosenbrock-type methods: the idea

Let us consider

u

⁰

(t) = f (u(t)) = Au(t) + g (u(t)), u(t

₀

) = u

₀

(3)

Exponential Rosenbrock-type methods: the idea

Let us consider

u

⁰

(t) = f (u(t)) = Au(t) + g (u(t)), u(t

₀

) = u

₀

For the numerical approximation u

_n

of u(t

_n

) at time t

_n

, first we linearise at each step

u

⁰

(t) = J

n

u(t) + g

n

(u(t)), t

_n

≤ t ≤ t

n+1

where

J

_n

= D

u

f (u

n

), g

_n

(u(t)) = f (u(t)) − J

n

u(t)

(4)

Exponential Rosenbrock-type methods: the idea

Let us consider

u

⁰

(t) = f (u(t)) = Au(t) + g (u(t)), u(t

₀

) = u

₀

For the numerical approximation u

_n

of u(t

_n

) at time t

_n

, first we linearise at each step

u

⁰

(t) = J

n

u(t) + g

n

(u(t)), t

_n

≤ t ≤ t

n+1

where

J

_n

= D

u

f (u

n

), g

_n

(u(t)) = f (u(t)) − J

n

u(t) and then we consider the variation-of-constants formula.

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Exponential Rosenbrock–Euler method

u(t

n+1

) = exp(h

n

J

_n

)u

n

+ Z

hn

0

exp ((h

n

− τ )J

n

) g

n

(u(t

n

+ τ ))dτ

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Exponential Rosenbrock–Euler method

u(t

n+1

) = exp(h

n

J

_n

)u

n

+ Z

hn

0

exp ((h

n

− τ )J

n

) g

n

(u(t

n

+ τ ))dτ and with g

_n

(u(t

n

+ τ )) ≈ g

n

(u

n

) we get the exponential

Rosenbrock–Euler method

u

_n+1

= exp(h

n

J

_n

)u

n

+ h

n

ϕ

1

(h

n

J

_n

)g

n

(u

n

) where

ϕ

₁

(h

_n

J

_n

) = 1 h

_n

Z

hn

0

exp ((h

_n

− τ )J

_n

) dτ, ϕ

₁

(z) = e

^z

− 1 z

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s -stage exponential Rosenbrock methods

More generally, we consider the s-stage exponential method

U

_ni

= exp(c

_i

h

_n

J

_n

)u

_n

+ h

_n

i −1

X

j=1

a

_ij

(h

_n

J

_n

)g

_n

(U

_nj

),

u

_n+1

= exp(h

_n

J

_n

)u

_n

+ h

_n

s

X

i=1

b

_i

(h

_n

J

_n

)g

_n

(U

_ni

).

where b

_i

and a

_ij

are linear combinations of ϕ

_k

(h

_n

J

_n

), ϕ

_k

(h

n

J

_n

) = 1

h

_n^k

Z

hn

0

exp ((h

n

− τ )J

n

) τ

^k−1

(k − 1)! d τ

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Methods of order 2, 3 and 4

It is easy to construct a 2-stage third order method (with a second order error estimator embedded) and a 3-stage fourth order

method (with a third order error estimator embedded).

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Methods of order 2, 3 and 4

It is easy to construct a 2-stage third order method (with a second order error estimator embedded) and a 3-stage fourth order

method (with a third order error estimator embedded).

The exponential Rosenbrock–Euler method

u

_n+1

= exp(h

_n

J

_n

)u

_n

+ h

_n

ϕ

₁

(h

_n

J

_n

)g

_n

(u

_n

)

is computationally attractive since it achieves second order with one stage only. With the additional ϕ

₁

(h

_n

J

_n

)g

_n

(u

_n+1

) evaluation, it is possible to have a third order error estimator.

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Matrix functions evaluation

In the applications, these methods need the efficient evaluation of the underlying matrix functions ϕ

_k

(hJ)v (J ∈ R

^N×N

, v ∈ R

^N

).

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Matrix functions evaluation

In the applications, these methods need the efficient evaluation of the underlying matrix functions ϕ

_k

(hJ)v (J ∈ R

^N×N

, v ∈ R

^N

).

Most authors regard Krylov-like as the methods of choice.

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Matrix functions evaluation

In the applications, these methods need the efficient evaluation of the underlying matrix functions ϕ

_k

(hJ)v (J ∈ R

^N×N

, v ∈ R

^N

).

Most authors regard Krylov-like as the methods of choice.

An alternative class of polynomial methods is based on direct interpolation or approximation of the ϕ

k

functions on the spectrum (or the field of values) of the relevant matrix.

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ReLPM: Real Leja Points Method

Among others, the ReLPM has shown very attractive computational features:

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ReLPM: Real Leja Points Method

Among others, the ReLPM has shown very attractive computational features:

I

it rests on Newton interpolation of the exponential functions at a sequence of real Leja points;

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ReLPM: Real Leja Points Method

Among others, the ReLPM has shown very attractive computational features:

I

it rests on Newton interpolation of the exponential functions at a sequence of real Leja points;

I

they guarantee maximal convergence of the interpolant and thus superlinear convergence of the corresponding matrix polynomials;

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ReLPM: Real Leja Points Method

Among others, the ReLPM has shown very attractive computational features:

I

it rests on Newton interpolation of the exponential functions at a sequence of real Leja points;

I

they guarantee maximal convergence of the interpolant and thus superlinear convergence of the corresponding matrix polynomials;

I

with respect to other sets of interpolation points, (Chebyshev, e.g.), the Leja points allow to increase the degree of

interpolation just by adding new nodes of the same sequence.

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The first 13 Leja points on [−2 , 2]

-3 -2 -1 0 1 2 3

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The first 13 Leja points on [−2 , 2]

-3 -2 -1 0 1 2 3

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The first 13 Leja points on [−2 , 2]

-3 -2 -1 0 1 2 3

(20)

The first 13 Leja points on [−2 , 2]

-3 -2 -1 0 1 2 3

(21)

The first 13 Leja points on [−2 , 2]

-3 -2 -1 0 1 2 3

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The first 13 Leja points on [−2 , 2]

-3 -2 -1 0 1 2 3

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The first 13 Leja points on [−2 , 2]

-3 -2 -1 0 1 2 3

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The first 13 Leja points on [−2 , 2]

-3 -2 -1 0 1 2 3

(25)

The first 13 Leja points on [−2 , 2]

-3 -2 -1 0 1 2 3

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The first 13 Leja points on [−2 , 2]

-3 -2 -1 0 1 2 3

(27)

The first 13 Leja points on [−2 , 2]

-3 -2 -1 0 1 2 3

(28)

The first 13 Leja points on [−2 , 2]

-3 -2 -1 0 1 2 3

(29)

The first 13 Leja points on [−2 , 2]

-3 -2 -1 0 1 2 3

(30)

The first 13 Leja points on [−2 , 2]

-3 -2 -1 0 1 2 3

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The first 13 Leja points on [−2 , 2]

-3 -2 -1 0 1 2 3

(32)

Estimating the spectrum: Gershgorin’s circles

A key step is a cheap estimate of a real focal interval [a, b] such that the minimal ellipse of the confocal family which contains the spectrum has a small capacity (half sum of the semi-axes).

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Estimating the spectrum: Gershgorin’s circles

A key step is a cheap estimate of a real focal interval [a, b] such that the minimal ellipse of the confocal family which contains the spectrum has a small capacity (half sum of the semi-axes).

The numerical experience with matrices arising from stable spatial discretizations of parabolic equations has shown that good results can be obtained simply by intersecting the Gershgorin’s circles of the matrix with the real axis.

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Kernel of ReLPM

The kernel of the ReLPM code is given by the interpolation of ϕ

_k

(hz) at Leja points of the real focal interval

[a, b] = [c − 2γ, c + 2γ].

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Kernel of ReLPM

The kernel of the ReLPM code is given by the interpolation of ϕ

_k

(hz) at Leja points of the real focal interval

[a, b] = [c − 2γ, c + 2γ].

In practice, it is numerically convenient to interpolate the function ϕ

_k

(h(c + γξ)) at Leja points {ξ

s

} of the reference interval [−2, 2].

(36)

Kernel of ReLPM

The kernel of the ReLPM code is given by the interpolation of ϕ

_k

(hz) at Leja points of the real focal interval

[a, b] = [c − 2γ, c + 2γ].

In practice, it is numerically convenient to interpolate the function ϕ

_k

(h(c + γξ)) at Leja points {ξ

s

} of the reference interval [−2, 2].

Then, given the corresponding divided differences {d

i

} for such a function, the matrix Newton polynomial of degree m is

ϕ

_k

(hJ)v ≈ p

m

(J)v =

m

X

i=0

d

_i

Ω

i

, Ω

i

= ((J − cI )/γ − ξ

_{i −1}

I ) Ω

_{i −1}

Ω

0

= v

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ReLPM: key features

I

real arithmetic;

(38)

ReLPM: key features

I

real arithmetic;

I

no Krylov subspace to store;

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ReLPM: key features

I

real arithmetic;

I

no Krylov subspace to store;

I

linear complexity due to the 2-term recurrence;

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ReLPM: key features

I

real arithmetic;

I

no Krylov subspace to store;

I

linear complexity due to the 2-term recurrence;

I

no linear system to solve;

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ReLPM: key features

I

real arithmetic;

I

no Krylov subspace to store;

I

linear complexity due to the 2-term recurrence;

I

no linear system to solve;

I

well structured for a parallel implementation.

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Time sub-steps

In general, it is not feasible to interpolate with the original time step h, which has to be split. This happens, e.g., when the expected degree for convergence is too large.

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Time sub-steps

In general, it is not feasible to interpolate with the original time step h, which has to be split. This happens, e.g., when the expected degree for convergence is too large.

Setting K = hJ, ϕ

_k

(K )v (k > 0) is the solution at t = 1 of y

⁰

(t) = Ky (t) + t

^k−1

(k − 1)! v, y(0) = y

₀

= 0

(44)

Time sub-steps

In general, it is not feasible to interpolate with the original time step h, which has to be split. This happens, e.g., when the expected degree for convergence is too large.

Setting K = hJ, ϕ

_k

(K )v (k > 0) is the solution at t = 1 of y

⁰

(t) = Ky (t) + t

^k−1

(k − 1)! v, y(0) = y

₀

= 0 Given δ = 1/L, then y

L

= ϕ

k

(K )v , where

y

_`+1

= exp(δK )y

`

+δ

^k

k−1

X

i=0

`

^k−1−i

(k − 1 − i)! ϕ

i+1

(δK )v , ` = 0, . . . , L−1.

(45)

Time sub-steps

In general, it is not feasible to interpolate with the original time step h, which has to be split. This happens, e.g., when the expected degree for convergence is too large.

Setting K = hJ, ϕ

_k

(K )v (k > 0) is the solution at t = 1 of y

⁰

(t) = Ky (t) + t

^k−1

(k − 1)! v, y(0) = y

₀

= 0 Given δ = 1/L, then y

L

= ϕ

k

(K )v , where

y

_`+1

= exp(δK )y

`

+δ

^k

k−1

X

i=0

`

^k−1−i

(k − 1 − i)! ϕ

i+1

(δK )v , ` = 0, . . . , L−1.

It requires k evaluations of a matrix function for ` = 0. When increasing ` by one, only one additional evaluation, exp(δK )y

_`

, is required. The other terms are scalar multiples of the previous evaluations, which have to be stored.

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Diffusion-advection-reaction equations

We consider

∂

_t

u = ε(∂

_xx

u + ∂

_yy

u ) − α(∂

_x

u + ∂

_y

u) + γ u

u − 1

2 (1 − u) on the unit square subject to homogeneous Neumann b. c.

(47)

Diffusion-advection-reaction equations

We consider

∂

_t

u = ε(∂

_xx

u + ∂

_yy

u ) − α(∂

_x

u + ∂

_y

u) + γ u

u − 1

2 (1 − u) on the unit square subject to homogeneous Neumann b. c.

Standard finite differences in space with ∆x = ∆y = 0.005 (d.o.f.

40401), final time T = 0.3.

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Time integrators

We consider 5 second order time integrators:

(49)

Time integrators

We consider 5 second order time integrators:

erow2l

exponential Rosenbrock–Euler method with ϕ

k

evaluations by ReLPM;

(50)

Time integrators

We consider 5 second order time integrators:

erow2l

exponential Rosenbrock–Euler method with ϕ

k

evaluations by ReLPM;

erow2k

exponential Rosenbrock–Euler method with ϕ

k

evaluations by Krylov method (phipro routine by Y. Saad);

(51)

Time integrators

We consider 5 second order time integrators:

erow2l

exponential Rosenbrock–Euler method with ϕ

k

evaluations by ReLPM;

erow2k

exponential Rosenbrock–Euler method with ϕ

k

evaluations by Krylov method (phipro routine by Y. Saad);

rkc

explicit Runge–Kutta–Chebyshev method (code by Sommeijer, Shampine and Verwer);

(52)

Time integrators

We consider 5 second order time integrators:

erow2l

exponential Rosenbrock–Euler method with ϕ

k

evaluations by ReLPM;

erow2k

exponential Rosenbrock–Euler method with ϕ

k

evaluations by Krylov method (phipro routine by Y. Saad);

rkc

explicit Runge–Kutta–Chebyshev method (code by Sommeijer, Shampine and Verwer);

rock2

explicit Orthogonal Runge–Kutta–Chebyshev (code by Abdulle and Medovikov);

(53)

Time integrators

We consider 5 second order time integrators:

erow2l

exponential Rosenbrock–Euler method with ϕ

k

evaluations by ReLPM;

erow2k

exponential Rosenbrock–Euler method with ϕ

k

evaluations by Krylov method (phipro routine by Y. Saad);

rkc

explicit Runge–Kutta–Chebyshev method (code by Sommeijer, Shampine and Verwer);

rock2

explicit Orthogonal Runge–Kutta–Chebyshev (code by Abdulle and Medovikov);

cn

classical Crank–Nicolson scheme (inexact Newton nonlinear solver, BiCGStab linear solver, ILU with no fill-in perconditioner).

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Results: small advection, small reaction

ε = 0.1, α = −1, γ = 1 (tolerances: 10

⁻³

–10

⁻⁶

)

1 2 4 8 16 32

CPU seconds 10^-6

10^-5 10^-4 10^-3 10^-2

relative error in 2-norm

rkcrock2 erow2k erow2l

(55)

Results: large reaction

ε = 0.1, α = −1, γ = 100 (tolerances: 10

⁻³

–10

⁻⁶

)

2 4 8 16 32 64

CPU seconds 10^-6

10^-5 10^-4 10^-3 10^-2

(56)

Results: large advection

ε = 0.1, α = −10, γ = 1 (tolerances: 10

⁻³

–10

⁻⁶

)

2 4 8 16 32 64

CPU seconds 10^-7

10^-6 10^-5 10^-4 10^-3 10^-2

(57)

Exp. meshless Numerical experiment

Collocation with cardinal functions

Given the PDE

∂

_t

u(t, x) = Lu(t, x)

(with L a second order, linear operator), we consider the interpolation

u(t, x) ≈

N

X

j=1

φ

_j

(x)u(t; x

_j

) = φ(x) ◦ u(t)

where φ

_j

(x

_i

) = δ

_ij

and u(t; x

_j

) = u(t, x

_j

).

(58)

Collocation with cardinal functions

Given the PDE

∂

_t

u(t, x) = Lu(t, x)

(with L a second order, linear operator), we consider the interpolation

u(t, x) ≈

N

X

j=1

φ

_j

(x)u(t; x

_j

) = φ(x) ◦ u(t)

where φ

_j

(x

_i

) = δ

_ij

and u(t; x

_j

) = u(t, x

_j

). Then, φ(x) ◦ u

⁰

(t) = L (φ(x) ◦ u(t))

(59)

Collocation with cardinal functions

Given the PDE

∂

_t

u(t, x) = Lu(t, x)

(with L a second order, linear operator), we consider the interpolation

u(t, x) ≈

N

X

j=1

φ

_j

(x)u(t; x

_j

) = φ(x) ◦ u(t)

where φ

_j

(x

_i

) = δ

_ij

and u(t; x

_j

) = u(t, x

_j

). Then, φ(x) ◦ u

⁰

(t) = L (φ(x) ◦ u(t)) Now, we can collocate at the same {x

i

}.

(60)

Exponential meshless method

^∗

We get

u

⁰

(t) = L (φ(x) ◦ u(t)) |

_(x₁_,...,x_N₎T

= (L(φ)(x) ◦ u(t)) |

_(x₁_,...,x_N₎T

= Au(t) where A = a

_ij

= L(φ

_j

)(x

_i

).

∗In collaboration with M. Vianello, University of Padua

(61)

Exponential meshless method

^∗

We get

u

⁰

(t) = L (φ(x) ◦ u(t)) |

_(x₁_,...,x_N₎T

= (L(φ)(x) ◦ u(t)) |

_(x₁_,...,x_N₎T

= Au(t) where A = a

_ij

= L(φ

_j

)(x

_i

).

An iterative method (such as ReLPM or Krylov method) for the approximation of exp(h

n

A)u

n

is based on successive products

Au

^(m)_n

, m = 0, 1, . . . , u

⁽⁰⁾_n

= u

n

(62)

Exponential meshless method

^∗

We get

u

⁰

(t) = L (φ(x) ◦ u(t)) |

_(x₁_,...,x_N₎T

= (L(φ)(x) ◦ u(t)) |

_(x₁_,...,x_N₎T

= Au(t) where A = a

_ij

= L(φ

_j

)(x

_i

).

An iterative method (such as ReLPM or Krylov method) for the approximation of exp(h

n

A)u

n

is based on successive products

Au

^(m)_n

, m = 0, 1, . . . , u

⁽⁰⁾_n

= u

n

If the nodes {x

_i

} are arbitrary and we have a tool to compute Au

_n^(m)

directly as L

φ(x) ◦ u

^(m)n

|

_(x₁_,...,x_N₎T

, we get an exponential meshless method.

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2D examples: CShep2D and QShep2D

CShep2D, by R. J. Renka, is among the most accurate and

efficient scattered data interpolation algorithms. It constructs a C

²

interpolant in a moving least-square fashion. The costs range between O(N) and O(N

²

) (preprocessing) and O(1) and O(N) (evaluation), depending on the distribution of the nodes. It can compute also the first and the second derivative. The required storage is 9N.

(64)

2D examples: CShep2D and QShep2D

CShep2D, by R. J. Renka, is among the most accurate and

efficient scattered data interpolation algorithms. It constructs a C

²

interpolant in a moving least-square fashion. The costs range between O(N) and O(N

²

) (preprocessing) and O(1) and O(N) (evaluation), depending on the distribution of the nodes. It can compute also the first and the second derivative. The required storage is 9N.

QShep2D constructs a C

¹

interpolant. It can compute also the first and the second derivative (although not continuous). The required storage is 5N.

(65)

A monitor for the check points

Since the collocation, or center, points are arbitrary, we can add or remove points at each time step, depending on the behaviour of the solution on a set of check points.

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A monitor for the check points

Since the collocation, or center, points are arbitrary, we can add or remove points at each time step, depending on the behaviour of the solution on a set of check points.

For example, we can consider a curvature monitor

|∆x

²

u

_xx

(t; (ˇ x

_i

, ˇ y

_i

))| + |∆y

²

u

_yy

(t; (ˇ x

_i

, ˇ y

_i

))|

which has to be evaluated at each check point (ˇ x

_i

, ˇ y

_i

).

(67)

Center and check points

check point center point

(68)

Center and check points

center point check point

(69)

Unstable derivatives

Derivatives computed directly by {C,Q}Shep2D give raise to instabilities during time integration (under investigation).

(70)

Unstable derivatives

Derivatives computed directly by {C,Q}Shep2D give raise to instabilities during time integration (under investigation).

Idea: derivatives in a FD fashion. They are much more stable.

center point check point FD stencil

(71)

The Molenkamp–Crowley test

Let us consider

∂

_t

u + (au)

_x

+ (bu)

_y

= 0 with

a(x, y ) = −2π

y − 1

2 , b(x, y ) = 2π

x − 1

2

(72)

The Molenkamp–Crowley test

Let us consider

∂

_t

u + (au)

_x

+ (bu)

_y

= 0 with

a(x, y ) = −2π

y − 1

2 , b(x, y ) = 2π

x − 1

2 The initial profile is rotated around the center of the domain. At time t = 1 one rotation will be completed. Homogeneous Dirichlet conditions are prescribed at the inflow boundaries.

(73)

The Molenkamp–Crowley test

0 0.2

0.4 0.6

0.8 1 0

0.2 0.4

0.6 0.8

1 0

0.2 0.4 0.6 0.8 1

(74)

The Molenkamp–Crowley test

0 0.2

0.4 0.6

0.8 1 0

0.2 0.4

0.6 0.8

1 0

0.2 0.4 0.6 0.8 1

(75)

The Molenkamp–Crowley test

0 0.2

0.4 0.6

0.8 1 0

0.2 0.4

0.6 0.8

1 0

0.2 0.4 0.6 0.8 1

(76)

The Molenkamp–Crowley test

0 0.2

0.4 0.6

0.8 1 0

0.2 0.4

0.6 0.8

1 0

0.2 0.4 0.6 0.8 1

(77)

The Molenkamp–Crowley test

0 0.2

0.4 0.6

0.8 1 0

0.2 0.4

0.6 0.8

1 0

0.2 0.4 0.6 0.8 1

(78)

The Molenkamp–Crowley test

0 0.2

0.4 0.6

0.8 1 0

0.2 0.4

0.6 0.8

1 0

0.2 0.4 0.6 0.8 1

(79)

The Molenkamp–Crowley test

0 0.2

0.4 0.6

0.8 1 0

0.2 0.4

0.6 0.8

1 0

0.2 0.4 0.6 0.8 1

(80)

The Molenkamp–Crowley test

0 0.2

0.4 0.6

0.8 1 0

0.2 0.4

0.6 0.8

1 0

0.2 0.4 0.6 0.8 1

(81)

The Molenkamp–Crowley test

0 0.2

0.4 0.6

0.8 1 0

0.2 0.4

0.6 0.8

1 0

0.2 0.4 0.6 0.8 1

(82)

The Molenkamp–Crowley test

0 0.2

0.4 0.6

0.8 1 0

0.2 0.4

0.6 0.8

1 0

0.2 0.4 0.6 0.8 1

(83)

The Molenkamp–Crowley test

0 0.2

0.4 0.6

0.8 1 0

0.2 0.4

0.6 0.8

1 0

0.2 0.4 0.6 0.8 1

(84)

The Molenkamp–Crowley test

0 0.2

0.4 0.6

0.8 1 0

0.2 0.4

0.6 0.8

1 0

0.2 0.4 0.6 0.8 1

(85)

The Molenkamp–Crowley test

0 0.2

0.4 0.6

0.8 1 0

0.2 0.4

0.6 0.8

1 0

0.2 0.4 0.6 0.8 1

(86)

The Molenkamp–Crowley test

0 0.2

0.4 0.6

0.8 1 0

0.2 0.4

0.6 0.8

1 0

0.2 0.4 0.6 0.8 1

(87)

The Molenkamp–Crowley test

0 0.2

0.4 0.6

0.8 1 0

0.2 0.4

0.6 0.8

1 0

0.2 0.4 0.6 0.8 1

(88)

The Molenkamp–Crowley test

0 0.2

0.4 0.6

0.8 1 0

0.2 0.4

0.6 0.8

1 0

0.2 0.4 0.6 0.8 1

(89)

The Molenkamp–Crowley test

0 0.2

0.4 0.6

0.8 1 0

0.2 0.4

0.6 0.8

1 0

0.2 0.4 0.6 0.8 1

(90)

The Molenkamp–Crowley test

0 0.2

0.4 0.6

0.8 1 0

0.2 0.4

0.6 0.8

1 0

0.2 0.4 0.6 0.8 1

(91)

The Molenkamp–Crowley test

0 0.2

0.4 0.6

0.8 1 0

0.2 0.4

0.6 0.8

1 0

0.2 0.4 0.6 0.8 1

(92)

The Molenkamp–Crowley test

0 0.2

0.4 0.6

0.8 1 0

0.2 0.4

0.6 0.8

1 0

0.2 0.4 0.6 0.8 1

(93)

The first symmetrized 13 Leja points on i [−2 , 2]

-3 -2 -1 0 1 2 3

(94)

The first symmetrized 13 Leja points on i [−2 , 2]

-3 -2 -1 0 1 2 3

(95)

The first symmetrized 13 Leja points on i [−2 , 2]

-3 -2 -1 0 1 2 3

(96)

The first symmetrized 13 Leja points on i [−2 , 2]

-3 -2 -1 0 1 2 3

(97)

The first symmetrized 13 Leja points on i [−2 , 2]

-3 -2 -1 0 1 2 3

(98)

The first symmetrized 13 Leja points on i [−2 , 2]

-3 -2 -1 0 1 2 3

(99)

The first symmetrized 13 Leja points on i [−2 , 2]

-3 -2 -1 0 1 2 3

(100)

The first symmetrized 13 Leja points on i [−2 , 2]

-3 -2 -1 0 1 2 3

(101)

The first symmetrized 13 Leja points on i [−2 , 2]

-3 -2 -1 0 1 2 3