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IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Experiments with Order Arrows
Ali Abdi, Tabriz University
John Butcher, University of Auckland
SciCADE 2011
Toronto
11–15 July 2011
0 5 10 15 20 251
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
1 Introduction
Order and stability
Three Runge–Kutta methods
Relative stability regions2 Order stars and order arrows
Example 1: the implicit Euler method
Example 2: a third order implicit method
Properties of order arrows3 Applications
The Ehle “conjecture”
The Daniel-Moore “conjecture”
The Butcher-Chipman conjecture4 Drawing pictures
Why arrow pictures are hard to draw
Why arrow pictures are easy to draw
A differential equation for order arrows5 Gallery
0 5 10 15 20 252
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
1 Introduction
Order and stability
Three Runge–Kutta methods
Relative stability regions2 Order stars and order arrows
Example 1: the implicit Euler method
Example 2: a third order implicit method
Properties of order arrows3 Applications
The Ehle “conjecture”
The Daniel-Moore “conjecture”
The Butcher-Chipman conjecture4 Drawing pictures
Why arrow pictures are hard to draw
Why arrow pictures are easy to draw
A differential equation for order arrows5 Gallery
0 5 10 15 20 253
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
1 Introduction
Order and stability
Three Runge–Kutta methods
Relative stability regions2 Order stars and order arrows
Example 1: the implicit Euler method
Example 2: a third order implicit method
Properties of order arrows3 Applications
The Ehle “conjecture”
The Daniel-Moore “conjecture”
The Butcher-Chipman conjecture4 Drawing pictures
Why arrow pictures are hard to draw
Why arrow pictures are easy to draw
A differential equation for order arrows5 Gallery
0 5 10 15 20 254
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
1 Introduction
Order and stability
Three Runge–Kutta methods
Relative stability regions2 Order stars and order arrows
Example 1: the implicit Euler method
Example 2: a third order implicit method
Properties of order arrows3 Applications
The Ehle “conjecture”
The Daniel-Moore “conjecture”
The Butcher-Chipman conjecture4 Drawing pictures
Why arrow pictures are hard to draw
Why arrow pictures are easy to draw
A differential equation for order arrows5 Gallery
0 5 10 15 20 255
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
1 Introduction
Order and stability
Three Runge–Kutta methods
Relative stability regions2 Order stars and order arrows
Example 1: the implicit Euler method
Example 2: a third order implicit method
Properties of order arrows3 Applications
The Ehle “conjecture”
The Daniel-Moore “conjecture”
The Butcher-Chipman conjecture4 Drawing pictures
Why arrow pictures are hard to draw
Why arrow pictures are easy to draw
A differential equation for order arrows5 Gallery
0 5 10 15 20 256
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Order and stabilityThree Runge–Kutta methodsRelative stability regions
Order and stability
In the solution of stiff problems there are many aims such as
High orderGood stabilityEconomical implementation
These attributes are not independent and there may be a conflict betweenthem.
Order arrows are a tool for exploring restrictions on order for methods that arerequired to be A-stable.
0 5 10 15 20 257
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Order and stabilityThree Runge–Kutta methodsRelative stability regions
Order and stability
In the solution of stiff problems there are many aims such as
High orderGood stabilityEconomical implementation
These attributes are not independent and there may be a conflict betweenthem.
Order arrows are a tool for exploring restrictions on order for methods that arerequired to be A-stable.
0 5 10 15 20 258
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Order and stabilityThree Runge–Kutta methodsRelative stability regions
Order and stability
In the solution of stiff problems there are many aims such as
High orderGood stabilityEconomical implementation
These attributes are not independent and there may be a conflict betweenthem.
Order arrows are a tool for exploring restrictions on order for methods that arerequired to be A-stable.
0 5 10 15 20 259
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Order and stabilityThree Runge–Kutta methodsRelative stability regions
Three Runge–Kutta methods
I would like to talk about three special methods which I will call Eu,Im andTh.
0 0
1(Eu)
1 1
1(Im)
0 16 −4
376
12
16
23 −1
3
1 16
23
16
16
23
16
(Th)
Eu is the Euler method,Im is the Implicit Euler method andTh is a Third order implicit Runge–Kutta method
0 5 10 15 20 2510
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Order and stabilityThree Runge–Kutta methodsRelative stability regions
Three Runge–Kutta methods
I would like to talk about three special methods which I will call Eu,Im andTh.
0 0
1(Eu)
1 1
1(Im)
0 16 −4
376
12
16
23 −1
3
1 16
23
16
16
23
16
(Th)
Eu is the Euler method,Im is the Implicit Euler method andTh is a Third order implicit Runge–Kutta method
0 5 10 15 20 2511
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Order and stabilityThree Runge–Kutta methodsRelative stability regions
The stability functions for the three methods are
R(z) = 1+z = exp(z)− 12z2 +O(z3) (Eu)
R(z) =1
1−z= exp(z)+ 1
2z2 +O(z3) (Im)
R(z) =1
1−z+ 12z2− 1
6z3= exp(z)+ 1
24z4 +O(z5) (Th)
0 5 10 15 20 2512
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Order and stabilityThree Runge–Kutta methodsRelative stability regions
The stability functions for the three methods are
R(z) = 1+z = exp(z)− 12z2 +O(z3) (Eu)
R(z) =1
1−z= exp(z)+ 1
2z2 +O(z3) (Im)
R(z) =1
1−z+ 12z2− 1
6z3= exp(z)+ 1
24z4 +O(z5) (Th)
and the stability regions are
Eu Im Th0 5 10 15 20 25
13
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Order and stabilityThree Runge–Kutta methodsRelative stability regions
The Euler method has very poor stability and it should not be usedwith stiff problems.
The Implicit Euler method is A-stable and it can safely be used withmost stiff problems.
The special third order method is not A-stable but it is A(α)-stablewith α ≈ 88◦.
For many problems, such as pure diffusion problems, the special thirdorder method will be completely satisfactory.
However, in this talk, we are interested in A-stability alone and thecompetition this property has with the order of methods.
0 5 10 15 20 2514
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Order and stabilityThree Runge–Kutta methodsRelative stability regions
The Euler method has very poor stability and it should not be usedwith stiff problems.
The Implicit Euler method is A-stable and it can safely be used withmost stiff problems.
The special third order method is not A-stable but it is A(α)-stablewith α ≈ 88◦.
For many problems, such as pure diffusion problems, the special thirdorder method will be completely satisfactory.
However, in this talk, we are interested in A-stability alone and thecompetition this property has with the order of methods.
0 5 10 15 20 2515
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Order and stabilityThree Runge–Kutta methodsRelative stability regions
The Euler method has very poor stability and it should not be usedwith stiff problems.
The Implicit Euler method is A-stable and it can safely be used withmost stiff problems.
The special third order method is not A-stable but it is A(α)-stablewith α ≈ 88◦.
For many problems, such as pure diffusion problems, the special thirdorder method will be completely satisfactory.
However, in this talk, we are interested in A-stability alone and thecompetition this property has with the order of methods.
0 5 10 15 20 2516
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Order and stabilityThree Runge–Kutta methodsRelative stability regions
The Euler method has very poor stability and it should not be usedwith stiff problems.
The Implicit Euler method is A-stable and it can safely be used withmost stiff problems.
The special third order method is not A-stable but it is A(α)-stablewith α ≈ 88◦.
For many problems, such as pure diffusion problems, the special thirdorder method will be completely satisfactory.
However, in this talk, we are interested in A-stability alone and thecompetition this property has with the order of methods.
0 5 10 15 20 2517
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Order and stabilityThree Runge–Kutta methodsRelative stability regions
The Euler method has very poor stability and it should not be usedwith stiff problems.
The Implicit Euler method is A-stable and it can safely be used withmost stiff problems.
The special third order method is not A-stable but it is A(α)-stablewith α ≈ 88◦.
For many problems, such as pure diffusion problems, the special thirdorder method will be completely satisfactory.
However, in this talk, we are interested in A-stability alone and thecompetition this property has with the order of methods.
0 5 10 15 20 2518
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Order and stabilityThree Runge–Kutta methodsRelative stability regions
Relative stability regions
The “instability region” associated with a stability function R(z) is theset of points in the complex plane such that
|R(z)| > 1.Therelative instability region is the set such that
|exp(−z)R(z)| > 1.The stability region and the relative stability regions aredefined in asimilar way but with> replaced by<.
The relative instability region is known as the “order star”and therelative stability region is known as the dual order star.
These, and the closely related “order arrows”, are introduced in thenext section for theIm andTh methods.
Just as order stars are defined in terms of|exp(−z)R(z)|, order arrowsare defined as the paths traced out by the points for which exp(−z)R(z)is real and positive.
0 5 10 15 20 2519
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Order and stabilityThree Runge–Kutta methodsRelative stability regions
Relative stability regions
The “instability region” associated with a stability function R(z) is theset of points in the complex plane such that
|R(z)| > 1.Therelative instability region is the set such that
|exp(−z)R(z)| > 1.The stability region and the relative stability regions aredefined in asimilar way but with> replaced by<.
The relative instability region is known as the “order star”and therelative stability region is known as the dual order star.
These, and the closely related “order arrows”, are introduced in thenext section for theIm andTh methods.
Just as order stars are defined in terms of|exp(−z)R(z)|, order arrowsare defined as the paths traced out by the points for which exp(−z)R(z)is real and positive.
0 5 10 15 20 2520
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Order and stabilityThree Runge–Kutta methodsRelative stability regions
Relative stability regions
The “instability region” associated with a stability function R(z) is theset of points in the complex plane such that
|R(z)| > 1.Therelative instability region is the set such that
|exp(−z)R(z)| > 1.The stability region and the relative stability regions aredefined in asimilar way but with> replaced by<.
The relative instability region is known as the “order star”and therelative stability region is known as the dual order star.
These, and the closely related “order arrows”, are introduced in thenext section for theIm andTh methods.
Just as order stars are defined in terms of|exp(−z)R(z)|, order arrowsare defined as the paths traced out by the points for which exp(−z)R(z)is real and positive.
0 5 10 15 20 2521
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Order and stabilityThree Runge–Kutta methodsRelative stability regions
Relative stability regions
The “instability region” associated with a stability function R(z) is theset of points in the complex plane such that
|R(z)| > 1.Therelative instability region is the set such that
|exp(−z)R(z)| > 1.The stability region and the relative stability regions aredefined in asimilar way but with> replaced by<.
The relative instability region is known as the “order star”and therelative stability region is known as the dual order star.
These, and the closely related “order arrows”, are introduced in thenext section for theIm andTh methods.
Just as order stars are defined in terms of|exp(−z)R(z)|, order arrowsare defined as the paths traced out by the points for which exp(−z)R(z)is real and positive.
0 5 10 15 20 2522
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Order and stabilityThree Runge–Kutta methodsRelative stability regions
Relative stability regions
The “instability region” associated with a stability function R(z) is theset of points in the complex plane such that
|R(z)| > 1.Therelative instability region is the set such that
|exp(−z)R(z)| > 1.The stability region and the relative stability regions aredefined in asimilar way but with> replaced by<.
The relative instability region is known as the “order star”and therelative stability region is known as the dual order star.
These, and the closely related “order arrows”, are introduced in thenext section for theIm andTh methods.
Just as order stars are defined in terms of|exp(−z)R(z)|, order arrowsare defined as the paths traced out by the points for which exp(−z)R(z)is real and positive.
0 5 10 15 20 2523
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Order and stabilityThree Runge–Kutta methodsRelative stability regions
Relative stability regions
The “instability region” associated with a stability function R(z) is theset of points in the complex plane such that
|R(z)| > 1.Therelative instability region is the set such that
|exp(−z)R(z)| > 1.The stability region and the relative stability regions aredefined in asimilar way but with> replaced by<.
The relative instability region is known as the “order star”and therelative stability region is known as the dual order star.
These, and the closely related “order arrows”, are introduced in thenext section for theIm andTh methods.
Just as order stars are defined in terms of|exp(−z)R(z)|, order arrowsare defined as the paths traced out by the points for which exp(−z)R(z)is real and positive.
0 5 10 15 20 2524
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Example 1: the implicit Euler methodExample 2: a third order implicit methodProperties of order arrows
1 Introduction
Order and stability
Three Runge–Kutta methods
Relative stability regions2 Order stars and order arrows
Example 1: the implicit Euler method
Example 2: a third order implicit method
Properties of order arrows3 Applications
The Ehle “conjecture”
The Daniel-Moore “conjecture”
The Butcher-Chipman conjecture4 Drawing pictures
Why arrow pictures are hard to draw
Why arrow pictures are easy to draw
A differential equation for order arrows5 Gallery
0 5 10 15 20 2525
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Example 1: the implicit Euler methodExample 2: a third order implicit methodProperties of order arrows
Example: 1 theIm method (order star)
0 5 10 15 20 2526
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Example 1: the implicit Euler methodExample 2: a third order implicit methodProperties of order arrows
Example: 1 theIm method (order star)
Note the behaviour near zero:becauseR(z)exp(−z)=1+ 1
2z2+O(z3),the dashed red lines are tangent to the order star boundary atzero and theangle between these tangents is exactlyπ/2.
0 5 10 15 20 2527
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Example 1: the implicit Euler methodExample 2: a third order implicit methodProperties of order arrows
Example: 1 theIm method (order star)
Note the behaviour near zero:becauseR(z)exp(−z)=1+ 1
2z2+O(z3),the dashed red lines are tangent to the order star boundary atzero and theangle between these tangents is exactlyπ/2.
Note also the existence of a pole in the “bounded finger”0 5 10 15 20 25
28
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Example 1: the implicit Euler methodExample 2: a third order implicit methodProperties of order arrows
Example 1: theIm method (order arrows)
0 5 10 15 20 2529
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Example 1: the implicit Euler methodExample 2: a third order implicit methodProperties of order arrows
Example 1: theIm method (order arrows)
Note the behaviour near zero:becauseR(z)exp(−z)= 1+1
2z2+O(z3), the dashed red lines are tangent to theorder arrows at zero and the angle between these tangents is exactly π/2.
0 5 10 15 20 2530
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Example 1: the implicit Euler methodExample 2: a third order implicit methodProperties of order arrows
Example 1: theIm method (order arrows)
Note the behaviour near zero:becauseR(z)exp(−z)= 1+1
2z2+O(z3), the dashed red lines are tangent to theorder arrows at zero and the angle between these tangents is exactly π/2.
Up-arrows (increasing values ofR(z)e−z) alternate with down-arrows.
0 5 10 15 20 2531
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Example 1: the implicit Euler methodExample 2: a third order implicit methodProperties of order arrows
Example 1: theIm method (order arrows)
Note the behaviour near zero:becauseR(z)exp(−z)= 1+1
2z2+O(z3), the dashed red lines are tangent to theorder arrows at zero and the angle between these tangents is exactly π/2.
Up-arrows (increasing values ofR(z)e−z) alternate with down-arrows.
Note the existence of a pole as the termination point of an up-arrow.
0 5 10 15 20 2532
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Example 1: the implicit Euler methodExample 2: a third order implicit methodProperties of order arrows
Example 2: theTh method (order star)
0 5 10 15 20 2533
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Example 1: the implicit Euler methodExample 2: a third order implicit methodProperties of order arrows
Example 2: theTh method (order star)
Fingers subtend anglesπ/4 at 0 because exp(−z)R(z)≈1+ 124z4.
0 5 10 15 20 2534
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Example 1: the implicit Euler methodExample 2: a third order implicit methodProperties of order arrows
Example 2: theTh method (order star)
Fingers subtend anglesπ/4 at 0 because exp(−z)R(z)≈1+ 124z4.
Bounded fingers contain poles.
0 5 10 15 20 2535
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Example 1: the implicit Euler methodExample 2: a third order implicit methodProperties of order arrows
Example 2: theTh method (order star)
Fingers subtend anglesπ/4 at 0 because exp(−z)R(z)≈1+ 124z4.
Bounded fingers contain poles.
Fingers overlap imaginary axis. Hence the method is not A-stable.0 5 10 15 20 25
36
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Example 1: the implicit Euler methodExample 2: a third order implicit methodProperties of order arrows
Example 2: theTh method (order arrows)
0 5 10 15 20 2537
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Example 1: the implicit Euler methodExample 2: a third order implicit methodProperties of order arrows
Example 2: theTh method (order arrows)
Anglesπ/4 between arrows at 0, because exp(−z)R(z)≈1+ 124z4.
0 5 10 15 20 2538
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Example 1: the implicit Euler methodExample 2: a third order implicit methodProperties of order arrows
Example 2: theTh method (order arrows)
Anglesπ/4 between arrows at 0, because exp(−z)R(z)≈1+ 124z4.
Poles are at the ends of up-arrows from 0.
0 5 10 15 20 2539
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Example 1: the implicit Euler methodExample 2: a third order implicit methodProperties of order arrows
Example 2: theTh method (order arrows)
Anglesπ/4 between arrows at 0, because exp(−z)R(z)≈1+ 124z4.
Poles are at the ends of up-arrows from 0.
There is an up-arrow tangential to the imaginary axis.Hence the method is not A-stable.
0 5 10 15 20 2540
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Example 1: the implicit Euler methodExample 2: a third order implicit methodProperties of order arrows
Properties of order arrows
Order Stars were introduced in the classic 1978 paper1 and are thesubject of a 1991 monograph2.
Order arrows have related properties such as
Up-arrows from zero terminate at poles or at−∞Down-arrows from zero terminate at zeros or at+∞There is an arrow from zero in the direction of the positive realaxis (up- or down- depending on the sign of the error constant)
For an orderp approximation, there arep+1 down-arrows fromzero alternating withp+1 up-arrows from zero
The angle between one arrow from zero and the next isπ/(p+1)
1Wanner G., Hairer E. and Nørsett S. P., Order stars and stability theorems, BIT18 (1978), 475–489
2Iserles A. and Nørsett S. P., Order Stars, Chapman & Hall, (1991)
0 5 10 15 20 2541
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Example 1: the implicit Euler methodExample 2: a third order implicit methodProperties of order arrows
Properties of order arrows
Order Stars were introduced in the classic 1978 paper1 and are thesubject of a 1991 monograph2.
Order arrows have related properties such as
Up-arrows from zero terminate at poles or at−∞Down-arrows from zero terminate at zeros or at+∞There is an arrow from zero in the direction of the positive realaxis (up- or down- depending on the sign of the error constant)
For an orderp approximation, there arep+1 down-arrows fromzero alternating withp+1 up-arrows from zero
The angle between one arrow from zero and the next isπ/(p+1)
1Wanner G., Hairer E. and Nørsett S. P., Order stars and stability theorems, BIT18 (1978), 475–489
2Iserles A. and Nørsett S. P., Order Stars, Chapman & Hall, (1991)
0 5 10 15 20 2542
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Example 1: the implicit Euler methodExample 2: a third order implicit methodProperties of order arrows
Properties of order arrows
Order Stars were introduced in the classic 1978 paper1 and are thesubject of a 1991 monograph2.
Order arrows have related properties such as
Up-arrows from zero terminate at poles or at−∞Down-arrows from zero terminate at zeros or at+∞There is an arrow from zero in the direction of the positive realaxis (up- or down- depending on the sign of the error constant)
For an orderp approximation, there arep+1 down-arrows fromzero alternating withp+1 up-arrows from zero
The angle between one arrow from zero and the next isπ/(p+1)
1Wanner G., Hairer E. and Nørsett S. P., Order stars and stability theorems, BIT18 (1978), 475–489
2Iserles A. and Nørsett S. P., Order Stars, Chapman & Hall, (1991)
0 5 10 15 20 2543
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Example 1: the implicit Euler methodExample 2: a third order implicit methodProperties of order arrows
Properties of order arrows
Order Stars were introduced in the classic 1978 paper1 and are thesubject of a 1991 monograph2.
Order arrows have related properties such as
Up-arrows from zero terminate at poles or at−∞Down-arrows from zero terminate at zeros or at+∞There is an arrow from zero in the direction of the positive realaxis (up- or down- depending on the sign of the error constant)
For an orderp approximation, there arep+1 down-arrows fromzero alternating withp+1 up-arrows from zero
The angle between one arrow from zero and the next isπ/(p+1)
1Wanner G., Hairer E. and Nørsett S. P., Order stars and stability theorems, BIT18 (1978), 475–489
2Iserles A. and Nørsett S. P., Order Stars, Chapman & Hall, (1991)
0 5 10 15 20 2544
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Example 1: the implicit Euler methodExample 2: a third order implicit methodProperties of order arrows
Properties of order arrows
Order Stars were introduced in the classic 1978 paper1 and are thesubject of a 1991 monograph2.
Order arrows have related properties such as
Up-arrows from zero terminate at poles or at−∞Down-arrows from zero terminate at zeros or at+∞There is an arrow from zero in the direction of the positive realaxis (up- or down- depending on the sign of the error constant)
For an orderp approximation, there arep+1 down-arrows fromzero alternating withp+1 up-arrows from zero
The angle between one arrow from zero and the next isπ/(p+1)
1Wanner G., Hairer E. and Nørsett S. P., Order stars and stability theorems, BIT18 (1978), 475–489
2Iserles A. and Nørsett S. P., Order Stars, Chapman & Hall, (1991)
0 5 10 15 20 2545
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Example 1: the implicit Euler methodExample 2: a third order implicit methodProperties of order arrows
Properties of order arrows
Order Stars were introduced in the classic 1978 paper1 and are thesubject of a 1991 monograph2.
Order arrows have related properties such as
Up-arrows from zero terminate at poles or at−∞Down-arrows from zero terminate at zeros or at+∞There is an arrow from zero in the direction of the positive realaxis (up- or down- depending on the sign of the error constant)
For an orderp approximation, there arep+1 down-arrows fromzero alternating withp+1 up-arrows from zero
The angle between one arrow from zero and the next isπ/(p+1)
1Wanner G., Hairer E. and Nørsett S. P., Order stars and stability theorems, BIT18 (1978), 475–489
2Iserles A. and Nørsett S. P., Order Stars, Chapman & Hall, (1991)
0 5 10 15 20 2546
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Example 1: the implicit Euler methodExample 2: a third order implicit methodProperties of order arrows
If R(z) is an A-function (the stability function of an A-stable method),then we can make further statements
Theorem
Let R(z) be an A-function then1 exp(−z)R(z) has no poles in the left half-plane.2 No up-arrow from zero can be tangential to the imaginary axis.3 No up-arrow from zero can cross the imaginary axis.
These properties follow from similar facts aboutR(z) and theobservation that multiplication by exp(−z) does not affect the poles orthe behaviour of|R(z)| on the imaginary axis.
0 5 10 15 20 2547
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Example 1: the implicit Euler methodExample 2: a third order implicit methodProperties of order arrows
If R(z) is an A-function (the stability function of an A-stable method),then we can make further statements
Theorem
Let R(z) be an A-function then1 exp(−z)R(z) has no poles in the left half-plane.2 No up-arrow from zero can be tangential to the imaginary axis.3 No up-arrow from zero can cross the imaginary axis.
These properties follow from similar facts aboutR(z) and theobservation that multiplication by exp(−z) does not affect the poles orthe behaviour of|R(z)| on the imaginary axis.
0 5 10 15 20 2548
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Example 1: the implicit Euler methodExample 2: a third order implicit methodProperties of order arrows
If R(z) is an A-function (the stability function of an A-stable method),then we can make further statements
Theorem
Let R(z) be an A-function then1 exp(−z)R(z) has no poles in the left half-plane.2 No up-arrow from zero can be tangential to the imaginary axis.3 No up-arrow from zero can cross the imaginary axis.
These properties follow from similar facts aboutR(z) and theobservation that multiplication by exp(−z) does not affect the poles orthe behaviour of|R(z)| on the imaginary axis.
0 5 10 15 20 2549
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Example 1: the implicit Euler methodExample 2: a third order implicit methodProperties of order arrows
If R(z) is an A-function (the stability function of an A-stable method),then we can make further statements
Theorem
Let R(z) be an A-function then1 exp(−z)R(z) has no poles in the left half-plane.2 No up-arrow from zero can be tangential to the imaginary axis.3 No up-arrow from zero can cross the imaginary axis.
These properties follow from similar facts aboutR(z) and theobservation that multiplication by exp(−z) does not affect the poles orthe behaviour of|R(z)| on the imaginary axis.
0 5 10 15 20 2550
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Example 1: the implicit Euler methodExample 2: a third order implicit methodProperties of order arrows
If R(z) is an A-function (the stability function of an A-stable method),then we can make further statements
Theorem
Let R(z) be an A-function then1 exp(−z)R(z) has no poles in the left half-plane.2 No up-arrow from zero can be tangential to the imaginary axis.3 No up-arrow from zero can cross the imaginary axis.
These properties follow from similar facts aboutR(z) and theobservation that multiplication by exp(−z) does not affect the poles orthe behaviour of|R(z)| on the imaginary axis.
0 5 10 15 20 2551
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
The Ehle “conjecture”The Daniel-Moore “conjecture”The Butcher-Chipman conjecture
1 Introduction
Order and stability
Three Runge–Kutta methods
Relative stability regions2 Order stars and order arrows
Example 1: the implicit Euler method
Example 2: a third order implicit method
Properties of order arrows3 Applications
The Ehle “conjecture”
The Daniel-Moore “conjecture”
The Butcher-Chipman conjecture4 Drawing pictures
Why arrow pictures are hard to draw
Why arrow pictures are easy to draw
A differential equation for order arrows5 Gallery
0 5 10 15 20 2552
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
The Ehle “conjecture”The Daniel-Moore “conjecture”The Butcher-Chipman conjecture
The Ehle “conjecture”
A Pade approximation is a rational functionR(z) = N(z)/D(z) such thatR(z) = exp(z)+O(|z|p+1),
where the order isp = n+d, with n = deg(N) andd = deg(D).
0 5 10 15 20 2553
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
The Ehle “conjecture”The Daniel-Moore “conjecture”The Butcher-Chipman conjecture
The Ehle “conjecture”
A Pade approximation is a rational functionR(z) = N(z)/D(z) such thatR(z) = exp(z)+O(|z|p+1),
where the order isp = n+d, with n = deg(N) andd = deg(D).
Some Pade approximations correspond to A-stable numerical methods andsome don’t. See the table:
0 5 10 15 20 2554
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
The Ehle “conjecture”The Daniel-Moore “conjecture”The Butcher-Chipman conjecture
The Ehle “conjecture”
A Pade approximation is a rational functionR(z) = N(z)/D(z) such thatR(z) = exp(z)+O(|z|p+1),
where the order isp = n+d, with n = deg(N) andd = deg(D).
Some Pade approximations correspond to A-stable numerical methods andsome don’t. See the table:
dn 0 1 2 3 4 5
0 1 1+z 1+z+ 12z2
1 11−z
1+ 12z
1− 12z
1+ 23z+ 1
6z2
1− 13z
2 11−z+ 1
2z2
1+ 13z
1− 23z+ 1
6z2
1+ 12z+ 1
12z2
1− 12z+ 1
12z2
3 11−z+ 1
2z2− 16z3
1+ 14z
1− 34z+ 1
4z2− 124z3
1+ 25z+ 1
20z2
1− 35z+ 3
20z2− 160z3
4
5
0 5 10 15 20 2555
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
The Ehle “conjecture”The Daniel-Moore “conjecture”The Butcher-Chipman conjecture
The Ehle “conjecture”
A Pade approximation is a rational functionR(z) = N(z)/D(z) such thatR(z) = exp(z)+O(|z|p+1),
where the order isp = n+d, with n = deg(N) andd = deg(D).
Some Pade approximations correspond to A-stable numerical methods andsome don’t. See the table:
dn 0 1 2 3 4 5
0 1 1+z 1+z+ 12z2
1 11−z
1+ 12z
1− 12z
1+ 23z+ 1
6z2
1− 13z
2 11−z+ 1
2z2
1+ 13z
1− 23z+ 1
6z2
1+ 12z+ 1
12z2
1− 12z+ 1
12z2
3 11−z+ 1
2z2− 16z3
1+ 14z
1− 34z+ 1
4z2− 124z3
1+ 25z+ 1
20z2
1− 35z+ 3
20z2− 160z3
4
5
Byron Ehle conjectured that A-stability impliesd≤ n+2.0 5 10 15 20 25
56
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
The Ehle “conjecture”The Daniel-Moore “conjecture”The Butcher-Chipman conjecture
Theorem
Let R(z) = N(z)/D(z) be an[d,n] approximation with order p= n+d.Then each of the poles and each of the zeros is a terminal pointof anarrow from zero.
0 5 10 15 20 2557
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
The Ehle “conjecture”The Daniel-Moore “conjecture”The Butcher-Chipman conjecture
Theorem
Let R(z) = N(z)/D(z) be an[d,n] approximation with order p= n+d.Then each of the poles and each of the zeros is a terminal pointof anarrow from zero.
This result is illustrated in the case[d,n] = [3,2].
0 5 10 15 20 2558
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
The Ehle “conjecture”The Daniel-Moore “conjecture”The Butcher-Chipman conjecture
Theorem
Let R(z) = N(z)/D(z) be an[d,n] approximation with order p= n+d.Then each of the poles and each of the zeros is a terminal pointof anarrow from zero.
This result is illustrated in the case[d,n] = [3,2].(1) Let n be the number of down-arrows terminating atzeros.
0 5 10 15 20 2559
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
The Ehle “conjecture”The Daniel-Moore “conjecture”The Butcher-Chipman conjecture
Theorem
Let R(z) = N(z)/D(z) be an[d,n] approximation with order p= n+d.Then each of the poles and each of the zeros is a terminal pointof anarrow from zero.
This result is illustrated in the case[d,n] = [3,2].(1) Let n be the number of down-arrows terminating atzeros.(2) Let d be the number of up-arrows terminating atpoles.
0 5 10 15 20 2560
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
The Ehle “conjecture”The Daniel-Moore “conjecture”The Butcher-Chipman conjecture
Theorem
Let R(z) = N(z)/D(z) be an[d,n] approximation with order p= n+d.Then each of the poles and each of the zeros is a terminal pointof anarrow from zero.
This result is illustrated in the case[d,n] = [3,2].(1) Let n be the number of down-arrows terminating atzeros.(2) Let d be the number of up-arrows terminating atpoles.(3) From (1), 1+ n+ d− n down-arrows terminate at+∞.
0 5 10 15 20 2561
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
The Ehle “conjecture”The Daniel-Moore “conjecture”The Butcher-Chipman conjecture
Theorem
Let R(z) = N(z)/D(z) be an[d,n] approximation with order p= n+d.Then each of the poles and each of the zeros is a terminal pointof anarrow from zero.
This result is illustrated in the case[d,n] = [3,2].(1) Let n be the number of down-arrows terminating atzeros.(2) Let d be the number of up-arrows terminating atpoles.(3) From (1), 1+ n+ d− n down-arrows terminate at+∞.(4) Because down-arrows and up-arrows cannot cross there are at leastn+d− n up-arrows terminating at poles.
0 5 10 15 20 2562
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
The Ehle “conjecture”The Daniel-Moore “conjecture”The Butcher-Chipman conjecture
Theorem
Let R(z) = N(z)/D(z) be an[d,n] approximation with order p= n+d.Then each of the poles and each of the zeros is a terminal pointof anarrow from zero.
This result is illustrated in the case[d,n] = [3,2].(1) Let n be the number of down-arrows terminating atzeros.(2) Let d be the number of up-arrows terminating atpoles.(3) From (1), 1+ n+ d− n down-arrows terminate at+∞.(4) Because down-arrows and up-arrows cannot cross there are at leastn+d− n up-arrows terminating at poles.(5) From (2) and (4),n+d− n≤ d.
0 5 10 15 20 2563
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
The Ehle “conjecture”The Daniel-Moore “conjecture”The Butcher-Chipman conjecture
Theorem
Let R(z) = N(z)/D(z) be an[d,n] approximation with order p= n+d.Then each of the poles and each of the zeros is a terminal pointof anarrow from zero.
This result is illustrated in the case[d,n] = [3,2].(1) Let n be the number of down-arrows terminating atzeros.(2) Let d be the number of up-arrows terminating atpoles.(3) From (1), 1+ n+ d− n down-arrows terminate at+∞.(4) Because down-arrows and up-arrows cannot cross there are at leastn+d− n up-arrows terminating at poles.(5) From (2) and (4),n+d− n≤ d.Hence the sum of the two non-negative integersn− n andd− d isnon-positive. Thereforen = n andd = d.
0 5 10 15 20 2564
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
The Ehle “conjecture”The Daniel-Moore “conjecture”The Butcher-Chipman conjecture
Theorem (Hairer-Nørsett-Wanner Theorem)
A [d,n] Pade approximation is an A-function only if
d−n≤ 2.
Let Θ be the set of angles in(−π,π] at which an up-arrow leaves zeroand terminates at a pole.
Because successive members ofΘ differ by at least 2π/(n+d+1), itfollows that
max(Θ)−min(Θ) ≥ 2π(d−1)
n+d+1.
This angle must be less thanπ, otherwise there will be up-arrows fromzero which do one of the following
1 are tangential to the imaginary axis,2 terminate at a pole in the left half-plane3 cross back over the imaginary axis to the right half-plane.
The last two of these options are illustrated on the next slide.0 5 10 15 20 25
65
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
The Ehle “conjecture”The Daniel-Moore “conjecture”The Butcher-Chipman conjecture
Theorem (Hairer-Nørsett-Wanner Theorem)
A [d,n] Pade approximation is an A-function only if
d−n≤ 2.
Let Θ be the set of angles in(−π,π] at which an up-arrow leaves zeroand terminates at a pole.
Because successive members ofΘ differ by at least 2π/(n+d+1), itfollows that
max(Θ)−min(Θ) ≥ 2π(d−1)
n+d+1.
This angle must be less thanπ, otherwise there will be up-arrows fromzero which do one of the following
1 are tangential to the imaginary axis,2 terminate at a pole in the left half-plane3 cross back over the imaginary axis to the right half-plane.
The last two of these options are illustrated on the next slide.0 5 10 15 20 25
66
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
The Ehle “conjecture”The Daniel-Moore “conjecture”The Butcher-Chipman conjecture
Theorem (Hairer-Nørsett-Wanner Theorem)
A [d,n] Pade approximation is an A-function only if
d−n≤ 2.
Let Θ be the set of angles in(−π,π] at which an up-arrow leaves zeroand terminates at a pole.
Because successive members ofΘ differ by at least 2π/(n+d+1), itfollows that
max(Θ)−min(Θ) ≥ 2π(d−1)
n+d+1.
This angle must be less thanπ, otherwise there will be up-arrows fromzero which do one of the following
1 are tangential to the imaginary axis,2 terminate at a pole in the left half-plane3 cross back over the imaginary axis to the right half-plane.
The last two of these options are illustrated on the next slide.0 5 10 15 20 25
67
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
The Ehle “conjecture”The Daniel-Moore “conjecture”The Butcher-Chipman conjecture
Theorem (Hairer-Nørsett-Wanner Theorem)
A [d,n] Pade approximation is an A-function only if
d−n≤ 2.
Let Θ be the set of angles in(−π,π] at which an up-arrow leaves zeroand terminates at a pole.
Because successive members ofΘ differ by at least 2π/(n+d+1), itfollows that
max(Θ)−min(Θ) ≥ 2π(d−1)
n+d+1.
This angle must be less thanπ, otherwise there will be up-arrows fromzero which do one of the following
1 are tangential to the imaginary axis,2 terminate at a pole in the left half-plane3 cross back over the imaginary axis to the right half-plane.
The last two of these options are illustrated on the next slide.0 5 10 15 20 25
68
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
The Ehle “conjecture”The Daniel-Moore “conjecture”The Butcher-Chipman conjecture
Theorem (Hairer-Nørsett-Wanner Theorem)
A [d,n] Pade approximation is an A-function only if
d−n≤ 2.
Let Θ be the set of angles in(−π,π] at which an up-arrow leaves zeroand terminates at a pole.
Because successive members ofΘ differ by at least 2π/(n+d+1), itfollows that
max(Θ)−min(Θ) ≥ 2π(d−1)
n+d+1.
This angle must be less thanπ, otherwise there will be up-arrows fromzero which do one of the following
1 are tangential to the imaginary axis,2 terminate at a pole in the left half-plane3 cross back over the imaginary axis to the right half-plane.
The last two of these options are illustrated on the next slide.0 5 10 15 20 25
69
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
The Ehle “conjecture”The Daniel-Moore “conjecture”The Butcher-Chipman conjecture
2πp+1
0 5 10 15 20 2570
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
The Ehle “conjecture”The Daniel-Moore “conjecture”The Butcher-Chipman conjecture
2πp+1
0 5 10 15 20 2571
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
The Ehle “conjecture”The Daniel-Moore “conjecture”The Butcher-Chipman conjecture
2πp+1
Since2π(d−1)
n+d+1< π,
we deduce thatd−n < 3.
0 5 10 15 20 2572
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
The Ehle “conjecture”The Daniel-Moore “conjecture”The Butcher-Chipman conjecture
The Daniel-Moore “conjecture”
This former conjecture was first proved using order stars in [1], buttoday I will outline an order arrow proof.
It concerns a more general type of approximation, in whichR(z) isreplaced by a solution to a polynomial equation
Φ(w,z) = wmP0(z)+wm−1P1(z)+ · · ·+Pm(z) = 0,and the degrees ofP0,P1, . . . ,Pm are[d0,d1, . . . ,dm].
Order arrows now live on a Riemann surface
Theorem
A multivalue approximation with degree vector[d0,d1, . . . ,dm] is anA-function only if
p≤ 2d0.
0 5 10 15 20 2573
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
The Ehle “conjecture”The Daniel-Moore “conjecture”The Butcher-Chipman conjecture
The Daniel-Moore “conjecture”
This former conjecture was first proved using order stars in [1], buttoday I will outline an order arrow proof.
It concerns a more general type of approximation, in whichR(z) isreplaced by a solution to a polynomial equation
Φ(w,z) = wmP0(z)+wm−1P1(z)+ · · ·+Pm(z) = 0,and the degrees ofP0,P1, . . . ,Pm are[d0,d1, . . . ,dm].
Order arrows now live on a Riemann surface
Theorem
A multivalue approximation with degree vector[d0,d1, . . . ,dm] is anA-function only if
p≤ 2d0.
0 5 10 15 20 2574
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
The Ehle “conjecture”The Daniel-Moore “conjecture”The Butcher-Chipman conjecture
The Daniel-Moore “conjecture”
This former conjecture was first proved using order stars in [1], buttoday I will outline an order arrow proof.
It concerns a more general type of approximation, in whichR(z) isreplaced by a solution to a polynomial equation
Φ(w,z) = wmP0(z)+wm−1P1(z)+ · · ·+Pm(z) = 0,and the degrees ofP0,P1, . . . ,Pm are[d0,d1, . . . ,dm].
Order arrows now live on a Riemann surface
Theorem
A multivalue approximation with degree vector[d0,d1, . . . ,dm] is anA-function only if
p≤ 2d0.
0 5 10 15 20 2575
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
The Ehle “conjecture”The Daniel-Moore “conjecture”The Butcher-Chipman conjecture
The Daniel-Moore “conjecture”
This former conjecture was first proved using order stars in [1], buttoday I will outline an order arrow proof.
It concerns a more general type of approximation, in whichR(z) isreplaced by a solution to a polynomial equation
Φ(w,z) = wmP0(z)+wm−1P1(z)+ · · ·+Pm(z) = 0,and the degrees ofP0,P1, . . . ,Pm are[d0,d1, . . . ,dm].
Order arrows now live on a Riemann surface
Theorem
A multivalue approximation with degree vector[d0,d1, . . . ,dm] is anA-function only if
p≤ 2d0.
0 5 10 15 20 2576
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
The Ehle “conjecture”The Daniel-Moore “conjecture”The Butcher-Chipman conjecture
For an A-approximation, we must avoid an arrow diagram like thefollowing, because we cannot have an up-arrow from zero crossing theimaginary axis on its way to−∞.
2πp+1
0 5 10 15 20 2577
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
The Ehle “conjecture”The Daniel-Moore “conjecture”The Butcher-Chipman conjecture
For an A-approximation, we must avoid an arrow diagram like thefollowing, because we cannot have an up-arrow from zero crossing theimaginary axis on its way to−∞.
2πp+1
Hence,2π(d0 +1)
p+1> π,
which impliesp≤ 2d0.0 5 10 15 20 25
78
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
The Ehle “conjecture”The Daniel-Moore “conjecture”The Butcher-Chipman conjecture
The Butcher-Chipman conjecture
This conjecture concerns a multivalue generalization of Padeapproximations in which the order is
p = d0 +d1 + · · ·+dm+m−1.The conjecture surmised that a necessary condition for anA-approximation is that
2d0−p≤ 2,
just as for the Ehle result.
It is not possible to include a proof of this result here but itcan at leastbe noted that the crucial part of the proof is that every pole is at the endof an up-arrow from zero.
The simple argument which worked in the casem= 1 cannot be usedin the more general situation because it might be possible forup-arrows from zero to cross down-arrows from zero if they lie ondifferent sheets of the Riemann surface.
0 5 10 15 20 2579
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
The Ehle “conjecture”The Daniel-Moore “conjecture”The Butcher-Chipman conjecture
The Butcher-Chipman conjecture
This conjecture concerns a multivalue generalization of Padeapproximations in which the order is
p = d0 +d1 + · · ·+dm+m−1.The conjecture surmised that a necessary condition for anA-approximation is that
2d0−p≤ 2,
just as for the Ehle result.
It is not possible to include a proof of this result here but itcan at leastbe noted that the crucial part of the proof is that every pole is at the endof an up-arrow from zero.
The simple argument which worked in the casem= 1 cannot be usedin the more general situation because it might be possible forup-arrows from zero to cross down-arrows from zero if they lie ondifferent sheets of the Riemann surface.
0 5 10 15 20 2580
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
The Ehle “conjecture”The Daniel-Moore “conjecture”The Butcher-Chipman conjecture
The Butcher-Chipman conjecture
This conjecture concerns a multivalue generalization of Padeapproximations in which the order is
p = d0 +d1 + · · ·+dm+m−1.The conjecture surmised that a necessary condition for anA-approximation is that
2d0−p≤ 2,
just as for the Ehle result.
It is not possible to include a proof of this result here but itcan at leastbe noted that the crucial part of the proof is that every pole is at the endof an up-arrow from zero.
The simple argument which worked in the casem= 1 cannot be usedin the more general situation because it might be possible forup-arrows from zero to cross down-arrows from zero if they lie ondifferent sheets of the Riemann surface.
0 5 10 15 20 2581
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
The Ehle “conjecture”The Daniel-Moore “conjecture”The Butcher-Chipman conjecture
The Butcher-Chipman conjecture
This conjecture concerns a multivalue generalization of Padeapproximations in which the order is
p = d0 +d1 + · · ·+dm+m−1.The conjecture surmised that a necessary condition for anA-approximation is that
2d0−p≤ 2,
just as for the Ehle result.
It is not possible to include a proof of this result here but itcan at leastbe noted that the crucial part of the proof is that every pole is at the endof an up-arrow from zero.
The simple argument which worked in the casem= 1 cannot be usedin the more general situation because it might be possible forup-arrows from zero to cross down-arrows from zero if they lie ondifferent sheets of the Riemann surface.
0 5 10 15 20 2582
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Why arrow pictures are hard to drawWhy arrow pictures are easy to drawA differential equation for order arrows
1 Introduction
Order and stability
Three Runge–Kutta methods
Relative stability regions2 Order stars and order arrows
Example 1: the implicit Euler method
Example 2: a third order implicit method
Properties of order arrows3 Applications
The Ehle “conjecture”
The Daniel-Moore “conjecture”
The Butcher-Chipman conjecture4 Drawing pictures
Why arrow pictures are hard to draw
Why arrow pictures are easy to draw
A differential equation for order arrows5 Gallery
0 5 10 15 20 2583
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Why arrow pictures are hard to drawWhy arrow pictures are easy to drawA differential equation for order arrows
Why arrow pictures are hard to draw
Becausew(z) = R(z)exp(−z) is very close to 1 whenz is close tozero, it is difficult to determine accurately whenw(z) is real.
To illustrate this, consider the[5,5] Pade approximation
R(z) =1+ 1
2z+ 19z2 + 1
72z3 + 11008z
4 + 130240z
5
1− 12z+ 1
9z2− 172z3 + 1
1008z4− 1
30240z5
In the next slide, we will present a figure constructed by evaluatingthe imaginary part ofR(z)exp(−z) over a grid of points superimposedon the unit circle centred at 0.
The centre of each small square was regarded as a point on an orderarrow if the sum of the signs of the imaginary parts ofw(z) at thecorners was−1, 0 or+1 and the real part ofw(z) is positive.
0 5 10 15 20 2584
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Why arrow pictures are hard to drawWhy arrow pictures are easy to drawA differential equation for order arrows
Why arrow pictures are hard to draw
Becausew(z) = R(z)exp(−z) is very close to 1 whenz is close tozero, it is difficult to determine accurately whenw(z) is real.
To illustrate this, consider the[5,5] Pade approximation
R(z) =1+ 1
2z+ 19z2 + 1
72z3 + 11008z
4 + 130240z
5
1− 12z+ 1
9z2− 172z3 + 1
1008z4− 1
30240z5
In the next slide, we will present a figure constructed by evaluatingthe imaginary part ofR(z)exp(−z) over a grid of points superimposedon the unit circle centred at 0.
The centre of each small square was regarded as a point on an orderarrow if the sum of the signs of the imaginary parts ofw(z) at thecorners was−1, 0 or+1 and the real part ofw(z) is positive.
0 5 10 15 20 2585
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Why arrow pictures are hard to drawWhy arrow pictures are easy to drawA differential equation for order arrows
Why arrow pictures are hard to draw
Becausew(z) = R(z)exp(−z) is very close to 1 whenz is close tozero, it is difficult to determine accurately whenw(z) is real.
To illustrate this, consider the[5,5] Pade approximation
R(z) =1+ 1
2z+ 19z2 + 1
72z3 + 11008z
4 + 130240z
5
1− 12z+ 1
9z2− 172z3 + 1
1008z4− 1
30240z5
In the next slide, we will present a figure constructed by evaluatingthe imaginary part ofR(z)exp(−z) over a grid of points superimposedon the unit circle centred at 0.
The centre of each small square was regarded as a point on an orderarrow if the sum of the signs of the imaginary parts ofw(z) at thecorners was−1, 0 or+1 and the real part ofw(z) is positive.
0 5 10 15 20 2586
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Why arrow pictures are hard to drawWhy arrow pictures are easy to drawA differential equation for order arrows
Why arrow pictures are hard to draw
Becausew(z) = R(z)exp(−z) is very close to 1 whenz is close tozero, it is difficult to determine accurately whenw(z) is real.
To illustrate this, consider the[5,5] Pade approximation
R(z) =1+ 1
2z+ 19z2 + 1
72z3 + 11008z
4 + 130240z
5
1− 12z+ 1
9z2− 172z3 + 1
1008z4− 1
30240z5
In the next slide, we will present a figure constructed by evaluatingthe imaginary part ofR(z)exp(−z) over a grid of points superimposedon the unit circle centred at 0.
The centre of each small square was regarded as a point on an orderarrow if the sum of the signs of the imaginary parts ofw(z) at thecorners was−1, 0 or+1 and the real part ofw(z) is positive.
0 5 10 15 20 2587
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Why arrow pictures are hard to drawWhy arrow pictures are easy to drawA differential equation for order arrows
|z| = 1
0 5 10 15 20 2588
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Why arrow pictures are hard to drawWhy arrow pictures are easy to drawA differential equation for order arrows
Why arrow pictures are easy to draw
Even though we cannot obtain a clear picture of the arrows by thistechnique, we can take into account that, near zero, we know that thelines we want are approximately radial with arguments
θk = 2πk/22, k = 0,1,2, . . . ,21.
In the next figure, lines in these directions are shown but with eachpoint replaced by the closest point arising in the previous figure.
0 5 10 15 20 2589
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Why arrow pictures are hard to drawWhy arrow pictures are easy to drawA differential equation for order arrows
Why arrow pictures are easy to draw
Even though we cannot obtain a clear picture of the arrows by thistechnique, we can take into account that, near zero, we know that thelines we want are approximately radial with arguments
θk = 2πk/22, k = 0,1,2, . . . ,21.
In the next figure, lines in these directions are shown but with eachpoint replaced by the closest point arising in the previous figure.
0 5 10 15 20 2590
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Why arrow pictures are hard to drawWhy arrow pictures are easy to drawA differential equation for order arrows
b b b bb
bb
b
b
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This gives a reasonable looking effect which can be improvedbystarting with a much finer grid and extending the region covered to alarge rectangle centred at the origin.
Most of the pictures in this talk were drawn in this way.0 5 10 15 20 25
91
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Why arrow pictures are hard to drawWhy arrow pictures are easy to drawA differential equation for order arrows
b b b bb
bb
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
bb
b
bbbb b
bb
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
bb
b
This gives a reasonable looking effect which can be improvedbystarting with a much finer grid and extending the region covered to alarge rectangle centred at the origin.
Most of the pictures in this talk were drawn in this way.0 5 10 15 20 25
92
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Why arrow pictures are hard to drawWhy arrow pictures are easy to drawA differential equation for order arrows
b b b bb
bb
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
bb
b
bbbb b
bb
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
bb
b
This gives a reasonable looking effect which can be improvedbystarting with a much finer grid and extending the region covered to alarge rectangle centred at the origin.
Most of the pictures in this talk were drawn in this way.0 5 10 15 20 25
93
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Why arrow pictures are hard to drawWhy arrow pictures are easy to drawA differential equation for order arrows
A differential equation for order arrows
The difficulty caused by machine arithmetic can be eliminated inanother way, in the case of a Pade approximationN(z)/D(z).
Write w = exp(t) in the relationshipwD(z)exp(z)−N(z) = 0, so that
exp(z+ t)D(z)−N(z) = 0. (1)We can now construct a differential equation expressing thedependence ofz on t. Differentiate (1) and it is found that
exp(z+ t)(z′(t)(D′(z)+D(z))+D(z))−z′(t)N′(z) = 0. (2)
Eliminate exp(z+ t) from (1) and (2) to yield the differential equation
z′(t)F = N(z)D(z), (3)where
F = D(z)N′(z)−D′(z)N(z)−D(z)N(z). (4)
0 5 10 15 20 2594
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Why arrow pictures are hard to drawWhy arrow pictures are easy to drawA differential equation for order arrows
A differential equation for order arrows
The difficulty caused by machine arithmetic can be eliminated inanother way, in the case of a Pade approximationN(z)/D(z).
Write w = exp(t) in the relationshipwD(z)exp(z)−N(z) = 0, so that
exp(z+ t)D(z)−N(z) = 0. (1)We can now construct a differential equation expressing thedependence ofz on t. Differentiate (1) and it is found that
exp(z+ t)(z′(t)(D′(z)+D(z))+D(z))−z′(t)N′(z) = 0. (2)
Eliminate exp(z+ t) from (1) and (2) to yield the differential equation
z′(t)F = N(z)D(z), (3)where
F = D(z)N′(z)−D′(z)N(z)−D(z)N(z). (4)
0 5 10 15 20 2595
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Why arrow pictures are hard to drawWhy arrow pictures are easy to drawA differential equation for order arrows
A differential equation for order arrows
The difficulty caused by machine arithmetic can be eliminated inanother way, in the case of a Pade approximationN(z)/D(z).
Write w = exp(t) in the relationshipwD(z)exp(z)−N(z) = 0, so that
exp(z+ t)D(z)−N(z) = 0. (1)We can now construct a differential equation expressing thedependence ofz on t. Differentiate (1) and it is found that
exp(z+ t)(z′(t)(D′(z)+D(z))+D(z))−z′(t)N′(z) = 0. (2)
Eliminate exp(z+ t) from (1) and (2) to yield the differential equation
z′(t)F = N(z)D(z), (3)where
F = D(z)N′(z)−D′(z)N(z)−D(z)N(z). (4)
0 5 10 15 20 2596
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Why arrow pictures are hard to drawWhy arrow pictures are easy to drawA differential equation for order arrows
A differential equation for order arrows
The difficulty caused by machine arithmetic can be eliminated inanother way, in the case of a Pade approximationN(z)/D(z).
Write w = exp(t) in the relationshipwD(z)exp(z)−N(z) = 0, so that
exp(z+ t)D(z)−N(z) = 0. (1)We can now construct a differential equation expressing thedependence ofz on t. Differentiate (1) and it is found that
exp(z+ t)(z′(t)(D′(z)+D(z))+D(z))−z′(t)N′(z) = 0. (2)
Eliminate exp(z+ t) from (1) and (2) to yield the differential equation
z′(t)F = N(z)D(z), (3)where
F = D(z)N′(z)−D′(z)N(z)−D(z)N(z). (4)
0 5 10 15 20 2597
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Why arrow pictures are hard to drawWhy arrow pictures are easy to drawA differential equation for order arrows
Interpret exp(z+ t)D(z)−N(z) = 0 as definingt as a function ofz for|z| small, so thatt = −Czp+1+O(zp+2), whereC is the error constant.
We can now say something aboutF. It is equal to
D(z)N′(z)−D′(z)N(z)−D(z)N(z) = −(p+1)Czp +O(zp+1).
TheO(zp+1) term can be deleted becauseF is a polynomial of degreep.
Hence ast moves from zero in a positive direction (up-arrows) or anegative direction (down-arrows) thezvalue traces out a pathsatisfying the differential equation
Cd(zp+1)
dt= −N(z)D(z),
This differential equation can be used to draw order arrows accurately.
0 5 10 15 20 2598
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Why arrow pictures are hard to drawWhy arrow pictures are easy to drawA differential equation for order arrows
Interpret exp(z+ t)D(z)−N(z) = 0 as definingt as a function ofz for|z| small, so thatt = −Czp+1+O(zp+2), whereC is the error constant.
We can now say something aboutF. It is equal to
D(z)N′(z)−D′(z)N(z)−D(z)N(z) = −(p+1)Czp +O(zp+1).
TheO(zp+1) term can be deleted becauseF is a polynomial of degreep.
Hence ast moves from zero in a positive direction (up-arrows) or anegative direction (down-arrows) thezvalue traces out a pathsatisfying the differential equation
Cd(zp+1)
dt= −N(z)D(z),
This differential equation can be used to draw order arrows accurately.
0 5 10 15 20 2599
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Why arrow pictures are hard to drawWhy arrow pictures are easy to drawA differential equation for order arrows
Interpret exp(z+ t)D(z)−N(z) = 0 as definingt as a function ofz for|z| small, so thatt = −Czp+1+O(zp+2), whereC is the error constant.
We can now say something aboutF. It is equal to
D(z)N′(z)−D′(z)N(z)−D(z)N(z) = −(p+1)Czp +O(zp+1).
TheO(zp+1) term can be deleted becauseF is a polynomial of degreep.
Hence ast moves from zero in a positive direction (up-arrows) or anegative direction (down-arrows) thezvalue traces out a pathsatisfying the differential equation
Cd(zp+1)
dt= −N(z)D(z),
This differential equation can be used to draw order arrows accurately.
0 5 10 15 20 25100
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Why arrow pictures are hard to drawWhy arrow pictures are easy to drawA differential equation for order arrows
Interpret exp(z+ t)D(z)−N(z) = 0 as definingt as a function ofz for|z| small, so thatt = −Czp+1+O(zp+2), whereC is the error constant.
We can now say something aboutF. It is equal to
D(z)N′(z)−D′(z)N(z)−D(z)N(z) = −(p+1)Czp +O(zp+1).
TheO(zp+1) term can be deleted becauseF is a polynomial of degreep.
Hence ast moves from zero in a positive direction (up-arrows) or anegative direction (down-arrows) thezvalue traces out a pathsatisfying the differential equation
Cd(zp+1)
dt= −N(z)D(z),
This differential equation can be used to draw order arrows accurately.
0 5 10 15 20 25101
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Why arrow pictures are hard to drawWhy arrow pictures are easy to drawA differential equation for order arrows
Interpret exp(z+ t)D(z)−N(z) = 0 as definingt as a function ofz for|z| small, so thatt = −Czp+1+O(zp+2), whereC is the error constant.
We can now say something aboutF. It is equal to
D(z)N′(z)−D′(z)N(z)−D(z)N(z) = −(p+1)Czp +O(zp+1).
TheO(zp+1) term can be deleted becauseF is a polynomial of degreep.
Hence ast moves from zero in a positive direction (up-arrows) or anegative direction (down-arrows) thezvalue traces out a pathsatisfying the differential equation
Cd(zp+1)
dt= −N(z)D(z),
This differential equation can be used to draw order arrows accurately.
0 5 10 15 20 25102
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
1 Introduction
Order and stability
Three Runge–Kutta methods
Relative stability regions2 Order stars and order arrows
Example 1: the implicit Euler method
Example 2: a third order implicit method
Properties of order arrows3 Applications
The Ehle “conjecture”
The Daniel-Moore “conjecture”
The Butcher-Chipman conjecture4 Drawing pictures
Why arrow pictures are hard to draw
Why arrow pictures are easy to draw
A differential equation for order arrows5 Gallery
0 5 10 15 20 25103
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Gallery
I would now like to show you a gallery of some order arrows picturesbased on both rational and quadratic approximations.
Where an approximation is given in the form[
P0(z), P1(z), P2(z)]
this denotes the quadratic function
Φ(w,z) = P0(z)w2 +P1(z)w+P2(z).
The first picture will be for the[5,5] Pade approximation.
This was constructed using the differential equation approach.
0 5 10 15 20 25104
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Gallery
I would now like to show you a gallery of some order arrows picturesbased on both rational and quadratic approximations.
Where an approximation is given in the form[
P0(z), P1(z), P2(z)]
this denotes the quadratic function
Φ(w,z) = P0(z)w2 +P1(z)w+P2(z).
The first picture will be for the[5,5] Pade approximation.
This was constructed using the differential equation approach.
0 5 10 15 20 25105
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Gallery
I would now like to show you a gallery of some order arrows picturesbased on both rational and quadratic approximations.
Where an approximation is given in the form[
P0(z), P1(z), P2(z)]
this denotes the quadratic function
Φ(w,z) = P0(z)w2 +P1(z)w+P2(z).
The first picture will be for the[5,5] Pade approximation.
This was constructed using the differential equation approach.
0 5 10 15 20 25106
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Gallery
I would now like to show you a gallery of some order arrows picturesbased on both rational and quadratic approximations.
Where an approximation is given in the form[
P0(z), P1(z), P2(z)]
this denotes the quadratic function
Φ(w,z) = P0(z)w2 +P1(z)w+P2(z).
The first picture will be for the[5,5] Pade approximation.
This was constructed using the differential equation approach.
0 5 10 15 20 25107
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Pade approximation[5,5]
0 5 10 15 20 25108
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Pade approximation[5,5]
10
−10
10−10
Sixth order with poles{4,5,6,7,8,9}
0 5 10 15 20 25109
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Pade approximation[5,5]
10
−10
10−10
Sixth order with poles{4,5,6,7,8,9}
Sixth order with poles{6,7,8,9,10,11}
10
−10
10−5
0 5 10 15 20 25110
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Pade approximation[5,5]
10
−10
10−10
Sixth order with poles{4,5,6,7,8,9}
Sixth order with poles{6,7,8,9,10,11}
10
−10
10−5
Seventh order with poles{ 52, 7
2, 72, 9
2, 92, 2621
533 }
5
−5
5−5
0 5 10 15 20 25111
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Pade approximation[5,5]
10
−10
10−10
Sixth order with poles{4,5,6,7,8,9}
Sixth order with poles{6,7,8,9,10,11}
10
−10
10−5
Seventh order with poles{ 52, 7
2, 72, 9
2, 92, 2621
533 }
5
−5
5−5
A second derivative approximationp = 65
−5
10−5
0 5 10 15 20 25112
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Pade approximation[5,5]
Sixth order with poles{6,7,8,9,10,11}
10
−10
10−5
Seventh order with poles{ 52, 7
2, 72, 9
2, 92, 2621
533 }
5
−5
5−5
A second derivative approximationp = 65
−5
10−5
An A-acceptable second derivative approximationp = 6
4
−4
4−4
0 5 10 15 20 25113
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Pade approximation[5,5]Seventh order with poles{ 5
2, 72, 7
2, 92, 9
2, 2621533 }
5
−5
5−5
A second derivative approximationp = 65
−5
10−5
An A-acceptable second derivative approximationp = 6
4
−4
4−4
A ninth order approximation
8
−8
8−8
0 5 10 15 20 25114
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Pade approximation[5,5]
A second derivative approximationp = 65
−5
10−5
An A-acceptable second derivative approximationp = 6
4
−4
4−4
A ninth order approximation
8
−8
8−8
A fifteenth order approximation
10
−10
10−10
0 5 10 15 20 25115
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Pade approximation[5,5]
An A-acceptable second derivative approximationp = 6
4
−4
4−4
A ninth order approximation
8
−8
8−8
A fifteenth order approximation
10
−10
10−10
Fifth order method[(1− 1
4 z+ 112 z2)2,− 61
3 , 583 + 113
6 z+ 14316 z2 + 95
36 z3 + 67144z4
]
4
−4
4−4+
+
+
+
+
+
+
+
0 5 10 15 20 25116
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Pade approximation[5,5]
A ninth order approximation
8
−8
8−8
A fifteenth order approximation
10
−10
10−10
Fifth order method[(1− 1
4 z+ 112 z2)2,− 61
3 , 583 + 113
6 z+ 14316 z2 + 95
36 z3 + 67144z4
]
4
−4
4−4+
+
+
+
+
+
+
+
p=7:[(1− 3
10z+120z
2)3,− 79350 , 743
50 + 36925z+ 731
100z2+ 2387
1000z3+ 3403
6000z4+ 399
4000z5+ 829
72000z6]
4
−4
4−4
+
+
+
+
+
+
+
+
+
+
+
+
0 5 10 15 20 25117
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Pade approximation[5,5]
A fifteenth order approximation
10
−10
10−10
Fifth order method[(1− 1
4 z+ 112 z2)2,− 61
3 , 583 + 113
6 z+ 14316 z2 + 95
36 z3 + 67144z4
]
4
−4
4−4+
+
+
+
+
+
+
+
p=7:[(1− 3
10z+120z
2)3,− 79350 , 743
50 + 36925z+ 731
100z2+ 2387
1000z3+ 3403
6000z4+ 399
4000z5+ 829
72000z6]
4
−4
4−4
+
+
+
+
+
+
+
+
+
+
+
+
[(1−4
7z+( 221−
√31930840 )z2)2,−3
2, 12+
914z+(529
588−√
31930420 )z2+(23
88−√
31930294 )z3+(26497
70560−4√
319302205 )z4
]
3
−3
3−3
+
+
+
+
++
++
0 5 10 15 20 25118
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Pade approximation[5,5]
Fifth order method[(1− 1
4 z+ 112 z2)2,− 61
3 , 583 + 113
6 z+ 14316 z2 + 95
36 z3 + 67144z4
]
4
−4
4−4+
+
+
+
+
+
+
+
p=7:[(1− 3
10z+120z
2)3,− 79350 , 743
50 + 36925z+ 731
100z2+ 2387
1000z3+ 3403
6000z4+ 399
4000z5+ 829
72000z6]
4
−4
4−4
+
+
+
+
+
+
+
+
+
+
+
+
[(1−4
7z+( 221−
√31930840 )z2)2,−3
2, 12+
914z+(529
588−√
31930420 )z2+(23
88−√
31930294 )z3+(26497
70560−4√
319302205 )z4
]
3
−3
3−3
+
+
+
+
++
++
[(1− 3
10z+133z
2)3,−13109299825,
3126799825+
42569199650z+
38213399300z
2+ 54221311979000z
3+ 275321497375z
4+ 305995989500z
5+ 1451917968500z
6]
3
−3
3−3
+
+
+
+
+
+
+
+
+
+
++
0 5 10 15 20 25119
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Pade approximation[5,5]p=7:
[(1− 3
10z+120z
2)3,− 79350 , 743
50 + 36925z+ 731
100z2+ 2387
1000z3+ 3403
6000z4+ 399
4000z5+ 829
72000z6]
4
−4
4−4
+
+
+
+
+
+
+
+
+
+
+
+
[(1−4
7z+( 221−
√31930840 )z2)2,−3
2, 12+
914z+(529
588−√
31930420 )z2+(23
88−√
31930294 )z3+(26497
70560−4√
319302205 )z4
]
3
−3
3−3
+
+
+
+
++
++
[(1− 3
10z+133z
2)3,−13109299825,
3126799825+
42569199650z+
38213399300z
2+ 54221311979000z
3+ 275321497375z
4+ 305995989500z
5+ 1451917968500z
6]
3
−3
3−3
+
+
+
+
+
+
+
+
+
+
++
[(1− 1
6z+ 112z2)2(1− 1
2z)2,− 32, 1
2 + 56z+ 23
36z2 + 13z3 + 7
48z4 + 1212160z
5 + 492880z
6]
3
−3
3−3
+
+
+
+
+
+
+
+
+
+++
0 5 10 15 20 25120
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Pade approximation[5,5][(1−4
7z+( 221−
√31930840 )z2)2,−3
2, 12+
914z+(529
588−√
31930420 )z2+(23
88−√
31930294 )z3+(26497
70560−4√
319302205 )z4
]
3
−3
3−3
+
+
+
+
++
++
[(1− 3
10z+133z
2)3,−13109299825,
3126799825+
42569199650z+
38213399300z
2+ 54221311979000z
3+ 275321497375z
4+ 305995989500z
5+ 1451917968500z
6]
3
−3
3−3
+
+
+
+
+
+
+
+
+
+
++
[(1− 1
6z+ 112z2)2(1− 1
2z)2,− 32, 1
2 + 56z+ 23
36z2 + 13z3 + 7
48z4 + 1212160z
5 + 492880z
6]
3
−3
3−3
+
+
+
+
+
+
+
+
+
+++
[(1− 1
5z+ 215z2)2(1− 1
2z)2,− 32, 1
2+ 910z+ 89
150z2+ 79300z3+ 437
3600z4+ 71
1200z5+ 7
288z6]
3
−3
3−3
+
+
+
+
+
+
+
+
+
+
++
0 5 10 15 20 25121
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
Pade approximation[5,5]
[(1− 1
6z+ 112z2)2(1− 1
2z)2,− 32, 1
2 + 56z+ 23
36z2 + 13z3 + 7
48z4 + 1212160z
5 + 492880z
6]
3
−3
3−3
+
+
+
+
+
+
+
+
+
+++
[(1− 1
5z+ 215z2)2(1− 1
2z)2,− 32, 1
2+ 910z+ 89
150z2+ 79300z3+ 437
3600z4+ 71
1200z5+ 7
288z6]
3
−3
3−3
+
+
+
+
+
+
+
+
+
+
++
[(1− 1
5z+ 215z2)2(1− 1
3z)2,−32, 1
2+ 1730z+ 179
900z2− 1
60z3− 732160z
4− 8910800z
5+ 19964800z
6]
4
−4
4−2+
+
+
+
+
+
+
+
+
+
+
+
0 5 10 15 20 25122
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
10
−10
10−10
10
−10
10−5
5
−5
5−5
5
−5
10−5
4
−4
4−4
8
−8
8−8
10
−10
10−10
4
−4
4−4+
+
+
+
+
+
+
+
4
−4
4−4
+
+
+
+
+
+
+
+
+
+
+
+
3
−3
3−3
+
+
+
+
++
++
3
−3
3−3
+
+
+
+
+
+
+
+
+
+
++
3
−3
3−3
+
+
+
+
+
+
+
+
+
+++
3
−3
3−3
+
+
+
+
+
+
+
+
+
+
++
4
−4
4−2+
+
+
+
+
+
+
+
+
+
+
+
0 5 10 15 20 25123
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
4
−4
4−4
8
−8
8−8
10
−10
10−10
4
−4
4−4+
+
+
+
+
+
+
+
4
−4
4−4
+
+
+
+
+
+
+
+
+
+
+
+
3
−3
3−3
+
+
+
+
++
++
3
−3
3−3
+
+
+
+
+
+
+
+
+
+
++
3
−3
3−3
+
+
+
+
+
+
+
+
+
+++
3
−3
3−3
+
+
+
+
+
+
+
+
+
+
++
4
−4
4−2+
+
+
+
+
+
+
+
+
+
+
+
THANK YOU FOR
0 5 10 15 20 25124
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
3
−3
3−3
+
+
+
+
++
++
3
−3
3−3
+
+
+
+
+
+
+
+
+
+
++
3
−3
3−3
+
+
+
+
+
+
+
+
+
+++
3
−3
3−3
+
+
+
+
+
+
+
+
+
+
++
4
−4
4−2+
+
+
+
+
+
+
+
+
+
+
+
THANK YOU FOR
YOUR KIND
0 5 10 15 20 25125
IntroductionOrder stars and order arrows
ApplicationsDrawing pictures
Gallery
THANK YOU FOR
YOUR KIND
ATTENTION
0 5 10 15 20 25126
