1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
|
\documentclass{article}
\usepackage{axiom}
\begin{document}
\title{\$SPAD/src/input images2a.input}
\author{The Axiom Team}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{License}
<<license>>=
--Copyright The Numerical Algorithms Group Limited 1994.
@
<<*>>=
<<license>>
-- Newton's Iteration function
-- newtonStep(f) returns a newton's iteration function for the
-- expression f.
newtonStep(f) ==
fun := complexNumericFunction f
deriv := complexDerivativeFunction(f,1)
(b:Complex DoubleFloat):Complex DoubleFloat +->
b - fun(b)/deriv(b)
-- create complex numeric functions from an expression
complexFunPack := MakeUnaryCompiledFunction(EXPR INT, Complex DoubleFloat, Complex DoubleFloat)
-- create a complex numeric function from an expression
complexNumericFunction x ==
v := theVariable x
compiledFunction(x, v)$complexFunPack
-- create a complex numeric derivatiave function from an expression
complexDerivativeFunction(x,n) ==
v := theVariable x
df := differentiate(x,v,n)
compiledFunction(df, v)$complexFunPack
-- return the unique variable in x, or an error if it is multivariate
theVariable x ==
vl := variables x
nv := # vl
nv > 1 => error "Expression is not univariate."
nv = 0 => 'x
first vl
-- complex surface and vector field drawing by SCM
-- complex surface vector field drawing
C := Complex DoubleFloat
S := Segment DoubleFloat
PC := Record(rr:DoubleFloat, th:DoubleFloat)
realSteps: PI := 25 -- the number of steps in the real direction
imagSteps: PI := 25 -- the number of steps in the imaginary direction
clipValue: DoubleFloat := 10 -- the maximum length of a vector to draw
-- Draw a complex function as a height field
-- uses the complex norm as the height and the complex argument as the color
-- optionally it will draw arrows on the surface indicating the direction
-- of the complex argument
-- sample call:
-- f: C -> C
-- f z == exp(1/z)
-- drawComplex(f, 0.3..3, 0..2*%pi, false)
-- parameter descriptions:
-- f: the function to draw
-- rRange: the range of the real values
-- imagRange: the range of imaginary values
drawComplex(f: C -> C, realRange: S, imagRange: S): VIEW3D ==
free realSteps, imagSteps
delReal := (hi(realRange) - lo(realRange))/realSteps
delImag := (hi(imagRange) - lo(imagRange))/imagSteps
funTable: ARRAY2(PC) := new(realSteps+1, imagSteps+1, [0,0]$PC)
real := lo(realRange)
for i in 1..realSteps+1 repeat
imag := lo(imagRange)
for j in 1..imagSteps+1 repeat
z := f complex(real, imag)
funTable(i,j) := [clipFun(sqrt norm z), argument(z)]$PC
imag := imag + delImag
real := real + delReal
llp:List List Point DoubleFloat := []
real := lo(realRange)
for i in 1..realSteps+1 repeat
imag := lo(imagRange)
lp:List Point DoubleFloat := []
for j in 1..imagSteps+1 repeat
lp := cons(point [real,imag, funTable(i,j).rr,
funTable(i,j).th] ,lp)
imag := imag + delImag
real := real + delReal
llp := cons(reverse! lp, llp)
llp := reverse! llp
space := mesh(llp)$ThreeSpace(DoubleFloat)
makeViewport3D(space, "Complex Function")$VIEW3D
-- draw a complex vector field
-- these vector fields should be viewed from the top by pressing the
-- "XY" translate button on the VIEW3D control panel
-- parameters:
-- f: the mapping from C to C which we will draw
-- realRange: the range of the reals
-- tRange: the range of the imaginaries
-- sample call:
-- f z == sin z
-- drawComplexVectorField(f, -2..2, -2..2)
-- call the functions 'setRealSteps' and 'setImagSteps' to change the
-- number of arrows drawn in each direction.
drawComplexVectorField(f: C -> C, realRange: S, imagRange: S): VIEW3D ==
-- compute the steps size of the grid
delReal := (hi(realRange) - lo(realRange))/realSteps
delImag := (hi(imagRange) - lo(imagRange))/imagSteps
-- create the space to hold the arrows
space := create3Space()$ThreeSpace DoubleFloat
real := lo(realRange)
for i in 1..realSteps+1 repeat
imag := lo(imagRange)
for j in 1..imagSteps+1 repeat
-- compute the function
z := f complex(real, imag)
-- get the direction of the arrow
arg := argument z
-- get the length of the arrow
len := clipFun(sqrt norm z)
-- create point at the base of the arrow
p1 := point [real, imag, 0.0@DoubleFloat, arg]
-- scale the arrow length so it isn't too long
scaleLen := delReal * len
-- create the point at the top of the arrow
p2 := point [p1.1 + scaleLen*cos(arg), p1.2 + scaleLen*sin(arg),
0.0@DoubleFloat, arg]
-- make the pointer at the top of the arrow
arrow := makeArrow(p1, p2, scaleLen, arg)
-- add the line segments in the arrow to the space
for a in arrow repeat curve(space, a)$ThreeSpace DoubleFloat
imag := imag + delImag
real := real + delReal
-- draw the vector feild
makeViewport3D(space, "Complex Vector Field")$VIEW3D
-- relative size of the arrow head compared to the length of the arrow
arrowScale := 0.25@DoubleFloat
-- angle of the arrow head
arrowAngle := %pi-%pi/10.0@DoubleFloat
-- Add an arrow head to a line segment, which starts at 'p1', ends at 'p2',
-- has length 'len', and and angle 'arg'. We pass 'len' and 'arg' as
-- arguments since thet were already computed by the calling program
makeArrow(p1, p2, len, arg) ==
c1 := cos(arg + arrowAngle)
s1 := sin(arg + arrowAngle)
c2 := cos(arg - arrowAngle)
s2 := sin(arg - arrowAngle)
p3 := point [p2.1 + c1*arrowScale*len, p2.2 + s1*arrowScale*len,
p2.3, p2.4]
p4 := point [p2.1 + c2*arrowScale*len, p2.2 + s2*arrowScale*len,
p2.3, p2.4]
[[p1, p2, p3], [p2, p4]]
-- set the number of steps to use in the real direction
setRealSteps(n) ==
free realSteps
realSteps := n
-- set the number of steps to use in the imaginary direction
setImagSteps(n) ==
free imagSteps
imagSteps := n
-- set the maximum length of a vector
setClipValue clip ==
free clipValue
clipValue := clip
-- clip a value in the interval (-clip...clip)
clipFun(x:DoubleFloat):DoubleFloat ==
min(max(x, -clipValue), clipValue)
-- create a Newton's iteration function for the equation x**3 = 2.
f := newtonStep(x**3 - 2)
setClipValue(4)
drawComplexVectorField(f**3, -3..3, -3..3)
drawComplex(f**3, -3..3, -3..3)
drawComplex(f**4, -3..3, -3..3)
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
|
