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
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
|
;;;"root.scm" Newton's and Laguerre's methods for finding roots.
;Copyright (C) 1996, 1997 Aubrey Jaffer
;
;Permission to copy this software, to modify it, to redistribute it,
;to distribute modified versions, and to use it for any purpose is
;granted, subject to the following restrictions and understandings.
;
;1. Any copy made of this software must include this copyright notice
;in full.
;
;2. I have made no warranty or representation that the operation of
;this software will be error-free, and I am under no obligation to
;provide any services, by way of maintenance, update, or otherwise.
;
;3. In conjunction with products arising from the use of this
;material, there shall be no use of my name in any advertising,
;promotional, or sales literature without prior written consent in
;each case.
(require 'logical)
;;;; Newton's Method explained in:
;;; D. E. Knuth, "The Art of Computer Programming", Vol 2 /
;;; Seminumerical Algorithms, Reading Massachusetts, Addison-Wesley
;;; Publishing Company, 2nd Edition, p. 510
;@
(define (newton:find-integer-root f df/dx x_0)
(let loop ((x x_0) (fx (f x_0)))
(cond
((zero? fx) x)
(else
(let ((df (df/dx x)))
(cond
((zero? df) #f) ; stuck at local min/max
(else
(let* ((delta (quotient (+ fx (quotient df 2)) df))
(next-x (cond ((not (zero? delta)) (- x delta))
((positive? fx) (- x 1))
(else (- x -1))))
(next-fx (f next-x)))
(cond ((>= (abs next-fx) (abs fx)) x)
(else (loop next-x next-fx)))))))))))
;;(define (integer-sqrt y)
;; (newton:find-integer-root (lambda (x) (- (* x x) y))
;; (lambda (x) (* 2 x))
;; (ash 1 (quotient (integer-length y) 2))))
;;;; http://www.cs.cmu.edu/afs/cs/project/ai-repository/ai/lang/lisp/code/math/isqrt/isqrt.txt
;;; Akira Kurihara
;;; School of Mathematics
;;; Japan Women's University
;@
(define integer-sqrt
(let ((table '#(0
1 1 1
2 2 2 2 2
3 3 3 3 3 3 3
4 4 4 4 4 4 4 4 4))
(square (lambda (x) (* x x))))
(lambda (n)
(define (isqrt n)
(if (> n 24)
(let* ((len/4 (quotient (- (integer-length n) 1) 4))
(top (isqrt (ash n (* -2 len/4))))
(init (ash top len/4))
(q (quotient n init))
(iter (quotient (+ init q) 2)))
(cond ((odd? q) iter)
((< (remainder n init) (square (- iter init))) (- iter 1))
(else iter)))
(vector-ref table n)))
(if (and (exact? n) (integer? n) (not (negative? n)))
(isqrt n)
(type-error 'integer-sqrt n)))))
;@
(define (newton:find-root f df/dx x_0 prec)
(if (and (negative? prec) (integer? prec))
(let loop ((x x_0) (fx (f x_0)) (count prec))
(cond ((zero? count) x)
(else (let ((df (df/dx x)))
(cond ((zero? df) #f) ; stuck at local min/max
(else (let* ((next-x (- x (/ fx df)))
(next-fx (f next-x)))
(cond ((= next-x x) x)
((> (abs next-fx) (abs fx)) #f)
(else (loop next-x next-fx
(+ 1 count)))))))))))
(let loop ((x x_0) (fx (f x_0)))
(cond ((< (abs fx) prec) x)
(else (let ((df (df/dx x)))
(cond ((zero? df) #f) ; stuck at local min/max
(else (let* ((next-x (- x (/ fx df)))
(next-fx (f next-x)))
(cond ((= next-x x) x)
((> (abs next-fx) (abs fx)) #f)
(else (loop next-x next-fx))))))))))))
;;; H. J. Orchard, "The Laguerre Method for Finding the Zeros of
;;; Polynomials", IEEE Transactions on Circuits and Systems, Vol. 36,
;;; No. 11, November 1989, pp 1377-1381.
;@
(define (laguerre:find-root f df/dz ddf/dz^2 z_0 prec)
(if (and (negative? prec) (integer? prec))
(let loop ((z z_0) (fz (f z_0)) (count prec))
(cond ((zero? count) z)
(else
(let* ((df (df/dz z))
(ddf (ddf/dz^2 z))
(disc (sqrt (- (* df df) (* fz ddf)))))
(if (zero? disc)
#f
(let* ((next-z
(- z (/ fz (if (negative? (+ (* (real-part df)
(real-part disc))
(* (imag-part df)
(imag-part disc))))
(- disc) disc))))
(next-fz (f next-z)))
(cond ((>= (magnitude next-fz) (magnitude fz)) z)
(else (loop next-z next-fz (+ 1 count))))))))))
(let loop ((z z_0) (fz (f z_0)) (delta-z #f))
(cond ((< (magnitude fz) prec) z)
(else
(let* ((df (df/dz z))
(ddf (ddf/dz^2 z))
(disc (sqrt (- (* df df) (* fz ddf)))))
;;(print 'disc disc)
(if (zero? disc)
#f
(let* ((next-z
(- z (/ fz (if (negative? (+ (* (real-part df)
(real-part disc))
(* (imag-part df)
(imag-part disc))))
(- disc) disc))))
(next-delta-z (magnitude (- next-z z))))
;;(print 'next-z next-z )
;;(print '(f next-z) (f next-z))
;;(print 'delta-z delta-z 'next-delta-z next-delta-z)
(cond ((zero? next-delta-z) z)
((and delta-z (>= next-delta-z delta-z)) z)
(else
(loop next-z (f next-z) next-delta-z)))))))))))
;@
(define (laguerre:find-polynomial-root deg f df/dz ddf/dz^2 z_0 prec)
(if (and (negative? prec) (integer? prec))
(let loop ((z z_0) (fz (f z_0)) (count prec))
(cond ((zero? count) z)
(else
(let* ((df (df/dz z))
(ddf (ddf/dz^2 z))
(tmp (* (+ deg -1) df))
(sqrt-H (sqrt (- (* tmp tmp) (* deg (+ deg -1) fz ddf))))
(df+sqrt-H (+ df sqrt-H))
(df-sqrt-H (- df sqrt-H))
(next-z
(- z (/ (* deg fz)
(if (>= (magnitude df+sqrt-H)
(magnitude df-sqrt-H))
df+sqrt-H
df-sqrt-H)))))
(loop next-z (f next-z) (+ 1 count))))))
(let loop ((z z_0) (fz (f z_0)))
(cond ((< (magnitude fz) prec) z)
(else
(let* ((df (df/dz z))
(ddf (ddf/dz^2 z))
(tmp (* (+ deg -1) df))
(sqrt-H (sqrt (- (* tmp tmp) (* deg (+ deg -1) fz ddf))))
(df+sqrt-H (+ df sqrt-H))
(df-sqrt-H (- df sqrt-H))
(next-z
(- z (/ (* deg fz)
(if (>= (magnitude df+sqrt-H)
(magnitude df-sqrt-H))
df+sqrt-H
df-sqrt-H)))))
(loop next-z (f next-z))))))))
(define (secant:find-root-1 f x0 x1 prec must-bracket?)
(letrec ((stop?
(cond ((procedure? prec) prec)
((and (integer? prec) (negative? prec))
(lambda (x0 f0 x1 f1 count)
(>= count (- prec))))
(else
(lambda (x0 f0 x1 f1 count)
(and (< (abs f0) prec)
(< (abs f1) prec))))))
(bracket-iter
(lambda (xlo flo glo xhi fhi ghi count)
(define (step xnew fnew)
(cond ((or (= xnew xlo)
(= xnew xhi))
(let ((xmid (+ xlo (* 1/2 (- xhi xlo)))))
(if (= xnew xmid)
xmid
(step xmid (f xmid)))))
((positive? fnew)
(bracket-iter xlo flo (if glo (* 0.5 glo) 1)
xnew fnew #f
(+ count 1)))
(else
(bracket-iter xnew fnew #f
xhi fhi (if ghi (* 0.5 ghi) 1)
(+ count 1)))))
(if (stop? xlo flo xhi fhi count)
(if (> (abs flo) (abs fhi)) xhi xlo)
(let* ((fflo (if glo (* glo flo) flo))
(ffhi (if ghi (* ghi fhi) fhi))
(del (- (/ fflo (- ffhi fflo))))
(xnew (+ xlo (* del (- xhi xlo))))
(fnew (f xnew)))
(step xnew fnew))))))
(let ((f0 (f x0))
(f1 (f x1)))
(cond ((<= f0 0 f1)
(bracket-iter x0 f0 #f x1 f1 #f 0))
((<= f1 0 f0)
(bracket-iter x1 f1 #f x0 f0 #f 0))
(must-bracket? #f)
(else
(let secant-iter ((x0 x0)
(f0 f0)
(x1 x1)
(f1 f1)
(count 0))
(cond ((stop? x0 f0 x1 f1 count)
(if (> (abs f0) (abs f1)) x1 x0))
((<= f0 0 f1)
(bracket-iter x0 f0 #f x1 f1 #f count))
((>= f0 0 f1)
(bracket-iter x1 f1 #f x0 f0 #f count))
((= f0 f1) #f)
(else
(let* ((xnew (+ x0 (* (- (/ f0 (- f1 f0))) (- x1 x0))))
(fnew (f xnew))
(fmax (max (abs f1) (abs fnew))))
(secant-iter x1 f1 xnew fnew (+ count 1)))))))))))
;@
(define (secant:find-root f x0 x1 prec)
(secant:find-root-1 f x0 x1 prec #f))
(define (secant:find-bracketed-root f x0 x1 prec)
(secant:find-root-1 f x0 x1 prec #t))
|
