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;;;"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))))
;@
(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 x1 fmax 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))
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