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;; complex_dynamics.lisp - functions julia, mandelbrot and rk
;;
;; Copyright (C) 2006-2021 Jaime E. Villate <villate@fe.up.pt>
;;
;; This program is free software; you can redistribute it and/or modify
;; it under the terms of the GNU General Public License as published by
;; the Free Software Foundation; either version 2 of the License, or
;; (at your option) any later version.
;;
;; This program is distributed in the hope that it will be useful,
;; but WITHOUT ANY WARRANTY; without even the implied warranty of
;; MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
;; GNU General Public License for more details.
;;
;; You should have received a copy of the GNU General Public License
;; along with this program; if not, write to the Free Software
;; Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston,
;; MA 02110-1301, USA.
;;
(in-package :maxima)
(macsyma-module dynamics)
;; Function $mandelbrot displays the Mandelbrot set
;; on the region: xmin < x <xmax, ymin < y <ymax.
(defmfun $mandelbrot (&rest extra-options)
(let (plot output-file (options (copy-tree *plot-options*))
a b c d e num dx dy x xmax xmin y ymax ymin m nx ny)
(setf (getf options :type) "plot2d")
(unless (getf options :x) (setf (getf options :x) '(-2 2)))
(unless (getf options :y) (setf (getf options :y) '(-2 2)))
(unless (getf options :iterations) (setf (getf options :iterations) 9))
(unless (getf options :xlabel) (setf (getf options :xlabel) "x"))
(unless (getf options :ylabel) (setf (getf options :ylabel) "y"))
(unless (member :color_bar options) (setf (getf options :color_bar) t))
(setf (getf options :palette)
'(((mlist) $gradient $magenta $violet $blue $cyan $green $yellow
$orange $red $brown $black)))
;; Parses the options given in the command line
(setq options (plot-options-parser extra-options options))
(unless (getf options :yx_ratio) (setf (getf options :same_xy) t))
(setq xmin (car (getf options :x)))
(setq xmax (cadr (getf options :x)))
(setq ymin (car (getf options :y)))
(setq ymax (cadr (getf options :y)))
(setq m (getf options :iterations))
(setq nx (or (car (getf options :grid)) 30))
(setq ny (or (cadr (getf options :grid)) 30))
(setq dx (/ (rationalize (- xmax xmin)) nx)) ; x incr. per pixel
(setq dy (/ (rationalize (- ymax ymin)) ny)) ; y incr. per pixel
;; Creates the object that will be passed to the external graphic program
(setq plot (make-instance 'gnuplot-plot))
(if (eq (getf options :plot_format) '$gnuplot_pipes)
(setf (slot-value plot 'pipe) T)
(setf (getf options :plot_format) '$gnuplot))
(setq output-file (plot-preamble plot options))
(setf
(slot-value plot 'data)
(concatenate
'string
(slot-value plot 'data)
(with-output-to-string (st)
(format st "set palette ~a~%"
(gnuplot-palette (rest (first (getf options :palette)))))
(format st "unset key~%")
(format st "plot '-' with image~%")
;; iterates through all grid points
(dotimes (i ny)
(setq y (+ ymin (/ dy 2) (* i dy)))
(dotimes (j nx)
(setq x (+ xmin (/ dx 2) (* j dx)))
(setq a 0) (setq b 0)
(setq num m)
(dotimes (l m)
(setq c (* a a))
(setq d (* b b))
(setq e (* 2 a b))
(when (> (+ c d) 4) (progn (setq num l) (return)))
(setq a (+ (float x) (- c d)))
(setq b (+ (float y) e)))
(format st "~f ~f ~d~%" x y num)))
(format st "e~%"))))
(plot-shipout plot options output-file)))
;; Function $julia(x,y) displays the Julia set for the
;; point (x,y) of the complex plane, on the region: xmin < x <xmax,
;; ymin < y <ymax.
(defmfun $julia (x y &rest extra-options)
(let (plot output-file (options (copy-tree *plot-options*))
num dx dy xmax xmin ymax ymin a b c d e a0 b0 m nx ny)
(setf (getf options :type) "plot2d")
(unless (getf options :x) (setf (getf options :x) '(-2 2)))
(unless (getf options :y) (setf (getf options :y) '(-2 2)))
(unless (getf options :iterations) (setf (getf options :iterations) 9))
(unless (getf options :xlabel) (setf (getf options :xlabel) "x"))
(unless (getf options :ylabel) (setf (getf options :ylabel) "y"))
(unless (member :color_bar options) (setf (getf options :color_bar) t))
(setf (getf options :palette)
'(((mlist) $gradient $magenta $violet $blue $cyan $green $yellow
$orange $red $brown $black)))
;; Parses the options given in the command line
(setq options (plot-options-parser extra-options options))
(unless (getf options :yx_ratio) (setf (getf options :same_xy) t))
(setq xmin (car (getf options :x)))
(setq xmax (cadr (getf options :x)))
(setq ymin (car (getf options :y)))
(setq ymax (cadr (getf options :y)))
(setq m (getf options :iterations))
(setq nx (or (car (getf options :grid)) 30))
(setq ny (or (cadr (getf options :grid)) 30))
(setq dx (/ (rationalize (- xmax xmin)) nx)) ; x incr. per pixel
(setq dy (/ (rationalize (- ymax ymin)) ny)) ; y incr. per pixel
;; Creates the object that will be passed to the external graphic program
(setq plot (make-instance 'gnuplot-plot))
(if (eq (getf options :plot_format) '$gnuplot_pipes)
(setf (slot-value plot 'pipe) T)
(setf (getf options :plot_format) '$gnuplot))
(setq output-file (plot-preamble plot options))
(setf
(slot-value plot 'data)
(concatenate
'string
(slot-value plot 'data)
(with-output-to-string (st)
(format st "set palette ~a~%"
(gnuplot-palette (rest (first (getf options :palette)))))
(format st "unset key~%")
(format st "plot '-' with image~%")
;; iterates through all grid points
(dotimes (i ny)
(setq b0 (+ ymin (/ dy 2) (* i dy)))
(dotimes (j nx)
(setq a0 (+ xmin (/ dx 2) (* j dx)))
(setq a (float a0))
(setq b (float b0))
(setq num m)
(dotimes (l m)
(setq c (* a a))
(setq d (* b b))
(setq e (* 2 a b))
(when (> (+ c d) 4) (progn (setq num l) (return)))
(setq a (+ x (- c d)))
(setq b (+ y e)))
(format st "~f ~f ~d~%" a0 b0 num)))
(format st "e~%"))))
(plot-shipout plot options output-file)))
;; Function $rk implements the 4th order Runge-Kutta numerical method.
;; exprs: an expression or maxima list with n expressions
;; vars: name of (or list of names of) the independent variable(s)
;; initial: initial value for the variable (or list of initial values)
;; domain: maxima list with four elements (name of the independent
;; variable, its initial and final values and its increment)
(defmfun $rk (exprs vars initial domain
&aux d u fun k1 k2 k3 k4 r1 r2 r3 traj r
(it (mapcar #'coerce-float (cddr domain))))
(unless ($listp exprs) (setq exprs `((mlist) ,exprs)))
(unless ($listp initial) (setq initial `((mlist) ,initial)))
(unless ($listp vars) (setq vars `((mlist) ,vars)))
(dolist (var (cdr vars))
(unless (symbolp var)
(merror (intl:gettext "rk: variable name expected; found: ~M") var)))
(unless (symbolp (cadr domain))
(merror (intl:gettext "rk: variable name expected; found: ~M")
(cadr domain)))
(setq vars (concatenate 'list '((mlist)) (list (cadr domain)) (cdr vars)))
(setq r (concatenate 'list `(,(car it)) (mapcar #'coerce-float (cdr initial))))
(setq fun (mapcar #'(lambda (x) (coerce-float-fun x vars)) (cdr exprs)))
(setq d (/ (- (cadr it) (car it)) (caddr it)))
(setq traj (list (cons '(mlist) r)))
(do ((m 1 (1+ m))) ((> m d))
(ignore-errors
(setq k1 (mapcar #'(lambda (x) (apply x r)) fun))
(setq r1 (map 'list #'+ (cdr r) (mapcar #'(lambda (x) (* (/ (caddr it) 2) x)) k1)))
(push (+ (car r) (/ (caddr it) 2)) r1)
(setq k2 (mapcar #'(lambda (x) (apply x r1)) fun))
(setq r2 (map 'list #'+ (cdr r) (mapcar #'(lambda (x) (* (/ (caddr it) 2) x)) k2)))
(push (+ (car r) (/ (caddr it) 2)) r2)
(setq k3 (mapcar #'(lambda (x) (apply x r2)) fun))
(setq r3 (map 'list #'+ (cdr r) (mapcar #'(lambda (x) (* (caddr it) x)) k3)))
(push (+ (car r) (caddr it)) r3)
(setq k4 (mapcar #'(lambda (x) (apply x r3)) fun))
(setq u (map 'list #'+
(mapcar #'(lambda (x) (* 1/6 x)) k1)
(mapcar #'(lambda (x) (* 1/3 x)) k2)
(mapcar #'(lambda (x) (* 1/3 x)) k3)
(mapcar #'(lambda (x) (* 1/6 x)) k4)))
(setq r
(concatenate 'list
`(,(+ (car it) (* m (caddr it))))
(map 'list #'+ (cdr r) (mapcar #'(lambda (x) (* (caddr it) x)) u))))
(push (cons '(mlist) r) traj)))
(when (< (car r) (cadr it))
(let ((s (- (cadr it) (car r))))
(ignore-errors
(setq k1 (mapcar #'(lambda (x) (apply x r)) fun))
(setq r1 (map 'list #'+ (cdr r) (mapcar #'(lambda (x) (* (/ s 2) x)) k1)))
(push (+ (car r) (/ s 2)) r1)
(setq k2 (mapcar #'(lambda (x) (apply x r1)) fun))
(setq r2 (map 'list #'+ (cdr r) (mapcar #'(lambda (x) (* (/ s 2) x)) k2)))
(push (+ (car r) (/ s 2)) r2)
(setq k3 (mapcar #'(lambda (x) (apply x r2)) fun))
(setq r3 (map 'list #'+ (cdr r) (mapcar #'(lambda (x) (* s x)) k3)))
(push (+ (car r) s) r3)
(setq k4 (mapcar #'(lambda (x) (apply x r3)) fun))
(setq u (map 'list #'+
(mapcar #'(lambda (x) (* 1/6 x)) k1)
(mapcar #'(lambda (x) (* 1/3 x)) k2)
(mapcar #'(lambda (x) (* 1/3 x)) k3)
(mapcar #'(lambda (x) (* 1/6 x)) k4)))
(setq r
(concatenate 'list
`(,(cadr it))
(map 'list #'+ (cdr r) (mapcar #'(lambda (x) (* s x)) u))))
(push (cons '(mlist) r) traj))))
(cons '(mlist) (nreverse traj)))
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