HCoop / bpt / emacs.git / blame
| Commit | Line | Data |
|---|---|---|
| 3132f345 CW |
1 | ;;; calcalg3.el --- more algebraic functions for Calc |
| 2 | ||
| bf77c646 | 3 | ;; Copyright (C) 1990, 1991, 1992, 1993, 2001 Free Software Foundation, Inc. |
| 3132f345 CW |
4 | |
| 5 | ;; Author: David Gillespie <daveg@synaptics.com> | |
| a1506d29 | 6 | ;; Maintainers: D. Goel <deego@gnufans.org> |
| 6e1c888a | 7 | ;; Colin Walters <walters@debian.org> |
| 136211a9 EZ |
8 | |
| 9 | ;; This file is part of GNU Emacs. | |
| 10 | ||
| 11 | ;; GNU Emacs is distributed in the hope that it will be useful, | |
| 12 | ;; but WITHOUT ANY WARRANTY. No author or distributor | |
| 13 | ;; accepts responsibility to anyone for the consequences of using it | |
| 14 | ;; or for whether it serves any particular purpose or works at all, | |
| 15 | ;; unless he says so in writing. Refer to the GNU Emacs General Public | |
| 16 | ;; License for full details. | |
| 17 | ||
| 18 | ;; Everyone is granted permission to copy, modify and redistribute | |
| 19 | ;; GNU Emacs, but only under the conditions described in the | |
| 20 | ;; GNU Emacs General Public License. A copy of this license is | |
| 21 | ;; supposed to have been given to you along with GNU Emacs so you | |
| 22 | ;; can know your rights and responsibilities. It should be in a | |
| 23 | ;; file named COPYING. Among other things, the copyright notice | |
| 24 | ;; and this notice must be preserved on all copies. | |
| 25 | ||
| 3132f345 | 26 | ;;; Commentary: |
| 136211a9 | 27 | |
| 3132f345 | 28 | ;;; Code: |
| 136211a9 EZ |
29 | |
| 30 | ;; This file is autoloaded from calc-ext.el. | |
| 31 | (require 'calc-ext) | |
| 32 | ||
| 33 | (require 'calc-macs) | |
| 34 | ||
| 35 | (defun calc-Need-calc-alg-3 () nil) | |
| 36 | ||
| 37 | ||
| 38 | (defun calc-find-root (var) | |
| 39 | (interactive "sVariable(s) to solve for: ") | |
| 40 | (calc-slow-wrapper | |
| 41 | (let ((func (if (calc-is-hyperbolic) 'calcFunc-wroot 'calcFunc-root))) | |
| 42 | (if (or (equal var "") (equal var "$")) | |
| 43 | (calc-enter-result 2 "root" (list func | |
| 44 | (calc-top-n 3) | |
| 45 | (calc-top-n 1) | |
| 46 | (calc-top-n 2))) | |
| 47 | (let ((var (if (and (string-match ",\\|[^ ] +[^ ]" var) | |
| 48 | (not (string-match "\\[" var))) | |
| 49 | (math-read-expr (concat "[" var "]")) | |
| 50 | (math-read-expr var)))) | |
| 51 | (if (eq (car-safe var) 'error) | |
| 52 | (error "Bad format in expression: %s" (nth 1 var))) | |
| 53 | (calc-enter-result 1 "root" (list func | |
| 54 | (calc-top-n 2) | |
| 55 | var | |
| bf77c646 | 56 | (calc-top-n 1)))))))) |
| 136211a9 EZ |
57 | |
| 58 | (defun calc-find-minimum (var) | |
| 59 | (interactive "sVariable(s) to minimize over: ") | |
| 60 | (calc-slow-wrapper | |
| 61 | (let ((func (if (calc-is-inverse) | |
| 62 | (if (calc-is-hyperbolic) | |
| 63 | 'calcFunc-wmaximize 'calcFunc-maximize) | |
| 64 | (if (calc-is-hyperbolic) | |
| 65 | 'calcFunc-wminimize 'calcFunc-minimize))) | |
| 66 | (tag (if (calc-is-inverse) "max" "min"))) | |
| 67 | (if (or (equal var "") (equal var "$")) | |
| 68 | (calc-enter-result 2 tag (list func | |
| 69 | (calc-top-n 3) | |
| 70 | (calc-top-n 1) | |
| 71 | (calc-top-n 2))) | |
| 72 | (let ((var (if (and (string-match ",\\|[^ ] +[^ ]" var) | |
| 73 | (not (string-match "\\[" var))) | |
| 74 | (math-read-expr (concat "[" var "]")) | |
| 75 | (math-read-expr var)))) | |
| 76 | (if (eq (car-safe var) 'error) | |
| 77 | (error "Bad format in expression: %s" (nth 1 var))) | |
| 78 | (calc-enter-result 1 tag (list func | |
| 79 | (calc-top-n 2) | |
| 80 | var | |
| bf77c646 | 81 | (calc-top-n 1)))))))) |
| 136211a9 EZ |
82 | |
| 83 | (defun calc-find-maximum (var) | |
| 84 | (interactive "sVariable to maximize over: ") | |
| 85 | (calc-invert-func) | |
| bf77c646 | 86 | (calc-find-minimum var)) |
| 136211a9 EZ |
87 | |
| 88 | ||
| 89 | (defun calc-poly-interp (arg) | |
| 90 | (interactive "P") | |
| 91 | (calc-slow-wrapper | |
| 92 | (let ((data (calc-top 2))) | |
| 93 | (if (or (consp arg) (eq arg 0) (eq arg 2)) | |
| 94 | (setq data (cons 'vec (calc-top-list 2 2))) | |
| 95 | (or (null arg) | |
| 96 | (error "Bad prefix argument"))) | |
| 97 | (if (calc-is-hyperbolic) | |
| 98 | (calc-enter-result 1 "rati" (list 'calcFunc-ratint data (calc-top 1))) | |
| 99 | (calc-enter-result 1 "poli" (list 'calcFunc-polint data | |
| bf77c646 | 100 | (calc-top 1))))))) |
| 136211a9 EZ |
101 | |
| 102 | ||
| 103 | (defun calc-curve-fit (arg &optional model coefnames varnames) | |
| 104 | (interactive "P") | |
| 105 | (calc-slow-wrapper | |
| 106 | (setq calc-aborted-prefix nil) | |
| 107 | (let ((func (if (calc-is-inverse) 'calcFunc-xfit | |
| 108 | (if (calc-is-hyperbolic) 'calcFunc-efit | |
| 109 | 'calcFunc-fit))) | |
| 110 | key (which 0) | |
| 111 | n nvars temp data | |
| 112 | (homog nil) | |
| 113 | (msgs '( "(Press ? for help)" | |
| 114 | "1 = linear or multilinear" | |
| 115 | "2-9 = polynomial fits; i = interpolating polynomial" | |
| 116 | "p = a x^b, ^ = a b^x" | |
| 117 | "e = a exp(b x), x = exp(a + b x), l = a + b ln(x)" | |
| 118 | "E = a 10^(b x), X = 10^(a + b x), L = a + b log10(x)" | |
| 119 | "q = a + b (x-c)^2" | |
| 120 | "g = (a/b sqrt(2 pi)) exp(-0.5*((x-c)/b)^2)" | |
| 121 | "h prefix = homogeneous model (no constant term)" | |
| 122 | "' = alg entry, $ = stack, u = Model1, U = Model2"))) | |
| 123 | (while (not model) | |
| 124 | (message "Fit to model: %s:%s" | |
| 125 | (nth which msgs) | |
| 126 | (if homog " h" "")) | |
| 127 | (setq key (read-char)) | |
| 128 | (cond ((= key ?\C-g) | |
| 129 | (keyboard-quit)) | |
| 130 | ((= key ??) | |
| 131 | (setq which (% (1+ which) (length msgs)))) | |
| 132 | ((memq key '(?h ?H)) | |
| 133 | (setq homog (not homog))) | |
| 134 | ((progn | |
| 135 | (if (eq key ?\$) | |
| 136 | (setq n 1) | |
| 137 | (setq n 0)) | |
| 138 | (cond ((null arg) | |
| 139 | (setq n (1+ n) | |
| 140 | data (calc-top n))) | |
| 141 | ((or (consp arg) (eq arg 0)) | |
| 142 | (setq n (+ n 2) | |
| 143 | data (calc-top n) | |
| 144 | data (if (math-matrixp data) | |
| 145 | (append data (list (calc-top (1- n)))) | |
| 146 | (list 'vec data (calc-top (1- n)))))) | |
| 147 | ((> (setq arg (prefix-numeric-value arg)) 0) | |
| 148 | (setq data (cons 'vec (calc-top-list arg (1+ n))) | |
| 149 | n (+ n arg))) | |
| 150 | (t (error "Bad prefix argument"))) | |
| 151 | (or (math-matrixp data) (not (cdr (cdr data))) | |
| 152 | (error "Data matrix is not a matrix!")) | |
| 153 | (setq nvars (- (length data) 2) | |
| 154 | coefnames nil | |
| 155 | varnames nil) | |
| 156 | nil)) | |
| 157 | ((= key ?1) ; linear or multilinear | |
| 158 | (calc-get-fit-variables nvars (1+ nvars) (and homog 0)) | |
| 159 | (setq model (math-mul coefnames | |
| 160 | (cons 'vec (cons 1 (cdr varnames)))))) | |
| 161 | ((and (>= key ?2) (<= key ?9)) ; polynomial | |
| 162 | (calc-get-fit-variables 1 (- key ?0 -1) (and homog 0)) | |
| 163 | (setq model (math-build-polynomial-expr (cdr coefnames) | |
| 164 | (nth 1 varnames)))) | |
| 165 | ((= key ?i) ; exact polynomial | |
| 166 | (calc-get-fit-variables 1 (1- (length (nth 1 data))) | |
| 167 | (and homog 0)) | |
| 168 | (setq model (math-build-polynomial-expr (cdr coefnames) | |
| 169 | (nth 1 varnames)))) | |
| 170 | ((= key ?p) ; power law | |
| 171 | (calc-get-fit-variables nvars (1+ nvars) (and homog 1)) | |
| 172 | (setq model (math-mul (nth 1 coefnames) | |
| 173 | (calcFunc-reduce | |
| 174 | '(var mul var-mul) | |
| 175 | (calcFunc-map | |
| 176 | '(var pow var-pow) | |
| 177 | varnames | |
| 178 | (cons 'vec (cdr (cdr coefnames)))))))) | |
| 179 | ((= key ?^) ; exponential law | |
| 180 | (calc-get-fit-variables nvars (1+ nvars) (and homog 1)) | |
| 181 | (setq model (math-mul (nth 1 coefnames) | |
| 182 | (calcFunc-reduce | |
| 183 | '(var mul var-mul) | |
| 184 | (calcFunc-map | |
| 185 | '(var pow var-pow) | |
| 186 | (cons 'vec (cdr (cdr coefnames))) | |
| 187 | varnames))))) | |
| 188 | ((memq key '(?e ?E)) | |
| 189 | (calc-get-fit-variables nvars (1+ nvars) (and homog 1)) | |
| 190 | (setq model (math-mul (nth 1 coefnames) | |
| 191 | (calcFunc-reduce | |
| 192 | '(var mul var-mul) | |
| 193 | (calcFunc-map | |
| 194 | (if (eq key ?e) | |
| 195 | '(var exp var-exp) | |
| 196 | '(calcFunc-lambda | |
| 197 | (var a var-a) | |
| 198 | (^ 10 (var a var-a)))) | |
| 199 | (calcFunc-map | |
| 200 | '(var mul var-mul) | |
| 201 | (cons 'vec (cdr (cdr coefnames))) | |
| 202 | varnames)))))) | |
| 203 | ((memq key '(?x ?X)) | |
| 204 | (calc-get-fit-variables nvars (1+ nvars) (and homog 0)) | |
| 205 | (setq model (math-mul coefnames | |
| 206 | (cons 'vec (cons 1 (cdr varnames))))) | |
| 207 | (setq model (if (eq key ?x) | |
| 208 | (list 'calcFunc-exp model) | |
| 209 | (list '^ 10 model)))) | |
| 210 | ((memq key '(?l ?L)) | |
| 211 | (calc-get-fit-variables nvars (1+ nvars) (and homog 0)) | |
| 212 | (setq model (math-mul coefnames | |
| 213 | (cons 'vec | |
| 214 | (cons 1 (cdr (calcFunc-map | |
| 215 | (if (eq key ?l) | |
| 216 | '(var ln var-ln) | |
| 217 | '(var log10 | |
| 218 | var-log10)) | |
| 219 | varnames))))))) | |
| 220 | ((= key ?q) | |
| 221 | (calc-get-fit-variables nvars (1+ (* 2 nvars)) (and homog 0)) | |
| 222 | (let ((c coefnames) | |
| 223 | (v varnames)) | |
| 224 | (setq model (nth 1 c)) | |
| 225 | (while (setq v (cdr v) c (cdr (cdr c))) | |
| 226 | (setq model (math-add | |
| 227 | model | |
| 228 | (list '* | |
| 229 | (car c) | |
| 230 | (list '^ | |
| 231 | (list '- (car v) (nth 1 c)) | |
| 232 | 2))))))) | |
| 233 | ((= key ?g) | |
| 234 | (setq model (math-read-expr "(AFit / BFit sqrt(2 pi)) exp(-0.5 * ((XFit - CFit) / BFit)^2)") | |
| 235 | varnames '(vec (var XFit var-XFit)) | |
| 236 | coefnames '(vec (var AFit var-AFit) | |
| 237 | (var BFit var-BFit) | |
| 238 | (var CFit var-CFit))) | |
| 239 | (calc-get-fit-variables 1 (1- (length coefnames)) (and homog 1))) | |
| 240 | ((memq key '(?\$ ?\' ?u ?U)) | |
| 241 | (let* ((defvars nil) | |
| 242 | (record-entry nil)) | |
| 243 | (if (eq key ?\') | |
| 244 | (let* ((calc-dollar-values calc-arg-values) | |
| 245 | (calc-dollar-used 0) | |
| 246 | (calc-hashes-used 0)) | |
| 247 | (setq model (calc-do-alg-entry "" "Model formula: ")) | |
| 248 | (if (/= (length model) 1) | |
| 249 | (error "Bad format")) | |
| 250 | (setq model (car model) | |
| 251 | record-entry t) | |
| 252 | (if (> calc-dollar-used 0) | |
| 253 | (setq coefnames | |
| 254 | (cons 'vec | |
| 255 | (nthcdr (- (length calc-arg-values) | |
| 256 | calc-dollar-used) | |
| 257 | (reverse calc-arg-values)))) | |
| 258 | (if (> calc-hashes-used 0) | |
| 259 | (setq coefnames | |
| 260 | (cons 'vec (calc-invent-args | |
| 261 | calc-hashes-used)))))) | |
| 262 | (progn | |
| 263 | (setq model (cond ((eq key ?u) | |
| 264 | (calc-var-value 'var-Model1)) | |
| 265 | ((eq key ?U) | |
| 266 | (calc-var-value 'var-Model2)) | |
| 267 | (t (calc-top 1)))) | |
| 268 | (or model (error "User model not yet defined")) | |
| 269 | (if (math-vectorp model) | |
| 270 | (if (and (memq (length model) '(3 4)) | |
| 271 | (not (math-objvecp (nth 1 model))) | |
| 272 | (math-vectorp (nth 2 model)) | |
| 273 | (or (null (nth 3 model)) | |
| 274 | (math-vectorp (nth 3 model)))) | |
| 275 | (setq varnames (nth 2 model) | |
| 276 | coefnames (or (nth 3 model) | |
| 277 | (cons 'vec | |
| 278 | (math-all-vars-but | |
| 279 | model varnames))) | |
| 280 | model (nth 1 model)) | |
| 281 | (error "Incorrect model specifier"))))) | |
| 282 | (or varnames | |
| 283 | (let ((with-y (eq (car-safe model) 'calcFunc-eq))) | |
| 284 | (if coefnames | |
| 285 | (calc-get-fit-variables (if with-y (1+ nvars) nvars) | |
| 286 | (1- (length coefnames)) | |
| 287 | (math-all-vars-but | |
| 288 | model coefnames) | |
| 289 | nil with-y) | |
| 290 | (let* ((coefs (math-all-vars-but model nil)) | |
| 291 | (vars nil) | |
| 292 | (n (- (length coefs) nvars (if with-y 2 1))) | |
| 293 | p) | |
| 294 | (if (< n 0) | |
| 295 | (error "Not enough variables in model")) | |
| 296 | (setq p (nthcdr n coefs)) | |
| 297 | (setq vars (cdr p)) | |
| 298 | (setcdr p nil) | |
| 299 | (calc-get-fit-variables (if with-y (1+ nvars) nvars) | |
| 300 | (length coefs) | |
| 301 | vars coefs with-y))))) | |
| 302 | (if record-entry | |
| 303 | (calc-record (list 'vec model varnames coefnames) | |
| 304 | "modl")))) | |
| 305 | (t (beep)))) | |
| 306 | (let ((calc-fit-to-trail t)) | |
| 307 | (calc-enter-result n (substring (symbol-name func) 9) | |
| 308 | (list func model | |
| 309 | (if (= (length varnames) 2) | |
| 310 | (nth 1 varnames) | |
| 311 | varnames) | |
| 312 | (if (= (length coefnames) 2) | |
| 313 | (nth 1 coefnames) | |
| 314 | coefnames) | |
| 315 | data)) | |
| 316 | (if (consp calc-fit-to-trail) | |
| bf77c646 | 317 | (calc-record (calc-normalize calc-fit-to-trail) "parm")))))) |
| 136211a9 EZ |
318 | |
| 319 | (defun calc-invent-independent-variables (n &optional but) | |
| bf77c646 | 320 | (calc-invent-variables n but '(x y z t) "x")) |
| 136211a9 EZ |
321 | |
| 322 | (defun calc-invent-parameter-variables (n &optional but) | |
| bf77c646 | 323 | (calc-invent-variables n but '(a b c d) "a")) |
| 136211a9 EZ |
324 | |
| 325 | (defun calc-invent-variables (num but names base) | |
| 326 | (let ((vars nil) | |
| 327 | (n num) (nn 0) | |
| 328 | var) | |
| 329 | (while (and (> n 0) names) | |
| 330 | (setq var (math-build-var-name (if (consp names) | |
| 331 | (car names) | |
| 67dbc846 CW |
332 | (concat base (int-to-string |
| 333 | (setq nn (1+ nn))))))) | |
| 136211a9 EZ |
334 | (or (math-expr-contains (cons 'vec but) var) |
| 335 | (setq vars (cons var vars) | |
| 336 | n (1- n))) | |
| 337 | (or (symbolp names) (setq names (cdr names)))) | |
| 338 | (if (= n 0) | |
| 339 | (nreverse vars) | |
| bf77c646 | 340 | (calc-invent-variables num but t base)))) |
| 136211a9 EZ |
341 | |
| 342 | (defun calc-get-fit-variables (nv nc &optional defv defc with-y homog) | |
| 343 | (or (= nv (if with-y (1+ nvars) nvars)) | |
| 344 | (error "Wrong number of data vectors for this type of model")) | |
| 345 | (if (integerp defv) | |
| 346 | (setq homog defv | |
| 347 | defv nil)) | |
| 348 | (if homog | |
| 349 | (setq nc (1- nc))) | |
| 350 | (or defv | |
| 351 | (setq defv (calc-invent-independent-variables nv))) | |
| 352 | (or defc | |
| 353 | (setq defc (calc-invent-parameter-variables nc defv))) | |
| 354 | (let ((vars (read-string (format "Fitting variables: (default %s; %s) " | |
| 355 | (mapconcat 'symbol-name | |
| 356 | (mapcar (function (lambda (v) | |
| 357 | (nth 1 v))) | |
| 358 | defv) | |
| 359 | ",") | |
| 360 | (mapconcat 'symbol-name | |
| 361 | (mapcar (function (lambda (v) | |
| 362 | (nth 1 v))) | |
| 363 | defc) | |
| 364 | ",")))) | |
| 365 | (coefs nil)) | |
| 366 | (setq vars (if (string-match "\\[" vars) | |
| 367 | (math-read-expr vars) | |
| 368 | (math-read-expr (concat "[" vars "]")))) | |
| 369 | (if (eq (car-safe vars) 'error) | |
| 370 | (error "Bad format in expression: %s" (nth 2 vars))) | |
| 371 | (or (math-vectorp vars) | |
| 372 | (error "Expected a variable or vector of variables")) | |
| 373 | (if (equal vars '(vec)) | |
| 374 | (setq vars (cons 'vec defv) | |
| 375 | coefs (cons 'vec defc)) | |
| 376 | (if (math-vectorp (nth 1 vars)) | |
| 377 | (if (and (= (length vars) 3) | |
| 378 | (math-vectorp (nth 2 vars))) | |
| 379 | (setq coefs (nth 2 vars) | |
| 380 | vars (nth 1 vars)) | |
| 381 | (error | |
| 382 | "Expected independent variables vector, then parameters vector")) | |
| 383 | (setq coefs (cons 'vec defc)))) | |
| 384 | (or (= nv (1- (length vars))) | |
| 385 | (and (not with-y) (= (1+ nv) (1- (length vars)))) | |
| 386 | (error "Expected %d independent variable%s" nv (if (= nv 1) "" "s"))) | |
| 387 | (or (= nc (1- (length coefs))) | |
| 388 | (error "Expected %d parameter variable%s" nc (if (= nc 1) "" "s"))) | |
| 389 | (if homog | |
| 390 | (setq coefs (cons 'vec (cons homog (cdr coefs))))) | |
| 391 | (if varnames | |
| 392 | (setq model (math-multi-subst model (cdr varnames) (cdr vars)))) | |
| 393 | (if coefnames | |
| 394 | (setq model (math-multi-subst model (cdr coefnames) (cdr coefs)))) | |
| 395 | (setq varnames vars | |
| bf77c646 | 396 | coefnames coefs))) |
| 136211a9 EZ |
397 | |
| 398 | ||
| 399 | ||
| 400 | ||
| 401 | ;;; The following algorithms are from Numerical Recipes chapter 9. | |
| 402 | ||
| 403 | ;;; "rtnewt" with safety kludges | |
| 404 | (defun math-newton-root (expr deriv guess orig-guess limit) | |
| 405 | (math-working "newton" guess) | |
| 406 | (let* ((var-DUMMY guess) | |
| 407 | next dval) | |
| 408 | (setq next (math-evaluate-expr expr) | |
| 409 | dval (math-evaluate-expr deriv)) | |
| 410 | (if (and (Math-numberp next) | |
| 411 | (Math-numberp dval) | |
| 412 | (not (Math-zerop dval))) | |
| 413 | (progn | |
| 414 | (setq next (math-sub guess (math-div next dval))) | |
| 415 | (if (math-nearly-equal guess (setq next (math-float next))) | |
| 416 | (progn | |
| 417 | (setq var-DUMMY next) | |
| 418 | (list 'vec next (math-evaluate-expr expr))) | |
| 419 | (if (Math-lessp (math-abs-approx (math-sub next orig-guess)) | |
| 420 | limit) | |
| 421 | (math-newton-root expr deriv next orig-guess limit) | |
| 422 | (math-reject-arg next "*Newton's method failed to converge")))) | |
| bf77c646 | 423 | (math-reject-arg next "*Newton's method encountered a singularity")))) |
| 136211a9 EZ |
424 | |
| 425 | ;;; Inspired by "rtsafe" | |
| 426 | (defun math-newton-search-root (expr deriv guess vguess ostep oostep | |
| 427 | low vlow high vhigh) | |
| 428 | (let ((var-DUMMY guess) | |
| 429 | (better t) | |
| 430 | pos step next vnext) | |
| 431 | (if guess | |
| 432 | (math-working "newton" (list 'intv 0 low high)) | |
| 433 | (math-working "bisect" (list 'intv 0 low high)) | |
| 434 | (setq ostep (math-mul-float (math-sub-float high low) | |
| 435 | '(float 5 -1)) | |
| 436 | guess (math-add-float low ostep) | |
| 437 | var-DUMMY guess | |
| 438 | vguess (math-evaluate-expr expr)) | |
| 439 | (or (Math-realp vguess) | |
| 440 | (progn | |
| 441 | (setq ostep (math-mul-float ostep '(float 6 -1)) | |
| 442 | guess (math-add-float low ostep) | |
| 443 | var-DUMMY guess | |
| 444 | vguess (math-evaluate-expr expr)) | |
| 445 | (or (math-realp vguess) | |
| 446 | (progn | |
| 447 | (setq ostep (math-mul-float ostep '(float 123456 -5)) | |
| 448 | guess (math-add-float low ostep) | |
| 449 | var-DUMMY guess | |
| 450 | vguess nil)))))) | |
| 451 | (or vguess | |
| 452 | (setq vguess (math-evaluate-expr expr))) | |
| 453 | (or (Math-realp vguess) | |
| 454 | (math-reject-arg guess "*Newton's method encountered a singularity")) | |
| 455 | (setq vguess (math-float vguess)) | |
| 456 | (if (eq (Math-negp vlow) (setq pos (Math-posp vguess))) | |
| 457 | (setq high guess | |
| 458 | vhigh vguess) | |
| 459 | (if (eq (Math-negp vhigh) pos) | |
| 460 | (setq low guess | |
| 461 | vlow vguess) | |
| 462 | (setq better nil))) | |
| 463 | (if (or (Math-zerop vguess) | |
| 464 | (math-nearly-equal low high)) | |
| 465 | (list 'vec guess vguess) | |
| 466 | (setq step (math-evaluate-expr deriv)) | |
| 467 | (if (and (Math-realp step) | |
| 468 | (not (Math-zerop step)) | |
| 469 | (setq step (math-div-float vguess (math-float step)) | |
| 470 | next (math-sub-float guess step)) | |
| 471 | (not (math-lessp-float high next)) | |
| 472 | (not (math-lessp-float next low))) | |
| 473 | (progn | |
| 474 | (setq var-DUMMY next | |
| 475 | vnext (math-evaluate-expr expr)) | |
| 476 | (if (or (Math-zerop vnext) | |
| 477 | (math-nearly-equal next guess)) | |
| 478 | (list 'vec next vnext) | |
| 479 | (if (and better | |
| 480 | (math-lessp-float (math-abs (or oostep | |
| 481 | (math-sub-float | |
| 482 | high low))) | |
| 483 | (math-abs | |
| 484 | (math-mul-float '(float 2 0) | |
| 485 | step)))) | |
| 486 | (math-newton-search-root expr deriv nil nil nil ostep | |
| 487 | low vlow high vhigh) | |
| 488 | (math-newton-search-root expr deriv next vnext step ostep | |
| 489 | low vlow high vhigh)))) | |
| 490 | (if (or (and (Math-posp vlow) (Math-posp vhigh)) | |
| 491 | (and (Math-negp vlow) (Math-negp vhigh))) | |
| 492 | (math-search-root expr deriv low vlow high vhigh) | |
| 493 | (math-newton-search-root expr deriv nil nil nil ostep | |
| bf77c646 | 494 | low vlow high vhigh)))))) |
| 136211a9 EZ |
495 | |
| 496 | ;;; Search for a root in an interval with no overt zero crossing. | |
| 497 | (defun math-search-root (expr deriv low vlow high vhigh) | |
| 498 | (let (found) | |
| 499 | (if root-widen | |
| 500 | (let ((iters 0) | |
| 501 | (iterlim (if (eq root-widen 'point) | |
| 502 | (+ calc-internal-prec 10) | |
| 503 | 20)) | |
| 504 | (factor (if (eq root-widen 'point) | |
| 505 | '(float 9 0) | |
| 506 | '(float 16 -1))) | |
| 507 | (prev nil) vprev waslow | |
| 508 | diff) | |
| 509 | (while (or (and (math-posp vlow) (math-posp vhigh)) | |
| 510 | (and (math-negp vlow) (math-negp vhigh))) | |
| 511 | (math-working "widen" (list 'intv 0 low high)) | |
| 512 | (if (> (setq iters (1+ iters)) iterlim) | |
| 513 | (math-reject-arg (list 'intv 0 low high) | |
| 514 | "*Unable to bracket root")) | |
| 515 | (if (= iters calc-internal-prec) | |
| 516 | (setq factor '(float 16 -1))) | |
| 517 | (setq diff (math-mul-float (math-sub-float high low) factor)) | |
| 518 | (if (Math-zerop diff) | |
| 519 | (setq high (calcFunc-incr high 10)) | |
| 520 | (if (math-lessp-float (math-abs vlow) (math-abs vhigh)) | |
| 521 | (setq waslow t | |
| 522 | prev low | |
| 523 | low (math-sub low diff) | |
| 524 | var-DUMMY low | |
| 525 | vprev vlow | |
| 526 | vlow (math-evaluate-expr expr)) | |
| 527 | (setq waslow nil | |
| 528 | prev high | |
| 529 | high (math-add high diff) | |
| 530 | var-DUMMY high | |
| 531 | vprev vhigh | |
| 532 | vhigh (math-evaluate-expr expr))))) | |
| 533 | (if prev | |
| 534 | (if waslow | |
| 535 | (setq high prev vhigh vprev) | |
| 536 | (setq low prev vlow vprev))) | |
| 537 | (setq found t)) | |
| 538 | (or (Math-realp vlow) | |
| 539 | (math-reject-arg vlow 'realp)) | |
| 540 | (or (Math-realp vhigh) | |
| 541 | (math-reject-arg vhigh 'realp)) | |
| 542 | (let ((xvals (list low high)) | |
| 543 | (yvals (list vlow vhigh)) | |
| 544 | (pos (Math-posp vlow)) | |
| 545 | (levels 0) | |
| 546 | (step (math-sub-float high low)) | |
| 547 | xp yp var-DUMMY) | |
| 548 | (while (and (<= (setq levels (1+ levels)) 5) | |
| 549 | (not found)) | |
| 550 | (setq xp xvals | |
| 551 | yp yvals | |
| 552 | step (math-mul-float step '(float 497 -3))) | |
| 553 | (while (and (cdr xp) (not found)) | |
| 554 | (if (Math-realp (car yp)) | |
| 555 | (setq low (car xp) | |
| 556 | vlow (car yp))) | |
| 557 | (setq high (math-add-float (car xp) step) | |
| 558 | var-DUMMY high | |
| 559 | vhigh (math-evaluate-expr expr)) | |
| 560 | (math-working "search" high) | |
| 561 | (if (and (Math-realp vhigh) | |
| 562 | (eq (math-negp vhigh) pos)) | |
| 563 | (setq found t) | |
| 564 | (setcdr xp (cons high (cdr xp))) | |
| 565 | (setcdr yp (cons vhigh (cdr yp))) | |
| 566 | (setq xp (cdr (cdr xp)) | |
| 567 | yp (cdr (cdr yp)))))))) | |
| 568 | (if found | |
| 569 | (if (Math-zerop vhigh) | |
| 570 | (list 'vec high vhigh) | |
| 571 | (if (Math-zerop vlow) | |
| 572 | (list 'vec low vlow) | |
| 573 | (if deriv | |
| 574 | (math-newton-search-root expr deriv nil nil nil nil | |
| 575 | low vlow high vhigh) | |
| 576 | (math-bisect-root expr low vlow high vhigh)))) | |
| 577 | (math-reject-arg (list 'intv 3 low high) | |
| bf77c646 | 578 | "*Unable to find a sign change in this interval")))) |
| 136211a9 EZ |
579 | |
| 580 | ;;; "rtbis" (but we should be using Brent's method) | |
| 581 | (defun math-bisect-root (expr low vlow high vhigh) | |
| 582 | (let ((step (math-sub-float high low)) | |
| 583 | (pos (Math-posp vhigh)) | |
| 584 | var-DUMMY | |
| 585 | mid vmid) | |
| 586 | (while (not (or (math-nearly-equal low | |
| 587 | (setq step (math-mul-float | |
| 588 | step '(float 5 -1)) | |
| 589 | mid (math-add-float low step))) | |
| 590 | (progn | |
| 591 | (setq var-DUMMY mid | |
| 592 | vmid (math-evaluate-expr expr)) | |
| 593 | (Math-zerop vmid)))) | |
| 594 | (math-working "bisect" mid) | |
| 595 | (if (eq (Math-posp vmid) pos) | |
| 596 | (setq high mid | |
| 597 | vhigh vmid) | |
| 598 | (setq low mid | |
| 599 | vlow vmid))) | |
| bf77c646 | 600 | (list 'vec mid vmid))) |
| 136211a9 EZ |
601 | |
| 602 | ;;; "mnewt" | |
| 603 | (defun math-newton-multi (expr jacob n guess orig-guess limit) | |
| 604 | (let ((m -1) | |
| 605 | (p guess) | |
| 606 | p2 expr-val jacob-val next) | |
| 607 | (while (< (setq p (cdr p) m (1+ m)) n) | |
| 608 | (set (nth 2 (aref math-root-vars m)) (car p))) | |
| 609 | (setq expr-val (math-evaluate-expr expr) | |
| 610 | jacob-val (math-evaluate-expr jacob)) | |
| 3132f345 CW |
611 | (unless (and (math-constp expr-val) |
| 612 | (math-constp jacob-val)) | |
| 613 | (math-reject-arg guess "*Newton's method encountered a singularity")) | |
| 136211a9 EZ |
614 | (setq next (math-add guess (math-div (math-float (math-neg expr-val)) |
| 615 | (math-float jacob-val))) | |
| 616 | p guess p2 next) | |
| 617 | (math-working "newton" next) | |
| 618 | (while (and (setq p (cdr p) p2 (cdr p2)) | |
| 619 | (math-nearly-equal (car p) (car p2)))) | |
| 620 | (if p | |
| 621 | (if (Math-lessp (math-abs-approx (math-sub next orig-guess)) | |
| 622 | limit) | |
| 623 | (math-newton-multi expr jacob n next orig-guess limit) | |
| 624 | (math-reject-arg nil "*Newton's method failed to converge")) | |
| bf77c646 | 625 | (list 'vec next expr-val)))) |
| 136211a9 EZ |
626 | |
| 627 | (defvar math-root-vars [(var DUMMY var-DUMMY)]) | |
| 628 | ||
| 629 | (defun math-find-root (expr var guess root-widen) | |
| 630 | (if (eq (car-safe expr) 'vec) | |
| 631 | (let ((n (1- (length expr))) | |
| 632 | (calc-symbolic-mode nil) | |
| 633 | (var-DUMMY nil) | |
| 634 | (jacob (list 'vec)) | |
| 635 | p p2 m row) | |
| 3132f345 CW |
636 | (unless (eq (car-safe var) 'vec) |
| 637 | (math-reject-arg var 'vectorp)) | |
| 638 | (unless (= (length var) (1+ n)) | |
| 639 | (math-dimension-error)) | |
| 136211a9 EZ |
640 | (setq expr (copy-sequence expr)) |
| 641 | (while (>= n (length math-root-vars)) | |
| 642 | (let ((symb (intern (concat "math-root-v" | |
| 643 | (int-to-string | |
| 644 | (length math-root-vars)))))) | |
| 645 | (setq math-root-vars (vconcat math-root-vars | |
| 646 | (vector (list 'var symb symb)))))) | |
| 647 | (setq m -1) | |
| 648 | (while (< (setq m (1+ m)) n) | |
| 649 | (set (nth 2 (aref math-root-vars m)) nil)) | |
| 650 | (setq m -1 p var) | |
| 651 | (while (setq m (1+ m) p (cdr p)) | |
| 652 | (or (eq (car-safe (car p)) 'var) | |
| 653 | (math-reject-arg var "*Expected a variable")) | |
| 654 | (setq p2 expr) | |
| 655 | (while (setq p2 (cdr p2)) | |
| 656 | (setcar p2 (math-expr-subst (car p2) (car p) | |
| 657 | (aref math-root-vars m))))) | |
| 3132f345 CW |
658 | (unless (eq (car-safe guess) 'vec) |
| 659 | (math-reject-arg guess 'vectorp)) | |
| 660 | (unless (= (length guess) (1+ n)) | |
| 661 | (math-dimension-error)) | |
| 136211a9 EZ |
662 | (setq guess (copy-sequence guess) |
| 663 | p guess) | |
| 664 | (while (setq p (cdr p)) | |
| 665 | (or (Math-numberp (car guess)) | |
| 666 | (math-reject-arg guess 'numberp)) | |
| 667 | (setcar p (math-float (car p)))) | |
| 668 | (setq p expr) | |
| 669 | (while (setq p (cdr p)) | |
| 670 | (if (assq (car-safe (car p)) calc-tweak-eqn-table) | |
| 671 | (setcar p (math-sub (nth 1 (car p)) (nth 2 (car p))))) | |
| 672 | (setcar p (math-evaluate-expr (car p))) | |
| 673 | (setq row (list 'vec) | |
| 674 | m -1) | |
| 675 | (while (< (setq m (1+ m)) n) | |
| 676 | (nconc row (list (math-evaluate-expr | |
| 677 | (or (calcFunc-deriv (car p) | |
| 678 | (aref math-root-vars m) | |
| 679 | nil t) | |
| 680 | (math-reject-arg | |
| 681 | expr | |
| 682 | "*Formulas must be differentiable")))))) | |
| 683 | (nconc jacob (list row))) | |
| 684 | (setq m (math-abs-approx guess)) | |
| 685 | (math-newton-multi expr jacob n guess guess | |
| 686 | (if (math-zerop m) '(float 1 3) (math-mul m 10)))) | |
| 3132f345 CW |
687 | (unless (eq (car-safe var) 'var) |
| 688 | (math-reject-arg var "*Expected a variable")) | |
| 689 | (unless (math-expr-contains expr var) | |
| 690 | (math-reject-arg expr "*Formula does not contain specified variable")) | |
| 136211a9 EZ |
691 | (if (assq (car expr) calc-tweak-eqn-table) |
| 692 | (setq expr (math-sub (nth 1 expr) (nth 2 expr)))) | |
| 693 | (math-with-extra-prec 2 | |
| 694 | (setq expr (math-expr-subst expr var '(var DUMMY var-DUMMY))) | |
| 695 | (let* ((calc-symbolic-mode nil) | |
| 696 | (var-DUMMY nil) | |
| 697 | (expr (math-evaluate-expr expr)) | |
| 698 | (deriv (calcFunc-deriv expr '(var DUMMY var-DUMMY) nil t)) | |
| 699 | low high vlow vhigh) | |
| 700 | (and deriv (setq deriv (math-evaluate-expr deriv))) | |
| 701 | (setq guess (math-float guess)) | |
| 702 | (if (and (math-numberp guess) | |
| 703 | deriv) | |
| 704 | (math-newton-root expr deriv guess guess | |
| 705 | (if (math-zerop guess) '(float 1 6) | |
| 706 | (math-mul (math-abs-approx guess) 100))) | |
| 707 | (if (Math-realp guess) | |
| 708 | (setq low guess | |
| 709 | high guess | |
| 710 | var-DUMMY guess | |
| 711 | vlow (math-evaluate-expr expr) | |
| 712 | vhigh vlow | |
| 713 | root-widen 'point) | |
| 714 | (if (eq (car guess) 'intv) | |
| 715 | (progn | |
| 716 | (or (math-constp guess) (math-reject-arg guess 'constp)) | |
| 717 | (setq low (nth 2 guess) | |
| 718 | high (nth 3 guess)) | |
| 719 | (if (memq (nth 1 guess) '(0 1)) | |
| 720 | (setq low (calcFunc-incr low 1 high))) | |
| 721 | (if (memq (nth 1 guess) '(0 2)) | |
| 722 | (setq high (calcFunc-incr high -1 low))) | |
| 723 | (setq var-DUMMY low | |
| 724 | vlow (math-evaluate-expr expr) | |
| 725 | var-DUMMY high | |
| 726 | vhigh (math-evaluate-expr expr))) | |
| 727 | (if (math-complexp guess) | |
| 728 | (math-reject-arg "*Complex root finder must have derivative") | |
| 729 | (math-reject-arg guess 'realp)))) | |
| 730 | (if (Math-zerop vlow) | |
| 731 | (list 'vec low vlow) | |
| 732 | (if (Math-zerop vhigh) | |
| 733 | (list 'vec high vhigh) | |
| 734 | (if (and deriv (Math-numberp vlow) (Math-numberp vhigh)) | |
| 735 | (math-newton-search-root expr deriv nil nil nil nil | |
| 736 | low vlow high vhigh) | |
| 737 | (if (or (and (Math-posp vlow) (Math-posp vhigh)) | |
| 738 | (and (Math-negp vlow) (Math-negp vhigh)) | |
| 739 | (not (Math-numberp vlow)) | |
| 740 | (not (Math-numberp vhigh))) | |
| 741 | (math-search-root expr deriv low vlow high vhigh) | |
| bf77c646 | 742 | (math-bisect-root expr low vlow high vhigh)))))))))) |
| 136211a9 EZ |
743 | |
| 744 | (defun calcFunc-root (expr var guess) | |
| bf77c646 | 745 | (math-find-root expr var guess nil)) |
| 136211a9 EZ |
746 | |
| 747 | (defun calcFunc-wroot (expr var guess) | |
| bf77c646 | 748 | (math-find-root expr var guess t)) |
| 136211a9 EZ |
749 | |
| 750 | ||
| 751 | ||
| 752 | ||
| 753 | ;;; The following algorithms come from Numerical Recipes, chapter 10. | |
| 754 | ||
| 755 | (defun math-min-eval (expr a) | |
| 756 | (if (Math-vectorp a) | |
| 757 | (let ((m -1)) | |
| 758 | (while (setq m (1+ m) a (cdr a)) | |
| 759 | (set (nth 2 (aref math-min-vars m)) (car a)))) | |
| 760 | (setq var-DUMMY a)) | |
| 761 | (setq a (math-evaluate-expr expr)) | |
| 762 | (if (Math-ratp a) | |
| 763 | (math-float a) | |
| 764 | (if (eq (car a) 'float) | |
| 765 | a | |
| bf77c646 | 766 | (math-reject-arg a 'realp)))) |
| 136211a9 | 767 | |
| 3132f345 | 768 | (defvar math-min-or-max "minimum") |
| 136211a9 EZ |
769 | |
| 770 | ;;; A bracket for a minimum is a < b < c where f(b) < f(a) and f(b) < f(c). | |
| 771 | ||
| 772 | ;;; "mnbrak" | |
| 773 | (defun math-widen-min (expr a b) | |
| 774 | (let ((done nil) | |
| 775 | (iters 30) | |
| 776 | incr c va vb vc u vu r q ulim bc ba qr) | |
| 777 | (or b (setq b (math-mul a '(float 101 -2)))) | |
| 778 | (setq va (math-min-eval expr a) | |
| 779 | vb (math-min-eval expr b)) | |
| 780 | (if (math-lessp-float va vb) | |
| 781 | (setq u a a b b u | |
| 782 | vu va va vb vb vu)) | |
| 783 | (setq c (math-add-float b (math-mul-float '(float 161803 -5) | |
| 784 | (math-sub-float b a))) | |
| 785 | vc (math-min-eval expr c)) | |
| 786 | (while (and (not done) (math-lessp-float vc vb)) | |
| 787 | (math-working "widen" (list 'intv 0 a c)) | |
| 788 | (if (= (setq iters (1- iters)) 0) | |
| 789 | (math-reject-arg nil (format "*Unable to find a %s near the interval" | |
| 790 | math-min-or-max))) | |
| 791 | (setq bc (math-sub-float b c) | |
| 792 | ba (math-sub-float b a) | |
| 793 | r (math-mul-float ba (math-sub-float vb vc)) | |
| 794 | q (math-mul-float bc (math-sub-float vb va)) | |
| 795 | qr (math-sub-float q r)) | |
| 796 | (if (math-lessp-float (math-abs qr) '(float 1 -20)) | |
| 797 | (setq qr (if (math-negp qr) '(float -1 -20) '(float 1 -20)))) | |
| 798 | (setq u (math-sub-float | |
| 799 | b | |
| 800 | (math-div-float (math-sub-float (math-mul-float bc q) | |
| 801 | (math-mul-float ba r)) | |
| 802 | (math-mul-float '(float 2 0) qr))) | |
| 803 | ulim (math-add-float b (math-mul-float '(float -1 2) bc)) | |
| 804 | incr (math-negp bc)) | |
| 805 | (if (if incr (math-lessp-float b u) (math-lessp-float u b)) | |
| 806 | (if (if incr (math-lessp-float u c) (math-lessp-float c u)) | |
| 807 | (if (math-lessp-float (setq vu (math-min-eval expr u)) vc) | |
| 808 | (setq a b va vb | |
| 809 | b u vb vu | |
| 810 | done t) | |
| 811 | (if (math-lessp-float vb vu) | |
| 812 | (setq c u vc vu | |
| 813 | done t) | |
| 814 | (setq u (math-add-float c (math-mul-float '(float -161803 -5) | |
| 815 | bc)) | |
| 816 | vu (math-min-eval expr u)))) | |
| 817 | (if (if incr (math-lessp-float u ulim) (math-lessp-float ulim u)) | |
| 818 | (if (math-lessp-float (setq vu (math-min-eval expr u)) vc) | |
| 819 | (setq b c vb vc | |
| 820 | c u vc vu | |
| 821 | u (math-add-float c (math-mul-float | |
| 822 | '(float -161803 -5) | |
| 823 | (math-sub-float b c))) | |
| 824 | vu (math-min-eval expr u))) | |
| 825 | (setq u ulim | |
| 826 | vu (math-min-eval expr u)))) | |
| 827 | (setq u (math-add-float c (math-mul-float '(float -161803 -5) | |
| 828 | bc)) | |
| 829 | vu (math-min-eval expr u))) | |
| 830 | (setq a b va vb | |
| 831 | b c vb vc | |
| 832 | c u vc vu)) | |
| 833 | (if (math-lessp-float a c) | |
| 834 | (list a va b vb c vc) | |
| bf77c646 | 835 | (list c vc b vb a va)))) |
| 136211a9 EZ |
836 | |
| 837 | (defun math-narrow-min (expr a c intv) | |
| 838 | (let ((xvals (list a c)) | |
| 839 | (yvals (list (math-min-eval expr a) | |
| 840 | (math-min-eval expr c))) | |
| 841 | (levels 0) | |
| 842 | (step (math-sub-float c a)) | |
| 843 | (found nil) | |
| 844 | xp yp b) | |
| 845 | (while (and (<= (setq levels (1+ levels)) 5) | |
| 846 | (not found)) | |
| 847 | (setq xp xvals | |
| 848 | yp yvals | |
| 849 | step (math-mul-float step '(float 497 -3))) | |
| 850 | (while (and (cdr xp) (not found)) | |
| 851 | (setq b (math-add-float (car xp) step)) | |
| 852 | (math-working "search" b) | |
| 853 | (setcdr xp (cons b (cdr xp))) | |
| 854 | (setcdr yp (cons (math-min-eval expr b) (cdr yp))) | |
| 855 | (if (and (math-lessp-float (nth 1 yp) (car yp)) | |
| 856 | (math-lessp-float (nth 1 yp) (nth 2 yp))) | |
| 857 | (setq found t) | |
| 858 | (setq xp (cdr xp) | |
| 859 | yp (cdr yp)) | |
| 860 | (if (and (cdr (cdr yp)) | |
| 861 | (math-lessp-float (nth 1 yp) (car yp)) | |
| 862 | (math-lessp-float (nth 1 yp) (nth 2 yp))) | |
| 863 | (setq found t) | |
| 864 | (setq xp (cdr xp) | |
| 865 | yp (cdr yp)))))) | |
| 866 | (if found | |
| 867 | (list (car xp) (car yp) | |
| 868 | (nth 1 xp) (nth 1 yp) | |
| 869 | (nth 2 xp) (nth 2 yp)) | |
| 870 | (or (if (math-lessp-float (car yvals) (nth 1 yvals)) | |
| 871 | (and (memq (nth 1 intv) '(2 3)) | |
| 872 | (let ((min (car yvals))) | |
| 873 | (while (and (setq yvals (cdr yvals)) | |
| 874 | (math-lessp-float min (car yvals)))) | |
| 875 | (and (not yvals) | |
| 876 | (list (nth 2 intv) min)))) | |
| 877 | (and (memq (nth 1 intv) '(1 3)) | |
| 878 | (setq yvals (nreverse yvals)) | |
| 879 | (let ((min (car yvals))) | |
| 880 | (while (and (setq yvals (cdr yvals)) | |
| 881 | (math-lessp-float min (car yvals)))) | |
| 882 | (and (not yvals) | |
| 883 | (list (nth 3 intv) min))))) | |
| 884 | (math-reject-arg nil (format "*Unable to find a %s in the interval" | |
| bf77c646 | 885 | math-min-or-max)))))) |
| 136211a9 EZ |
886 | |
| 887 | ;;; "brent" | |
| 888 | (defun math-brent-min (expr prec a va x vx b vb) | |
| 889 | (let ((iters (+ 20 (* 5 prec))) | |
| 890 | (w x) | |
| 891 | (vw vx) | |
| 892 | (v x) | |
| 893 | (vv vx) | |
| 894 | (tol (list 'float 1 (- -1 prec))) | |
| 895 | (zeps (list 'float 1 (- -5 prec))) | |
| 896 | (e '(float 0 0)) | |
| 897 | u vu xm tol1 tol2 etemp p q r xv xw) | |
| 898 | (while (progn | |
| 899 | (setq xm (math-mul-float '(float 5 -1) | |
| 900 | (math-add-float a b)) | |
| 901 | tol1 (math-add-float | |
| 902 | zeps | |
| 903 | (math-mul-float tol (math-abs x))) | |
| 904 | tol2 (math-mul-float tol1 '(float 2 0))) | |
| 905 | (math-lessp-float (math-sub-float tol2 | |
| 906 | (math-mul-float | |
| 907 | '(float 5 -1) | |
| 908 | (math-sub-float b a))) | |
| 909 | (math-abs (math-sub-float x xm)))) | |
| 910 | (if (= (setq iters (1- iters)) 0) | |
| 911 | (math-reject-arg nil (format "*Unable to converge on a %s" | |
| 912 | math-min-or-max))) | |
| 913 | (math-working "brent" x) | |
| 914 | (if (math-lessp-float (math-abs e) tol1) | |
| 915 | (setq e (if (math-lessp-float x xm) | |
| 916 | (math-sub-float b x) | |
| 917 | (math-sub-float a x)) | |
| 918 | d (math-mul-float '(float 381966 -6) e)) | |
| 919 | (setq xw (math-sub-float x w) | |
| 920 | r (math-mul-float xw (math-sub-float vx vv)) | |
| 921 | xv (math-sub-float x v) | |
| 922 | q (math-mul-float xv (math-sub-float vx vw)) | |
| 923 | p (math-sub-float (math-mul-float xv q) | |
| 924 | (math-mul-float xw r)) | |
| 925 | q (math-mul-float '(float 2 0) (math-sub-float q r))) | |
| 926 | (if (math-posp q) | |
| 927 | (setq p (math-neg-float p)) | |
| 928 | (setq q (math-neg-float q))) | |
| 929 | (setq etemp e | |
| 930 | e d) | |
| 931 | (if (and (math-lessp-float (math-abs p) | |
| 932 | (math-abs (math-mul-float | |
| 933 | '(float 5 -1) | |
| 934 | (math-mul-float q etemp)))) | |
| 935 | (math-lessp-float (math-mul-float | |
| 936 | q (math-sub-float a x)) p) | |
| 937 | (math-lessp-float p (math-mul-float | |
| 938 | q (math-sub-float b x)))) | |
| 939 | (progn | |
| 940 | (setq d (math-div-float p q) | |
| 941 | u (math-add-float x d)) | |
| 942 | (if (or (math-lessp-float (math-sub-float u a) tol2) | |
| 943 | (math-lessp-float (math-sub-float b u) tol2)) | |
| 944 | (setq d (if (math-lessp-float xm x) | |
| 945 | (math-neg-float tol1) | |
| 946 | tol1)))) | |
| 947 | (setq e (if (math-lessp-float x xm) | |
| 948 | (math-sub-float b x) | |
| 949 | (math-sub-float a x)) | |
| 950 | d (math-mul-float '(float 381966 -6) e)))) | |
| 951 | (setq u (math-add-float x | |
| 952 | (if (math-lessp-float (math-abs d) tol1) | |
| 953 | (if (math-negp d) | |
| 954 | (math-neg-float tol1) | |
| 955 | tol1) | |
| 956 | d)) | |
| 957 | vu (math-min-eval expr u)) | |
| 958 | (if (math-lessp-float vx vu) | |
| 959 | (progn | |
| 960 | (if (math-lessp-float u x) | |
| 961 | (setq a u) | |
| 962 | (setq b u)) | |
| 963 | (if (or (equal w x) | |
| 964 | (not (math-lessp-float vw vu))) | |
| 965 | (setq v w vv vw | |
| 966 | w u vw vu) | |
| 967 | (if (or (equal v x) | |
| 968 | (equal v w) | |
| 969 | (not (math-lessp-float vv vu))) | |
| 970 | (setq v u vv vu)))) | |
| 971 | (if (math-lessp-float u x) | |
| 972 | (setq b x) | |
| 973 | (setq a x)) | |
| 974 | (setq v w vv vw | |
| 975 | w x vw vx | |
| 976 | x u vx vu))) | |
| bf77c646 | 977 | (list 'vec x vx))) |
| 136211a9 EZ |
978 | |
| 979 | ;;; "powell" | |
| 980 | (defun math-powell-min (expr n guesses prec) | |
| 981 | (let* ((f1dim (math-line-min-func expr n)) | |
| 982 | (xi (calcFunc-idn 1 n)) | |
| 983 | (p (cons 'vec (mapcar 'car guesses))) | |
| 984 | (pt p) | |
| 985 | (ftol (list 'float 1 (- prec))) | |
| 986 | (fret (math-min-eval expr p)) | |
| 987 | fp ptt fptt xit i ibig del diff res) | |
| 988 | (while (progn | |
| 989 | (setq fp fret | |
| 990 | ibig 0 | |
| 991 | del '(float 0 0) | |
| 992 | i 0) | |
| 993 | (while (<= (setq i (1+ i)) n) | |
| 994 | (setq fptt fret | |
| 995 | res (math-line-min f1dim p | |
| 996 | (math-mat-col xi i) | |
| 997 | n prec) | |
| 998 | p (let ((calc-internal-prec prec)) | |
| 999 | (math-normalize (car res))) | |
| 1000 | fret (nth 2 res) | |
| 1001 | diff (math-abs (math-sub-float fptt fret))) | |
| 1002 | (if (math-lessp-float del diff) | |
| 1003 | (setq del diff | |
| 1004 | ibig i))) | |
| 1005 | (math-lessp-float | |
| 1006 | (math-mul-float ftol | |
| 1007 | (math-add-float (math-abs fp) | |
| 1008 | (math-abs fret))) | |
| 1009 | (math-mul-float '(float 2 0) | |
| 1010 | (math-abs (math-sub-float fp | |
| 1011 | fret))))) | |
| 1012 | (setq ptt (math-sub (math-mul '(float 2 0) p) pt) | |
| 1013 | xit (math-sub p pt) | |
| 1014 | pt p | |
| 1015 | fptt (math-min-eval expr ptt)) | |
| 1016 | (if (and (math-lessp-float fptt fp) | |
| 1017 | (math-lessp-float | |
| 1018 | (math-mul-float | |
| 1019 | (math-mul-float '(float 2 0) | |
| 1020 | (math-add-float | |
| 1021 | (math-sub-float fp | |
| 1022 | (math-mul-float '(float 2 0) | |
| 1023 | fret)) | |
| 1024 | fptt)) | |
| 1025 | (math-sqr-float (math-sub-float | |
| 1026 | (math-sub-float fp fret) del))) | |
| 1027 | (math-mul-float del | |
| 1028 | (math-sqr-float (math-sub-float fp fptt))))) | |
| 1029 | (progn | |
| 1030 | (setq res (math-line-min f1dim p xit n prec) | |
| 1031 | p (car res) | |
| 1032 | fret (nth 2 res) | |
| 1033 | i 0) | |
| 1034 | (while (<= (setq i (1+ i)) n) | |
| 1035 | (setcar (nthcdr ibig (nth i xi)) | |
| 1036 | (nth i (nth 1 res))))))) | |
| bf77c646 | 1037 | (list 'vec p fret))) |
| 136211a9 EZ |
1038 | |
| 1039 | (defun math-line-min-func (expr n) | |
| 1040 | (let ((m -1)) | |
| 1041 | (while (< (setq m (1+ m)) n) | |
| 1042 | (set (nth 2 (aref math-min-vars m)) | |
| 1043 | (list '+ | |
| 1044 | (list '* | |
| 1045 | '(var DUMMY var-DUMMY) | |
| 1046 | (list 'calcFunc-mrow '(var line-xi line-xi) (1+ m))) | |
| 1047 | (list 'calcFunc-mrow '(var line-p line-p) (1+ m))))) | |
| bf77c646 | 1048 | (math-evaluate-expr expr))) |
| 136211a9 EZ |
1049 | |
| 1050 | (defun math-line-min (f1dim line-p line-xi n prec) | |
| 1051 | (let* ((var-DUMMY nil) | |
| 1052 | (expr (math-evaluate-expr f1dim)) | |
| 1053 | (params (math-widen-min expr '(float 0 0) '(float 1 0))) | |
| 1054 | (res (apply 'math-brent-min expr prec params)) | |
| 1055 | (xi (math-mul (nth 1 res) line-xi))) | |
| bf77c646 | 1056 | (list (math-add line-p xi) xi (nth 2 res)))) |
| 136211a9 EZ |
1057 | |
| 1058 | ||
| 1059 | (defvar math-min-vars [(var DUMMY var-DUMMY)]) | |
| 1060 | ||
| 1061 | (defun math-find-minimum (expr var guess min-widen) | |
| 1062 | (let* ((calc-symbolic-mode nil) | |
| 1063 | (n 0) | |
| 1064 | (var-DUMMY nil) | |
| 1065 | (isvec (math-vectorp var)) | |
| 1066 | g guesses) | |
| 1067 | (or (math-vectorp var) | |
| 1068 | (setq var (list 'vec var))) | |
| 1069 | (or (math-vectorp guess) | |
| 1070 | (setq guess (list 'vec guess))) | |
| 1071 | (or (= (length var) (length guess)) | |
| 1072 | (math-dimension-error)) | |
| 1073 | (while (setq var (cdr var) guess (cdr guess)) | |
| 1074 | (or (eq (car-safe (car var)) 'var) | |
| 1075 | (math-reject-arg (car vg) "*Expected a variable")) | |
| 1076 | (or (math-expr-contains expr (car var)) | |
| 1077 | (math-reject-arg (car var) | |
| 1078 | "*Formula does not contain specified variable")) | |
| 1079 | (while (>= (1+ n) (length math-min-vars)) | |
| 1080 | (let ((symb (intern (concat "math-min-v" | |
| 1081 | (int-to-string | |
| 1082 | (length math-min-vars)))))) | |
| 1083 | (setq math-min-vars (vconcat math-min-vars | |
| 1084 | (vector (list 'var symb symb)))))) | |
| 1085 | (set (nth 2 (aref math-min-vars n)) nil) | |
| 1086 | (set (nth 2 (aref math-min-vars (1+ n))) nil) | |
| 1087 | (if (math-complexp (car guess)) | |
| 1088 | (setq expr (math-expr-subst expr | |
| 1089 | (car var) | |
| 1090 | (list '+ (aref math-min-vars n) | |
| 1091 | (list '* | |
| 1092 | (aref math-min-vars (1+ n)) | |
| 1093 | '(cplx 0 1)))) | |
| 1094 | guesses (let ((g (math-float (math-complex (car guess))))) | |
| 1095 | (cons (list (nth 2 g) nil nil) | |
| 1096 | (cons (list (nth 1 g) nil nil t) | |
| 1097 | guesses))) | |
| 1098 | n (+ n 2)) | |
| 1099 | (setq expr (math-expr-subst expr | |
| 1100 | (car var) | |
| 1101 | (aref math-min-vars n)) | |
| 1102 | guesses (cons (if (math-realp (car guess)) | |
| 1103 | (list (math-float (car guess)) nil nil) | |
| 1104 | (if (and (eq (car-safe (car guess)) 'intv) | |
| 1105 | (math-constp (car guess))) | |
| 1106 | (list (math-mul | |
| 1107 | (math-add (nth 2 (car guess)) | |
| 1108 | (nth 3 (car guess))) | |
| 1109 | '(float 5 -1)) | |
| 1110 | (math-float (nth 2 (car guess))) | |
| 1111 | (math-float (nth 3 (car guess))) | |
| 1112 | (car guess)) | |
| 1113 | (math-reject-arg (car guess) 'realp))) | |
| 1114 | guesses) | |
| 1115 | n (1+ n)))) | |
| 1116 | (setq guesses (nreverse guesses) | |
| 1117 | expr (math-evaluate-expr expr)) | |
| 1118 | (if (= n 1) | |
| 1119 | (let* ((params (if (nth 1 (car guesses)) | |
| 1120 | (if min-widen | |
| 1121 | (math-widen-min expr | |
| 1122 | (nth 1 (car guesses)) | |
| 1123 | (nth 2 (car guesses))) | |
| 1124 | (math-narrow-min expr | |
| 1125 | (nth 1 (car guesses)) | |
| 1126 | (nth 2 (car guesses)) | |
| 1127 | (nth 3 (car guesses)))) | |
| 1128 | (math-widen-min expr | |
| 1129 | (car (car guesses)) | |
| 1130 | nil))) | |
| 1131 | (prec calc-internal-prec) | |
| 1132 | (res (if (cdr (cdr params)) | |
| 1133 | (math-with-extra-prec (+ calc-internal-prec 2) | |
| 1134 | (apply 'math-brent-min expr prec params)) | |
| 1135 | (cons 'vec params)))) | |
| 1136 | (if isvec | |
| 1137 | (list 'vec (list 'vec (nth 1 res)) (nth 2 res)) | |
| 1138 | res)) | |
| 1139 | (let* ((prec calc-internal-prec) | |
| 1140 | (res (math-with-extra-prec (+ calc-internal-prec 2) | |
| 1141 | (math-powell-min expr n guesses prec))) | |
| 1142 | (p (nth 1 res)) | |
| 1143 | (vec (list 'vec))) | |
| 1144 | (while (setq p (cdr p)) | |
| 1145 | (if (nth 3 (car guesses)) | |
| 1146 | (progn | |
| 1147 | (nconc vec (list (math-normalize | |
| 1148 | (list 'cplx (car p) (nth 1 p))))) | |
| 1149 | (setq p (cdr p) | |
| 1150 | guesses (cdr guesses))) | |
| 1151 | (nconc vec (list (car p)))) | |
| 1152 | (setq guesses (cdr guesses))) | |
| 1153 | (if isvec | |
| 1154 | (list 'vec vec (nth 2 res)) | |
| bf77c646 | 1155 | (list 'vec (nth 1 vec) (nth 2 res))))))) |
| 136211a9 EZ |
1156 | |
| 1157 | (defun calcFunc-minimize (expr var guess) | |
| 1158 | (let ((calc-internal-prec (max (/ calc-internal-prec 2) 3)) | |
| 1159 | (math-min-or-max "minimum")) | |
| 1160 | (math-find-minimum (math-normalize expr) | |
| 1161 | (math-normalize var) | |
| bf77c646 | 1162 | (math-normalize guess) nil))) |
| 136211a9 EZ |
1163 | |
| 1164 | (defun calcFunc-wminimize (expr var guess) | |
| 1165 | (let ((calc-internal-prec (max (/ calc-internal-prec 2) 3)) | |
| 1166 | (math-min-or-max "minimum")) | |
| 1167 | (math-find-minimum (math-normalize expr) | |
| 1168 | (math-normalize var) | |
| bf77c646 | 1169 | (math-normalize guess) t))) |
| 136211a9 EZ |
1170 | |
| 1171 | (defun calcFunc-maximize (expr var guess) | |
| 1172 | (let* ((calc-internal-prec (max (/ calc-internal-prec 2) 3)) | |
| 1173 | (math-min-or-max "maximum") | |
| 1174 | (res (math-find-minimum (math-normalize (math-neg expr)) | |
| 1175 | (math-normalize var) | |
| 1176 | (math-normalize guess) nil))) | |
| bf77c646 | 1177 | (list 'vec (nth 1 res) (math-neg (nth 2 res))))) |
| 136211a9 EZ |
1178 | |
| 1179 | (defun calcFunc-wmaximize (expr var guess) | |
| 1180 | (let* ((calc-internal-prec (max (/ calc-internal-prec 2) 3)) | |
| 1181 | (math-min-or-max "maximum") | |
| 1182 | (res (math-find-minimum (math-normalize (math-neg expr)) | |
| 1183 | (math-normalize var) | |
| 1184 | (math-normalize guess) t))) | |
| bf77c646 | 1185 | (list 'vec (nth 1 res) (math-neg (nth 2 res))))) |
| 136211a9 EZ |
1186 | |
| 1187 | ||
| 1188 | ||
| 1189 | ||
| 1190 | ;;; The following algorithms come from Numerical Recipes, chapter 3. | |
| 1191 | ||
| 1192 | (defun calcFunc-polint (data x) | |
| 1193 | (or (math-matrixp data) (math-reject-arg data 'matrixp)) | |
| 1194 | (or (= (length data) 3) | |
| 1195 | (math-reject-arg data "*Wrong number of data rows")) | |
| 1196 | (or (> (length (nth 1 data)) 2) | |
| 1197 | (math-reject-arg data "*Too few data points")) | |
| 1198 | (if (and (math-vectorp x) (or (math-constp x) math-expand-formulas)) | |
| 1199 | (cons 'vec (mapcar (function (lambda (x) (calcFunc-polint data x))) | |
| 1200 | (cdr x))) | |
| 1201 | (or (math-objectp x) math-expand-formulas (math-reject-arg x 'objectp)) | |
| 1202 | (math-with-extra-prec 2 | |
| 1203 | (cons 'vec (math-poly-interp (cdr (nth 1 data)) (cdr (nth 2 data)) x | |
| bf77c646 | 1204 | nil))))) |
| 136211a9 EZ |
1205 | (put 'calcFunc-polint 'math-expandable t) |
| 1206 | ||
| 1207 | ||
| 1208 | (defun calcFunc-ratint (data x) | |
| 1209 | (or (math-matrixp data) (math-reject-arg data 'matrixp)) | |
| 1210 | (or (= (length data) 3) | |
| 1211 | (math-reject-arg data "*Wrong number of data rows")) | |
| 1212 | (or (> (length (nth 1 data)) 2) | |
| 1213 | (math-reject-arg data "*Too few data points")) | |
| 1214 | (if (and (math-vectorp x) (or (math-constp x) math-expand-formulas)) | |
| 1215 | (cons 'vec (mapcar (function (lambda (x) (calcFunc-ratint data x))) | |
| 1216 | (cdr x))) | |
| 1217 | (or (math-objectp x) math-expand-formulas (math-reject-arg x 'objectp)) | |
| 1218 | (math-with-extra-prec 2 | |
| 1219 | (cons 'vec (math-poly-interp (cdr (nth 1 data)) (cdr (nth 2 data)) x | |
| bf77c646 | 1220 | (cdr (cdr (cdr (nth 1 data))))))))) |
| 136211a9 EZ |
1221 | (put 'calcFunc-ratint 'math-expandable t) |
| 1222 | ||
| 1223 | ||
| 1224 | (defun math-poly-interp (xa ya x ratp) | |
| 1225 | (let ((n (length xa)) | |
| 1226 | (dif nil) | |
| 1227 | (ns nil) | |
| 1228 | (xax nil) | |
| 1229 | (c (copy-sequence ya)) | |
| 1230 | (d (copy-sequence ya)) | |
| 1231 | (i 0) | |
| 1232 | (m 0) | |
| 1233 | y dy (xp xa) xpm cp dp temp) | |
| 1234 | (while (<= (setq i (1+ i)) n) | |
| 1235 | (setq xax (cons (math-sub (car xp) x) xax) | |
| 1236 | xp (cdr xp) | |
| 1237 | temp (math-abs (car xax))) | |
| 1238 | (if (or (null dif) (math-lessp temp dif)) | |
| 1239 | (setq dif temp | |
| 1240 | ns i))) | |
| 1241 | (setq xax (nreverse xax) | |
| 1242 | ns (1- ns) | |
| 1243 | y (nth ns ya)) | |
| 1244 | (if (math-zerop dif) | |
| 1245 | (list y 0) | |
| 1246 | (while (< (setq m (1+ m)) n) | |
| 1247 | (setq i 0 | |
| 1248 | xp xax | |
| 1249 | xpm (nthcdr m xax) | |
| 1250 | cp c | |
| 1251 | dp d) | |
| 1252 | (while (<= (setq i (1+ i)) (- n m)) | |
| 1253 | (if ratp | |
| 1254 | (let ((t2 (math-div (math-mul (car xp) (car dp)) (car xpm)))) | |
| 1255 | (setq temp (math-div (math-sub (nth 1 cp) (car dp)) | |
| 1256 | (math-sub t2 (nth 1 cp)))) | |
| 1257 | (setcar dp (math-mul (nth 1 cp) temp)) | |
| 1258 | (setcar cp (math-mul t2 temp))) | |
| 1259 | (if (math-equal (car xp) (car xpm)) | |
| 1260 | (math-reject-arg (cons 'vec xa) "*Duplicate X values")) | |
| 1261 | (setq temp (math-div (math-sub (nth 1 cp) (car dp)) | |
| 1262 | (math-sub (car xp) (car xpm)))) | |
| 1263 | (setcar dp (math-mul (car xpm) temp)) | |
| 1264 | (setcar cp (math-mul (car xp) temp))) | |
| 1265 | (setq cp (cdr cp) | |
| 1266 | dp (cdr dp) | |
| 1267 | xp (cdr xp) | |
| 1268 | xpm (cdr xpm))) | |
| 1269 | (if (< (+ ns ns) (- n m)) | |
| 1270 | (setq dy (nth ns c)) | |
| 1271 | (setq ns (1- ns) | |
| 1272 | dy (nth ns d))) | |
| 1273 | (setq y (math-add y dy))) | |
| bf77c646 | 1274 | (list y dy)))) |
| 136211a9 EZ |
1275 | |
| 1276 | ||
| 1277 | ||
| 1278 | ;;; The following algorithms come from Numerical Recipes, chapter 4. | |
| 1279 | ||
| 1280 | (defun calcFunc-ninteg (expr var lo hi) | |
| 1281 | (setq lo (math-evaluate-expr lo) | |
| 1282 | hi (math-evaluate-expr hi)) | |
| 1283 | (or (math-numberp lo) (math-infinitep lo) (math-reject-arg lo 'numberp)) | |
| 1284 | (or (math-numberp hi) (math-infinitep hi) (math-reject-arg hi 'numberp)) | |
| 1285 | (if (math-lessp hi lo) | |
| 1286 | (math-neg (calcFunc-ninteg expr var hi lo)) | |
| 1287 | (setq expr (math-expr-subst expr var '(var DUMMY var-DUMMY))) | |
| 1288 | (let ((var-DUMMY nil) | |
| 1289 | (calc-symbolic-mode nil) | |
| 1290 | (calc-prefer-frac nil) | |
| 1291 | (sum 0)) | |
| 1292 | (setq expr (math-evaluate-expr expr)) | |
| 1293 | (if (equal lo '(neg (var inf var-inf))) | |
| 1294 | (let ((thi (if (math-lessp hi '(float -2 0)) | |
| 1295 | hi '(float -2 0)))) | |
| 1296 | (setq sum (math-ninteg-romberg | |
| 1297 | 'math-ninteg-midpoint expr | |
| 1298 | (math-float lo) (math-float thi) 'inf) | |
| 1299 | lo thi))) | |
| 1300 | (if (equal hi '(var inf var-inf)) | |
| 1301 | (let ((tlo (if (math-lessp '(float 2 0) lo) | |
| 1302 | lo '(float 2 0)))) | |
| 1303 | (setq sum (math-add sum | |
| 1304 | (math-ninteg-romberg | |
| 1305 | 'math-ninteg-midpoint expr | |
| 1306 | (math-float tlo) (math-float hi) 'inf)) | |
| 1307 | hi tlo))) | |
| 1308 | (or (math-equal lo hi) | |
| 1309 | (setq sum (math-add sum | |
| 1310 | (math-ninteg-romberg | |
| 1311 | 'math-ninteg-midpoint expr | |
| 1312 | (math-float lo) (math-float hi) nil)))) | |
| bf77c646 | 1313 | sum))) |
| 136211a9 EZ |
1314 | |
| 1315 | ||
| 1316 | ;;; Open Romberg method; "qromo" in section 4.4. | |
| a1506d29 | 1317 | (defun math-ninteg-romberg (func expr lo hi mode) |
| 136211a9 EZ |
1318 | (let ((curh '(float 1 0)) |
| 1319 | (h nil) | |
| 1320 | (s nil) | |
| 1321 | (j 0) | |
| 1322 | (ss nil) | |
| 1323 | (prec calc-internal-prec) | |
| 1324 | (integ-temp nil)) | |
| 1325 | (math-with-extra-prec 2 | |
| 1326 | ;; Limit on "j" loop must be 14 or less to keep "it" from overflowing. | |
| 1327 | (or (while (and (null ss) (<= (setq j (1+ j)) 8)) | |
| 1328 | (setq s (nconc s (list (funcall func expr lo hi mode))) | |
| 1329 | h (nconc h (list curh))) | |
| 1330 | (if (>= j 3) | |
| 1331 | (let ((res (math-poly-interp h s '(float 0 0) nil))) | |
| 1332 | (if (math-lessp (math-abs (nth 1 res)) | |
| 1333 | (calcFunc-scf (math-abs (car res)) | |
| 1334 | (- prec))) | |
| 1335 | (setq math-ninteg-convergence j | |
| 1336 | ss (car res))))) | |
| 1337 | (if (>= j 5) | |
| 1338 | (setq s (cdr s) | |
| 1339 | h (cdr h))) | |
| 1340 | (setq curh (math-div-float curh '(float 9 0)))) | |
| 1341 | ss | |
| bf77c646 | 1342 | (math-reject-arg nil (format "*Integral failed to converge")))))) |
| 136211a9 EZ |
1343 | |
| 1344 | ||
| 1345 | (defun math-ninteg-evaluate (expr x mode) | |
| 1346 | (if (eq mode 'inf) | |
| 1347 | (setq x (math-div '(float 1 0) x))) | |
| 1348 | (let* ((var-DUMMY x) | |
| 1349 | (res (math-evaluate-expr expr))) | |
| 1350 | (or (Math-numberp res) | |
| 1351 | (math-reject-arg res "*Integrand does not evaluate to a number")) | |
| 1352 | (if (eq mode 'inf) | |
| 1353 | (setq res (math-mul res (math-sqr x)))) | |
| bf77c646 | 1354 | res)) |
| 136211a9 EZ |
1355 | |
| 1356 | ||
| 1357 | (defun math-ninteg-midpoint (expr lo hi mode) ; uses "integ-temp" | |
| 1358 | (if (eq mode 'inf) | |
| 1359 | (let ((math-infinite-mode t) temp) | |
| 1360 | (setq temp (math-div 1 lo) | |
| 1361 | lo (math-div 1 hi) | |
| 1362 | hi temp))) | |
| 1363 | (if integ-temp | |
| 1364 | (let* ((it3 (* 3 (car integ-temp))) | |
| 1365 | (math-working-step-2 (* 2 (car integ-temp))) | |
| 1366 | (math-working-step 0) | |
| 1367 | (range (math-sub hi lo)) | |
| 1368 | (del (math-div range (math-float it3))) | |
| 1369 | (del2 (math-add del del)) | |
| 1370 | (del3 (math-add del del2)) | |
| 1371 | (x (math-add lo (math-mul '(float 5 -1) del))) | |
| 1372 | (sum '(float 0 0)) | |
| 1373 | (j 0) temp) | |
| 1374 | (while (<= (setq j (1+ j)) (car integ-temp)) | |
| 1375 | (setq math-working-step (1+ math-working-step) | |
| 1376 | temp (math-ninteg-evaluate expr x mode) | |
| 1377 | math-working-step (1+ math-working-step) | |
| 1378 | sum (math-add sum (math-add temp (math-ninteg-evaluate | |
| 1379 | expr (math-add x del2) | |
| 1380 | mode))) | |
| 1381 | x (math-add x del3))) | |
| 1382 | (setq integ-temp (list it3 | |
| 1383 | (math-add (math-div (nth 1 integ-temp) | |
| 1384 | '(float 3 0)) | |
| 1385 | (math-mul sum del))))) | |
| 1386 | (setq integ-temp (list 1 (math-mul | |
| 1387 | (math-sub hi lo) | |
| 1388 | (math-ninteg-evaluate | |
| 1389 | expr | |
| 1390 | (math-mul (math-add lo hi) '(float 5 -1)) | |
| 1391 | mode))))) | |
| bf77c646 | 1392 | (nth 1 integ-temp)) |
| 136211a9 EZ |
1393 | |
| 1394 | ||
| 1395 | ||
| 1396 | ||
| 1397 | ||
| 1398 | ;;; The following algorithms come from Numerical Recipes, chapter 14. | |
| 1399 | ||
| 3132f345 CW |
1400 | (defvar math-dummy-vars [(var DUMMY var-DUMMY)]) |
| 1401 | (defvar math-dummy-counter 0) | |
| 136211a9 EZ |
1402 | (defun math-dummy-variable () |
| 1403 | (if (= math-dummy-counter (length math-dummy-vars)) | |
| 1404 | (let ((symb (intern (format "math-dummy-%d" math-dummy-counter)))) | |
| 1405 | (setq math-dummy-vars (vconcat math-dummy-vars | |
| 1406 | (vector (list 'var symb symb)))))) | |
| 1407 | (set (nth 2 (aref math-dummy-vars math-dummy-counter)) nil) | |
| 1408 | (prog1 | |
| 1409 | (aref math-dummy-vars math-dummy-counter) | |
| bf77c646 | 1410 | (setq math-dummy-counter (1+ math-dummy-counter)))) |
| 136211a9 | 1411 | |
| 3132f345 CW |
1412 | (defvar math-in-fit 0) |
| 1413 | (defvar calc-fit-to-trail nil) | |
| 136211a9 EZ |
1414 | |
| 1415 | (defun calcFunc-fit (expr vars &optional coefs data) | |
| 1416 | (let ((math-in-fit 10)) | |
| 1417 | (math-with-extra-prec 2 | |
| bf77c646 | 1418 | (math-general-fit expr vars coefs data nil)))) |
| 136211a9 EZ |
1419 | |
| 1420 | (defun calcFunc-efit (expr vars &optional coefs data) | |
| 1421 | (let ((math-in-fit 10)) | |
| 1422 | (math-with-extra-prec 2 | |
| bf77c646 | 1423 | (math-general-fit expr vars coefs data 'sdev)))) |
| 136211a9 EZ |
1424 | |
| 1425 | (defun calcFunc-xfit (expr vars &optional coefs data) | |
| 1426 | (let ((math-in-fit 10)) | |
| 1427 | (math-with-extra-prec 2 | |
| bf77c646 | 1428 | (math-general-fit expr vars coefs data 'full)))) |
| 136211a9 EZ |
1429 | |
| 1430 | (defun math-general-fit (expr vars coefs data mode) | |
| 1431 | (let ((calc-simplify-mode nil) | |
| 1432 | (math-dummy-counter math-dummy-counter) | |
| 1433 | (math-in-fit 1) | |
| 1434 | (extended (eq mode 'full)) | |
| 1435 | (first-coef math-dummy-counter) | |
| 1436 | first-var | |
| 1437 | (plain-expr expr) | |
| 1438 | orig-expr | |
| 1439 | have-sdevs need-chisq chisq | |
| 1440 | (x-funcs nil) | |
| 1441 | (y-filter nil) | |
| 1442 | y-dummy | |
| 1443 | (coef-filters nil) | |
| 1444 | new-coefs | |
| 1445 | (xy-values nil) | |
| 1446 | (weights nil) | |
| 1447 | (var-YVAL nil) (var-YVALX nil) | |
| 1448 | covar beta | |
| 1449 | n nn m mm v dummy p) | |
| 1450 | ||
| 1451 | ;; Validate and parse arguments. | |
| 1452 | (or data | |
| 1453 | (if coefs | |
| 1454 | (setq data coefs | |
| 1455 | coefs nil) | |
| 1456 | (if (math-vectorp expr) | |
| 1457 | (if (memq (length expr) '(3 4)) | |
| 1458 | (setq data vars | |
| 1459 | vars (nth 2 expr) | |
| 1460 | coefs (nth 3 expr) | |
| 1461 | expr (nth 1 expr)) | |
| 1462 | (math-dimension-error)) | |
| 1463 | (setq data vars | |
| 1464 | vars nil | |
| 1465 | coefs nil)))) | |
| 1466 | (or (math-matrixp data) (math-reject-arg data 'matrixp)) | |
| 1467 | (setq v (1- (length data)) | |
| 1468 | n (1- (length (nth 1 data)))) | |
| 1469 | (or (math-vectorp vars) (null vars) | |
| 1470 | (setq vars (list 'vec vars))) | |
| 1471 | (or (math-vectorp coefs) (null coefs) | |
| 1472 | (setq coefs (list 'vec coefs))) | |
| 1473 | (or coefs | |
| 1474 | (setq coefs (cons 'vec (math-all-vars-but expr vars)))) | |
| 1475 | (or vars | |
| 1476 | (if (<= (1- (length coefs)) v) | |
| 1477 | (math-reject-arg coefs "*Not enough variables in model") | |
| 1478 | (setq coefs (copy-sequence coefs)) | |
| 1479 | (let ((p (nthcdr (- (length coefs) v | |
| 1480 | (if (eq (car-safe expr) 'calcFunc-eq) 1 0)) | |
| 1481 | coefs))) | |
| 1482 | (setq vars (cons 'vec (cdr p))) | |
| 1483 | (setcdr p nil)))) | |
| 1484 | (or (= (1- (length vars)) v) | |
| 1485 | (= (length vars) v) | |
| 1486 | (math-reject-arg vars "*Number of variables does not match data")) | |
| 1487 | (setq m (1- (length coefs))) | |
| 1488 | (if (< m 1) | |
| 1489 | (math-reject-arg coefs "*Need at least one parameter")) | |
| 1490 | ||
| 1491 | ;; Rewrite expr in terms of fitparam and fitvar, make into an equation. | |
| 1492 | (setq p coefs) | |
| 1493 | (while (setq p (cdr p)) | |
| 1494 | (or (eq (car-safe (car p)) 'var) | |
| 1495 | (math-reject-arg (car p) "*Expected a variable")) | |
| 1496 | (setq dummy (math-dummy-variable) | |
| 1497 | expr (math-expr-subst expr (car p) | |
| 1498 | (list 'calcFunc-fitparam | |
| 1499 | (- math-dummy-counter first-coef))))) | |
| 1500 | (setq first-var math-dummy-counter | |
| 1501 | p vars) | |
| 1502 | (while (setq p (cdr p)) | |
| 1503 | (or (eq (car-safe (car p)) 'var) | |
| 1504 | (math-reject-arg (car p) "*Expected a variable")) | |
| 1505 | (setq dummy (math-dummy-variable) | |
| 1506 | expr (math-expr-subst expr (car p) | |
| 1507 | (list 'calcFunc-fitvar | |
| 1508 | (- math-dummy-counter first-var))))) | |
| 1509 | (if (< math-dummy-counter (+ first-var v)) | |
| 1510 | (setq dummy (math-dummy-variable))) ; dependent variable may be unnamed | |
| 1511 | (setq y-dummy dummy | |
| 1512 | orig-expr expr) | |
| 1513 | (or (eq (car-safe expr) 'calcFunc-eq) | |
| 1514 | (setq expr (list 'calcFunc-eq (list 'calcFunc-fitvar v) expr))) | |
| 1515 | ||
| 1516 | (let ((calc-symbolic-mode nil)) | |
| 1517 | ||
| 1518 | ;; Apply rewrites to put expr into a linear-like form. | |
| 1519 | (setq expr (math-evaluate-expr expr) | |
| 1520 | expr (math-rewrite (list 'calcFunc-fitmodel expr) | |
| 1521 | '(var FitRules var-FitRules)) | |
| 1522 | math-in-fit 2 | |
| 1523 | expr (math-evaluate-expr expr)) | |
| 1524 | (or (and (eq (car-safe expr) 'calcFunc-fitsystem) | |
| 1525 | (= (length expr) 4) | |
| 1526 | (math-vectorp (nth 2 expr)) | |
| 1527 | (math-vectorp (nth 3 expr)) | |
| 1528 | (> (length (nth 2 expr)) 1) | |
| 1529 | (= (length (nth 3 expr)) (1+ m))) | |
| 1530 | (math-reject-arg plain-expr "*Model expression is too complex")) | |
| 1531 | (setq y-filter (nth 1 expr) | |
| 1532 | x-funcs (vconcat (cdr (nth 2 expr))) | |
| 1533 | coef-filters (nth 3 expr) | |
| 1534 | mm (length x-funcs)) | |
| 1535 | (if (equal y-filter y-dummy) | |
| 1536 | (setq y-filter nil)) | |
| 1537 | ||
| 1538 | ;; Build the (square) system of linear equations to be solved. | |
| 1539 | (setq beta (cons 'vec (make-list mm 0)) | |
| 1540 | covar (cons 'vec (mapcar 'copy-sequence (make-list mm beta)))) | |
| 1541 | (let* ((ptrs (vconcat (cdr data))) | |
| 1542 | (isigsq 1) | |
| 1543 | (xvals (make-vector mm 0)) | |
| 1544 | (i 0) | |
| 1545 | j k xval yval sigmasqr wt covj covjk covk betaj lud) | |
| 1546 | (while (<= (setq i (1+ i)) n) | |
| 1547 | ||
| 1548 | ;; Assign various independent variables for this data point. | |
| 1549 | (setq j 0 | |
| 1550 | sigmasqr nil) | |
| 1551 | (while (< j v) | |
| 1552 | (aset ptrs j (cdr (aref ptrs j))) | |
| 1553 | (setq xval (car (aref ptrs j))) | |
| 1554 | (if (= j (1- v)) | |
| 1555 | (if sigmasqr | |
| 1556 | (progn | |
| 1557 | (if (eq (car-safe xval) 'sdev) | |
| 1558 | (setq sigmasqr (math-add (math-sqr (nth 2 xval)) | |
| 1559 | sigmasqr) | |
| 1560 | xval (nth 1 xval))) | |
| 1561 | (if y-filter | |
| 1562 | (setq xval (math-make-sdev xval | |
| 1563 | (math-sqrt sigmasqr)))))) | |
| 1564 | (if (eq (car-safe xval) 'sdev) | |
| 1565 | (setq sigmasqr (math-add (math-sqr (nth 2 xval)) | |
| 1566 | (or sigmasqr 0)) | |
| 1567 | xval (nth 1 xval)))) | |
| 1568 | (set (nth 2 (aref math-dummy-vars (+ first-var j))) xval) | |
| 1569 | (setq j (1+ j))) | |
| 1570 | ||
| 1571 | ;; Compute Y value for this data point. | |
| 1572 | (if y-filter | |
| 1573 | (setq yval (math-evaluate-expr y-filter)) | |
| 1574 | (setq yval (symbol-value (nth 2 y-dummy)))) | |
| 1575 | (if (eq (car-safe yval) 'sdev) | |
| 1576 | (setq sigmasqr (math-sqr (nth 2 yval)) | |
| 1577 | yval (nth 1 yval))) | |
| 1578 | (if (= i 1) | |
| 1579 | (setq have-sdevs sigmasqr | |
| 1580 | need-chisq (or extended | |
| 1581 | (and (eq mode 'sdev) (not have-sdevs))))) | |
| 1582 | (if have-sdevs | |
| 1583 | (if sigmasqr | |
| 1584 | (progn | |
| 1585 | (setq isigsq (math-div 1 sigmasqr)) | |
| 1586 | (if need-chisq | |
| 1587 | (setq weights (cons isigsq weights)))) | |
| 1588 | (math-reject-arg yval "*Mixed error forms and plain numbers")) | |
| 1589 | (if sigmasqr | |
| 1590 | (math-reject-arg yval "*Mixed error forms and plain numbers"))) | |
| 1591 | ||
| 1592 | ;; Compute X values for this data point and update covar and beta. | |
| 1593 | (if (eq (car-safe xval) 'sdev) | |
| 1594 | (set (nth 2 y-dummy) (nth 1 xval))) | |
| 1595 | (setq j 0 | |
| 1596 | covj covar | |
| 1597 | betaj beta) | |
| 1598 | (while (< j mm) | |
| 1599 | (setq wt (math-evaluate-expr (aref x-funcs j))) | |
| 1600 | (aset xvals j wt) | |
| 1601 | (setq wt (math-mul wt isigsq) | |
| 1602 | betaj (cdr betaj) | |
| 1603 | covjk (car (setq covj (cdr covj))) | |
| 1604 | k 0) | |
| 1605 | (while (<= k j) | |
| 1606 | (setq covjk (cdr covjk)) | |
| 1607 | (setcar covjk (math-add (car covjk) | |
| 1608 | (math-mul wt (aref xvals k)))) | |
| 1609 | (setq k (1+ k))) | |
| 1610 | (setcar betaj (math-add (car betaj) (math-mul wt yval))) | |
| 1611 | (setq j (1+ j))) | |
| 1612 | (if need-chisq | |
| 1613 | (setq xy-values (cons (append xvals (list yval)) xy-values)))) | |
| 1614 | ||
| 1615 | ;; Fill in symmetric half of covar matrix. | |
| 1616 | (setq j 0 | |
| 1617 | covj covar) | |
| 1618 | (while (< j (1- mm)) | |
| 1619 | (setq k j | |
| 1620 | j (1+ j) | |
| 1621 | covjk (nthcdr j (car (setq covj (cdr covj)))) | |
| 1622 | covk (nthcdr j covar)) | |
| 1623 | (while (< (setq k (1+ k)) mm) | |
| 1624 | (setq covjk (cdr covjk) | |
| 1625 | covk (cdr covk)) | |
| 1626 | (setcar covjk (nth j (car covk)))))) | |
| 1627 | ||
| 1628 | ;; Solve the linear system. | |
| 1629 | (if mode | |
| 1630 | (progn | |
| 1631 | (setq covar (math-matrix-inv-raw covar)) | |
| 1632 | (if covar | |
| 1633 | (setq beta (math-mul covar beta)) | |
| 1634 | (if (math-zerop (math-abs beta)) | |
| 1635 | (setq covar (calcFunc-diag 0 (1- (length beta)))) | |
| 1636 | (math-reject-arg orig-expr "*Singular matrix"))) | |
| 1637 | (or (math-vectorp covar) | |
| 1638 | (setq covar (list 'vec (list 'vec covar))))) | |
| 1639 | (setq beta (math-div beta covar))) | |
| 1640 | ||
| 1641 | ;; Compute chi-square statistic if necessary. | |
| 1642 | (if need-chisq | |
| 1643 | (let (bp xp sum) | |
| 1644 | (setq chisq 0) | |
| 1645 | (while xy-values | |
| 1646 | (setq bp beta | |
| 1647 | xp (car xy-values) | |
| 1648 | sum 0) | |
| 1649 | (while (setq bp (cdr bp)) | |
| 1650 | (setq sum (math-add sum (math-mul (car bp) (car xp))) | |
| 1651 | xp (cdr xp))) | |
| 1652 | (setq sum (math-sqr (math-sub (car xp) sum))) | |
| 1653 | (if weights (setq sum (math-mul sum (car weights)))) | |
| 1654 | (setq chisq (math-add chisq sum) | |
| 1655 | weights (cdr weights) | |
| 1656 | xy-values (cdr xy-values))))) | |
| 1657 | ||
| 1658 | ;; Convert coefficients back into original terms. | |
| 1659 | (setq new-coefs (copy-sequence beta)) | |
| 1660 | (let* ((bp new-coefs) | |
| 1661 | (cp covar) | |
| 1662 | (sigdat 1) | |
| 1663 | (math-in-fit 3) | |
| 1664 | (j 0)) | |
| 1665 | (and mode (not have-sdevs) | |
| 1666 | (setq sigdat (if (<= n mm) | |
| 1667 | 0 | |
| 1668 | (math-div chisq (- n mm))))) | |
| 1669 | (if mode | |
| 1670 | (while (setq bp (cdr bp)) | |
| 1671 | (setcar bp (math-make-sdev | |
| 1672 | (car bp) | |
| 1673 | (math-sqrt (math-mul (nth (setq j (1+ j)) | |
| 1674 | (car (setq cp (cdr cp)))) | |
| 1675 | sigdat)))))) | |
| 1676 | (setq new-coefs (math-evaluate-expr coef-filters)) | |
| 1677 | (if calc-fit-to-trail | |
| 1678 | (let ((bp new-coefs) | |
| 1679 | (cp coefs) | |
| 1680 | (vec nil)) | |
| 1681 | (while (setq bp (cdr bp) cp (cdr cp)) | |
| 1682 | (setq vec (cons (list 'calcFunc-eq (car cp) (car bp)) vec))) | |
| 1683 | (setq calc-fit-to-trail (cons 'vec (nreverse vec))))))) | |
| 1684 | ||
| 1685 | ;; Substitute best-fit coefficients back into original formula. | |
| 1686 | (setq expr (math-multi-subst | |
| 1687 | orig-expr | |
| 1688 | (let ((n v) | |
| 1689 | (vec nil)) | |
| 1690 | (while (>= n 1) | |
| 1691 | (setq vec (cons (list 'calcFunc-fitvar n) vec) | |
| 1692 | n (1- n))) | |
| 1693 | (setq n m) | |
| 1694 | (while (>= n 1) | |
| 1695 | (setq vec (cons (list 'calcFunc-fitparam n) vec) | |
| 1696 | n (1- n))) | |
| 1697 | vec) | |
| 1698 | (append (cdr new-coefs) (cdr vars)))) | |
| 1699 | ||
| 1700 | ;; Package the result. | |
| 1701 | (math-normalize | |
| 1702 | (if extended | |
| 1703 | (list 'vec expr beta covar | |
| 1704 | (let ((p coef-filters) | |
| 1705 | (n 0)) | |
| 1706 | (while (and (setq n (1+ n) p (cdr p)) | |
| 1707 | (eq (car-safe (car p)) 'calcFunc-fitdummy) | |
| 1708 | (eq (nth 1 (car p)) n))) | |
| 1709 | (if p | |
| 1710 | coef-filters | |
| 1711 | (list 'vec))) | |
| 1712 | chisq | |
| 1713 | (if (and have-sdevs (> n mm)) | |
| 1714 | (list 'calcFunc-utpc chisq (- n mm)) | |
| 1715 | '(var nan var-nan))) | |
| bf77c646 | 1716 | expr)))) |
| 136211a9 | 1717 | |
| 136211a9 EZ |
1718 | |
| 1719 | (defun calcFunc-fitvar (x) | |
| 1720 | (if (>= math-in-fit 2) | |
| 1721 | (progn | |
| 1722 | (setq x (aref math-dummy-vars (+ first-var x -1))) | |
| 1723 | (or (calc-var-value (nth 2 x)) x)) | |
| bf77c646 | 1724 | (math-reject-arg x))) |
| 136211a9 EZ |
1725 | |
| 1726 | (defun calcFunc-fitparam (x) | |
| 1727 | (if (>= math-in-fit 2) | |
| 1728 | (progn | |
| 1729 | (setq x (aref math-dummy-vars (+ first-coef x -1))) | |
| 1730 | (or (calc-var-value (nth 2 x)) x)) | |
| bf77c646 | 1731 | (math-reject-arg x))) |
| 136211a9 EZ |
1732 | |
| 1733 | (defun calcFunc-fitdummy (x) | |
| 1734 | (if (= math-in-fit 3) | |
| 1735 | (nth x new-coefs) | |
| bf77c646 | 1736 | (math-reject-arg x))) |
| 136211a9 EZ |
1737 | |
| 1738 | (defun calcFunc-hasfitvars (expr) | |
| 1739 | (if (Math-primp expr) | |
| 1740 | 0 | |
| 1741 | (if (eq (car expr) 'calcFunc-fitvar) | |
| 1742 | (nth 1 expr) | |
| bf77c646 | 1743 | (apply 'max (mapcar 'calcFunc-hasfitvars (cdr expr)))))) |
| 136211a9 EZ |
1744 | |
| 1745 | (defun calcFunc-hasfitparams (expr) | |
| 1746 | (if (Math-primp expr) | |
| 1747 | 0 | |
| 1748 | (if (eq (car expr) 'calcFunc-fitparam) | |
| 1749 | (nth 1 expr) | |
| bf77c646 | 1750 | (apply 'max (mapcar 'calcFunc-hasfitparams (cdr expr)))))) |
| 136211a9 EZ |
1751 | |
| 1752 | ||
| 1753 | (defun math-all-vars-but (expr but) | |
| 1754 | (let* ((vars (math-all-vars-in expr)) | |
| 1755 | (p but)) | |
| 1756 | (while p | |
| 1757 | (setq vars (delq (assoc (car-safe p) vars) vars) | |
| 1758 | p (cdr p))) | |
| 1759 | (sort (mapcar 'car vars) | |
| bf77c646 | 1760 | (function (lambda (x y) (string< (nth 1 x) (nth 1 y))))))) |
| 136211a9 EZ |
1761 | |
| 1762 | (defun math-all-vars-in (expr) | |
| 1763 | (let ((vars nil) | |
| 1764 | found) | |
| 1765 | (math-all-vars-rec expr) | |
| bf77c646 | 1766 | vars)) |
| 136211a9 EZ |
1767 | |
| 1768 | (defun math-all-vars-rec (expr) | |
| 1769 | (if (Math-primp expr) | |
| 1770 | (if (eq (car-safe expr) 'var) | |
| 1771 | (or (math-const-var expr) | |
| 1772 | (if (setq found (assoc expr vars)) | |
| 1773 | (setcdr found (1+ (cdr found))) | |
| 1774 | (setq vars (cons (cons expr 1) vars))))) | |
| 1775 | (while (setq expr (cdr expr)) | |
| bf77c646 | 1776 | (math-all-vars-rec (car expr))))) |
| 136211a9 | 1777 | |
| ab5796a9 | 1778 | ;;; arch-tag: ff9f2920-8111-48b5-b3fa-b0682c3e44a6 |
| bf77c646 | 1779 | ;;; calcalg3.el ends here |