| 1 | ;;; calcalg3.el --- more algebraic functions for Calc |
| 2 | |
| 3 | ;; Copyright (C) 1990, 1991, 1992, 1993, 2001 Free Software Foundation, Inc. |
| 4 | |
| 5 | ;; Author: David Gillespie <daveg@synaptics.com> |
| 6 | ;; Maintainer: Jay Belanger <belanger@truman.edu> |
| 7 | |
| 8 | ;; This file is part of GNU Emacs. |
| 9 | |
| 10 | ;; GNU Emacs is distributed in the hope that it will be useful, |
| 11 | ;; but WITHOUT ANY WARRANTY. No author or distributor |
| 12 | ;; accepts responsibility to anyone for the consequences of using it |
| 13 | ;; or for whether it serves any particular purpose or works at all, |
| 14 | ;; unless he says so in writing. Refer to the GNU Emacs General Public |
| 15 | ;; License for full details. |
| 16 | |
| 17 | ;; Everyone is granted permission to copy, modify and redistribute |
| 18 | ;; GNU Emacs, but only under the conditions described in the |
| 19 | ;; GNU Emacs General Public License. A copy of this license is |
| 20 | ;; supposed to have been given to you along with GNU Emacs so you |
| 21 | ;; can know your rights and responsibilities. It should be in a |
| 22 | ;; file named COPYING. Among other things, the copyright notice |
| 23 | ;; and this notice must be preserved on all copies. |
| 24 | |
| 25 | ;;; Commentary: |
| 26 | |
| 27 | ;;; Code: |
| 28 | |
| 29 | ;; This file is autoloaded from calc-ext.el. |
| 30 | |
| 31 | (require 'calc-ext) |
| 32 | (require 'calc-macs) |
| 33 | |
| 34 | (defun calc-find-root (var) |
| 35 | (interactive "sVariable(s) to solve for: ") |
| 36 | (calc-slow-wrapper |
| 37 | (let ((func (if (calc-is-hyperbolic) 'calcFunc-wroot 'calcFunc-root))) |
| 38 | (if (or (equal var "") (equal var "$")) |
| 39 | (calc-enter-result 2 "root" (list func |
| 40 | (calc-top-n 3) |
| 41 | (calc-top-n 1) |
| 42 | (calc-top-n 2))) |
| 43 | (let ((var (if (and (string-match ",\\|[^ ] +[^ ]" var) |
| 44 | (not (string-match "\\[" var))) |
| 45 | (math-read-expr (concat "[" var "]")) |
| 46 | (math-read-expr var)))) |
| 47 | (if (eq (car-safe var) 'error) |
| 48 | (error "Bad format in expression: %s" (nth 1 var))) |
| 49 | (calc-enter-result 1 "root" (list func |
| 50 | (calc-top-n 2) |
| 51 | var |
| 52 | (calc-top-n 1)))))))) |
| 53 | |
| 54 | (defun calc-find-minimum (var) |
| 55 | (interactive "sVariable(s) to minimize over: ") |
| 56 | (calc-slow-wrapper |
| 57 | (let ((func (if (calc-is-inverse) |
| 58 | (if (calc-is-hyperbolic) |
| 59 | 'calcFunc-wmaximize 'calcFunc-maximize) |
| 60 | (if (calc-is-hyperbolic) |
| 61 | 'calcFunc-wminimize 'calcFunc-minimize))) |
| 62 | (tag (if (calc-is-inverse) "max" "min"))) |
| 63 | (if (or (equal var "") (equal var "$")) |
| 64 | (calc-enter-result 2 tag (list func |
| 65 | (calc-top-n 3) |
| 66 | (calc-top-n 1) |
| 67 | (calc-top-n 2))) |
| 68 | (let ((var (if (and (string-match ",\\|[^ ] +[^ ]" var) |
| 69 | (not (string-match "\\[" var))) |
| 70 | (math-read-expr (concat "[" var "]")) |
| 71 | (math-read-expr var)))) |
| 72 | (if (eq (car-safe var) 'error) |
| 73 | (error "Bad format in expression: %s" (nth 1 var))) |
| 74 | (calc-enter-result 1 tag (list func |
| 75 | (calc-top-n 2) |
| 76 | var |
| 77 | (calc-top-n 1)))))))) |
| 78 | |
| 79 | (defun calc-find-maximum (var) |
| 80 | (interactive "sVariable to maximize over: ") |
| 81 | (calc-invert-func) |
| 82 | (calc-find-minimum var)) |
| 83 | |
| 84 | |
| 85 | (defun calc-poly-interp (arg) |
| 86 | (interactive "P") |
| 87 | (calc-slow-wrapper |
| 88 | (let ((data (calc-top 2))) |
| 89 | (if (or (consp arg) (eq arg 0) (eq arg 2)) |
| 90 | (setq data (cons 'vec (calc-top-list 2 2))) |
| 91 | (or (null arg) |
| 92 | (error "Bad prefix argument"))) |
| 93 | (if (calc-is-hyperbolic) |
| 94 | (calc-enter-result 1 "rati" (list 'calcFunc-ratint data (calc-top 1))) |
| 95 | (calc-enter-result 1 "poli" (list 'calcFunc-polint data |
| 96 | (calc-top 1))))))) |
| 97 | |
| 98 | ;; The variables calc-curve-nvars, calc-curve-varnames, calc-curve-model and calc-curve-coefnames are local to calc-curve-fit, but are |
| 99 | ;; used by calc-get-fit-variables which is called by calc-curve-fit. |
| 100 | (defvar calc-curve-nvars) |
| 101 | (defvar calc-curve-varnames) |
| 102 | (defvar calc-curve-model) |
| 103 | (defvar calc-curve-coefnames) |
| 104 | |
| 105 | (defun calc-curve-fit (arg &optional calc-curve-model |
| 106 | calc-curve-coefnames calc-curve-varnames) |
| 107 | (interactive "P") |
| 108 | (calc-slow-wrapper |
| 109 | (setq calc-aborted-prefix nil) |
| 110 | (let ((func (if (calc-is-inverse) 'calcFunc-xfit |
| 111 | (if (calc-is-hyperbolic) 'calcFunc-efit |
| 112 | 'calcFunc-fit))) |
| 113 | key (which 0) |
| 114 | n calc-curve-nvars temp data |
| 115 | (homog nil) |
| 116 | (msgs '( "(Press ? for help)" |
| 117 | "1 = linear or multilinear" |
| 118 | "2-9 = polynomial fits; i = interpolating polynomial" |
| 119 | "p = a x^b, ^ = a b^x" |
| 120 | "e = a exp(b x), x = exp(a + b x), l = a + b ln(x)" |
| 121 | "E = a 10^(b x), X = 10^(a + b x), L = a + b log10(x)" |
| 122 | "q = a + b (x-c)^2" |
| 123 | "g = (a/b sqrt(2 pi)) exp(-0.5*((x-c)/b)^2)" |
| 124 | "h prefix = homogeneous model (no constant term)" |
| 125 | "' = alg entry, $ = stack, u = Model1, U = Model2"))) |
| 126 | (while (not calc-curve-model) |
| 127 | (message "Fit to model: %s:%s" |
| 128 | (nth which msgs) |
| 129 | (if homog " h" "")) |
| 130 | (setq key (read-char)) |
| 131 | (cond ((= key ?\C-g) |
| 132 | (keyboard-quit)) |
| 133 | ((= key ??) |
| 134 | (setq which (% (1+ which) (length msgs)))) |
| 135 | ((memq key '(?h ?H)) |
| 136 | (setq homog (not homog))) |
| 137 | ((progn |
| 138 | (if (eq key ?\$) |
| 139 | (setq n 1) |
| 140 | (setq n 0)) |
| 141 | (cond ((null arg) |
| 142 | (setq n (1+ n) |
| 143 | data (calc-top n))) |
| 144 | ((or (consp arg) (eq arg 0)) |
| 145 | (setq n (+ n 2) |
| 146 | data (calc-top n) |
| 147 | data (if (math-matrixp data) |
| 148 | (append data (list (calc-top (1- n)))) |
| 149 | (list 'vec data (calc-top (1- n)))))) |
| 150 | ((> (setq arg (prefix-numeric-value arg)) 0) |
| 151 | (setq data (cons 'vec (calc-top-list arg (1+ n))) |
| 152 | n (+ n arg))) |
| 153 | (t (error "Bad prefix argument"))) |
| 154 | (or (math-matrixp data) (not (cdr (cdr data))) |
| 155 | (error "Data matrix is not a matrix!")) |
| 156 | (setq calc-curve-nvars (- (length data) 2) |
| 157 | calc-curve-coefnames nil |
| 158 | calc-curve-varnames nil) |
| 159 | nil)) |
| 160 | ((= key ?1) ; linear or multilinear |
| 161 | (calc-get-fit-variables calc-curve-nvars |
| 162 | (1+ calc-curve-nvars) (and homog 0)) |
| 163 | (setq calc-curve-model (math-mul calc-curve-coefnames |
| 164 | (cons 'vec (cons 1 (cdr calc-curve-varnames)))))) |
| 165 | ((and (>= key ?2) (<= key ?9)) ; polynomial |
| 166 | (calc-get-fit-variables 1 (- key ?0 -1) (and homog 0)) |
| 167 | (setq calc-curve-model |
| 168 | (math-build-polynomial-expr (cdr calc-curve-coefnames) |
| 169 | (nth 1 calc-curve-varnames)))) |
| 170 | ((= key ?i) ; exact polynomial |
| 171 | (calc-get-fit-variables 1 (1- (length (nth 1 data))) |
| 172 | (and homog 0)) |
| 173 | (setq calc-curve-model |
| 174 | (math-build-polynomial-expr (cdr calc-curve-coefnames) |
| 175 | (nth 1 calc-curve-varnames)))) |
| 176 | ((= key ?p) ; power law |
| 177 | (calc-get-fit-variables calc-curve-nvars |
| 178 | (1+ calc-curve-nvars) (and homog 1)) |
| 179 | (setq calc-curve-model (math-mul (nth 1 calc-curve-coefnames) |
| 180 | (calcFunc-reduce |
| 181 | '(var mul var-mul) |
| 182 | (calcFunc-map |
| 183 | '(var pow var-pow) |
| 184 | calc-curve-varnames |
| 185 | (cons 'vec (cdr (cdr calc-curve-coefnames)))))))) |
| 186 | ((= key ?^) ; exponential law |
| 187 | (calc-get-fit-variables calc-curve-nvars |
| 188 | (1+ calc-curve-nvars) (and homog 1)) |
| 189 | (setq calc-curve-model (math-mul (nth 1 calc-curve-coefnames) |
| 190 | (calcFunc-reduce |
| 191 | '(var mul var-mul) |
| 192 | (calcFunc-map |
| 193 | '(var pow var-pow) |
| 194 | (cons 'vec (cdr (cdr calc-curve-coefnames))) |
| 195 | calc-curve-varnames))))) |
| 196 | ((memq key '(?e ?E)) |
| 197 | (calc-get-fit-variables calc-curve-nvars |
| 198 | (1+ calc-curve-nvars) (and homog 1)) |
| 199 | (setq calc-curve-model (math-mul (nth 1 calc-curve-coefnames) |
| 200 | (calcFunc-reduce |
| 201 | '(var mul var-mul) |
| 202 | (calcFunc-map |
| 203 | (if (eq key ?e) |
| 204 | '(var exp var-exp) |
| 205 | '(calcFunc-lambda |
| 206 | (var a var-a) |
| 207 | (^ 10 (var a var-a)))) |
| 208 | (calcFunc-map |
| 209 | '(var mul var-mul) |
| 210 | (cons 'vec (cdr (cdr calc-curve-coefnames))) |
| 211 | calc-curve-varnames)))))) |
| 212 | ((memq key '(?x ?X)) |
| 213 | (calc-get-fit-variables calc-curve-nvars |
| 214 | (1+ calc-curve-nvars) (and homog 0)) |
| 215 | (setq calc-curve-model (math-mul calc-curve-coefnames |
| 216 | (cons 'vec (cons 1 (cdr calc-curve-varnames))))) |
| 217 | (setq calc-curve-model (if (eq key ?x) |
| 218 | (list 'calcFunc-exp calc-curve-model) |
| 219 | (list '^ 10 calc-curve-model)))) |
| 220 | ((memq key '(?l ?L)) |
| 221 | (calc-get-fit-variables calc-curve-nvars |
| 222 | (1+ calc-curve-nvars) (and homog 0)) |
| 223 | (setq calc-curve-model (math-mul calc-curve-coefnames |
| 224 | (cons 'vec |
| 225 | (cons 1 (cdr (calcFunc-map |
| 226 | (if (eq key ?l) |
| 227 | '(var ln var-ln) |
| 228 | '(var log10 |
| 229 | var-log10)) |
| 230 | calc-curve-varnames))))))) |
| 231 | ((= key ?q) |
| 232 | (calc-get-fit-variables calc-curve-nvars |
| 233 | (1+ (* 2 calc-curve-nvars)) (and homog 0)) |
| 234 | (let ((c calc-curve-coefnames) |
| 235 | (v calc-curve-varnames)) |
| 236 | (setq calc-curve-model (nth 1 c)) |
| 237 | (while (setq v (cdr v) c (cdr (cdr c))) |
| 238 | (setq calc-curve-model (math-add |
| 239 | calc-curve-model |
| 240 | (list '* |
| 241 | (car c) |
| 242 | (list '^ |
| 243 | (list '- (car v) (nth 1 c)) |
| 244 | 2))))))) |
| 245 | ((= key ?g) |
| 246 | (setq calc-curve-model |
| 247 | (math-read-expr "(AFit / BFit sqrt(2 pi)) exp(-0.5 * ((XFit - CFit) / BFit)^2)") |
| 248 | calc-curve-varnames '(vec (var XFit var-XFit)) |
| 249 | calc-curve-coefnames '(vec (var AFit var-AFit) |
| 250 | (var BFit var-BFit) |
| 251 | (var CFit var-CFit))) |
| 252 | (calc-get-fit-variables 1 (1- (length calc-curve-coefnames)) |
| 253 | (and homog 1))) |
| 254 | ((memq key '(?\$ ?\' ?u ?U)) |
| 255 | (let* ((defvars nil) |
| 256 | (record-entry nil)) |
| 257 | (if (eq key ?\') |
| 258 | (let* ((calc-dollar-values calc-arg-values) |
| 259 | (calc-dollar-used 0) |
| 260 | (calc-hashes-used 0)) |
| 261 | (setq calc-curve-model (calc-do-alg-entry "" "Model formula: ")) |
| 262 | (if (/= (length calc-curve-model) 1) |
| 263 | (error "Bad format")) |
| 264 | (setq calc-curve-model (car calc-curve-model) |
| 265 | record-entry t) |
| 266 | (if (> calc-dollar-used 0) |
| 267 | (setq calc-curve-coefnames |
| 268 | (cons 'vec |
| 269 | (nthcdr (- (length calc-arg-values) |
| 270 | calc-dollar-used) |
| 271 | (reverse calc-arg-values)))) |
| 272 | (if (> calc-hashes-used 0) |
| 273 | (setq calc-curve-coefnames |
| 274 | (cons 'vec (calc-invent-args |
| 275 | calc-hashes-used)))))) |
| 276 | (progn |
| 277 | (setq calc-curve-model (cond ((eq key ?u) |
| 278 | (calc-var-value 'var-Model1)) |
| 279 | ((eq key ?U) |
| 280 | (calc-var-value 'var-Model2)) |
| 281 | (t (calc-top 1)))) |
| 282 | (or calc-curve-model (error "User model not yet defined")) |
| 283 | (if (math-vectorp calc-curve-model) |
| 284 | (if (and (memq (length calc-curve-model) '(3 4)) |
| 285 | (not (math-objvecp (nth 1 calc-curve-model))) |
| 286 | (math-vectorp (nth 2 calc-curve-model)) |
| 287 | (or (null (nth 3 calc-curve-model)) |
| 288 | (math-vectorp (nth 3 calc-curve-model)))) |
| 289 | (setq calc-curve-varnames (nth 2 calc-curve-model) |
| 290 | calc-curve-coefnames |
| 291 | (or (nth 3 calc-curve-model) |
| 292 | (cons 'vec |
| 293 | (math-all-vars-but |
| 294 | calc-curve-model calc-curve-varnames))) |
| 295 | calc-curve-model (nth 1 calc-curve-model)) |
| 296 | (error "Incorrect model specifier"))))) |
| 297 | (or calc-curve-varnames |
| 298 | (let ((with-y (eq (car-safe calc-curve-model) 'calcFunc-eq))) |
| 299 | (if calc-curve-coefnames |
| 300 | (calc-get-fit-variables |
| 301 | (if with-y (1+ calc-curve-nvars) calc-curve-nvars) |
| 302 | (1- (length calc-curve-coefnames)) |
| 303 | (math-all-vars-but |
| 304 | calc-curve-model calc-curve-coefnames) |
| 305 | nil with-y) |
| 306 | (let* ((coefs (math-all-vars-but calc-curve-model nil)) |
| 307 | (vars nil) |
| 308 | (n (- (length coefs) calc-curve-nvars (if with-y 2 1))) |
| 309 | p) |
| 310 | (if (< n 0) |
| 311 | (error "Not enough variables in model")) |
| 312 | (setq p (nthcdr n coefs)) |
| 313 | (setq vars (cdr p)) |
| 314 | (setcdr p nil) |
| 315 | (calc-get-fit-variables |
| 316 | (if with-y (1+ calc-curve-nvars) calc-curve-nvars) |
| 317 | (length coefs) |
| 318 | vars coefs with-y))))) |
| 319 | (if record-entry |
| 320 | (calc-record (list 'vec calc-curve-model |
| 321 | calc-curve-varnames calc-curve-coefnames) |
| 322 | "modl")))) |
| 323 | (t (beep)))) |
| 324 | (let ((calc-fit-to-trail t)) |
| 325 | (calc-enter-result n (substring (symbol-name func) 9) |
| 326 | (list func calc-curve-model |
| 327 | (if (= (length calc-curve-varnames) 2) |
| 328 | (nth 1 calc-curve-varnames) |
| 329 | calc-curve-varnames) |
| 330 | (if (= (length calc-curve-coefnames) 2) |
| 331 | (nth 1 calc-curve-coefnames) |
| 332 | calc-curve-coefnames) |
| 333 | data)) |
| 334 | (if (consp calc-fit-to-trail) |
| 335 | (calc-record (calc-normalize calc-fit-to-trail) "parm")))))) |
| 336 | |
| 337 | (defun calc-invent-independent-variables (n &optional but) |
| 338 | (calc-invent-variables n but '(x y z t) "x")) |
| 339 | |
| 340 | (defun calc-invent-parameter-variables (n &optional but) |
| 341 | (calc-invent-variables n but '(a b c d) "a")) |
| 342 | |
| 343 | (defun calc-invent-variables (num but names base) |
| 344 | (let ((vars nil) |
| 345 | (n num) (nn 0) |
| 346 | var) |
| 347 | (while (and (> n 0) names) |
| 348 | (setq var (math-build-var-name (if (consp names) |
| 349 | (car names) |
| 350 | (concat base (int-to-string |
| 351 | (setq nn (1+ nn))))))) |
| 352 | (or (math-expr-contains (cons 'vec but) var) |
| 353 | (setq vars (cons var vars) |
| 354 | n (1- n))) |
| 355 | (or (symbolp names) (setq names (cdr names)))) |
| 356 | (if (= n 0) |
| 357 | (nreverse vars) |
| 358 | (calc-invent-variables num but t base)))) |
| 359 | |
| 360 | (defun calc-get-fit-variables (nv nc &optional defv defc with-y homog) |
| 361 | (or (= nv (if with-y (1+ calc-curve-nvars) calc-curve-nvars)) |
| 362 | (error "Wrong number of data vectors for this type of model")) |
| 363 | (if (integerp defv) |
| 364 | (setq homog defv |
| 365 | defv nil)) |
| 366 | (if homog |
| 367 | (setq nc (1- nc))) |
| 368 | (or defv |
| 369 | (setq defv (calc-invent-independent-variables nv))) |
| 370 | (or defc |
| 371 | (setq defc (calc-invent-parameter-variables nc defv))) |
| 372 | (let ((vars (read-string (format "Fitting variables: (default %s; %s) " |
| 373 | (mapconcat 'symbol-name |
| 374 | (mapcar (function (lambda (v) |
| 375 | (nth 1 v))) |
| 376 | defv) |
| 377 | ",") |
| 378 | (mapconcat 'symbol-name |
| 379 | (mapcar (function (lambda (v) |
| 380 | (nth 1 v))) |
| 381 | defc) |
| 382 | ",")))) |
| 383 | (coefs nil)) |
| 384 | (setq vars (if (string-match "\\[" vars) |
| 385 | (math-read-expr vars) |
| 386 | (math-read-expr (concat "[" vars "]")))) |
| 387 | (if (eq (car-safe vars) 'error) |
| 388 | (error "Bad format in expression: %s" (nth 2 vars))) |
| 389 | (or (math-vectorp vars) |
| 390 | (error "Expected a variable or vector of variables")) |
| 391 | (if (equal vars '(vec)) |
| 392 | (setq vars (cons 'vec defv) |
| 393 | coefs (cons 'vec defc)) |
| 394 | (if (math-vectorp (nth 1 vars)) |
| 395 | (if (and (= (length vars) 3) |
| 396 | (math-vectorp (nth 2 vars))) |
| 397 | (setq coefs (nth 2 vars) |
| 398 | vars (nth 1 vars)) |
| 399 | (error |
| 400 | "Expected independent variables vector, then parameters vector")) |
| 401 | (setq coefs (cons 'vec defc)))) |
| 402 | (or (= nv (1- (length vars))) |
| 403 | (and (not with-y) (= (1+ nv) (1- (length vars)))) |
| 404 | (error "Expected %d independent variable%s" nv (if (= nv 1) "" "s"))) |
| 405 | (or (= nc (1- (length coefs))) |
| 406 | (error "Expected %d parameter variable%s" nc (if (= nc 1) "" "s"))) |
| 407 | (if homog |
| 408 | (setq coefs (cons 'vec (cons homog (cdr coefs))))) |
| 409 | (if calc-curve-varnames |
| 410 | (setq calc-curve-model (math-multi-subst calc-curve-model (cdr calc-curve-varnames) (cdr vars)))) |
| 411 | (if calc-curve-coefnames |
| 412 | (setq calc-curve-model (math-multi-subst calc-curve-model (cdr calc-curve-coefnames) (cdr coefs)))) |
| 413 | (setq calc-curve-varnames vars |
| 414 | calc-curve-coefnames coefs))) |
| 415 | |
| 416 | |
| 417 | |
| 418 | |
| 419 | ;;; The following algorithms are from Numerical Recipes chapter 9. |
| 420 | |
| 421 | ;;; "rtnewt" with safety kludges |
| 422 | |
| 423 | (defvar var-DUMMY) |
| 424 | |
| 425 | (defun math-newton-root (expr deriv guess orig-guess limit) |
| 426 | (math-working "newton" guess) |
| 427 | (let* ((var-DUMMY guess) |
| 428 | next dval) |
| 429 | (setq next (math-evaluate-expr expr) |
| 430 | dval (math-evaluate-expr deriv)) |
| 431 | (if (and (Math-numberp next) |
| 432 | (Math-numberp dval) |
| 433 | (not (Math-zerop dval))) |
| 434 | (progn |
| 435 | (setq next (math-sub guess (math-div next dval))) |
| 436 | (if (math-nearly-equal guess (setq next (math-float next))) |
| 437 | (progn |
| 438 | (setq var-DUMMY next) |
| 439 | (list 'vec next (math-evaluate-expr expr))) |
| 440 | (if (Math-lessp (math-abs-approx (math-sub next orig-guess)) |
| 441 | limit) |
| 442 | (math-newton-root expr deriv next orig-guess limit) |
| 443 | (math-reject-arg next "*Newton's method failed to converge")))) |
| 444 | (math-reject-arg next "*Newton's method encountered a singularity")))) |
| 445 | |
| 446 | ;;; Inspired by "rtsafe" |
| 447 | (defun math-newton-search-root (expr deriv guess vguess ostep oostep |
| 448 | low vlow high vhigh) |
| 449 | (let ((var-DUMMY guess) |
| 450 | (better t) |
| 451 | pos step next vnext) |
| 452 | (if guess |
| 453 | (math-working "newton" (list 'intv 0 low high)) |
| 454 | (math-working "bisect" (list 'intv 0 low high)) |
| 455 | (setq ostep (math-mul-float (math-sub-float high low) |
| 456 | '(float 5 -1)) |
| 457 | guess (math-add-float low ostep) |
| 458 | var-DUMMY guess |
| 459 | vguess (math-evaluate-expr expr)) |
| 460 | (or (Math-realp vguess) |
| 461 | (progn |
| 462 | (setq ostep (math-mul-float ostep '(float 6 -1)) |
| 463 | guess (math-add-float low ostep) |
| 464 | var-DUMMY guess |
| 465 | vguess (math-evaluate-expr expr)) |
| 466 | (or (math-realp vguess) |
| 467 | (progn |
| 468 | (setq ostep (math-mul-float ostep '(float 123456 -5)) |
| 469 | guess (math-add-float low ostep) |
| 470 | var-DUMMY guess |
| 471 | vguess nil)))))) |
| 472 | (or vguess |
| 473 | (setq vguess (math-evaluate-expr expr))) |
| 474 | (or (Math-realp vguess) |
| 475 | (math-reject-arg guess "*Newton's method encountered a singularity")) |
| 476 | (setq vguess (math-float vguess)) |
| 477 | (if (eq (Math-negp vlow) (setq pos (Math-posp vguess))) |
| 478 | (setq high guess |
| 479 | vhigh vguess) |
| 480 | (if (eq (Math-negp vhigh) pos) |
| 481 | (setq low guess |
| 482 | vlow vguess) |
| 483 | (setq better nil))) |
| 484 | (if (or (Math-zerop vguess) |
| 485 | (math-nearly-equal low high)) |
| 486 | (list 'vec guess vguess) |
| 487 | (setq step (math-evaluate-expr deriv)) |
| 488 | (if (and (Math-realp step) |
| 489 | (not (Math-zerop step)) |
| 490 | (setq step (math-div-float vguess (math-float step)) |
| 491 | next (math-sub-float guess step)) |
| 492 | (not (math-lessp-float high next)) |
| 493 | (not (math-lessp-float next low))) |
| 494 | (progn |
| 495 | (setq var-DUMMY next |
| 496 | vnext (math-evaluate-expr expr)) |
| 497 | (if (or (Math-zerop vnext) |
| 498 | (math-nearly-equal next guess)) |
| 499 | (list 'vec next vnext) |
| 500 | (if (and better |
| 501 | (math-lessp-float (math-abs (or oostep |
| 502 | (math-sub-float |
| 503 | high low))) |
| 504 | (math-abs |
| 505 | (math-mul-float '(float 2 0) |
| 506 | step)))) |
| 507 | (math-newton-search-root expr deriv nil nil nil ostep |
| 508 | low vlow high vhigh) |
| 509 | (math-newton-search-root expr deriv next vnext step ostep |
| 510 | low vlow high vhigh)))) |
| 511 | (if (or (and (Math-posp vlow) (Math-posp vhigh)) |
| 512 | (and (Math-negp vlow) (Math-negp vhigh))) |
| 513 | (math-search-root expr deriv low vlow high vhigh) |
| 514 | (math-newton-search-root expr deriv nil nil nil ostep |
| 515 | low vlow high vhigh)))))) |
| 516 | |
| 517 | ;;; Search for a root in an interval with no overt zero crossing. |
| 518 | |
| 519 | ;; The variable math-root-widen is local to math-find-root, but |
| 520 | ;; is used by math-search-root, which is called (directly and |
| 521 | ;; indirectly) by math-find-root. |
| 522 | (defvar math-root-widen) |
| 523 | |
| 524 | (defun math-search-root (expr deriv low vlow high vhigh) |
| 525 | (let (found) |
| 526 | (if math-root-widen |
| 527 | (let ((iters 0) |
| 528 | (iterlim (if (eq math-root-widen 'point) |
| 529 | (+ calc-internal-prec 10) |
| 530 | 20)) |
| 531 | (factor (if (eq math-root-widen 'point) |
| 532 | '(float 9 0) |
| 533 | '(float 16 -1))) |
| 534 | (prev nil) vprev waslow |
| 535 | diff) |
| 536 | (while (or (and (math-posp vlow) (math-posp vhigh)) |
| 537 | (and (math-negp vlow) (math-negp vhigh))) |
| 538 | (math-working "widen" (list 'intv 0 low high)) |
| 539 | (if (> (setq iters (1+ iters)) iterlim) |
| 540 | (math-reject-arg (list 'intv 0 low high) |
| 541 | "*Unable to bracket root")) |
| 542 | (if (= iters calc-internal-prec) |
| 543 | (setq factor '(float 16 -1))) |
| 544 | (setq diff (math-mul-float (math-sub-float high low) factor)) |
| 545 | (if (Math-zerop diff) |
| 546 | (setq high (calcFunc-incr high 10)) |
| 547 | (if (math-lessp-float (math-abs vlow) (math-abs vhigh)) |
| 548 | (setq waslow t |
| 549 | prev low |
| 550 | low (math-sub low diff) |
| 551 | var-DUMMY low |
| 552 | vprev vlow |
| 553 | vlow (math-evaluate-expr expr)) |
| 554 | (setq waslow nil |
| 555 | prev high |
| 556 | high (math-add high diff) |
| 557 | var-DUMMY high |
| 558 | vprev vhigh |
| 559 | vhigh (math-evaluate-expr expr))))) |
| 560 | (if prev |
| 561 | (if waslow |
| 562 | (setq high prev vhigh vprev) |
| 563 | (setq low prev vlow vprev))) |
| 564 | (setq found t)) |
| 565 | (or (Math-realp vlow) |
| 566 | (math-reject-arg vlow 'realp)) |
| 567 | (or (Math-realp vhigh) |
| 568 | (math-reject-arg vhigh 'realp)) |
| 569 | (let ((xvals (list low high)) |
| 570 | (yvals (list vlow vhigh)) |
| 571 | (pos (Math-posp vlow)) |
| 572 | (levels 0) |
| 573 | (step (math-sub-float high low)) |
| 574 | xp yp var-DUMMY) |
| 575 | (while (and (<= (setq levels (1+ levels)) 5) |
| 576 | (not found)) |
| 577 | (setq xp xvals |
| 578 | yp yvals |
| 579 | step (math-mul-float step '(float 497 -3))) |
| 580 | (while (and (cdr xp) (not found)) |
| 581 | (if (Math-realp (car yp)) |
| 582 | (setq low (car xp) |
| 583 | vlow (car yp))) |
| 584 | (setq high (math-add-float (car xp) step) |
| 585 | var-DUMMY high |
| 586 | vhigh (math-evaluate-expr expr)) |
| 587 | (math-working "search" high) |
| 588 | (if (and (Math-realp vhigh) |
| 589 | (eq (math-negp vhigh) pos)) |
| 590 | (setq found t) |
| 591 | (setcdr xp (cons high (cdr xp))) |
| 592 | (setcdr yp (cons vhigh (cdr yp))) |
| 593 | (setq xp (cdr (cdr xp)) |
| 594 | yp (cdr (cdr yp)))))))) |
| 595 | (if found |
| 596 | (if (Math-zerop vhigh) |
| 597 | (list 'vec high vhigh) |
| 598 | (if (Math-zerop vlow) |
| 599 | (list 'vec low vlow) |
| 600 | (if deriv |
| 601 | (math-newton-search-root expr deriv nil nil nil nil |
| 602 | low vlow high vhigh) |
| 603 | (math-bisect-root expr low vlow high vhigh)))) |
| 604 | (math-reject-arg (list 'intv 3 low high) |
| 605 | "*Unable to find a sign change in this interval")))) |
| 606 | |
| 607 | ;;; "rtbis" (but we should be using Brent's method) |
| 608 | (defun math-bisect-root (expr low vlow high vhigh) |
| 609 | (let ((step (math-sub-float high low)) |
| 610 | (pos (Math-posp vhigh)) |
| 611 | var-DUMMY |
| 612 | mid vmid) |
| 613 | (while (not (or (math-nearly-equal low |
| 614 | (setq step (math-mul-float |
| 615 | step '(float 5 -1)) |
| 616 | mid (math-add-float low step))) |
| 617 | (progn |
| 618 | (setq var-DUMMY mid |
| 619 | vmid (math-evaluate-expr expr)) |
| 620 | (Math-zerop vmid)))) |
| 621 | (math-working "bisect" mid) |
| 622 | (if (eq (Math-posp vmid) pos) |
| 623 | (setq high mid |
| 624 | vhigh vmid) |
| 625 | (setq low mid |
| 626 | vlow vmid))) |
| 627 | (list 'vec mid vmid))) |
| 628 | |
| 629 | ;;; "mnewt" |
| 630 | |
| 631 | (defvar math-root-vars [(var DUMMY var-DUMMY)]) |
| 632 | |
| 633 | (defun math-newton-multi (expr jacob n guess orig-guess limit) |
| 634 | (let ((m -1) |
| 635 | (p guess) |
| 636 | p2 expr-val jacob-val next) |
| 637 | (while (< (setq p (cdr p) m (1+ m)) n) |
| 638 | (set (nth 2 (aref math-root-vars m)) (car p))) |
| 639 | (setq expr-val (math-evaluate-expr expr) |
| 640 | jacob-val (math-evaluate-expr jacob)) |
| 641 | (unless (and (math-constp expr-val) |
| 642 | (math-constp jacob-val)) |
| 643 | (math-reject-arg guess "*Newton's method encountered a singularity")) |
| 644 | (setq next (math-add guess (math-div (math-float (math-neg expr-val)) |
| 645 | (math-float jacob-val))) |
| 646 | p guess p2 next) |
| 647 | (math-working "newton" next) |
| 648 | (while (and (setq p (cdr p) p2 (cdr p2)) |
| 649 | (math-nearly-equal (car p) (car p2)))) |
| 650 | (if p |
| 651 | (if (Math-lessp (math-abs-approx (math-sub next orig-guess)) |
| 652 | limit) |
| 653 | (math-newton-multi expr jacob n next orig-guess limit) |
| 654 | (math-reject-arg nil "*Newton's method failed to converge")) |
| 655 | (list 'vec next expr-val)))) |
| 656 | |
| 657 | |
| 658 | (defun math-find-root (expr var guess math-root-widen) |
| 659 | (if (eq (car-safe expr) 'vec) |
| 660 | (let ((n (1- (length expr))) |
| 661 | (calc-symbolic-mode nil) |
| 662 | (var-DUMMY nil) |
| 663 | (jacob (list 'vec)) |
| 664 | p p2 m row) |
| 665 | (unless (eq (car-safe var) 'vec) |
| 666 | (math-reject-arg var 'vectorp)) |
| 667 | (unless (= (length var) (1+ n)) |
| 668 | (math-dimension-error)) |
| 669 | (setq expr (copy-sequence expr)) |
| 670 | (while (>= n (length math-root-vars)) |
| 671 | (let ((symb (intern (concat "math-root-v" |
| 672 | (int-to-string |
| 673 | (length math-root-vars)))))) |
| 674 | (setq math-root-vars (vconcat math-root-vars |
| 675 | (vector (list 'var symb symb)))))) |
| 676 | (setq m -1) |
| 677 | (while (< (setq m (1+ m)) n) |
| 678 | (set (nth 2 (aref math-root-vars m)) nil)) |
| 679 | (setq m -1 p var) |
| 680 | (while (setq m (1+ m) p (cdr p)) |
| 681 | (or (eq (car-safe (car p)) 'var) |
| 682 | (math-reject-arg var "*Expected a variable")) |
| 683 | (setq p2 expr) |
| 684 | (while (setq p2 (cdr p2)) |
| 685 | (setcar p2 (math-expr-subst (car p2) (car p) |
| 686 | (aref math-root-vars m))))) |
| 687 | (unless (eq (car-safe guess) 'vec) |
| 688 | (math-reject-arg guess 'vectorp)) |
| 689 | (unless (= (length guess) (1+ n)) |
| 690 | (math-dimension-error)) |
| 691 | (setq guess (copy-sequence guess) |
| 692 | p guess) |
| 693 | (while (setq p (cdr p)) |
| 694 | (or (Math-numberp (car guess)) |
| 695 | (math-reject-arg guess 'numberp)) |
| 696 | (setcar p (math-float (car p)))) |
| 697 | (setq p expr) |
| 698 | (while (setq p (cdr p)) |
| 699 | (if (assq (car-safe (car p)) calc-tweak-eqn-table) |
| 700 | (setcar p (math-sub (nth 1 (car p)) (nth 2 (car p))))) |
| 701 | (setcar p (math-evaluate-expr (car p))) |
| 702 | (setq row (list 'vec) |
| 703 | m -1) |
| 704 | (while (< (setq m (1+ m)) n) |
| 705 | (nconc row (list (math-evaluate-expr |
| 706 | (or (calcFunc-deriv (car p) |
| 707 | (aref math-root-vars m) |
| 708 | nil t) |
| 709 | (math-reject-arg |
| 710 | expr |
| 711 | "*Formulas must be differentiable")))))) |
| 712 | (nconc jacob (list row))) |
| 713 | (setq m (math-abs-approx guess)) |
| 714 | (math-newton-multi expr jacob n guess guess |
| 715 | (if (math-zerop m) '(float 1 3) (math-mul m 10)))) |
| 716 | (unless (eq (car-safe var) 'var) |
| 717 | (math-reject-arg var "*Expected a variable")) |
| 718 | (unless (math-expr-contains expr var) |
| 719 | (math-reject-arg expr "*Formula does not contain specified variable")) |
| 720 | (if (assq (car expr) calc-tweak-eqn-table) |
| 721 | (setq expr (math-sub (nth 1 expr) (nth 2 expr)))) |
| 722 | (math-with-extra-prec 2 |
| 723 | (setq expr (math-expr-subst expr var '(var DUMMY var-DUMMY))) |
| 724 | (let* ((calc-symbolic-mode nil) |
| 725 | (var-DUMMY nil) |
| 726 | (expr (math-evaluate-expr expr)) |
| 727 | (deriv (calcFunc-deriv expr '(var DUMMY var-DUMMY) nil t)) |
| 728 | low high vlow vhigh) |
| 729 | (and deriv (setq deriv (math-evaluate-expr deriv))) |
| 730 | (setq guess (math-float guess)) |
| 731 | (if (and (math-numberp guess) |
| 732 | deriv) |
| 733 | (math-newton-root expr deriv guess guess |
| 734 | (if (math-zerop guess) '(float 1 6) |
| 735 | (math-mul (math-abs-approx guess) 100))) |
| 736 | (if (Math-realp guess) |
| 737 | (setq low guess |
| 738 | high guess |
| 739 | var-DUMMY guess |
| 740 | vlow (math-evaluate-expr expr) |
| 741 | vhigh vlow |
| 742 | math-root-widen 'point) |
| 743 | (if (eq (car guess) 'intv) |
| 744 | (progn |
| 745 | (or (math-constp guess) (math-reject-arg guess 'constp)) |
| 746 | (setq low (nth 2 guess) |
| 747 | high (nth 3 guess)) |
| 748 | (if (memq (nth 1 guess) '(0 1)) |
| 749 | (setq low (calcFunc-incr low 1 high))) |
| 750 | (if (memq (nth 1 guess) '(0 2)) |
| 751 | (setq high (calcFunc-incr high -1 low))) |
| 752 | (setq var-DUMMY low |
| 753 | vlow (math-evaluate-expr expr) |
| 754 | var-DUMMY high |
| 755 | vhigh (math-evaluate-expr expr))) |
| 756 | (if (math-complexp guess) |
| 757 | (math-reject-arg "*Complex root finder must have derivative") |
| 758 | (math-reject-arg guess 'realp)))) |
| 759 | (if (Math-zerop vlow) |
| 760 | (list 'vec low vlow) |
| 761 | (if (Math-zerop vhigh) |
| 762 | (list 'vec high vhigh) |
| 763 | (if (and deriv (Math-numberp vlow) (Math-numberp vhigh)) |
| 764 | (math-newton-search-root expr deriv nil nil nil nil |
| 765 | low vlow high vhigh) |
| 766 | (if (or (and (Math-posp vlow) (Math-posp vhigh)) |
| 767 | (and (Math-negp vlow) (Math-negp vhigh)) |
| 768 | (not (Math-numberp vlow)) |
| 769 | (not (Math-numberp vhigh))) |
| 770 | (math-search-root expr deriv low vlow high vhigh) |
| 771 | (math-bisect-root expr low vlow high vhigh)))))))))) |
| 772 | |
| 773 | (defun calcFunc-root (expr var guess) |
| 774 | (math-find-root expr var guess nil)) |
| 775 | |
| 776 | (defun calcFunc-wroot (expr var guess) |
| 777 | (math-find-root expr var guess t)) |
| 778 | |
| 779 | |
| 780 | |
| 781 | |
| 782 | ;;; The following algorithms come from Numerical Recipes, chapter 10. |
| 783 | |
| 784 | (defvar math-min-vars [(var DUMMY var-DUMMY)]) |
| 785 | |
| 786 | (defun math-min-eval (expr a) |
| 787 | (if (Math-vectorp a) |
| 788 | (let ((m -1)) |
| 789 | (while (setq m (1+ m) a (cdr a)) |
| 790 | (set (nth 2 (aref math-min-vars m)) (car a)))) |
| 791 | (setq var-DUMMY a)) |
| 792 | (setq a (math-evaluate-expr expr)) |
| 793 | (if (Math-ratp a) |
| 794 | (math-float a) |
| 795 | (if (eq (car a) 'float) |
| 796 | a |
| 797 | (math-reject-arg a 'realp)))) |
| 798 | |
| 799 | (defvar math-min-or-max "minimum") |
| 800 | |
| 801 | ;;; A bracket for a minimum is a < b < c where f(b) < f(a) and f(b) < f(c). |
| 802 | |
| 803 | ;;; "mnbrak" |
| 804 | (defun math-widen-min (expr a b) |
| 805 | (let ((done nil) |
| 806 | (iters 30) |
| 807 | incr c va vb vc u vu r q ulim bc ba qr) |
| 808 | (or b (setq b (math-mul a '(float 101 -2)))) |
| 809 | (setq va (math-min-eval expr a) |
| 810 | vb (math-min-eval expr b)) |
| 811 | (if (math-lessp-float va vb) |
| 812 | (setq u a a b b u |
| 813 | vu va va vb vb vu)) |
| 814 | (setq c (math-add-float b (math-mul-float '(float 161803 -5) |
| 815 | (math-sub-float b a))) |
| 816 | vc (math-min-eval expr c)) |
| 817 | (while (and (not done) (math-lessp-float vc vb)) |
| 818 | (math-working "widen" (list 'intv 0 a c)) |
| 819 | (if (= (setq iters (1- iters)) 0) |
| 820 | (math-reject-arg nil (format "*Unable to find a %s near the interval" |
| 821 | math-min-or-max))) |
| 822 | (setq bc (math-sub-float b c) |
| 823 | ba (math-sub-float b a) |
| 824 | r (math-mul-float ba (math-sub-float vb vc)) |
| 825 | q (math-mul-float bc (math-sub-float vb va)) |
| 826 | qr (math-sub-float q r)) |
| 827 | (if (math-lessp-float (math-abs qr) '(float 1 -20)) |
| 828 | (setq qr (if (math-negp qr) '(float -1 -20) '(float 1 -20)))) |
| 829 | (setq u (math-sub-float |
| 830 | b |
| 831 | (math-div-float (math-sub-float (math-mul-float bc q) |
| 832 | (math-mul-float ba r)) |
| 833 | (math-mul-float '(float 2 0) qr))) |
| 834 | ulim (math-add-float b (math-mul-float '(float -1 2) bc)) |
| 835 | incr (math-negp bc)) |
| 836 | (if (if incr (math-lessp-float b u) (math-lessp-float u b)) |
| 837 | (if (if incr (math-lessp-float u c) (math-lessp-float c u)) |
| 838 | (if (math-lessp-float (setq vu (math-min-eval expr u)) vc) |
| 839 | (setq a b va vb |
| 840 | b u vb vu |
| 841 | done t) |
| 842 | (if (math-lessp-float vb vu) |
| 843 | (setq c u vc vu |
| 844 | done t) |
| 845 | (setq u (math-add-float c (math-mul-float '(float -161803 -5) |
| 846 | bc)) |
| 847 | vu (math-min-eval expr u)))) |
| 848 | (if (if incr (math-lessp-float u ulim) (math-lessp-float ulim u)) |
| 849 | (if (math-lessp-float (setq vu (math-min-eval expr u)) vc) |
| 850 | (setq b c vb vc |
| 851 | c u vc vu |
| 852 | u (math-add-float c (math-mul-float |
| 853 | '(float -161803 -5) |
| 854 | (math-sub-float b c))) |
| 855 | vu (math-min-eval expr u))) |
| 856 | (setq u ulim |
| 857 | vu (math-min-eval expr u)))) |
| 858 | (setq u (math-add-float c (math-mul-float '(float -161803 -5) |
| 859 | bc)) |
| 860 | vu (math-min-eval expr u))) |
| 861 | (setq a b va vb |
| 862 | b c vb vc |
| 863 | c u vc vu)) |
| 864 | (if (math-lessp-float a c) |
| 865 | (list a va b vb c vc) |
| 866 | (list c vc b vb a va)))) |
| 867 | |
| 868 | (defun math-narrow-min (expr a c intv) |
| 869 | (let ((xvals (list a c)) |
| 870 | (yvals (list (math-min-eval expr a) |
| 871 | (math-min-eval expr c))) |
| 872 | (levels 0) |
| 873 | (step (math-sub-float c a)) |
| 874 | (found nil) |
| 875 | xp yp b) |
| 876 | (while (and (<= (setq levels (1+ levels)) 5) |
| 877 | (not found)) |
| 878 | (setq xp xvals |
| 879 | yp yvals |
| 880 | step (math-mul-float step '(float 497 -3))) |
| 881 | (while (and (cdr xp) (not found)) |
| 882 | (setq b (math-add-float (car xp) step)) |
| 883 | (math-working "search" b) |
| 884 | (setcdr xp (cons b (cdr xp))) |
| 885 | (setcdr yp (cons (math-min-eval expr b) (cdr yp))) |
| 886 | (if (and (math-lessp-float (nth 1 yp) (car yp)) |
| 887 | (math-lessp-float (nth 1 yp) (nth 2 yp))) |
| 888 | (setq found t) |
| 889 | (setq xp (cdr xp) |
| 890 | yp (cdr yp)) |
| 891 | (if (and (cdr (cdr yp)) |
| 892 | (math-lessp-float (nth 1 yp) (car yp)) |
| 893 | (math-lessp-float (nth 1 yp) (nth 2 yp))) |
| 894 | (setq found t) |
| 895 | (setq xp (cdr xp) |
| 896 | yp (cdr yp)))))) |
| 897 | (if found |
| 898 | (list (car xp) (car yp) |
| 899 | (nth 1 xp) (nth 1 yp) |
| 900 | (nth 2 xp) (nth 2 yp)) |
| 901 | (or (if (math-lessp-float (car yvals) (nth 1 yvals)) |
| 902 | (and (memq (nth 1 intv) '(2 3)) |
| 903 | (let ((min (car yvals))) |
| 904 | (while (and (setq yvals (cdr yvals)) |
| 905 | (math-lessp-float min (car yvals)))) |
| 906 | (and (not yvals) |
| 907 | (list (nth 2 intv) min)))) |
| 908 | (and (memq (nth 1 intv) '(1 3)) |
| 909 | (setq yvals (nreverse yvals)) |
| 910 | (let ((min (car yvals))) |
| 911 | (while (and (setq yvals (cdr yvals)) |
| 912 | (math-lessp-float min (car yvals)))) |
| 913 | (and (not yvals) |
| 914 | (list (nth 3 intv) min))))) |
| 915 | (math-reject-arg nil (format "*Unable to find a %s in the interval" |
| 916 | math-min-or-max)))))) |
| 917 | |
| 918 | ;;; "brent" |
| 919 | (defun math-brent-min (expr prec a va x vx b vb) |
| 920 | (let ((iters (+ 20 (* 5 prec))) |
| 921 | (w x) |
| 922 | (vw vx) |
| 923 | (v x) |
| 924 | (vv vx) |
| 925 | (tol (list 'float 1 (- -1 prec))) |
| 926 | (zeps (list 'float 1 (- -5 prec))) |
| 927 | (e '(float 0 0)) |
| 928 | d u vu xm tol1 tol2 etemp p q r xv xw) |
| 929 | (while (progn |
| 930 | (setq xm (math-mul-float '(float 5 -1) |
| 931 | (math-add-float a b)) |
| 932 | tol1 (math-add-float |
| 933 | zeps |
| 934 | (math-mul-float tol (math-abs x))) |
| 935 | tol2 (math-mul-float tol1 '(float 2 0))) |
| 936 | (math-lessp-float (math-sub-float tol2 |
| 937 | (math-mul-float |
| 938 | '(float 5 -1) |
| 939 | (math-sub-float b a))) |
| 940 | (math-abs (math-sub-float x xm)))) |
| 941 | (if (= (setq iters (1- iters)) 0) |
| 942 | (math-reject-arg nil (format "*Unable to converge on a %s" |
| 943 | math-min-or-max))) |
| 944 | (math-working "brent" x) |
| 945 | (if (math-lessp-float (math-abs e) tol1) |
| 946 | (setq e (if (math-lessp-float x xm) |
| 947 | (math-sub-float b x) |
| 948 | (math-sub-float a x)) |
| 949 | d (math-mul-float '(float 381966 -6) e)) |
| 950 | (setq xw (math-sub-float x w) |
| 951 | r (math-mul-float xw (math-sub-float vx vv)) |
| 952 | xv (math-sub-float x v) |
| 953 | q (math-mul-float xv (math-sub-float vx vw)) |
| 954 | p (math-sub-float (math-mul-float xv q) |
| 955 | (math-mul-float xw r)) |
| 956 | q (math-mul-float '(float 2 0) (math-sub-float q r))) |
| 957 | (if (math-posp q) |
| 958 | (setq p (math-neg-float p)) |
| 959 | (setq q (math-neg-float q))) |
| 960 | (setq etemp e |
| 961 | e d) |
| 962 | (if (and (math-lessp-float (math-abs p) |
| 963 | (math-abs (math-mul-float |
| 964 | '(float 5 -1) |
| 965 | (math-mul-float q etemp)))) |
| 966 | (math-lessp-float (math-mul-float |
| 967 | q (math-sub-float a x)) p) |
| 968 | (math-lessp-float p (math-mul-float |
| 969 | q (math-sub-float b x)))) |
| 970 | (progn |
| 971 | (setq d (math-div-float p q) |
| 972 | u (math-add-float x d)) |
| 973 | (if (or (math-lessp-float (math-sub-float u a) tol2) |
| 974 | (math-lessp-float (math-sub-float b u) tol2)) |
| 975 | (setq d (if (math-lessp-float xm x) |
| 976 | (math-neg-float tol1) |
| 977 | tol1)))) |
| 978 | (setq e (if (math-lessp-float x xm) |
| 979 | (math-sub-float b x) |
| 980 | (math-sub-float a x)) |
| 981 | d (math-mul-float '(float 381966 -6) e)))) |
| 982 | (setq u (math-add-float x |
| 983 | (if (math-lessp-float (math-abs d) tol1) |
| 984 | (if (math-negp d) |
| 985 | (math-neg-float tol1) |
| 986 | tol1) |
| 987 | d)) |
| 988 | vu (math-min-eval expr u)) |
| 989 | (if (math-lessp-float vx vu) |
| 990 | (progn |
| 991 | (if (math-lessp-float u x) |
| 992 | (setq a u) |
| 993 | (setq b u)) |
| 994 | (if (or (equal w x) |
| 995 | (not (math-lessp-float vw vu))) |
| 996 | (setq v w vv vw |
| 997 | w u vw vu) |
| 998 | (if (or (equal v x) |
| 999 | (equal v w) |
| 1000 | (not (math-lessp-float vv vu))) |
| 1001 | (setq v u vv vu)))) |
| 1002 | (if (math-lessp-float u x) |
| 1003 | (setq b x) |
| 1004 | (setq a x)) |
| 1005 | (setq v w vv vw |
| 1006 | w x vw vx |
| 1007 | x u vx vu))) |
| 1008 | (list 'vec x vx))) |
| 1009 | |
| 1010 | ;;; "powell" |
| 1011 | (defun math-powell-min (expr n guesses prec) |
| 1012 | (let* ((f1dim (math-line-min-func expr n)) |
| 1013 | (xi (calcFunc-idn 1 n)) |
| 1014 | (p (cons 'vec (mapcar 'car guesses))) |
| 1015 | (pt p) |
| 1016 | (ftol (list 'float 1 (- prec))) |
| 1017 | (fret (math-min-eval expr p)) |
| 1018 | fp ptt fptt xit i ibig del diff res) |
| 1019 | (while (progn |
| 1020 | (setq fp fret |
| 1021 | ibig 0 |
| 1022 | del '(float 0 0) |
| 1023 | i 0) |
| 1024 | (while (<= (setq i (1+ i)) n) |
| 1025 | (setq fptt fret |
| 1026 | res (math-line-min f1dim p |
| 1027 | (math-mat-col xi i) |
| 1028 | n prec) |
| 1029 | p (let ((calc-internal-prec prec)) |
| 1030 | (math-normalize (car res))) |
| 1031 | fret (nth 2 res) |
| 1032 | diff (math-abs (math-sub-float fptt fret))) |
| 1033 | (if (math-lessp-float del diff) |
| 1034 | (setq del diff |
| 1035 | ibig i))) |
| 1036 | (math-lessp-float |
| 1037 | (math-mul-float ftol |
| 1038 | (math-add-float (math-abs fp) |
| 1039 | (math-abs fret))) |
| 1040 | (math-mul-float '(float 2 0) |
| 1041 | (math-abs (math-sub-float fp |
| 1042 | fret))))) |
| 1043 | (setq ptt (math-sub (math-mul '(float 2 0) p) pt) |
| 1044 | xit (math-sub p pt) |
| 1045 | pt p |
| 1046 | fptt (math-min-eval expr ptt)) |
| 1047 | (if (and (math-lessp-float fptt fp) |
| 1048 | (math-lessp-float |
| 1049 | (math-mul-float |
| 1050 | (math-mul-float '(float 2 0) |
| 1051 | (math-add-float |
| 1052 | (math-sub-float fp |
| 1053 | (math-mul-float '(float 2 0) |
| 1054 | fret)) |
| 1055 | fptt)) |
| 1056 | (math-sqr-float (math-sub-float |
| 1057 | (math-sub-float fp fret) del))) |
| 1058 | (math-mul-float del |
| 1059 | (math-sqr-float (math-sub-float fp fptt))))) |
| 1060 | (progn |
| 1061 | (setq res (math-line-min f1dim p xit n prec) |
| 1062 | p (car res) |
| 1063 | fret (nth 2 res) |
| 1064 | i 0) |
| 1065 | (while (<= (setq i (1+ i)) n) |
| 1066 | (setcar (nthcdr ibig (nth i xi)) |
| 1067 | (nth i (nth 1 res))))))) |
| 1068 | (list 'vec p fret))) |
| 1069 | |
| 1070 | (defun math-line-min-func (expr n) |
| 1071 | (let ((m -1)) |
| 1072 | (while (< (setq m (1+ m)) n) |
| 1073 | (set (nth 2 (aref math-min-vars m)) |
| 1074 | (list '+ |
| 1075 | (list '* |
| 1076 | '(var DUMMY var-DUMMY) |
| 1077 | (list 'calcFunc-mrow '(var line-xi line-xi) (1+ m))) |
| 1078 | (list 'calcFunc-mrow '(var line-p line-p) (1+ m))))) |
| 1079 | (math-evaluate-expr expr))) |
| 1080 | |
| 1081 | (defun math-line-min (f1dim line-p line-xi n prec) |
| 1082 | (let* ((var-DUMMY nil) |
| 1083 | (expr (math-evaluate-expr f1dim)) |
| 1084 | (params (math-widen-min expr '(float 0 0) '(float 1 0))) |
| 1085 | (res (apply 'math-brent-min expr prec params)) |
| 1086 | (xi (math-mul (nth 1 res) line-xi))) |
| 1087 | (list (math-add line-p xi) xi (nth 2 res)))) |
| 1088 | |
| 1089 | |
| 1090 | (defun math-find-minimum (expr var guess min-widen) |
| 1091 | (let* ((calc-symbolic-mode nil) |
| 1092 | (n 0) |
| 1093 | (var-DUMMY nil) |
| 1094 | (isvec (math-vectorp var)) |
| 1095 | g guesses) |
| 1096 | (or (math-vectorp var) |
| 1097 | (setq var (list 'vec var))) |
| 1098 | (or (math-vectorp guess) |
| 1099 | (setq guess (list 'vec guess))) |
| 1100 | (or (= (length var) (length guess)) |
| 1101 | (math-dimension-error)) |
| 1102 | (while (setq var (cdr var) guess (cdr guess)) |
| 1103 | (or (eq (car-safe (car var)) 'var) |
| 1104 | (math-reject-arg (car var) "*Expected a variable")) |
| 1105 | (or (math-expr-contains expr (car var)) |
| 1106 | (math-reject-arg (car var) |
| 1107 | "*Formula does not contain specified variable")) |
| 1108 | (while (>= (1+ n) (length math-min-vars)) |
| 1109 | (let ((symb (intern (concat "math-min-v" |
| 1110 | (int-to-string |
| 1111 | (length math-min-vars)))))) |
| 1112 | (setq math-min-vars (vconcat math-min-vars |
| 1113 | (vector (list 'var symb symb)))))) |
| 1114 | (set (nth 2 (aref math-min-vars n)) nil) |
| 1115 | (set (nth 2 (aref math-min-vars (1+ n))) nil) |
| 1116 | (if (math-complexp (car guess)) |
| 1117 | (setq expr (math-expr-subst expr |
| 1118 | (car var) |
| 1119 | (list '+ (aref math-min-vars n) |
| 1120 | (list '* |
| 1121 | (aref math-min-vars (1+ n)) |
| 1122 | '(cplx 0 1)))) |
| 1123 | guesses (let ((g (math-float (math-complex (car guess))))) |
| 1124 | (cons (list (nth 2 g) nil nil) |
| 1125 | (cons (list (nth 1 g) nil nil t) |
| 1126 | guesses))) |
| 1127 | n (+ n 2)) |
| 1128 | (setq expr (math-expr-subst expr |
| 1129 | (car var) |
| 1130 | (aref math-min-vars n)) |
| 1131 | guesses (cons (if (math-realp (car guess)) |
| 1132 | (list (math-float (car guess)) nil nil) |
| 1133 | (if (and (eq (car-safe (car guess)) 'intv) |
| 1134 | (math-constp (car guess))) |
| 1135 | (list (math-mul |
| 1136 | (math-add (nth 2 (car guess)) |
| 1137 | (nth 3 (car guess))) |
| 1138 | '(float 5 -1)) |
| 1139 | (math-float (nth 2 (car guess))) |
| 1140 | (math-float (nth 3 (car guess))) |
| 1141 | (car guess)) |
| 1142 | (math-reject-arg (car guess) 'realp))) |
| 1143 | guesses) |
| 1144 | n (1+ n)))) |
| 1145 | (setq guesses (nreverse guesses) |
| 1146 | expr (math-evaluate-expr expr)) |
| 1147 | (if (= n 1) |
| 1148 | (let* ((params (if (nth 1 (car guesses)) |
| 1149 | (if min-widen |
| 1150 | (math-widen-min expr |
| 1151 | (nth 1 (car guesses)) |
| 1152 | (nth 2 (car guesses))) |
| 1153 | (math-narrow-min expr |
| 1154 | (nth 1 (car guesses)) |
| 1155 | (nth 2 (car guesses)) |
| 1156 | (nth 3 (car guesses)))) |
| 1157 | (math-widen-min expr |
| 1158 | (car (car guesses)) |
| 1159 | nil))) |
| 1160 | (prec calc-internal-prec) |
| 1161 | (res (if (cdr (cdr params)) |
| 1162 | (math-with-extra-prec (+ calc-internal-prec 2) |
| 1163 | (apply 'math-brent-min expr prec params)) |
| 1164 | (cons 'vec params)))) |
| 1165 | (if isvec |
| 1166 | (list 'vec (list 'vec (nth 1 res)) (nth 2 res)) |
| 1167 | res)) |
| 1168 | (let* ((prec calc-internal-prec) |
| 1169 | (res (math-with-extra-prec (+ calc-internal-prec 2) |
| 1170 | (math-powell-min expr n guesses prec))) |
| 1171 | (p (nth 1 res)) |
| 1172 | (vec (list 'vec))) |
| 1173 | (while (setq p (cdr p)) |
| 1174 | (if (nth 3 (car guesses)) |
| 1175 | (progn |
| 1176 | (nconc vec (list (math-normalize |
| 1177 | (list 'cplx (car p) (nth 1 p))))) |
| 1178 | (setq p (cdr p) |
| 1179 | guesses (cdr guesses))) |
| 1180 | (nconc vec (list (car p)))) |
| 1181 | (setq guesses (cdr guesses))) |
| 1182 | (if isvec |
| 1183 | (list 'vec vec (nth 2 res)) |
| 1184 | (list 'vec (nth 1 vec) (nth 2 res))))))) |
| 1185 | |
| 1186 | (defun calcFunc-minimize (expr var guess) |
| 1187 | (let ((calc-internal-prec (max (/ calc-internal-prec 2) 3)) |
| 1188 | (math-min-or-max "minimum")) |
| 1189 | (math-find-minimum (math-normalize expr) |
| 1190 | (math-normalize var) |
| 1191 | (math-normalize guess) nil))) |
| 1192 | |
| 1193 | (defun calcFunc-wminimize (expr var guess) |
| 1194 | (let ((calc-internal-prec (max (/ calc-internal-prec 2) 3)) |
| 1195 | (math-min-or-max "minimum")) |
| 1196 | (math-find-minimum (math-normalize expr) |
| 1197 | (math-normalize var) |
| 1198 | (math-normalize guess) t))) |
| 1199 | |
| 1200 | (defun calcFunc-maximize (expr var guess) |
| 1201 | (let* ((calc-internal-prec (max (/ calc-internal-prec 2) 3)) |
| 1202 | (math-min-or-max "maximum") |
| 1203 | (res (math-find-minimum (math-normalize (math-neg expr)) |
| 1204 | (math-normalize var) |
| 1205 | (math-normalize guess) nil))) |
| 1206 | (list 'vec (nth 1 res) (math-neg (nth 2 res))))) |
| 1207 | |
| 1208 | (defun calcFunc-wmaximize (expr var guess) |
| 1209 | (let* ((calc-internal-prec (max (/ calc-internal-prec 2) 3)) |
| 1210 | (math-min-or-max "maximum") |
| 1211 | (res (math-find-minimum (math-normalize (math-neg expr)) |
| 1212 | (math-normalize var) |
| 1213 | (math-normalize guess) t))) |
| 1214 | (list 'vec (nth 1 res) (math-neg (nth 2 res))))) |
| 1215 | |
| 1216 | |
| 1217 | |
| 1218 | |
| 1219 | ;;; The following algorithms come from Numerical Recipes, chapter 3. |
| 1220 | |
| 1221 | (defun calcFunc-polint (data x) |
| 1222 | (or (math-matrixp data) (math-reject-arg data 'matrixp)) |
| 1223 | (or (= (length data) 3) |
| 1224 | (math-reject-arg data "*Wrong number of data rows")) |
| 1225 | (or (> (length (nth 1 data)) 2) |
| 1226 | (math-reject-arg data "*Too few data points")) |
| 1227 | (if (and (math-vectorp x) (or (math-constp x) math-expand-formulas)) |
| 1228 | (cons 'vec (mapcar (function (lambda (x) (calcFunc-polint data x))) |
| 1229 | (cdr x))) |
| 1230 | (or (math-objectp x) math-expand-formulas (math-reject-arg x 'objectp)) |
| 1231 | (math-with-extra-prec 2 |
| 1232 | (cons 'vec (math-poly-interp (cdr (nth 1 data)) (cdr (nth 2 data)) x |
| 1233 | nil))))) |
| 1234 | (put 'calcFunc-polint 'math-expandable t) |
| 1235 | |
| 1236 | |
| 1237 | (defun calcFunc-ratint (data x) |
| 1238 | (or (math-matrixp data) (math-reject-arg data 'matrixp)) |
| 1239 | (or (= (length data) 3) |
| 1240 | (math-reject-arg data "*Wrong number of data rows")) |
| 1241 | (or (> (length (nth 1 data)) 2) |
| 1242 | (math-reject-arg data "*Too few data points")) |
| 1243 | (if (and (math-vectorp x) (or (math-constp x) math-expand-formulas)) |
| 1244 | (cons 'vec (mapcar (function (lambda (x) (calcFunc-ratint data x))) |
| 1245 | (cdr x))) |
| 1246 | (or (math-objectp x) math-expand-formulas (math-reject-arg x 'objectp)) |
| 1247 | (math-with-extra-prec 2 |
| 1248 | (cons 'vec (math-poly-interp (cdr (nth 1 data)) (cdr (nth 2 data)) x |
| 1249 | (cdr (cdr (cdr (nth 1 data))))))))) |
| 1250 | (put 'calcFunc-ratint 'math-expandable t) |
| 1251 | |
| 1252 | |
| 1253 | (defun math-poly-interp (xa ya x ratp) |
| 1254 | (let ((n (length xa)) |
| 1255 | (dif nil) |
| 1256 | (ns nil) |
| 1257 | (xax nil) |
| 1258 | (c (copy-sequence ya)) |
| 1259 | (d (copy-sequence ya)) |
| 1260 | (i 0) |
| 1261 | (m 0) |
| 1262 | y dy (xp xa) xpm cp dp temp) |
| 1263 | (while (<= (setq i (1+ i)) n) |
| 1264 | (setq xax (cons (math-sub (car xp) x) xax) |
| 1265 | xp (cdr xp) |
| 1266 | temp (math-abs (car xax))) |
| 1267 | (if (or (null dif) (math-lessp temp dif)) |
| 1268 | (setq dif temp |
| 1269 | ns i))) |
| 1270 | (setq xax (nreverse xax) |
| 1271 | ns (1- ns) |
| 1272 | y (nth ns ya)) |
| 1273 | (if (math-zerop dif) |
| 1274 | (list y 0) |
| 1275 | (while (< (setq m (1+ m)) n) |
| 1276 | (setq i 0 |
| 1277 | xp xax |
| 1278 | xpm (nthcdr m xax) |
| 1279 | cp c |
| 1280 | dp d) |
| 1281 | (while (<= (setq i (1+ i)) (- n m)) |
| 1282 | (if ratp |
| 1283 | (let ((t2 (math-div (math-mul (car xp) (car dp)) (car xpm)))) |
| 1284 | (setq temp (math-div (math-sub (nth 1 cp) (car dp)) |
| 1285 | (math-sub t2 (nth 1 cp)))) |
| 1286 | (setcar dp (math-mul (nth 1 cp) temp)) |
| 1287 | (setcar cp (math-mul t2 temp))) |
| 1288 | (if (math-equal (car xp) (car xpm)) |
| 1289 | (math-reject-arg (cons 'vec xa) "*Duplicate X values")) |
| 1290 | (setq temp (math-div (math-sub (nth 1 cp) (car dp)) |
| 1291 | (math-sub (car xp) (car xpm)))) |
| 1292 | (setcar dp (math-mul (car xpm) temp)) |
| 1293 | (setcar cp (math-mul (car xp) temp))) |
| 1294 | (setq cp (cdr cp) |
| 1295 | dp (cdr dp) |
| 1296 | xp (cdr xp) |
| 1297 | xpm (cdr xpm))) |
| 1298 | (if (< (+ ns ns) (- n m)) |
| 1299 | (setq dy (nth ns c)) |
| 1300 | (setq ns (1- ns) |
| 1301 | dy (nth ns d))) |
| 1302 | (setq y (math-add y dy))) |
| 1303 | (list y dy)))) |
| 1304 | |
| 1305 | |
| 1306 | |
| 1307 | ;;; The following algorithms come from Numerical Recipes, chapter 4. |
| 1308 | |
| 1309 | (defun calcFunc-ninteg (expr var lo hi) |
| 1310 | (setq lo (math-evaluate-expr lo) |
| 1311 | hi (math-evaluate-expr hi)) |
| 1312 | (or (math-numberp lo) (math-infinitep lo) (math-reject-arg lo 'numberp)) |
| 1313 | (or (math-numberp hi) (math-infinitep hi) (math-reject-arg hi 'numberp)) |
| 1314 | (if (math-lessp hi lo) |
| 1315 | (math-neg (calcFunc-ninteg expr var hi lo)) |
| 1316 | (setq expr (math-expr-subst expr var '(var DUMMY var-DUMMY))) |
| 1317 | (let ((var-DUMMY nil) |
| 1318 | (calc-symbolic-mode nil) |
| 1319 | (calc-prefer-frac nil) |
| 1320 | (sum 0)) |
| 1321 | (setq expr (math-evaluate-expr expr)) |
| 1322 | (if (equal lo '(neg (var inf var-inf))) |
| 1323 | (let ((thi (if (math-lessp hi '(float -2 0)) |
| 1324 | hi '(float -2 0)))) |
| 1325 | (setq sum (math-ninteg-romberg |
| 1326 | 'math-ninteg-midpoint expr |
| 1327 | (math-float lo) (math-float thi) 'inf) |
| 1328 | lo thi))) |
| 1329 | (if (equal hi '(var inf var-inf)) |
| 1330 | (let ((tlo (if (math-lessp '(float 2 0) lo) |
| 1331 | lo '(float 2 0)))) |
| 1332 | (setq sum (math-add sum |
| 1333 | (math-ninteg-romberg |
| 1334 | 'math-ninteg-midpoint expr |
| 1335 | (math-float tlo) (math-float hi) 'inf)) |
| 1336 | hi tlo))) |
| 1337 | (or (math-equal lo hi) |
| 1338 | (setq sum (math-add sum |
| 1339 | (math-ninteg-romberg |
| 1340 | 'math-ninteg-midpoint expr |
| 1341 | (math-float lo) (math-float hi) nil)))) |
| 1342 | sum))) |
| 1343 | |
| 1344 | |
| 1345 | ;;; Open Romberg method; "qromo" in section 4.4. |
| 1346 | |
| 1347 | ;; The variable math-ninteg-temp is local to math-ninteg-romberg, |
| 1348 | ;; but is used by math-ninteg-midpoint, which is used by |
| 1349 | ;; math-ninteg-romberg. |
| 1350 | (defvar math-ninteg-temp) |
| 1351 | |
| 1352 | (defun math-ninteg-romberg (func expr lo hi mode) |
| 1353 | (let ((curh '(float 1 0)) |
| 1354 | (h nil) |
| 1355 | (s nil) |
| 1356 | (j 0) |
| 1357 | (ss nil) |
| 1358 | (prec calc-internal-prec) |
| 1359 | (math-ninteg-temp nil)) |
| 1360 | (math-with-extra-prec 2 |
| 1361 | ;; Limit on "j" loop must be 14 or less to keep "it" from overflowing. |
| 1362 | (or (while (and (null ss) (<= (setq j (1+ j)) 8)) |
| 1363 | (setq s (nconc s (list (funcall func expr lo hi mode))) |
| 1364 | h (nconc h (list curh))) |
| 1365 | (if (>= j 3) |
| 1366 | (let ((res (math-poly-interp h s '(float 0 0) nil))) |
| 1367 | (if (math-lessp (math-abs (nth 1 res)) |
| 1368 | (calcFunc-scf (math-abs (car res)) |
| 1369 | (- prec))) |
| 1370 | (setq ss (car res))))) |
| 1371 | (if (>= j 5) |
| 1372 | (setq s (cdr s) |
| 1373 | h (cdr h))) |
| 1374 | (setq curh (math-div-float curh '(float 9 0)))) |
| 1375 | ss |
| 1376 | (math-reject-arg nil (format "*Integral failed to converge")))))) |
| 1377 | |
| 1378 | |
| 1379 | (defun math-ninteg-evaluate (expr x mode) |
| 1380 | (if (eq mode 'inf) |
| 1381 | (setq x (math-div '(float 1 0) x))) |
| 1382 | (let* ((var-DUMMY x) |
| 1383 | (res (math-evaluate-expr expr))) |
| 1384 | (or (Math-numberp res) |
| 1385 | (math-reject-arg res "*Integrand does not evaluate to a number")) |
| 1386 | (if (eq mode 'inf) |
| 1387 | (setq res (math-mul res (math-sqr x)))) |
| 1388 | res)) |
| 1389 | |
| 1390 | |
| 1391 | (defun math-ninteg-midpoint (expr lo hi mode) ; uses "math-ninteg-temp" |
| 1392 | (if (eq mode 'inf) |
| 1393 | (let ((math-infinite-mode t) temp) |
| 1394 | (setq temp (math-div 1 lo) |
| 1395 | lo (math-div 1 hi) |
| 1396 | hi temp))) |
| 1397 | (if math-ninteg-temp |
| 1398 | (let* ((it3 (* 3 (car math-ninteg-temp))) |
| 1399 | (math-working-step-2 (* 2 (car math-ninteg-temp))) |
| 1400 | (math-working-step 0) |
| 1401 | (range (math-sub hi lo)) |
| 1402 | (del (math-div range (math-float it3))) |
| 1403 | (del2 (math-add del del)) |
| 1404 | (del3 (math-add del del2)) |
| 1405 | (x (math-add lo (math-mul '(float 5 -1) del))) |
| 1406 | (sum '(float 0 0)) |
| 1407 | (j 0) temp) |
| 1408 | (while (<= (setq j (1+ j)) (car math-ninteg-temp)) |
| 1409 | (setq math-working-step (1+ math-working-step) |
| 1410 | temp (math-ninteg-evaluate expr x mode) |
| 1411 | math-working-step (1+ math-working-step) |
| 1412 | sum (math-add sum (math-add temp (math-ninteg-evaluate |
| 1413 | expr (math-add x del2) |
| 1414 | mode))) |
| 1415 | x (math-add x del3))) |
| 1416 | (setq math-ninteg-temp (list it3 |
| 1417 | (math-add (math-div (nth 1 math-ninteg-temp) |
| 1418 | '(float 3 0)) |
| 1419 | (math-mul sum del))))) |
| 1420 | (setq math-ninteg-temp (list 1 (math-mul |
| 1421 | (math-sub hi lo) |
| 1422 | (math-ninteg-evaluate |
| 1423 | expr |
| 1424 | (math-mul (math-add lo hi) '(float 5 -1)) |
| 1425 | mode))))) |
| 1426 | (nth 1 math-ninteg-temp)) |
| 1427 | |
| 1428 | |
| 1429 | |
| 1430 | |
| 1431 | |
| 1432 | ;;; The following algorithms come from Numerical Recipes, chapter 14. |
| 1433 | |
| 1434 | (defvar math-dummy-vars [(var DUMMY var-DUMMY)]) |
| 1435 | (defvar math-dummy-counter 0) |
| 1436 | (defun math-dummy-variable () |
| 1437 | (if (= math-dummy-counter (length math-dummy-vars)) |
| 1438 | (let ((symb (intern (format "math-dummy-%d" math-dummy-counter)))) |
| 1439 | (setq math-dummy-vars (vconcat math-dummy-vars |
| 1440 | (vector (list 'var symb symb)))))) |
| 1441 | (set (nth 2 (aref math-dummy-vars math-dummy-counter)) nil) |
| 1442 | (prog1 |
| 1443 | (aref math-dummy-vars math-dummy-counter) |
| 1444 | (setq math-dummy-counter (1+ math-dummy-counter)))) |
| 1445 | |
| 1446 | (defvar math-in-fit 0) |
| 1447 | (defvar calc-fit-to-trail nil) |
| 1448 | |
| 1449 | (defun calcFunc-fit (expr vars &optional coefs data) |
| 1450 | (let ((math-in-fit 10)) |
| 1451 | (math-with-extra-prec 2 |
| 1452 | (math-general-fit expr vars coefs data nil)))) |
| 1453 | |
| 1454 | (defun calcFunc-efit (expr vars &optional coefs data) |
| 1455 | (let ((math-in-fit 10)) |
| 1456 | (math-with-extra-prec 2 |
| 1457 | (math-general-fit expr vars coefs data 'sdev)))) |
| 1458 | |
| 1459 | (defun calcFunc-xfit (expr vars &optional coefs data) |
| 1460 | (let ((math-in-fit 10)) |
| 1461 | (math-with-extra-prec 2 |
| 1462 | (math-general-fit expr vars coefs data 'full)))) |
| 1463 | |
| 1464 | ;; The variables math-fit-first-var, math-fit-first-coef and |
| 1465 | ;; math-fit-new-coefs are local to math-general-fit, but are used by |
| 1466 | ;; calcFunc-fitvar, calcFunc-fitparam and calcFunc-fitdummy |
| 1467 | ;; (respectively), which are used by math-general-fit. |
| 1468 | (defvar math-fit-first-var) |
| 1469 | (defvar math-fit-first-coef) |
| 1470 | (defvar math-fit-new-coefs) |
| 1471 | |
| 1472 | (defun math-general-fit (expr vars coefs data mode) |
| 1473 | (let ((calc-simplify-mode nil) |
| 1474 | (math-dummy-counter math-dummy-counter) |
| 1475 | (math-in-fit 1) |
| 1476 | (extended (eq mode 'full)) |
| 1477 | (math-fit-first-coef math-dummy-counter) |
| 1478 | math-fit-first-var |
| 1479 | (plain-expr expr) |
| 1480 | orig-expr |
| 1481 | have-sdevs need-chisq chisq |
| 1482 | (x-funcs nil) |
| 1483 | (y-filter nil) |
| 1484 | y-dummy |
| 1485 | (coef-filters nil) |
| 1486 | math-fit-new-coefs |
| 1487 | (xy-values nil) |
| 1488 | (weights nil) |
| 1489 | (var-YVAL nil) (var-YVALX nil) |
| 1490 | covar beta |
| 1491 | n nn m mm v dummy p) |
| 1492 | |
| 1493 | ;; Validate and parse arguments. |
| 1494 | (or data |
| 1495 | (if coefs |
| 1496 | (setq data coefs |
| 1497 | coefs nil) |
| 1498 | (if (math-vectorp expr) |
| 1499 | (if (memq (length expr) '(3 4)) |
| 1500 | (setq data vars |
| 1501 | vars (nth 2 expr) |
| 1502 | coefs (nth 3 expr) |
| 1503 | expr (nth 1 expr)) |
| 1504 | (math-dimension-error)) |
| 1505 | (setq data vars |
| 1506 | vars nil |
| 1507 | coefs nil)))) |
| 1508 | (or (math-matrixp data) (math-reject-arg data 'matrixp)) |
| 1509 | (setq v (1- (length data)) |
| 1510 | n (1- (length (nth 1 data)))) |
| 1511 | (or (math-vectorp vars) (null vars) |
| 1512 | (setq vars (list 'vec vars))) |
| 1513 | (or (math-vectorp coefs) (null coefs) |
| 1514 | (setq coefs (list 'vec coefs))) |
| 1515 | (or coefs |
| 1516 | (setq coefs (cons 'vec (math-all-vars-but expr vars)))) |
| 1517 | (or vars |
| 1518 | (if (<= (1- (length coefs)) v) |
| 1519 | (math-reject-arg coefs "*Not enough variables in model") |
| 1520 | (setq coefs (copy-sequence coefs)) |
| 1521 | (let ((p (nthcdr (- (length coefs) v |
| 1522 | (if (eq (car-safe expr) 'calcFunc-eq) 1 0)) |
| 1523 | coefs))) |
| 1524 | (setq vars (cons 'vec (cdr p))) |
| 1525 | (setcdr p nil)))) |
| 1526 | (or (= (1- (length vars)) v) |
| 1527 | (= (length vars) v) |
| 1528 | (math-reject-arg vars "*Number of variables does not match data")) |
| 1529 | (setq m (1- (length coefs))) |
| 1530 | (if (< m 1) |
| 1531 | (math-reject-arg coefs "*Need at least one parameter")) |
| 1532 | |
| 1533 | ;; Rewrite expr in terms of fitparam and fitvar, make into an equation. |
| 1534 | (setq p coefs) |
| 1535 | (while (setq p (cdr p)) |
| 1536 | (or (eq (car-safe (car p)) 'var) |
| 1537 | (math-reject-arg (car p) "*Expected a variable")) |
| 1538 | (setq dummy (math-dummy-variable) |
| 1539 | expr (math-expr-subst expr (car p) |
| 1540 | (list 'calcFunc-fitparam |
| 1541 | (- math-dummy-counter math-fit-first-coef))))) |
| 1542 | (setq math-fit-first-var math-dummy-counter |
| 1543 | p vars) |
| 1544 | (while (setq p (cdr p)) |
| 1545 | (or (eq (car-safe (car p)) 'var) |
| 1546 | (math-reject-arg (car p) "*Expected a variable")) |
| 1547 | (setq dummy (math-dummy-variable) |
| 1548 | expr (math-expr-subst expr (car p) |
| 1549 | (list 'calcFunc-fitvar |
| 1550 | (- math-dummy-counter math-fit-first-var))))) |
| 1551 | (if (< math-dummy-counter (+ math-fit-first-var v)) |
| 1552 | (setq dummy (math-dummy-variable))) ; dependent variable may be unnamed |
| 1553 | (setq y-dummy dummy |
| 1554 | orig-expr expr) |
| 1555 | (or (eq (car-safe expr) 'calcFunc-eq) |
| 1556 | (setq expr (list 'calcFunc-eq (list 'calcFunc-fitvar v) expr))) |
| 1557 | |
| 1558 | (let ((calc-symbolic-mode nil)) |
| 1559 | |
| 1560 | ;; Apply rewrites to put expr into a linear-like form. |
| 1561 | (setq expr (math-evaluate-expr expr) |
| 1562 | expr (math-rewrite (list 'calcFunc-fitmodel expr) |
| 1563 | '(var FitRules var-FitRules)) |
| 1564 | math-in-fit 2 |
| 1565 | expr (math-evaluate-expr expr)) |
| 1566 | (or (and (eq (car-safe expr) 'calcFunc-fitsystem) |
| 1567 | (= (length expr) 4) |
| 1568 | (math-vectorp (nth 2 expr)) |
| 1569 | (math-vectorp (nth 3 expr)) |
| 1570 | (> (length (nth 2 expr)) 1) |
| 1571 | (= (length (nth 3 expr)) (1+ m))) |
| 1572 | (math-reject-arg plain-expr "*Model expression is too complex")) |
| 1573 | (setq y-filter (nth 1 expr) |
| 1574 | x-funcs (vconcat (cdr (nth 2 expr))) |
| 1575 | coef-filters (nth 3 expr) |
| 1576 | mm (length x-funcs)) |
| 1577 | (if (equal y-filter y-dummy) |
| 1578 | (setq y-filter nil)) |
| 1579 | |
| 1580 | ;; Build the (square) system of linear equations to be solved. |
| 1581 | (setq beta (cons 'vec (make-list mm 0)) |
| 1582 | covar (cons 'vec (mapcar 'copy-sequence (make-list mm beta)))) |
| 1583 | (let* ((ptrs (vconcat (cdr data))) |
| 1584 | (isigsq 1) |
| 1585 | (xvals (make-vector mm 0)) |
| 1586 | (i 0) |
| 1587 | j k xval yval sigmasqr wt covj covjk covk betaj lud) |
| 1588 | (while (<= (setq i (1+ i)) n) |
| 1589 | |
| 1590 | ;; Assign various independent variables for this data point. |
| 1591 | (setq j 0 |
| 1592 | sigmasqr nil) |
| 1593 | (while (< j v) |
| 1594 | (aset ptrs j (cdr (aref ptrs j))) |
| 1595 | (setq xval (car (aref ptrs j))) |
| 1596 | (if (= j (1- v)) |
| 1597 | (if sigmasqr |
| 1598 | (progn |
| 1599 | (if (eq (car-safe xval) 'sdev) |
| 1600 | (setq sigmasqr (math-add (math-sqr (nth 2 xval)) |
| 1601 | sigmasqr) |
| 1602 | xval (nth 1 xval))) |
| 1603 | (if y-filter |
| 1604 | (setq xval (math-make-sdev xval |
| 1605 | (math-sqrt sigmasqr)))))) |
| 1606 | (if (eq (car-safe xval) 'sdev) |
| 1607 | (setq sigmasqr (math-add (math-sqr (nth 2 xval)) |
| 1608 | (or sigmasqr 0)) |
| 1609 | xval (nth 1 xval)))) |
| 1610 | (set (nth 2 (aref math-dummy-vars (+ math-fit-first-var j))) xval) |
| 1611 | (setq j (1+ j))) |
| 1612 | |
| 1613 | ;; Compute Y value for this data point. |
| 1614 | (if y-filter |
| 1615 | (setq yval (math-evaluate-expr y-filter)) |
| 1616 | (setq yval (symbol-value (nth 2 y-dummy)))) |
| 1617 | (if (eq (car-safe yval) 'sdev) |
| 1618 | (setq sigmasqr (math-sqr (nth 2 yval)) |
| 1619 | yval (nth 1 yval))) |
| 1620 | (if (= i 1) |
| 1621 | (setq have-sdevs sigmasqr |
| 1622 | need-chisq (or extended |
| 1623 | (and (eq mode 'sdev) (not have-sdevs))))) |
| 1624 | (if have-sdevs |
| 1625 | (if sigmasqr |
| 1626 | (progn |
| 1627 | (setq isigsq (math-div 1 sigmasqr)) |
| 1628 | (if need-chisq |
| 1629 | (setq weights (cons isigsq weights)))) |
| 1630 | (math-reject-arg yval "*Mixed error forms and plain numbers")) |
| 1631 | (if sigmasqr |
| 1632 | (math-reject-arg yval "*Mixed error forms and plain numbers"))) |
| 1633 | |
| 1634 | ;; Compute X values for this data point and update covar and beta. |
| 1635 | (if (eq (car-safe xval) 'sdev) |
| 1636 | (set (nth 2 y-dummy) (nth 1 xval))) |
| 1637 | (setq j 0 |
| 1638 | covj covar |
| 1639 | betaj beta) |
| 1640 | (while (< j mm) |
| 1641 | (setq wt (math-evaluate-expr (aref x-funcs j))) |
| 1642 | (aset xvals j wt) |
| 1643 | (setq wt (math-mul wt isigsq) |
| 1644 | betaj (cdr betaj) |
| 1645 | covjk (car (setq covj (cdr covj))) |
| 1646 | k 0) |
| 1647 | (while (<= k j) |
| 1648 | (setq covjk (cdr covjk)) |
| 1649 | (setcar covjk (math-add (car covjk) |
| 1650 | (math-mul wt (aref xvals k)))) |
| 1651 | (setq k (1+ k))) |
| 1652 | (setcar betaj (math-add (car betaj) (math-mul wt yval))) |
| 1653 | (setq j (1+ j))) |
| 1654 | (if need-chisq |
| 1655 | (setq xy-values (cons (append xvals (list yval)) xy-values)))) |
| 1656 | |
| 1657 | ;; Fill in symmetric half of covar matrix. |
| 1658 | (setq j 0 |
| 1659 | covj covar) |
| 1660 | (while (< j (1- mm)) |
| 1661 | (setq k j |
| 1662 | j (1+ j) |
| 1663 | covjk (nthcdr j (car (setq covj (cdr covj)))) |
| 1664 | covk (nthcdr j covar)) |
| 1665 | (while (< (setq k (1+ k)) mm) |
| 1666 | (setq covjk (cdr covjk) |
| 1667 | covk (cdr covk)) |
| 1668 | (setcar covjk (nth j (car covk)))))) |
| 1669 | |
| 1670 | ;; Solve the linear system. |
| 1671 | (if mode |
| 1672 | (progn |
| 1673 | (setq covar (math-matrix-inv-raw covar)) |
| 1674 | (if covar |
| 1675 | (setq beta (math-mul covar beta)) |
| 1676 | (if (math-zerop (math-abs beta)) |
| 1677 | (setq covar (calcFunc-diag 0 (1- (length beta)))) |
| 1678 | (math-reject-arg orig-expr "*Singular matrix"))) |
| 1679 | (or (math-vectorp covar) |
| 1680 | (setq covar (list 'vec (list 'vec covar))))) |
| 1681 | (setq beta (math-div beta covar))) |
| 1682 | |
| 1683 | ;; Compute chi-square statistic if necessary. |
| 1684 | (if need-chisq |
| 1685 | (let (bp xp sum) |
| 1686 | (setq chisq 0) |
| 1687 | (while xy-values |
| 1688 | (setq bp beta |
| 1689 | xp (car xy-values) |
| 1690 | sum 0) |
| 1691 | (while (setq bp (cdr bp)) |
| 1692 | (setq sum (math-add sum (math-mul (car bp) (car xp))) |
| 1693 | xp (cdr xp))) |
| 1694 | (setq sum (math-sqr (math-sub (car xp) sum))) |
| 1695 | (if weights (setq sum (math-mul sum (car weights)))) |
| 1696 | (setq chisq (math-add chisq sum) |
| 1697 | weights (cdr weights) |
| 1698 | xy-values (cdr xy-values))))) |
| 1699 | |
| 1700 | ;; Convert coefficients back into original terms. |
| 1701 | (setq math-fit-new-coefs (copy-sequence beta)) |
| 1702 | (let* ((bp math-fit-new-coefs) |
| 1703 | (cp covar) |
| 1704 | (sigdat 1) |
| 1705 | (math-in-fit 3) |
| 1706 | (j 0)) |
| 1707 | (and mode (not have-sdevs) |
| 1708 | (setq sigdat (if (<= n mm) |
| 1709 | 0 |
| 1710 | (math-div chisq (- n mm))))) |
| 1711 | (if mode |
| 1712 | (while (setq bp (cdr bp)) |
| 1713 | (setcar bp (math-make-sdev |
| 1714 | (car bp) |
| 1715 | (math-sqrt (math-mul (nth (setq j (1+ j)) |
| 1716 | (car (setq cp (cdr cp)))) |
| 1717 | sigdat)))))) |
| 1718 | (setq math-fit-new-coefs (math-evaluate-expr coef-filters)) |
| 1719 | (if calc-fit-to-trail |
| 1720 | (let ((bp math-fit-new-coefs) |
| 1721 | (cp coefs) |
| 1722 | (vec nil)) |
| 1723 | (while (setq bp (cdr bp) cp (cdr cp)) |
| 1724 | (setq vec (cons (list 'calcFunc-eq (car cp) (car bp)) vec))) |
| 1725 | (setq calc-fit-to-trail (cons 'vec (nreverse vec))))))) |
| 1726 | |
| 1727 | ;; Substitute best-fit coefficients back into original formula. |
| 1728 | (setq expr (math-multi-subst |
| 1729 | orig-expr |
| 1730 | (let ((n v) |
| 1731 | (vec nil)) |
| 1732 | (while (>= n 1) |
| 1733 | (setq vec (cons (list 'calcFunc-fitvar n) vec) |
| 1734 | n (1- n))) |
| 1735 | (setq n m) |
| 1736 | (while (>= n 1) |
| 1737 | (setq vec (cons (list 'calcFunc-fitparam n) vec) |
| 1738 | n (1- n))) |
| 1739 | vec) |
| 1740 | (append (cdr math-fit-new-coefs) (cdr vars)))) |
| 1741 | |
| 1742 | ;; Package the result. |
| 1743 | (math-normalize |
| 1744 | (if extended |
| 1745 | (list 'vec expr beta covar |
| 1746 | (let ((p coef-filters) |
| 1747 | (n 0)) |
| 1748 | (while (and (setq n (1+ n) p (cdr p)) |
| 1749 | (eq (car-safe (car p)) 'calcFunc-fitdummy) |
| 1750 | (eq (nth 1 (car p)) n))) |
| 1751 | (if p |
| 1752 | coef-filters |
| 1753 | (list 'vec))) |
| 1754 | chisq |
| 1755 | (if (and have-sdevs (> n mm)) |
| 1756 | (list 'calcFunc-utpc chisq (- n mm)) |
| 1757 | '(var nan var-nan))) |
| 1758 | expr)))) |
| 1759 | |
| 1760 | |
| 1761 | (defun calcFunc-fitvar (x) |
| 1762 | (if (>= math-in-fit 2) |
| 1763 | (progn |
| 1764 | (setq x (aref math-dummy-vars (+ math-fit-first-var x -1))) |
| 1765 | (or (calc-var-value (nth 2 x)) x)) |
| 1766 | (math-reject-arg x))) |
| 1767 | |
| 1768 | (defun calcFunc-fitparam (x) |
| 1769 | (if (>= math-in-fit 2) |
| 1770 | (progn |
| 1771 | (setq x (aref math-dummy-vars (+ math-fit-first-coef x -1))) |
| 1772 | (or (calc-var-value (nth 2 x)) x)) |
| 1773 | (math-reject-arg x))) |
| 1774 | |
| 1775 | (defun calcFunc-fitdummy (x) |
| 1776 | (if (= math-in-fit 3) |
| 1777 | (nth x math-fit-new-coefs) |
| 1778 | (math-reject-arg x))) |
| 1779 | |
| 1780 | (defun calcFunc-hasfitvars (expr) |
| 1781 | (if (Math-primp expr) |
| 1782 | 0 |
| 1783 | (if (eq (car expr) 'calcFunc-fitvar) |
| 1784 | (nth 1 expr) |
| 1785 | (apply 'max (mapcar 'calcFunc-hasfitvars (cdr expr)))))) |
| 1786 | |
| 1787 | (defun calcFunc-hasfitparams (expr) |
| 1788 | (if (Math-primp expr) |
| 1789 | 0 |
| 1790 | (if (eq (car expr) 'calcFunc-fitparam) |
| 1791 | (nth 1 expr) |
| 1792 | (apply 'max (mapcar 'calcFunc-hasfitparams (cdr expr)))))) |
| 1793 | |
| 1794 | |
| 1795 | (defun math-all-vars-but (expr but) |
| 1796 | (let* ((vars (math-all-vars-in expr)) |
| 1797 | (p but)) |
| 1798 | (while p |
| 1799 | (setq vars (delq (assoc (car-safe p) vars) vars) |
| 1800 | p (cdr p))) |
| 1801 | (sort (mapcar 'car vars) |
| 1802 | (function (lambda (x y) (string< (nth 1 x) (nth 1 y))))))) |
| 1803 | |
| 1804 | ;; The variables math-all-vars-vars (the vars for math-all-vars) and |
| 1805 | ;; math-all-vars-found are local to math-all-vars-in, but are used by |
| 1806 | ;; math-all-vars-rec which is called by math-all-vars-in. |
| 1807 | (defvar math-all-vars-vars) |
| 1808 | (defvar math-all-vars-found) |
| 1809 | |
| 1810 | (defun math-all-vars-in (expr) |
| 1811 | (let ((math-all-vars-vars nil) |
| 1812 | math-all-vars-found) |
| 1813 | (math-all-vars-rec expr) |
| 1814 | math-all-vars-vars)) |
| 1815 | |
| 1816 | (defun math-all-vars-rec (expr) |
| 1817 | (if (Math-primp expr) |
| 1818 | (if (eq (car-safe expr) 'var) |
| 1819 | (or (math-const-var expr) |
| 1820 | (if (setq math-all-vars-found (assoc expr math-all-vars-vars)) |
| 1821 | (setcdr math-all-vars-found (1+ (cdr math-all-vars-found))) |
| 1822 | (setq math-all-vars-vars (cons (cons expr 1) math-all-vars-vars))))) |
| 1823 | (while (setq expr (cdr expr)) |
| 1824 | (math-all-vars-rec (car expr))))) |
| 1825 | |
| 1826 | (provide 'calcalg3) |
| 1827 | |
| 1828 | ;;; arch-tag: ff9f2920-8111-48b5-b3fa-b0682c3e44a6 |
| 1829 | ;;; calcalg3.el ends here |