1 ;;; calc-mtx.el --- matrix functions for Calc
3 ;; Copyright (C) 1990, 1991, 1992, 1993, 2001, 2002, 2003, 2004,
4 ;; 2005, 2006, 2007, 2008, 2009, 2010 Free Software Foundation, Inc.
6 ;; Author: David Gillespie <daveg@synaptics.com>
7 ;; Maintainer: Jay Belanger <jay.p.belanger@gmail.com>
9 ;; This file is part of GNU Emacs.
11 ;; GNU Emacs is free software: you can redistribute it and/or modify
12 ;; it under the terms of the GNU General Public License as published by
13 ;; the Free Software Foundation, either version 3 of the License, or
14 ;; (at your option) any later version.
16 ;; GNU Emacs is distributed in the hope that it will be useful,
17 ;; but WITHOUT ANY WARRANTY; without even the implied warranty of
18 ;; MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
19 ;; GNU General Public License for more details.
21 ;; You should have received a copy of the GNU General Public License
22 ;; along with GNU Emacs. If not, see <http://www.gnu.org/licenses/>.
28 ;; This file is autoloaded from calc-ext.el.
33 (defun calc-mdet (arg)
36 (calc-unary-op "mdet" 'calcFunc-det arg)))
38 (defun calc-mtrace (arg)
41 (calc-unary-op "mtr" 'calcFunc-tr arg)))
43 (defun calc-mlud (arg)
46 (calc-unary-op "mlud" 'calcFunc-lud arg)))
49 ;;; Coerce row vector A to be a matrix. [V V]
50 (defun math-row-matrix (a)
51 (if (and (Math-vectorp a)
52 (not (math-matrixp a)))
56 ;;; Coerce column vector A to be a matrix. [V V]
57 (defun math-col-matrix (a)
58 (if (and (Math-vectorp a)
59 (not (math-matrixp a)))
60 (cons 'vec (mapcar (function (lambda (x) (list 'vec x))) (cdr a)))
65 ;;; Multiply matrices A and B. [V V V]
66 (defun math-mul-mats (a b)
68 (cols (length (nth 1 b)))
70 (while (setq a (cdr a))
73 (while (> (setq col (1- col)) 0)
74 (setq ap (cdr (car a))
76 accum (math-mul (car ap) (nth col (car bp))))
77 (while (setq ap (cdr ap) bp (cdr bp))
78 (setq accum (math-add accum (math-mul (car ap) (nth col (car bp))))))
79 (setq row (cons accum row)))
80 (setq mat (cons (cons 'vec row) mat)))
81 (cons 'vec (nreverse mat))))
83 (defun math-mul-mat-vec (a b)
84 (cons 'vec (mapcar (function (lambda (row)
85 (math-dot-product row b)))
90 (defun calcFunc-tr (mat) ; [Public]
91 (if (math-square-matrixp mat)
92 (math-matrix-trace-step 2 (1- (length mat)) mat (nth 1 (nth 1 mat)))
93 (math-reject-arg mat 'square-matrixp)))
95 (defun math-matrix-trace-step (n size mat sum)
97 (math-matrix-trace-step (1+ n) size mat
98 (math-add sum (nth n (nth n mat))))
102 ;;; Matrix inverse and determinant.
103 (defun math-matrix-inv-raw (m)
104 (let ((n (1- (length m))))
106 (let ((det (math-det-raw m)))
107 (and (not (math-zerop det))
114 (math-neg (nth 2 (nth 1 m))))
116 (math-neg (nth 1 (nth 2 m)))
121 (math-sub (math-mul (nth 3 (nth 3 m))
123 (math-mul (nth 3 (nth 2 m))
125 (math-sub (math-mul (nth 3 (nth 1 m))
127 (math-mul (nth 3 (nth 3 m))
129 (math-sub (math-mul (nth 3 (nth 2 m))
131 (math-mul (nth 3 (nth 1 m))
134 (math-sub (math-mul (nth 3 (nth 2 m))
136 (math-mul (nth 3 (nth 3 m))
138 (math-sub (math-mul (nth 3 (nth 3 m))
140 (math-mul (nth 3 (nth 1 m))
142 (math-sub (math-mul (nth 3 (nth 1 m))
144 (math-mul (nth 3 (nth 2 m))
147 (math-sub (math-mul (nth 2 (nth 3 m))
149 (math-mul (nth 2 (nth 2 m))
151 (math-sub (math-mul (nth 2 (nth 1 m))
153 (math-mul (nth 2 (nth 3 m))
155 (math-sub (math-mul (nth 2 (nth 2 m))
157 (math-mul (nth 2 (nth 1 m))
158 (nth 1 (nth 2 m))))))))
160 (let ((lud (math-matrix-lud m)))
162 (math-lud-solve lud (calcFunc-idn 1 n)))))))
164 (defun calcFunc-det (m)
165 (if (math-square-matrixp m)
166 (math-with-extra-prec 2 (math-det-raw m))
167 (if (and (eq (car-safe m) 'calcFunc-idn)
168 (or (math-zerop (nth 1 m))
169 (math-equal-int (nth 1 m) 1)))
171 (math-reject-arg m 'square-matrixp))))
173 ;; The variable math-det-lu is local to math-det-raw, but is
174 ;; used by math-det-step, which is called by math-det-raw.
177 (defun math-det-raw (m)
178 (let ((n (1- (length m))))
182 (math-sub (math-mul (nth 1 (nth 1 m))
184 (math-mul (nth 2 (nth 1 m))
192 (math-mul (nth 1 (nth 1 m))
193 (math-mul (nth 2 (nth 2 m))
195 (math-mul (nth 2 (nth 1 m))
196 (math-mul (nth 3 (nth 2 m))
198 (math-mul (nth 3 (nth 1 m))
199 (math-mul (nth 1 (nth 2 m))
201 (math-mul (nth 3 (nth 1 m))
202 (math-mul (nth 2 (nth 2 m))
204 (math-mul (nth 1 (nth 1 m))
205 (math-mul (nth 3 (nth 2 m))
207 (math-mul (nth 2 (nth 1 m))
208 (math-mul (nth 1 (nth 2 m))
209 (nth 3 (nth 3 m))))))
210 (t (let ((lud (math-matrix-lud m)))
212 (let ((math-det-lu (car lud)))
213 (math-det-step n (nth 2 lud)))
216 (defun math-det-step (n prod)
218 (math-det-step (1- n) (math-mul prod (nth n (nth n math-det-lu))))
221 ;;; This returns a list (LU index d), or nil if not possible.
222 ;;; Argument M must be a square matrix.
223 (defvar math-lud-cache nil)
224 (defun math-matrix-lud (m)
225 (let ((old (assoc m math-lud-cache))
226 (context (list calc-internal-prec calc-prefer-frac)))
227 (if (and old (equal (nth 1 old) context))
229 (let* ((lud (catch 'singular (math-do-matrix-lud m)))
230 (entry (cons context lud)))
233 (setq math-lud-cache (cons (cons m entry) math-lud-cache)))
236 ;;; Numerical Recipes section 2.3; implicit pivoting omitted.
237 (defun math-do-matrix-lud (m)
238 (let* ((lu (math-copy-matrix m))
240 i (j 1) k imax sum big
247 (math-working "LUD step" (format "%d/%d" j i))
248 (setq sum (nth j (nth i lu))
251 (setq sum (math-sub sum (math-mul (nth k (nth i lu))
254 (setcar (nthcdr j (nth i lu)) sum)
257 (math-working "LUD step" (format "%d/%d" j i))
258 (setq sum (nth j (nth i lu))
261 (setq sum (math-sub sum (math-mul (nth k (nth i lu))
264 (setcar (nthcdr j (nth i lu)) sum)
265 (let ((dum (math-abs-approx sum)))
266 (if (Math-lessp big dum)
271 (setq lu (math-swap-rows lu j imax)
273 (setq index (cons imax index))
274 (let ((pivot (nth j (nth j lu))))
275 (if (math-zerop pivot)
276 (throw 'singular nil)
278 (while (<= (setq i (1+ i)) n)
279 (setcar (nthcdr j (nth i lu))
280 (math-div (nth j (nth i lu)) pivot)))))
282 (list lu (nreverse index) d)))
284 (defun math-swap-rows (m r1 r2)
286 (let* ((r1prev (nthcdr (1- r1) m))
288 (r2prev (nthcdr (1- r2) m))
293 (setcdr row2 (cdr row1))
294 (setcdr row1 r2next)))
298 (defun math-lud-solve (lud b &optional need)
300 (let* ((x (math-copy-matrix b))
302 (m (1- (length (nth 1 x))))
307 (math-working "LUD solver step" col)
314 sum (nth col (nth ip x)))
315 (setcar (nthcdr col (nth ip x)) (nth col (nth i x)))
321 (setq sum (math-sub sum (math-mul (nth j (nth i lu))
322 (nth col (nth j x))))
324 (setcar (nthcdr col (nth i x)) sum)
326 (while (>= (setq i (1- i)) 1)
327 (setq sum (nth col (nth i x))
329 (while (<= (setq j (1+ j)) n)
330 (setq sum (math-sub sum (math-mul (nth j (nth i lu))
331 (nth col (nth j x))))))
332 (setcar (nthcdr col (nth i x))
333 (math-div sum (nth i (nth i lu)))))
337 (math-reject-arg need "*Singular matrix"))))
339 (defun calcFunc-lud (m)
340 (if (math-square-matrixp m)
341 (or (math-with-extra-prec 2
342 (let ((lud (math-matrix-lud m)))
344 (let* ((lmat (math-copy-matrix (car lud)))
345 (umat (math-copy-matrix (car lud)))
346 (n (1- (length (car lud))))
347 (perm (calcFunc-idn 1 n))
352 (setcar (nthcdr j (nth i lmat)) 0)
354 (setcar (nthcdr j (nth j lmat)) 1)
355 (while (<= (setq i (1+ i)) n)
356 (setcar (nthcdr j (nth i umat)) 0))
358 (while (>= (setq j (1- j)) 1)
359 (let ((pos (nth (1- j) (nth 1 lud))))
361 (setq perm (math-swap-rows perm j pos)))))
362 (list 'vec perm lmat umat)))))
363 (math-reject-arg m "*Singular matrix"))
364 (math-reject-arg m 'square-matrixp)))
368 ;; arch-tag: fc0947b1-90e1-4a23-8950-d8ead9c3a306
369 ;;; calc-mtx.el ends here