1 ;;; calc-mtx.el --- matrix functions for Calc
3 ;; Copyright (C) 1990, 1991, 1992, 1993, 2001 Free Software Foundation, Inc.
5 ;; Author: David Gillespie <daveg@synaptics.com>
6 ;; Maintainer: Jay Belanger <belanger@truman.edu>
8 ;; This file is part of GNU Emacs.
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.
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.
30 ;; This file is autoloaded from calc-ext.el.
35 (defun calc-Need-calc-mat () nil)
38 (defun calc-mdet (arg)
41 (calc-unary-op "mdet" 'calcFunc-det arg)))
43 (defun calc-mtrace (arg)
46 (calc-unary-op "mtr" 'calcFunc-tr arg)))
48 (defun calc-mlud (arg)
51 (calc-unary-op "mlud" 'calcFunc-lud arg)))
54 ;;; Coerce row vector A to be a matrix. [V V]
55 (defun math-row-matrix (a)
56 (if (and (Math-vectorp a)
57 (not (math-matrixp a)))
61 ;;; Coerce column vector A to be a matrix. [V V]
62 (defun math-col-matrix (a)
63 (if (and (Math-vectorp a)
64 (not (math-matrixp a)))
65 (cons 'vec (mapcar (function (lambda (x) (list 'vec x))) (cdr a)))
70 ;;; Multiply matrices A and B. [V V V]
71 (defun math-mul-mats (a b)
73 (cols (length (nth 1 b)))
75 (while (setq a (cdr a))
78 (while (> (setq col (1- col)) 0)
79 (setq ap (cdr (car a))
81 accum (math-mul (car ap) (nth col (car bp))))
82 (while (setq ap (cdr ap) bp (cdr bp))
83 (setq accum (math-add accum (math-mul (car ap) (nth col (car bp))))))
84 (setq row (cons accum row)))
85 (setq mat (cons (cons 'vec row) mat)))
86 (cons 'vec (nreverse mat))))
88 (defun math-mul-mat-vec (a b)
89 (cons 'vec (mapcar (function (lambda (row)
90 (math-dot-product row b)))
95 (defun calcFunc-tr (mat) ; [Public]
96 (if (math-square-matrixp mat)
97 (math-matrix-trace-step 2 (1- (length mat)) mat (nth 1 (nth 1 mat)))
98 (math-reject-arg mat 'square-matrixp)))
100 (defun math-matrix-trace-step (n size mat sum)
102 (math-matrix-trace-step (1+ n) size mat
103 (math-add sum (nth n (nth n mat))))
107 ;;; Matrix inverse and determinant.
108 (defun math-matrix-inv-raw (m)
109 (let ((n (1- (length m))))
111 (let ((det (math-det-raw m)))
112 (and (not (math-zerop det))
119 (math-neg (nth 2 (nth 1 m))))
121 (math-neg (nth 1 (nth 2 m)))
126 (math-sub (math-mul (nth 3 (nth 3 m))
128 (math-mul (nth 3 (nth 2 m))
130 (math-sub (math-mul (nth 3 (nth 1 m))
132 (math-mul (nth 3 (nth 3 m))
134 (math-sub (math-mul (nth 3 (nth 2 m))
136 (math-mul (nth 3 (nth 1 m))
139 (math-sub (math-mul (nth 3 (nth 2 m))
141 (math-mul (nth 3 (nth 3 m))
143 (math-sub (math-mul (nth 3 (nth 3 m))
145 (math-mul (nth 3 (nth 1 m))
147 (math-sub (math-mul (nth 3 (nth 1 m))
149 (math-mul (nth 3 (nth 2 m))
152 (math-sub (math-mul (nth 2 (nth 3 m))
154 (math-mul (nth 2 (nth 2 m))
156 (math-sub (math-mul (nth 2 (nth 1 m))
158 (math-mul (nth 2 (nth 3 m))
160 (math-sub (math-mul (nth 2 (nth 2 m))
162 (math-mul (nth 2 (nth 1 m))
163 (nth 1 (nth 2 m))))))))
165 (let ((lud (math-matrix-lud m)))
167 (math-lud-solve lud (calcFunc-idn 1 n)))))))
169 (defun calcFunc-det (m)
170 (if (math-square-matrixp m)
171 (math-with-extra-prec 2 (math-det-raw m))
172 (if (and (eq (car-safe m) 'calcFunc-idn)
173 (or (math-zerop (nth 1 m))
174 (math-equal-int (nth 1 m) 1)))
176 (math-reject-arg m 'square-matrixp))))
178 ;; The variable math-det-lu is local to math-det-raw, but is
179 ;; used by math-det-step, which is called by math-det-raw.
182 (defun math-det-raw (m)
183 (let ((n (1- (length m))))
187 (math-sub (math-mul (nth 1 (nth 1 m))
189 (math-mul (nth 2 (nth 1 m))
197 (math-mul (nth 1 (nth 1 m))
198 (math-mul (nth 2 (nth 2 m))
200 (math-mul (nth 2 (nth 1 m))
201 (math-mul (nth 3 (nth 2 m))
203 (math-mul (nth 3 (nth 1 m))
204 (math-mul (nth 1 (nth 2 m))
206 (math-mul (nth 3 (nth 1 m))
207 (math-mul (nth 2 (nth 2 m))
209 (math-mul (nth 1 (nth 1 m))
210 (math-mul (nth 3 (nth 2 m))
212 (math-mul (nth 2 (nth 1 m))
213 (math-mul (nth 1 (nth 2 m))
214 (nth 3 (nth 3 m))))))
215 (t (let ((lud (math-matrix-lud m)))
217 (let ((math-det-lu (car lud)))
218 (math-det-step n (nth 2 lud)))
221 (defun math-det-step (n prod)
223 (math-det-step (1- n) (math-mul prod (nth n (nth n math-det-lu))))
226 ;;; This returns a list (LU index d), or nil if not possible.
227 ;;; Argument M must be a square matrix.
228 (defvar math-lud-cache nil)
229 (defun math-matrix-lud (m)
230 (let ((old (assoc m math-lud-cache))
231 (context (list calc-internal-prec calc-prefer-frac)))
232 (if (and old (equal (nth 1 old) context))
234 (let* ((lud (catch 'singular (math-do-matrix-lud m)))
235 (entry (cons context lud)))
238 (setq math-lud-cache (cons (cons m entry) math-lud-cache)))
241 ;;; Numerical Recipes section 2.3; implicit pivoting omitted.
242 (defun math-do-matrix-lud (m)
243 (let* ((lu (math-copy-matrix m))
245 i (j 1) k imax sum big
252 (math-working "LUD step" (format "%d/%d" j i))
253 (setq sum (nth j (nth i lu))
256 (setq sum (math-sub sum (math-mul (nth k (nth i lu))
259 (setcar (nthcdr j (nth i lu)) sum)
262 (math-working "LUD step" (format "%d/%d" j i))
263 (setq sum (nth j (nth i lu))
266 (setq sum (math-sub sum (math-mul (nth k (nth i lu))
269 (setcar (nthcdr j (nth i lu)) sum)
270 (let ((dum (math-abs-approx sum)))
271 (if (Math-lessp big dum)
276 (setq lu (math-swap-rows lu j imax)
278 (setq index (cons imax index))
279 (let ((pivot (nth j (nth j lu))))
280 (if (math-zerop pivot)
281 (throw 'singular nil)
283 (while (<= (setq i (1+ i)) n)
284 (setcar (nthcdr j (nth i lu))
285 (math-div (nth j (nth i lu)) pivot)))))
287 (list lu (nreverse index) d)))
289 (defun math-swap-rows (m r1 r2)
291 (let* ((r1prev (nthcdr (1- r1) m))
293 (r2prev (nthcdr (1- r2) m))
298 (setcdr row2 (cdr row1))
299 (setcdr row1 r2next)))
303 (defun math-lud-solve (lud b &optional need)
305 (let* ((x (math-copy-matrix b))
307 (m (1- (length (nth 1 x))))
312 (math-working "LUD solver step" col)
319 sum (nth col (nth ip x)))
320 (setcar (nthcdr col (nth ip x)) (nth col (nth i x)))
326 (setq sum (math-sub sum (math-mul (nth j (nth i lu))
327 (nth col (nth j x))))
329 (setcar (nthcdr col (nth i x)) sum)
331 (while (>= (setq i (1- i)) 1)
332 (setq sum (nth col (nth i x))
334 (while (<= (setq j (1+ j)) n)
335 (setq sum (math-sub sum (math-mul (nth j (nth i lu))
336 (nth col (nth j x))))))
337 (setcar (nthcdr col (nth i x))
338 (math-div sum (nth i (nth i lu)))))
342 (math-reject-arg need "*Singular matrix"))))
344 (defun calcFunc-lud (m)
345 (if (math-square-matrixp m)
346 (or (math-with-extra-prec 2
347 (let ((lud (math-matrix-lud m)))
349 (let* ((lmat (math-copy-matrix (car lud)))
350 (umat (math-copy-matrix (car lud)))
351 (n (1- (length (car lud))))
352 (perm (calcFunc-idn 1 n))
357 (setcar (nthcdr j (nth i lmat)) 0)
359 (setcar (nthcdr j (nth j lmat)) 1)
360 (while (<= (setq i (1+ i)) n)
361 (setcar (nthcdr j (nth i umat)) 0))
363 (while (>= (setq j (1- j)) 1)
364 (let ((pos (nth (1- j) (nth 1 lud))))
366 (setq perm (math-swap-rows perm j pos)))))
367 (list 'vec perm lmat umat)))))
368 (math-reject-arg m "*Singular matrix"))
369 (math-reject-arg m 'square-matrixp)))
371 ;;; arch-tag: fc0947b1-90e1-4a23-8950-d8ead9c3a306
372 ;;; calc-mtx.el ends here