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.
29 ;; This file is autoloaded from calc-ext.el.
34 (defun calc-mdet (arg)
37 (calc-unary-op "mdet" 'calcFunc-det arg)))
39 (defun calc-mtrace (arg)
42 (calc-unary-op "mtr" 'calcFunc-tr arg)))
44 (defun calc-mlud (arg)
47 (calc-unary-op "mlud" 'calcFunc-lud arg)))
50 ;;; Coerce row vector A to be a matrix. [V V]
51 (defun math-row-matrix (a)
52 (if (and (Math-vectorp a)
53 (not (math-matrixp a)))
57 ;;; Coerce column vector A to be a matrix. [V V]
58 (defun math-col-matrix (a)
59 (if (and (Math-vectorp a)
60 (not (math-matrixp a)))
61 (cons 'vec (mapcar (function (lambda (x) (list 'vec x))) (cdr a)))
66 ;;; Multiply matrices A and B. [V V V]
67 (defun math-mul-mats (a b)
69 (cols (length (nth 1 b)))
71 (while (setq a (cdr a))
74 (while (> (setq col (1- col)) 0)
75 (setq ap (cdr (car a))
77 accum (math-mul (car ap) (nth col (car bp))))
78 (while (setq ap (cdr ap) bp (cdr bp))
79 (setq accum (math-add accum (math-mul (car ap) (nth col (car bp))))))
80 (setq row (cons accum row)))
81 (setq mat (cons (cons 'vec row) mat)))
82 (cons 'vec (nreverse mat))))
84 (defun math-mul-mat-vec (a b)
85 (cons 'vec (mapcar (function (lambda (row)
86 (math-dot-product row b)))
91 (defun calcFunc-tr (mat) ; [Public]
92 (if (math-square-matrixp mat)
93 (math-matrix-trace-step 2 (1- (length mat)) mat (nth 1 (nth 1 mat)))
94 (math-reject-arg mat 'square-matrixp)))
96 (defun math-matrix-trace-step (n size mat sum)
98 (math-matrix-trace-step (1+ n) size mat
99 (math-add sum (nth n (nth n mat))))
103 ;;; Matrix inverse and determinant.
104 (defun math-matrix-inv-raw (m)
105 (let ((n (1- (length m))))
107 (let ((det (math-det-raw m)))
108 (and (not (math-zerop det))
115 (math-neg (nth 2 (nth 1 m))))
117 (math-neg (nth 1 (nth 2 m)))
122 (math-sub (math-mul (nth 3 (nth 3 m))
124 (math-mul (nth 3 (nth 2 m))
126 (math-sub (math-mul (nth 3 (nth 1 m))
128 (math-mul (nth 3 (nth 3 m))
130 (math-sub (math-mul (nth 3 (nth 2 m))
132 (math-mul (nth 3 (nth 1 m))
135 (math-sub (math-mul (nth 3 (nth 2 m))
137 (math-mul (nth 3 (nth 3 m))
139 (math-sub (math-mul (nth 3 (nth 3 m))
141 (math-mul (nth 3 (nth 1 m))
143 (math-sub (math-mul (nth 3 (nth 1 m))
145 (math-mul (nth 3 (nth 2 m))
148 (math-sub (math-mul (nth 2 (nth 3 m))
150 (math-mul (nth 2 (nth 2 m))
152 (math-sub (math-mul (nth 2 (nth 1 m))
154 (math-mul (nth 2 (nth 3 m))
156 (math-sub (math-mul (nth 2 (nth 2 m))
158 (math-mul (nth 2 (nth 1 m))
159 (nth 1 (nth 2 m))))))))
161 (let ((lud (math-matrix-lud m)))
163 (math-lud-solve lud (calcFunc-idn 1 n)))))))
165 (defun calcFunc-det (m)
166 (if (math-square-matrixp m)
167 (math-with-extra-prec 2 (math-det-raw m))
168 (if (and (eq (car-safe m) 'calcFunc-idn)
169 (or (math-zerop (nth 1 m))
170 (math-equal-int (nth 1 m) 1)))
172 (math-reject-arg m 'square-matrixp))))
174 ;; The variable math-det-lu is local to math-det-raw, but is
175 ;; used by math-det-step, which is called by math-det-raw.
178 (defun math-det-raw (m)
179 (let ((n (1- (length m))))
183 (math-sub (math-mul (nth 1 (nth 1 m))
185 (math-mul (nth 2 (nth 1 m))
193 (math-mul (nth 1 (nth 1 m))
194 (math-mul (nth 2 (nth 2 m))
196 (math-mul (nth 2 (nth 1 m))
197 (math-mul (nth 3 (nth 2 m))
199 (math-mul (nth 3 (nth 1 m))
200 (math-mul (nth 1 (nth 2 m))
202 (math-mul (nth 3 (nth 1 m))
203 (math-mul (nth 2 (nth 2 m))
205 (math-mul (nth 1 (nth 1 m))
206 (math-mul (nth 3 (nth 2 m))
208 (math-mul (nth 2 (nth 1 m))
209 (math-mul (nth 1 (nth 2 m))
210 (nth 3 (nth 3 m))))))
211 (t (let ((lud (math-matrix-lud m)))
213 (let ((math-det-lu (car lud)))
214 (math-det-step n (nth 2 lud)))
217 (defun math-det-step (n prod)
219 (math-det-step (1- n) (math-mul prod (nth n (nth n math-det-lu))))
222 ;;; This returns a list (LU index d), or nil if not possible.
223 ;;; Argument M must be a square matrix.
224 (defvar math-lud-cache nil)
225 (defun math-matrix-lud (m)
226 (let ((old (assoc m math-lud-cache))
227 (context (list calc-internal-prec calc-prefer-frac)))
228 (if (and old (equal (nth 1 old) context))
230 (let* ((lud (catch 'singular (math-do-matrix-lud m)))
231 (entry (cons context lud)))
234 (setq math-lud-cache (cons (cons m entry) math-lud-cache)))
237 ;;; Numerical Recipes section 2.3; implicit pivoting omitted.
238 (defun math-do-matrix-lud (m)
239 (let* ((lu (math-copy-matrix m))
241 i (j 1) k imax sum big
248 (math-working "LUD step" (format "%d/%d" j i))
249 (setq sum (nth j (nth i lu))
252 (setq sum (math-sub sum (math-mul (nth k (nth i lu))
255 (setcar (nthcdr j (nth i lu)) sum)
258 (math-working "LUD step" (format "%d/%d" j i))
259 (setq sum (nth j (nth i lu))
262 (setq sum (math-sub sum (math-mul (nth k (nth i lu))
265 (setcar (nthcdr j (nth i lu)) sum)
266 (let ((dum (math-abs-approx sum)))
267 (if (Math-lessp big dum)
272 (setq lu (math-swap-rows lu j imax)
274 (setq index (cons imax index))
275 (let ((pivot (nth j (nth j lu))))
276 (if (math-zerop pivot)
277 (throw 'singular nil)
279 (while (<= (setq i (1+ i)) n)
280 (setcar (nthcdr j (nth i lu))
281 (math-div (nth j (nth i lu)) pivot)))))
283 (list lu (nreverse index) d)))
285 (defun math-swap-rows (m r1 r2)
287 (let* ((r1prev (nthcdr (1- r1) m))
289 (r2prev (nthcdr (1- r2) m))
294 (setcdr row2 (cdr row1))
295 (setcdr row1 r2next)))
299 (defun math-lud-solve (lud b &optional need)
301 (let* ((x (math-copy-matrix b))
303 (m (1- (length (nth 1 x))))
308 (math-working "LUD solver step" col)
315 sum (nth col (nth ip x)))
316 (setcar (nthcdr col (nth ip x)) (nth col (nth i x)))
322 (setq sum (math-sub sum (math-mul (nth j (nth i lu))
323 (nth col (nth j x))))
325 (setcar (nthcdr col (nth i x)) sum)
327 (while (>= (setq i (1- i)) 1)
328 (setq sum (nth col (nth i x))
330 (while (<= (setq j (1+ j)) n)
331 (setq sum (math-sub sum (math-mul (nth j (nth i lu))
332 (nth col (nth j x))))))
333 (setcar (nthcdr col (nth i x))
334 (math-div sum (nth i (nth i lu)))))
338 (math-reject-arg need "*Singular matrix"))))
340 (defun calcFunc-lud (m)
341 (if (math-square-matrixp m)
342 (or (math-with-extra-prec 2
343 (let ((lud (math-matrix-lud m)))
345 (let* ((lmat (math-copy-matrix (car lud)))
346 (umat (math-copy-matrix (car lud)))
347 (n (1- (length (car lud))))
348 (perm (calcFunc-idn 1 n))
353 (setcar (nthcdr j (nth i lmat)) 0)
355 (setcar (nthcdr j (nth j lmat)) 1)
356 (while (<= (setq i (1+ i)) n)
357 (setcar (nthcdr j (nth i umat)) 0))
359 (while (>= (setq j (1- j)) 1)
360 (let ((pos (nth (1- j) (nth 1 lud))))
362 (setq perm (math-swap-rows perm j pos)))))
363 (list 'vec perm lmat umat)))))
364 (math-reject-arg m "*Singular matrix"))
365 (math-reject-arg m 'square-matrixp)))
369 ;;; arch-tag: fc0947b1-90e1-4a23-8950-d8ead9c3a306
370 ;;; calc-mtx.el ends here