| blob | 339993f98a57b62bd37268c62b9e7518f74e8939 |
1 ;;; calc-rules.el --- rules for simplifying algebraic expressions in Calc
3 ;; Copyright (C) 1990, 1991, 1992, 1993, 2001, 2002, 2003, 2004,
4 ;; 2005, 2006 Free Software Foundation, Inc.
6 ;; Author: David Gillespie <daveg@synaptics.com>
7 ;; Maintainer: Jay Belanger <belanger@truman.edu>
9 ;; This file is part of GNU Emacs.
11 ;; GNU Emacs is distributed in the hope that it will be useful,
12 ;; but WITHOUT ANY WARRANTY. No author or distributor
13 ;; accepts responsibility to anyone for the consequences of using it
14 ;; or for whether it serves any particular purpose or works at all,
15 ;; unless he says so in writing. Refer to the GNU Emacs General Public
16 ;; License for full details.
18 ;; Everyone is granted permission to copy, modify and redistribute
19 ;; GNU Emacs, but only under the conditions described in the
20 ;; GNU Emacs General Public License. A copy of this license is
21 ;; supposed to have been given to you along with GNU Emacs so you
22 ;; can know your rights and responsibilities. It should be in a
23 ;; file named COPYING. Among other things, the copyright notice
24 ;; and this notice must be preserved on all copies.
26 ;;; Commentary:
28 ;;; Code:
30 ;; This file is autoloaded from calc-ext.el.
42 "CommuteRules"
45 iterations(1),
46 select(plain(a + b)) := select(plain(b + a)),
47 select(plain(a - b)) := select(plain((-b) + a)),
48 select(plain((1/a) * b)) := select(b / a),
49 select(plain(a * b)) := select(b * a),
50 select((1/a) / b) := select((1/b) / a),
51 select(a / b) := select((1/b) * a),
52 select((a^b) ^ c) := select((a^c) ^ b),
53 select(log(a, b)) := select(1 / log(b, a)),
54 select(plain(a && b)) := select(b && a),
55 select(plain(a || b)) := select(b || a),
56 select(plain(a = b)) := select(b = a),
57 select(plain(a != b)) := select(b != a),
58 select(a < b) := select(b > a),
59 select(a > b) := select(b < a),
60 select(a <= b) := select(b >= a),
64 "JumpRules"
67 iterations(1),
68 plain(select(x) = y) := 0 = select(-x) + y,
69 plain(a + select(x) = y) := a = select(-x) + y,
70 plain(a - select(x) = y) := a = select(x) + y,
71 plain(select(x) + a = y) := a = select(-x) + y,
72 plain(a * select(x) = y) := a = y / select(x),
73 plain(a / select(x) = y) := a = select(x) * y,
74 plain(select(x) / a = y) := 1/a = y / select(x),
75 plain(a ^ select(2) = y) := a = select(sqrt(y)),
76 plain(a ^ select(x) = y) := a = y ^ select(1/x),
77 plain(select(x) ^ a = y) := a = log(y, select(x)),
78 plain(log(a, select(x)) = y) := a = select(x) ^ y,
79 plain(log(select(x), a) = y) := a = select(x) ^ (1/y),
80 plain(y = select(x)) := y - select(x) = 0,
81 plain(y = a + select(x)) := y - select(x) = a,
82 plain(y = a - select(x)) := y + select(x) = a,
83 plain(y = select(x) + a) := y - select(x) = a,
84 plain(y = a * select(x)) := y / select(x) = a,
85 plain(y = a / select(x)) := y * select(x) = a,
86 plain(y = select(x) / a) := y / select(x) = 1/a,
87 plain(y = a ^ select(2)) := select(sqrt(y)) = a,
88 plain(y = a ^ select(x)) := y ^ select(1/x) = a,
89 plain(y = select(x) ^ a) := log(y, select(x)) = a,
90 plain(y = log(a, select(x))) := select(x) ^ y = a,
94 "DistribRules"
97 iterations(1),
98 x * select(a + b) := x*select(a) + x*b,
99 x * select(sum(a,b,c,d)) := sum(x*select(a),b,c,d),
100 x / select(a + b) := 1 / (select(a)/x + b/x),
101 select(a + b) / x := select(a)/x + b/x,
102 sum(select(a),b,c,d) / x := sum(select(a)/x,b,c,d),
103 x ^ select(a + b) := x^select(a) * x^b,
104 x ^ select(sum(a,b,c,d)) := prod(x^select(a),b,c,d),
105 x ^ select(a * b) := (x^a)^select(b),
106 x ^ select(a / b) := (x^a)^select(1/b),
107 select(a + b) ^ n := select(x)
108 :: integer(n) :: n >= 2
109 :: let(x, expandpow(a+b,n))
110 :: quote(matches(x,y+z)),
111 select(a + b) ^ x := a*select(a+b)^(x-1) + b*select(a+b)^(x-1),
112 select(a * b) ^ x := a^x * select(b)^x,
113 select(prod(a,b,c,d)) ^ x := prod(select(a)^x,b,c,d),
114 select(a / b) ^ x := select(a)^x / b^x,
115 select(- a) ^ x := (-1)^x * select(a)^x,
116 plain(-select(a + b)) := select(-a) - b,
117 plain(-select(sum(a,b,c,d))) := sum(select(-a),b,c,d),
118 plain(-select(a * b)) := select(-a) * b,
119 plain(-select(a / b)) := select(-a) / b,
120 sqrt(select(a * b)) := sqrt(select(a)) * sqrt(b),
121 sqrt(select(prod(a,b,c,d))) := prod(sqrt(select(a)),b,c,d),
122 sqrt(select(a / b)) := sqrt(select(a)) / sqrt(b),
123 sqrt(select(- a)) := sqrt(-1) sqrt(select(a)),
124 exp(select(a + b)) := exp(select(a)) / exp(-b) :: negative(b),
125 exp(select(a + b)) := exp(select(a)) * exp(b),
126 exp(select(sum(a,b,c,d))) := prod(exp(select(a)),b,c,d),
127 exp(select(a * b)) := exp(select(a)) ^ b :: constant(b),
128 exp(select(a * b)) := exp(select(a)) ^ b,
129 exp(select(a / b)) := exp(select(a)) ^ (1/b),
130 ln(select(a * b)) := ln(select(a)) + ln(b),
131 ln(select(prod(a,b,c,d))) := sum(ln(select(a)),b,c,d),
132 ln(select(a / b)) := ln(select(a)) - ln(b),
133 ln(select(a ^ b)) := ln(select(a)) * b,
134 log10(select(a * b)) := log10(select(a)) + log10(b),
135 log10(select(prod(a,b,c,d))) := sum(log10(select(a)),b,c,d),
136 log10(select(a / b)) := log10(select(a)) - log10(b),
137 log10(select(a ^ b)) := log10(select(a)) * b,
138 log(select(a * b), x) := log(select(a), x) + log(b,x),
139 log(select(prod(a,b,c,d)),x) := sum(log(select(a),x),b,c,d),
140 log(select(a / b), x) := log(select(a), x) - log(b,x),
141 log(select(a ^ b), x) := log(select(a), x) * b,
142 log(a, select(b)) := ln(a) / select(ln(b)),
143 sin(select(a + b)) := sin(select(a)) cos(b) + cos(a) sin(b),
144 sin(select(2 a)) := 2 sin(select(a)) cos(a),
145 sin(select(n a)) := 2sin((n-1) select(a)) cos(a) - sin((n-2) a)
146 :: integer(n) :: n > 2,
147 cos(select(a + b)) := cos(select(a)) cos(b) - sin(a) sin(b),
148 cos(select(2 a)) := 2 cos(select(a))^2 - 1,
149 cos(select(n a)) := 2cos((n-1) select(a)) cos(a) - cos((n-2) a)
150 :: integer(n) :: n > 2,
151 tan(select(a + b)) := (tan(select(a)) + tan(b)) /
152 (1 - tan(a) tan(b)),
153 tan(select(2 a)) := 2 tan(select(a)) / (1 - tan(a)^2),
154 tan(select(n a)) := (tan((n-1) select(a)) + tan(a)) /
155 (1 - tan((n-1) a) tan(a))
156 :: integer(n) :: n > 2,
157 cot(select(a + b)) := (cot(select(a)) cot(b) - 1) /
158 (cot(a) + cot(b)),
159 sinh(select(a + b)) := sinh(select(a)) cosh(b) + cosh(a) sinh(b),
160 cosh(select(a + b)) := cosh(select(a)) cosh(b) + sinh(a) sinh(b),
161 tanh(select(a + b)) := (tanh(select(a)) + tanh(b)) /
162 (1 + tanh(a) tanh(b)),
163 coth(select(a + b)) := (coth(select(a)) coth(b) + 1) /
164 (coth(a) + coth(b)),
165 x && select(a || b) := (x && select(a)) || (x && b),
166 select(a || b) && x := (select(a) && x) || (b && x),
167 ! select(a && b) := (!a) || (!b),
171 "MergeRules"
174 iterations(1),
175 (x*opt(a)) + select(x*b) := x * (a + select(b)),
176 (x*opt(a)) - select(x*b) := x * (a - select(b)),
177 sum(select(x)*a,b,c,d) := x * sum(select(a),b,c,d),
178 (a/x) + select(b/x) := (a + select(b)) / x,
179 (a/x) - select(b/x) := (a - select(b)) / x,
180 sum(a/select(x),b,c,d) := sum(select(a),b,c,d) / x,
181 (a/opt(b)) + select(c/d) := ((select(a)*d) + (b*c)) / (b*d),
182 (a/opt(b)) - select(c/d) := ((select(a)*d) - (b*c)) / (b*d),
183 (x^opt(a)) * select(x^b) := x ^ (a + select(b)),
184 (x^opt(a)) / select(x^b) := x ^ (a - select(b)),
185 select(x^a) / (x^opt(b)) := x ^ (select(a) - b),
186 prod(select(x)^a,b,c,d) := x ^ sum(select(a),b,c,d),
187 select(x^a) / (x^opt(b)) := x ^ (select(a) - b),
188 (a^x) * select(b^x) := select((a * b) ^x),
189 (a^x) / select(b^x) := select((b / b) ^ x),
190 select(a^x) / (b^x) := select((a / b) ^ x),
191 prod(a^select(x),b,c,d) := select(prod(a,b,c,d) ^ x),
192 (a^x) * select(b^y) := select((a * b^(y-x)) ^x),
193 (a^x) / select(b^y) := select((b / b^(y-x)) ^ x),
194 select(a^x) / (b^y) := select((a / b^(y-x)) ^ x),
195 select(x^a) ^ b := x ^ select(a * b),
196 (x^a) ^ select(b) := x ^ select(a * b),
197 select(sqrt(a)) ^ b := select(a ^ (b / 2)),
198 sqrt(a) ^ select(b) := select(a ^ (b / 2)),
199 sqrt(select(a) ^ b) := select(a ^ (b / 2)),
200 sqrt(a ^ select(b)) := select(a ^ (b / 2)),
201 sqrt(a) * select(sqrt(b)) := select(sqrt(a * b)),
202 sqrt(a) / select(sqrt(b)) := select(sqrt(a / b)),
203 select(sqrt(a)) / sqrt(b) := select(sqrt(a / b)),
204 prod(select(sqrt(a)),b,c,d) := select(sqrt(prod(a,b,c,d))),
205 exp(a) * select(exp(b)) := select(exp(a + b)),
206 exp(a) / select(exp(b)) := select(exp(a - b)),
207 select(exp(a)) / exp(b) := select(exp(a - b)),
208 prod(select(exp(a)),b,c,d) := select(exp(sum(a,b,c,d))),
209 select(exp(a)) ^ b := select(exp(a * b)),
210 exp(a) ^ select(b) := select(exp(a * b)),
211 ln(a) + select(ln(b)) := select(ln(a * b)),
212 ln(a) - select(ln(b)) := select(ln(a / b)),
213 select(ln(a)) - ln(b) := select(ln(a / b)),
214 sum(select(ln(a)),b,c,d) := select(ln(prod(a,b,c,d))),
215 b * select(ln(a)) := select(ln(a ^ b)),
216 select(b) * ln(a) := select(ln(a ^ b)),
217 select(ln(a)) / ln(b) := select(log(a, b)),
218 ln(a) / select(ln(b)) := select(log(a, b)),
219 select(ln(a)) / b := select(ln(a ^ (1/b))),
220 ln(a) / select(b) := select(ln(a ^ (1/b))),
221 log10(a) + select(log10(b)) := select(log10(a * b)),
222 log10(a) - select(log10(b)) := select(log10(a / b)),
223 select(log10(a)) - log10(b) := select(log10(a / b)),
224 sum(select(log10(a)),b,c,d) := select(log10(prod(a,b,c,d))),
225 b * select(log10(a)) := select(log10(a ^ b)),
226 select(b) * log10(a) := select(log10(a ^ b)),
227 select(log10(a)) / log10(b) := select(log(a, b)),
228 log10(a) / select(log10(b)) := select(log(a, b)),
229 select(log10(a)) / b := select(log10(a ^ (1/b))),
230 log10(a) / select(b) := select(log10(a ^ (1/b))),
231 log(a,x) + select(log(b,x)) := select(log(a * b,x)),
232 log(a,x) - select(log(b,x)) := select(log(a / b,x)),
233 select(log(a,x)) - log(b,x) := select(log(a / b,x)),
234 sum(select(log(a,x)),b,c,d) := select(log(prod(a,b,c,d),x)),
235 b * select(log(a,x)) := select(log(a ^ b,x)),
236 select(b) * log(a,x) := select(log(a ^ b,x)),
237 select(log(a,x)) / log(b,x) := select(log(a, b)),
238 log(a,x) / select(log(b,x)) := select(log(a, b)),
239 select(log(a,x)) / b := select(log(a ^ (1/b),x)),
240 log(a,x) / select(b) := select(log(a ^ (1/b),x)),
244 "NegateRules"
247 iterations(1),
248 a + select(x) := a - select(-x),
249 a - select(x) := a + select(-x),
250 sum(select(x),b,c,d) := -sum(select(-x),b,c,d),
251 a * select(x) := -a * select(-x),
252 a / select(x) := -a / select(-x),
253 select(x) / a := -select(-x) / a,
254 prod(select(x),b,c,d) := (-1)^(d-c+1) * prod(select(-x),b,c,d),
255 select(x) ^ n := select(-x) ^ a :: integer(n) :: n%2 = 0,
256 select(x) ^ n := -(select(-x) ^ a) :: integer(n) :: n%2 = 1,
257 select(x) ^ a := (-select(-x)) ^ a,
258 a ^ select(x) := (1 / a)^select(-x),
259 abs(select(x)) := abs(select(-x)),
260 i sqrt(select(x)) := -sqrt(select(-x)),
261 sqrt(select(x)) := i sqrt(select(-x)),
262 re(select(x)) := -re(select(-x)),
263 im(select(x)) := -im(select(-x)),
264 conj(select(x)) := -conj(select(-x)),
265 trunc(select(x)) := -trunc(select(-x)),
266 round(select(x)) := -round(select(-x)),
267 floor(select(x)) := -ceil(select(-x)),
268 ceil(select(x)) := -floor(select(-x)),
269 ftrunc(select(x)) := -ftrunc(select(-x)),
270 fround(select(x)) := -fround(select(-x)),
271 ffloor(select(x)) := -fceil(select(-x)),
272 fceil(select(x)) := -ffloor(select(-x)),
273 exp(select(x)) := 1 / exp(select(-x)),
274 sin(select(x)) := -sin(select(-x)),
275 cos(select(x)) := cos(select(-x)),
276 tan(select(x)) := -tan(select(-x)),
277 sec(select(x)) := sec(select(-x)),
278 csc(select(x)) := -csc(select(-x)),
279 cot(select(x)) := -cot(select(-x)),
280 arcsin(select(x)) := -arcsin(select(-x)),
281 arccos(select(x)) := 4 arctan(1) - arccos(select(-x)),
282 arctan(select(x)) := -arctan(select(-x)),
283 sinh(select(x)) := -sinh(select(-x)),
284 cosh(select(x)) := cosh(select(-x)),
285 tanh(select(x)) := -tanh(select(-x)),
286 sech(select(x)) := sech(select(-x)),
287 csch(select(x)) := -csch(select(-x)),
288 coth(select(x)) := -coth(select(-x)),
289 arcsinh(select(x)) := -arcsinh(select(-x)),
290 arctanh(select(x)) := -arctanh(select(-x)),
291 select(x) = a := select(-x) = -a,
292 select(x) != a := select(-x) != -a,
293 select(x) < a := select(-x) > -a,
294 select(x) > a := select(-x) < -a,
295 select(x) <= a := select(-x) >= -a,
296 select(x) >= a := select(-x) <= -a,
297 a < select(x) := -a > select(-x),
298 a > select(x) := -a < select(-x),
299 a <= select(x) := -a >= select(-x),
300 a >= select(x) := -a <= select(-x),
304 "InvertRules"
307 iterations(1),
308 a * select(x) := a / select(1/x),
309 a / select(x) := a * select(1/x),
310 select(x) / a := 1 / (select(1/x) a),
311 prod(select(x),b,c,d) := 1 / prod(select(1/x),b,c,d),
312 abs(select(x)) := 1 / abs(select(1/x)),
313 sqrt(select(x)) := 1 / sqrt(select(1/x)),
314 ln(select(x)) := -ln(select(1/x)),
315 log10(select(x)) := -log10(select(1/x)),
316 log(select(x), a) := -log(select(1/x), a),
317 log(a, select(x)) := -log(a, select(1/x)),
318 arctan(select(x)) := simplify(2 arctan(1))-arctan(select(1/x)),
319 select(x) = a := select(1/x) = 1/a,
320 select(x) != a := select(1/x) != 1/a,
321 select(x) < a := select(1/x) > 1/a,
322 select(x) > a := select(1/x) < 1/a,
323 select(x) <= a := select(1/x) >= 1/a,
324 select(x) >= a := select(1/x) <= 1/a,
325 a < select(x) := 1/a > select(1/x),
326 a > select(x) := 1/a < select(1/x),
327 a <= select(x) := 1/a >= select(1/x),
328 a >= select(x) := 1/a <= select(1/x),
333 "FactorRules"
336 thecoefs(x, [z, a+b, c]) := thefactors(x, [d x + d a/c, (c/d) x + (b/d)])
337 :: z = a b/c :: let(d := pgcd(pcont(c), pcont(b))),
338 thecoefs(x, [z, a, c]) := thefactors(x, [(r x + a/(2 r))^2])
339 :: z = (a/2)^2/c :: let(r := esimplify(sqrt(c)))
340 :: !matches(r, sqrt(rr)),
341 thecoefs(x, [z, 0, c]) := thefactors(x, [rc x + rz, rc x - rz])
342 :: negative(z)
343 :: let(rz := esimplify(sqrt(-z))) :: !matches(rz, sqrt(rzz))
344 :: let(rc := esimplify(sqrt(c))) :: !matches(rc, sqrt(rcc)),
345 thecoefs(x, [z, 0, c]) := thefactors(x, [rz + rc x, rz - rc x])
346 :: negative(c)
347 :: let(rz := esimplify(sqrt(z))) :: !matches(rz, sqrt(rzz))
348 :: let(rc := esimplify(sqrt(-c))) :: !matches(rc, sqrt(rcc))
350 ;;(setq var-FactorRules 'calc-FactorRules)
354 "IntegAfterRules"
357 opt(a) ln(x) + opt(b) ln(y) := 2 a esimplify(arctanh(x-1))
358 :: a + b = 0 :: nrat(x + y) = 2 || nrat(x - y) = 2,
359 a * (b + c) := a b + a c :: constant(a)
362 ;;(setq var-IntegAfterRules 'calc-IntegAfterRules)
366 "FitRules"
370 schedule(1,2,3,4),
371 iterations(inf),
373 phase(1),
374 e^x := exp(x),
375 x^y := exp(y ln(x)) :: !istrue(constant(y)),
376 x/y := x fitinv(y),
377 fitinv(x y) := fitinv(x) fitinv(y),
378 exp(a) exp(b) := exp(a + b),
379 a exp(b) := exp(ln(a) + b) :: !hasfitvars(a),
380 fitinv(exp(a)) := exp(-a),
381 ln(a b) := ln(a) + ln(b),
382 ln(fitinv(a)) := -ln(a),
383 log10(a b) := log10(a) + log10(b),
384 log10(fitinv(a)) := -log10(a),
385 log(a,b) := ln(a)/ln(b),
386 ln(exp(a)) := a,
387 a*(b+c) := a*b + a*c,
388 (a+b)^n := x :: integer(n) :: n >= 2
389 :: let(x, expandpow(a+b,n))
390 :: quote(matches(x,y+z)),
392 phase(1,2),
393 fitmodel(y = x) := fitmodel(0, y - x),
394 fitmodel(y, x+c) := fitmodel(y-c, x) :: !hasfitparams(c),
395 fitmodel(y, x c) := fitmodel(y/c, x) :: !hasfitparams(c),
396 fitmodel(y, x/(c opt(d))) := fitmodel(y c, x/d) :: !hasfitparams(c),
397 fitmodel(y, apply(f,[x])) := fitmodel(yy, x)
398 :: hasfitparams(x)
399 :: let(FTemp() = yy,
400 solve(apply(f,[FTemp()]) = y,
401 FTemp())),
402 fitmodel(y, apply(f,[x,c])) := fitmodel(yy, x)
403 :: !hasfitparams(c)
404 :: let(FTemp() = yy,
405 solve(apply(f,[FTemp(),c]) = y,
406 FTemp())),
407 fitmodel(y, apply(f,[c,x])) := fitmodel(yy, x)
408 :: !hasfitparams(c)
409 :: let(FTemp() = yy,
410 solve(apply(f,[c,FTemp()]) = y,
411 FTemp())),
413 phase(2,3),
414 fitmodel(y, x) := fitsystem(y, [], [], fitpart(1,1,x)),
415 fitpart(a,b,plain(x + y)) := fitpart(a,b,x) + fitpart(a,b,y),
416 fitpart(a,b,plain(x - y)) := fitpart(a,b,x) + fitpart(-a,b,y),
417 fitpart(a,b,plain(-x)) := fitpart(-a,b,x),
418 fitpart(a,b,x opt(c)) := fitpart(a,x b,c) :: !hasfitvars(x),
419 fitpart(a,x opt(b),c) := fitpart(x a,b,c) :: !hasfitparams(x),
420 fitpart(a,x y + x opt(z),c) := fitpart(a,x*(y+z),c),
421 fitpart(a,b,c) := fitpart2(a,b,c),
423 phase(3),
424 fitpart2(a1,b1,x) + fitpart2(a2,b2,x) := fitpart(1, a1 b1 + a2 b2, x),
425 fitpart2(a1,x,c1) + fitpart2(a2,x,c2) := fitpart2(1, x, a1 c1 + a2 c2),
427 phase(4),
428 fitinv(x) := 1 / x,
429 exp(x + ln(y)) := y exp(x),
430 exp(x ln(y)) := y^x,
431 ln(x) + ln(y) := ln(x y),
432 ln(x) - ln(y) := ln(x/y),
433 x*y + x*z := x*(y+z),
434 fitsystem(y, xv, pv, fitpart2(a,fitparam(b),c) + opt(d))
435 := fitsystem(y, rcons(xv, a c),
436 rcons(pv, fitdummy(b) = fitparam(b)), d)
437 :: b = vlen(pv)+1,
438 fitsystem(y, xv, pv, fitpart2(a,b,c) + opt(d))
439 := fitsystem(y, rcons(xv, a c),
440 rcons(pv, fitdummy(vlen(pv)+1) = b), d),
441 fitsystem(y, xv, pv, 0) := fitsystem(y, xv, cons(fvh,fvt))
442 :: !hasfitparams(xv)
443 :: let(cons(fvh,fvt),
444 solve(pv, table(fitparam(j), j, 1,
445 hasfitparams(pv)))),
450 ;;; arch-tag: 0ed54a52-38f3-4ed7-9ca7-b8ecf8f2febe
451 ;;; calc-rules.el ends here