| blob | 81994497591f7bc1f804f1cd683014db1bb161a9 |
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, 2007, 2008, 2009, 2010, 2011 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/>.
24 ;;; Commentary:
26 ;;; Code:
28 ;; This file is autoloaded from calc-ext.el.
40 "CommuteRules"
43 iterations(1),
44 select(plain(a + b)) := select(plain(b + a)),
45 select(plain(a - b)) := select(plain((-b) + a)),
46 select(plain((1/a) * b)) := select(b / a),
47 select(plain(a * b)) := select(b * a),
48 select((1/a) / b) := select((1/b) / a),
49 select(a / b) := select((1/b) * a),
50 select((a^b) ^ c) := select((a^c) ^ b),
51 select(log(a, b)) := select(1 / log(b, a)),
52 select(plain(a && b)) := select(b && a),
53 select(plain(a || b)) := select(b || a),
54 select(plain(a = b)) := select(b = a),
55 select(plain(a != b)) := select(b != a),
56 select(a < b) := select(b > a),
57 select(a > b) := select(b < a),
58 select(a <= b) := select(b >= a),
62 "JumpRules"
65 iterations(1),
66 plain(select(x) = y) := 0 = select(-x) + y,
67 plain(a + select(x) = y) := a = select(-x) + y,
68 plain(a - select(x) = y) := a = select(x) + y,
69 plain(select(x) + a = y) := a = select(-x) + y,
70 plain(a * select(x) = y) := a = y / select(x),
71 plain(a / select(x) = y) := a = select(x) * y,
72 plain(select(x) / a = y) := 1/a = y / select(x),
73 plain(a ^ select(2) = y) := a = select(sqrt(y)),
74 plain(a ^ select(x) = y) := a = y ^ select(1/x),
75 plain(select(x) ^ a = y) := a = log(y, select(x)),
76 plain(log(a, select(x)) = y) := a = select(x) ^ y,
77 plain(log(select(x), a) = y) := a = select(x) ^ (1/y),
78 plain(y = select(x)) := y - select(x) = 0,
79 plain(y = a + select(x)) := y - select(x) = a,
80 plain(y = a - select(x)) := y + select(x) = a,
81 plain(y = select(x) + a) := y - select(x) = a,
82 plain(y = a * select(x)) := y / select(x) = a,
83 plain(y = a / select(x)) := y * select(x) = a,
84 plain(y = select(x) / a) := y / select(x) = 1/a,
85 plain(y = a ^ select(2)) := select(sqrt(y)) = a,
86 plain(y = a ^ select(x)) := y ^ select(1/x) = a,
87 plain(y = select(x) ^ a) := log(y, select(x)) = a,
88 plain(y = log(a, select(x))) := select(x) ^ y = a,
92 "DistribRules"
95 iterations(1),
96 x * select(a + b) := x*select(a) + x*b,
97 x * select(sum(a,b,c,d)) := sum(x*select(a),b,c,d),
98 x / select(a + b) := 1 / (select(a)/x + b/x),
99 select(a + b) / x := select(a)/x + b/x,
100 sum(select(a),b,c,d) / x := sum(select(a)/x,b,c,d),
101 x ^ select(a + b) := x^select(a) * x^b,
102 x ^ select(sum(a,b,c,d)) := prod(x^select(a),b,c,d),
103 x ^ select(a * b) := (x^a)^select(b),
104 x ^ select(a / b) := (x^a)^select(1/b),
105 select(a + b) ^ n := select(x)
106 :: integer(n) :: n >= 2
107 :: let(x, expandpow(a+b,n))
108 :: quote(matches(x,y+z)),
109 select(a + b) ^ x := a*select(a+b)^(x-1) + b*select(a+b)^(x-1),
110 select(a * b) ^ x := a^x * select(b)^x,
111 select(prod(a,b,c,d)) ^ x := prod(select(a)^x,b,c,d),
112 select(a / b) ^ x := select(a)^x / b^x,
113 select(- a) ^ x := (-1)^x * select(a)^x,
114 plain(-select(a + b)) := select(-a) - b,
115 plain(-select(sum(a,b,c,d))) := sum(select(-a),b,c,d),
116 plain(-select(a * b)) := select(-a) * b,
117 plain(-select(a / b)) := select(-a) / b,
118 sqrt(select(a * b)) := sqrt(select(a)) * sqrt(b),
119 sqrt(select(prod(a,b,c,d))) := prod(sqrt(select(a)),b,c,d),
120 sqrt(select(a / b)) := sqrt(select(a)) / sqrt(b),
121 sqrt(select(- a)) := sqrt(-1) sqrt(select(a)),
122 exp(select(a + b)) := exp(select(a)) / exp(-b) :: negative(b),
123 exp(select(a + b)) := exp(select(a)) * exp(b),
124 exp(select(sum(a,b,c,d))) := prod(exp(select(a)),b,c,d),
125 exp(select(a * b)) := exp(select(a)) ^ b :: constant(b),
126 exp(select(a * b)) := exp(select(a)) ^ b,
127 exp(select(a / b)) := exp(select(a)) ^ (1/b),
128 ln(select(a * b)) := ln(select(a)) + ln(b),
129 ln(select(prod(a,b,c,d))) := sum(ln(select(a)),b,c,d),
130 ln(select(a / b)) := ln(select(a)) - ln(b),
131 ln(select(a ^ b)) := ln(select(a)) * b,
132 log10(select(a * b)) := log10(select(a)) + log10(b),
133 log10(select(prod(a,b,c,d))) := sum(log10(select(a)),b,c,d),
134 log10(select(a / b)) := log10(select(a)) - log10(b),
135 log10(select(a ^ b)) := log10(select(a)) * b,
136 log(select(a * b), x) := log(select(a), x) + log(b,x),
137 log(select(prod(a,b,c,d)),x) := sum(log(select(a),x),b,c,d),
138 log(select(a / b), x) := log(select(a), x) - log(b,x),
139 log(select(a ^ b), x) := log(select(a), x) * b,
140 log(a, select(b)) := ln(a) / select(ln(b)),
141 sin(select(a + b)) := sin(select(a)) cos(b) + cos(a) sin(b),
142 sin(select(2 a)) := 2 sin(select(a)) cos(a),
143 sin(select(n a)) := 2sin((n-1) select(a)) cos(a) - sin((n-2) a)
144 :: integer(n) :: n > 2,
145 cos(select(a + b)) := cos(select(a)) cos(b) - sin(a) sin(b),
146 cos(select(2 a)) := 2 cos(select(a))^2 - 1,
147 cos(select(n a)) := 2cos((n-1) select(a)) cos(a) - cos((n-2) a)
148 :: integer(n) :: n > 2,
149 tan(select(a + b)) := (tan(select(a)) + tan(b)) /
150 (1 - tan(a) tan(b)),
151 tan(select(2 a)) := 2 tan(select(a)) / (1 - tan(a)^2),
152 tan(select(n a)) := (tan((n-1) select(a)) + tan(a)) /
153 (1 - tan((n-1) a) tan(a))
154 :: integer(n) :: n > 2,
155 cot(select(a + b)) := (cot(select(a)) cot(b) - 1) /
156 (cot(a) + cot(b)),
157 sinh(select(a + b)) := sinh(select(a)) cosh(b) + cosh(a) sinh(b),
158 cosh(select(a + b)) := cosh(select(a)) cosh(b) + sinh(a) sinh(b),
159 tanh(select(a + b)) := (tanh(select(a)) + tanh(b)) /
160 (1 + tanh(a) tanh(b)),
161 coth(select(a + b)) := (coth(select(a)) coth(b) + 1) /
162 (coth(a) + coth(b)),
163 x && select(a || b) := (x && select(a)) || (x && b),
164 select(a || b) && x := (select(a) && x) || (b && x),
165 ! select(a && b) := (!a) || (!b),
169 "MergeRules"
172 iterations(1),
173 (x*opt(a)) + select(x*b) := x * (a + select(b)),
174 (x*opt(a)) - select(x*b) := x * (a - select(b)),
175 sum(select(x)*a,b,c,d) := x * sum(select(a),b,c,d),
176 (a/x) + select(b/x) := (a + select(b)) / x,
177 (a/x) - select(b/x) := (a - select(b)) / x,
178 sum(a/select(x),b,c,d) := sum(select(a),b,c,d) / x,
179 (a/opt(b)) + select(c/d) := ((select(a)*d) + (b*c)) / (b*d),
180 (a/opt(b)) - select(c/d) := ((select(a)*d) - (b*c)) / (b*d),
181 (x^opt(a)) * select(x^b) := x ^ (a + select(b)),
182 (x^opt(a)) / select(x^b) := x ^ (a - select(b)),
183 select(x^a) / (x^opt(b)) := x ^ (select(a) - b),
184 prod(select(x)^a,b,c,d) := x ^ sum(select(a),b,c,d),
185 select(x^a) / (x^opt(b)) := x ^ (select(a) - b),
186 (a^x) * select(b^x) := select((a * b) ^x),
187 (a^x) / select(b^x) := select((b / b) ^ x),
188 select(a^x) / (b^x) := select((a / b) ^ x),
189 prod(a^select(x),b,c,d) := select(prod(a,b,c,d) ^ x),
190 (a^x) * select(b^y) := select((a * b^(y-x)) ^x),
191 (a^x) / select(b^y) := select((b / b^(y-x)) ^ x),
192 select(a^x) / (b^y) := select((a / b^(y-x)) ^ x),
193 select(x^a) ^ b := x ^ select(a * b),
194 (x^a) ^ select(b) := x ^ select(a * b),
195 select(sqrt(a)) ^ b := select(a ^ (b / 2)),
196 sqrt(a) ^ select(b) := select(a ^ (b / 2)),
197 sqrt(select(a) ^ b) := select(a ^ (b / 2)),
198 sqrt(a ^ select(b)) := select(a ^ (b / 2)),
199 sqrt(a) * select(sqrt(b)) := select(sqrt(a * b)),
200 sqrt(a) / select(sqrt(b)) := select(sqrt(a / b)),
201 select(sqrt(a)) / sqrt(b) := select(sqrt(a / b)),
202 prod(select(sqrt(a)),b,c,d) := select(sqrt(prod(a,b,c,d))),
203 exp(a) * select(exp(b)) := select(exp(a + b)),
204 exp(a) / select(exp(b)) := select(exp(a - b)),
205 select(exp(a)) / exp(b) := select(exp(a - b)),
206 prod(select(exp(a)),b,c,d) := select(exp(sum(a,b,c,d))),
207 select(exp(a)) ^ b := select(exp(a * b)),
208 exp(a) ^ select(b) := select(exp(a * b)),
209 ln(a) + select(ln(b)) := select(ln(a * b)),
210 ln(a) - select(ln(b)) := select(ln(a / b)),
211 select(ln(a)) - ln(b) := select(ln(a / b)),
212 sum(select(ln(a)),b,c,d) := select(ln(prod(a,b,c,d))),
213 b * select(ln(a)) := select(ln(a ^ b)),
214 select(b) * ln(a) := select(ln(a ^ b)),
215 select(ln(a)) / ln(b) := select(log(a, b)),
216 ln(a) / select(ln(b)) := select(log(a, b)),
217 select(ln(a)) / b := select(ln(a ^ (1/b))),
218 ln(a) / select(b) := select(ln(a ^ (1/b))),
219 log10(a) + select(log10(b)) := select(log10(a * b)),
220 log10(a) - select(log10(b)) := select(log10(a / b)),
221 select(log10(a)) - log10(b) := select(log10(a / b)),
222 sum(select(log10(a)),b,c,d) := select(log10(prod(a,b,c,d))),
223 b * select(log10(a)) := select(log10(a ^ b)),
224 select(b) * log10(a) := select(log10(a ^ b)),
225 select(log10(a)) / log10(b) := select(log(a, b)),
226 log10(a) / select(log10(b)) := select(log(a, b)),
227 select(log10(a)) / b := select(log10(a ^ (1/b))),
228 log10(a) / select(b) := select(log10(a ^ (1/b))),
229 log(a,x) + select(log(b,x)) := select(log(a * b,x)),
230 log(a,x) - select(log(b,x)) := select(log(a / b,x)),
231 select(log(a,x)) - log(b,x) := select(log(a / b,x)),
232 sum(select(log(a,x)),b,c,d) := select(log(prod(a,b,c,d),x)),
233 b * select(log(a,x)) := select(log(a ^ b,x)),
234 select(b) * log(a,x) := select(log(a ^ b,x)),
235 select(log(a,x)) / log(b,x) := select(log(a, b)),
236 log(a,x) / select(log(b,x)) := select(log(a, b)),
237 select(log(a,x)) / b := select(log(a ^ (1/b),x)),
238 log(a,x) / select(b) := select(log(a ^ (1/b),x)),
242 "NegateRules"
245 iterations(1),
246 a + select(x) := a - select(-x),
247 a - select(x) := a + select(-x),
248 sum(select(x),b,c,d) := -sum(select(-x),b,c,d),
249 a * select(x) := -a * select(-x),
250 a / select(x) := -a / select(-x),
251 select(x) / a := -select(-x) / a,
252 prod(select(x),b,c,d) := (-1)^(d-c+1) * prod(select(-x),b,c,d),
253 select(x) ^ n := select(-x) ^ a :: integer(n) :: n%2 = 0,
254 select(x) ^ n := -(select(-x) ^ a) :: integer(n) :: n%2 = 1,
255 select(x) ^ a := (-select(-x)) ^ a,
256 a ^ select(x) := (1 / a)^select(-x),
257 abs(select(x)) := abs(select(-x)),
258 i sqrt(select(x)) := -sqrt(select(-x)),
259 sqrt(select(x)) := i sqrt(select(-x)),
260 re(select(x)) := -re(select(-x)),
261 im(select(x)) := -im(select(-x)),
262 conj(select(x)) := -conj(select(-x)),
263 trunc(select(x)) := -trunc(select(-x)),
264 round(select(x)) := -round(select(-x)),
265 floor(select(x)) := -ceil(select(-x)),
266 ceil(select(x)) := -floor(select(-x)),
267 ftrunc(select(x)) := -ftrunc(select(-x)),
268 fround(select(x)) := -fround(select(-x)),
269 ffloor(select(x)) := -fceil(select(-x)),
270 fceil(select(x)) := -ffloor(select(-x)),
271 exp(select(x)) := 1 / exp(select(-x)),
272 sin(select(x)) := -sin(select(-x)),
273 cos(select(x)) := cos(select(-x)),
274 tan(select(x)) := -tan(select(-x)),
275 sec(select(x)) := sec(select(-x)),
276 csc(select(x)) := -csc(select(-x)),
277 cot(select(x)) := -cot(select(-x)),
278 arcsin(select(x)) := -arcsin(select(-x)),
279 arccos(select(x)) := 4 arctan(1) - arccos(select(-x)),
280 arctan(select(x)) := -arctan(select(-x)),
281 sinh(select(x)) := -sinh(select(-x)),
282 cosh(select(x)) := cosh(select(-x)),
283 tanh(select(x)) := -tanh(select(-x)),
284 sech(select(x)) := sech(select(-x)),
285 csch(select(x)) := -csch(select(-x)),
286 coth(select(x)) := -coth(select(-x)),
287 arcsinh(select(x)) := -arcsinh(select(-x)),
288 arctanh(select(x)) := -arctanh(select(-x)),
289 select(x) = a := select(-x) = -a,
290 select(x) != a := select(-x) != -a,
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 a < select(x) := -a > select(-x),
296 a > select(x) := -a < select(-x),
297 a <= select(x) := -a >= select(-x),
298 a >= select(x) := -a <= select(-x),
302 "InvertRules"
305 iterations(1),
306 a * select(x) := a / select(1/x),
307 a / select(x) := a * select(1/x),
308 select(x) / a := 1 / (select(1/x) a),
309 prod(select(x),b,c,d) := 1 / prod(select(1/x),b,c,d),
310 abs(select(x)) := 1 / abs(select(1/x)),
311 sqrt(select(x)) := 1 / sqrt(select(1/x)),
312 ln(select(x)) := -ln(select(1/x)),
313 log10(select(x)) := -log10(select(1/x)),
314 log(select(x), a) := -log(select(1/x), a),
315 log(a, select(x)) := -log(a, select(1/x)),
316 arctan(select(x)) := simplify(2 arctan(1))-arctan(select(1/x)),
317 select(x) = a := select(1/x) = 1/a,
318 select(x) != a := select(1/x) != 1/a,
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 a < select(x) := 1/a > select(1/x),
324 a > select(x) := 1/a < select(1/x),
325 a <= select(x) := 1/a >= select(1/x),
326 a >= select(x) := 1/a <= select(1/x),
331 "FactorRules"
334 thecoefs(x, [z, a+b, c]) := thefactors(x, [d x + d a/c, (c/d) x + (b/d)])
335 :: z = a b/c :: let(d := pgcd(pcont(c), pcont(b))),
336 thecoefs(x, [z, a, c]) := thefactors(x, [(r x + a/(2 r))^2])
337 :: z = (a/2)^2/c :: let(r := esimplify(sqrt(c)))
338 :: !matches(r, sqrt(rr)),
339 thecoefs(x, [z, 0, c]) := thefactors(x, [rc x + rz, rc x - rz])
340 :: negative(z)
341 :: let(rz := esimplify(sqrt(-z))) :: !matches(rz, sqrt(rzz))
342 :: let(rc := esimplify(sqrt(c))) :: !matches(rc, sqrt(rcc)),
343 thecoefs(x, [z, 0, c]) := thefactors(x, [rz + rc x, rz - rc x])
344 :: negative(c)
345 :: let(rz := esimplify(sqrt(z))) :: !matches(rz, sqrt(rzz))
346 :: let(rc := esimplify(sqrt(-c))) :: !matches(rc, sqrt(rcc))
348 ;;(setq var-FactorRules 'calc-FactorRules)
352 "IntegAfterRules"
355 opt(a) ln(x) + opt(b) ln(y) := 2 a esimplify(arctanh(x-1))
356 :: a + b = 0 :: nrat(x + y) = 2 || nrat(x - y) = 2,
357 a * (b + c) := a b + a c :: constant(a)
360 ;;(setq var-IntegAfterRules 'calc-IntegAfterRules)
364 "FitRules"
368 schedule(1,2,3,4),
369 iterations(inf),
371 phase(1),
372 e^x := exp(x),
373 x^y := exp(y ln(x)) :: !istrue(constant(y)),
374 x/y := x fitinv(y),
375 fitinv(x y) := fitinv(x) fitinv(y),
376 exp(a) exp(b) := exp(a + b),
377 a exp(b) := exp(ln(a) + b) :: !hasfitvars(a),
378 fitinv(exp(a)) := exp(-a),
379 ln(a b) := ln(a) + ln(b),
380 ln(fitinv(a)) := -ln(a),
381 log10(a b) := log10(a) + log10(b),
382 log10(fitinv(a)) := -log10(a),
383 log(a,b) := ln(a)/ln(b),
384 ln(exp(a)) := a,
385 a*(b+c) := a*b + a*c,
386 (a+b)^n := x :: integer(n) :: n >= 2
387 :: let(x, expandpow(a+b,n))
388 :: quote(matches(x,y+z)),
390 phase(1,2),
391 fitmodel(y = x) := fitmodel(0, y - x),
392 fitmodel(y, x+c) := fitmodel(y-c, x) :: !hasfitparams(c),
393 fitmodel(y, x c) := fitmodel(y/c, x) :: !hasfitparams(c),
394 fitmodel(y, x/(c opt(d))) := fitmodel(y c, x/d) :: !hasfitparams(c),
395 fitmodel(y, apply(f,[x])) := fitmodel(yy, x)
396 :: hasfitparams(x)
397 :: let(FTemp() = yy,
398 solve(apply(f,[FTemp()]) = y,
399 FTemp())),
400 fitmodel(y, apply(f,[x,c])) := fitmodel(yy, x)
401 :: !hasfitparams(c)
402 :: let(FTemp() = yy,
403 solve(apply(f,[FTemp(),c]) = y,
404 FTemp())),
405 fitmodel(y, apply(f,[c,x])) := fitmodel(yy, x)
406 :: !hasfitparams(c)
407 :: let(FTemp() = yy,
408 solve(apply(f,[c,FTemp()]) = y,
409 FTemp())),
411 phase(2,3),
412 fitmodel(y, x) := fitsystem(y, [], [], fitpart(1,1,x)),
413 fitpart(a,b,plain(x + y)) := fitpart(a,b,x) + fitpart(a,b,y),
414 fitpart(a,b,plain(x - y)) := fitpart(a,b,x) + fitpart(-a,b,y),
415 fitpart(a,b,plain(-x)) := fitpart(-a,b,x),
416 fitpart(a,b,x opt(c)) := fitpart(a,x b,c) :: !hasfitvars(x),
417 fitpart(a,x opt(b),c) := fitpart(x a,b,c) :: !hasfitparams(x),
418 fitpart(a,x y + x opt(z),c) := fitpart(a,x*(y+z),c),
419 fitpart(a,b,c) := fitpart2(a,b,c),
421 phase(3),
422 fitpart2(a1,b1,x) + fitpart2(a2,b2,x) := fitpart(1, a1 b1 + a2 b2, x),
423 fitpart2(a1,x,c1) + fitpart2(a2,x,c2) := fitpart2(1, x, a1 c1 + a2 c2),
425 phase(4),
426 fitinv(x) := 1 / x,
427 exp(x + ln(y)) := y exp(x),
428 exp(x ln(y)) := y^x,
429 ln(x) + ln(y) := ln(x y),
430 ln(x) - ln(y) := ln(x/y),
431 x*y + x*z := x*(y+z),
432 fitsystem(y, xv, pv, fitpart2(a,fitparam(b),c) + opt(d))
433 := fitsystem(y, rcons(xv, a c),
434 rcons(pv, fitdummy(b) = fitparam(b)), d)
435 :: b = vlen(pv)+1,
436 fitsystem(y, xv, pv, fitpart2(a,b,c) + opt(d))
437 := fitsystem(y, rcons(xv, a c),
438 rcons(pv, fitdummy(vlen(pv)+1) = b), d),
439 fitsystem(y, xv, pv, 0) := fitsystem(y, xv, cons(fvh,fvt))
440 :: !hasfitparams(xv)
441 :: let(cons(fvh,fvt),
442 solve(pv, table(fitparam(j), j, 1,
443 hasfitparams(pv)))),
448 ;;; calc-rules.el ends here