calcalg2.el 123 KB
Newer Older
1 2
;;; calcalg2.el --- more algebraic functions for Calc

Jay Belanger's avatar
Jay Belanger committed
3 4
;; Copyright (C) 1990, 1991, 1992, 1993, 2001, 2002, 2003, 2004,
;;   2005 Free Software Foundation, Inc.
5 6

;; Author: David Gillespie <daveg@synaptics.com>
7
;; Maintainer: Jay Belanger <belanger@truman.edu>
Eli Zaretskii's avatar
Eli Zaretskii committed
8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

;; This file is part of GNU Emacs.

;; GNU Emacs is distributed in the hope that it will be useful,
;; but WITHOUT ANY WARRANTY.  No author or distributor
;; accepts responsibility to anyone for the consequences of using it
;; or for whether it serves any particular purpose or works at all,
;; unless he says so in writing.  Refer to the GNU Emacs General Public
;; License for full details.

;; Everyone is granted permission to copy, modify and redistribute
;; GNU Emacs, but only under the conditions described in the
;; GNU Emacs General Public License.   A copy of this license is
;; supposed to have been given to you along with GNU Emacs so you
;; can know your rights and responsibilities.  It should be in a
;; file named COPYING.  Among other things, the copyright notice
;; and this notice must be preserved on all copies.

26
;;; Commentary:
Eli Zaretskii's avatar
Eli Zaretskii committed
27

28
;;; Code:
Eli Zaretskii's avatar
Eli Zaretskii committed
29 30 31

;; This file is autoloaded from calc-ext.el.

Jay Belanger's avatar
Jay Belanger committed
32
(require 'calc-ext)
Eli Zaretskii's avatar
Eli Zaretskii committed
33 34 35 36 37
(require 'calc-macs)

(defun calc-derivative (var num)
  (interactive "sDifferentiate with respect to: \np")
  (calc-slow-wrapper
38 39
   (when (< num 0)
     (error "Order of derivative must be positive"))
Eli Zaretskii's avatar
Eli Zaretskii committed
40 41 42 43 44 45 46
   (let ((func (if (calc-is-hyperbolic) 'calcFunc-tderiv 'calcFunc-deriv))
	 n expr)
     (if (or (equal var "") (equal var "$"))
	 (setq n 2
	       expr (calc-top-n 2)
	       var (calc-top-n 1))
       (setq var (math-read-expr var))
47 48
       (when (eq (car-safe var) 'error)
	 (error "Bad format in expression: %s" (nth 1 var)))
Eli Zaretskii's avatar
Eli Zaretskii committed
49 50 51 52
       (setq n 1
	     expr (calc-top-n 1)))
     (while (>= (setq num (1- num)) 0)
       (setq expr (list func expr var)))
53
     (calc-enter-result n "derv" expr))))
Eli Zaretskii's avatar
Eli Zaretskii committed
54 55 56 57 58 59 60 61 62 63 64 65 66

(defun calc-integral (var)
  (interactive "sIntegration variable: ")
  (calc-slow-wrapper
   (if (or (equal var "") (equal var "$"))
       (calc-enter-result 2 "intg" (list 'calcFunc-integ
					 (calc-top-n 2)
					 (calc-top-n 1)))
     (let ((var (math-read-expr var)))
       (if (eq (car-safe var) 'error)
	   (error "Bad format in expression: %s" (nth 1 var)))
       (calc-enter-result 1 "intg" (list 'calcFunc-integ
					 (calc-top-n 1)
67
					 var))))))
Eli Zaretskii's avatar
Eli Zaretskii committed
68 69 70 71

(defun calc-num-integral (&optional varname lowname highname)
  (interactive "sIntegration variable: ")
  (calc-tabular-command 'calcFunc-ninteg "Integration" "nint"
72
			nil varname lowname highname))
Eli Zaretskii's avatar
Eli Zaretskii committed
73 74 75 76

(defun calc-summation (arg &optional varname lowname highname)
  (interactive "P\nsSummation variable: ")
  (calc-tabular-command 'calcFunc-sum "Summation" "sum"
77
			arg varname lowname highname))
Eli Zaretskii's avatar
Eli Zaretskii committed
78 79 80 81

(defun calc-alt-summation (arg &optional varname lowname highname)
  (interactive "P\nsSummation variable: ")
  (calc-tabular-command 'calcFunc-asum "Summation" "asum"
82
			arg varname lowname highname))
Eli Zaretskii's avatar
Eli Zaretskii committed
83 84 85 86

(defun calc-product (arg &optional varname lowname highname)
  (interactive "P\nsIndex variable: ")
  (calc-tabular-command 'calcFunc-prod "Index" "prod"
87
			arg varname lowname highname))
Eli Zaretskii's avatar
Eli Zaretskii committed
88 89 90 91

(defun calc-tabulate (arg &optional varname lowname highname)
  (interactive "P\nsIndex variable: ")
  (calc-tabular-command 'calcFunc-table "Index" "tabl"
92
			arg varname lowname highname))
Eli Zaretskii's avatar
Eli Zaretskii committed
93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149

(defun calc-tabular-command (func prompt prefix arg varname lowname highname)
  (calc-slow-wrapper
   (let (var (low nil) (high nil) (step nil) stepname stepnum (num 1) expr)
     (if (consp arg)
	 (setq stepnum 1)
       (setq stepnum 0))
     (if (or (equal varname "") (equal varname "$") (null varname))
	 (setq high (calc-top-n (+ stepnum 1))
	       low (calc-top-n (+ stepnum 2))
	       var (calc-top-n (+ stepnum 3))
	       num (+ stepnum 4))
       (setq var (if (stringp varname) (math-read-expr varname) varname))
       (if (eq (car-safe var) 'error)
	   (error "Bad format in expression: %s" (nth 1 var)))
       (or lowname
	   (setq lowname (read-string (concat prompt " variable: " varname
					      ", from: "))))
       (if (or (equal lowname "") (equal lowname "$"))
	   (setq high (calc-top-n (+ stepnum 1))
		 low (calc-top-n (+ stepnum 2))
		 num (+ stepnum 3))
	 (setq low (if (stringp lowname) (math-read-expr lowname) lowname))
	 (if (eq (car-safe low) 'error)
	     (error "Bad format in expression: %s" (nth 1 low)))
	 (or highname
	     (setq highname (read-string (concat prompt " variable: " varname
						 ", from: " lowname
						 ", to: "))))
	 (if (or (equal highname "") (equal highname "$"))
	     (setq high (calc-top-n (+ stepnum 1))
		   num (+ stepnum 2))
	   (setq high (if (stringp highname) (math-read-expr highname)
			highname))
	   (if (eq (car-safe high) 'error)
	       (error "Bad format in expression: %s" (nth 1 high)))
	   (if (consp arg)
	       (progn
		 (setq stepname (read-string (concat prompt " variable: "
						     varname
						     ", from: " lowname
						     ", to: " highname
						     ", step: ")))
		 (if (or (equal stepname "") (equal stepname "$"))
		     (setq step (calc-top-n 1)
			   num 2)
		   (setq step (math-read-expr stepname))
		   (if (eq (car-safe step) 'error)
		       (error "Bad format in expression: %s"
			      (nth 1 step)))))))))
     (or step
	 (if (consp arg)
	     (setq step (calc-top-n 1))
	   (if arg
	       (setq step (prefix-numeric-value arg)))))
     (setq expr (calc-top-n num))
     (calc-enter-result num prefix (append (list func expr var low high)
150
					   (and step (list step)))))))
Eli Zaretskii's avatar
Eli Zaretskii committed
151 152

(defun calc-solve-for (var)
153
  (interactive "sVariable(s) to solve for: ")
Eli Zaretskii's avatar
Eli Zaretskii committed
154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169
  (calc-slow-wrapper
   (let ((func (if (calc-is-inverse)
		   (if (calc-is-hyperbolic) 'calcFunc-ffinv 'calcFunc-finv)
		 (if (calc-is-hyperbolic) 'calcFunc-fsolve 'calcFunc-solve))))
     (if (or (equal var "") (equal var "$"))
	 (calc-enter-result 2 "solv" (list func
					   (calc-top-n 2)
					   (calc-top-n 1)))
       (let ((var (if (and (string-match ",\\|[^ ] +[^ ]" var)
			   (not (string-match "\\[" var)))
		      (math-read-expr (concat "[" var "]"))
		    (math-read-expr var))))
	 (if (eq (car-safe var) 'error)
	     (error "Bad format in expression: %s" (nth 1 var)))
	 (calc-enter-result 1 "solv" (list func
					   (calc-top-n 1)
170
					   var)))))))
Eli Zaretskii's avatar
Eli Zaretskii committed
171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186

(defun calc-poly-roots (var)
  (interactive "sVariable to solve for: ")
  (calc-slow-wrapper
   (if (or (equal var "") (equal var "$"))
       (calc-enter-result 2 "prts" (list 'calcFunc-roots
					 (calc-top-n 2)
					 (calc-top-n 1)))
     (let ((var (if (and (string-match ",\\|[^ ] +[^ ]" var)
			 (not (string-match "\\[" var)))
		    (math-read-expr (concat "[" var "]"))
		  (math-read-expr var))))
       (if (eq (car-safe var) 'error)
	   (error "Bad format in expression: %s" (nth 1 var)))
       (calc-enter-result 1 "prts" (list 'calcFunc-roots
					 (calc-top-n 1)
187
					 var))))))
Eli Zaretskii's avatar
Eli Zaretskii committed
188 189 190 191 192 193 194 195 196 197

(defun calc-taylor (var nterms)
  (interactive "sTaylor expansion variable: \nNNumber of terms: ")
  (calc-slow-wrapper
   (let ((var (math-read-expr var)))
     (if (eq (car-safe var) 'error)
	 (error "Bad format in expression: %s" (nth 1 var)))
     (calc-enter-result 1 "tylr" (list 'calcFunc-taylor
				       (calc-top-n 1)
				       var
198
				       (prefix-numeric-value nterms))))))
Eli Zaretskii's avatar
Eli Zaretskii committed
199 200


201 202 203 204 205
;; The following are global variables used by math-derivative and some 
;; related functions
(defvar math-deriv-var)
(defvar math-deriv-total)
(defvar math-deriv-symb)
206 207
(defvar math-decls-cache)
(defvar math-decls-all)
208 209 210

(defun math-derivative (expr)
  (cond ((equal expr math-deriv-var)
Eli Zaretskii's avatar
Eli Zaretskii committed
211 212 213 214
	 1)
	((or (Math-scalarp expr)
	     (eq (car expr) 'sdev)
	     (and (eq (car expr) 'var)
215
		  (or (not math-deriv-total)
Eli Zaretskii's avatar
Eli Zaretskii committed
216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286
		      (math-const-var expr)
		      (progn
			(math-setup-declarations)
			(memq 'const (nth 1 (or (assq (nth 2 expr)
						      math-decls-cache)
						math-decls-all)))))))
	 0)
	((eq (car expr) '+)
	 (math-add (math-derivative (nth 1 expr))
		   (math-derivative (nth 2 expr))))
	((eq (car expr) '-)
	 (math-sub (math-derivative (nth 1 expr))
		   (math-derivative (nth 2 expr))))
	((memq (car expr) '(calcFunc-eq calcFunc-neq calcFunc-lt
					calcFunc-gt calcFunc-leq calcFunc-geq))
	 (list (car expr)
	       (math-derivative (nth 1 expr))
	       (math-derivative (nth 2 expr))))
	((eq (car expr) 'neg)
	 (math-neg (math-derivative (nth 1 expr))))
	((eq (car expr) '*)
	 (math-add (math-mul (nth 2 expr)
			     (math-derivative (nth 1 expr)))
		   (math-mul (nth 1 expr)
			     (math-derivative (nth 2 expr)))))
	((eq (car expr) '/)
	 (math-sub (math-div (math-derivative (nth 1 expr))
			     (nth 2 expr))
		   (math-div (math-mul (nth 1 expr)
				       (math-derivative (nth 2 expr)))
			     (math-sqr (nth 2 expr)))))
	((eq (car expr) '^)
	 (let ((du (math-derivative (nth 1 expr)))
	       (dv (math-derivative (nth 2 expr))))
	   (or (Math-zerop du)
	       (setq du (math-mul (nth 2 expr)
				  (math-mul (math-normalize
					     (list '^
						   (nth 1 expr)
						   (math-add (nth 2 expr) -1)))
					    du))))
	   (or (Math-zerop dv)
	       (setq dv (math-mul (math-normalize
				   (list 'calcFunc-ln (nth 1 expr)))
				  (math-mul expr dv))))
	   (math-add du dv)))
	((eq (car expr) '%)
	 (math-derivative (nth 1 expr)))   ; a reasonable definition
	((eq (car expr) 'vec)
	 (math-map-vec 'math-derivative expr))
	((and (memq (car expr) '(calcFunc-conj calcFunc-re calcFunc-im))
	      (= (length expr) 2))
	 (list (car expr) (math-derivative (nth 1 expr))))
	((and (memq (car expr) '(calcFunc-subscr calcFunc-mrow calcFunc-mcol))
	      (= (length expr) 3))
	 (let ((d (math-derivative (nth 1 expr))))
	   (if (math-numberp d)
	       0    ; assume x and x_1 are independent vars
	     (list (car expr) d (nth 2 expr)))))
	(t (or (and (symbolp (car expr))
		    (if (= (length expr) 2)
			(let ((handler (get (car expr) 'math-derivative)))
			  (and handler
			       (let ((deriv (math-derivative (nth 1 expr))))
				 (if (Math-zerop deriv)
				     deriv
				   (math-mul (funcall handler (nth 1 expr))
					     deriv)))))
		      (let ((handler (get (car expr) 'math-derivative-n)))
			(and handler
			     (funcall handler expr)))))
287
	       (and (not (eq math-deriv-symb 'pre-expand))
Eli Zaretskii's avatar
Eli Zaretskii committed
288 289
		    (let ((exp (math-expand-formula expr)))
		      (and exp
290
			   (or (let ((math-deriv-symb 'pre-expand))
Eli Zaretskii's avatar
Eli Zaretskii committed
291 292 293 294 295
				 (catch 'math-deriv (math-derivative expr)))
			       (math-derivative exp)))))
	       (if (or (Math-objvecp expr)
		       (eq (car expr) 'var)
		       (not (symbolp (car expr))))
296
		   (if math-deriv-symb
Eli Zaretskii's avatar
Eli Zaretskii committed
297
		       (throw 'math-deriv nil)
298
		     (list (if math-deriv-total 'calcFunc-tderiv 'calcFunc-deriv)
Eli Zaretskii's avatar
Eli Zaretskii committed
299
			   expr
300
			   math-deriv-var))
Eli Zaretskii's avatar
Eli Zaretskii committed
301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329
		 (let ((accum 0)
		       (arg expr)
		       (n 1)
		       derv)
		   (while (setq arg (cdr arg))
		     (or (Math-zerop (setq derv (math-derivative (car arg))))
			 (let ((func (intern (concat (symbol-name (car expr))
						     "'"
						     (if (> n 1)
							 (int-to-string n)
						       ""))))
			       (prop (cond ((= (length expr) 2)
					    'math-derivative-1)
					   ((= (length expr) 3)
					    'math-derivative-2)
					   ((= (length expr) 4)
					    'math-derivative-3)
					   ((= (length expr) 5)
					    'math-derivative-4)
					   ((= (length expr) 6)
					    'math-derivative-5))))
			   (setq accum
				 (math-add
				  accum
				  (math-mul
				   derv
				   (let ((handler (get func prop)))
				     (or (and prop handler
					      (apply handler (cdr expr)))
330
					 (if (and math-deriv-symb
Eli Zaretskii's avatar
Eli Zaretskii committed
331 332 333 334 335
						  (not (get func
							    'calc-user-defn)))
					     (throw 'math-deriv nil)
					   (cons func (cdr expr))))))))))
		     (setq n (1+ n)))
336
		   accum))))))
Eli Zaretskii's avatar
Eli Zaretskii committed
337

338 339
(defun calcFunc-deriv (expr math-deriv-var &optional deriv-value math-deriv-symb)
  (let* ((math-deriv-total nil)
Eli Zaretskii's avatar
Eli Zaretskii committed
340 341 342 343 344 345
	 (res (catch 'math-deriv (math-derivative expr))))
    (or (eq (car-safe res) 'calcFunc-deriv)
	(null res)
	(setq res (math-normalize res)))
    (and res
	 (if deriv-value
346
	     (math-expr-subst res math-deriv-var deriv-value)
347
	   res))))
Eli Zaretskii's avatar
Eli Zaretskii committed
348

349
(defun calcFunc-tderiv (expr math-deriv-var &optional deriv-value math-deriv-symb)
Eli Zaretskii's avatar
Eli Zaretskii committed
350
  (math-setup-declarations)
351
  (let* ((math-deriv-total t)
Eli Zaretskii's avatar
Eli Zaretskii committed
352 353 354 355 356 357
	 (res (catch 'math-deriv (math-derivative expr))))
    (or (eq (car-safe res) 'calcFunc-tderiv)
	(null res)
	(setq res (math-normalize res)))
    (and res
	 (if deriv-value
358
	     (math-expr-subst res math-deriv-var deriv-value)
359
	   res))))
Eli Zaretskii's avatar
Eli Zaretskii committed
360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413

(put 'calcFunc-inv\' 'math-derivative-1
     (function (lambda (u) (math-neg (math-div 1 (math-sqr u))))))

(put 'calcFunc-sqrt\' 'math-derivative-1
     (function (lambda (u) (math-div 1 (math-mul 2 (list 'calcFunc-sqrt u))))))

(put 'calcFunc-deg\' 'math-derivative-1
     (function (lambda (u) (math-div-float '(float 18 1) (math-pi)))))

(put 'calcFunc-rad\' 'math-derivative-1
     (function (lambda (u) (math-pi-over-180))))

(put 'calcFunc-ln\' 'math-derivative-1
     (function (lambda (u) (math-div 1 u))))

(put 'calcFunc-log10\' 'math-derivative-1
     (function (lambda (u)
		 (math-div (math-div 1 (math-normalize '(calcFunc-ln 10)))
			   u))))

(put 'calcFunc-lnp1\' 'math-derivative-1
     (function (lambda (u) (math-div 1 (math-add u 1)))))

(put 'calcFunc-log\' 'math-derivative-2
     (function (lambda (x b)
		 (and (not (Math-zerop b))
		      (let ((lnv (math-normalize
				  (list 'calcFunc-ln b))))
			(math-div 1 (math-mul lnv x)))))))

(put 'calcFunc-log\'2 'math-derivative-2
     (function (lambda (x b)
		 (let ((lnv (list 'calcFunc-ln b)))
		   (math-neg (math-div (list 'calcFunc-log x b)
				       (math-mul lnv b)))))))

(put 'calcFunc-exp\' 'math-derivative-1
     (function (lambda (u) (math-normalize (list 'calcFunc-exp u)))))

(put 'calcFunc-expm1\' 'math-derivative-1
     (function (lambda (u) (math-normalize (list 'calcFunc-expm1 u)))))

(put 'calcFunc-sin\' 'math-derivative-1
     (function (lambda (u) (math-to-radians-2 (math-normalize
					       (list 'calcFunc-cos u))))))

(put 'calcFunc-cos\' 'math-derivative-1
     (function (lambda (u) (math-neg (math-to-radians-2
				      (math-normalize
				       (list 'calcFunc-sin u)))))))

(put 'calcFunc-tan\' 'math-derivative-1
     (function (lambda (u) (math-to-radians-2
414 415 416
			    (math-sqr
                             (math-normalize
                              (list 'calcFunc-sec u)))))))
Eli Zaretskii's avatar
Eli Zaretskii committed
417

418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437
(put 'calcFunc-sec\' 'math-derivative-1
     (function (lambda (u) (math-to-radians-2 
                            (math-mul
                             (math-normalize
                              (list 'calcFunc-sec u))
                             (math-normalize
                              (list 'calcFunc-tan u)))))))

(put 'calcFunc-csc\' 'math-derivative-1
     (function (lambda (u) (math-neg 
                            (math-to-radians-2
                             (math-mul
                              (math-normalize
                               (list 'calcFunc-csc u))
                              (math-normalize
                               (list 'calcFunc-cot u))))))))

(put 'calcFunc-cot\' 'math-derivative-1
     (function (lambda (u) (math-neg
                            (math-to-radians-2
438 439 440
                             (math-sqr
                              (math-normalize
                               (list 'calcFunc-csc u))))))))
441

Eli Zaretskii's avatar
Eli Zaretskii committed
442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466
(put 'calcFunc-arcsin\' 'math-derivative-1
     (function (lambda (u)
		 (math-from-radians-2
		  (math-div 1 (math-normalize
			       (list 'calcFunc-sqrt
				     (math-sub 1 (math-sqr u)))))))))

(put 'calcFunc-arccos\' 'math-derivative-1
     (function (lambda (u)
		 (math-from-radians-2
		  (math-div -1 (math-normalize
				(list 'calcFunc-sqrt
				      (math-sub 1 (math-sqr u)))))))))

(put 'calcFunc-arctan\' 'math-derivative-1
     (function (lambda (u) (math-from-radians-2
			    (math-div 1 (math-add 1 (math-sqr u)))))))

(put 'calcFunc-sinh\' 'math-derivative-1
     (function (lambda (u) (math-normalize (list 'calcFunc-cosh u)))))

(put 'calcFunc-cosh\' 'math-derivative-1
     (function (lambda (u) (math-normalize (list 'calcFunc-sinh u)))))

(put 'calcFunc-tanh\' 'math-derivative-1
467 468 469
     (function (lambda (u) (math-sqr
                            (math-normalize
                             (list 'calcFunc-sech u))))))
Eli Zaretskii's avatar
Eli Zaretskii committed
470

471 472 473 474 475 476 477 478 479 480 481 482
(put 'calcFunc-sech\' 'math-derivative-1
     (function (lambda (u) (math-neg
                            (math-mul
                             (math-normalize (list 'calcFunc-sech u))
                             (math-normalize (list 'calcFunc-tanh u)))))))

(put 'calcFunc-csch\' 'math-derivative-1
     (function (lambda (u) (math-neg
                            (math-mul
                             (math-normalize (list 'calcFunc-csch u))
                             (math-normalize (list 'calcFunc-coth u)))))))

483
(put 'calcFunc-coth\' 'math-derivative-1
484
     (function (lambda (u) (math-neg
485 486 487
                            (math-sqr
                             (math-normalize
                              (list 'calcFunc-csch u)))))))
488

Eli Zaretskii's avatar
Eli Zaretskii committed
489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534
(put 'calcFunc-arcsinh\' 'math-derivative-1
     (function (lambda (u)
		 (math-div 1 (math-normalize
			      (list 'calcFunc-sqrt
				    (math-add (math-sqr u) 1)))))))

(put 'calcFunc-arccosh\' 'math-derivative-1
     (function (lambda (u)
		  (math-div 1 (math-normalize
			       (list 'calcFunc-sqrt
				     (math-add (math-sqr u) -1)))))))

(put 'calcFunc-arctanh\' 'math-derivative-1
     (function (lambda (u) (math-div 1 (math-sub 1 (math-sqr u))))))

(put 'calcFunc-bern\'2 'math-derivative-2
     (function (lambda (n x)
		 (math-mul n (list 'calcFunc-bern (math-add n -1) x)))))

(put 'calcFunc-euler\'2 'math-derivative-2
     (function (lambda (n x)
		 (math-mul n (list 'calcFunc-euler (math-add n -1) x)))))

(put 'calcFunc-gammag\'2 'math-derivative-2
     (function (lambda (a x) (math-deriv-gamma a x 1))))

(put 'calcFunc-gammaG\'2 'math-derivative-2
     (function (lambda (a x) (math-deriv-gamma a x -1))))

(put 'calcFunc-gammaP\'2 'math-derivative-2
     (function (lambda (a x) (math-deriv-gamma a x
					       (math-div
						1 (math-normalize
						   (list 'calcFunc-gamma
							 a)))))))

(put 'calcFunc-gammaQ\'2 'math-derivative-2
     (function (lambda (a x) (math-deriv-gamma a x
					       (math-div
						-1 (math-normalize
						    (list 'calcFunc-gamma
							  a)))))))

(defun math-deriv-gamma (a x scale)
  (math-mul scale
	    (math-mul (math-pow x (math-add a -1))
535
		      (list 'calcFunc-exp (math-neg x)))))
Eli Zaretskii's avatar
Eli Zaretskii committed
536 537 538 539 540 541 542 543 544 545 546 547 548

(put 'calcFunc-betaB\' 'math-derivative-3
     (function (lambda (x a b) (math-deriv-beta x a b 1))))

(put 'calcFunc-betaI\' 'math-derivative-3
     (function (lambda (x a b) (math-deriv-beta x a b
						(math-div
						 1 (list 'calcFunc-beta
							 a b))))))

(defun math-deriv-beta (x a b scale)
  (math-mul (math-mul (math-pow x (math-add a -1))
		      (math-pow (math-sub 1 x) (math-add b -1)))
549
	    scale))
Eli Zaretskii's avatar
Eli Zaretskii committed
550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589

(put 'calcFunc-erf\' 'math-derivative-1
     (function (lambda (x) (math-div 2
				     (math-mul (list 'calcFunc-exp
						     (math-sqr x))
					       (if calc-symbolic-mode
						   '(calcFunc-sqrt
						     (var pi var-pi))
						 (math-sqrt-pi)))))))

(put 'calcFunc-erfc\' 'math-derivative-1
     (function (lambda (x) (math-div -2
				     (math-mul (list 'calcFunc-exp
						     (math-sqr x))
					       (if calc-symbolic-mode
						   '(calcFunc-sqrt
						     (var pi var-pi))
						 (math-sqrt-pi)))))))

(put 'calcFunc-besJ\'2 'math-derivative-2
     (function (lambda (v z) (math-div (math-sub (list 'calcFunc-besJ
						       (math-add v -1)
						       z)
						 (list 'calcFunc-besJ
						       (math-add v 1)
						       z))
				       2))))

(put 'calcFunc-besY\'2 'math-derivative-2
     (function (lambda (v z) (math-div (math-sub (list 'calcFunc-besY
						       (math-add v -1)
						       z)
						 (list 'calcFunc-besY
						       (math-add v 1)
						       z))
				       2))))

(put 'calcFunc-sum 'math-derivative-n
     (function
      (lambda (expr)
590
	(if (math-expr-contains (cons 'vec (cdr (cdr expr))) math-deriv-var)
Eli Zaretskii's avatar
Eli Zaretskii committed
591 592 593 594 595 596 597 598
	    (throw 'math-deriv nil)
	  (cons 'calcFunc-sum
		(cons (math-derivative (nth 1 expr))
		      (cdr (cdr expr))))))))

(put 'calcFunc-prod 'math-derivative-n
     (function
      (lambda (expr)
599
	(if (math-expr-contains (cons 'vec (cdr (cdr expr))) math-deriv-var)
Eli Zaretskii's avatar
Eli Zaretskii committed
600 601 602 603 604 605 606 607 608 609 610
	    (throw 'math-deriv nil)
	  (math-mul expr
		    (cons 'calcFunc-sum
			  (cons (math-div (math-derivative (nth 1 expr))
					  (nth 1 expr))
				(cdr (cdr expr)))))))))

(put 'calcFunc-integ 'math-derivative-n
     (function
      (lambda (expr)
	(if (= (length expr) 3)
611
	    (if (equal (nth 2 expr) math-deriv-var)
Eli Zaretskii's avatar
Eli Zaretskii committed
612 613 614 615 616 617 618 619 620 621 622 623 624 625
		(nth 1 expr)
	      (math-normalize
	       (list 'calcFunc-integ
		     (math-derivative (nth 1 expr))
		     (nth 2 expr))))
	  (if (= (length expr) 5)
	      (let ((lower (math-expr-subst (nth 1 expr) (nth 2 expr)
					    (nth 3 expr)))
		    (upper (math-expr-subst (nth 1 expr) (nth 2 expr)
					    (nth 4 expr))))
		(math-add (math-sub (math-mul upper
					      (math-derivative (nth 4 expr)))
				    (math-mul lower
					      (math-derivative (nth 3 expr))))
626
			  (if (equal (nth 2 expr) math-deriv-var)
Eli Zaretskii's avatar
Eli Zaretskii committed
627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648
			      0
			    (math-normalize
			     (list 'calcFunc-integ
				   (math-derivative (nth 1 expr)) (nth 2 expr)
				   (nth 3 expr) (nth 4 expr)))))))))))

(put 'calcFunc-if 'math-derivative-n
     (function
      (lambda (expr)
	(and (= (length expr) 4)
	     (list 'calcFunc-if (nth 1 expr)
		   (math-derivative (nth 2 expr))
		   (math-derivative (nth 3 expr)))))))

(put 'calcFunc-subscr 'math-derivative-n
     (function
      (lambda (expr)
	(and (= (length expr) 3)
	     (list 'calcFunc-subscr (nth 1 expr)
		   (math-derivative (nth 2 expr)))))))


649 650 651 652 653
(defvar math-integ-var '(var X ---))
(defvar math-integ-var-2 '(var Y ---))
(defvar math-integ-vars (list 'f math-integ-var math-integ-var-2))
(defvar math-integ-var-list (list math-integ-var))
(defvar math-integ-var-list-list (list math-integ-var-list))
Eli Zaretskii's avatar
Eli Zaretskii committed
654

655 656 657 658 659 660 661 662 663 664 665 666 667 668 669
;; math-integ-depth is a local variable for math-try-integral, but is used
;; by math-integral and math-tracing-integral
;; which are called (directly or indirectly) by math-try-integral.
(defvar math-integ-depth)
;; math-integ-level is a local variable for math-try-integral, but is used
;; by math-integral, math-do-integral, math-tracing-integral, 
;; math-sub-integration, math-integrate-by-parts and 
;; math-integrate-by-substitution, which are called (directly or 
;; indirectly) by math-try-integral.
(defvar math-integ-level)
;; math-integral-limit is a local variable for calcFunc-integ, but is
;; used by math-tracing-integral, math-sub-integration and 
;; math-try-integration. 
(defvar math-integral-limit)

Eli Zaretskii's avatar
Eli Zaretskii committed
670 671 672 673 674 675 676 677 678 679 680 681 682 683 684
(defmacro math-tracing-integral (&rest parts)
  (list 'and
	'trace-buffer
	(list 'save-excursion
	      '(set-buffer trace-buffer)
	      '(goto-char (point-max))
	      (list 'and
		    '(bolp)
		    '(insert (make-string (- math-integral-limit
					     math-integ-level) 32)
			     (format "%2d " math-integ-depth)
			     (make-string math-integ-level 32)))
	      ;;(list 'condition-case 'err
		    (cons 'insert parts)
		;;    '(error (insert (prin1-to-string err))))
685
	      '(sit-for 0))))
Eli Zaretskii's avatar
Eli Zaretskii committed
686 687 688 689 690 691 692 693

;;; The following wrapper caches results and avoids infinite recursion.
;;; Each cache entry is: ( A B )          Integral of A is B;
;;;			 ( A N )          Integral of A failed at level N;
;;;			 ( A busy )	  Currently working on integral of A;
;;;			 ( A parts )	  Currently working, integ-by-parts;
;;;			 ( A parts2 )	  Currently working, integ-by-parts;
;;;			 ( A cancelled )  Ignore this cache entry;
694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712
;;;			 ( A [B] )        Same result as for math-cur-record = B.

;; math-cur-record is a local variable for math-try-integral, but is used
;; by math-integral, math-replace-integral-parts and math-integrate-by-parts
;; which are called (directly or indirectly) by math-try-integral, as well as
;; by calc-dump-integral-cache
(defvar math-cur-record)
;; math-enable-subst and math-any-substs are local variables for
;; calcFunc-integ, but are used by math-integral and math-try-integral.
(defvar math-enable-subst)
(defvar math-any-substs)

;; math-integ-msg is a local variable for math-try-integral, but is
;; used (both locally and non-locally) by math-integral.
(defvar math-integ-msg)

(defvar math-integral-cache nil)
(defvar math-integral-cache-state nil)

Eli Zaretskii's avatar
Eli Zaretskii committed
713
(defun math-integral (expr &optional simplify same-as-above)
714 715
  (let* ((simp math-cur-record)
	 (math-cur-record (assoc expr math-integral-cache))
Eli Zaretskii's avatar
Eli Zaretskii committed
716 717 718 719 720
	 (math-integ-depth (1+ math-integ-depth))
	 (val 'cancelled))
    (math-tracing-integral "Integrating "
			   (math-format-value expr 1000)
			   "...\n")
721
    (and math-cur-record
Eli Zaretskii's avatar
Eli Zaretskii committed
722 723
	 (progn
	   (math-tracing-integral "Found "
724 725 726
				  (math-format-value (nth 1 math-cur-record) 1000))
	   (and (consp (nth 1 math-cur-record))
		(math-replace-integral-parts math-cur-record))
Eli Zaretskii's avatar
Eli Zaretskii committed
727
	   (math-tracing-integral " => "
728
				  (math-format-value (nth 1 math-cur-record) 1000)
Eli Zaretskii's avatar
Eli Zaretskii committed
729
				  "\n")))
730 731 732 733
    (or (and math-cur-record
	     (not (eq (nth 1 math-cur-record) 'cancelled))
	     (or (not (integerp (nth 1 math-cur-record)))
		 (>= (nth 1 math-cur-record) math-integ-level)))
Eli Zaretskii's avatar
Eli Zaretskii committed
734 735 736 737 738 739 740 741 742 743 744 745 746 747
	(and (math-integral-contains-parts expr)
	     (progn
	       (setq val nil)
	       t))
	(unwind-protect
	    (progn
	      (let (math-integ-msg)
		(if (eq calc-display-working-message 'lots)
		    (progn
		      (calc-set-command-flag 'clear-message)
		      (setq math-integ-msg (format
					    "Working... Integrating %s"
					    (math-format-flat-expr expr 0)))
		      (message math-integ-msg)))
748 749
		(if math-cur-record
		    (setcar (cdr math-cur-record)
Eli Zaretskii's avatar
Eli Zaretskii committed
750
			    (if same-as-above (vector simp) 'busy))
751
		  (setq math-cur-record
Eli Zaretskii's avatar
Eli Zaretskii committed
752
			(list expr (if same-as-above (vector simp) 'busy))
753
			math-integral-cache (cons math-cur-record
Eli Zaretskii's avatar
Eli Zaretskii committed
754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774
						  math-integral-cache)))
		(if (eq simplify 'yes)
		    (progn
		      (math-tracing-integral "Simplifying...")
		      (setq simp (math-simplify expr))
		      (setq val (if (equal simp expr)
				    (progn
				      (math-tracing-integral " no change\n")
				      (math-do-integral expr))
				  (math-tracing-integral " simplified\n")
				  (math-integral simp 'no t))))
		  (or (setq val (math-do-integral expr))
		      (eq simplify 'no)
		      (let ((simp (math-simplify expr)))
			(or (equal simp expr)
			    (progn
			      (math-tracing-integral "Trying again after "
						     "simplification...\n")
			      (setq val (math-integral simp 'no t))))))))
	      (if (eq calc-display-working-message 'lots)
		  (message math-integ-msg)))
775
	  (setcar (cdr math-cur-record) (or val
Eli Zaretskii's avatar
Eli Zaretskii committed
776 777 778 779
				       (if (or math-enable-subst
					       (not math-any-substs))
					   math-integ-level
					 'cancelled)))))
780
    (setq val math-cur-record)
Eli Zaretskii's avatar
Eli Zaretskii committed
781 782 783 784 785 786 787 788 789 790 791 792 793
    (while (vectorp (nth 1 val))
      (setq val (aref (nth 1 val) 0)))
    (setq val (if (memq (nth 1 val) '(parts parts2))
		  (progn
		    (setcar (cdr val) 'parts2)
		    (list 'var 'PARTS val))
		(and (consp (nth 1 val))
		     (nth 1 val))))
    (math-tracing-integral "Integral of "
			   (math-format-value expr 1000)
			   "  is  "
			   (math-format-value val 1000)
			   "\n")
794
    val))
Eli Zaretskii's avatar
Eli Zaretskii committed
795 796 797 798 799 800 801 802

(defun math-integral-contains-parts (expr)
  (if (Math-primp expr)
      (and (eq (car-safe expr) 'var)
	   (eq (nth 1 expr) 'PARTS)
	   (listp (nth 2 expr)))
    (while (and (setq expr (cdr expr))
		(not (math-integral-contains-parts (car expr)))))
803
    expr))
Eli Zaretskii's avatar
Eli Zaretskii committed
804 805 806 807 808 809 810 811 812 813 814 815

(defun math-replace-integral-parts (expr)
  (or (Math-primp expr)
      (while (setq expr (cdr expr))
	(and (consp (car expr))
	     (if (eq (car (car expr)) 'var)
		 (and (eq (nth 1 (car expr)) 'PARTS)
		      (consp (nth 2 (car expr)))
		      (if (listp (nth 1 (nth 2 (car expr))))
			  (progn
			    (setcar expr (nth 1 (nth 2 (car expr))))
			    (math-replace-integral-parts (cons 'foo expr)))
816
			(setcar (cdr math-cur-record) 'cancelled)))
817
	       (math-replace-integral-parts (car expr)))))))
Eli Zaretskii's avatar
Eli Zaretskii committed
818

819 820 821
(defvar math-linear-subst-tried t
  "Non-nil means that a linear substitution has been tried.")

822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842
;; The variable math-has-rules is a local variable for math-try-integral,
;; but is used by math-do-integral, which is called (non-directly) by
;; math-try-integral.
(defvar math-has-rules)

;; math-old-integ is a local variable for math-do-integral, but is
;; used by math-sub-integration.
(defvar math-old-integ)

;; The variables math-t1, math-t2 and math-t3 are local to 
;; math-do-integral, math-try-solve-for and math-decompose-poly, but
;; are used by functions they call (directly or indirectly); 
;; math-do-integral calls math-do-integral-methods;
;; math-try-solve-for calls math-try-solve-prod, 
;; math-solve-find-root-term and math-solve-find-root-in-prod;
;; math-decompose-poly calls math-solve-poly-funny-powers and
;; math-solve-crunch-poly.
(defvar math-t1)
(defvar math-t2)
(defvar math-t3)

Eli Zaretskii's avatar
Eli Zaretskii committed
843
(defun math-do-integral (expr)
844
  (let ((math-linear-subst-tried nil)
845
        math-t1 math-t2)
Eli Zaretskii's avatar
Eli Zaretskii committed
846 847 848 849 850
    (or (cond ((not (math-expr-contains expr math-integ-var))
	       (math-mul expr math-integ-var))
	      ((equal expr math-integ-var)
	       (math-div (math-sqr expr) 2))
	      ((eq (car expr) '+)
851 852 853
	       (and (setq math-t1 (math-integral (nth 1 expr)))
		    (setq math-t2 (math-integral (nth 2 expr)))
		    (math-add math-t1 math-t2)))
Eli Zaretskii's avatar
Eli Zaretskii committed
854
	      ((eq (car expr) '-)
855 856 857
	       (and (setq math-t1 (math-integral (nth 1 expr)))
		    (setq math-t2 (math-integral (nth 2 expr)))
		    (math-sub math-t1 math-t2)))
Eli Zaretskii's avatar
Eli Zaretskii committed
858
	      ((eq (car expr) 'neg)
859 860
	       (and (setq math-t1 (math-integral (nth 1 expr)))
		    (math-neg math-t1)))
Eli Zaretskii's avatar
Eli Zaretskii committed
861 862
	      ((eq (car expr) '*)
	       (cond ((not (math-expr-contains (nth 1 expr) math-integ-var))
863 864
		      (and (setq math-t1 (math-integral (nth 2 expr)))
			   (math-mul (nth 1 expr) math-t1)))
Eli Zaretskii's avatar
Eli Zaretskii committed
865
		     ((not (math-expr-contains (nth 2 expr) math-integ-var))
866 867
		      (and (setq math-t1 (math-integral (nth 1 expr)))
			   (math-mul math-t1 (nth 2 expr))))
Eli Zaretskii's avatar
Eli Zaretskii committed
868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885
		     ((memq (car-safe (nth 1 expr)) '(+ -))
		      (math-integral (list (car (nth 1 expr))
					   (math-mul (nth 1 (nth 1 expr))
						     (nth 2 expr))
					   (math-mul (nth 2 (nth 1 expr))
						     (nth 2 expr)))
				     'yes t))
		     ((memq (car-safe (nth 2 expr)) '(+ -))
		      (math-integral (list (car (nth 2 expr))
					   (math-mul (nth 1 (nth 2 expr))
						     (nth 1 expr))
					   (math-mul (nth 2 (nth 2 expr))
						     (nth 1 expr)))
				     'yes t))))
	      ((eq (car expr) '/)
	       (cond ((and (not (math-expr-contains (nth 1 expr)
						    math-integ-var))
			   (not (math-equal-int (nth 1 expr) 1)))
886 887
		      (and (setq math-t1 (math-integral (math-div 1 (nth 2 expr))))
			   (math-mul (nth 1 expr) math-t1)))
Eli Zaretskii's avatar
Eli Zaretskii committed
888
		     ((not (math-expr-contains (nth 2 expr) math-integ-var))
889 890
		      (and (setq math-t1 (math-integral (nth 1 expr)))
			   (math-div math-t1 (nth 2 expr))))
Eli Zaretskii's avatar
Eli Zaretskii committed
891 892 893
		     ((and (eq (car-safe (nth 1 expr)) '*)
			   (not (math-expr-contains (nth 1 (nth 1 expr))
						    math-integ-var)))
894
		      (and (setq math-t1 (math-integral
Eli Zaretskii's avatar
Eli Zaretskii committed
895 896
				     (math-div (nth 2 (nth 1 expr))
					       (nth 2 expr))))
897
			   (math-mul math-t1 (nth 1 (nth 1 expr)))))
Eli Zaretskii's avatar
Eli Zaretskii committed
898 899 900
		     ((and (eq (car-safe (nth 1 expr)) '*)
			   (not (math-expr-contains (nth 2 (nth 1 expr))
						    math-integ-var)))
901
		      (and (setq math-t1 (math-integral
Eli Zaretskii's avatar
Eli Zaretskii committed
902 903
				     (math-div (nth 1 (nth 1 expr))
					       (nth 2 expr))))
904
			   (math-mul math-t1 (nth 2 (nth 1 expr)))))
Eli Zaretskii's avatar
Eli Zaretskii committed
905 906 907
		     ((and (eq (car-safe (nth 2 expr)) '*)
			   (not (math-expr-contains (nth 1 (nth 2 expr))
						    math-integ-var)))
908
		      (and (setq math-t1 (math-integral
Eli Zaretskii's avatar
Eli Zaretskii committed
909 910
				     (math-div (nth 1 expr)
					       (nth 2 (nth 2 expr)))))
911
			   (math-div math-t1 (nth 1 (nth 2 expr)))))
Eli Zaretskii's avatar
Eli Zaretskii committed
912 913 914
		     ((and (eq (car-safe (nth 2 expr)) '*)
			   (not (math-expr-contains (nth 2 (nth 2 expr))
						    math-integ-var)))
915
		      (and (setq math-t1 (math-integral
Eli Zaretskii's avatar
Eli Zaretskii committed
916 917
				     (math-div (nth 1 expr)
					       (nth 1 (nth 2 expr)))))
918
			   (math-div math-t1 (nth 2 (nth 2 expr)))))
Eli Zaretskii's avatar
Eli Zaretskii committed
919 920 921 922 923 924 925
		     ((eq (car-safe (nth 2 expr)) 'calcFunc-exp)
		      (math-integral
		       (math-mul (nth 1 expr)
				 (list 'calcFunc-exp
				       (math-neg (nth 1 (nth 2 expr)))))))))
	      ((eq (car expr) '^)
	       (cond ((not (math-expr-contains (nth 1 expr) math-integ-var))
926
		      (or (and (setq math-t1 (math-is-polynomial (nth 2 expr)
Eli Zaretskii's avatar
Eli Zaretskii committed
927 928
							    math-integ-var 1))
			       (math-div expr
929
					 (math-mul (nth 1 math-t1)
Eli Zaretskii's avatar
Eli Zaretskii committed
930 931 932 933 934 935 936 937 938 939 940 941 942 943 944
						   (math-normalize
						    (list 'calcFunc-ln
							  (nth 1 expr))))))
			  (math-integral
			   (list 'calcFunc-exp
				 (math-mul (nth 2 expr)
					   (math-normalize
					    (list 'calcFunc-ln
						  (nth 1 expr)))))
			   'yes t)))
		     ((not (math-expr-contains (nth 2 expr) math-integ-var))
		      (if (and (integerp (nth 2 expr)) (< (nth 2 expr) 0))
			  (math-integral
			   (list '/ 1 (math-pow (nth 1 expr) (- (nth 2 expr))))
			   nil t)
945
			(or (and (setq math-t1 (math-is-polynomial (nth 1 expr)
Eli Zaretskii's avatar
Eli Zaretskii committed
946 947
							      math-integ-var
							      1))
948 949 950
				 (setq math-t2 (math-add (nth 2 expr) 1))
				 (math-div (math-pow (nth 1 expr) math-t2)
					   (math-mul math-t2 (nth 1 math-t1))))
Eli Zaretskii's avatar
Eli Zaretskii committed
951 952 953 954 955 956 957 958 959 960
			    (and (Math-negp (nth 2 expr))
				 (math-integral
				  (math-div 1
					    (math-pow (nth 1 expr)
						      (math-neg
						       (nth 2 expr))))
				  nil t))
			    nil))))))

	;; Integral of a polynomial.
961
	(and (setq math-t1 (math-is-polynomial expr math-integ-var 20))
Eli Zaretskii's avatar
Eli Zaretskii committed
962 963
	     (let ((accum 0)
		   (n 1))
964
	       (while math-t1
Eli Zaretskii's avatar
Eli Zaretskii committed
965
		 (if (setq accum (math-add accum
966
					   (math-div (math-mul (car math-t1)
Eli Zaretskii's avatar
Eli Zaretskii committed
967 968 969 970
							       (math-pow
								math-integ-var
								n))
						     n))
971
			   math-t1 (cdr math-t1))
Eli Zaretskii's avatar
Eli Zaretskii committed
972 973 974 975 976 977
		     (setq n (1+ n))))
	       accum))

	;; Try looking it up!
	(cond ((= (length expr) 2)
	       (and (symbolp (car expr))
978
		    (setq math-t1 (get (car expr) 'math-integral))
Eli Zaretskii's avatar
Eli Zaretskii committed
979
		    (progn
980 981
		      (while (and math-t1
				  (not (setq math-t2 (funcall (car math-t1)
Eli Zaretskii's avatar
Eli Zaretskii committed
982
							 (nth 1 expr)))))
983 984
			(setq math-t1 (cdr math-t1)))
		      (and math-t2 (math-normalize math-t2)))))
Eli Zaretskii's avatar
Eli Zaretskii committed
985 986
	      ((= (length expr) 3)
	       (and (symbolp (car expr))
987
		    (setq math-t1 (get (car expr) 'math-integral-2))
Eli Zaretskii's avatar
Eli Zaretskii committed
988
		    (progn
989 990
		      (while (and math-t1
				  (not (setq math-t2 (funcall (car math-t1)
Eli Zaretskii's avatar
Eli Zaretskii committed
991 992
							 (nth 1 expr)
							 (nth 2 expr)))))
993 994
			(setq math-t1 (cdr math-t1)))
		      (and math-t2 (math-normalize math-t2))))))
Eli Zaretskii's avatar
Eli Zaretskii committed
995 996 997

	;; Integral of a rational function.
	(and (math-ratpoly-p expr math-integ-var)
998 999 1000
	     (setq math-t1 (calcFunc-apart expr math-integ-var))
	     (not (equal math-t1 expr))
	     (math-integral math-t1))
Eli Zaretskii's avatar
Eli Zaretskii committed
1001 1002

	;; Try user-defined integration rules.
1003
	(and math-has-rules
Eli Zaretskii's avatar
Eli Zaretskii committed
1004 1005 1006 1007 1008 1009 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 1024 1025 1026 1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040 1041 1042 1043 1044 1045 1046 1047 1048 1049 1050 1051 1052 1053 1054 1055 1056 1057 1058 1059 1060 1061 1062 1063
	     (let ((math-old-integ (symbol-function 'calcFunc-integ))
		   (input (list 'calcFunc-integtry expr math-integ-var))
		   res part)
	       (unwind-protect
		   (progn
		     (fset 'calcFunc-integ 'math-sub-integration)
		     (setq res (math-rewrite input
					     '(var IntegRules var-IntegRules)
					     1))
		     (fset 'calcFunc-integ math-old-integ)
		     (and (not (equal res input))
			  (if (setq part (math-expr-calls
					  res '(calcFunc-integsubst)))
			      (and (memq (length part) '(3 4 5))
				   (let ((parts (mapcar
						 (function
						  (lambda (x)
						    (math-expr-subst
						     x (nth 2 part)
						     math-integ-var)))
						 (cdr part))))
				     (math-integrate-by-substitution
				      expr (car parts) t
				      (or (nth 2 parts)
					  (list 'calcFunc-integfailed
						math-integ-var))
				      (nth 3 parts))))
			    (if (not (math-expr-calls res
						      '(calcFunc-integtry
							calcFunc-integfailed)))
				res))))
		 (fset 'calcFunc-integ math-old-integ))))

	;; See if the function is a symbolic derivative.
	(and (string-match "'" (symbol-name (car expr)))
	     (let ((name (symbol-name (car expr)))
		   (p expr) (n 0) (which nil) (bad nil))
	       (while (setq n (1+ n) p (cdr p))
		 (if (equal (car p) math-integ-var)
		     (if which (setq bad t) (setq which n))
		   (if (math-expr-contains (car p) math-integ-var)
		       (setq bad t))))
	       (and which (not bad)
		    (let ((prime (if (= which 1) "'" (format "'%d" which))))
		      (and (string-match (concat prime "\\('['0-9]*\\|$\\)")
					 name)
			   (cons (intern
				  (concat
				   (substring name 0 (match-beginning 0))
				   (substring name (+ (match-beginning 0)
						      (length prime)))))
				 (cdr expr)))))))

	;; Try transformation methods (parts, substitutions).
	(and (> math-integ-level 0)
	     (math-do-integral-methods expr))

	;; Try expanding the function's definition.
	(let ((res (math-expand-formula expr)))
	  (and res
1064
	       (math-integral res))))))
Eli Zaretskii's avatar
Eli Zaretskii committed
1065 1066 1067 1068 1069 1070 1071 1072 1073 1074

(defun math-sub-integration (expr &rest rest)
  (or (if (or (not rest)
	      (and (< math-integ-level math-integral-limit)
		   (eq (car rest) math-integ-var)))
	  (math-integral expr)
	(let ((res (apply math-old-integ expr rest)))
	  (and (or (= math-integ-level math-integral-limit)
		   (not (math-expr-calls res 'calcFunc-integ)))
	       res)))
1075
      (list 'calcFunc-integfailed expr)))
Eli Zaretskii's avatar
Eli Zaretskii committed
1076

1077 1078 1079 1080 1081 1082 1083 1084 1085 1086 1087 1088
;; math-so-far is a local variable for math-do-integral-methods, but
;; is used by math-integ-try-linear-substitutions and 
;; math-integ-try-substitutions.
(defvar math-so-far)

;; math-integ-expr is a local variable for math-do-integral-methods,
;; but is used by math-integ-try-linear-substitutions and 
;; math-integ-try-substitutions.
(defvar math-integ-expr)

(defun math-do-integral-methods (math-integ-expr)
  (let ((math-so-far math-integ-var-list-list)
Eli Zaretskii's avatar
Eli Zaretskii committed
1089 1090 1091 1092
	rat-in)

    ;; Integration by substitution, for various likely sub-expressions.
    ;; (In first pass, we look only for sub-exprs that are linear in X.)
1093 1094
    (or (math-integ-try-linear-substitutions math-integ-expr)
        (math-integ-try-substitutions math-integ-expr)
Eli Zaretskii's avatar
Eli Zaretskii committed
1095 1096

	;; If function has sines and cosines, try tan(x/2) substitution.
1097
	(and (let ((p (setq rat-in (math-expr-rational-in math-integ-expr))))
Eli Zaretskii's avatar
Eli Zaretskii committed
1098 1099 1100
	       (while (and p
			   (memq (car (car p)) '(calcFunc-sin
						 calcFunc-cos
1101 1102 1103 1104
						 calcFunc-tan
                                                 calcFunc-sec
                                                 calcFunc-csc
                                                 calcFunc-cot))
Eli Zaretskii's avatar
Eli Zaretskii committed
1105 1106 1107
			   (equal (nth 1 (car p)) math-integ-var))
		 (setq p (cdr p)))
	       (null p))
1108 1109
	     (or (and (math-integ-parts-easy math-integ-expr)
		      (math-integ-try-parts math-integ-expr t))
Eli Zaretskii's avatar
Eli Zaretskii committed
1110
		 (math-integrate-by-good-substitution
1111
		  math-integ-expr (list 'calcFunc-tan (math-div math-integ-var 2)))))
Eli Zaretskii's avatar
Eli Zaretskii committed
1112 1113 1114 1115 1116 1117 1118

	;; If function has sinh and cosh, try tanh(x/2) substitution.
	(and (let ((p rat-in))
	       (while (and p
			   (memq (car (car p)) '(calcFunc-sinh
						 calcFunc-cosh
						 calcFunc-tanh
1119 1120 1121
                                                 calcFunc-sech
                                                 calcFunc-csch
                                                 calcFunc-coth
Eli Zaretskii's avatar
Eli Zaretskii committed
1122 1123 1124 1125
						 calcFunc-exp))
			   (equal (nth 1 (car p)) math-integ-var))
		 (setq p (cdr p)))
	       (null p))
1126 1127
	     (or (and (math-integ-parts-easy math-integ-expr)
		      (math-integ-try-parts math-integ-expr t))
Eli Zaretskii's avatar
Eli Zaretskii committed
1128
		 (math-integrate-by-good-substitution
1129
		  math-integ-expr (list 'calcFunc-tanh (math-div math-integ-var 2)))))
Eli Zaretskii's avatar
Eli Zaretskii committed
1130 1131 1132

	;; If function has square roots, try sin, tan, or sec substitution.
	(and (let ((p rat-in))
1133
	       (setq math-t1 nil)
Eli Zaretskii's avatar
Eli Zaretskii committed
1134 1135 1136
	       (while (and p
			   (or (equal (car p) math-integ-var)
			       (and (eq (car (car p)) 'calcFunc-sqrt)
1137 1138
				    (setq math-t1 (math-is-polynomial
					      (nth 1 (setq math-t2 (car p)))
Eli Zaretskii's avatar
Eli Zaretskii committed
1139 1140
					      math-integ-var 2)))))
		 (setq p (cdr p)))
1141 1142 1143 1144 1145 1146
	       (and (null p) math-t1))
	     (if (cdr (cdr math-t1))
		 (if (math-guess-if-neg (nth 2 math-t1))
		     (let* ((c (math-sqrt (math-neg (nth 2 math-t1))))
			    (d (math-div (nth 1 math-t1) (math-mul -2 c)))
			    (a (math-sqrt (math-add (car math-t1) (math-sqr d)))))
Eli Zaretskii's avatar
Eli Zaretskii committed
1147
		       (math-integrate-by-good-substitution
1148
			math-integ-expr (list 'calcFunc-arcsin
Eli Zaretskii's avatar
Eli Zaretskii committed
1149 1150 1151
				   (math-div-thru
				    (math-add (math-mul c math-integ-var) d)
				    a))))
1152 1153 1154
		   (let* ((c (math-sqrt (nth 2 math-t1)))
			  (d (math-div (nth 1 math-t1) (math-mul 2 c)))
			  (aa (math-sub (car math-t1) (math-sqr d))))
Eli Zaretskii's avatar
Eli Zaretskii committed
1155 1156
		     (if (and nil (not (and (eq d 0) (eq c 1))))
			 (math-integrate-by-good-substitution
1157
			  math-integ-expr (math-add (math-mul c math-integ-var) d))
Eli Zaretskii's avatar
Eli Zaretskii committed
1158 1159
		       (if (math-guess-if-neg aa)
			   (math-integrate-by-good-substitution
1160
			    math-integ-expr (list 'calcFunc-arccosh
Eli Zaretskii's avatar
Eli Zaretskii committed
1161 1162 1163 1164 1165
				       (math-div-thru
					(math-add (math-mul c math-integ-var)
						  d)
					(math-sqrt (math-neg aa)))))
			 (math-integrate-by-good-substitution
1166
			  math-integ-expr (list 'calcFunc-arcsinh
Eli Zaretskii's avatar
Eli Zaretskii committed
1167 1168 1169 1170
				     (math-div-thru
				      (math-add (math-mul c math-integ-var)
						d)
				      (math-sqrt aa))))))))
1171
	       (math-integrate-by-good-substitution math-integ-expr math-t2)) )
Eli Zaretskii's avatar
Eli Zaretskii committed
1172 1173

	;; Try integration by parts.
1174
	(math-integ-try-parts math-integ-expr)
Eli Zaretskii's avatar
Eli Zaretskii committed
1175 1176

	;; Give up.
1177
	nil)))
Eli Zaretskii's avatar
Eli Zaretskii committed
1178 1179 1180 1181 1182 1183 1184 1185 1186 1187 1188 1189 1190 1191

(defun math-integ-parts-easy (expr)
  (cond ((Math-primp expr) t)
	((memq (car expr) '(+ - *))
	 (and (math-integ-parts-easy (nth 1 expr))
	      (math-integ-parts-easy (nth 2 expr))))
	((eq (car expr) '/)
	 (and (math-integ-parts-easy (nth 1 expr))
	      (math-atomic-factorp (nth 2 expr))))
	((eq (car expr) '^)
	 (and (natnump (nth 2 expr))
	      (math-integ-parts-easy (nth 1 expr))))
	((eq (car expr) 'neg)
	 (math-integ-parts-easy (nth 1 expr)))
1192
	(t t)))
Eli Zaretskii's avatar
Eli Zaretskii committed
1193

1194 1195 1196 1197 1198 1199 1200 1201 1202
;; math-prev-parts-v is local to calcFunc-integ (as well as
;; math-integrate-by-parts), but is used by math-integ-try-parts.
(defvar math-prev-parts-v)

;; math-good-parts is local to calcFunc-integ (as well as
;; math-integ-try-parts), but is used by math-integrate-by-parts.
(defvar math-good-parts)


Eli Zaretskii's avatar
Eli Zaretskii committed
1203 1204 1205 1206 1207 1208 1209 1210 1211 1212 1213 1214 1215 1216 1217 1218 1219 1220 1221 1222 1223 1224 1225 1226
(defun math-integ-try-parts (expr &optional math-good-parts)
  ;; Integration by parts:
  ;;   integ(f(x) g(x),x) = f(x) h(x) - integ(h(x) f'(x),x)
  ;;     where h(x) = integ(g(x),x).
  (or (let ((exp (calcFunc-expand expr)))
	(and (not (equal exp expr))
	     (math-integral exp)))
      (and (eq (car expr) '*)
	   (let ((first-bad (or (math-polynomial-p (nth 1 expr)
						   math-integ-var)
				(equal (nth 2 expr) math-prev-parts-v))))
	     (or (and first-bad   ; so try this one first
		      (math-integrate-by-parts (nth 1 expr) (nth 2 expr)))
		 (math-integrate-by-parts (nth 2 expr) (nth 1 expr))
		 (and (not first-bad)
		      (math-integrate-by-parts (nth 1 expr) (nth 2 expr))))))
      (and (eq (car expr) '/)
	   (math-expr-contains (nth 1 expr) math-integ-var)
	   (let ((recip (math-div 1 (nth 2 expr))))
	     (or (math-integrate-by-parts (nth 1 expr) recip)
		 (math-integrate-by-parts recip (nth 1 expr)))))
      (and (eq (car expr) '^)
	   (math-integrate-by-parts (math-pow (nth 1 expr)
					      (math-sub (nth 2 expr) 1))
1227
				    (nth 1 expr)))))
Eli Zaretskii's avatar
Eli Zaretskii committed
1228 1229 1230 1231 1232 1233 1234 1235 1236 1237 1238

(defun math-integrate-by-parts (u vprime)
  (let ((math-integ-level (if (or math-good-parts
				  (math-polynomial-p u math-integ-var))
			      math-integ-level
			    (1- math-integ-level)))
	(math-doing-parts t)
	v temp)
    (and (>= math-integ-level 0)
	 (unwind-protect
	     (progn
1239
	       (setcar (cdr math-cur-record) 'parts)
Eli Zaretskii's avatar
Eli Zaretskii committed
1240 1241 1242 1243 1244 1245 1246 1247 1248 1249
	       (math-tracing-integral "Integrating by parts, u = "
				      (math-format-value u 1000)
				      ", v' = "
				      (math-format-value vprime 1000)
				      "\n")
	       (and (setq v (math-integral vprime))
		    (setq temp (calcFunc-deriv u math-integ-var nil t))
		    (setq temp (let ((math-prev-parts-v v))
				 (math-integral (math-mul v temp) 'yes)))
		    (setq temp (math-sub (math-mul u v) temp))
1250
		    (if (eq (nth 1 math-cur-record) 'parts)
Eli Zaretskii's avatar
Eli Zaretskii committed
1251
			(calcFunc-expand temp)
1252
		      (setq v (list 'var 'PARTS math-cur-record)
Eli Zaretskii's avatar
Eli Zaretskii committed
1253 1254 1255 1256
			    temp (let (calc-next-why)
				   (math-solve-for (math-sub v temp) 0 v nil)))
		      (and temp (not (integerp temp))
			   (math-simplify-extended temp)))))
1257
	   (setcar (cdr math-cur-record) 'busy)))))
Eli Zaretskii's avatar
Eli Zaretskii committed
1258 1259 1260 1261 1262 1263

;;; This tries two different formulations, hoping the algebraic simplifier
;;; will be strong enough to handle at least one.
(defun math-integrate-by-substitution (expr u &optional user uinv uinvprime)
  (and (> math-integ-level 0)
       (let ((math-integ-level (max (- math-integ-level 2) 0)))
1264
	 (math-integrate-by-good-substitution expr u user uinv uinvprime))))
Eli Zaretskii's avatar
Eli Zaretskii committed
1265 1266 1267 1268 1269 1270 1271 1272 1273 1274 1275 1276 1277 1278 1279 1280 1281 1282 1283 1284 1285 1286 1287 1288 1289 1290 1291 1292 1293 1294 1295 1296 1297 1298 1299 1300 1301 1302 1303 1304 1305 1306 1307 1308 1309 1310 1311 1312 1313

(defun math-integrate-by-good-substitution (expr u &optional user
						 uinv uinvprime)
  (let ((math-living-dangerously t)
	deriv temp)
    (and (setq uinv (if uinv
			(math-expr-subst uinv math-integ-var
					 math-integ-var-2)
		      (let (calc-next-why)
			(math-solve-for u
					math-integ-var-2
					math-integ-var nil))))
	 (progn
	   (math-tracing-integral "Integrating by substitution, u = "
				  (math-format-value u 1000)
				  "\n")
	   (or (and (setq deriv (calcFunc-deriv u
						math-integ-var nil
						(not user)))
		    (setq temp (math-integral (math-expr-subst
					       (math-expr-subst
						(math-expr-subst
						 (math-div expr deriv)
						 u
						 math-integ-var-2)
						math-integ-var
						uinv)
					       math-integ-var-2
					       math-integ-var)
					      'yes)))
	       (and (setq deriv (or uinvprime
				    (calcFunc-deriv uinv
						    math-integ-var-2
						    math-integ-var
						    (not user))))
		    (setq temp (math-integral (math-mul
					       (math-expr-subst
						(math-expr-subst
						 (math-expr-subst
						  expr
						  u
						  math-integ-var-2)
						 math-integ-var
						 uinv)
						math-integ-var-2
						math-integ-var)
					       deriv)
					      'yes)))))
	 (math-simplify-extended
1314
	  (math-expr-subst temp math-integ-var u)))))
Eli Zaretskii's avatar
Eli Zaretskii committed
1315 1316 1317

;;; Look for substitutions of the form u = a x + b.
(defun math-integ-try-linear-substitutions (sub-expr)
1318
  (setq math-linear-subst-tried t)
Eli Zaretskii's avatar
Eli Zaretskii committed
1319 1320 1321 1322 1323 1324 1325 1326 1327
  (and (not (Math-primp sub-expr))
       (or (and (not (memq (car sub-expr) '(+ - * / neg)))
		(not (and (eq (car sub-expr) '^)
			  (integerp (nth 2 sub-expr))))
		(math-expr-contains sub-expr math-integ-var)
		(let ((res nil))
		  (while (and (setq sub-expr (cdr sub-expr))
			      (or (not (math-linear-in (car sub-expr)
						       math-integ-var))
1328
				  (assoc (car sub-expr) math-so-far)
Eli Zaretskii's avatar
Eli Zaretskii committed
1329
				  (progn
1330 1331
				    (setq math-so-far (cons (list (car sub-expr))
						       math-so-far))
Eli Zaretskii's avatar
Eli Zaretskii committed
1332 1333
				    (not (setq res
					       (math-integrate-by-substitution
1334
						math-integ-expr (car sub-expr))))))))
Eli Zaretskii's avatar
Eli Zaretskii committed
1335 1336 1337 1338 1339
		  res))
	   (let ((res nil))
	     (while (and (setq sub-expr (cdr sub-expr))
			 (not (setq res (math-integ-try-linear-substitutions
					 (car sub-expr))))))
1340
	     res))))
Eli Zaretskii's avatar
Eli Zaretskii committed
1341 1342 1343 1344

;;; Recursively try different substitutions based on various sub-expressions.
(defun math-integ-try-substitutions (sub-expr &optional allow-rat)
  (and (not (Math-primp sub-expr))
1345
       (not (assoc sub-expr math-so-far))
Eli Zaretskii's avatar
Eli Zaretskii committed
1346 1347 1348 1349 1350 1351
       (math-expr-contains sub-expr math-integ-var)
       (or (and (if (and (not (memq (car sub-expr) '(+ - * / neg)))
			 (not (and (eq (car sub-expr) '^)
				   (integerp (nth 2 sub-expr)))))
		    (setq allow-rat t)
		  (prog1 allow-rat (setq allow-rat nil)))
1352 1353
		(not (eq sub-expr math-integ-expr))
		(or (math-integrate-by-substitution math-integ-expr sub-expr)
Eli Zaretskii's avatar
Eli Zaretskii committed
1354 1355 1356 1357 1358 1359 1360
		    (and (eq (car sub-expr) '^)
			 (integerp (nth 2 sub-expr))
			 (< (nth 2 sub-expr) 0)
			 (math-integ-try-substitutions
			  (math-pow (nth 1 sub-expr) (- (nth 2 sub-expr)))
			  t))))
	   (let ((res nil))
1361
	     (setq math-so-far (cons (list sub-expr) math-so-far))
Eli Zaretskii's avatar
Eli Zaretskii committed
1362 1363 1364
	     (while (and (setq sub-expr (cdr sub-expr))
			 (not (setq res (math-integ-try-substitutions
					 (car sub-expr) allow-rat)))))
1365
	     res))))
Eli Zaretskii's avatar
Eli Zaretskii committed
1366

1367 1368
;; The variable math-expr-parts is local to math-expr-rational-in,
;; but is used by math-expr-rational-in-rec
1369
(defvar math-expr-parts)
1370

Eli Zaretskii's avatar
Eli Zaretskii committed
1371
(defun math-expr-rational-in (expr)
1372
  (let ((math-expr-parts nil))
Eli Zaretskii's avatar
Eli Zaretskii committed
1373
    (math-expr-rational-in-rec expr)
1374
    (mapcar 'car math-expr-parts)))
Eli Zaretskii's avatar
Eli Zaretskii committed
1375 1376 1377 1378

(defun math-expr-rational-in-rec (expr)
  (cond ((Math-primp expr)
	 (and (equal expr math-integ-var)
1379 1380
	      (not (assoc expr math-expr-parts))
	      (setq math-expr-parts (cons (list expr) math-expr-parts))))
Eli Zaretskii's avatar
Eli Zaretskii committed
1381 1382 1383 1384 1385 1386 1387 1388
	((or (memq (car expr) '(+ - * / neg))
	     (and (eq (car expr) '^) (integerp (nth 2 expr))))
	 (math-expr-rational-in-rec (nth 1 expr))
	 (and (nth 2 expr) (math-expr-rational-in-rec (nth 2 expr))))
	((and (eq (car expr) '^)
	      (eq (math-quarter-integer (nth 2 expr)) 2))
	 (math-expr-rational-in-rec (list 'calcFunc-sqrt (nth 1 expr))))
	(t
1389
	 (and (not (assoc expr math-expr-parts))
Eli Zaretskii's avatar
Eli Zaretskii committed
1390
	      (math-expr-contains expr math-integ-var)
1391
	      (setq math-expr-parts (cons (list expr) math-expr-parts))))))
Eli Zaretskii's avatar
Eli Zaretskii committed
1392 1393 1394 1395 1396 1397 1398 1399 1400 1401 1402 1403 1404 1405

(defun math-expr-calls (expr funcs &optional arg-contains)
  (if (consp expr)
      (if (or (memq (car expr) funcs)
	      (and (eq (car expr) '^) (eq (car funcs) 'calcFunc-sqrt)
		   (eq (math-quarter-integer (nth 2 expr)) 2)))
	  (and (or (not arg-contains)
		   (math-expr-contains expr arg-contains))
	       expr)
	(and (not (Math-primp expr))
	     (let ((res nil))
	       (while (and (setq expr (cdr expr))
			   (not (setq res (math-expr-calls
					   (car expr) funcs arg-contains)))))
1406
	       res)))))
Eli Zaretskii's avatar
Eli Zaretskii committed
1407 1408 1409 1410 1411 1412 1413 1414 1415 1416

(defun math-fix-const-terms (expr except-vars)
  (cond ((not (math-expr-depends expr except-vars)) 0)
	((Math-primp expr) expr)
	((eq (car expr) '+)
	 (math-add (math-fix-const-terms (nth 1 expr) except-vars)
		   (math-fix-const-terms (nth 2 expr) except-vars)))
	((eq (car expr) '-)
	 (math-sub (math-fix-const-terms (nth 1 expr) except-vars)
		   (math-fix-const-terms (nth 2 expr) except-vars)))
1417
	(t expr)))
Eli Zaretskii's avatar
Eli Zaretskii committed
1418 1419 1420 1421 1422 1423 1424

;; Command for debugging the Calculator's symbolic integrator.
(defun calc-dump-integral-cache (&optional arg)
  (interactive "P")
  (let ((buf (current-buffer)))
    (unwind-protect
	(let ((p math-integral-cache)
1425
	      math-cur-record)
1426
	  (display-buffer (get-buffer-create "*Integral Cache*"))
Eli Zaretskii's avatar
Eli Zaretskii committed
1427 1428 1429
	  (set-buffer (get-buffer "*Integral Cache*"))
	  (erase-buffer)
	  (while p
1430 1431 1432
	    (setq math-cur-record (car p))
	    (or arg (math-replace-integral-parts math-cur-record))
	    (insert (math-format-flat-expr (car math-cur-record) 0)
Eli Zaretskii's avatar
Eli Zaretskii committed
1433
		    " --> "
1434 1435 1436
		    (if (symbolp (nth 1 math-cur-record))
			(concat "(" (symbol-name (nth 1 math-cur-record)) ")")
		      (math-format-flat-expr (nth 1 math-cur-record) 0))
Eli Zaretskii's avatar
Eli Zaretskii committed
1437 1438 1439
		    "\n")
	    (setq p (cdr p)))
	  (goto-char (point-min)))
1440
      (set-buffer buf))))
Eli Zaretskii's avatar
Eli Zaretskii committed