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;; "anti-diff.scm" rational-function anti-derivative. -*-scheme-*-
;; Copyright 2020, 2023 Aubrey Jaffer
;;
;; This program is free software; you can redistribute it and/or modify
;; it under the terms of the GNU General Public License as published by
;; the Free Software Foundation, either version 3 of the License, or (at
;; your option) any later version.
;;
;; This program is distributed in the hope that it will be useful, but
;; WITHOUT ANY WARRANTY; without even the implied warranty of
;; MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
;; General Public License for more details.
;;
;; You should have received a copy of the GNU General Public License
;; along with this program; if not, write to the Free Software
;; Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA.
(require 'common-list-functions)
;; returns a list of unsquared factors of increasing power.
(define (sqfr-splits c var)
(define splitter (poly:diff c var))
(cond ((not (number? splitter))
(let ((d '())
(aj '())
(bi (poly:gcd c splitter)))
(do ((b bi (poly:/ b d))
(a (poly:/ c bi) d))
((number? b)
(reverse (cons a aj)))
(set! d (poly:gcd a b))
(set! aj (cons (poly:/ a d) aj)))))
((number? c) (list c))
(else (list (univ:demote c)))))
;; The inverse of sqfr-splits. Returns the product of increasing
;; powers of list of factors fcts.
(define (powfacts fcts)
(let lp ((fcts fcts) (n 1) (acc 1))
(cond ((null? fcts) acc)
(else
(lp (cdr fcts)
(+ 1 n)
(poly:* (ipow-by-squaring (car fcts) n 1 poly:*)
acc))))))
(define (rat:integrate dnmf L var ver)
(define dY (normalize (diff (var->expl ver) var)))
(define Q (denom dY))
(define Qxd (poly:degree Q var))
(define N (reduce-init poly:* 1 dnmf))
(define M
(let ((M 0) (j 0))
(if math:debug (math:print 'dnmf= dnmf))
(for-each (lambda (p)
(set! M (poly:+ (poly:*
(reduce-init poly:* 1 (butnth j dnmf))
(poly:* (- j) (poly:diff p var)))
M))
(set! j (+ 1 j)))
dnmf)
M))
(let ((A 0) (R L) (Nyd (poly:degree N ver)))
(define NyC (poly:coeff N ver Nyd))
(define Nxd (poly:degree NyC var))
(define (fnl) (app* $1/$2 A (powfacts (cdr dnmf))))
(define (fail . args)
(math:warn 'could-not-find-algebraic-anti-derivative)
(apply math:print args)
novalue)
(if math:debug (math:print 'M= M 'N= N 'NyC= NyC 'Nxd= Nxd))
(let lp ((Ryd (poly:degree (num R) ver)))
(define RyC (poly:coeff (num R) ver Ryd))
(define Rxd (poly:degree RyC var))
(define g (+ 1 (- Rxd Nxd)))
(define h (- Ryd Nyd))
(define T
(cond ((negative? g) (math:warn 'g= g '<0) 1)
((number? Q) (univ:monomial 1 g var))
((<= Qxd g)
(poly:* (ipow-by-squaring Q (quotient g Qxd) 1 poly:*)
(poly:* (univ:monomial 1 (remainder g Qxd) var)
(univ:monomial 1 h ver))))
(else (poly:* (univ:monomial 1 g var)
(univ:monomial 1 h ver)))))
(define dT (normalize (diff T var)))
(define B (normalize (app* $1*$2+$3 N dT (poly:* M T))))
(define C (expr:normalize
(app* $1/$2
(poly:* (denom B) (poly:coeff RyC var Rxd))
(poly:* (poly:coeff (poly:coeff (num B) ver Ryd) var Rxd)
(denom R)))))
(cond (math:debug
(math:print 'Ryd= Ryd 'Rxd= Rxd
'Byd= (poly:degree (num B) ver)
'ByC= (poly:coeff (num B) ver (poly:degree (num B) ver))
'Bxd= (poly:degree (poly:coeff (num B) ver (poly:degree (num B) ver)) var)
'Qxd= Qxd 'g= g 'h= h)
(math:print 'R= R)
(math:print 'T= T)
(math:print 'dT= dT)
(math:print 'B= B)
(math:print 'C= C)))
(set! A (expr:normalize (app* $1*$2+$3 C T A)))
(set! R (expr:normalize (app* $1-$2*$3 R C B)))
(cond ((and math:debug (number? R) (not (zero? R)))
(math:print 'nonzero-number-R= R)))
(cond ((if (number? R) (zero? R) (univ:zero? R))
(fnl))
;; ((number? R) (math:warn 'non-zero-constant-part R) (fnl))
((> (poly:degree (num R) ver) Ryd)
(fail 'increasing-Ryd (poly:degree (num R) ver) 'vs Ryd))
((and (= (poly:degree (num R) ver) Ryd)
(>= (poly:degree (poly:coeff (num R) ver (poly:degree (num R) ver)) var)
Rxd))
(fail 'non-decreasing-Rxd
(poly:degree (poly:coeff (num R) ver (poly:degree (num R) ver)) var)
'vs Rxd))
(else (lp (poly:degree (num R) ver)))))))
(define (poly:cont2 ve v p)
(apply poly:gcd*
(map (lambda (c) (univ:cont (promote v c)))
(cdr (promote ve p)))))
(define (independent-of-var? poly var)
(and (zero? (poly:degree poly var))
(null? (poly:find-var-exts poly var))))
(define (indef-integrate p v)
(define nm (num p))
(define dnm (denom p))
(define verlst (poly:find-var-exts nm v))
(define ver (if (null? verlst) _$ (car verlst)))
(define cnm (poly:cont2 ver v nm))
(define cdnm (univ:cont (promote v dnm)))
(define nm/cnm (poly:/ nm cnm))
(define dnm/cdnm (poly:/ dnm cdnm))
(cond ((> (length verlst) 1)
(math:warn 'too-many-extensions-involving v ': verlst)
novalue)
(else
(let ((ans (normalize
(app* $1*$2
(rat:integrate (sqfr-splits dnm/cdnm v) nm/cnm v ver)
(app* $1/$2 cnm cdnm)))))
(let ((chk (expr:normalize (diff ans v))))
(cond ((novalue? ans) ans)
((independent-of-var? (normalize (app* $1-$2 chk p)) v)
ans)
((independent-of-var? (normalize (app* $1+$2 chk p)) v)
(if math:debug (math:warn 'integration-was-negated))
(app* _-$1 ans))
(else
(math:warn 'diff-of-integral-mismatch chk)
(math:print 'mr-diff-of-integral-mismatch p)
novalue)))))))
(define (integrate . args)
(if (not (<= 2 (length args) 4)) (bltn:error 'integrate 'wna args))
(let ((expr (normalize (car args)))
(var (expl->var (cadr args)))
(lo (if (null? (cddr args)) #f (caddr args)))
(hi (and (= 4 (length args)) (cadddr args))))
(cond ((= 2 (length args))
(indef-integrate expr var))
(else
(let ((sexp (sexp:alpha-convert (list (var:sexp var))
(cano->sexp expr horner))))
(define ifun (indef-integrate (sexp->math sexp) $1))
(cond ((novalue? ifun) ifun)
((case (length args)
((3) (app* ifun lo))
((4) (app* $1-$2 (app* ifun hi) (app* ifun lo)))))))))))
(defbltn 'integrate 2 4 integrate)
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