;;; -*- Mode: Lisp; Package: Maxima; Syntax: Common-Lisp; Base: 10 -*- ;;;;
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;; The data in this file contains enhancments. ;;;;;
;;; ;;;;;
;;; Copyright (c) 1984,1987 by William Schelter,University of Texas ;;;;;
;;; All rights reserved ;;;;;
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;; (c) Copyright 1982 Massachusetts Institute of Technology
(in-package "MAXIMA")
;; Non-commutative product and exponentiation simplifier
;; Written: July 1978 by CWH
;; Flags to control simplification:
(macsyma-module mdot)
(DEFMVAR $DOTCONSTRULES T
"Causes a non-commutative product of a constant and
another term to be simplified to a commutative product. Turning on this
flag effectively turns on DOT0SIMP, DOT0NSCSIMP, and DOT1SIMP as well.")
(DEFMVAR $DOT0SIMP T
"Causes a non-commutative product of zero and a scalar term to
be simplified to a commutative product.")
(DEFMVAR $DOT0NSCSIMP T
"Causes a non-commutative product of zero and a nonscalar term
to be simplified to a commutative product.")
(DEFMVAR $DOT1SIMP T
"Causes a non-commutative product of one and another term to be
simplified to a commutative product.")
(DEFMVAR $DOTSCRULES NIL
"Causes a non-commutative product of a scalar and another term to
be simplified to a commutative product. Scalars and constants are carried
to the front of the expression.")
(DEFMVAR $DOTDISTRIB NIL
"Causes every non-commutative product to be expanded each time it
is simplified, i.e. A . (B + C) will simplify to A . B + A . C.")
(DEFMVAR $DOTEXPTSIMP T "Causes A . A to be simplified to A ^^ 2.")
(DEFMVAR $DOTASSOC T
"Causes a non-commutative product to be considered associative, so
that A . (B . C) is simplified to A . B . C. If this flag is off, dot is
taken to be right associative, i.e. A . B . C is simplified to A . (B . C).")
(DEFMVAR $DOALLMXOPS T
"Causes all operations relating to matrices (and lists) to be
carried out. For example, the product of two matrices will actually be
computed rather than simply being returned. Turning on this switch
effectively turns on the following three.")
(DEFMVAR $DOMXMXOPS T "Causes matrix-matrix operations to be carried out.")
(DEFMVAR $DOSCMXOPS NIL "Causes scalar-matrix operations to be carried out.")
(DEFMVAR $DOMXNCTIMES NIL
"Causes non-commutative products of matrices to be carried out.")
(DEFMVAR $SCALARMATRIXP T
"Causes a square matrix of dimension one to be converted to a
scalar, i.e. its only element.")
(DEFMVAR $DOTIDENT 1 "The value to be returned by X^^0.")
(DEFMVAR $ASSUMESCALAR T
"This governs whether unknown expressions 'exp' are assumed to behave
like scalars for combinations of the form 'exp op matrix' where op is one of
{+, *, ^, .}. It has three settings:
FALSE -- such expressions behave like non-scalars.
TRUE -- such expressions behave like scalars only for the commutative
operators but not for non-commutative multiplication.
ALL -- such expressions will behave like scalars for all operators
listed above.
Note: This switch is primarily for the benefit of old code. If possible,
you should declare your variables to be SCALAR or NONSCALAR so that there
is no need to rely on the setting of this switch.")
;; Specials defined elsewhere.
(DECLARE-TOP(SPECIAL $EXPOP $EXPON ; Controls behavior of EXPAND
SIGN ; Something to do with BBSORT1
ERRORSW)
(FIXNUM $EXPOP $EXPON)
(*EXPR FIRSTN $IDENT POWERX MXORLISTP1 ONEP1
SCALAR-OR-CONSTANT-P EQTEST BBSORT1 OUTERMAP1 TIMEX))
(defun simpnct (exp vestigial simp-flag)
vestigial ;ignored
(let ((check exp)
(first-factor (simpcheck (cadr exp) simp-flag))
(remainder (if (cdddr exp)
(ncmuln (cddr exp) simp-flag)
(simpcheck (caddr exp) simp-flag))))
(cond ((null (cdr exp)) $dotident)
((null (cddr exp)) first-factor)
; This does (. sc m) --> (f* sc m) and (. (f* sc m1) m2) --> (f* sc (. m1 m2))
; and (. m1 (f* sc m2)) --> (f* sc (. m1 m2)) where sc can be a scalar
; or constant, and m1 and m2 are non-constant, non-scalar expressions.
((commutative-productp first-factor remainder)
(mul2 first-factor remainder))
((product-with-inner-scalarp first-factor)
(let ((p-p (partition-product first-factor)))
(outer-constant (car p-p) (cdr p-p) remainder)))
((product-with-inner-scalarp remainder)
(let ((p-p (partition-product remainder)))
(outer-constant (car p-p) first-factor (cdr p-p))))
; This code does distribution when flags are set and when called by
; $EXPAND. The way we recognize if we are called by $EXPAND is to look at
; the value of $EXPOP, but this is a kludge since $EXPOP has nothing to do
; with expanding (. A (f+ B C)) --> (f+ (. A B) (. A C)). I think that
; $EXPAND wants to have two flags: one which says to convert
; exponentiations to repeated products, and another which says to
; distribute products over sums.
((and (mplusp first-factor) (or $dotdistrib (not (zerop $expop))))
(addn (mapcar #'(lambda (x) (ncmul x remainder))
(cdr first-factor))
t))
((and (mplusp remainder) (or $dotdistrib (not (zerop $expop))))
(addn (mapcar #'(lambda (x) (ncmul first-factor x))
(cdr remainder))
t))
; This code carries out matrix operations when flags are set.
((matrix-matrix-productp first-factor remainder)
(timex first-factor remainder))
((or (scalar-matrix-productp first-factor remainder)
(scalar-matrix-productp remainder first-factor))
(simplifya (outermap1 'mnctimes first-factor remainder) t))
; (. (^^ x n) (^^ x m)) --> (^^ x (f+ n m))
((and (simpnct-alike first-factor remainder) $dotexptsimp)
(simpnct-merge-factors first-factor remainder))
; (. (. x y) z) --> (. x y z)
((and (mnctimesp first-factor) $dotassoc)
(ncmuln (append (cdr first-factor)
(if (mnctimesp remainder)
(cdr remainder)
(ncons remainder)))
t))
; (. (^^ (. x y) m) (^^ (. x y) n) z) --> (. (^^ (. x y) m+n) z)
; (. (^^ (. x y) m) x y z) --> (. (^^ (. x y) m+1) z)
; (. x y (^^ (. x y) m) z) --> (. (^^ (. x y) m+1) z)
; (. x y x y z) --> (. (^^ (. x y) 2) z)
((and (mnctimesp remainder) $dotassoc $dotexptsimp)
(setq exp (simpnct-merge-product first-factor (cdr remainder)))
(if (and (mnctimesp exp) $dotassoc)
(simpnct-antisym-check (cdr exp) check)
(eqtest exp check)))
; (. x (. y z)) --> (. x y z)
((and (mnctimesp remainder) $dotassoc)
(simpnct-antisym-check (cons first-factor (cdr remainder)) check))
(t (eqtest (list '(mnctimes) first-factor remainder) check)))))
; Predicate functions for simplifying a non-commutative product to a
; commutative one. SIMPNCT-CONSTANTP actually determines if a term is a
; constant and is not a nonscalar, i.e. not declared nonscalar and not a
; constant list or matrix. The function CONSTANTP determines if its argument
; is a number or a variable declared constant.
(defun commutative-productp (first-factor remainder)
(or (simpnct-sc-or-const-p first-factor)
(simpnct-sc-or-const-p remainder)
(simpnct-onep first-factor)
(simpnct-onep remainder)
(zero-productp first-factor remainder)
(zero-productp remainder first-factor)))
(defun simpnct-sc-or-const-p (term)
(or (simpnct-constantp term) (simpnct-assumescalarp term)))
(defun simpnct-constantp (term)
(and $dotconstrules
(or (mnump term)
(and ($constantp term) (not ($nonscalarp term))))))
(defun simpnct-assumescalarp (term)
(and $dotscrules (scalar-or-constant-p term (eq $assumescalar '$all))))
(defun simpnct-onep (term) (and $dot1simp (onep1 term)))
(defun zero-productp (one-term other-term)
(and (zerop1 one-term)
$dot0simp
(or $dot0nscsimp (not ($nonscalarp other-term)))))
; This function takes a form and determines if it is a product
; containing a constant or a declared scalar. Note that in the
; next three functions, the word "scalar" is used to refer to a constant
; or a declared scalar. This is a bad way of doing things since we have
; to cdr down an expression twice: once to determine if a scalar is there
; and once again to pull it out.
(defun product-with-inner-scalarp (product)
(and (mtimesp product)
(or $dotconstrules $dotscrules)
(do ((factor-list (cdr product) (cdr factor-list)))
((null factor-list) nil)
(if (simpnct-sc-or-const-p (car factor-list))
(return t)))))
; This function takes a commutative product and separates it into a scalar
; part and a non-scalar part.
(defun partition-product (product)
(do ((factor-list (cdr product) (cdr factor-list))
(scalar-list nil)
(nonscalar-list nil))
((null factor-list) (cons (nreverse scalar-list)
(muln (nreverse nonscalar-list) t)))
(if (simpnct-sc-or-const-p (car factor-list))
(push (car factor-list) scalar-list)
(push (car factor-list) nonscalar-list))))
; This function takes a list of constants and scalars, and two nonscalar
; expressions and forms a non-commutative product of the nonscalar
; expressions, and a commutative product of the constants and scalars and
; the non-commutative product.
(defun outer-constant (constant nonscalar1 nonscalar2)
(muln (nconc constant (ncons (ncmul nonscalar1 nonscalar2))) t))
(defun simpnct-base (term) (if (mncexptp term) (cadr term) term))
(defun simpnct-power (term) (if (mncexptp term) (caddr term) 1))
(defun simpnct-alike (term1 term2)
(alike1 (simpnct-base term1) (simpnct-base term2)))
(defun simpnct-merge-factors (term1 term2)
(ncpower (simpnct-base term1)
(add2 (simpnct-power term1) (simpnct-power term2))))
(defun matrix-matrix-productp (term1 term2)
(and (or $doallmxops $domxmxops $domxnctimes)
(mxorlistp1 term1)
(mxorlistp1 term2)))
(defun scalar-matrix-productp (term1 term2)
(and (or $doallmxops $doscmxops)
(mxorlistp1 term1)
(scalar-or-constant-p term2 (eq $assumescalar '$all))))
(declare-top(muzzled t))
(defun simpncexpt (exp vestigial simp-flag)
vestigial ;ignored
(let ((factor (simpcheck (cadr exp) simp-flag))
(power (simpcheck (caddr exp) simp-flag))
(check exp))
(twoargcheck exp)
(cond ((zerop1 power)
(if (zerop1 factor)
(if (not errorsw)
(merror "~M has been generated"
(list '(mncexpt) factor power))
(throw 'errorsw t)))
(if (mxorlistp1 factor) (identitymx factor) $dotident))
((onep1 power) factor)
((simpnct-sc-or-const-p factor) (power factor power))
((and (zerop1 factor) $dot0simp) factor)
((and (onep1 factor) $dot1simp) factor)
((and (or $doallmxops $domxmxops)
(mxorlistp1 factor)
(fixnump power) 'fixnum)
(let (($scalarmatrixp (or ($listp factor) $scalarmatrixp)))
(simplify (powerx factor power))))
;; This does (A+B)^^2 --> A^^2 + A.B + B.A + B^^2
;; and (A.B)^^2 --> A.B.A.B
((and (or (mplusp factor)
(and (not $dotexptsimp) (mnctimesp factor)))
(fixnump power)
(not (greaterp power $expop))
(plusp power))
(ncmul factor (ncpower factor (f1- power))))
;; This does the same thing as above for (A+B)^^(-2)
;; and (A.B)^^(-2). Here the "-" operator does the trick
;; for us.
((and (or (mplusp factor)
(and (not $dotexptsimp) (mnctimesp factor)))
(fixnump power)
(not (greaterp (minus power) $expon))
(minusp power))
(ncmul (simpnct-invert factor) (ncpower factor (f1+ power))))
((product-with-inner-scalarp factor)
(let ((p-p (partition-product factor)))
(mul2 (power (muln (car p-p) t) power)
(ncpower (cdr p-p) power))))
((and $dotassoc (mncexptp factor))
(ncpower (cadr factor) (mul2 (caddr factor) power)))
(t (eqtest (list '(mncexpt) factor power) check)))))
(declare-top(muzzled nil))
(defun simpnct-invert (exp)
(cond ((mnctimesp exp)
(ncmuln (nreverse (mapcar #'simpnct-invert (cdr exp))) t))
((and (mncexptp exp) (integerp (caddr exp)))
(ncpower (cadr exp) (minus (caddr exp))))
(t (list '(mncexpt simp) exp -1))))
(defun identitymx (x)
(if (and ($listp (cadr x)) (= (length (cdr x)) (length (cdadr x))))
(simplifya (cons (car x) (cdr ($ident (length (cdr x))))) t)
$dotident))
; This function incorporates the hairy search which enables such
; simplifications as (. a b a b) --> (^^ (. a b) 2). It assumes
; that FIRST-FACTOR is not a dot product and that REMAINDER is.
; For the product (. a b c d e), three basic types of comparisons
; are done:
;
; 1) a <---> b first-factor <---> inner-product
; a <---> (. b c)
; a <---> (. b c d)
; a <---> (. b c d e) (this case handled in SIMPNCT)
;
; 2) (. a b) <---> c outer-product <---> (car rest)
; (. a b c) <---> d
; (. a b c d) <---> e
;
; 3) (. a b) <---> (. c d) outer-product <---> (firstn rest)
;
; Note that INNER-PRODUCT and OUTER-PRODUCT share list structure which
; is clobbered as new terms are added.
(defun simpnct-merge-product (first-factor remainder)
(let ((half-product-length (// (f1+ (length remainder)) 2))
(inner-product (car remainder))
(outer-product (list '(mnctimes) first-factor (car remainder))))
(do ((merge-length 2 (f1+ merge-length))
(rest (cdr remainder) (cdr rest)))
((null rest) outer-product)
(cond ((simpnct-alike first-factor inner-product)
(return
(ncmuln
(cons (simpnct-merge-factors first-factor inner-product)
rest)
t)))
((simpnct-alike outer-product (car rest))
(return
(ncmuln
(cons (simpnct-merge-factors outer-product (car rest))
(cdr rest))
t)))
((and (not (> merge-length half-product-length))
(alike1 outer-product
(cons '(mnctimes)
(firstn merge-length rest))))
(return
(ncmuln (cons (ncpower outer-product 2)
(nthcdr merge-length rest))
t)))
((= merge-length 2)
(setq inner-product
(cons '(mnctimes) (cddr outer-product)))))
(rplacd (last inner-product) (ncons (car rest))))))
(defun simpnct-antisym-check (l check)
(let (sign)
(cond ((and (get 'mnctimes '$antisymmetric) (cddr l))
(setq l (bbsort1 l))
(cond ((equal l 0) 0)
((prog1 (null sign)
(setq l (eqtest (cons '(mnctimes) l) check)))
l)
(t (neg l))))
(t (eqtest (cons '(mnctimes) l) check)))))