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;;; -*- 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)
(macsyma-module mat)
;; this is the mat package
(declare-top (special *ech* *tri* $algebraic $multiplicities equations
mul* $dispflag $nolabels errrjfflag *det*
xm* xn* varlist ax linelable sol))
;;these are arrays.
(defvar *row*)
(defvar *col*)
(defvar *colinv*)
(defmvar $globalsolve nil)
(defmvar $sparse nil)
(defmvar $backsubst t)
(defmvar *rank* nil)
(defmvar *inv* nil)
(defun solcoef (m *c varl flag)
(prog (cc answer leftover)
(setq cc (cdr (ratrep* *c)))
(if (or (atom (car cc))
(not (equal (cdar cc) '(1 1)))
(not (equal 1 (cdr cc))))
;; NOTE TO TRANSLATORS: NOT CLEAR WHAT IS "UNACCEPTABLE" HERE
(merror (intl:gettext "solve: unacceptable variable: ~M") *c))
(setq answer (ratreduce (prodcoef (car cc) (car m)) (cdr m)))
(if (not flag) (return answer))
(setq leftover
(rdis (ratplus m (rattimes (ratminus answer) cc t))))
(if (or (not (freeof *c leftover))
(dependsall (rdis answer) varl))
(errrjf "`non-linear'"))
(return answer)))
(defun formx (flag nam eql varl)
(prog (b ax x ix j)
(setq varlist varl)
(mapc #'newvar eql)
(and (not $algebraic)
(some #'algp varlist)
(setq $algebraic t))
(setf (symbol-value nam) (make-array (list (1+ (setq xn* (length eql)))
(1+ (setq xm* (1+ (length varl)))))))
(setq nam (get-array-pointer nam))
(setq ix 0)
loop1
(cond ((null eql) (return varlist)))
(setq ax (car eql))
(setq eql (cdr eql))
(incf ix)
(setf (aref nam ix xm*) (const ax varl))
(setq j 0)
(setq b varl) (setq ax (cdr (ratrep* ax)))
loop2
(setq x (car b))
(setq b (cdr b))
(incf j)
(setf (aref nam ix j) (solcoef ax x varl flag))
(cond (b (go loop2)))
(go loop1)))
(defun dependsall (exp l)
(cond ((null l) nil)
((or (not (freeof (car l) exp)) (dependsall exp (cdr l))) t)
(t nil)))
(setq *det* nil *ech* nil *tri* nil)
(defun ptorat (ax m n)
(prog (i j)
(setq ax (get-array-pointer ax))
(setq i (1+ m))
(incf n)
loop1
(when (equal i 1) (return nil))
(decf i)
(setq j n)
loop2
(when (equal j 1) (go loop1))
(decf j)
(setf (aref ax i j) (cons (aref ax i j) 1))
(go loop2)))
(defun meqhk (z)
(if (and (not (atom z)) (eq (caar z) 'mequal))
(simplus (list '(mplus) (cadr z) (list '(mtimes) -1 (caddr z))) 1 nil)
z))
(defun const (e varl)
(setq varl (mapcar #'(lambda(x) (caadr (ratrep* x))) varl))
(setq e (cdr (ratrep* e)))
(let ((zl (make-list (length varl) :initial-element 0)))
(ratreduce (pctimes -1 (pcsubsty zl varl (car e)))
(pcsubsty zl varl (cdr e)))))
(defvar *mosesflag nil)
(defmvar $%rnum 0)
(defmfun make-param ()
(let ((param (intern (format nil "~A~D" '$%r (incf $%rnum)))))
(tuchus $%rnum_list param)
param))
(defmvar $linsolve_params t "`linsolve' generates %Rnums")
(defun ith (x n)
(if (atom x) nil (nth (1- n) x)))
(defun polyize (ax r m mul)
(declare (fixnum m))
(do ((c 1 (1+ c)) (d))
((> c m) nil)
(declare (fixnum c))
(setq d (aref ax r c))
(setq d (cond ((equal mul 1) (car d))
(t (ptimes (car d)
(pquotientchk mul (cdr d))))))
(setf (aref ax r c) (if $sparse (cons d 1) d))))
;; TWO-STEP FRACTION-FREE GAUSSIAN ELIMINATION ROUTINE
(defun tfgeli (ax n m &aux ($sparse (and $sparse (or *det* *inv*))))
;;$sparse is also controlling whether polyize stores polys or ratforms
(setq ax (get-array-pointer ax))
(setq mul* 1)
(do ((r 1 (1+ r)))
((> r n) (cond ((and $sparse *det*)(sprdet ax n))
((and *inv* $sparse)(newinv ax n m))
(t (tfgeli1 ax n m))))
(do ((c 1 (1+ c))
(d)
(mul 1))
((> c m)
(and *det* (setq mul* (ptimes mul* mul)))
(polyize ax r m mul))
(cond ((equal 1 (setq d (cdr (aref ax r c)))) nil)
(t (setq mul (ptimes mul (pquotient d (pgcd mul d)))))))))
;; The author of the following programs is Tadatoshi Minamikawa (TM).
;; This program is one-step fraction-free Gaussian elimination with
;; optimal pivotting. DRB claims the hair in this program is not
;; necessary and that straightforward Gaussian elimination is sufficient,
;; for sake of future implementors.
;; To debug, delete the comments around PRINT and BREAK statements.
(declare-top (special permsign a rank delta nrow nvar n m variableorder
dependentrows inconsistentrows l k))
(defun tfgeli1 (ax n m)
(prog (k l delta variableorder inconsistentrows
dependentrows nrow nvar rank permsign result)
(setq ax (get-array-pointer ax))
(setq *col* (make-array (1+ m) :initial-element 0))
(setq *row* (make-array (1+ n) :initial-element 0))
(setq *colinv* (make-array (1+ m) :initial-element 0))
;; (PRINT 'ONESTEP-LIPSON-WITH-PIVOTTING)
(setq nrow n)
(setq nvar (cond (*rank* m) (*det* m) (*inv* n) (*ech* m) (*tri* m) (t (1- m))))
(do ((i 1 (1+ i)))
((> i n))
(setf (aref *row* i) i))
(do ((i 1 (1+ i)))
((> i m))
(setf (aref *col* i) i) (setf (aref *colinv* i) i))
(setq result
(cond (*rank* (forward t) rank)
(*det* (forward t)
(cond ((= nrow n) (cond (permsign (pminus delta))
(t delta)))
(t 0)))
(*inv* (forward t) (backward) (recoverorder1))
(*ech* (forward nil) (recoverorder2))
(*tri* (forward nil) (recoverorder2))
(t (forward t) (cond ($backsubst (backward)))
(recoverorder2)
(list dependentrows inconsistentrows variableorder))))
(return result)))
;;FORWARD ELIMINATION
;;IF THE SWITCH *CPIVOT IS NIL, IT AVOIDS THE COLUMN PIVOTTING.
(defun forward (*cpivot)
(setq delta 1) ;DELTA HOLDS THE CURRENT DETERMINANT
(do ((k 1 (1+ k))
(nvar nvar) ;PROTECTS AGAINST TEMPORARAY RESETS DONE IN PIVOT
(m m))
((or (> k nrow) (> k nvar)))
(cond ((pivot ax k *cpivot) (return nil)))
;; PIVOT IS T IF THERE IS NO MORE NON-ZERO ROW LEFT. THEN GET OUT OF THE LOOP
(do ((i (1+ k) (1+ i)))
((> i nrow))
(do ((j (1+ k) (1+ j)))
((> j m))
(setf (aref ax (aref *row* i) (aref *col* j))
(pquotient (pdifference (ptimes (aref ax (aref *row* k) (aref *col* k))
(aref ax (aref *row* i) (aref *col* j)))
(ptimes (aref ax (aref *row* i) (aref *col* k))
(aref ax (aref *row* k) (aref *col* j))))
delta))))
(do ((i (1+ k) (1+ i)))
((> i nrow))
(setf (aref ax (aref *row* i) (aref *col* k)) 0))
(setq delta (aref ax (aref *row* k) (aref *col* k))))
;; UNDOES COLUMN HACK IN PIVOT.
(or *cpivot (do ((i 1 (1+ i))) ((> i m)) (setf (aref *col* i) i)))
(setq rank (min nrow nvar)))
;; BACKWARD SUBSTITUTION
(defun backward ()
(do ((i (1- rank) (1- i)))
((< i 1))
(do ((l (1+ rank) (1+ l)))
((> l m))
(setf (aref ax (aref *row* i) (aref *col* l))
(pquotient (pdifference
(ptimes (aref ax (aref *row* i) (aref *col* l))
(aref ax (aref *row* rank) (aref *col* rank)))
(do ((j (1+ i) (1+ j)) (sum 0))
((> j rank) sum)
(setq sum (pplus sum (ptimes (aref ax (aref *row* i) (aref *col* j))
(aref ax (aref *row* j) (aref *col* l)))))))
(aref ax (aref *row* i) (aref *col* i)))))
(do ((l (1+ i) (1+ l)))
((> l rank))
(setf (aref ax (aref *row* i) (aref *col* l)) 0)))
;; PUT DELTA INTO THE DIAGONAL MATRIX
(setq delta (aref ax (aref *row* rank) (aref *col* rank)))
(do ((i 1 (1+ i)))
((> i rank))
(setf (aref ax (aref *row* i) (aref *col* i)) delta)))
;;RECOVER THE ORDER OF ROWS AND COLUMNS.
(defun recoverorder1 ()
;;(PRINT 'REARRANGE)
(do ((i nvar (1- i)))
((= i 0))
(setq variableorder (cons i variableorder)))
(do ((i (1+ rank) (1+ i)))
((> i n))
(cond ((equal (aref ax (aref *row* i) (aref *col* m)) 0)
(setq dependentrows (cons (aref *row* i) dependentrows)))
(t (setq inconsistentrows (cons (aref *row* i) inconsistentrows)))))
(do ((i 1 (1+ i)))
((> i n))
(cond ((not (= (aref *row* (aref *colinv* i)) i))
(prog ()
(moverow ax n m i 0)
(setq l i)
loop
(setq k (aref *row* (aref *colinv* l)))
(setf (aref *row* (aref *colinv* l)) l)
(cond ((= k i) (moverow ax n m 0 l))
(t (moverow ax n m k l)
(setq l k)
(go loop))))))))
(defun recoverorder2 ()
(do ((i nvar (1- i)))
((= i 0))
(setq variableorder (cons (aref *col* i) variableorder)))
(do ((i (1+ rank) (1+ i)))
((> i n))
(cond ((equal (aref ax (aref *row* i) (aref *col* m)) 0)
(setq dependentrows (cons (aref *row* i) dependentrows)))
(t (setq inconsistentrows (cons (aref *row* i) inconsistentrows)))))
(do ((i 1 (1+ i)))
((> i n))
(cond ((not (= (aref *row* i) i))
(prog ()
(moverow ax n m i 0)
(setq l i)
loop
(setq k (aref *row* l))
(setf (aref *row* l) l)
(cond ((= k i) (moverow ax n m 0 l))
(t (moverow ax n m k l)
(setq l k)
(go loop)))))))
(do ((i 1 (1+ i)))
((> i nvar))
(cond ((not (= (aref *col* i) i))
(prog ()
(movecol ax n m i 0)
(setq l i)
loop2
(setq k (aref *col* l))
(setf (aref *col* l) l)
(cond ((= k i) (movecol ax n m 0 l))
(t (movecol ax n m k l)
(setq l k)
(go loop2))))))))
;;THIS PROGRAM IS USED IN REARRANGEMENT
(defun moverow (ax n m i j)
(do ((k 1 (1+ k))) ((> k m))
(setf (aref ax j k) (aref ax i k))))
(defun movecol (ax n m i j)
(do ((k 1 (1+ k))) ((> k n))
(setf (aref ax k j) (aref ax k i))))
;;COMPLEXITY IS DEFINED AS FOLLOWS
;; COMPLEXITY(0)=0
;; COMPLEXITY(CONSTANT)=1
;; COMPLEXITY(POLYNOMIAL)=1+SUM(COMPLEXITY(C(N))+COMPLEXITY(E(N)), FOR N=0,1 ...M)
;; WHERE POLYNOMIAL IS OF THE FORM
;; SUM(C(N)*X^E(N), FOR N=0,1 ... M) X IS THE VARIABLE
(defun complexity (exp)
(cond ((null exp) 0)
((equal exp 0) 0)
((atom exp) 1)
(t (+ (complexity (car exp)) (complexity (cdr exp))))))
(defun complexity/row (ax i j1 j2)
(do ((j j1 (1+ j)) (sum 0))
((> j j2) sum)
(incf sum (complexity (aref ax (aref *row* i) (aref *col* j))))))
(defun complexity/col (ax j i1 i2)
(do ((i i1 (1+ i)) (sum 0))
((> i i2) sum)
(incf sum (complexity (aref ax (aref *row* i) (aref *col* j))))))
(defun zerop/row (ax i j1 j2)
(do ((j j1 (1+ j)))
((> j j2) t)
(cond ((not (equal (aref ax (aref *row* i) (aref *col* j)) 0)) (return nil)))))
;;PIVOTTING ALGORITHM
(defun pivot (ax k *cpivot)
(prog (row/optimal col/optimal complexity/i/min complexity/j/min
complexity/i complexity/j complexity/det complexity/det/min dummy)
(setq row/optimal k complexity/i/min 1000000. complexity/j/min 1000000.)
;;TEST THE SINGULARITY
(cond ((do ((i k (1+ i)) (isallzero t))
((> i nrow) isallzero)
loop (cond ((zerop/row ax i k nvar)
(cond (*inv* (merror (intl:gettext "solve: singular matrix.")))
(t (exchangerow i nrow)
(decf nrow)))
(unless (> i nrow) (go loop)))
(t (setq isallzero nil))))
(return t)))
;; FIND AN OPTIMAL ROW
;; IF *CPIVOT IS NIL, (AX I K) WHICH IS TO BE THE PIVOT MUST BE NONZERO.
;; BUT IF *CPIVOT IS T, IT IS UNNECESSARY BECAUSE WE CAN DO THE COLUMN PIVOT.
findrow
(do ((i k (1+ i)))
((> i nrow))
(cond ((or *cpivot (not (equal (aref ax (aref *row* i) (aref *col* k)) 0)))
(cond ((> complexity/i/min
(setq complexity/i (complexity/row ax i k m)))
(setq row/optimal i complexity/i/min complexity/i))))))
;; EXCHANGE THE ROWS K AND ROW/OPTIMAL
(exchangerow k row/optimal)
;; IF THE FLAG *CPIVOT IS NIL, THE FOLLOWING STEPS ARE NOT EXECUTED.
;; THIS TREATMENT WAS DONE FOR THE LSA AND ECHELONTHINGS WHICH ARE NOT
;; HAPPY WITH THE COLUMN OPERATIONS.
(cond ((null *cpivot)
(cond ((not (equal (aref ax (aref *row* k) (aref *col* k)) 0))
(return nil))
(t (do ((i k (1+ i))) ((= i nvar))
(setf (aref *col* i) (aref *col* (1+ i))))
(setq nvar (1- nvar) m (1- m))
(go findrow)))))
;;STEP3 ... FIND THE OPTIMAL COLUMN
(setq col/optimal 0
complexity/det/min 1000000.
complexity/j/min 1000000.)
(do ((j k (1+ j)))
((> j nvar))
(cond ((not (equal (aref ax (aref *row* k) (aref *col* j)) 0))
(cond ((> complexity/det/min
(setq complexity/det
(complexity (aref ax (aref *row* k) (aref *col* j)))))
(setq col/optimal j
complexity/det/min complexity/det
complexity/j/min (complexity/col ax j (1+ k) n)))
((equal complexity/det/min complexity/det)
(cond ((> complexity/j/min
(setq complexity/j
(complexity/col ax j (1+ k) n)))
(setq col/optimal j
complexity/det/min complexity/det
complexity/j/min complexity/j))))))))
;; EXCHANGE THE COLUMNS K AND COL/OPTIMAL
(exchangecol k col/optimal)
(setq dummy (aref *colinv* (aref *col* k)))
(setf (aref *colinv* (aref *col* k)) (aref *colinv* (aref *col* col/optimal)))
(setf (aref *colinv* (aref *col* col/optimal)) dummy)
(return nil)))
(defun exchangerow (i j)
(prog (dummy)
(setq dummy (aref *row* i))
(setf (aref *row* i) (aref *row* j))
(setf (aref *row* j) dummy)
(cond ((= i j) (return nil))
(t (setq permsign (not permsign))))))
(defun exchangecol (i j)
(prog (dummy)
(setq dummy (aref *col* i))
(setf (aref *col* i) (aref *col* j))
(setf (aref *col* j) dummy)
(cond ((= i j) (return nil))
(t (setq permsign (not permsign))))))
;; Displays list of solutions.
(defun solve2 (llist)
(setq $multiplicities nil)
(map2c #'(lambda (equatn multipl)
(setq equations
(nconc equations (list (displine equatn))))
(push multipl $multiplicities)
(if (and (> multipl 1) $dispflag)
(mtell (intl:gettext "solve: multiplicity ~A~%") multipl)))
llist)
(setq $multiplicities (cons '(mlist simp) (nreverse $multiplicities))))
;; Displays an expression and returns its linelabel.
(defmfun displine (exp)
(let ($nolabels (tim 0))
(elabel exp)
(cond ($dispflag (remprop linelable 'nodisp)
(setq tim (get-internal-run-time))
(mterpri)
(displa (list '(mlable) linelable exp))
(timeorg tim))
(t (putprop linelable t 'nodisp)))
linelable))
(declare-top (unspecial permsign a rank delta nrow nvar n m variableorder
dependentrows inconsistentrows l k))
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