;;; -*- Mode: Lisp -*- ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;; ;;; Copyright (C) 1999, 2002, 2009, 2015 Marek Rychlik ;;; ;;; 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 2 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. ;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;; ;; Run tests using 5am unit testing framework ;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;; We assume that QuickLisp package manager is installed. ;; See : ;; https://www.quicklisp.org/beta/ ;; ;; The following is unnecessary after running: ;; * (ql:add-to-init-file) ;; at lisp prompt: ;;(load "~/quicklisp/setup") (ql:quickload :fiveam) (defpackage #:5am-division (:use :cl :it.bese.fiveam :copy :monom :polynomial :infix :symbolic-polynomial :division :ring)) (in-package :5am-division) (def-suite division-suite :description "Division algorithm suite") (in-suite division-suite) ;; Manual calculation supporting the test below. ;; We divide X^2 by [X+Y,X-2*Y] with LEX> as order. ;; LM(X^2)=X^2 is divisible by LM(X+Y)=X so the first partial quotient is X. ;; Next, X^2 - X*(X+Y) = -X*Y. ;; LM(-X*Y)=X*Y is divibile by LM(X+Y)=X so the second partial quotient is -Y. ;; Next, -X*Y-(-Y)*(X+Y) = Y^2. ;; LM(Y^2)=Y^2 is not divisible by LM(X+Y)=X or LM(X-2*Y)=X. Hence, division ;; ends. The list of quotients is [X-Y,0]. The remainder is Y^2 (def-fixture division-context () (let* ((f (string->poly "x^2" '(x y))) (y-sq (string->poly "y^2" '(x y))) (fl (cdr (string->poly "[x+y,x-2*y]" '(x y)))) (quotients (cdr (string->poly "[x-y,0]" '(x y)))) (one (make-instance 'rational-field :value 1))) (&body))) (test normal-form "Normal form" (with-fixture division-context () (is (universal-equalp (multiple-value-list (normal-form f fl)) (list y-sq one 2))) (is (universal-equalp (multiple-value-list (poly-pseudo-divide f fl)) (list quotients y-sq one 2))) (is-false (buchberger-criterion fl)) ) ) ;; Maxima ;;poly_normal_form(3*x^3*y+2*z, [x^2*y+x,x-y^2-z^3],[x,y,z]); ;;Result: (-3*z^6)-6*y^2*z^3+2*z-3*y^4 (test normal-form-simple (let ((vars '(x y z))) (is (universal-equalp (normal-form (string->poly "3*x^3*y+2*z" vars) (cdr (string->poly "[x^2*y+x,x-y^2-z^3]" vars))) (string->poly "(-3*z^6)-6*y^2*z^3+2*z-3*y^4" vars))))) (test normal-form-easy "Easy normal form tests" (is (universal-zerop (normal-form (string->poly "0" '(x y)) (cdr (string->poly "[x,y]" '(x y)))))) ;; Maxima equivalent: poly_normal_form(3*x^2*y-x*y-1,[x-y,x+y],[x,y]); (is (universal-equalp (normal-form (string->poly "3*x^2*y-x*y-1" '(x y)) (cdr (string->poly "[x-y,x+y]" '(x y)))) (string->poly "3*y^3-y^2-1" '(x y)))) ;; Maxima equivalent: poly_normal_form(3*x^2*y*z-x*y^3-1,[x^2-2*y,x+y*z],[x,y,z]); (is (universal-equalp (normal-form (string->poly "3*x^2*y*z-x*y^3-1" '(x y z)) (cdr (string->poly "[x^2-2*y,x+y*z]" '(x y z)))) (string->poly "y^4*z+6*y^2*z-1" '(x y z))))) (def-fixture exact-division-context () (let* ((f (string->poly "x^2-4*y^2" '(x y))) (g (string->poly "x+2*y" '(x y))) (h (string->poly "x-2*y" '(x y)))) (&body))) (test exact-division "Exact division in polynomial ring" (with-fixture exact-division-context () (is (universal-equalp (poly-exact-divide f g) h)) (is (universal-zerop (subtract-from (poly-exact-divide f g) h))))) ;; Check if a set of generators satisfies the Buchberger criterion ;; The example is the Enneper surface ideal. Run this in Maxima to ;; obtain the Grobner basis: ;; poly_grobner([x-3*u-3*u*v^2+u^3,y-3*v-3*u^2*v+v^3,z-3*u^2+3*v^2],[u,v,x,y,z]); (def-fixture buchberger-criterion-context () (let ((fl (cdr (string->poly "[x-3*u-3*u*v^2+u^3,y-3*v-3*u^2*v+v^3,z-3*u^2+3*v^2]" '(u v x y z)))) (gb (cdr (string->poly "[x-3*u*v^2+u^3-3*u,y+v^3-3*u^2*v-3*v,z+3*v^2-3*u^2, (-u*z)-3*x+6*u*v^2+9*u,(-v*z)+y-2*v^3-3*v,z^2+6*v^2*z-9*z-9*v*y+9*u*x, 4*u*v*z-3*u*y+3*v*x,2*u*z^2+6*x*z-18*u*z-9*u*v*y+9*v^2*x, (-8*u*z^3)-24*x*z^2+72*u*z^2-36*v^2*x*z+27*u*y^2-27*v*x*y, z^3+18*v^2*z^2-18*z^2-54*v*y*z+54*v^2*z+81*z+27*y^2-27*x^2, (-4*z^4)+48*z^3-108*v*y*z^2+108*z^2+135*y^2*z+324*v*y*z+108*x^2*z -1296*v^2*z-1944*z-243*v^2*y^2-648*y^2+243*v^2*x^2+648*x^2, 8*v*z^3-9*y*z^2+72*v*z^2+54*v^2*y*z-27*y*z-27*v*y^2+27*v*x^2, (-8*v*z^4)+12*y*z^3-96*v*z^3-216*v*z^2-135*v*y^2*z+324*y*z-27*v*x^2*z +81*y^3+81*v*y^2-81*x^2*y-81*v*x^2, (-64*v*z^6)+120*y*z^5-1152*v*z^5+288*y*z^4-5184*v*z^4-648*v*y^2*z^3 -216*y*z^3+6912*v*z^3+81*y^3*z^2-9720*v*y^2*z^2 -1539*x^2*y*z^2+31104*y*z^2+62208*v*z^2+8505*y^3*z +46656*v*y^2*z-8505*x^2*y*z-93312*y*z+729*v*y^4-23328*y^3 -1458*v*x^2*y^2-23328*v*y^2+23328*x^2*y+729*v*x^4 +23328*v*x^2, 8*z^6-72*z^5+648*v*y*z^4-648*z^4-945*y^2*z^3+5184*v*y*z^3-189*x^2*z^3 +5832*z^3+972*y^2*z^2+17496*v*y*z^2-2430*x^2*z^2+8748*v*y^3*z -19683*y^2*z+2187*x^2*z-5103*y^4-4374*v*y^3+5832*x^2*y^2 +4374*v*x^2*y-729*x^4, 8*z^7-48*z^6+648*v*y*z^5-864*z^5-945*y^2*z^4+5832*v*y*z^4-189*x^2*z^4 +3888*z^4+81*y^2*z^3+17496*v*y*z^3-2997*x^2*z^3+17496*z^3 +8748*v*y^3*z^2-16767*y^2*z^2+17496*v*y*z^2-5103*x^2*z^2 -5103*y^4*z+5832*x^2*y^2*z-6561*y^2*z-729*x^4*z+6561*x^2*z -2187*y^4+4374*x^2*y^2-2187*x^4, 64*z^9-10368*z^7+1296*y^2*z^6-1296*x^2*z^6-34992*y^2*z^5-34992*x^2*z^5 +419904*z^5+174960*y^2*z^4-174960*x^2*z^4-10935*y^4*z^3 -56862*x^2*y^2*z^3+314928*y^2*z^3-10935*x^4*z^3+314928*x^2*z^3 +118098*y^4*z^2-118098*x^4*z^2+59049*y^4*z-118098*x^2*y^2*z +59049*x^4*z+19683*y^6-59049*x^2*y^4+59049*x^4*y^2-19683*x^6]" '(u v x y z))))) (&body))) (test buchberger-containment "Check ideal containment" (with-fixture buchberger-criterion-context () (let ((fl-copy (mapcar #'copy-instance fl)) (gb-copy (mapcar #'copy-instance gb))) (loop for i from 0 below (length fl) do (is (universal-zerop (normal-form (elt fl i) gb)) "Failed with I=~S~%" I)) ;; GB should not change in the process (is (universal-equalp gb gb-copy)) ;; FL should not change either (is (universal-equalp fl fl-copy))))) (test buchberger-criterion-with-normal-form "Buchberger criterion using normal form" (with-fixture buchberger-criterion-context () (let ((fl-copy (mapcar #'copy-instance fl)) (gb-copy (mapcar #'copy-instance gb))) (loop for i from 0 below (length gb) do (loop for j from (1+ i) below (length gb) do (is (universal-zerop (normal-form (s-polynomial (elt gb i) (elt gb j)) gb)) "Failed with I=~S, J=~S~%" I J))) (is (universal-equalp fl fl-copy)) (is (universal-equalp gb gb-copy))))) (test buchberger-criterion-with-pseudo-division "Buchberger criterion using pseudo-division" (with-fixture buchberger-criterion-context () (let ((fl-copy (mapcar #'copy-instance fl)) (gb-copy (mapcar #'copy-instance gb))) (loop for i from 0 below (length gb) do (loop for j from (1+ i) below (length gb) do (is (universal-zerop (second (multiple-value-list (poly-pseudo-divide (s-polynomial (elt gb i) (elt gb j)) gb-copy)))) "Failed with I=~S, J=~S~%" I J))) (is (universal-equalp fl fl-copy)) (is (universal-equalp gb gb-copy))))) (test buchberger-criterion-with-grobner-test "Buchberger criterion using grobner-test" (with-fixture buchberger-criterion-context () (is-true (grobner-test gb fl)) ) ) (run! 'division-suite) (format t "All tests done!~%")