Annotation of OpenXM_contrib2/asir2018/lib/sturm, Revision 1.1
1.1 ! noro 1: /*
! 2: * Copyright (c) 1994-2000 FUJITSU LABORATORIES LIMITED
! 3: * All rights reserved.
! 4: *
! 5: * FUJITSU LABORATORIES LIMITED ("FLL") hereby grants you a limited,
! 6: * non-exclusive and royalty-free license to use, copy, modify and
! 7: * redistribute, solely for non-commercial and non-profit purposes, the
! 8: * computer program, "Risa/Asir" ("SOFTWARE"), subject to the terms and
! 9: * conditions of this Agreement. For the avoidance of doubt, you acquire
! 10: * only a limited right to use the SOFTWARE hereunder, and FLL or any
! 11: * third party developer retains all rights, including but not limited to
! 12: * copyrights, in and to the SOFTWARE.
! 13: *
! 14: * (1) FLL does not grant you a license in any way for commercial
! 15: * purposes. You may use the SOFTWARE only for non-commercial and
! 16: * non-profit purposes only, such as academic, research and internal
! 17: * business use.
! 18: * (2) The SOFTWARE is protected by the Copyright Law of Japan and
! 19: * international copyright treaties. If you make copies of the SOFTWARE,
! 20: * with or without modification, as permitted hereunder, you shall affix
! 21: * to all such copies of the SOFTWARE the above copyright notice.
! 22: * (3) An explicit reference to this SOFTWARE and its copyright owner
! 23: * shall be made on your publication or presentation in any form of the
! 24: * results obtained by use of the SOFTWARE.
! 25: * (4) In the event that you modify the SOFTWARE, you shall notify FLL by
! 26: * e-mail at risa-admin@sec.flab.fujitsu.co.jp of the detailed specification
! 27: * for such modification or the source code of the modified part of the
! 28: * SOFTWARE.
! 29: *
! 30: * THE SOFTWARE IS PROVIDED AS IS WITHOUT ANY WARRANTY OF ANY KIND. FLL
! 31: * MAKES ABSOLUTELY NO WARRANTIES, EXPRESSED, IMPLIED OR STATUTORY, AND
! 32: * EXPRESSLY DISCLAIMS ANY IMPLIED WARRANTY OF MERCHANTABILITY, FITNESS
! 33: * FOR A PARTICULAR PURPOSE OR NONINFRINGEMENT OF THIRD PARTIES'
! 34: * RIGHTS. NO FLL DEALER, AGENT, EMPLOYEES IS AUTHORIZED TO MAKE ANY
! 35: * MODIFICATIONS, EXTENSIONS, OR ADDITIONS TO THIS WARRANTY.
! 36: * UNDER NO CIRCUMSTANCES AND UNDER NO LEGAL THEORY, TORT, CONTRACT,
! 37: * OR OTHERWISE, SHALL FLL BE LIABLE TO YOU OR ANY OTHER PERSON FOR ANY
! 38: * DIRECT, INDIRECT, SPECIAL, INCIDENTAL, PUNITIVE OR CONSEQUENTIAL
! 39: * DAMAGES OF ANY CHARACTER, INCLUDING, WITHOUT LIMITATION, DAMAGES
! 40: * ARISING OUT OF OR RELATING TO THE SOFTWARE OR THIS AGREEMENT, DAMAGES
! 41: * FOR LOSS OF GOODWILL, WORK STOPPAGE, OR LOSS OF DATA, OR FOR ANY
! 42: * DAMAGES, EVEN IF FLL SHALL HAVE BEEN INFORMED OF THE POSSIBILITY OF
! 43: * SUCH DAMAGES, OR FOR ANY CLAIM BY ANY OTHER PARTY. EVEN IF A PART
! 44: * OF THE SOFTWARE HAS BEEN DEVELOPED BY A THIRD PARTY, THE THIRD PARTY
! 45: * DEVELOPER SHALL HAVE NO LIABILITY IN CONNECTION WITH THE USE,
! 46: * PERFORMANCE OR NON-PERFORMANCE OF THE SOFTWARE.
! 47: *
! 48: * $OpenXM$
! 49: */
! 50: /* find intervals which include roots of a polynomial */
! 51:
! 52: #include "defs.h"
! 53:
! 54: def slice(P,XR,YR,WH) {
! 55: X = FIRST(XR); XMIN = SECOND(XR); XMAX = THIRD(XR);
! 56: Y = FIRST(YR); YMIN = SECOND(YR); YMAX = THIRD(YR);
! 57: W = FIRST(WH); H = SECOND(WH);
! 58: XS = (XMAX-XMIN)/W; YS = (YMAX-YMIN)/H;
! 59: T = ptozp(subst(P,X,X*XS+XMIN,Y,Y*YS+YMIN));
! 60: R = newvect(W+1);
! 61: for ( I = 0; I <= W; I++ ) {
! 62: S = sturm(subst(T,X,I));
! 63: R[I] = numch(S,Y,0)-numch(S,Y,H);
! 64: }
! 65: return R;
! 66: }
! 67:
! 68: def slice1(P,XR,YR,WH) {
! 69: X = FIRST(XR); XMIN = SECOND(XR); XMAX = THIRD(XR);
! 70: Y = FIRST(YR); YMIN = SECOND(YR); YMAX = THIRD(YR);
! 71: W = FIRST(WH); H = SECOND(WH);
! 72: XS = (XMAX-XMIN)/W; YS = (YMAX-YMIN)/H;
! 73: T = ptozp(subst(P,X,X*XS+XMIN,Y,Y*YS+YMIN));
! 74: R = newvect(W+1);
! 75: for ( I = 0; I <= W; I++ ) {
! 76: S = sturm(subst(T,X,I));
! 77: R[I] = newvect(H+1);
! 78: seproot(S,Y,0,H,R[I]);
! 79: }
! 80: return R;
! 81: }
! 82:
! 83: def seproot(S,X,MI,MA,R) {
! 84: N = car(size(S));
! 85: for ( I = MI; I <= MA; I++ )
! 86: if ( !(T = subst(S[0],X,I)) )
! 87: R[I] = -1;
! 88: else
! 89: break;
! 90: if ( I > MA )
! 91: return;
! 92: for ( J = MA; J >= MI; J-- )
! 93: if ( !(T = subst(S[0],X,J)) )
! 94: R[J] = -1;
! 95: else
! 96: break;
! 97: R[I] = numch(S,X,I); R[J] = numch(S,X,J);
! 98: if ( J <= I+1 )
! 99: return;
! 100: if ( R[I] == R[J] ) {
! 101: for ( K = I + 1; K < J; K++ )
! 102: R[K] = R[I];
! 103: return;
! 104: }
! 105: T = idiv(I+J,2);
! 106: seproot(S,X,I,T,R);
! 107: seproot(S,X,T,J,R);
! 108: }
! 109:
! 110: /* compute the sturm sequence of P */
! 111:
! 112: def sturm(P) {
! 113: V = var(P); N = deg(P,V); T = newvect(N+1);
! 114: G1 = T[0] = P; G2 = T[1] = ptozp(diff(P,var(P)));
! 115: for ( I = 1, H = 1, X = 1; ; ) {
! 116: if ( !deg(G2,V) )
! 117: break;
! 118: D = deg(G1,V)-deg(G2,V);
! 119: if ( (L = LCOEF(G2)) < 0 )
! 120: L = -L;
! 121: if ( !(R = -srem(L^(D+1)*G1,G2)) )
! 122: break;
! 123: if ( type(R) == 1 ) {
! 124: T[++I] = (R>0?1:-1); break;
! 125: }
! 126: M = H^D; G1 = G2;
! 127: G2 = T[++I] = sdiv(R,M*X);
! 128: if ( (X = LCOEF(G1)) < 0 )
! 129: X = -X;
! 130: H = X^D*H/M;
! 131: }
! 132: S = newvect(I+1);
! 133: for ( J = 0; J <= I; J++ )
! 134: S[J] = T[J];
! 135: return S;
! 136: }
! 137:
! 138: def numch(S,V,A) {
! 139: N = car(size(S));
! 140: for ( T = subst(S[0],V,A), I = 1, C = 0; I < N; I++ ) {
! 141: T1 = subst(S[I],V,A);
! 142: if ( !T1 )
! 143: continue;
! 144: if ( (T1 > 0 && T < 0) || (T1 < 0 && T > 0) )
! 145: C++;
! 146: T = T1;
! 147: }
! 148: return C;
! 149: }
! 150:
! 151: def numch0(S,V,A,T) {
! 152: N = car(size(S));
! 153: for ( I = 1, C = 0; I < N; I++ ) {
! 154: T1 = subst(S[I],V,A);
! 155: if ( !T1 )
! 156: continue;
! 157: if ( (T1 > 0 && T < 0) || (T1 < 0 && T > 0) )
! 158: C++;
! 159: T = T1;
! 160: }
! 161: return C;
! 162: }
! 163:
! 164: def count_real_roots(F)
! 165: {
! 166: if ( type(F) == 1 )
! 167: return 0;
! 168: V = var(F);
! 169: R = 0;
! 170: /* remove three roots : -1, 0, 1 */
! 171: if ( Q = tdiv(F,V) ) {
! 172: F = Q; R++;
! 173: while ( Q = tdiv(F,V) )
! 174: F = Q;
! 175: }
! 176: if ( Q = tdiv(F,V-1) ) {
! 177: F = Q; R++;
! 178: while ( Q = tdiv(F,V-1) )
! 179: F = Q;
! 180: }
! 181: if ( Q = tdiv(F,V+1) ) {
! 182: F = Q; R++;
! 183: while ( Q = tdiv(F,V+1) )
! 184: F = Q;
! 185: }
! 186: if ( type(F) == 1 )
! 187: return R;
! 188: S = sturm(F);
! 189: /* number of roots in [-1,1] */
! 190: R += numch(S,V,-1)-numch(S,V,1);
! 191: RS = sturm(ureverse(F));
! 192: /* number of roots in [-inf,-1] \cup [1,inf] */
! 193: R += numch(RS,V,-1)-numch(RS,V,1);
! 194: return R;
! 195: }
! 196: end;
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