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Annotation of OpenXM_contrib2/asir2000/lib/sturm, Revision 1.4

1.2       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
1.3       noro       26:  * e-mail at risa-admin@sec.flab.fujitsu.co.jp of the detailed specification
1.2       noro       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'
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                     35:  * MODIFICATIONS, EXTENSIONS, OR ADDITIONS TO THIS WARRANTY.
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                     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:  *
1.4     ! noro       48:  * $OpenXM: OpenXM_contrib2/asir2000/lib/sturm,v 1.3 2000/08/22 05:04:23 noro Exp $
1.2       noro       49: */
1.1       noro       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;
1.4     ! noro      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;
1.1       noro      195: }
                    196: end;

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