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'
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: *
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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