Annotation of OpenXM_contrib2/asir2000/lib/sturm, Revision 1.1.1.1
1.1 noro 1: /* $OpenXM: OpenXM/src/asir99/lib/sturm,v 1.1.1.1 1999/11/10 08:12:31 noro Exp $ */
2: /* find intervals which include roots of a polynomial */
3:
4: #include "defs.h"
5:
6: def slice(P,XR,YR,WH) {
7: X = FIRST(XR); XMIN = SECOND(XR); XMAX = THIRD(XR);
8: Y = FIRST(YR); YMIN = SECOND(YR); YMAX = THIRD(YR);
9: W = FIRST(WH); H = SECOND(WH);
10: XS = (XMAX-XMIN)/W; YS = (YMAX-YMIN)/H;
11: T = ptozp(subst(P,X,X*XS+XMIN,Y,Y*YS+YMIN));
12: R = newvect(W+1);
13: for ( I = 0; I <= W; I++ ) {
14: S = sturm(subst(T,X,I));
15: R[I] = numch(S,Y,0)-numch(S,Y,H);
16: }
17: return R;
18: }
19:
20: def slice1(P,XR,YR,WH) {
21: X = FIRST(XR); XMIN = SECOND(XR); XMAX = THIRD(XR);
22: Y = FIRST(YR); YMIN = SECOND(YR); YMAX = THIRD(YR);
23: W = FIRST(WH); H = SECOND(WH);
24: XS = (XMAX-XMIN)/W; YS = (YMAX-YMIN)/H;
25: T = ptozp(subst(P,X,X*XS+XMIN,Y,Y*YS+YMIN));
26: R = newvect(W+1);
27: for ( I = 0; I <= W; I++ ) {
28: S = sturm(subst(T,X,I));
29: R[I] = newvect(H+1);
30: seproot(S,Y,0,H,R[I]);
31: }
32: return R;
33: }
34:
35: def seproot(S,X,MI,MA,R) {
36: N = car(size(S));
37: for ( I = MI; I <= MA; I++ )
38: if ( !(T = subst(S[0],X,I)) )
39: R[I] = -1;
40: else
41: break;
42: if ( I > MA )
43: return;
44: for ( J = MA; J >= MI; J-- )
45: if ( !(T = subst(S[0],X,J)) )
46: R[J] = -1;
47: else
48: break;
49: R[I] = numch(S,X,I); R[J] = numch(S,X,J);
50: if ( J <= I+1 )
51: return;
52: if ( R[I] == R[J] ) {
53: for ( K = I + 1; K < J; K++ )
54: R[K] = R[I];
55: return;
56: }
57: T = idiv(I+J,2);
58: seproot(S,X,I,T,R);
59: seproot(S,X,T,J,R);
60: }
61:
62: /* compute the sturm sequence of P */
63:
64: def sturm(P) {
65: V = var(P); N = deg(P,V); T = newvect(N+1);
66: G1 = T[0] = P; G2 = T[1] = ptozp(diff(P,var(P)));
67: for ( I = 1, H = 1, X = 1; ; ) {
68: if ( !deg(G2,V) )
69: break;
70: D = deg(G1,V)-deg(G2,V);
71: if ( (L = LCOEF(G2)) < 0 )
72: L = -L;
73: if ( !(R = -srem(L^(D+1)*G1,G2)) )
74: break;
75: if ( type(R) == 1 ) {
76: T[++I] = (R>0?1:-1); break;
77: }
78: M = H^D; G1 = G2;
79: G2 = T[++I] = sdiv(R,M*X);
80: if ( (X = LCOEF(G1)) < 0 )
81: X = -X;
82: H = X^D*H/M;
83: }
84: S = newvect(I+1);
85: for ( J = 0; J <= I; J++ )
86: S[J] = T[J];
87: return S;
88: }
89:
90: def numch(S,V,A) {
91: N = car(size(S));
92: for ( T = subst(S[0],V,A), I = 1, C = 0; I < N; I++ ) {
93: T1 = subst(S[I],V,A);
94: if ( !T1 )
95: continue;
96: if ( (T1 > 0 && T < 0) || (T1 < 0 && T > 0) )
97: C++;
98: T = T1;
99: }
100: return C;
101: }
102:
103: def numch0(S,V,A,T) {
104: N = car(size(S));
105: for ( I = 1, C = 0; I < N; I++ ) {
106: T1 = subst(S[I],V,A);
107: if ( !T1 )
108: continue;
109: if ( (T1 > 0 && T < 0) || (T1 < 0 && T > 0) )
110: C++;
111: T = T1;
112: }
113: return C;
114: }
115: end;
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