/* $OpenXM: OpenXM_contrib2/asir2000/lib/sturm,v 1.1 1999/12/03 07:39:11 noro Exp $ */
/* find intervals which include roots of a polynomial */
#include "defs.h"
def slice(P,XR,YR,WH) {
X = FIRST(XR); XMIN = SECOND(XR); XMAX = THIRD(XR);
Y = FIRST(YR); YMIN = SECOND(YR); YMAX = THIRD(YR);
W = FIRST(WH); H = SECOND(WH);
XS = (XMAX-XMIN)/W; YS = (YMAX-YMIN)/H;
T = ptozp(subst(P,X,X*XS+XMIN,Y,Y*YS+YMIN));
R = newvect(W+1);
for ( I = 0; I <= W; I++ ) {
S = sturm(subst(T,X,I));
R[I] = numch(S,Y,0)-numch(S,Y,H);
}
return R;
}
def slice1(P,XR,YR,WH) {
X = FIRST(XR); XMIN = SECOND(XR); XMAX = THIRD(XR);
Y = FIRST(YR); YMIN = SECOND(YR); YMAX = THIRD(YR);
W = FIRST(WH); H = SECOND(WH);
XS = (XMAX-XMIN)/W; YS = (YMAX-YMIN)/H;
T = ptozp(subst(P,X,X*XS+XMIN,Y,Y*YS+YMIN));
R = newvect(W+1);
for ( I = 0; I <= W; I++ ) {
S = sturm(subst(T,X,I));
R[I] = newvect(H+1);
seproot(S,Y,0,H,R[I]);
}
return R;
}
def seproot(S,X,MI,MA,R) {
N = car(size(S));
for ( I = MI; I <= MA; I++ )
if ( !(T = subst(S[0],X,I)) )
R[I] = -1;
else
break;
if ( I > MA )
return;
for ( J = MA; J >= MI; J-- )
if ( !(T = subst(S[0],X,J)) )
R[J] = -1;
else
break;
R[I] = numch(S,X,I); R[J] = numch(S,X,J);
if ( J <= I+1 )
return;
if ( R[I] == R[J] ) {
for ( K = I + 1; K < J; K++ )
R[K] = R[I];
return;
}
T = idiv(I+J,2);
seproot(S,X,I,T,R);
seproot(S,X,T,J,R);
}
/* compute the sturm sequence of P */
def sturm(P) {
V = var(P); N = deg(P,V); T = newvect(N+1);
G1 = T[0] = P; G2 = T[1] = ptozp(diff(P,var(P)));
for ( I = 1, H = 1, X = 1; ; ) {
if ( !deg(G2,V) )
break;
D = deg(G1,V)-deg(G2,V);
if ( (L = LCOEF(G2)) < 0 )
L = -L;
if ( !(R = -srem(L^(D+1)*G1,G2)) )
break;
if ( type(R) == 1 ) {
T[++I] = (R>0?1:-1); break;
}
M = H^D; G1 = G2;
G2 = T[++I] = sdiv(R,M*X);
if ( (X = LCOEF(G1)) < 0 )
X = -X;
H = X^D*H/M;
}
S = newvect(I+1);
for ( J = 0; J <= I; J++ )
S[J] = T[J];
return S;
}
def numch(S,V,A) {
N = car(size(S));
for ( T = subst(S[0],V,A), I = 1, C = 0; I < N; I++ ) {
T1 = subst(S[I],V,A);
if ( !T1 )
continue;
if ( (T1 > 0 && T < 0) || (T1 < 0 && T > 0) )
C++;
T = T1;
}
return C;
}
def numch0(S,V,A,T) {
N = car(size(S));
for ( I = 1, C = 0; I < N; I++ ) {
T1 = subst(S[I],V,A);
if ( !T1 )
continue;
if ( (T1 > 0 && T < 0) || (T1 < 0 && T > 0) )
C++;
T = T1;
}
return C;
}
end;