source: git/factory/cf_resultant.cc @ 72dd6e

spielwiese
Last change on this file since 72dd6e was 72dd6e, checked in by Jens Schmidt <schmidt@…>, 27 years ago
* canonicalform.h: declarations of functions from cf_algorithm.cc moved to cf_algorithm.h. In all files which refer to these functions #include of cf_algorithm.h added. git-svn-id: file:///usr/local/Singular/svn/trunk@658 2c84dea3-7e68-4137-9b89-c4e89433aadc
  • Property mode set to 100644
File size: 6.1 KB
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1/* emacs edit mode for this file is -*- C++ -*- */
2/* $Id: cf_resultant.cc,v 1.3 1997-09-04 14:49:29 schmidt Exp $ */
3
4//{{{ docu
5//
6// cf_resultant.cc - algorithms for calculating resultants.
7//
8// Header file: cf_algorithm.h
9//
10//}}}
11
12#include <config.h>
13
14#include "assert.h"
15
16#include "canonicalform.h"
17#include "variable.h"
18#include "cf_algorithm.h"
19
20//{{{ CFArray subResChain ( const CanonicalForm & f, const CanonicalForm & g, const Variable & x )
21//{{{ docu
22//
23// subResChain() - caculate extended subresultant chain.
24//
25// The chain is calculated from f and g with respect to variable
26// x which should not be an algebraic variable.  If f or q equals
27// zero, an array consisting of one zero entry is returned.
28//
29// Note: this is not the standard subresultant chain but the
30// *extended* chain!
31//
32// This algorithm is from the article of R. Loos - 'Generalized
33// Polynomial Remainder Sequences' in B. Buchberger - 'Computer
34// Algebra - Symbolic and Algebraic Computation' with some
35// necessary extensions concerning the calculation of the first
36// step.
37//
38//}}}
39CFArray
40subResChain ( const CanonicalForm & f, const CanonicalForm & g, const Variable & x )
41{
42    ASSERT( x.level() > 0, "cannot calculate subresultant sequence with respect to algebraic variables" );
43
44    CFArray trivialResult( 0, 0 );
45    CanonicalForm F, G;
46    Variable X;
47
48    // some checks on triviality
49    if ( f.isZero() || g.isZero() ) {
50        trivialResult[0] = 0;
51        return trivialResult;
52    }
53
54    // make x main variable
55    if ( f.mvar() > x || g.mvar() > x ) {
56        if ( f.mvar() > g.mvar() )
57            X = f.mvar();
58        else
59            X = g.mvar();
60        F = swapvar( f, X, x );
61        G = swapvar( g, X, x );
62    }
63    else {
64        X = x;
65        F = f;
66        G = g;
67    }
68    // at this point, we have to calculate the sequence of F and
69    // G in respect to X where X is equal to or greater than the
70    // main variables of F and G
71
72    // initialization of chain
73    int m = degree( F, X );
74    int n = degree( G, X );
75
76    int j = (m <= n) ? n : m-1;
77    int r;
78
79    CFArray S( 0, j+1 );
80    CanonicalForm R;
81    S[j+1] = F; S[j] = G;
82
83    // make sure that S[j+1] is regular and j < n
84    if ( m == n && j > 0 ) {
85        S[j-1] = LC( S[j], X ) * psr( S[j+1], S[j], X );
86        j--;
87    } else if ( m < n ) {
88        S[j-1] = LC( S[j], X ) * LC( S[j], X ) * S[j+1];
89        j--;
90    } else if ( m > n && j > 0 ) {
91        // calculate first step
92        r = degree( S[j], X );
93        R = LC( S[j+1], X );
94
95        // if there was a gap calculate similar polynomial
96        if ( j > r && r >= 0 )
97            S[r] = power( LC( S[j], X ), j - r ) * S[j] * power( R, j - r );
98
99        if ( r > 0 ) {
100            // calculate remainder
101            S[r-1] = psr( S[j+1], S[j], X ) * power( -R, j - r );
102            j = r-1;
103        }
104    }
105
106    while ( j > 0 ) {
107        // at this point, 0 < j < n and S[j+1] is regular
108        r = degree( S[j], X );
109        R = LC( S[j+1], X );
110       
111        // if there was a gap calculate similar polynomial
112        if ( j > r && r >= 0 )
113            S[r] = (power( LC( S[j], X ), j - r ) * S[j]) / power( R, j - r );
114
115        if ( r <= 0 ) break;
116        // calculate remainder
117        S[r-1] = psr( S[j+1], S[j], X ) / power( -R, j - r + 2 );
118
119        j = r-1;
120        // again 0 <= j < r <= jOld and S[j+1] is regular
121    }
122
123    for ( j = 0; j <= S.max(); j++ ) {
124        // reswap variables if necessary
125        if ( X != x ) {
126            S[j] = swapvar( S[j], X, x );
127        }
128    }
129
130    return S;
131}
132//}}}
133
134//{{{ static CanonicalForm trivialResultant( const CanonicalForm & f, const CanonicalForm & g, const Variable & x )
135//{{{ docu
136//
137// trivialResultant - calculate trivial resultants.
138//
139// x's level should be larger than f's and g's levels.  Either f
140// or g should be constant or both linear.
141//
142//}}}
143static CanonicalForm
144trivialResultant( const CanonicalForm & f, const CanonicalForm & g, const Variable & x )
145{
146    // f or g in R
147    if ( degree( f, x ) == 0 )
148        return power( f, degree( g, x ) );
149    if ( degree( g, x ) == 0 )
150        return power( g, degree( f, x ) );
151
152    // f and g are linear polynomials
153    return LC( f, x ) * g - LC( g, x ) * f;
154}
155//}}}
156
157//{{{ CanonicalForm resultant( const CanonicalForm & f, const CanonicalForm & g, const Variable & x )
158//{{{ docu
159//
160// resultant() - return resultant of f and g with respect to x.
161//
162// The chain is calculated from f and g with respect to variable
163// x which should not be an algebraic variable.  If f or q equals
164// zero, zero is returned.  If f is a coefficient with respect to
165// x, f^degree(g, x) is returned, analogously for g.
166//
167// This algorithm serves as a wrapper around other resultant
168// algorithms which do the real work.  Here we use standard
169// properties of resultants only.
170//
171//}}}
172CanonicalForm
173resultant( const CanonicalForm & f, const CanonicalForm & g, const Variable & x )
174{
175    ASSERT( x.level() > 0, "cannot calculate resultant with respect to algebraic variables" );
176
177    // some checks on triviality.  We will not use degree( v )
178    // here because this may involve variable swapping.
179    if ( f.isZero() || g.isZero() )
180        return 0;
181    if ( f.mvar() < x )
182        return power( f, g.degree( x ) );
183    if ( g.mvar() < x )
184        return power( g, f.degree( x ) );
185
186    // make x main variale
187    CanonicalForm F, G;
188    Variable X;
189    if ( f.mvar() > x || g.mvar() > x ) {
190        if ( f.mvar() > g.mvar() )
191            X = f.mvar();
192        else
193            X = g.mvar();
194        F = swapvar( f, X, x );
195        G = swapvar( g, X, x );
196    }
197    else {
198        X = x;
199        F = f;
200        G = g;
201    }
202    // at this point, we have to calculate resultant( F, G, X )
203    // where X is equal to or greater than the main variables
204    // of F and G
205
206    int m = degree( F, X );
207    int n = degree( G, X );
208    // catch trivial cases
209    if ( m+n <= 2 || m == 0 || n == 0 )
210        return swapvar( trivialResultant( F, G, X ), X, x );
211
212    // exchange F and G if necessary
213    int flipFactor;
214    if ( m < n ) {
215        CanonicalForm swap = F;
216        F = G; G = swap;
217        int degswap = m;
218        m = n; n = degswap;
219        if ( m & 1 && n & 1 )
220            flipFactor = -1;
221        else
222            flipFactor = 1;
223    } else
224        flipFactor = 1;
225
226    // this is not an effective way to calculate the resultant!
227    CanonicalForm extFactor;
228    if ( m == n ) {
229        if ( n & 1 )
230            extFactor = -LC( G, X );
231        else
232            extFactor = LC( G, X );
233    } else
234        extFactor = power( LC( F, X ), m-n-1 );
235
236    CanonicalForm result;
237    result = subResChain( F, G, X )[0] / extFactor;
238
239    return swapvar( result, X, x ) * flipFactor;
240}
241//}}}
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