[9bab9f] | 1 | // emacs edit mode for this file is -*- C++ -*- |
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[edb4893] | 2 | // $Id: cf_gcd.cc,v 1.1 1996-06-03 08:32:56 stobbe Exp $ |
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[9bab9f] | 3 | |
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| 4 | /* |
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| 5 | $Log: not supported by cvs2svn $ |
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[edb4893] | 6 | Revision 1.0 1996/05/17 11:56:37 stobbe |
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| 7 | Initial revision |
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| 8 | |
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[9bab9f] | 9 | */ |
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| 10 | |
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| 11 | #include "assert.h" |
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| 12 | #include "cf_defs.h" |
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| 13 | #include "canonicalform.h" |
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| 14 | #include "cf_iter.h" |
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| 15 | #include "cf_reval.h" |
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[edb4893] | 16 | #include "cf_primes.h" |
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| 17 | #include "cf_chinese.h" |
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| 18 | #include "templates/functions.h" |
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| 19 | |
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| 20 | static CanonicalForm gcd_poly( const CanonicalForm & f, const CanonicalForm& g, bool modularflag ); |
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| 21 | |
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[9bab9f] | 22 | |
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[edb4893] | 23 | static int |
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| 24 | isqrt ( int a ) |
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[9bab9f] | 25 | { |
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[edb4893] | 26 | int h, x0, x1 = a; |
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| 27 | do { |
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| 28 | x0 = x1; |
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| 29 | h = x0 * x0 + a - 1; |
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| 30 | if ( h % (2 * x0) == 0 ) |
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| 31 | x1 = h / (2 * x0); |
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| 32 | else |
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| 33 | x1 = (h - 1) / (2 * x0); |
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| 34 | } while ( x1 < x0 ); |
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| 35 | return x1; |
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[9bab9f] | 36 | } |
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| 37 | |
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| 38 | static bool |
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| 39 | gcd_test_one ( const CanonicalForm & f, const CanonicalForm & g ) |
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| 40 | { |
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| 41 | int count = 0; |
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| 42 | // assume polys have same level; |
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| 43 | CFRandom * sample = CFRandomFactory::generate(); |
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| 44 | REvaluation e( 2, f.level(), *sample ); |
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| 45 | delete sample; |
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| 46 | CanonicalForm lcf = swapvar( LC( f ), Variable(1), f.mvar() ); |
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| 47 | CanonicalForm lcg = swapvar( LC( g ), Variable(1), f.mvar() ); |
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| 48 | while ( ( e( lcf ).isZero() || e( lcg ).isZero() ) && count < 100 ) { |
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| 49 | e.nextpoint(); |
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| 50 | count++; |
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| 51 | } |
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| 52 | if ( count == 100 ) |
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| 53 | return false; |
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| 54 | CanonicalForm F=swapvar( f, Variable(1), f.mvar() ); |
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| 55 | CanonicalForm G=swapvar( g, Variable(1), g.mvar() ); |
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| 56 | if ( e(F).taildegree() > 0 && e(G).taildegree() > 0 ) |
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| 57 | return false; |
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| 58 | return gcd( e( F ), e( G ) ).degree() < 1; |
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| 59 | } |
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| 60 | |
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[edb4893] | 61 | static CanonicalForm |
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| 62 | maxnorm ( const CanonicalForm & f ) |
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| 63 | { |
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| 64 | CanonicalForm m = 0, h; |
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| 65 | CFIterator i; |
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| 66 | for ( i = f; i.hasTerms(); i++ ) |
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| 67 | m = tmax( m, abs( i.coeff() ) ); |
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| 68 | return m; |
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| 69 | } |
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| 70 | |
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| 71 | static void |
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| 72 | chinesePoly ( const CanonicalForm & f1, const CanonicalForm & q1, const CanonicalForm & f2, const CanonicalForm & q2, CanonicalForm & f, CanonicalForm & q ) |
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| 73 | { |
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| 74 | CFIterator i1 = f1, i2 = f2; |
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| 75 | CanonicalForm c; |
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| 76 | Variable x = f1.mvar(); |
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| 77 | f = 0; |
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| 78 | while ( i1.hasTerms() && i2.hasTerms() ) { |
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| 79 | if ( i1.exp() == i2.exp() ) { |
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| 80 | chineseRemainder( i1.coeff(), q1, i2.coeff(), q2, c, q ); |
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| 81 | f += power( x, i1.exp() ) * c; |
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| 82 | i1++; i2++; |
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| 83 | } |
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| 84 | else if ( i1.exp() > i2.exp() ) { |
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| 85 | chineseRemainder( 0, q1, i2.coeff(), q2, c, q ); |
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| 86 | f += power( x, i2.exp() ) * c; |
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| 87 | i2++; |
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| 88 | } |
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| 89 | else { |
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| 90 | chineseRemainder( i1.coeff(), q1, 0, q2, c, q ); |
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| 91 | f += power( x, i1.exp() ) * c; |
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| 92 | i1++; |
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| 93 | } |
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| 94 | } |
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| 95 | while ( i1.hasTerms() ) { |
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| 96 | chineseRemainder( i1.coeff(), q1, 0, q2, c, q ); |
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| 97 | f += power( x, i1.exp() ) * c; |
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| 98 | i1++; |
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| 99 | } |
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| 100 | while ( i2.hasTerms() ) { |
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| 101 | chineseRemainder( 0, q1, i2.coeff(), q2, c, q ); |
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| 102 | f += power( x, i2.exp() ) * c; |
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| 103 | i2++; |
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| 104 | } |
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| 105 | } |
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| 106 | |
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| 107 | static CanonicalForm |
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| 108 | balance ( const CanonicalForm & f, const CanonicalForm & q ) |
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| 109 | { |
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| 110 | CFIterator i; |
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| 111 | CanonicalForm result = 0, qh = q / 2; |
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| 112 | Variable x = f.mvar(); |
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| 113 | for ( i = f; i.hasTerms(); i++ ) { |
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| 114 | if ( i.coeff() > qh ) |
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| 115 | result += power( x, i.exp() ) * (i.coeff() - q); |
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| 116 | else |
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| 117 | result += power( x, i.exp() ) * i.coeff(); |
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| 118 | } |
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| 119 | return result; |
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| 120 | } |
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| 121 | |
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[9bab9f] | 122 | CanonicalForm |
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| 123 | igcd ( const CanonicalForm & f, const CanonicalForm & g ) |
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| 124 | { |
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| 125 | CanonicalForm a, b, c, dummy; |
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| 126 | |
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| 127 | if ( f.inZ() && g.inZ() && ! isOn( SW_RATIONAL ) ) { |
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| 128 | if ( f.sign() < 0 ) a = -f; else a = f; |
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| 129 | if ( g.sign() < 0 ) b = -g; else b = g; |
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| 130 | while ( ! b.isZero() ) { |
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| 131 | divrem( a, b, dummy, c ); |
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| 132 | a = b; |
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| 133 | b = c; |
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| 134 | } |
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| 135 | return a; |
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| 136 | } |
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| 137 | else |
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| 138 | return 1; |
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| 139 | } |
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| 140 | |
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| 141 | static CanonicalForm |
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| 142 | icontent ( const CanonicalForm & f, const CanonicalForm & c ) |
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| 143 | { |
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| 144 | if ( f.inCoeffDomain() ) |
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| 145 | return gcd( f, c ); |
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| 146 | else { |
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| 147 | CanonicalForm g = c; |
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| 148 | for ( CFIterator i = f; i.hasTerms() && ! g.isOne(); i++ ) |
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| 149 | g = icontent( i.coeff(), g ); |
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| 150 | return g; |
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| 151 | } |
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| 152 | } |
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| 153 | |
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| 154 | CanonicalForm |
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| 155 | icontent ( const CanonicalForm & f ) |
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| 156 | { |
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| 157 | return icontent( f, 0 ); |
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| 158 | } |
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| 159 | |
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| 160 | CanonicalForm |
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| 161 | iextgcd ( const CanonicalForm & f, const CanonicalForm & g, CanonicalForm & a, CanonicalForm & b ) |
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| 162 | { |
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| 163 | CanonicalForm p0 = f, p1 = g; |
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| 164 | CanonicalForm f0 = 1, f1 = 0, g0 = 0, g1 = 1, q, r; |
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| 165 | |
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| 166 | while ( ! p1.isZero() ) { |
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| 167 | divrem( p0, p1, q, r ); |
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| 168 | p0 = p1; p1 = r; |
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| 169 | r = g0 - g1 * q; |
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| 170 | g0 = g1; g1 = r; |
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| 171 | r = f0 - f1 * q; |
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| 172 | f0 = f1; f1 = r; |
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| 173 | } |
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| 174 | a = f0; |
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| 175 | b = g0; |
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| 176 | if ( p0.sign() < 0 ) { |
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| 177 | p0 = -p0; |
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| 178 | a = -a; |
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| 179 | b = -b; |
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| 180 | } |
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| 181 | return p0; |
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| 182 | } |
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| 183 | |
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| 184 | CanonicalForm |
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| 185 | extgcd ( const CanonicalForm & f, const CanonicalForm & g, CanonicalForm & a, CanonicalForm & b ) |
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| 186 | { |
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| 187 | CanonicalForm contf = content( f ); |
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| 188 | CanonicalForm contg = content( g ); |
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| 189 | |
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| 190 | CanonicalForm p0 = f / contf, p1 = g / contg; |
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| 191 | CanonicalForm f0 = 1, f1 = 0, g0 = 0, g1 = 1, q, r; |
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| 192 | |
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| 193 | while ( ! p1.isZero() ) { |
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| 194 | divrem( p0, p1, q, r ); |
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| 195 | p0 = p1; p1 = r; |
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| 196 | r = g0 - g1 * q; |
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| 197 | g0 = g1; g1 = r; |
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| 198 | r = f0 - f1 * q; |
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| 199 | f0 = f1; f1 = r; |
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| 200 | } |
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| 201 | CanonicalForm contp0 = content( p0 ); |
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| 202 | a = f0 / ( contf * contp0 ); |
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| 203 | b = g0 / ( contg * contp0 ); |
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| 204 | p0 /= contp0; |
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| 205 | if ( p0.sign() < 0 ) { |
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| 206 | p0 = -p0; |
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| 207 | a = -a; |
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| 208 | b = -b; |
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| 209 | } |
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| 210 | return p0; |
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| 211 | } |
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| 212 | |
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[edb4893] | 213 | static CanonicalForm |
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| 214 | gcd_poly_univar0( const CanonicalForm & F, const CanonicalForm & G, bool primitive ) |
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| 215 | { |
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| 216 | CanonicalForm f, g, c, cg, cl, BB, B, M, q, Dp, newD, D, newq; |
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| 217 | int p, i, n; |
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| 218 | |
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| 219 | if ( primitive ) { |
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| 220 | f = F; |
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| 221 | g = G; |
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| 222 | c = 1; |
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| 223 | } |
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| 224 | else { |
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| 225 | CanonicalForm cF = content( F ), cG = content( G ); |
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| 226 | f = F / cF; |
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| 227 | g = G / cG; |
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| 228 | c = igcd( cF, cG ); |
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| 229 | } |
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| 230 | cg = gcd( f.lc(), g.lc() ); |
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| 231 | cl = ( f.lc() / cg ) * g.lc(); |
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| 232 | // B = 2 * cg * tmin( |
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| 233 | // maxnorm(f)*power(CanonicalForm(2),f.degree())*isqrt(f.degree()+1), |
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| 234 | // maxnorm(g)*power(CanonicalForm(2),g.degree())*isqrt(g.degree()+1) |
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| 235 | // )+1; |
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| 236 | M = tmin( maxnorm(f), maxnorm(g) ); |
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| 237 | BB = power(CanonicalForm(2),tmin(f.degree(),g.degree()))*M; |
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| 238 | q = 0; |
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| 239 | i = 1; |
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| 240 | n = cf_getNumBigPrimes(); |
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| 241 | while ( true ) { |
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| 242 | B = BB; |
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| 243 | while ( i < n && q < B ) { |
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| 244 | p = cf_getBigPrime( i ); |
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| 245 | i++; |
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| 246 | while ( i < n && mod( cl, p ) == 0 ) { |
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| 247 | p = cf_getBigPrime( i ); |
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| 248 | i++; |
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| 249 | } |
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| 250 | setCharacteristic( p ); |
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| 251 | Dp = gcd( mapinto( f ), mapinto( g ) ); |
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| 252 | Dp = ( Dp / Dp.lc() ) * mapinto( cg ); |
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| 253 | setCharacteristic( 0 ); |
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| 254 | if ( Dp.degree() == 0 ) return c; |
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| 255 | if ( q.isZero() ) { |
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| 256 | D = mapinto( Dp ); |
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| 257 | q = p; |
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| 258 | B = power(CanonicalForm(2),D.degree())*M+1; |
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| 259 | } |
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| 260 | else { |
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| 261 | if ( Dp.degree() == D.degree() ) { |
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| 262 | chinesePoly( D, q, mapinto( Dp ), p, newD, newq ); |
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| 263 | q = newq; |
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| 264 | D = newD; |
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| 265 | } |
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| 266 | else if ( Dp.degree() < D.degree() ) { |
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| 267 | // all previous p's are bad primes |
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| 268 | q = p; |
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| 269 | D = mapinto( Dp ); |
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| 270 | B = power(CanonicalForm(2),D.degree())*M+1; |
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| 271 | } |
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| 272 | // else p is a bad prime |
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| 273 | } |
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| 274 | } |
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| 275 | if ( i < n ) { |
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| 276 | // now balance D mod q |
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| 277 | D = pp( balance( cg * D, q ) ); |
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| 278 | if ( divides( D, f ) && divides( D, g ) ) |
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| 279 | return D * c; |
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| 280 | else |
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| 281 | q = 0; |
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| 282 | } |
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| 283 | else { |
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| 284 | return gcd_poly( F, G, false ); |
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| 285 | } |
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| 286 | } |
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| 287 | } |
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| 288 | |
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| 289 | |
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| 290 | static CanonicalForm |
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| 291 | gcd_poly( const CanonicalForm & f, const CanonicalForm& g, bool modularflag ) |
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[9bab9f] | 292 | { |
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| 293 | CanonicalForm C, Ci, Ci1, Hi, bi, pi, pi1, pi2; |
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| 294 | int delta; |
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| 295 | Variable v = f.mvar(); |
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| 296 | |
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| 297 | if ( f.degree( v ) >= g.degree( v ) ) { |
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| 298 | pi = f; pi1 = g; |
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| 299 | } |
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| 300 | else { |
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| 301 | pi = g; pi1 = f; |
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| 302 | } |
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| 303 | Ci = content( pi ); Ci1 = content( pi1 ); |
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| 304 | C = gcd( Ci, Ci1 ); |
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| 305 | pi1 = pi1 / Ci1; pi = pi / Ci; |
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[edb4893] | 306 | if ( ! pi.isUnivariate() ) { |
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[9bab9f] | 307 | if ( gcd_test_one( pi1, pi ) ) |
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| 308 | return C; |
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[edb4893] | 309 | } |
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| 310 | else { |
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| 311 | // pi is univariate |
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| 312 | if ( modularflag ) |
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| 313 | return gcd_poly_univar0( pi, pi1, true ) * C; |
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| 314 | } |
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[9bab9f] | 315 | delta = degree( pi, v ) - degree( pi1, v ); |
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| 316 | Hi = power( LC( pi1, v ), delta ); |
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| 317 | if ( (delta+1) % 2 ) |
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| 318 | bi = 1; |
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| 319 | else |
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| 320 | bi = -1; |
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| 321 | while ( degree( pi1, v ) > 0 ) { |
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| 322 | pi2 = psr( pi, pi1, v ); |
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| 323 | pi2 = pi2 / bi; |
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| 324 | pi = pi1; pi1 = pi2; |
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| 325 | if ( degree( pi1, v ) > 0 ) { |
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| 326 | delta = degree( pi, v ) - degree( pi1, v ); |
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| 327 | if ( (delta+1) % 2 ) |
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| 328 | bi = LC( pi, v ) * power( Hi, delta ); |
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| 329 | else |
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| 330 | bi = -LC( pi, v ) * power( Hi, delta ); |
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| 331 | Hi = power( LC( pi1, v ), delta ) / power( Hi, delta-1 ); |
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| 332 | } |
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| 333 | } |
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| 334 | if ( degree( pi1, v ) == 0 ) |
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| 335 | return C; |
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| 336 | else { |
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| 337 | return C * pp( pi ); |
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| 338 | } |
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| 339 | } |
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| 340 | |
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| 341 | |
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| 342 | static CanonicalForm |
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| 343 | cf_content ( const CanonicalForm & f, const CanonicalForm & g ) |
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| 344 | { |
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| 345 | if ( f.inPolyDomain() || ( f.inExtension() && ! getReduce( f.mvar() ) ) ) { |
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| 346 | CFIterator i = f; |
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| 347 | CanonicalForm result = g; |
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| 348 | while ( i.hasTerms() && ! result.isOne() ) { |
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| 349 | result = gcd( result, i.coeff() ); |
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| 350 | i++; |
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| 351 | } |
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| 352 | return result; |
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| 353 | } |
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| 354 | else |
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| 355 | if ( f.sign() < 0 ) |
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| 356 | return -f; |
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| 357 | else |
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| 358 | return f; |
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| 359 | } |
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| 360 | |
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| 361 | CanonicalForm |
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| 362 | content ( const CanonicalForm & f ) |
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| 363 | { |
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| 364 | return cf_content( f, 0 ); |
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| 365 | } |
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| 366 | |
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| 367 | CanonicalForm |
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| 368 | content ( const CanonicalForm & f, const Variable & x ) |
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| 369 | { |
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| 370 | if ( f.mvar() == x ) |
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| 371 | return cf_content( f, 0 ); |
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| 372 | else if ( f.mvar() < x ) |
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| 373 | return f; |
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| 374 | else |
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| 375 | return swapvar( content( swapvar( f, f.mvar(), x ), f.mvar() ), f.mvar(), x ); |
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| 376 | } |
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| 377 | |
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| 378 | CanonicalForm |
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| 379 | vcontent ( const CanonicalForm & f, const Variable & x ) |
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| 380 | { |
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| 381 | if ( f.mvar() <= x ) |
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| 382 | return content( f, x ); |
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| 383 | else { |
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| 384 | CFIterator i; |
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| 385 | CanonicalForm d = 0; |
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| 386 | for ( i = f; i.hasTerms() && ! d.isOne(); i++ ) |
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| 387 | d = gcd( d, vcontent( i.coeff(), x ) ); |
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| 388 | return d; |
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| 389 | } |
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| 390 | } |
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| 391 | |
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| 392 | CanonicalForm |
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| 393 | pp ( const CanonicalForm & f ) |
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| 394 | { |
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| 395 | if ( f.isZero() ) |
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| 396 | return f; |
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| 397 | else |
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| 398 | return f / content( f ); |
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| 399 | } |
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| 400 | |
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| 401 | CanonicalForm |
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| 402 | gcd ( const CanonicalForm & f, const CanonicalForm & g ) |
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| 403 | { |
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| 404 | if ( f.isZero() ) |
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| 405 | return g; |
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| 406 | else if ( g.isZero() ) |
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| 407 | return f; |
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| 408 | else if ( f.inBaseDomain() ) |
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| 409 | if ( g.inBaseDomain() ) |
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| 410 | return igcd( f, g ); |
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| 411 | else |
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| 412 | return cf_content( g, f ); |
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| 413 | else if ( g.inBaseDomain() ) |
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| 414 | return cf_content( f, g ); |
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| 415 | else if ( f.mvar() == g.mvar() ) |
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| 416 | if ( f.inExtension() && getReduce( f.mvar() ) ) |
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| 417 | return 1; |
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| 418 | else { |
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| 419 | if ( getCharacteristic() == 0 && isOn( SW_RATIONAL ) ) { |
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| 420 | Off( SW_RATIONAL ); |
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| 421 | CanonicalForm l = lcm( common_den( f ), common_den( g ) ); |
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| 422 | On( SW_RATIONAL ); |
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| 423 | CanonicalForm F = f * l, G = g * l; |
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| 424 | Off( SW_RATIONAL ); |
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[edb4893] | 425 | l = gcd_poly( F, G, true ); |
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[9bab9f] | 426 | On( SW_RATIONAL ); |
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| 427 | return l; |
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| 428 | } |
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| 429 | else |
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[edb4893] | 430 | return gcd_poly( f, g, getCharacteristic()==0 ); |
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[9bab9f] | 431 | } |
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| 432 | else if ( f.mvar() > g.mvar() ) |
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| 433 | return cf_content( f, g ); |
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| 434 | else |
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| 435 | return cf_content( g, f ); |
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| 436 | } |
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| 437 | |
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| 438 | CanonicalForm |
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| 439 | lcm ( const CanonicalForm & f, const CanonicalForm & g ) |
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| 440 | { |
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| 441 | return ( f / gcd( f, g ) ) * g; |
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| 442 | } |
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