[483c5c] | 1 | /* emacs edit mode for this file is -*- C++ -*- */ |
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[341696] | 2 | /* $Id$ */ |
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[483c5c] | 3 | |
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| 4 | //{{{ docu |
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| 5 | // |
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| 6 | // cf_algorithm.cc - simple mathematical algorithms. |
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| 7 | // |
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[a27df3] | 8 | // Hierarchy: mathematical algorithms on canonical forms |
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[483c5c] | 9 | // |
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[a27df3] | 10 | // Developers note: |
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| 11 | // ---------------- |
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| 12 | // A "mathematical" algorithm is an algorithm which calculates |
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| 13 | // some mathematical function in contrast to a "structural" |
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| 14 | // algorithm which gives structural information on polynomials. |
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[cbb3fa] | 15 | // |
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[a27df3] | 16 | // Compare these functions to the functions in `cf_ops.cc', which |
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| 17 | // are structural algorithms. |
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[483c5c] | 18 | // |
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| 19 | //}}} |
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| 20 | |
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| 21 | #include <config.h> |
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| 22 | |
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[cbb3fa] | 23 | #include "assert.h" |
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| 24 | |
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[a27df3] | 25 | #include "cf_factory.h" |
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[483c5c] | 26 | #include "cf_defs.h" |
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| 27 | #include "canonicalform.h" |
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[9719434] | 28 | #include "cf_algorithm.h" |
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[483c5c] | 29 | #include "variable.h" |
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| 30 | #include "cf_iter.h" |
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[a27df3] | 31 | #include "ftmpl_functions.h" |
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[483c5c] | 32 | |
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[27bb97f] | 33 | void out_cf(const char *s1,const CanonicalForm &f,const char *s2); |
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[ad82f6] | 34 | |
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[cbb3fa] | 35 | //{{{ CanonicalForm psr ( const CanonicalForm & f, const CanonicalForm & g, const Variable & x ) |
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| 36 | //{{{ docu |
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| 37 | // |
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[a27df3] | 38 | // psr() - return pseudo remainder of `f' and `g' with respect |
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[9719434] | 39 | // to `x'. |
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[cbb3fa] | 40 | // |
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[a27df3] | 41 | // `g' must not equal zero. |
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[cbb3fa] | 42 | // |
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[a27df3] | 43 | // For f and g in R[x], R an arbitrary ring, g != 0, there is a |
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| 44 | // representation |
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[cbb3fa] | 45 | // |
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[a27df3] | 46 | // LC(g)^s*f = g*q + r |
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[cbb3fa] | 47 | // |
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| 48 | // with r = 0 or deg(r) < deg(g) and s = 0 if f = 0 or |
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| 49 | // s = max( 0, deg(f)-deg(g)+1 ) otherwise. |
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[a27df3] | 50 | // r = psr(f, g) and q = psq(f, g) are called "pseudo remainder" |
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| 51 | // and "pseudo quotient", resp. They are uniquely determined if |
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| 52 | // LC(g) is not a zero divisor in R. |
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| 53 | // |
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| 54 | // See H.-J. Reiffen/G. Scheja/U. Vetter - "Algebra", 2nd ed., |
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| 55 | // par. 15, for a reference. |
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| 56 | // |
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| 57 | // Type info: |
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| 58 | // ---------- |
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| 59 | // f, g: Current |
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| 60 | // x: Polynomial |
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| 61 | // |
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| 62 | // Polynomials over prime power domains are admissible if |
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| 63 | // lc(LC(`g',`x')) is not a zero divisor. This is a slightly |
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| 64 | // stronger precondition than mathematically necessary since |
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| 65 | // pseudo remainder and quotient are well-defined if LC(`g',`x') |
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| 66 | // is not a zero divisor. |
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| 67 | // |
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| 68 | // For example, psr(y^2, (13*x+1)*y) is well-defined in |
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| 69 | // (Z/13^2[x])[y] since (13*x+1) is not a zero divisor. But |
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| 70 | // calculating it with Factory would fail since 13 is a zero |
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| 71 | // divisor in Z/13^2. |
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| 72 | // |
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| 73 | // Due to this inconsistency with mathematical notion, we decided |
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| 74 | // not to declare type `CurrentPP' for `f' and `g'. |
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| 75 | // |
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| 76 | // Developers note: |
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| 77 | // ---------------- |
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| 78 | // This is not an optimal implementation. Better would have been |
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| 79 | // an implementation in `InternalPoly' avoiding the |
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| 80 | // exponentiation of the leading coefficient of `g'. In contrast |
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| 81 | // to `psq()' and `psqr()' it definitely seems worth to implement |
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| 82 | // the pseudo remainder on the internal level. Should be done |
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| 83 | // soon. |
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[cbb3fa] | 84 | // |
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| 85 | //}}} |
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[483c5c] | 86 | CanonicalForm |
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[7fb3ba4] | 87 | #if 0 |
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[483c5c] | 88 | psr ( const CanonicalForm & f, const CanonicalForm & g, const Variable & x ) |
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| 89 | { |
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[6918d65] | 90 | |
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[a27df3] | 91 | ASSERT( x.level() > 0, "type error: polynomial variable expected" ); |
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| 92 | ASSERT( ! g.isZero(), "math error: division by zero" ); |
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| 93 | |
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| 94 | // swap variables such that x's level is larger or equal |
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| 95 | // than both f's and g's levels. |
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| 96 | Variable X = tmax( tmax( f.mvar(), g.mvar() ), x ); |
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| 97 | CanonicalForm F = swapvar( f, x, X ); |
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| 98 | CanonicalForm G = swapvar( g, x, X ); |
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[cbb3fa] | 99 | |
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[a27df3] | 100 | // now, we have to calculate the pseudo remainder of F and G |
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| 101 | // w.r.t. X |
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| 102 | int fDegree = degree( F, X ); |
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| 103 | int gDegree = degree( G, X ); |
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[5453c7] | 104 | if ( (fDegree < 0) || (fDegree < gDegree) ) |
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[6918d65] | 105 | return f; |
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[5453c7] | 106 | else |
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| 107 | { |
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[6918d65] | 108 | CanonicalForm xresult = (power( LC( G, X ), fDegree-gDegree+1 ) * F) ; |
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| 109 | CanonicalForm result = xresult -(xresult/G)*G; |
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| 110 | return swapvar( result, x, X ); |
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[a27df3] | 111 | } |
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[483c5c] | 112 | } |
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[5bb462] | 113 | #else |
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[ad82f6] | 114 | psr ( const CanonicalForm &rr, const CanonicalForm &vv, const Variable & x ) |
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| 115 | { |
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[5bb462] | 116 | CanonicalForm r=rr, v=vv, l, test, lu, lv, t, retvalue; |
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| 117 | int dr, dv, d,n=0; |
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| 118 | |
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| 119 | |
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| 120 | dr = degree( r, x ); |
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[ad82f6] | 121 | if (dr>0) |
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[dad0bc5] | 122 | { |
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[ad82f6] | 123 | dv = degree( v, x ); |
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| 124 | if (dv <= dr) {l=LC(v,x); v = v -l*power(x,dv);} |
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| 125 | else { l = 1; } |
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| 126 | d= dr-dv+1; |
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[e7f5ac] | 127 | //out_cf("psr(",rr," "); |
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| 128 | //out_cf("",vv," "); |
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| 129 | //printf(" var=%d\n",x.level()); |
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[ad82f6] | 130 | while ( ( dv <= dr ) && ( !r.isZero()) ) |
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| 131 | { |
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| 132 | test = power(x,dr-dv)*v*LC(r,x); |
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| 133 | if ( dr == 0 ) { r= CanonicalForm(0); } |
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| 134 | else { r= r - LC(r,x)*power(x,dr); } |
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| 135 | r= l*r -test; |
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| 136 | dr= degree(r,x); |
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| 137 | n+=1; |
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| 138 | } |
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| 139 | r= power(l, d-n)*r; |
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[5bb462] | 140 | } |
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| 141 | return r; |
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| 142 | } |
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| 143 | #endif |
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[cbb3fa] | 144 | //}}} |
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[483c5c] | 145 | |
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[cbb3fa] | 146 | //{{{ CanonicalForm psq ( const CanonicalForm & f, const CanonicalForm & g, const Variable & x ) |
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| 147 | //{{{ docu |
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| 148 | // |
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[a27df3] | 149 | // psq() - return pseudo quotient of `f' and `g' with respect |
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[9719434] | 150 | // to `x'. |
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[cbb3fa] | 151 | // |
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[a27df3] | 152 | // `g' must not equal zero. |
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| 153 | // |
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[9719434] | 154 | // See `psr()' for more detailed information. |
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[cbb3fa] | 155 | // |
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[a27df3] | 156 | // Type info: |
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| 157 | // ---------- |
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| 158 | // f, g: Current |
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| 159 | // x: Polynomial |
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| 160 | // |
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| 161 | // Developers note: |
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| 162 | // ---------------- |
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| 163 | // This is not an optimal implementation. Better would have been |
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| 164 | // an implementation in `InternalPoly' avoiding the |
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| 165 | // exponentiation of the leading coefficient of `g'. It seemed |
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| 166 | // not worth to do so. |
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| 167 | // |
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[cbb3fa] | 168 | //}}} |
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[483c5c] | 169 | CanonicalForm |
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| 170 | psq ( const CanonicalForm & f, const CanonicalForm & g, const Variable & x ) |
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| 171 | { |
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[a27df3] | 172 | ASSERT( x.level() > 0, "type error: polynomial variable expected" ); |
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| 173 | ASSERT( ! g.isZero(), "math error: division by zero" ); |
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[cbb3fa] | 174 | |
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[a27df3] | 175 | // swap variables such that x's level is larger or equal |
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| 176 | // than both f's and g's levels. |
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| 177 | Variable X = tmax( tmax( f.mvar(), g.mvar() ), x ); |
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| 178 | CanonicalForm F = swapvar( f, x, X ); |
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| 179 | CanonicalForm G = swapvar( g, x, X ); |
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| 180 | |
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| 181 | // now, we have to calculate the pseudo remainder of F and G |
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| 182 | // w.r.t. X |
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| 183 | int fDegree = degree( F, X ); |
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| 184 | int gDegree = degree( G, X ); |
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| 185 | if ( fDegree < 0 || fDegree < gDegree ) |
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[6918d65] | 186 | return 0; |
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[a27df3] | 187 | else { |
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[6918d65] | 188 | CanonicalForm result = (power( LC( G, X ), fDegree-gDegree+1 ) * F) / G; |
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| 189 | return swapvar( result, x, X ); |
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[a27df3] | 190 | } |
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[483c5c] | 191 | } |
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[cbb3fa] | 192 | //}}} |
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[483c5c] | 193 | |
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[cbb3fa] | 194 | //{{{ void psqr ( const CanonicalForm & f, const CanonicalForm & g, CanonicalForm & q, CanonicalForm & r, const Variable & x ) |
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| 195 | //{{{ docu |
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| 196 | // |
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[9719434] | 197 | // psqr() - calculate pseudo quotient and remainder of `f' and |
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| 198 | // `g' with respect to `x'. |
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[cbb3fa] | 199 | // |
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[a27df3] | 200 | // Returns the pseudo quotient of `f' and `g' in `q', the pseudo |
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| 201 | // remainder in `r'. `g' must not equal zero. |
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| 202 | // |
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[9719434] | 203 | // See `psr()' for more detailed information. |
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[cbb3fa] | 204 | // |
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[a27df3] | 205 | // Type info: |
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| 206 | // ---------- |
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| 207 | // f, g: Current |
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| 208 | // q, r: Anything |
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| 209 | // x: Polynomial |
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| 210 | // |
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| 211 | // Developers note: |
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| 212 | // ---------------- |
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| 213 | // This is not an optimal implementation. Better would have been |
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| 214 | // an implementation in `InternalPoly' avoiding the |
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| 215 | // exponentiation of the leading coefficient of `g'. It seemed |
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| 216 | // not worth to do so. |
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| 217 | // |
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[cbb3fa] | 218 | //}}} |
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[483c5c] | 219 | void |
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| 220 | psqr ( const CanonicalForm & f, const CanonicalForm & g, CanonicalForm & q, CanonicalForm & r, const Variable& x ) |
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| 221 | { |
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[a27df3] | 222 | ASSERT( x.level() > 0, "type error: polynomial variable expected" ); |
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| 223 | ASSERT( ! g.isZero(), "math error: division by zero" ); |
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| 224 | |
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| 225 | // swap variables such that x's level is larger or equal |
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| 226 | // than both f's and g's levels. |
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| 227 | Variable X = tmax( tmax( f.mvar(), g.mvar() ), x ); |
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| 228 | CanonicalForm F = swapvar( f, x, X ); |
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| 229 | CanonicalForm G = swapvar( g, x, X ); |
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[cbb3fa] | 230 | |
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[a27df3] | 231 | // now, we have to calculate the pseudo remainder of F and G |
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| 232 | // w.r.t. X |
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| 233 | int fDegree = degree( F, X ); |
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| 234 | int gDegree = degree( G, X ); |
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| 235 | if ( fDegree < 0 || fDegree < gDegree ) { |
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[6918d65] | 236 | q = 0; r = f; |
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[a27df3] | 237 | } else { |
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[6918d65] | 238 | divrem( power( LC( G, X ), fDegree-gDegree+1 ) * F, G, q, r ); |
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| 239 | q = swapvar( q, x, X ); |
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| 240 | r = swapvar( r, x, X ); |
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[cbb3fa] | 241 | } |
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[483c5c] | 242 | } |
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[cbb3fa] | 243 | //}}} |
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[483c5c] | 244 | |
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[9719434] | 245 | //{{{ static CanonicalForm internalBCommonDen ( const CanonicalForm & f ) |
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[cbb3fa] | 246 | //{{{ docu |
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| 247 | // |
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[9719434] | 248 | // internalBCommonDen() - recursively calculate multivariate |
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| 249 | // common denominator of coefficients of `f'. |
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[cbb3fa] | 250 | // |
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[9719434] | 251 | // Used by: bCommonDen() |
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[cbb3fa] | 252 | // |
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[a27df3] | 253 | // Type info: |
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| 254 | // ---------- |
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| 255 | // f: Poly( Q ) |
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| 256 | // Switches: isOff( SW_RATIONAL ) |
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| 257 | // |
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[cbb3fa] | 258 | //}}} |
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[483c5c] | 259 | static CanonicalForm |
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[9719434] | 260 | internalBCommonDen ( const CanonicalForm & f ) |
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[483c5c] | 261 | { |
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[cbb3fa] | 262 | if ( f.inBaseDomain() ) |
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[6918d65] | 263 | return f.den(); |
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[483c5c] | 264 | else { |
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[6918d65] | 265 | CanonicalForm result = 1; |
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| 266 | for ( CFIterator i = f; i.hasTerms(); i++ ) |
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| 267 | result = blcm( result, internalBCommonDen( i.coeff() ) ); |
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| 268 | return result; |
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[483c5c] | 269 | } |
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| 270 | } |
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[cbb3fa] | 271 | //}}} |
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[483c5c] | 272 | |
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[9719434] | 273 | //{{{ CanonicalForm bCommonDen ( const CanonicalForm & f ) |
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[cbb3fa] | 274 | //{{{ docu |
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| 275 | // |
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[9719434] | 276 | // bCommonDen() - calculate multivariate common denominator of |
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| 277 | // coefficients of `f'. |
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[cbb3fa] | 278 | // |
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| 279 | // The common denominator is calculated with respect to all |
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[9719434] | 280 | // coefficients of `f' which are in a base domain. In other |
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| 281 | // words, common_den( `f' ) * `f' is guaranteed to have integer |
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[a27df3] | 282 | // coefficients only. The common denominator of zero is one. |
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[cbb3fa] | 283 | // |
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[9719434] | 284 | // Returns something non-trivial iff the current domain is Q. |
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[cbb3fa] | 285 | // |
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[a27df3] | 286 | // Type info: |
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| 287 | // ---------- |
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| 288 | // f: CurrentPP |
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| 289 | // |
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[cbb3fa] | 290 | //}}} |
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[483c5c] | 291 | CanonicalForm |
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[9719434] | 292 | bCommonDen ( const CanonicalForm & f ) |
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[483c5c] | 293 | { |
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| 294 | if ( getCharacteristic() == 0 && isOn( SW_RATIONAL ) ) { |
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[6918d65] | 295 | // otherwise `bgcd()' returns one |
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| 296 | Off( SW_RATIONAL ); |
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| 297 | CanonicalForm result = internalBCommonDen( f ); |
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| 298 | On( SW_RATIONAL ); |
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| 299 | return result; |
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[9719434] | 300 | } else |
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[6918d65] | 301 | return CanonicalForm( 1 ); |
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[483c5c] | 302 | } |
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[cbb3fa] | 303 | //}}} |
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[dff969] | 304 | |
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[ebc602] | 305 | //{{{ bool fdivides ( const CanonicalForm & f, const CanonicalForm & g ) |
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[dff969] | 306 | //{{{ docu |
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| 307 | // |
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[ebc602] | 308 | // fdivides() - check whether `f' divides `g'. |
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[dff969] | 309 | // |
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[a27df3] | 310 | // Returns true iff `f' divides `g'. Uses some extra heuristic |
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| 311 | // to avoid polynomial division. Without the heuristic, the test |
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| 312 | // essentialy looks like `divremt(g, f, q, r) && r.isZero()'. |
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| 313 | // |
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| 314 | // Type info: |
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| 315 | // ---------- |
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| 316 | // f, g: Current |
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| 317 | // |
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| 318 | // Elements from prime power domains (or polynomials over such |
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| 319 | // domains) are admissible if `f' (or lc(`f'), resp.) is not a |
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| 320 | // zero divisor. This is a slightly stronger precondition than |
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| 321 | // mathematically necessary since divisibility is a well-defined |
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| 322 | // notion in arbitrary rings. Hence, we decided not to declare |
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| 323 | // the weaker type `CurrentPP'. |
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| 324 | // |
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| 325 | // Developers note: |
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| 326 | // ---------------- |
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[ebc602] | 327 | // One may consider the the test `fdivides( f.LC(), g.LC() )' in |
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[a27df3] | 328 | // the main `if'-test superfluous since `divremt()' in the |
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| 329 | // `if'-body repeats the test. However, `divremt()' does not use |
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| 330 | // any heuristic to do so. |
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| 331 | // |
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[ebc602] | 332 | // It seems not reasonable to call `fdivides()' from `divremt()' |
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| 333 | // to check divisibility of leading coefficients. `fdivides()' is |
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[a27df3] | 334 | // on a relatively high level compared to `divremt()'. |
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[dff969] | 335 | // |
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| 336 | //}}} |
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| 337 | bool |
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[ebc602] | 338 | fdivides ( const CanonicalForm & f, const CanonicalForm & g ) |
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[dff969] | 339 | { |
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[a27df3] | 340 | // trivial cases |
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| 341 | if ( g.isZero() ) |
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[6918d65] | 342 | return true; |
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[a27df3] | 343 | else if ( f.isZero() ) |
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[6918d65] | 344 | return false; |
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[a27df3] | 345 | |
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| 346 | if ( (f.inCoeffDomain() || g.inCoeffDomain()) |
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[6918d65] | 347 | && ((getCharacteristic() == 0 && isOn( SW_RATIONAL )) |
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| 348 | || (getCharacteristic() > 0 && CFFactory::gettype() != PrimePowerDomain)) ) |
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| 349 | // if we are in a field all elements not equal to zero are units |
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| 350 | if ( f.inCoeffDomain() ) |
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| 351 | return true; |
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| 352 | else |
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| 353 | // g.inCoeffDomain() |
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| 354 | return false; |
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[a27df3] | 355 | |
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| 356 | // we may assume now that both levels either equal LEVELBASE |
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| 357 | // or are greater zero |
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| 358 | int fLevel = f.level(); |
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| 359 | int gLevel = g.level(); |
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[94a967] | 360 | if ( (gLevel > 0) && (fLevel == gLevel) ) |
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[6918d65] | 361 | // f and g are polynomials in the same main variable |
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| 362 | if ( degree( f ) <= degree( g ) |
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| 363 | && fdivides( f.tailcoeff(), g.tailcoeff() ) |
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[94a967] | 364 | && fdivides( f.LC(), g.LC() ) ) |
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| 365 | { |
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[6918d65] | 366 | CanonicalForm q, r; |
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| 367 | return divremt( g, f, q, r ) && r.isZero(); |
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| 368 | } |
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| 369 | else |
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| 370 | return false; |
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[a27df3] | 371 | else if ( gLevel < fLevel ) |
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[6918d65] | 372 | // g is a coefficient w.r.t. f |
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| 373 | return false; |
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[94a967] | 374 | else |
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| 375 | { |
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[6918d65] | 376 | // either f is a coefficient w.r.t. polynomial g or both |
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| 377 | // f and g are from a base domain (should be Z or Z/p^n, |
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| 378 | // then) |
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| 379 | CanonicalForm q, r; |
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| 380 | return divremt( g, f, q, r ) && r.isZero(); |
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[9719434] | 381 | } |
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| 382 | } |
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| 383 | //}}} |
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| 384 | |
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| 385 | //{{{ CanonicalForm maxNorm ( const CanonicalForm & f ) |
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| 386 | //{{{ docu |
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| 387 | // |
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[a27df3] | 388 | // maxNorm() - return maximum norm of `f'. |
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[9719434] | 389 | // |
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| 390 | // That is, the base coefficient of `f' with the largest absolute |
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| 391 | // value. |
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| 392 | // |
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| 393 | // Valid for arbitrary polynomials over arbitrary domains, but |
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| 394 | // most useful for multivariate polynomials over Z. |
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| 395 | // |
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[a27df3] | 396 | // Type info: |
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| 397 | // ---------- |
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| 398 | // f: CurrentPP |
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| 399 | // |
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[9719434] | 400 | //}}} |
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| 401 | CanonicalForm |
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| 402 | maxNorm ( const CanonicalForm & f ) |
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| 403 | { |
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| 404 | if ( f.inBaseDomain() ) |
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[6918d65] | 405 | return abs( f ); |
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[9719434] | 406 | else { |
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[6918d65] | 407 | CanonicalForm result = 0; |
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| 408 | for ( CFIterator i = f; i.hasTerms(); i++ ) { |
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| 409 | CanonicalForm coeffMaxNorm = maxNorm( i.coeff() ); |
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| 410 | if ( coeffMaxNorm > result ) |
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| 411 | result = coeffMaxNorm; |
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| 412 | } |
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| 413 | return result; |
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[9719434] | 414 | } |
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| 415 | } |
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| 416 | //}}} |
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| 417 | |
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| 418 | //{{{ CanonicalForm euclideanNorm ( const CanonicalForm & f ) |
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| 419 | //{{{ docu |
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| 420 | // |
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[a27df3] | 421 | // euclideanNorm() - return Euclidean norm of `f'. |
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| 422 | // |
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| 423 | // Returns the largest integer smaller or equal norm(`f') = |
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| 424 | // sqrt(sum( `f'[i]^2 )). |
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[9719434] | 425 | // |
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[a27df3] | 426 | // Type info: |
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| 427 | // ---------- |
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| 428 | // f: UVPoly( Z ) |
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[9719434] | 429 | // |
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| 430 | //}}} |
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| 431 | CanonicalForm |
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| 432 | euclideanNorm ( const CanonicalForm & f ) |
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| 433 | { |
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| 434 | ASSERT( (f.inBaseDomain() || f.isUnivariate()) && f.LC().inZ(), |
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[6918d65] | 435 | "type error: univariate poly over Z expected" ); |
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[9719434] | 436 | |
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| 437 | CanonicalForm result = 0; |
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| 438 | for ( CFIterator i = f; i.hasTerms(); i++ ) { |
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[6918d65] | 439 | CanonicalForm coeff = i.coeff(); |
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| 440 | result += coeff*coeff; |
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[dff969] | 441 | } |
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[9719434] | 442 | return sqrt( result ); |
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[dff969] | 443 | } |
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[5cf721] | 444 | //}}} |
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