[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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[e4fe2b] | 21 | #include "config.h" |
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[483c5c] | 22 | |
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[650f2d8] | 23 | #include "cf_assert.h" |
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[cbb3fa] | 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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[6db552] | 31 | #include "templates/ftmpl_functions.h" |
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[42281d6] | 32 | #include "algext.h" |
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[483c5c] | 33 | |
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[27bb97f] | 34 | void out_cf(const char *s1,const CanonicalForm &f,const char *s2); |
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[ad82f6] | 35 | |
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[cbb3fa] | 36 | //{{{ CanonicalForm psr ( const CanonicalForm & f, const CanonicalForm & g, const Variable & x ) |
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| 37 | //{{{ docu |
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| 38 | // |
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[a27df3] | 39 | // psr() - return pseudo remainder of `f' and `g' with respect |
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[9719434] | 40 | // to `x'. |
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[cbb3fa] | 41 | // |
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[a27df3] | 42 | // `g' must not equal zero. |
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[cbb3fa] | 43 | // |
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[a27df3] | 44 | // For f and g in R[x], R an arbitrary ring, g != 0, there is a |
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| 45 | // representation |
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[cbb3fa] | 46 | // |
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[a27df3] | 47 | // LC(g)^s*f = g*q + r |
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[cbb3fa] | 48 | // |
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| 49 | // with r = 0 or deg(r) < deg(g) and s = 0 if f = 0 or |
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| 50 | // s = max( 0, deg(f)-deg(g)+1 ) otherwise. |
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[a27df3] | 51 | // r = psr(f, g) and q = psq(f, g) are called "pseudo remainder" |
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| 52 | // and "pseudo quotient", resp. They are uniquely determined if |
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| 53 | // LC(g) is not a zero divisor in R. |
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| 54 | // |
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| 55 | // See H.-J. Reiffen/G. Scheja/U. Vetter - "Algebra", 2nd ed., |
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| 56 | // par. 15, for a reference. |
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| 57 | // |
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| 58 | // Type info: |
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| 59 | // ---------- |
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| 60 | // f, g: Current |
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| 61 | // x: Polynomial |
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| 62 | // |
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| 63 | // Polynomials over prime power domains are admissible if |
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| 64 | // lc(LC(`g',`x')) is not a zero divisor. This is a slightly |
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| 65 | // stronger precondition than mathematically necessary since |
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| 66 | // pseudo remainder and quotient are well-defined if LC(`g',`x') |
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| 67 | // is not a zero divisor. |
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| 68 | // |
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| 69 | // For example, psr(y^2, (13*x+1)*y) is well-defined in |
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| 70 | // (Z/13^2[x])[y] since (13*x+1) is not a zero divisor. But |
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| 71 | // calculating it with Factory would fail since 13 is a zero |
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| 72 | // divisor in Z/13^2. |
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| 73 | // |
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| 74 | // Due to this inconsistency with mathematical notion, we decided |
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| 75 | // not to declare type `CurrentPP' for `f' and `g'. |
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| 76 | // |
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| 77 | // Developers note: |
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| 78 | // ---------------- |
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| 79 | // This is not an optimal implementation. Better would have been |
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| 80 | // an implementation in `InternalPoly' avoiding the |
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| 81 | // exponentiation of the leading coefficient of `g'. In contrast |
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| 82 | // to `psq()' and `psqr()' it definitely seems worth to implement |
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| 83 | // the pseudo remainder on the internal level. Should be done |
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| 84 | // soon. |
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[cbb3fa] | 85 | // |
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| 86 | //}}} |
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[483c5c] | 87 | CanonicalForm |
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[7fb3ba4] | 88 | #if 0 |
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[483c5c] | 89 | psr ( const CanonicalForm & f, const CanonicalForm & g, const Variable & x ) |
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| 90 | { |
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[6918d65] | 91 | |
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[a27df3] | 92 | ASSERT( x.level() > 0, "type error: polynomial variable expected" ); |
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| 93 | ASSERT( ! g.isZero(), "math error: division by zero" ); |
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| 94 | |
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| 95 | // swap variables such that x's level is larger or equal |
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| 96 | // than both f's and g's levels. |
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| 97 | Variable X = tmax( tmax( f.mvar(), g.mvar() ), x ); |
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| 98 | CanonicalForm F = swapvar( f, x, X ); |
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| 99 | CanonicalForm G = swapvar( g, x, X ); |
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[cbb3fa] | 100 | |
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[a27df3] | 101 | // now, we have to calculate the pseudo remainder of F and G |
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| 102 | // w.r.t. X |
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| 103 | int fDegree = degree( F, X ); |
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| 104 | int gDegree = degree( G, X ); |
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[5453c7] | 105 | if ( (fDegree < 0) || (fDegree < gDegree) ) |
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[6918d65] | 106 | return f; |
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[5453c7] | 107 | else |
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| 108 | { |
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[6918d65] | 109 | CanonicalForm xresult = (power( LC( G, X ), fDegree-gDegree+1 ) * F) ; |
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| 110 | CanonicalForm result = xresult -(xresult/G)*G; |
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| 111 | return swapvar( result, x, X ); |
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[a27df3] | 112 | } |
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[483c5c] | 113 | } |
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[5bb462] | 114 | #else |
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[ad82f6] | 115 | psr ( const CanonicalForm &rr, const CanonicalForm &vv, const Variable & x ) |
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| 116 | { |
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[5bb462] | 117 | CanonicalForm r=rr, v=vv, l, test, lu, lv, t, retvalue; |
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| 118 | int dr, dv, d,n=0; |
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| 119 | |
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| 120 | |
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| 121 | dr = degree( r, x ); |
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[ad82f6] | 122 | if (dr>0) |
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[dad0bc5] | 123 | { |
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[ad82f6] | 124 | dv = degree( v, x ); |
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| 125 | if (dv <= dr) {l=LC(v,x); v = v -l*power(x,dv);} |
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| 126 | else { l = 1; } |
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| 127 | d= dr-dv+1; |
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[e7f5ac] | 128 | //out_cf("psr(",rr," "); |
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| 129 | //out_cf("",vv," "); |
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| 130 | //printf(" var=%d\n",x.level()); |
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[ad82f6] | 131 | while ( ( dv <= dr ) && ( !r.isZero()) ) |
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| 132 | { |
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| 133 | test = power(x,dr-dv)*v*LC(r,x); |
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| 134 | if ( dr == 0 ) { r= CanonicalForm(0); } |
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| 135 | else { r= r - LC(r,x)*power(x,dr); } |
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| 136 | r= l*r -test; |
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| 137 | dr= degree(r,x); |
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| 138 | n+=1; |
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| 139 | } |
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| 140 | r= power(l, d-n)*r; |
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[5bb462] | 141 | } |
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| 142 | return r; |
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| 143 | } |
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| 144 | #endif |
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[cbb3fa] | 145 | //}}} |
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[483c5c] | 146 | |
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[cbb3fa] | 147 | //{{{ CanonicalForm psq ( const CanonicalForm & f, const CanonicalForm & g, const Variable & x ) |
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| 148 | //{{{ docu |
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| 149 | // |
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[a27df3] | 150 | // psq() - return pseudo quotient of `f' and `g' with respect |
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[9719434] | 151 | // to `x'. |
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[cbb3fa] | 152 | // |
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[a27df3] | 153 | // `g' must not equal zero. |
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| 154 | // |
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[9719434] | 155 | // See `psr()' for more detailed information. |
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[cbb3fa] | 156 | // |
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[a27df3] | 157 | // Type info: |
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| 158 | // ---------- |
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| 159 | // f, g: Current |
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| 160 | // x: Polynomial |
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| 161 | // |
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| 162 | // Developers note: |
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| 163 | // ---------------- |
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| 164 | // This is not an optimal implementation. Better would have been |
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| 165 | // an implementation in `InternalPoly' avoiding the |
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| 166 | // exponentiation of the leading coefficient of `g'. It seemed |
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| 167 | // not worth to do so. |
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| 168 | // |
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[cbb3fa] | 169 | //}}} |
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[483c5c] | 170 | CanonicalForm |
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| 171 | psq ( const CanonicalForm & f, const CanonicalForm & g, const Variable & x ) |
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| 172 | { |
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[a27df3] | 173 | ASSERT( x.level() > 0, "type error: polynomial variable expected" ); |
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| 174 | ASSERT( ! g.isZero(), "math error: division by zero" ); |
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[cbb3fa] | 175 | |
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[a27df3] | 176 | // swap variables such that x's level is larger or equal |
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| 177 | // than both f's and g's levels. |
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| 178 | Variable X = tmax( tmax( f.mvar(), g.mvar() ), x ); |
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| 179 | CanonicalForm F = swapvar( f, x, X ); |
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| 180 | CanonicalForm G = swapvar( g, x, X ); |
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| 181 | |
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| 182 | // now, we have to calculate the pseudo remainder of F and G |
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| 183 | // w.r.t. X |
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| 184 | int fDegree = degree( F, X ); |
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| 185 | int gDegree = degree( G, X ); |
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| 186 | if ( fDegree < 0 || fDegree < gDegree ) |
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[6918d65] | 187 | return 0; |
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[a27df3] | 188 | else { |
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[6918d65] | 189 | CanonicalForm result = (power( LC( G, X ), fDegree-gDegree+1 ) * F) / G; |
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| 190 | return swapvar( result, x, X ); |
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[a27df3] | 191 | } |
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[483c5c] | 192 | } |
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[cbb3fa] | 193 | //}}} |
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[483c5c] | 194 | |
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[cbb3fa] | 195 | //{{{ void psqr ( const CanonicalForm & f, const CanonicalForm & g, CanonicalForm & q, CanonicalForm & r, const Variable & x ) |
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| 196 | //{{{ docu |
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| 197 | // |
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[9719434] | 198 | // psqr() - calculate pseudo quotient and remainder of `f' and |
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| 199 | // `g' with respect to `x'. |
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[cbb3fa] | 200 | // |
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[a27df3] | 201 | // Returns the pseudo quotient of `f' and `g' in `q', the pseudo |
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| 202 | // remainder in `r'. `g' must not equal zero. |
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| 203 | // |
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[9719434] | 204 | // See `psr()' for more detailed information. |
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[cbb3fa] | 205 | // |
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[a27df3] | 206 | // Type info: |
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| 207 | // ---------- |
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| 208 | // f, g: Current |
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| 209 | // q, r: Anything |
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| 210 | // x: Polynomial |
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| 211 | // |
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| 212 | // Developers note: |
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| 213 | // ---------------- |
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| 214 | // This is not an optimal implementation. Better would have been |
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| 215 | // an implementation in `InternalPoly' avoiding the |
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| 216 | // exponentiation of the leading coefficient of `g'. It seemed |
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| 217 | // not worth to do so. |
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| 218 | // |
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[cbb3fa] | 219 | //}}} |
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[483c5c] | 220 | void |
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| 221 | psqr ( const CanonicalForm & f, const CanonicalForm & g, CanonicalForm & q, CanonicalForm & r, const Variable& x ) |
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| 222 | { |
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[a27df3] | 223 | ASSERT( x.level() > 0, "type error: polynomial variable expected" ); |
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| 224 | ASSERT( ! g.isZero(), "math error: division by zero" ); |
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| 225 | |
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| 226 | // swap variables such that x's level is larger or equal |
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| 227 | // than both f's and g's levels. |
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| 228 | Variable X = tmax( tmax( f.mvar(), g.mvar() ), x ); |
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| 229 | CanonicalForm F = swapvar( f, x, X ); |
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| 230 | CanonicalForm G = swapvar( g, x, X ); |
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[cbb3fa] | 231 | |
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[a27df3] | 232 | // now, we have to calculate the pseudo remainder of F and G |
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| 233 | // w.r.t. X |
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| 234 | int fDegree = degree( F, X ); |
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| 235 | int gDegree = degree( G, X ); |
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| 236 | if ( fDegree < 0 || fDegree < gDegree ) { |
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[6918d65] | 237 | q = 0; r = f; |
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[a27df3] | 238 | } else { |
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[6918d65] | 239 | divrem( power( LC( G, X ), fDegree-gDegree+1 ) * F, G, q, r ); |
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| 240 | q = swapvar( q, x, X ); |
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| 241 | r = swapvar( r, x, X ); |
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[cbb3fa] | 242 | } |
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[483c5c] | 243 | } |
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[cbb3fa] | 244 | //}}} |
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[483c5c] | 245 | |
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[9719434] | 246 | //{{{ static CanonicalForm internalBCommonDen ( const CanonicalForm & f ) |
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[cbb3fa] | 247 | //{{{ docu |
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| 248 | // |
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[9719434] | 249 | // internalBCommonDen() - recursively calculate multivariate |
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| 250 | // common denominator of coefficients of `f'. |
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[cbb3fa] | 251 | // |
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[9719434] | 252 | // Used by: bCommonDen() |
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[cbb3fa] | 253 | // |
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[a27df3] | 254 | // Type info: |
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| 255 | // ---------- |
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| 256 | // f: Poly( Q ) |
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| 257 | // Switches: isOff( SW_RATIONAL ) |
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| 258 | // |
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[cbb3fa] | 259 | //}}} |
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[483c5c] | 260 | static CanonicalForm |
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[9719434] | 261 | internalBCommonDen ( const CanonicalForm & f ) |
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[483c5c] | 262 | { |
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[cbb3fa] | 263 | if ( f.inBaseDomain() ) |
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[6918d65] | 264 | return f.den(); |
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[483c5c] | 265 | else { |
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[6918d65] | 266 | CanonicalForm result = 1; |
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| 267 | for ( CFIterator i = f; i.hasTerms(); i++ ) |
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| 268 | result = blcm( result, internalBCommonDen( i.coeff() ) ); |
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| 269 | return result; |
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[483c5c] | 270 | } |
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| 271 | } |
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[cbb3fa] | 272 | //}}} |
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[483c5c] | 273 | |
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[9719434] | 274 | //{{{ CanonicalForm bCommonDen ( const CanonicalForm & f ) |
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[cbb3fa] | 275 | //{{{ docu |
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| 276 | // |
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[9719434] | 277 | // bCommonDen() - calculate multivariate common denominator of |
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| 278 | // coefficients of `f'. |
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[cbb3fa] | 279 | // |
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| 280 | // The common denominator is calculated with respect to all |
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[9719434] | 281 | // coefficients of `f' which are in a base domain. In other |
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| 282 | // words, common_den( `f' ) * `f' is guaranteed to have integer |
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[a27df3] | 283 | // coefficients only. The common denominator of zero is one. |
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[cbb3fa] | 284 | // |
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[9719434] | 285 | // Returns something non-trivial iff the current domain is Q. |
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[cbb3fa] | 286 | // |
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[a27df3] | 287 | // Type info: |
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| 288 | // ---------- |
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| 289 | // f: CurrentPP |
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| 290 | // |
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[cbb3fa] | 291 | //}}} |
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[483c5c] | 292 | CanonicalForm |
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[9719434] | 293 | bCommonDen ( const CanonicalForm & f ) |
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[483c5c] | 294 | { |
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| 295 | if ( getCharacteristic() == 0 && isOn( SW_RATIONAL ) ) { |
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[6918d65] | 296 | // otherwise `bgcd()' returns one |
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| 297 | Off( SW_RATIONAL ); |
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| 298 | CanonicalForm result = internalBCommonDen( f ); |
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| 299 | On( SW_RATIONAL ); |
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| 300 | return result; |
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[9719434] | 301 | } else |
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[6918d65] | 302 | return CanonicalForm( 1 ); |
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[483c5c] | 303 | } |
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[cbb3fa] | 304 | //}}} |
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[dff969] | 305 | |
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[ebc602] | 306 | //{{{ bool fdivides ( const CanonicalForm & f, const CanonicalForm & g ) |
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[dff969] | 307 | //{{{ docu |
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| 308 | // |
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[ebc602] | 309 | // fdivides() - check whether `f' divides `g'. |
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[dff969] | 310 | // |
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[a27df3] | 311 | // Returns true iff `f' divides `g'. Uses some extra heuristic |
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| 312 | // to avoid polynomial division. Without the heuristic, the test |
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| 313 | // essentialy looks like `divremt(g, f, q, r) && r.isZero()'. |
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| 314 | // |
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| 315 | // Type info: |
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| 316 | // ---------- |
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| 317 | // f, g: Current |
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| 318 | // |
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| 319 | // Elements from prime power domains (or polynomials over such |
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| 320 | // domains) are admissible if `f' (or lc(`f'), resp.) is not a |
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| 321 | // zero divisor. This is a slightly stronger precondition than |
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| 322 | // mathematically necessary since divisibility is a well-defined |
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| 323 | // notion in arbitrary rings. Hence, we decided not to declare |
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| 324 | // the weaker type `CurrentPP'. |
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| 325 | // |
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| 326 | // Developers note: |
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| 327 | // ---------------- |
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[ebc602] | 328 | // One may consider the the test `fdivides( f.LC(), g.LC() )' in |
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[a27df3] | 329 | // the main `if'-test superfluous since `divremt()' in the |
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| 330 | // `if'-body repeats the test. However, `divremt()' does not use |
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| 331 | // any heuristic to do so. |
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| 332 | // |
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[ebc602] | 333 | // It seems not reasonable to call `fdivides()' from `divremt()' |
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| 334 | // to check divisibility of leading coefficients. `fdivides()' is |
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[a27df3] | 335 | // on a relatively high level compared to `divremt()'. |
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[dff969] | 336 | // |
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| 337 | //}}} |
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| 338 | bool |
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[ebc602] | 339 | fdivides ( const CanonicalForm & f, const CanonicalForm & g ) |
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[dff969] | 340 | { |
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[a27df3] | 341 | // trivial cases |
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| 342 | if ( g.isZero() ) |
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[6918d65] | 343 | return true; |
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[a27df3] | 344 | else if ( f.isZero() ) |
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[6918d65] | 345 | return false; |
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[a27df3] | 346 | |
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| 347 | if ( (f.inCoeffDomain() || g.inCoeffDomain()) |
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[6918d65] | 348 | && ((getCharacteristic() == 0 && isOn( SW_RATIONAL )) |
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| 349 | || (getCharacteristic() > 0 && CFFactory::gettype() != PrimePowerDomain)) ) |
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[c1b9927] | 350 | { |
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[6918d65] | 351 | // if we are in a field all elements not equal to zero are units |
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| 352 | if ( f.inCoeffDomain() ) |
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| 353 | return true; |
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| 354 | else |
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| 355 | // g.inCoeffDomain() |
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| 356 | return false; |
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[c1b9927] | 357 | } |
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[a27df3] | 358 | |
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| 359 | // we may assume now that both levels either equal LEVELBASE |
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| 360 | // or are greater zero |
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| 361 | int fLevel = f.level(); |
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| 362 | int gLevel = g.level(); |
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[94a967] | 363 | if ( (gLevel > 0) && (fLevel == gLevel) ) |
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[6918d65] | 364 | // f and g are polynomials in the same main variable |
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| 365 | if ( degree( f ) <= degree( g ) |
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| 366 | && fdivides( f.tailcoeff(), g.tailcoeff() ) |
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[94a967] | 367 | && fdivides( f.LC(), g.LC() ) ) |
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| 368 | { |
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[6918d65] | 369 | CanonicalForm q, r; |
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| 370 | return divremt( g, f, q, r ) && r.isZero(); |
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| 371 | } |
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| 372 | else |
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| 373 | return false; |
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[a27df3] | 374 | else if ( gLevel < fLevel ) |
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[6918d65] | 375 | // g is a coefficient w.r.t. f |
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| 376 | return false; |
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[94a967] | 377 | else |
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| 378 | { |
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[6918d65] | 379 | // either f is a coefficient w.r.t. polynomial g or both |
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| 380 | // f and g are from a base domain (should be Z or Z/p^n, |
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| 381 | // then) |
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| 382 | CanonicalForm q, r; |
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| 383 | return divremt( g, f, q, r ) && r.isZero(); |
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[9719434] | 384 | } |
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| 385 | } |
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| 386 | //}}} |
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| 387 | |
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[21b8f4c] | 388 | /// same as fdivides if true returns quotient quot of g by f otherwise quot == 0 |
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| 389 | bool |
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| 390 | fdivides ( const CanonicalForm & f, const CanonicalForm & g, CanonicalForm& quot ) |
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| 391 | { |
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| 392 | quot= 0; |
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| 393 | // trivial cases |
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| 394 | if ( g.isZero() ) |
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| 395 | return true; |
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| 396 | else if ( f.isZero() ) |
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| 397 | return false; |
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| 398 | |
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| 399 | if ( (f.inCoeffDomain() || g.inCoeffDomain()) |
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| 400 | && ((getCharacteristic() == 0 && isOn( SW_RATIONAL )) |
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| 401 | || (getCharacteristic() > 0 && CFFactory::gettype() != PrimePowerDomain)) ) |
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| 402 | { |
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| 403 | // if we are in a field all elements not equal to zero are units |
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| 404 | if ( f.inCoeffDomain() ) |
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| 405 | { |
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| 406 | quot= g/f; |
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| 407 | return true; |
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| 408 | } |
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| 409 | else |
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| 410 | // g.inCoeffDomain() |
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| 411 | return false; |
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| 412 | } |
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| 413 | |
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| 414 | // we may assume now that both levels either equal LEVELBASE |
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| 415 | // or are greater zero |
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| 416 | int fLevel = f.level(); |
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| 417 | int gLevel = g.level(); |
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| 418 | if ( (gLevel > 0) && (fLevel == gLevel) ) |
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| 419 | // f and g are polynomials in the same main variable |
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| 420 | if ( degree( f ) <= degree( g ) |
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| 421 | && fdivides( f.tailcoeff(), g.tailcoeff() ) |
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| 422 | && fdivides( f.LC(), g.LC() ) ) |
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| 423 | { |
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| 424 | CanonicalForm q, r; |
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| 425 | if (divremt( g, f, q, r ) && r.isZero()) |
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| 426 | { |
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| 427 | quot= q; |
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| 428 | return true; |
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| 429 | } |
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| 430 | else |
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| 431 | return false; |
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| 432 | } |
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| 433 | else |
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| 434 | return false; |
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| 435 | else if ( gLevel < fLevel ) |
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| 436 | // g is a coefficient w.r.t. f |
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| 437 | return false; |
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| 438 | else |
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| 439 | { |
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| 440 | // either f is a coefficient w.r.t. polynomial g or both |
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| 441 | // f and g are from a base domain (should be Z or Z/p^n, |
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| 442 | // then) |
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| 443 | CanonicalForm q, r; |
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| 444 | if (divremt( g, f, q, r ) && r.isZero()) |
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| 445 | { |
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| 446 | quot= q; |
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| 447 | return true; |
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| 448 | } |
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| 449 | else |
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| 450 | return false; |
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| 451 | } |
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| 452 | } |
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| 453 | |
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[42281d6] | 454 | /// same as fdivides but handles zero divisors in Z_p[t]/(f)[x1,...,xn] for reducible f |
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| 455 | bool |
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| 456 | tryFdivides ( const CanonicalForm & f, const CanonicalForm & g, const CanonicalForm& M, bool& fail ) |
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| 457 | { |
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| 458 | fail= false; |
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| 459 | // trivial cases |
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| 460 | if ( g.isZero() ) |
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| 461 | return true; |
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| 462 | else if ( f.isZero() ) |
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| 463 | return false; |
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| 464 | |
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| 465 | if (f.inCoeffDomain() || g.inCoeffDomain()) |
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| 466 | { |
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| 467 | // if we are in a field all elements not equal to zero are units |
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| 468 | if ( f.inCoeffDomain() ) |
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| 469 | { |
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| 470 | CanonicalForm inv; |
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| 471 | tryInvert (f, M, inv, fail); |
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| 472 | return !fail; |
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| 473 | } |
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| 474 | else |
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| 475 | { |
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| 476 | return false; |
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| 477 | } |
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| 478 | } |
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| 479 | |
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| 480 | // we may assume now that both levels either equal LEVELBASE |
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| 481 | // or are greater zero |
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| 482 | int fLevel = f.level(); |
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| 483 | int gLevel = g.level(); |
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| 484 | if ( (gLevel > 0) && (fLevel == gLevel) ) |
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| 485 | { |
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| 486 | if (degree( f ) > degree( g )) |
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| 487 | return false; |
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| 488 | bool dividestail= tryFdivides (f.tailcoeff(), g.tailcoeff(), M, fail); |
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| 489 | |
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| 490 | if (fail || !dividestail) |
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| 491 | return false; |
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| 492 | bool dividesLC= tryFdivides (f.LC(),g.LC(), M, fail); |
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| 493 | if (fail || !dividesLC) |
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| 494 | return false; |
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| 495 | CanonicalForm q,r; |
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| 496 | bool divides= tryDivremt (g, f, q, r, M, fail); |
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| 497 | if (fail || !divides) |
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| 498 | return false; |
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| 499 | return r.isZero(); |
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| 500 | } |
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| 501 | else if ( gLevel < fLevel ) |
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| 502 | { |
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| 503 | // g is a coefficient w.r.t. f |
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| 504 | return false; |
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| 505 | } |
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| 506 | else |
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| 507 | { |
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| 508 | // either f is a coefficient w.r.t. polynomial g or both |
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| 509 | // f and g are from a base domain (should be Z or Z/p^n, |
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| 510 | // then) |
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| 511 | CanonicalForm q, r; |
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| 512 | bool divides= tryDivremt (g, f, q, r, M, fail); |
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| 513 | if (fail || !divides) |
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| 514 | return false; |
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| 515 | return r.isZero(); |
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| 516 | } |
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| 517 | } |
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| 518 | |
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[9719434] | 519 | //{{{ CanonicalForm maxNorm ( const CanonicalForm & f ) |
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| 520 | //{{{ docu |
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| 521 | // |
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[a27df3] | 522 | // maxNorm() - return maximum norm of `f'. |
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[9719434] | 523 | // |
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| 524 | // That is, the base coefficient of `f' with the largest absolute |
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| 525 | // value. |
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| 526 | // |
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| 527 | // Valid for arbitrary polynomials over arbitrary domains, but |
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| 528 | // most useful for multivariate polynomials over Z. |
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| 529 | // |
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[a27df3] | 530 | // Type info: |
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| 531 | // ---------- |
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| 532 | // f: CurrentPP |
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| 533 | // |
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[9719434] | 534 | //}}} |
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| 535 | CanonicalForm |
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| 536 | maxNorm ( const CanonicalForm & f ) |
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| 537 | { |
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| 538 | if ( f.inBaseDomain() ) |
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[6918d65] | 539 | return abs( f ); |
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[9719434] | 540 | else { |
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[6918d65] | 541 | CanonicalForm result = 0; |
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| 542 | for ( CFIterator i = f; i.hasTerms(); i++ ) { |
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| 543 | CanonicalForm coeffMaxNorm = maxNorm( i.coeff() ); |
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| 544 | if ( coeffMaxNorm > result ) |
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| 545 | result = coeffMaxNorm; |
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| 546 | } |
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| 547 | return result; |
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[9719434] | 548 | } |
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| 549 | } |
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| 550 | //}}} |
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| 551 | |
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| 552 | //{{{ CanonicalForm euclideanNorm ( const CanonicalForm & f ) |
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| 553 | //{{{ docu |
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| 554 | // |
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[a27df3] | 555 | // euclideanNorm() - return Euclidean norm of `f'. |
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| 556 | // |
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| 557 | // Returns the largest integer smaller or equal norm(`f') = |
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| 558 | // sqrt(sum( `f'[i]^2 )). |
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[9719434] | 559 | // |
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[a27df3] | 560 | // Type info: |
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| 561 | // ---------- |
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| 562 | // f: UVPoly( Z ) |
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[9719434] | 563 | // |
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| 564 | //}}} |
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| 565 | CanonicalForm |
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| 566 | euclideanNorm ( const CanonicalForm & f ) |
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| 567 | { |
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| 568 | ASSERT( (f.inBaseDomain() || f.isUnivariate()) && f.LC().inZ(), |
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[6918d65] | 569 | "type error: univariate poly over Z expected" ); |
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[9719434] | 570 | |
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| 571 | CanonicalForm result = 0; |
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| 572 | for ( CFIterator i = f; i.hasTerms(); i++ ) { |
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[6918d65] | 573 | CanonicalForm coeff = i.coeff(); |
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| 574 | result += coeff*coeff; |
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[dff969] | 575 | } |
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[9719434] | 576 | return sqrt( result ); |
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[dff969] | 577 | } |
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[5cf721] | 578 | //}}} |
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