1 | /* emacs edit mode for this file is -*- C++ -*- */ |
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2 | /* $Id: cf_algorithm.cc,v 1.15 2008-09-23 11:46:54 Singular Exp $ */ |
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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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8 | // Hierarchy: mathematical algorithms on canonical forms |
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9 | // |
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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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15 | // |
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16 | // Compare these functions to the functions in `cf_ops.cc', which |
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17 | // are structural algorithms. |
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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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23 | #include "assert.h" |
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24 | |
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25 | #include "cf_factory.h" |
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26 | #include "cf_defs.h" |
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27 | #include "canonicalform.h" |
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28 | #include "cf_algorithm.h" |
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29 | #include "variable.h" |
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30 | #include "cf_iter.h" |
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31 | #include "ftmpl_functions.h" |
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32 | |
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33 | void out_cf(char *s1,const CanonicalForm &f,char *s2); |
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34 | |
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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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38 | // psr() - return pseudo remainder of `f' and `g' with respect |
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39 | // to `x'. |
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40 | // |
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41 | // `g' must not equal zero. |
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42 | // |
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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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45 | // |
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46 | // LC(g)^s*f = g*q + r |
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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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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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84 | // |
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85 | //}}} |
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86 | CanonicalForm |
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87 | #if 0 |
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88 | psr ( const CanonicalForm & f, const CanonicalForm & g, const Variable & x ) |
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89 | { |
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90 | |
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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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99 | |
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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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104 | if ( (fDegree < 0) || (fDegree < gDegree) ) |
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105 | return f; |
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106 | else |
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107 | { |
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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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111 | } |
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112 | } |
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113 | #else |
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114 | psr ( const CanonicalForm &rr, const CanonicalForm &vv, const Variable & x ) |
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115 | { |
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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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121 | if (dr>0) |
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122 | { |
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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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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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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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140 | } |
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141 | return r; |
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142 | } |
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143 | #endif |
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144 | //}}} |
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145 | |
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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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149 | // psq() - return pseudo quotient of `f' and `g' with respect |
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150 | // to `x'. |
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151 | // |
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152 | // `g' must not equal zero. |
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153 | // |
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154 | // See `psr()' for more detailed information. |
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155 | // |
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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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168 | //}}} |
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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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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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174 | |
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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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186 | return 0; |
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187 | else { |
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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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190 | } |
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191 | } |
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192 | //}}} |
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193 | |
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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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197 | // psqr() - calculate pseudo quotient and remainder of `f' and |
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198 | // `g' with respect to `x'. |
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199 | // |
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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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203 | // See `psr()' for more detailed information. |
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204 | // |
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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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218 | //}}} |
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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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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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230 | |
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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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236 | q = 0; r = f; |
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237 | } else { |
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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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241 | } |
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242 | } |
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243 | //}}} |
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244 | |
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245 | //{{{ static CanonicalForm internalBCommonDen ( const CanonicalForm & f ) |
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246 | //{{{ docu |
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247 | // |
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248 | // internalBCommonDen() - recursively calculate multivariate |
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249 | // common denominator of coefficients of `f'. |
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250 | // |
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251 | // Used by: bCommonDen() |
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252 | // |
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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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258 | //}}} |
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259 | static CanonicalForm |
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260 | internalBCommonDen ( const CanonicalForm & f ) |
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261 | { |
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262 | if ( f.inBaseDomain() ) |
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263 | return f.den(); |
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264 | else { |
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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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269 | } |
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270 | } |
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271 | //}}} |
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272 | |
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273 | //{{{ CanonicalForm bCommonDen ( const CanonicalForm & f ) |
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274 | //{{{ docu |
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275 | // |
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276 | // bCommonDen() - calculate multivariate common denominator of |
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277 | // coefficients of `f'. |
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278 | // |
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279 | // The common denominator is calculated with respect to all |
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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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282 | // coefficients only. The common denominator of zero is one. |
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283 | // |
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284 | // Returns something non-trivial iff the current domain is Q. |
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285 | // |
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286 | // Type info: |
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287 | // ---------- |
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288 | // f: CurrentPP |
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289 | // |
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290 | //}}} |
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291 | CanonicalForm |
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292 | bCommonDen ( const CanonicalForm & f ) |
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293 | { |
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294 | if ( getCharacteristic() == 0 && isOn( SW_RATIONAL ) ) { |
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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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300 | } else |
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301 | return CanonicalForm( 1 ); |
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302 | } |
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303 | //}}} |
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304 | |
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305 | //{{{ bool fdivides ( const CanonicalForm & f, const CanonicalForm & g ) |
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306 | //{{{ docu |
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307 | // |
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308 | // fdivides() - check whether `f' divides `g'. |
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309 | // |
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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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327 | // One may consider the the test `fdivides( f.LC(), g.LC() )' in |
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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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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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334 | // on a relatively high level compared to `divremt()'. |
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335 | // |
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336 | //}}} |
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337 | bool |
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338 | fdivides ( const CanonicalForm & f, const CanonicalForm & g ) |
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339 | { |
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340 | // trivial cases |
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341 | if ( g.isZero() ) |
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342 | return true; |
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343 | else if ( f.isZero() ) |
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344 | return false; |
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345 | |
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346 | if ( (f.inCoeffDomain() || g.inCoeffDomain()) |
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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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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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360 | if ( (gLevel > 0) && (fLevel == gLevel) ) |
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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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364 | && fdivides( f.LC(), g.LC() ) ) |
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365 | { |
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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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371 | else if ( gLevel < fLevel ) |
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372 | // g is a coefficient w.r.t. f |
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373 | return false; |
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374 | else |
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375 | { |
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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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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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388 | // maxNorm() - return maximum norm of `f'. |
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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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396 | // Type info: |
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397 | // ---------- |
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398 | // f: CurrentPP |
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399 | // |
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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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405 | return abs( f ); |
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406 | else { |
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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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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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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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425 | // |
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426 | // Type info: |
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427 | // ---------- |
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428 | // f: UVPoly( Z ) |
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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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435 | "type error: univariate poly over Z expected" ); |
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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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439 | CanonicalForm coeff = i.coeff(); |
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440 | result += coeff*coeff; |
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441 | } |
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442 | return sqrt( result ); |
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443 | } |
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444 | //}}} |
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