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Objects written by Bill Wheeler for representing polynomials.
1 #! /usr/local/bin/perl 2 3 package Polynomial; 4 5 #@ISA = qw(Function); 6 7 sub givenVariableScaleAndRoots { 8 my ($class, $variable,$scale,@roots) = @_; 9 # print @roots,"\n"; 10 my @coefficients; 11 my $degree = scalar(@roots); 12 foreach my $i (0..$degree-1) {$roots[$i] *= -1;} 13 foreach my $i (0..$degree) { 14 $coefficients[$i] = $scale * symmetric_function($degree-$i, \@roots); 15 } 16 # print @coefficients, "\n"; 17 my $self = {'degree' => $degree, 18 'variable' => $variable, 19 'scale' => $scale, 20 'roots' => \@roots, 21 'coefficients' => \@coefficients, 22 }; 23 bless $self, $class; 24 # print "OK\n"; 25 return($self); 26 } 27 28 sub givenVariableScaleVTranslateAndRoots { 29 my ($class, $variable,$scale,$vtranslate,@roots) = @_; 30 # print @roots,"\n"; 31 my @coefficients; 32 my $degree = scalar(@roots); 33 foreach my $i (0..$degree) { 34 $coefficients[$i] = $scale * symmetric_function($degree-$i, \@roots); 35 } 36 $coefficients[0] += $vtranslate; 37 # print @coefficients, "\n"; 38 my $self = {'degree' => $degree, 39 'variable' => $variable, 40 'scale' => $scale, 41 'roots' => \@roots, 42 'coefficients' => \@coefficients, 43 }; 44 bless $self, $class; 45 # print "OK\n"; 46 return($self); 47 } 48 49 sub givenVariableAndCoefficients { # Give Coefficients from a_0 to a_n 50 my ($class, $variable, @coefficients) = @_; 51 my $degree = scalar(@coefficients) -1; 52 while ($degree >= 0 and $coefficients[$degree] == 0) { 53 $degree--; 54 } 55 my $self = {'degree' => $degree, 56 'variable' => $variable, 57 'scale' => 1, 58 'roots' => [], 59 'coefficients' => \@coefficients, 60 }; 61 bless $self, $class; 62 return($self); 63 } 64 65 66 67 sub degree { 68 my $self=shift; 69 if (@_) {$self -> {'degree'} = shift;} 70 return ($self -> {'degree'}); 71 } 72 73 sub coefficients { 74 my $self = shift; 75 if (@_) {$self -> {'coefficients'} = shift;} 76 return ($self -> {'coefficients'}); 77 } 78 79 sub variable { 80 my $self = shift; 81 if (@_) {$self->{'variable'} = shift;} 82 return($self ->{'variable'}); 83 } 84 85 sub TeXizeHighFirst { 86 my $self = shift; 87 my $degree = $self->degree(); 88 my $variable = $self->variable(); 89 my $coefficients = $self->coefficients(); 90 my $found = 0; 91 my $string = ""; 92 my $coef; 93 my $jth; 94 foreach my $i (0..$degree-1) { 95 $jth = $degree - $i; 96 $vterm = ($jth == 1) ? "$variable" : "$variable^{${jth}}"; 97 if (($coef = $coefficients -> [$jth]) != 0) 98 {if ($found) { 99 $string .= 100 ($coef == 1) ? " + $vterm" 101 : ($coef > 0) ? " + $coef $vterm" 102 : ($coef == -1) ? " - $vterm" 103 : " $coef $vterm"; 104 } 105 else { 106 $found=1; 107 $string .= 108 ($coef == 1) ? " $vterm" 109 : ($coef > 0) ? " $coef $vterm" 110 : ($coef == -1) ? " - $vterm" 111 : " $coef $vterm"; 112 } 113 } 114 } 115 $jth = 0; 116 if (($coef = $coefficients -> [$jth]) != 0) 117 {if ($found) { 118 $string .= ($coef > 0) ? " + $coef " 119 : " $coef " ; 120 } 121 else { 122 $found=1; 123 $string .= ($coef > 0) ? " $coef " 124 : " $coef " ; 125 } 126 } 127 return($string); 128 } 129 130 sub TeXizeLowFirst { 131 my $self = shift; 132 my $degree = $self->degree(); 133 my $variable = $self->variable(); 134 my $coefficients = $self->coefficients(); 135 my $found = 0; 136 my $string = ""; 137 my $coef; 138 my $jth; 139 $jth = 0; 140 if (($coef = $coefficients -> [$jth]) != 0) { 141 $found=1; 142 $string .= $coef; 143 } 144 foreach my $i (1..$degree) { 145 $jth = $i; 146 $vterm = ($jth == 1) ? "$variable" : "$variable^{${jth}}"; 147 if (($coef = $coefficients -> [$jth]) != 0) 148 {if ($found) { 149 $string .= 150 ($coef == 1) ? " + $vterm" 151 : ($coef > 0) ? " + $coef $vterm" 152 : ($coef == -1) ? " - $vterm" 153 : " $coef $vterm"; 154 } 155 else { 156 $found=1; 157 $string .= 158 ($coef == 1) ? " $vterm" 159 : ($coef > 0) ? " $coef $vterm" 160 : ($coef == -1) ? " - $vterm" 161 : " $coef $vterm"; 162 } 163 } 164 } 165 return($string); 166 } 167 168 169 sub perlizeHighFirst { 170 my $self = shift; 171 my $degree = $self->degree(); 172 my $variable = $self->variable(); 173 my $coefficients = $self->coefficients(); 174 my $found = 0; 175 my $string = ""; 176 my $coef; 177 my $jth; 178 foreach my $i (0..$degree-1) { 179 $jth = $degree - $i; 180 $vterm = ($jth == 1) ? "$variable" : "$variable**(${jth})"; 181 if (($coef = $coefficients -> [$jth]) != 0) 182 {if ($found) { 183 $string .= 184 ($coef == 1) ? " + $vterm" 185 : ($coef > 0) ? " + $coef * $vterm" 186 : ($coef == -1) ? " - $vterm" 187 : " $coef * $vterm"; 188 } 189 else { 190 $found=1; 191 $string .= 192 ($coef == 1) ? " $vterm" 193 : ($coef > 0) ? " $coef* $vterm" 194 : ($coef == -1) ? " - $vterm" 195 : " $coef * $vterm"; 196 } 197 } 198 } 199 $jth = 0; 200 if (($coef = $coefficients -> [$jth]) != 0) 201 {if ($found) { 202 $string .= ($coef > 0) ? " + $coef " 203 : " $coef " ; 204 } 205 else { 206 $found=1; 207 $string .= ($coef > 0) ? " $coef " 208 : " $coef " ; 209 } 210 } 211 return($string); 212 } 213 214 sub ith_symmetric_function { 215 my ($K, $i, $arrayref) = @_; 216 my $N = scalar(@$arrayref); 217 if ($K < 0 or $i < 0 or $K > $N or $i > $N - $K ) 218 {return(0);} 219 elsif ($K == 0) 220 {return(1);} 221 elsif ($K == 1) 222 {return($arrayref -> [$i]);} 223 else { 224 my $sum = 0; 225 foreach my $j ($i+1 .. $N-($K-1)) { 226 $sum += ith_symmetric_function($K-1, $j, $arrayref); 227 } 228 return(($arrayref -> [$i]) * $sum); 229 } 230 } 231 232 sub symmetric_function { 233 my ($K, $arrayref) = @_; 234 my $N = scalar(@$arrayref); 235 if ($K < 0 or $K > $N) {return(0);} 236 if ($K == 0) {return(1);} 237 my $sum=0; 238 foreach my $i (0..$N-$K) { 239 $sum += ith_symmetric_function($K, $i, $arrayref); 240 } 241 return($sum); 242 } 243 244 sub changeVar { 245 my $self = shift; 246 my $newVar = shift; 247 if (!defined($newVar)) {return(undef());} 248 my $coefficients=$self->coefficients; 249 my $newP = givenVariableAndCoefficients Polynomial($newVar, @{$coefficients}); 250 return($newP); 251 } 252 253 sub scalarMult { 254 my $self = shift; 255 my $scalar = shift; 256 if (!(defined($scalar))) {return(undef());} 257 my $variable = $self -> variable; 258 my $coefficients = $self -> coefficients; 259 my $degree = $self -> degree; 260 my @scalarMultCoeffs = (); 261 foreach my $i (0..$degree) { 262 $scalarMultCoeffs[$i] = $scalar * $coefficients->[$i]; 263 } 264 my $scalarM = givenVariableAndCoefficients Polynomial($variable, @scalarMultCoeffs); 265 return($scalarM); 266 } 267 268 sub polyAddLike { 269 my $self = shift; 270 my $other = shift; 271 if (!(ref($other) eq "Polynomial")) { return(undef());} 272 if (($self -> variable) ne ($other -> variable)) {return(undef());} 273 my $variable = $self->variable; 274 my $mycoefficients = $self->coefficients; 275 my $othercoefficients = $other -> coefficients; 276 my $myDegree = $self -> degree; 277 my $otherDegree = $other -> degree; 278 my $degree = ($myDegree >= $otherDegree) ? $myDegree : $otherDegree; 279 my @coefficients; 280 foreach my $i (0..$degree) { 281 $coefficients[$i] = ( (defined($mycoefficients->[$i])) ? $mycoefficients->[$i] : 0 ) 282 + ( (defined($othercoefficients->[$i])) ? $othercoefficients->[$i] : 0 ); 283 } 284 my $sum = givenVariableAndCoefficients Polynomial($variable, @coefficients); 285 return($sum); 286 } 287 288 289 sub polyMultLike { 290 my $self = shift; 291 my $other = shift; 292 if (!(ref($other) eq "Polynomial")) { return(undef());} 293 if (($self -> variable) ne ($other -> variable)) {return(undef());} 294 my $variable = $self->variable; 295 my $mycoefficients = $self->coefficients; 296 my $othercoefficients = $other -> coefficients; 297 my $myDegree = $self -> degree; 298 my $otherDegree = $other -> degree; 299 my $degree = $myDegree+$otherDegree; 300 my @coefficients; 301 foreach my $i (0..$degree) { 302 $coefficients[$i]=0; 303 my $iB = ($i > $otherDegree ) ? $i-$otherDegree : 0; 304 my $iT = ($i > $myDegree ) ? $myDegree : $i; 305 foreach my $j ($iB .. $iT) { 306 $coefficients[$i] += $mycoefficients->[$j] * $othercoefficients->[$i-$j]; 307 } 308 } 309 my $polyM = givenVariableAndCoefficients Polynomial($variable, @coefficients); 310 return($polyM); 311 } 312 313 314 sub derivative { 315 my $self = shift; 316 my $variable = shift; 317 if (!(defined($variable))) { $variable = $self -> variable;} 318 my $degree = $self -> degree; 319 my $coefficients = $self->coefficients; 320 my @derivativeCoeffs = (); 321 foreach my $i (1..$degree) { 322 $derivativeCoeffs[$i-1] = $i * $coefficients->[$i]; 323 } 324 my $deriv = givenVariableAndCoefficients Polynomial($variable, @derivativeCoeffs); 325 return($deriv); 326 } 327 328 329 sub antiDerivative { 330 my $self = shift; 331 my $variable = shift; 332 if (!(defined($variable))) { $variable = $self -> variable;} 333 my $degree = $self -> degree; 334 my $coefficients = $self->coefficients; 335 my @antiDerivativeCoeffs = (); 336 $antiDerivativeCoeffs[0] = 0; 337 foreach my $i (0..$degree) { 338 $antiDerivativeCoeffs[$i+1] = ($coefficients->[$i])/($i+1); 339 } 340 my $antiDeriv = givenVariableAndCoefficients Polynomial($variable, @antiDerivativeCoeffs); 341 return($antiDeriv); 342 } 343 344 1;
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