projects / OpenSPARC-T2-SAM / blame
| Commit | Line | Data |
|---|---|---|
| 920dae64 AT |
1 | # |
| 2 | # Complex numbers and associated mathematical functions | |
| 3 | # -- Raphael Manfredi Since Sep 1996 | |
| 4 | # -- Jarkko Hietaniemi Since Mar 1997 | |
| 5 | # -- Daniel S. Lewart Since Sep 1997 | |
| 6 | # | |
| 7 | ||
| 8 | package Math::Complex; | |
| 9 | ||
| 10 | use vars qw($VERSION @ISA @EXPORT @EXPORT_OK %EXPORT_TAGS $Inf); | |
| 11 | ||
| 12 | $VERSION = 1.35; | |
| 13 | ||
| 14 | BEGIN { | |
| 15 | unless ($^O eq 'unicosmk') { | |
| 16 | my $e = $!; | |
| 17 | # We do want an arithmetic overflow, Inf INF inf Infinity:. | |
| 18 | undef $Inf unless eval <<'EOE' and $Inf =~ /^inf(?:inity)?$/i; | |
| 19 | local $SIG{FPE} = sub {die}; | |
| 20 | my $t = CORE::exp 30; | |
| 21 | $Inf = CORE::exp $t; | |
| 22 | EOE | |
| 23 | if (!defined $Inf) { # Try a different method | |
| 24 | undef $Inf unless eval <<'EOE' and $Inf =~ /^inf(?:inity)?$/i; | |
| 25 | local $SIG{FPE} = sub {die}; | |
| 26 | my $t = 1; | |
| 27 | $Inf = $t + "1e99999999999999999999999999999999"; | |
| 28 | EOE | |
| 29 | } | |
| 30 | $! = $e; # Clear ERANGE. | |
| 31 | } | |
| 32 | $Inf = "Inf" if !defined $Inf || !($Inf > 0); # Desperation. | |
| 33 | } | |
| 34 | ||
| 35 | use strict; | |
| 36 | ||
| 37 | my $i; | |
| 38 | my %LOGN; | |
| 39 | ||
| 40 | # Regular expression for floating point numbers. | |
| 41 | # These days we could use Scalar::Util::lln(), I guess. | |
| 42 | my $gre = qr'\s*([\+\-]?(?:(?:(?:\d+(?:_\d+)*(?:\.\d*(?:_\d+)*)?|\.\d+(?:_\d+)*)(?:[eE][\+\-]?\d+(?:_\d+)*)?))|inf)'i; | |
| 43 | ||
| 44 | require Exporter; | |
| 45 | ||
| 46 | @ISA = qw(Exporter); | |
| 47 | ||
| 48 | my @trig = qw( | |
| 49 | pi | |
| 50 | tan | |
| 51 | csc cosec sec cot cotan | |
| 52 | asin acos atan | |
| 53 | acsc acosec asec acot acotan | |
| 54 | sinh cosh tanh | |
| 55 | csch cosech sech coth cotanh | |
| 56 | asinh acosh atanh | |
| 57 | acsch acosech asech acoth acotanh | |
| 58 | ); | |
| 59 | ||
| 60 | @EXPORT = (qw( | |
| 61 | i Re Im rho theta arg | |
| 62 | sqrt log ln | |
| 63 | log10 logn cbrt root | |
| 64 | cplx cplxe | |
| 65 | atan2 | |
| 66 | ), | |
| 67 | @trig); | |
| 68 | ||
| 69 | @EXPORT_OK = qw(decplx); | |
| 70 | ||
| 71 | %EXPORT_TAGS = ( | |
| 72 | 'trig' => [@trig], | |
| 73 | ); | |
| 74 | ||
| 75 | use overload | |
| 76 | '+' => \&plus, | |
| 77 | '-' => \&minus, | |
| 78 | '*' => \&multiply, | |
| 79 | '/' => \÷, | |
| 80 | '**' => \&power, | |
| 81 | '==' => \&numeq, | |
| 82 | '<=>' => \&spaceship, | |
| 83 | 'neg' => \&negate, | |
| 84 | '~' => \&conjugate, | |
| 85 | 'abs' => \&abs, | |
| 86 | 'sqrt' => \&sqrt, | |
| 87 | 'exp' => \&exp, | |
| 88 | 'log' => \&log, | |
| 89 | 'sin' => \&sin, | |
| 90 | 'cos' => \&cos, | |
| 91 | 'tan' => \&tan, | |
| 92 | 'atan2' => \&atan2, | |
| 93 | qw("" stringify); | |
| 94 | ||
| 95 | # | |
| 96 | # Package "privates" | |
| 97 | # | |
| 98 | ||
| 99 | my %DISPLAY_FORMAT = ('style' => 'cartesian', | |
| 100 | 'polar_pretty_print' => 1); | |
| 101 | my $eps = 1e-14; # Epsilon | |
| 102 | ||
| 103 | # | |
| 104 | # Object attributes (internal): | |
| 105 | # cartesian [real, imaginary] -- cartesian form | |
| 106 | # polar [rho, theta] -- polar form | |
| 107 | # c_dirty cartesian form not up-to-date | |
| 108 | # p_dirty polar form not up-to-date | |
| 109 | # display display format (package's global when not set) | |
| 110 | # | |
| 111 | ||
| 112 | # Die on bad *make() arguments. | |
| 113 | ||
| 114 | sub _cannot_make { | |
| 115 | die "@{[(caller(1))[3]]}: Cannot take $_[0] of '$_[1]'.\n"; | |
| 116 | } | |
| 117 | ||
| 118 | sub _make { | |
| 119 | my $arg = shift; | |
| 120 | my ($p, $q); | |
| 121 | ||
| 122 | if ($arg =~ /^$gre$/) { | |
| 123 | ($p, $q) = ($1, 0); | |
| 124 | } elsif ($arg =~ /^(?:$gre)?$gre\s*i\s*$/) { | |
| 125 | ($p, $q) = ($1 || 0, $2); | |
| 126 | } elsif ($arg =~ /^\s*\(\s*$gre\s*(?:,\s*$gre\s*)?\)\s*$/) { | |
| 127 | ($p, $q) = ($1, $2 || 0); | |
| 128 | } | |
| 129 | ||
| 130 | if (defined $p) { | |
| 131 | $p =~ s/^\+//; | |
| 132 | $p =~ s/^(-?)inf$/"${1}9**9**9"/e; | |
| 133 | $q =~ s/^\+//; | |
| 134 | $q =~ s/^(-?)inf$/"${1}9**9**9"/e; | |
| 135 | } | |
| 136 | ||
| 137 | return ($p, $q); | |
| 138 | } | |
| 139 | ||
| 140 | sub _emake { | |
| 141 | my $arg = shift; | |
| 142 | my ($p, $q); | |
| 143 | ||
| 144 | if ($arg =~ /^\s*\[\s*$gre\s*(?:,\s*$gre\s*)?\]\s*$/) { | |
| 145 | ($p, $q) = ($1, $2 || 0); | |
| 146 | } elsif ($arg =~ m!^\s*\[\s*$gre\s*(?:,\s*([-+]?\d*\s*)?pi(?:/\s*(\d+))?\s*)?\]\s*$!) { | |
| 147 | ($p, $q) = ($1, ($2 eq '-' ? -1 : ($2 || 1)) * pi() / ($3 || 1)); | |
| 148 | } elsif ($arg =~ /^\s*\[\s*$gre\s*\]\s*$/) { | |
| 149 | ($p, $q) = ($1, 0); | |
| 150 | } elsif ($arg =~ /^\s*$gre\s*$/) { | |
| 151 | ($p, $q) = ($1, 0); | |
| 152 | } | |
| 153 | ||
| 154 | if (defined $p) { | |
| 155 | $p =~ s/^\+//; | |
| 156 | $q =~ s/^\+//; | |
| 157 | $p =~ s/^(-?)inf$/"${1}9**9**9"/e; | |
| 158 | $q =~ s/^(-?)inf$/"${1}9**9**9"/e; | |
| 159 | } | |
| 160 | ||
| 161 | return ($p, $q); | |
| 162 | } | |
| 163 | ||
| 164 | # | |
| 165 | # ->make | |
| 166 | # | |
| 167 | # Create a new complex number (cartesian form) | |
| 168 | # | |
| 169 | sub make { | |
| 170 | my $self = bless {}, shift; | |
| 171 | my ($re, $im); | |
| 172 | if (@_ == 0) { | |
| 173 | ($re, $im) = (0, 0); | |
| 174 | } elsif (@_ == 1) { | |
| 175 | return (ref $self)->emake($_[0]) | |
| 176 | if ($_[0] =~ /^\s*\[/); | |
| 177 | ($re, $im) = _make($_[0]); | |
| 178 | } elsif (@_ == 2) { | |
| 179 | ($re, $im) = @_; | |
| 180 | } | |
| 181 | if (defined $re) { | |
| 182 | _cannot_make("real part", $re) unless $re =~ /^$gre$/; | |
| 183 | } | |
| 184 | $im ||= 0; | |
| 185 | _cannot_make("imaginary part", $im) unless $im =~ /^$gre$/; | |
| 186 | $self->set_cartesian([$re, $im ]); | |
| 187 | $self->display_format('cartesian'); | |
| 188 | ||
| 189 | return $self; | |
| 190 | } | |
| 191 | ||
| 192 | # | |
| 193 | # ->emake | |
| 194 | # | |
| 195 | # Create a new complex number (exponential form) | |
| 196 | # | |
| 197 | sub emake { | |
| 198 | my $self = bless {}, shift; | |
| 199 | my ($rho, $theta); | |
| 200 | if (@_ == 0) { | |
| 201 | ($rho, $theta) = (0, 0); | |
| 202 | } elsif (@_ == 1) { | |
| 203 | return (ref $self)->make($_[0]) | |
| 204 | if ($_[0] =~ /^\s*\(/ || $_[0] =~ /i\s*$/); | |
| 205 | ($rho, $theta) = _emake($_[0]); | |
| 206 | } elsif (@_ == 2) { | |
| 207 | ($rho, $theta) = @_; | |
| 208 | } | |
| 209 | if (defined $rho && defined $theta) { | |
| 210 | if ($rho < 0) { | |
| 211 | $rho = -$rho; | |
| 212 | $theta = ($theta <= 0) ? $theta + pi() : $theta - pi(); | |
| 213 | } | |
| 214 | } | |
| 215 | if (defined $rho) { | |
| 216 | _cannot_make("rho", $rho) unless $rho =~ /^$gre$/; | |
| 217 | } | |
| 218 | $theta ||= 0; | |
| 219 | _cannot_make("theta", $theta) unless $theta =~ /^$gre$/; | |
| 220 | $self->set_polar([$rho, $theta]); | |
| 221 | $self->display_format('polar'); | |
| 222 | ||
| 223 | return $self; | |
| 224 | } | |
| 225 | ||
| 226 | sub new { &make } # For backward compatibility only. | |
| 227 | ||
| 228 | # | |
| 229 | # cplx | |
| 230 | # | |
| 231 | # Creates a complex number from a (re, im) tuple. | |
| 232 | # This avoids the burden of writing Math::Complex->make(re, im). | |
| 233 | # | |
| 234 | sub cplx { | |
| 235 | return __PACKAGE__->make(@_); | |
| 236 | } | |
| 237 | ||
| 238 | # | |
| 239 | # cplxe | |
| 240 | # | |
| 241 | # Creates a complex number from a (rho, theta) tuple. | |
| 242 | # This avoids the burden of writing Math::Complex->emake(rho, theta). | |
| 243 | # | |
| 244 | sub cplxe { | |
| 245 | return __PACKAGE__->emake(@_); | |
| 246 | } | |
| 247 | ||
| 248 | # | |
| 249 | # pi | |
| 250 | # | |
| 251 | # The number defined as pi = 180 degrees | |
| 252 | # | |
| 253 | sub pi () { 4 * CORE::atan2(1, 1) } | |
| 254 | ||
| 255 | # | |
| 256 | # pit2 | |
| 257 | # | |
| 258 | # The full circle | |
| 259 | # | |
| 260 | sub pit2 () { 2 * pi } | |
| 261 | ||
| 262 | # | |
| 263 | # pip2 | |
| 264 | # | |
| 265 | # The quarter circle | |
| 266 | # | |
| 267 | sub pip2 () { pi / 2 } | |
| 268 | ||
| 269 | # | |
| 270 | # deg1 | |
| 271 | # | |
| 272 | # One degree in radians, used in stringify_polar. | |
| 273 | # | |
| 274 | ||
| 275 | sub deg1 () { pi / 180 } | |
| 276 | ||
| 277 | # | |
| 278 | # uplog10 | |
| 279 | # | |
| 280 | # Used in log10(). | |
| 281 | # | |
| 282 | sub uplog10 () { 1 / CORE::log(10) } | |
| 283 | ||
| 284 | # | |
| 285 | # i | |
| 286 | # | |
| 287 | # The number defined as i*i = -1; | |
| 288 | # | |
| 289 | sub i () { | |
| 290 | return $i if ($i); | |
| 291 | $i = bless {}; | |
| 292 | $i->{'cartesian'} = [0, 1]; | |
| 293 | $i->{'polar'} = [1, pip2]; | |
| 294 | $i->{c_dirty} = 0; | |
| 295 | $i->{p_dirty} = 0; | |
| 296 | return $i; | |
| 297 | } | |
| 298 | ||
| 299 | # | |
| 300 | # ip2 | |
| 301 | # | |
| 302 | # Half of i. | |
| 303 | # | |
| 304 | sub ip2 () { i / 2 } | |
| 305 | ||
| 306 | # | |
| 307 | # Attribute access/set routines | |
| 308 | # | |
| 309 | ||
| 310 | sub cartesian {$_[0]->{c_dirty} ? | |
| 311 | $_[0]->update_cartesian : $_[0]->{'cartesian'}} | |
| 312 | sub polar {$_[0]->{p_dirty} ? | |
| 313 | $_[0]->update_polar : $_[0]->{'polar'}} | |
| 314 | ||
| 315 | sub set_cartesian { $_[0]->{p_dirty}++; $_[0]->{c_dirty} = 0; | |
| 316 | $_[0]->{'cartesian'} = $_[1] } | |
| 317 | sub set_polar { $_[0]->{c_dirty}++; $_[0]->{p_dirty} = 0; | |
| 318 | $_[0]->{'polar'} = $_[1] } | |
| 319 | ||
| 320 | # | |
| 321 | # ->update_cartesian | |
| 322 | # | |
| 323 | # Recompute and return the cartesian form, given accurate polar form. | |
| 324 | # | |
| 325 | sub update_cartesian { | |
| 326 | my $self = shift; | |
| 327 | my ($r, $t) = @{$self->{'polar'}}; | |
| 328 | $self->{c_dirty} = 0; | |
| 329 | return $self->{'cartesian'} = [$r * CORE::cos($t), $r * CORE::sin($t)]; | |
| 330 | } | |
| 331 | ||
| 332 | # | |
| 333 | # | |
| 334 | # ->update_polar | |
| 335 | # | |
| 336 | # Recompute and return the polar form, given accurate cartesian form. | |
| 337 | # | |
| 338 | sub update_polar { | |
| 339 | my $self = shift; | |
| 340 | my ($x, $y) = @{$self->{'cartesian'}}; | |
| 341 | $self->{p_dirty} = 0; | |
| 342 | return $self->{'polar'} = [0, 0] if $x == 0 && $y == 0; | |
| 343 | return $self->{'polar'} = [CORE::sqrt($x*$x + $y*$y), | |
| 344 | CORE::atan2($y, $x)]; | |
| 345 | } | |
| 346 | ||
| 347 | # | |
| 348 | # (plus) | |
| 349 | # | |
| 350 | # Computes z1+z2. | |
| 351 | # | |
| 352 | sub plus { | |
| 353 | my ($z1, $z2, $regular) = @_; | |
| 354 | my ($re1, $im1) = @{$z1->cartesian}; | |
| 355 | $z2 = cplx($z2) unless ref $z2; | |
| 356 | my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0); | |
| 357 | unless (defined $regular) { | |
| 358 | $z1->set_cartesian([$re1 + $re2, $im1 + $im2]); | |
| 359 | return $z1; | |
| 360 | } | |
| 361 | return (ref $z1)->make($re1 + $re2, $im1 + $im2); | |
| 362 | } | |
| 363 | ||
| 364 | # | |
| 365 | # (minus) | |
| 366 | # | |
| 367 | # Computes z1-z2. | |
| 368 | # | |
| 369 | sub minus { | |
| 370 | my ($z1, $z2, $inverted) = @_; | |
| 371 | my ($re1, $im1) = @{$z1->cartesian}; | |
| 372 | $z2 = cplx($z2) unless ref $z2; | |
| 373 | my ($re2, $im2) = @{$z2->cartesian}; | |
| 374 | unless (defined $inverted) { | |
| 375 | $z1->set_cartesian([$re1 - $re2, $im1 - $im2]); | |
| 376 | return $z1; | |
| 377 | } | |
| 378 | return $inverted ? | |
| 379 | (ref $z1)->make($re2 - $re1, $im2 - $im1) : | |
| 380 | (ref $z1)->make($re1 - $re2, $im1 - $im2); | |
| 381 | ||
| 382 | } | |
| 383 | ||
| 384 | # | |
| 385 | # (multiply) | |
| 386 | # | |
| 387 | # Computes z1*z2. | |
| 388 | # | |
| 389 | sub multiply { | |
| 390 | my ($z1, $z2, $regular) = @_; | |
| 391 | if ($z1->{p_dirty} == 0 and ref $z2 and $z2->{p_dirty} == 0) { | |
| 392 | # if both polar better use polar to avoid rounding errors | |
| 393 | my ($r1, $t1) = @{$z1->polar}; | |
| 394 | my ($r2, $t2) = @{$z2->polar}; | |
| 395 | my $t = $t1 + $t2; | |
| 396 | if ($t > pi()) { $t -= pit2 } | |
| 397 | elsif ($t <= -pi()) { $t += pit2 } | |
| 398 | unless (defined $regular) { | |
| 399 | $z1->set_polar([$r1 * $r2, $t]); | |
| 400 | return $z1; | |
| 401 | } | |
| 402 | return (ref $z1)->emake($r1 * $r2, $t); | |
| 403 | } else { | |
| 404 | my ($x1, $y1) = @{$z1->cartesian}; | |
| 405 | if (ref $z2) { | |
| 406 | my ($x2, $y2) = @{$z2->cartesian}; | |
| 407 | return (ref $z1)->make($x1*$x2-$y1*$y2, $x1*$y2+$y1*$x2); | |
| 408 | } else { | |
| 409 | return (ref $z1)->make($x1*$z2, $y1*$z2); | |
| 410 | } | |
| 411 | } | |
| 412 | } | |
| 413 | ||
| 414 | # | |
| 415 | # _divbyzero | |
| 416 | # | |
| 417 | # Die on division by zero. | |
| 418 | # | |
| 419 | sub _divbyzero { | |
| 420 | my $mess = "$_[0]: Division by zero.\n"; | |
| 421 | ||
| 422 | if (defined $_[1]) { | |
| 423 | $mess .= "(Because in the definition of $_[0], the divisor "; | |
| 424 | $mess .= "$_[1] " unless ("$_[1]" eq '0'); | |
| 425 | $mess .= "is 0)\n"; | |
| 426 | } | |
| 427 | ||
| 428 | my @up = caller(1); | |
| 429 | ||
| 430 | $mess .= "Died at $up[1] line $up[2].\n"; | |
| 431 | ||
| 432 | die $mess; | |
| 433 | } | |
| 434 | ||
| 435 | # | |
| 436 | # (divide) | |
| 437 | # | |
| 438 | # Computes z1/z2. | |
| 439 | # | |
| 440 | sub divide { | |
| 441 | my ($z1, $z2, $inverted) = @_; | |
| 442 | if ($z1->{p_dirty} == 0 and ref $z2 and $z2->{p_dirty} == 0) { | |
| 443 | # if both polar better use polar to avoid rounding errors | |
| 444 | my ($r1, $t1) = @{$z1->polar}; | |
| 445 | my ($r2, $t2) = @{$z2->polar}; | |
| 446 | my $t; | |
| 447 | if ($inverted) { | |
| 448 | _divbyzero "$z2/0" if ($r1 == 0); | |
| 449 | $t = $t2 - $t1; | |
| 450 | if ($t > pi()) { $t -= pit2 } | |
| 451 | elsif ($t <= -pi()) { $t += pit2 } | |
| 452 | return (ref $z1)->emake($r2 / $r1, $t); | |
| 453 | } else { | |
| 454 | _divbyzero "$z1/0" if ($r2 == 0); | |
| 455 | $t = $t1 - $t2; | |
| 456 | if ($t > pi()) { $t -= pit2 } | |
| 457 | elsif ($t <= -pi()) { $t += pit2 } | |
| 458 | return (ref $z1)->emake($r1 / $r2, $t); | |
| 459 | } | |
| 460 | } else { | |
| 461 | my ($d, $x2, $y2); | |
| 462 | if ($inverted) { | |
| 463 | ($x2, $y2) = @{$z1->cartesian}; | |
| 464 | $d = $x2*$x2 + $y2*$y2; | |
| 465 | _divbyzero "$z2/0" if $d == 0; | |
| 466 | return (ref $z1)->make(($x2*$z2)/$d, -($y2*$z2)/$d); | |
| 467 | } else { | |
| 468 | my ($x1, $y1) = @{$z1->cartesian}; | |
| 469 | if (ref $z2) { | |
| 470 | ($x2, $y2) = @{$z2->cartesian}; | |
| 471 | $d = $x2*$x2 + $y2*$y2; | |
| 472 | _divbyzero "$z1/0" if $d == 0; | |
| 473 | my $u = ($x1*$x2 + $y1*$y2)/$d; | |
| 474 | my $v = ($y1*$x2 - $x1*$y2)/$d; | |
| 475 | return (ref $z1)->make($u, $v); | |
| 476 | } else { | |
| 477 | _divbyzero "$z1/0" if $z2 == 0; | |
| 478 | return (ref $z1)->make($x1/$z2, $y1/$z2); | |
| 479 | } | |
| 480 | } | |
| 481 | } | |
| 482 | } | |
| 483 | ||
| 484 | # | |
| 485 | # (power) | |
| 486 | # | |
| 487 | # Computes z1**z2 = exp(z2 * log z1)). | |
| 488 | # | |
| 489 | sub power { | |
| 490 | my ($z1, $z2, $inverted) = @_; | |
| 491 | if ($inverted) { | |
| 492 | return 1 if $z1 == 0 || $z2 == 1; | |
| 493 | return 0 if $z2 == 0 && Re($z1) > 0; | |
| 494 | } else { | |
| 495 | return 1 if $z2 == 0 || $z1 == 1; | |
| 496 | return 0 if $z1 == 0 && Re($z2) > 0; | |
| 497 | } | |
| 498 | my $w = $inverted ? &exp($z1 * &log($z2)) | |
| 499 | : &exp($z2 * &log($z1)); | |
| 500 | # If both arguments cartesian, return cartesian, else polar. | |
| 501 | return $z1->{c_dirty} == 0 && | |
| 502 | (not ref $z2 or $z2->{c_dirty} == 0) ? | |
| 503 | cplx(@{$w->cartesian}) : $w; | |
| 504 | } | |
| 505 | ||
| 506 | # | |
| 507 | # (spaceship) | |
| 508 | # | |
| 509 | # Computes z1 <=> z2. | |
| 510 | # Sorts on the real part first, then on the imaginary part. Thus 2-4i < 3+8i. | |
| 511 | # | |
| 512 | sub spaceship { | |
| 513 | my ($z1, $z2, $inverted) = @_; | |
| 514 | my ($re1, $im1) = ref $z1 ? @{$z1->cartesian} : ($z1, 0); | |
| 515 | my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0); | |
| 516 | my $sgn = $inverted ? -1 : 1; | |
| 517 | return $sgn * ($re1 <=> $re2) if $re1 != $re2; | |
| 518 | return $sgn * ($im1 <=> $im2); | |
| 519 | } | |
| 520 | ||
| 521 | # | |
| 522 | # (numeq) | |
| 523 | # | |
| 524 | # Computes z1 == z2. | |
| 525 | # | |
| 526 | # (Required in addition to spaceship() because of NaNs.) | |
| 527 | sub numeq { | |
| 528 | my ($z1, $z2, $inverted) = @_; | |
| 529 | my ($re1, $im1) = ref $z1 ? @{$z1->cartesian} : ($z1, 0); | |
| 530 | my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0); | |
| 531 | return $re1 == $re2 && $im1 == $im2 ? 1 : 0; | |
| 532 | } | |
| 533 | ||
| 534 | # | |
| 535 | # (negate) | |
| 536 | # | |
| 537 | # Computes -z. | |
| 538 | # | |
| 539 | sub negate { | |
| 540 | my ($z) = @_; | |
| 541 | if ($z->{c_dirty}) { | |
| 542 | my ($r, $t) = @{$z->polar}; | |
| 543 | $t = ($t <= 0) ? $t + pi : $t - pi; | |
| 544 | return (ref $z)->emake($r, $t); | |
| 545 | } | |
| 546 | my ($re, $im) = @{$z->cartesian}; | |
| 547 | return (ref $z)->make(-$re, -$im); | |
| 548 | } | |
| 549 | ||
| 550 | # | |
| 551 | # (conjugate) | |
| 552 | # | |
| 553 | # Compute complex's conjugate. | |
| 554 | # | |
| 555 | sub conjugate { | |
| 556 | my ($z) = @_; | |
| 557 | if ($z->{c_dirty}) { | |
| 558 | my ($r, $t) = @{$z->polar}; | |
| 559 | return (ref $z)->emake($r, -$t); | |
| 560 | } | |
| 561 | my ($re, $im) = @{$z->cartesian}; | |
| 562 | return (ref $z)->make($re, -$im); | |
| 563 | } | |
| 564 | ||
| 565 | # | |
| 566 | # (abs) | |
| 567 | # | |
| 568 | # Compute or set complex's norm (rho). | |
| 569 | # | |
| 570 | sub abs { | |
| 571 | my ($z, $rho) = @_; | |
| 572 | unless (ref $z) { | |
| 573 | if (@_ == 2) { | |
| 574 | $_[0] = $_[1]; | |
| 575 | } else { | |
| 576 | return CORE::abs($z); | |
| 577 | } | |
| 578 | } | |
| 579 | if (defined $rho) { | |
| 580 | $z->{'polar'} = [ $rho, ${$z->polar}[1] ]; | |
| 581 | $z->{p_dirty} = 0; | |
| 582 | $z->{c_dirty} = 1; | |
| 583 | return $rho; | |
| 584 | } else { | |
| 585 | return ${$z->polar}[0]; | |
| 586 | } | |
| 587 | } | |
| 588 | ||
| 589 | sub _theta { | |
| 590 | my $theta = $_[0]; | |
| 591 | ||
| 592 | if ($$theta > pi()) { $$theta -= pit2 } | |
| 593 | elsif ($$theta <= -pi()) { $$theta += pit2 } | |
| 594 | } | |
| 595 | ||
| 596 | # | |
| 597 | # arg | |
| 598 | # | |
| 599 | # Compute or set complex's argument (theta). | |
| 600 | # | |
| 601 | sub arg { | |
| 602 | my ($z, $theta) = @_; | |
| 603 | return $z unless ref $z; | |
| 604 | if (defined $theta) { | |
| 605 | _theta(\$theta); | |
| 606 | $z->{'polar'} = [ ${$z->polar}[0], $theta ]; | |
| 607 | $z->{p_dirty} = 0; | |
| 608 | $z->{c_dirty} = 1; | |
| 609 | } else { | |
| 610 | $theta = ${$z->polar}[1]; | |
| 611 | _theta(\$theta); | |
| 612 | } | |
| 613 | return $theta; | |
| 614 | } | |
| 615 | ||
| 616 | # | |
| 617 | # (sqrt) | |
| 618 | # | |
| 619 | # Compute sqrt(z). | |
| 620 | # | |
| 621 | # It is quite tempting to use wantarray here so that in list context | |
| 622 | # sqrt() would return the two solutions. This, however, would | |
| 623 | # break things like | |
| 624 | # | |
| 625 | # print "sqrt(z) = ", sqrt($z), "\n"; | |
| 626 | # | |
| 627 | # The two values would be printed side by side without no intervening | |
| 628 | # whitespace, quite confusing. | |
| 629 | # Therefore if you want the two solutions use the root(). | |
| 630 | # | |
| 631 | sub sqrt { | |
| 632 | my ($z) = @_; | |
| 633 | my ($re, $im) = ref $z ? @{$z->cartesian} : ($z, 0); | |
| 634 | return $re < 0 ? cplx(0, CORE::sqrt(-$re)) : CORE::sqrt($re) | |
| 635 | if $im == 0; | |
| 636 | my ($r, $t) = @{$z->polar}; | |
| 637 | return (ref $z)->emake(CORE::sqrt($r), $t/2); | |
| 638 | } | |
| 639 | ||
| 640 | # | |
| 641 | # cbrt | |
| 642 | # | |
| 643 | # Compute cbrt(z) (cubic root). | |
| 644 | # | |
| 645 | # Why are we not returning three values? The same answer as for sqrt(). | |
| 646 | # | |
| 647 | sub cbrt { | |
| 648 | my ($z) = @_; | |
| 649 | return $z < 0 ? | |
| 650 | -CORE::exp(CORE::log(-$z)/3) : | |
| 651 | ($z > 0 ? CORE::exp(CORE::log($z)/3): 0) | |
| 652 | unless ref $z; | |
| 653 | my ($r, $t) = @{$z->polar}; | |
| 654 | return 0 if $r == 0; | |
| 655 | return (ref $z)->emake(CORE::exp(CORE::log($r)/3), $t/3); | |
| 656 | } | |
| 657 | ||
| 658 | # | |
| 659 | # _rootbad | |
| 660 | # | |
| 661 | # Die on bad root. | |
| 662 | # | |
| 663 | sub _rootbad { | |
| 664 | my $mess = "Root '$_[0]' illegal, root rank must be positive integer.\n"; | |
| 665 | ||
| 666 | my @up = caller(1); | |
| 667 | ||
| 668 | $mess .= "Died at $up[1] line $up[2].\n"; | |
| 669 | ||
| 670 | die $mess; | |
| 671 | } | |
| 672 | ||
| 673 | # | |
| 674 | # root | |
| 675 | # | |
| 676 | # Computes all nth root for z, returning an array whose size is n. | |
| 677 | # `n' must be a positive integer. | |
| 678 | # | |
| 679 | # The roots are given by (for k = 0..n-1): | |
| 680 | # | |
| 681 | # z^(1/n) = r^(1/n) (cos ((t+2 k pi)/n) + i sin ((t+2 k pi)/n)) | |
| 682 | # | |
| 683 | sub root { | |
| 684 | my ($z, $n, $k) = @_; | |
| 685 | _rootbad($n) if ($n < 1 or int($n) != $n); | |
| 686 | my ($r, $t) = ref $z ? | |
| 687 | @{$z->polar} : (CORE::abs($z), $z >= 0 ? 0 : pi); | |
| 688 | my $theta_inc = pit2 / $n; | |
| 689 | my $rho = $r ** (1/$n); | |
| 690 | my $cartesian = ref $z && $z->{c_dirty} == 0; | |
| 691 | if (@_ == 2) { | |
| 692 | my @root; | |
| 693 | for (my $i = 0, my $theta = $t / $n; | |
| 694 | $i < $n; | |
| 695 | $i++, $theta += $theta_inc) { | |
| 696 | my $w = cplxe($rho, $theta); | |
| 697 | # Yes, $cartesian is loop invariant. | |
| 698 | push @root, $cartesian ? cplx(@{$w->cartesian}) : $w; | |
| 699 | } | |
| 700 | return @root; | |
| 701 | } elsif (@_ == 3) { | |
| 702 | my $w = cplxe($rho, $t / $n + $k * $theta_inc); | |
| 703 | return $cartesian ? cplx(@{$w->cartesian}) : $w; | |
| 704 | } | |
| 705 | } | |
| 706 | ||
| 707 | # | |
| 708 | # Re | |
| 709 | # | |
| 710 | # Return or set Re(z). | |
| 711 | # | |
| 712 | sub Re { | |
| 713 | my ($z, $Re) = @_; | |
| 714 | return $z unless ref $z; | |
| 715 | if (defined $Re) { | |
| 716 | $z->{'cartesian'} = [ $Re, ${$z->cartesian}[1] ]; | |
| 717 | $z->{c_dirty} = 0; | |
| 718 | $z->{p_dirty} = 1; | |
| 719 | } else { | |
| 720 | return ${$z->cartesian}[0]; | |
| 721 | } | |
| 722 | } | |
| 723 | ||
| 724 | # | |
| 725 | # Im | |
| 726 | # | |
| 727 | # Return or set Im(z). | |
| 728 | # | |
| 729 | sub Im { | |
| 730 | my ($z, $Im) = @_; | |
| 731 | return 0 unless ref $z; | |
| 732 | if (defined $Im) { | |
| 733 | $z->{'cartesian'} = [ ${$z->cartesian}[0], $Im ]; | |
| 734 | $z->{c_dirty} = 0; | |
| 735 | $z->{p_dirty} = 1; | |
| 736 | } else { | |
| 737 | return ${$z->cartesian}[1]; | |
| 738 | } | |
| 739 | } | |
| 740 | ||
| 741 | # | |
| 742 | # rho | |
| 743 | # | |
| 744 | # Return or set rho(w). | |
| 745 | # | |
| 746 | sub rho { | |
| 747 | Math::Complex::abs(@_); | |
| 748 | } | |
| 749 | ||
| 750 | # | |
| 751 | # theta | |
| 752 | # | |
| 753 | # Return or set theta(w). | |
| 754 | # | |
| 755 | sub theta { | |
| 756 | Math::Complex::arg(@_); | |
| 757 | } | |
| 758 | ||
| 759 | # | |
| 760 | # (exp) | |
| 761 | # | |
| 762 | # Computes exp(z). | |
| 763 | # | |
| 764 | sub exp { | |
| 765 | my ($z) = @_; | |
| 766 | my ($x, $y) = @{$z->cartesian}; | |
| 767 | return (ref $z)->emake(CORE::exp($x), $y); | |
| 768 | } | |
| 769 | ||
| 770 | # | |
| 771 | # _logofzero | |
| 772 | # | |
| 773 | # Die on logarithm of zero. | |
| 774 | # | |
| 775 | sub _logofzero { | |
| 776 | my $mess = "$_[0]: Logarithm of zero.\n"; | |
| 777 | ||
| 778 | if (defined $_[1]) { | |
| 779 | $mess .= "(Because in the definition of $_[0], the argument "; | |
| 780 | $mess .= "$_[1] " unless ($_[1] eq '0'); | |
| 781 | $mess .= "is 0)\n"; | |
| 782 | } | |
| 783 | ||
| 784 | my @up = caller(1); | |
| 785 | ||
| 786 | $mess .= "Died at $up[1] line $up[2].\n"; | |
| 787 | ||
| 788 | die $mess; | |
| 789 | } | |
| 790 | ||
| 791 | # | |
| 792 | # (log) | |
| 793 | # | |
| 794 | # Compute log(z). | |
| 795 | # | |
| 796 | sub log { | |
| 797 | my ($z) = @_; | |
| 798 | unless (ref $z) { | |
| 799 | _logofzero("log") if $z == 0; | |
| 800 | return $z > 0 ? CORE::log($z) : cplx(CORE::log(-$z), pi); | |
| 801 | } | |
| 802 | my ($r, $t) = @{$z->polar}; | |
| 803 | _logofzero("log") if $r == 0; | |
| 804 | if ($t > pi()) { $t -= pit2 } | |
| 805 | elsif ($t <= -pi()) { $t += pit2 } | |
| 806 | return (ref $z)->make(CORE::log($r), $t); | |
| 807 | } | |
| 808 | ||
| 809 | # | |
| 810 | # ln | |
| 811 | # | |
| 812 | # Alias for log(). | |
| 813 | # | |
| 814 | sub ln { Math::Complex::log(@_) } | |
| 815 | ||
| 816 | # | |
| 817 | # log10 | |
| 818 | # | |
| 819 | # Compute log10(z). | |
| 820 | # | |
| 821 | ||
| 822 | sub log10 { | |
| 823 | return Math::Complex::log($_[0]) * uplog10; | |
| 824 | } | |
| 825 | ||
| 826 | # | |
| 827 | # logn | |
| 828 | # | |
| 829 | # Compute logn(z,n) = log(z) / log(n) | |
| 830 | # | |
| 831 | sub logn { | |
| 832 | my ($z, $n) = @_; | |
| 833 | $z = cplx($z, 0) unless ref $z; | |
| 834 | my $logn = $LOGN{$n}; | |
| 835 | $logn = $LOGN{$n} = CORE::log($n) unless defined $logn; # Cache log(n) | |
| 836 | return &log($z) / $logn; | |
| 837 | } | |
| 838 | ||
| 839 | # | |
| 840 | # (cos) | |
| 841 | # | |
| 842 | # Compute cos(z) = (exp(iz) + exp(-iz))/2. | |
| 843 | # | |
| 844 | sub cos { | |
| 845 | my ($z) = @_; | |
| 846 | return CORE::cos($z) unless ref $z; | |
| 847 | my ($x, $y) = @{$z->cartesian}; | |
| 848 | my $ey = CORE::exp($y); | |
| 849 | my $sx = CORE::sin($x); | |
| 850 | my $cx = CORE::cos($x); | |
| 851 | my $ey_1 = $ey ? 1 / $ey : $Inf; | |
| 852 | return (ref $z)->make($cx * ($ey + $ey_1)/2, | |
| 853 | $sx * ($ey_1 - $ey)/2); | |
| 854 | } | |
| 855 | ||
| 856 | # | |
| 857 | # (sin) | |
| 858 | # | |
| 859 | # Compute sin(z) = (exp(iz) - exp(-iz))/2. | |
| 860 | # | |
| 861 | sub sin { | |
| 862 | my ($z) = @_; | |
| 863 | return CORE::sin($z) unless ref $z; | |
| 864 | my ($x, $y) = @{$z->cartesian}; | |
| 865 | my $ey = CORE::exp($y); | |
| 866 | my $sx = CORE::sin($x); | |
| 867 | my $cx = CORE::cos($x); | |
| 868 | my $ey_1 = $ey ? 1 / $ey : $Inf; | |
| 869 | return (ref $z)->make($sx * ($ey + $ey_1)/2, | |
| 870 | $cx * ($ey - $ey_1)/2); | |
| 871 | } | |
| 872 | ||
| 873 | # | |
| 874 | # tan | |
| 875 | # | |
| 876 | # Compute tan(z) = sin(z) / cos(z). | |
| 877 | # | |
| 878 | sub tan { | |
| 879 | my ($z) = @_; | |
| 880 | my $cz = &cos($z); | |
| 881 | _divbyzero "tan($z)", "cos($z)" if $cz == 0; | |
| 882 | return &sin($z) / $cz; | |
| 883 | } | |
| 884 | ||
| 885 | # | |
| 886 | # sec | |
| 887 | # | |
| 888 | # Computes the secant sec(z) = 1 / cos(z). | |
| 889 | # | |
| 890 | sub sec { | |
| 891 | my ($z) = @_; | |
| 892 | my $cz = &cos($z); | |
| 893 | _divbyzero "sec($z)", "cos($z)" if ($cz == 0); | |
| 894 | return 1 / $cz; | |
| 895 | } | |
| 896 | ||
| 897 | # | |
| 898 | # csc | |
| 899 | # | |
| 900 | # Computes the cosecant csc(z) = 1 / sin(z). | |
| 901 | # | |
| 902 | sub csc { | |
| 903 | my ($z) = @_; | |
| 904 | my $sz = &sin($z); | |
| 905 | _divbyzero "csc($z)", "sin($z)" if ($sz == 0); | |
| 906 | return 1 / $sz; | |
| 907 | } | |
| 908 | ||
| 909 | # | |
| 910 | # cosec | |
| 911 | # | |
| 912 | # Alias for csc(). | |
| 913 | # | |
| 914 | sub cosec { Math::Complex::csc(@_) } | |
| 915 | ||
| 916 | # | |
| 917 | # cot | |
| 918 | # | |
| 919 | # Computes cot(z) = cos(z) / sin(z). | |
| 920 | # | |
| 921 | sub cot { | |
| 922 | my ($z) = @_; | |
| 923 | my $sz = &sin($z); | |
| 924 | _divbyzero "cot($z)", "sin($z)" if ($sz == 0); | |
| 925 | return &cos($z) / $sz; | |
| 926 | } | |
| 927 | ||
| 928 | # | |
| 929 | # cotan | |
| 930 | # | |
| 931 | # Alias for cot(). | |
| 932 | # | |
| 933 | sub cotan { Math::Complex::cot(@_) } | |
| 934 | ||
| 935 | # | |
| 936 | # acos | |
| 937 | # | |
| 938 | # Computes the arc cosine acos(z) = -i log(z + sqrt(z*z-1)). | |
| 939 | # | |
| 940 | sub acos { | |
| 941 | my $z = $_[0]; | |
| 942 | return CORE::atan2(CORE::sqrt(1-$z*$z), $z) | |
| 943 | if (! ref $z) && CORE::abs($z) <= 1; | |
| 944 | $z = cplx($z, 0) unless ref $z; | |
| 945 | my ($x, $y) = @{$z->cartesian}; | |
| 946 | return 0 if $x == 1 && $y == 0; | |
| 947 | my $t1 = CORE::sqrt(($x+1)*($x+1) + $y*$y); | |
| 948 | my $t2 = CORE::sqrt(($x-1)*($x-1) + $y*$y); | |
| 949 | my $alpha = ($t1 + $t2)/2; | |
| 950 | my $beta = ($t1 - $t2)/2; | |
| 951 | $alpha = 1 if $alpha < 1; | |
| 952 | if ($beta > 1) { $beta = 1 } | |
| 953 | elsif ($beta < -1) { $beta = -1 } | |
| 954 | my $u = CORE::atan2(CORE::sqrt(1-$beta*$beta), $beta); | |
| 955 | my $v = CORE::log($alpha + CORE::sqrt($alpha*$alpha-1)); | |
| 956 | $v = -$v if $y > 0 || ($y == 0 && $x < -1); | |
| 957 | return (ref $z)->make($u, $v); | |
| 958 | } | |
| 959 | ||
| 960 | # | |
| 961 | # asin | |
| 962 | # | |
| 963 | # Computes the arc sine asin(z) = -i log(iz + sqrt(1-z*z)). | |
| 964 | # | |
| 965 | sub asin { | |
| 966 | my $z = $_[0]; | |
| 967 | return CORE::atan2($z, CORE::sqrt(1-$z*$z)) | |
| 968 | if (! ref $z) && CORE::abs($z) <= 1; | |
| 969 | $z = cplx($z, 0) unless ref $z; | |
| 970 | my ($x, $y) = @{$z->cartesian}; | |
| 971 | return 0 if $x == 0 && $y == 0; | |
| 972 | my $t1 = CORE::sqrt(($x+1)*($x+1) + $y*$y); | |
| 973 | my $t2 = CORE::sqrt(($x-1)*($x-1) + $y*$y); | |
| 974 | my $alpha = ($t1 + $t2)/2; | |
| 975 | my $beta = ($t1 - $t2)/2; | |
| 976 | $alpha = 1 if $alpha < 1; | |
| 977 | if ($beta > 1) { $beta = 1 } | |
| 978 | elsif ($beta < -1) { $beta = -1 } | |
| 979 | my $u = CORE::atan2($beta, CORE::sqrt(1-$beta*$beta)); | |
| 980 | my $v = -CORE::log($alpha + CORE::sqrt($alpha*$alpha-1)); | |
| 981 | $v = -$v if $y > 0 || ($y == 0 && $x < -1); | |
| 982 | return (ref $z)->make($u, $v); | |
| 983 | } | |
| 984 | ||
| 985 | # | |
| 986 | # atan | |
| 987 | # | |
| 988 | # Computes the arc tangent atan(z) = i/2 log((i+z) / (i-z)). | |
| 989 | # | |
| 990 | sub atan { | |
| 991 | my ($z) = @_; | |
| 992 | return CORE::atan2($z, 1) unless ref $z; | |
| 993 | my ($x, $y) = ref $z ? @{$z->cartesian} : ($z, 0); | |
| 994 | return 0 if $x == 0 && $y == 0; | |
| 995 | _divbyzero "atan(i)" if ( $z == i); | |
| 996 | _logofzero "atan(-i)" if (-$z == i); # -i is a bad file test... | |
| 997 | my $log = &log((i + $z) / (i - $z)); | |
| 998 | return ip2 * $log; | |
| 999 | } | |
| 1000 | ||
| 1001 | # | |
| 1002 | # asec | |
| 1003 | # | |
| 1004 | # Computes the arc secant asec(z) = acos(1 / z). | |
| 1005 | # | |
| 1006 | sub asec { | |
| 1007 | my ($z) = @_; | |
| 1008 | _divbyzero "asec($z)", $z if ($z == 0); | |
| 1009 | return acos(1 / $z); | |
| 1010 | } | |
| 1011 | ||
| 1012 | # | |
| 1013 | # acsc | |
| 1014 | # | |
| 1015 | # Computes the arc cosecant acsc(z) = asin(1 / z). | |
| 1016 | # | |
| 1017 | sub acsc { | |
| 1018 | my ($z) = @_; | |
| 1019 | _divbyzero "acsc($z)", $z if ($z == 0); | |
| 1020 | return asin(1 / $z); | |
| 1021 | } | |
| 1022 | ||
| 1023 | # | |
| 1024 | # acosec | |
| 1025 | # | |
| 1026 | # Alias for acsc(). | |
| 1027 | # | |
| 1028 | sub acosec { Math::Complex::acsc(@_) } | |
| 1029 | ||
| 1030 | # | |
| 1031 | # acot | |
| 1032 | # | |
| 1033 | # Computes the arc cotangent acot(z) = atan(1 / z) | |
| 1034 | # | |
| 1035 | sub acot { | |
| 1036 | my ($z) = @_; | |
| 1037 | _divbyzero "acot(0)" if $z == 0; | |
| 1038 | return ($z >= 0) ? CORE::atan2(1, $z) : CORE::atan2(-1, -$z) | |
| 1039 | unless ref $z; | |
| 1040 | _divbyzero "acot(i)" if ($z - i == 0); | |
| 1041 | _logofzero "acot(-i)" if ($z + i == 0); | |
| 1042 | return atan(1 / $z); | |
| 1043 | } | |
| 1044 | ||
| 1045 | # | |
| 1046 | # acotan | |
| 1047 | # | |
| 1048 | # Alias for acot(). | |
| 1049 | # | |
| 1050 | sub acotan { Math::Complex::acot(@_) } | |
| 1051 | ||
| 1052 | # | |
| 1053 | # cosh | |
| 1054 | # | |
| 1055 | # Computes the hyperbolic cosine cosh(z) = (exp(z) + exp(-z))/2. | |
| 1056 | # | |
| 1057 | sub cosh { | |
| 1058 | my ($z) = @_; | |
| 1059 | my $ex; | |
| 1060 | unless (ref $z) { | |
| 1061 | $ex = CORE::exp($z); | |
| 1062 | return $ex ? ($ex + 1/$ex)/2 : $Inf; | |
| 1063 | } | |
| 1064 | my ($x, $y) = @{$z->cartesian}; | |
| 1065 | $ex = CORE::exp($x); | |
| 1066 | my $ex_1 = $ex ? 1 / $ex : $Inf; | |
| 1067 | return (ref $z)->make(CORE::cos($y) * ($ex + $ex_1)/2, | |
| 1068 | CORE::sin($y) * ($ex - $ex_1)/2); | |
| 1069 | } | |
| 1070 | ||
| 1071 | # | |
| 1072 | # sinh | |
| 1073 | # | |
| 1074 | # Computes the hyperbolic sine sinh(z) = (exp(z) - exp(-z))/2. | |
| 1075 | # | |
| 1076 | sub sinh { | |
| 1077 | my ($z) = @_; | |
| 1078 | my $ex; | |
| 1079 | unless (ref $z) { | |
| 1080 | return 0 if $z == 0; | |
| 1081 | $ex = CORE::exp($z); | |
| 1082 | return $ex ? ($ex - 1/$ex)/2 : "-$Inf"; | |
| 1083 | } | |
| 1084 | my ($x, $y) = @{$z->cartesian}; | |
| 1085 | my $cy = CORE::cos($y); | |
| 1086 | my $sy = CORE::sin($y); | |
| 1087 | $ex = CORE::exp($x); | |
| 1088 | my $ex_1 = $ex ? 1 / $ex : $Inf; | |
| 1089 | return (ref $z)->make(CORE::cos($y) * ($ex - $ex_1)/2, | |
| 1090 | CORE::sin($y) * ($ex + $ex_1)/2); | |
| 1091 | } | |
| 1092 | ||
| 1093 | # | |
| 1094 | # tanh | |
| 1095 | # | |
| 1096 | # Computes the hyperbolic tangent tanh(z) = sinh(z) / cosh(z). | |
| 1097 | # | |
| 1098 | sub tanh { | |
| 1099 | my ($z) = @_; | |
| 1100 | my $cz = cosh($z); | |
| 1101 | _divbyzero "tanh($z)", "cosh($z)" if ($cz == 0); | |
| 1102 | return sinh($z) / $cz; | |
| 1103 | } | |
| 1104 | ||
| 1105 | # | |
| 1106 | # sech | |
| 1107 | # | |
| 1108 | # Computes the hyperbolic secant sech(z) = 1 / cosh(z). | |
| 1109 | # | |
| 1110 | sub sech { | |
| 1111 | my ($z) = @_; | |
| 1112 | my $cz = cosh($z); | |
| 1113 | _divbyzero "sech($z)", "cosh($z)" if ($cz == 0); | |
| 1114 | return 1 / $cz; | |
| 1115 | } | |
| 1116 | ||
| 1117 | # | |
| 1118 | # csch | |
| 1119 | # | |
| 1120 | # Computes the hyperbolic cosecant csch(z) = 1 / sinh(z). | |
| 1121 | # | |
| 1122 | sub csch { | |
| 1123 | my ($z) = @_; | |
| 1124 | my $sz = sinh($z); | |
| 1125 | _divbyzero "csch($z)", "sinh($z)" if ($sz == 0); | |
| 1126 | return 1 / $sz; | |
| 1127 | } | |
| 1128 | ||
| 1129 | # | |
| 1130 | # cosech | |
| 1131 | # | |
| 1132 | # Alias for csch(). | |
| 1133 | # | |
| 1134 | sub cosech { Math::Complex::csch(@_) } | |
| 1135 | ||
| 1136 | # | |
| 1137 | # coth | |
| 1138 | # | |
| 1139 | # Computes the hyperbolic cotangent coth(z) = cosh(z) / sinh(z). | |
| 1140 | # | |
| 1141 | sub coth { | |
| 1142 | my ($z) = @_; | |
| 1143 | my $sz = sinh($z); | |
| 1144 | _divbyzero "coth($z)", "sinh($z)" if $sz == 0; | |
| 1145 | return cosh($z) / $sz; | |
| 1146 | } | |
| 1147 | ||
| 1148 | # | |
| 1149 | # cotanh | |
| 1150 | # | |
| 1151 | # Alias for coth(). | |
| 1152 | # | |
| 1153 | sub cotanh { Math::Complex::coth(@_) } | |
| 1154 | ||
| 1155 | # | |
| 1156 | # acosh | |
| 1157 | # | |
| 1158 | # Computes the arc hyperbolic cosine acosh(z) = log(z + sqrt(z*z-1)). | |
| 1159 | # | |
| 1160 | sub acosh { | |
| 1161 | my ($z) = @_; | |
| 1162 | unless (ref $z) { | |
| 1163 | $z = cplx($z, 0); | |
| 1164 | } | |
| 1165 | my ($re, $im) = @{$z->cartesian}; | |
| 1166 | if ($im == 0) { | |
| 1167 | return CORE::log($re + CORE::sqrt($re*$re - 1)) | |
| 1168 | if $re >= 1; | |
| 1169 | return cplx(0, CORE::atan2(CORE::sqrt(1 - $re*$re), $re)) | |
| 1170 | if CORE::abs($re) < 1; | |
| 1171 | } | |
| 1172 | my $t = &sqrt($z * $z - 1) + $z; | |
| 1173 | # Try Taylor if looking bad (this usually means that | |
| 1174 | # $z was large negative, therefore the sqrt is really | |
| 1175 | # close to abs(z), summing that with z...) | |
| 1176 | $t = 1/(2 * $z) - 1/(8 * $z**3) + 1/(16 * $z**5) - 5/(128 * $z**7) | |
| 1177 | if $t == 0; | |
| 1178 | my $u = &log($t); | |
| 1179 | $u->Im(-$u->Im) if $re < 0 && $im == 0; | |
| 1180 | return $re < 0 ? -$u : $u; | |
| 1181 | } | |
| 1182 | ||
| 1183 | # | |
| 1184 | # asinh | |
| 1185 | # | |
| 1186 | # Computes the arc hyperbolic sine asinh(z) = log(z + sqrt(z*z+1)) | |
| 1187 | # | |
| 1188 | sub asinh { | |
| 1189 | my ($z) = @_; | |
| 1190 | unless (ref $z) { | |
| 1191 | my $t = $z + CORE::sqrt($z*$z + 1); | |
| 1192 | return CORE::log($t) if $t; | |
| 1193 | } | |
| 1194 | my $t = &sqrt($z * $z + 1) + $z; | |
| 1195 | # Try Taylor if looking bad (this usually means that | |
| 1196 | # $z was large negative, therefore the sqrt is really | |
| 1197 | # close to abs(z), summing that with z...) | |
| 1198 | $t = 1/(2 * $z) - 1/(8 * $z**3) + 1/(16 * $z**5) - 5/(128 * $z**7) | |
| 1199 | if $t == 0; | |
| 1200 | return &log($t); | |
| 1201 | } | |
| 1202 | ||
| 1203 | # | |
| 1204 | # atanh | |
| 1205 | # | |
| 1206 | # Computes the arc hyperbolic tangent atanh(z) = 1/2 log((1+z) / (1-z)). | |
| 1207 | # | |
| 1208 | sub atanh { | |
| 1209 | my ($z) = @_; | |
| 1210 | unless (ref $z) { | |
| 1211 | return CORE::log((1 + $z)/(1 - $z))/2 if CORE::abs($z) < 1; | |
| 1212 | $z = cplx($z, 0); | |
| 1213 | } | |
| 1214 | _divbyzero 'atanh(1)', "1 - $z" if (1 - $z == 0); | |
| 1215 | _logofzero 'atanh(-1)' if (1 + $z == 0); | |
| 1216 | return 0.5 * &log((1 + $z) / (1 - $z)); | |
| 1217 | } | |
| 1218 | ||
| 1219 | # | |
| 1220 | # asech | |
| 1221 | # | |
| 1222 | # Computes the hyperbolic arc secant asech(z) = acosh(1 / z). | |
| 1223 | # | |
| 1224 | sub asech { | |
| 1225 | my ($z) = @_; | |
| 1226 | _divbyzero 'asech(0)', "$z" if ($z == 0); | |
| 1227 | return acosh(1 / $z); | |
| 1228 | } | |
| 1229 | ||
| 1230 | # | |
| 1231 | # acsch | |
| 1232 | # | |
| 1233 | # Computes the hyperbolic arc cosecant acsch(z) = asinh(1 / z). | |
| 1234 | # | |
| 1235 | sub acsch { | |
| 1236 | my ($z) = @_; | |
| 1237 | _divbyzero 'acsch(0)', $z if ($z == 0); | |
| 1238 | return asinh(1 / $z); | |
| 1239 | } | |
| 1240 | ||
| 1241 | # | |
| 1242 | # acosech | |
| 1243 | # | |
| 1244 | # Alias for acosh(). | |
| 1245 | # | |
| 1246 | sub acosech { Math::Complex::acsch(@_) } | |
| 1247 | ||
| 1248 | # | |
| 1249 | # acoth | |
| 1250 | # | |
| 1251 | # Computes the arc hyperbolic cotangent acoth(z) = 1/2 log((1+z) / (z-1)). | |
| 1252 | # | |
| 1253 | sub acoth { | |
| 1254 | my ($z) = @_; | |
| 1255 | _divbyzero 'acoth(0)' if ($z == 0); | |
| 1256 | unless (ref $z) { | |
| 1257 | return CORE::log(($z + 1)/($z - 1))/2 if CORE::abs($z) > 1; | |
| 1258 | $z = cplx($z, 0); | |
| 1259 | } | |
| 1260 | _divbyzero 'acoth(1)', "$z - 1" if ($z - 1 == 0); | |
| 1261 | _logofzero 'acoth(-1)', "1 + $z" if (1 + $z == 0); | |
| 1262 | return &log((1 + $z) / ($z - 1)) / 2; | |
| 1263 | } | |
| 1264 | ||
| 1265 | # | |
| 1266 | # acotanh | |
| 1267 | # | |
| 1268 | # Alias for acot(). | |
| 1269 | # | |
| 1270 | sub acotanh { Math::Complex::acoth(@_) } | |
| 1271 | ||
| 1272 | # | |
| 1273 | # (atan2) | |
| 1274 | # | |
| 1275 | # Compute atan(z1/z2), minding the right quadrant. | |
| 1276 | # | |
| 1277 | sub atan2 { | |
| 1278 | my ($z1, $z2, $inverted) = @_; | |
| 1279 | my ($re1, $im1, $re2, $im2); | |
| 1280 | if ($inverted) { | |
| 1281 | ($re1, $im1) = ref $z2 ? @{$z2->cartesian} : ($z2, 0); | |
| 1282 | ($re2, $im2) = ref $z1 ? @{$z1->cartesian} : ($z1, 0); | |
| 1283 | } else { | |
| 1284 | ($re1, $im1) = ref $z1 ? @{$z1->cartesian} : ($z1, 0); | |
| 1285 | ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0); | |
| 1286 | } | |
| 1287 | if ($im1 || $im2) { | |
| 1288 | # In MATLAB the imaginary parts are ignored. | |
| 1289 | # warn "atan2: Imaginary parts ignored"; | |
| 1290 | # http://documents.wolfram.com/mathematica/functions/ArcTan | |
| 1291 | # NOTE: Mathematica ArcTan[x,y] while atan2(y,x) | |
| 1292 | my $s = $z1 * $z1 + $z2 * $z2; | |
| 1293 | _divbyzero("atan2") if $s == 0; | |
| 1294 | my $i = &i; | |
| 1295 | my $r = $z2 + $z1 * $i; | |
| 1296 | return -$i * &log($r / &sqrt( $s )); | |
| 1297 | } | |
| 1298 | return CORE::atan2($re1, $re2); | |
| 1299 | } | |
| 1300 | ||
| 1301 | # | |
| 1302 | # display_format | |
| 1303 | # ->display_format | |
| 1304 | # | |
| 1305 | # Set (get if no argument) the display format for all complex numbers that | |
| 1306 | # don't happen to have overridden it via ->display_format | |
| 1307 | # | |
| 1308 | # When called as an object method, this actually sets the display format for | |
| 1309 | # the current object. | |
| 1310 | # | |
| 1311 | # Valid object formats are 'c' and 'p' for cartesian and polar. The first | |
| 1312 | # letter is used actually, so the type can be fully spelled out for clarity. | |
| 1313 | # | |
| 1314 | sub display_format { | |
| 1315 | my $self = shift; | |
| 1316 | my %display_format = %DISPLAY_FORMAT; | |
| 1317 | ||
| 1318 | if (ref $self) { # Called as an object method | |
| 1319 | if (exists $self->{display_format}) { | |
| 1320 | my %obj = %{$self->{display_format}}; | |
| 1321 | @display_format{keys %obj} = values %obj; | |
| 1322 | } | |
| 1323 | } | |
| 1324 | if (@_ == 1) { | |
| 1325 | $display_format{style} = shift; | |
| 1326 | } else { | |
| 1327 | my %new = @_; | |
| 1328 | @display_format{keys %new} = values %new; | |
| 1329 | } | |
| 1330 | ||
| 1331 | if (ref $self) { # Called as an object method | |
| 1332 | $self->{display_format} = { %display_format }; | |
| 1333 | return | |
| 1334 | wantarray ? | |
| 1335 | %{$self->{display_format}} : | |
| 1336 | $self->{display_format}->{style}; | |
| 1337 | } | |
| 1338 | ||
| 1339 | # Called as a class method | |
| 1340 | %DISPLAY_FORMAT = %display_format; | |
| 1341 | return | |
| 1342 | wantarray ? | |
| 1343 | %DISPLAY_FORMAT : | |
| 1344 | $DISPLAY_FORMAT{style}; | |
| 1345 | } | |
| 1346 | ||
| 1347 | # | |
| 1348 | # (stringify) | |
| 1349 | # | |
| 1350 | # Show nicely formatted complex number under its cartesian or polar form, | |
| 1351 | # depending on the current display format: | |
| 1352 | # | |
| 1353 | # . If a specific display format has been recorded for this object, use it. | |
| 1354 | # . Otherwise, use the generic current default for all complex numbers, | |
| 1355 | # which is a package global variable. | |
| 1356 | # | |
| 1357 | sub stringify { | |
| 1358 | my ($z) = shift; | |
| 1359 | ||
| 1360 | my $style = $z->display_format; | |
| 1361 | ||
| 1362 | $style = $DISPLAY_FORMAT{style} unless defined $style; | |
| 1363 | ||
| 1364 | return $z->stringify_polar if $style =~ /^p/i; | |
| 1365 | return $z->stringify_cartesian; | |
| 1366 | } | |
| 1367 | ||
| 1368 | # | |
| 1369 | # ->stringify_cartesian | |
| 1370 | # | |
| 1371 | # Stringify as a cartesian representation 'a+bi'. | |
| 1372 | # | |
| 1373 | sub stringify_cartesian { | |
| 1374 | my $z = shift; | |
| 1375 | my ($x, $y) = @{$z->cartesian}; | |
| 1376 | my ($re, $im); | |
| 1377 | ||
| 1378 | my %format = $z->display_format; | |
| 1379 | my $format = $format{format}; | |
| 1380 | ||
| 1381 | if ($x) { | |
| 1382 | if ($x =~ /^NaN[QS]?$/i) { | |
| 1383 | $re = $x; | |
| 1384 | } else { | |
| 1385 | if ($x =~ /^-?$Inf$/oi) { | |
| 1386 | $re = $x; | |
| 1387 | } else { | |
| 1388 | $re = defined $format ? sprintf($format, $x) : $x; | |
| 1389 | } | |
| 1390 | } | |
| 1391 | } else { | |
| 1392 | undef $re; | |
| 1393 | } | |
| 1394 | ||
| 1395 | if ($y) { | |
| 1396 | if ($y =~ /^(NaN[QS]?)$/i) { | |
| 1397 | $im = $y; | |
| 1398 | } else { | |
| 1399 | if ($y =~ /^-?$Inf$/oi) { | |
| 1400 | $im = $y; | |
| 1401 | } else { | |
| 1402 | $im = | |
| 1403 | defined $format ? | |
| 1404 | sprintf($format, $y) : | |
| 1405 | ($y == 1 ? "" : ($y == -1 ? "-" : $y)); | |
| 1406 | } | |
| 1407 | } | |
| 1408 | $im .= "i"; | |
| 1409 | } else { | |
| 1410 | undef $im; | |
| 1411 | } | |
| 1412 | ||
| 1413 | my $str = $re; | |
| 1414 | ||
| 1415 | if (defined $im) { | |
| 1416 | if ($y < 0) { | |
| 1417 | $str .= $im; | |
| 1418 | } elsif ($y > 0 || $im =~ /^NaN[QS]?i$/i) { | |
| 1419 | $str .= "+" if defined $re; | |
| 1420 | $str .= $im; | |
| 1421 | } | |
| 1422 | } elsif (!defined $re) { | |
| 1423 | $str = "0"; | |
| 1424 | } | |
| 1425 | ||
| 1426 | return $str; | |
| 1427 | } | |
| 1428 | ||
| 1429 | ||
| 1430 | # | |
| 1431 | # ->stringify_polar | |
| 1432 | # | |
| 1433 | # Stringify as a polar representation '[r,t]'. | |
| 1434 | # | |
| 1435 | sub stringify_polar { | |
| 1436 | my $z = shift; | |
| 1437 | my ($r, $t) = @{$z->polar}; | |
| 1438 | my $theta; | |
| 1439 | ||
| 1440 | my %format = $z->display_format; | |
| 1441 | my $format = $format{format}; | |
| 1442 | ||
| 1443 | if ($t =~ /^NaN[QS]?$/i || $t =~ /^-?$Inf$/oi) { | |
| 1444 | $theta = $t; | |
| 1445 | } elsif ($t == pi) { | |
| 1446 | $theta = "pi"; | |
| 1447 | } elsif ($r == 0 || $t == 0) { | |
| 1448 | $theta = defined $format ? sprintf($format, $t) : $t; | |
| 1449 | } | |
| 1450 | ||
| 1451 | return "[$r,$theta]" if defined $theta; | |
| 1452 | ||
| 1453 | # | |
| 1454 | # Try to identify pi/n and friends. | |
| 1455 | # | |
| 1456 | ||
| 1457 | $t -= int(CORE::abs($t) / pit2) * pit2; | |
| 1458 | ||
| 1459 | if ($format{polar_pretty_print} && $t) { | |
| 1460 | my ($a, $b); | |
| 1461 | for $a (2..9) { | |
| 1462 | $b = $t * $a / pi; | |
| 1463 | if ($b =~ /^-?\d+$/) { | |
| 1464 | $b = $b < 0 ? "-" : "" if CORE::abs($b) == 1; | |
| 1465 | $theta = "${b}pi/$a"; | |
| 1466 | last; | |
| 1467 | } | |
| 1468 | } | |
| 1469 | } | |
| 1470 | ||
| 1471 | if (defined $format) { | |
| 1472 | $r = sprintf($format, $r); | |
| 1473 | $theta = sprintf($format, $theta) unless defined $theta; | |
| 1474 | } else { | |
| 1475 | $theta = $t unless defined $theta; | |
| 1476 | } | |
| 1477 | ||
| 1478 | return "[$r,$theta]"; | |
| 1479 | } | |
| 1480 | ||
| 1481 | 1; | |
| 1482 | __END__ | |
| 1483 | ||
| 1484 | =pod | |
| 1485 | ||
| 1486 | =head1 NAME | |
| 1487 | ||
| 1488 | Math::Complex - complex numbers and associated mathematical functions | |
| 1489 | ||
| 1490 | =head1 SYNOPSIS | |
| 1491 | ||
| 1492 | use Math::Complex; | |
| 1493 | ||
| 1494 | $z = Math::Complex->make(5, 6); | |
| 1495 | $t = 4 - 3*i + $z; | |
| 1496 | $j = cplxe(1, 2*pi/3); | |
| 1497 | ||
| 1498 | =head1 DESCRIPTION | |
| 1499 | ||
| 1500 | This package lets you create and manipulate complex numbers. By default, | |
| 1501 | I<Perl> limits itself to real numbers, but an extra C<use> statement brings | |
| 1502 | full complex support, along with a full set of mathematical functions | |
| 1503 | typically associated with and/or extended to complex numbers. | |
| 1504 | ||
| 1505 | If you wonder what complex numbers are, they were invented to be able to solve | |
| 1506 | the following equation: | |
| 1507 | ||
| 1508 | x*x = -1 | |
| 1509 | ||
| 1510 | and by definition, the solution is noted I<i> (engineers use I<j> instead since | |
| 1511 | I<i> usually denotes an intensity, but the name does not matter). The number | |
| 1512 | I<i> is a pure I<imaginary> number. | |
| 1513 | ||
| 1514 | The arithmetics with pure imaginary numbers works just like you would expect | |
| 1515 | it with real numbers... you just have to remember that | |
| 1516 | ||
| 1517 | i*i = -1 | |
| 1518 | ||
| 1519 | so you have: | |
| 1520 | ||
| 1521 | 5i + 7i = i * (5 + 7) = 12i | |
| 1522 | 4i - 3i = i * (4 - 3) = i | |
| 1523 | 4i * 2i = -8 | |
| 1524 | 6i / 2i = 3 | |
| 1525 | 1 / i = -i | |
| 1526 | ||
| 1527 | Complex numbers are numbers that have both a real part and an imaginary | |
| 1528 | part, and are usually noted: | |
| 1529 | ||
| 1530 | a + bi | |
| 1531 | ||
| 1532 | where C<a> is the I<real> part and C<b> is the I<imaginary> part. The | |
| 1533 | arithmetic with complex numbers is straightforward. You have to | |
| 1534 | keep track of the real and the imaginary parts, but otherwise the | |
| 1535 | rules used for real numbers just apply: | |
| 1536 | ||
| 1537 | (4 + 3i) + (5 - 2i) = (4 + 5) + i(3 - 2) = 9 + i | |
| 1538 | (2 + i) * (4 - i) = 2*4 + 4i -2i -i*i = 8 + 2i + 1 = 9 + 2i | |
| 1539 | ||
| 1540 | A graphical representation of complex numbers is possible in a plane | |
| 1541 | (also called the I<complex plane>, but it's really a 2D plane). | |
| 1542 | The number | |
| 1543 | ||
| 1544 | z = a + bi | |
| 1545 | ||
| 1546 | is the point whose coordinates are (a, b). Actually, it would | |
| 1547 | be the vector originating from (0, 0) to (a, b). It follows that the addition | |
| 1548 | of two complex numbers is a vectorial addition. | |
| 1549 | ||
| 1550 | Since there is a bijection between a point in the 2D plane and a complex | |
| 1551 | number (i.e. the mapping is unique and reciprocal), a complex number | |
| 1552 | can also be uniquely identified with polar coordinates: | |
| 1553 | ||
| 1554 | [rho, theta] | |
| 1555 | ||
| 1556 | where C<rho> is the distance to the origin, and C<theta> the angle between | |
| 1557 | the vector and the I<x> axis. There is a notation for this using the | |
| 1558 | exponential form, which is: | |
| 1559 | ||
| 1560 | rho * exp(i * theta) | |
| 1561 | ||
| 1562 | where I<i> is the famous imaginary number introduced above. Conversion | |
| 1563 | between this form and the cartesian form C<a + bi> is immediate: | |
| 1564 | ||
| 1565 | a = rho * cos(theta) | |
| 1566 | b = rho * sin(theta) | |
| 1567 | ||
| 1568 | which is also expressed by this formula: | |
| 1569 | ||
| 1570 | z = rho * exp(i * theta) = rho * (cos theta + i * sin theta) | |
| 1571 | ||
| 1572 | In other words, it's the projection of the vector onto the I<x> and I<y> | |
| 1573 | axes. Mathematicians call I<rho> the I<norm> or I<modulus> and I<theta> | |
| 1574 | the I<argument> of the complex number. The I<norm> of C<z> will be | |
| 1575 | noted C<abs(z)>. | |
| 1576 | ||
| 1577 | The polar notation (also known as the trigonometric | |
| 1578 | representation) is much more handy for performing multiplications and | |
| 1579 | divisions of complex numbers, whilst the cartesian notation is better | |
| 1580 | suited for additions and subtractions. Real numbers are on the I<x> | |
| 1581 | axis, and therefore I<theta> is zero or I<pi>. | |
| 1582 | ||
| 1583 | All the common operations that can be performed on a real number have | |
| 1584 | been defined to work on complex numbers as well, and are merely | |
| 1585 | I<extensions> of the operations defined on real numbers. This means | |
| 1586 | they keep their natural meaning when there is no imaginary part, provided | |
| 1587 | the number is within their definition set. | |
| 1588 | ||
| 1589 | For instance, the C<sqrt> routine which computes the square root of | |
| 1590 | its argument is only defined for non-negative real numbers and yields a | |
| 1591 | non-negative real number (it is an application from B<R+> to B<R+>). | |
| 1592 | If we allow it to return a complex number, then it can be extended to | |
| 1593 | negative real numbers to become an application from B<R> to B<C> (the | |
| 1594 | set of complex numbers): | |
| 1595 | ||
| 1596 | sqrt(x) = x >= 0 ? sqrt(x) : sqrt(-x)*i | |
| 1597 | ||
| 1598 | It can also be extended to be an application from B<C> to B<C>, | |
| 1599 | whilst its restriction to B<R> behaves as defined above by using | |
| 1600 | the following definition: | |
| 1601 | ||
| 1602 | sqrt(z = [r,t]) = sqrt(r) * exp(i * t/2) | |
| 1603 | ||
| 1604 | Indeed, a negative real number can be noted C<[x,pi]> (the modulus | |
| 1605 | I<x> is always non-negative, so C<[x,pi]> is really C<-x>, a negative | |
| 1606 | number) and the above definition states that | |
| 1607 | ||
| 1608 | sqrt([x,pi]) = sqrt(x) * exp(i*pi/2) = [sqrt(x),pi/2] = sqrt(x)*i | |
| 1609 | ||
| 1610 | which is exactly what we had defined for negative real numbers above. | |
| 1611 | The C<sqrt> returns only one of the solutions: if you want the both, | |
| 1612 | use the C<root> function. | |
| 1613 | ||
| 1614 | All the common mathematical functions defined on real numbers that | |
| 1615 | are extended to complex numbers share that same property of working | |
| 1616 | I<as usual> when the imaginary part is zero (otherwise, it would not | |
| 1617 | be called an extension, would it?). | |
| 1618 | ||
| 1619 | A I<new> operation possible on a complex number that is | |
| 1620 | the identity for real numbers is called the I<conjugate>, and is noted | |
| 1621 | with a horizontal bar above the number, or C<~z> here. | |
| 1622 | ||
| 1623 | z = a + bi | |
| 1624 | ~z = a - bi | |
| 1625 | ||
| 1626 | Simple... Now look: | |
| 1627 | ||
| 1628 | z * ~z = (a + bi) * (a - bi) = a*a + b*b | |
| 1629 | ||
| 1630 | We saw that the norm of C<z> was noted C<abs(z)> and was defined as the | |
| 1631 | distance to the origin, also known as: | |
| 1632 | ||
| 1633 | rho = abs(z) = sqrt(a*a + b*b) | |
| 1634 | ||
| 1635 | so | |
| 1636 | ||
| 1637 | z * ~z = abs(z) ** 2 | |
| 1638 | ||
| 1639 | If z is a pure real number (i.e. C<b == 0>), then the above yields: | |
| 1640 | ||
| 1641 | a * a = abs(a) ** 2 | |
| 1642 | ||
| 1643 | which is true (C<abs> has the regular meaning for real number, i.e. stands | |
| 1644 | for the absolute value). This example explains why the norm of C<z> is | |
| 1645 | noted C<abs(z)>: it extends the C<abs> function to complex numbers, yet | |
| 1646 | is the regular C<abs> we know when the complex number actually has no | |
| 1647 | imaginary part... This justifies I<a posteriori> our use of the C<abs> | |
| 1648 | notation for the norm. | |
| 1649 | ||
| 1650 | =head1 OPERATIONS | |
| 1651 | ||
| 1652 | Given the following notations: | |
| 1653 | ||
| 1654 | z1 = a + bi = r1 * exp(i * t1) | |
| 1655 | z2 = c + di = r2 * exp(i * t2) | |
| 1656 | z = <any complex or real number> | |
| 1657 | ||
| 1658 | the following (overloaded) operations are supported on complex numbers: | |
| 1659 | ||
| 1660 | z1 + z2 = (a + c) + i(b + d) | |
| 1661 | z1 - z2 = (a - c) + i(b - d) | |
| 1662 | z1 * z2 = (r1 * r2) * exp(i * (t1 + t2)) | |
| 1663 | z1 / z2 = (r1 / r2) * exp(i * (t1 - t2)) | |
| 1664 | z1 ** z2 = exp(z2 * log z1) | |
| 1665 | ~z = a - bi | |
| 1666 | abs(z) = r1 = sqrt(a*a + b*b) | |
| 1667 | sqrt(z) = sqrt(r1) * exp(i * t/2) | |
| 1668 | exp(z) = exp(a) * exp(i * b) | |
| 1669 | log(z) = log(r1) + i*t | |
| 1670 | sin(z) = 1/2i (exp(i * z1) - exp(-i * z)) | |
| 1671 | cos(z) = 1/2 (exp(i * z1) + exp(-i * z)) | |
| 1672 | atan2(y, x) = atan(y / x) # Minding the right quadrant, note the order. | |
| 1673 | ||
| 1674 | The definition used for complex arguments of atan2() is | |
| 1675 | ||
| 1676 | -i log((x + iy)/sqrt(x*x+y*y)) | |
| 1677 | ||
| 1678 | The following extra operations are supported on both real and complex | |
| 1679 | numbers: | |
| 1680 | ||
| 1681 | Re(z) = a | |
| 1682 | Im(z) = b | |
| 1683 | arg(z) = t | |
| 1684 | abs(z) = r | |
| 1685 | ||
| 1686 | cbrt(z) = z ** (1/3) | |
| 1687 | log10(z) = log(z) / log(10) | |
| 1688 | logn(z, n) = log(z) / log(n) | |
| 1689 | ||
| 1690 | tan(z) = sin(z) / cos(z) | |
| 1691 | ||
| 1692 | csc(z) = 1 / sin(z) | |
| 1693 | sec(z) = 1 / cos(z) | |
| 1694 | cot(z) = 1 / tan(z) | |
| 1695 | ||
| 1696 | asin(z) = -i * log(i*z + sqrt(1-z*z)) | |
| 1697 | acos(z) = -i * log(z + i*sqrt(1-z*z)) | |
| 1698 | atan(z) = i/2 * log((i+z) / (i-z)) | |
| 1699 | ||
| 1700 | acsc(z) = asin(1 / z) | |
| 1701 | asec(z) = acos(1 / z) | |
| 1702 | acot(z) = atan(1 / z) = -i/2 * log((i+z) / (z-i)) | |
| 1703 | ||
| 1704 | sinh(z) = 1/2 (exp(z) - exp(-z)) | |
| 1705 | cosh(z) = 1/2 (exp(z) + exp(-z)) | |
| 1706 | tanh(z) = sinh(z) / cosh(z) = (exp(z) - exp(-z)) / (exp(z) + exp(-z)) | |
| 1707 | ||
| 1708 | csch(z) = 1 / sinh(z) | |
| 1709 | sech(z) = 1 / cosh(z) | |
| 1710 | coth(z) = 1 / tanh(z) | |
| 1711 | ||
| 1712 | asinh(z) = log(z + sqrt(z*z+1)) | |
| 1713 | acosh(z) = log(z + sqrt(z*z-1)) | |
| 1714 | atanh(z) = 1/2 * log((1+z) / (1-z)) | |
| 1715 | ||
| 1716 | acsch(z) = asinh(1 / z) | |
| 1717 | asech(z) = acosh(1 / z) | |
| 1718 | acoth(z) = atanh(1 / z) = 1/2 * log((1+z) / (z-1)) | |
| 1719 | ||
| 1720 | I<arg>, I<abs>, I<log>, I<csc>, I<cot>, I<acsc>, I<acot>, I<csch>, | |
| 1721 | I<coth>, I<acosech>, I<acotanh>, have aliases I<rho>, I<theta>, I<ln>, | |
| 1722 | I<cosec>, I<cotan>, I<acosec>, I<acotan>, I<cosech>, I<cotanh>, | |
| 1723 | I<acosech>, I<acotanh>, respectively. C<Re>, C<Im>, C<arg>, C<abs>, | |
| 1724 | C<rho>, and C<theta> can be used also as mutators. The C<cbrt> | |
| 1725 | returns only one of the solutions: if you want all three, use the | |
| 1726 | C<root> function. | |
| 1727 | ||
| 1728 | The I<root> function is available to compute all the I<n> | |
| 1729 | roots of some complex, where I<n> is a strictly positive integer. | |
| 1730 | There are exactly I<n> such roots, returned as a list. Getting the | |
| 1731 | number mathematicians call C<j> such that: | |
| 1732 | ||
| 1733 | 1 + j + j*j = 0; | |
| 1734 | ||
| 1735 | is a simple matter of writing: | |
| 1736 | ||
| 1737 | $j = ((root(1, 3))[1]; | |
| 1738 | ||
| 1739 | The I<k>th root for C<z = [r,t]> is given by: | |
| 1740 | ||
| 1741 | (root(z, n))[k] = r**(1/n) * exp(i * (t + 2*k*pi)/n) | |
| 1742 | ||
| 1743 | You can return the I<k>th root directly by C<root(z, n, k)>, | |
| 1744 | indexing starting from I<zero> and ending at I<n - 1>. | |
| 1745 | ||
| 1746 | The I<spaceship> comparison operator, E<lt>=E<gt>, is also defined. In | |
| 1747 | order to ensure its restriction to real numbers is conform to what you | |
| 1748 | would expect, the comparison is run on the real part of the complex | |
| 1749 | number first, and imaginary parts are compared only when the real | |
| 1750 | parts match. | |
| 1751 | ||
| 1752 | =head1 CREATION | |
| 1753 | ||
| 1754 | To create a complex number, use either: | |
| 1755 | ||
| 1756 | $z = Math::Complex->make(3, 4); | |
| 1757 | $z = cplx(3, 4); | |
| 1758 | ||
| 1759 | if you know the cartesian form of the number, or | |
| 1760 | ||
| 1761 | $z = 3 + 4*i; | |
| 1762 | ||
| 1763 | if you like. To create a number using the polar form, use either: | |
| 1764 | ||
| 1765 | $z = Math::Complex->emake(5, pi/3); | |
| 1766 | $x = cplxe(5, pi/3); | |
| 1767 | ||
| 1768 | instead. The first argument is the modulus, the second is the angle | |
| 1769 | (in radians, the full circle is 2*pi). (Mnemonic: C<e> is used as a | |
| 1770 | notation for complex numbers in the polar form). | |
| 1771 | ||
| 1772 | It is possible to write: | |
| 1773 | ||
| 1774 | $x = cplxe(-3, pi/4); | |
| 1775 | ||
| 1776 | but that will be silently converted into C<[3,-3pi/4]>, since the | |
| 1777 | modulus must be non-negative (it represents the distance to the origin | |
| 1778 | in the complex plane). | |
| 1779 | ||
| 1780 | It is also possible to have a complex number as either argument of the | |
| 1781 | C<make>, C<emake>, C<cplx>, and C<cplxe>: the appropriate component of | |
| 1782 | the argument will be used. | |
| 1783 | ||
| 1784 | $z1 = cplx(-2, 1); | |
| 1785 | $z2 = cplx($z1, 4); | |
| 1786 | ||
| 1787 | The C<new>, C<make>, C<emake>, C<cplx>, and C<cplxe> will also | |
| 1788 | understand a single (string) argument of the forms | |
| 1789 | ||
| 1790 | 2-3i | |
| 1791 | -3i | |
| 1792 | [2,3] | |
| 1793 | [2,-3pi/4] | |
| 1794 | [2] | |
| 1795 | ||
| 1796 | in which case the appropriate cartesian and exponential components | |
| 1797 | will be parsed from the string and used to create new complex numbers. | |
| 1798 | The imaginary component and the theta, respectively, will default to zero. | |
| 1799 | ||
| 1800 | The C<new>, C<make>, C<emake>, C<cplx>, and C<cplxe> will also | |
| 1801 | understand the case of no arguments: this means plain zero or (0, 0). | |
| 1802 | ||
| 1803 | =head1 DISPLAYING | |
| 1804 | ||
| 1805 | When printed, a complex number is usually shown under its cartesian | |
| 1806 | style I<a+bi>, but there are legitimate cases where the polar style | |
| 1807 | I<[r,t]> is more appropriate. The process of converting the complex | |
| 1808 | number into a string that can be displayed is known as I<stringification>. | |
| 1809 | ||
| 1810 | By calling the class method C<Math::Complex::display_format> and | |
| 1811 | supplying either C<"polar"> or C<"cartesian"> as an argument, you | |
| 1812 | override the default display style, which is C<"cartesian">. Not | |
| 1813 | supplying any argument returns the current settings. | |
| 1814 | ||
| 1815 | This default can be overridden on a per-number basis by calling the | |
| 1816 | C<display_format> method instead. As before, not supplying any argument | |
| 1817 | returns the current display style for this number. Otherwise whatever you | |
| 1818 | specify will be the new display style for I<this> particular number. | |
| 1819 | ||
| 1820 | For instance: | |
| 1821 | ||
| 1822 | use Math::Complex; | |
| 1823 | ||
| 1824 | Math::Complex::display_format('polar'); | |
| 1825 | $j = (root(1, 3))[1]; | |
| 1826 | print "j = $j\n"; # Prints "j = [1,2pi/3]" | |
| 1827 | $j->display_format('cartesian'); | |
| 1828 | print "j = $j\n"; # Prints "j = -0.5+0.866025403784439i" | |
| 1829 | ||
| 1830 | The polar style attempts to emphasize arguments like I<k*pi/n> | |
| 1831 | (where I<n> is a positive integer and I<k> an integer within [-9, +9]), | |
| 1832 | this is called I<polar pretty-printing>. | |
| 1833 | ||
| 1834 | For the reverse of stringifying, see the C<make> and C<emake>. | |
| 1835 | ||
| 1836 | =head2 CHANGED IN PERL 5.6 | |
| 1837 | ||
| 1838 | The C<display_format> class method and the corresponding | |
| 1839 | C<display_format> object method can now be called using | |
| 1840 | a parameter hash instead of just a one parameter. | |
| 1841 | ||
| 1842 | The old display format style, which can have values C<"cartesian"> or | |
| 1843 | C<"polar">, can be changed using the C<"style"> parameter. | |
| 1844 | ||
| 1845 | $j->display_format(style => "polar"); | |
| 1846 | ||
| 1847 | The one parameter calling convention also still works. | |
| 1848 | ||
| 1849 | $j->display_format("polar"); | |
| 1850 | ||
| 1851 | There are two new display parameters. | |
| 1852 | ||
| 1853 | The first one is C<"format">, which is a sprintf()-style format string | |
| 1854 | to be used for both numeric parts of the complex number(s). The is | |
| 1855 | somewhat system-dependent but most often it corresponds to C<"%.15g">. | |
| 1856 | You can revert to the default by setting the C<format> to C<undef>. | |
| 1857 | ||
| 1858 | # the $j from the above example | |
| 1859 | ||
| 1860 | $j->display_format('format' => '%.5f'); | |
| 1861 | print "j = $j\n"; # Prints "j = -0.50000+0.86603i" | |
| 1862 | $j->display_format('format' => undef); | |
| 1863 | print "j = $j\n"; # Prints "j = -0.5+0.86603i" | |
| 1864 | ||
| 1865 | Notice that this affects also the return values of the | |
| 1866 | C<display_format> methods: in list context the whole parameter hash | |
| 1867 | will be returned, as opposed to only the style parameter value. | |
| 1868 | This is a potential incompatibility with earlier versions if you | |
| 1869 | have been calling the C<display_format> method in list context. | |
| 1870 | ||
| 1871 | The second new display parameter is C<"polar_pretty_print">, which can | |
| 1872 | be set to true or false, the default being true. See the previous | |
| 1873 | section for what this means. | |
| 1874 | ||
| 1875 | =head1 USAGE | |
| 1876 | ||
| 1877 | Thanks to overloading, the handling of arithmetics with complex numbers | |
| 1878 | is simple and almost transparent. | |
| 1879 | ||
| 1880 | Here are some examples: | |
| 1881 | ||
| 1882 | use Math::Complex; | |
| 1883 | ||
| 1884 | $j = cplxe(1, 2*pi/3); # $j ** 3 == 1 | |
| 1885 | print "j = $j, j**3 = ", $j ** 3, "\n"; | |
| 1886 | print "1 + j + j**2 = ", 1 + $j + $j**2, "\n"; | |
| 1887 | ||
| 1888 | $z = -16 + 0*i; # Force it to be a complex | |
| 1889 | print "sqrt($z) = ", sqrt($z), "\n"; | |
| 1890 | ||
| 1891 | $k = exp(i * 2*pi/3); | |
| 1892 | print "$j - $k = ", $j - $k, "\n"; | |
| 1893 | ||
| 1894 | $z->Re(3); # Re, Im, arg, abs, | |
| 1895 | $j->arg(2); # (the last two aka rho, theta) | |
| 1896 | # can be used also as mutators. | |
| 1897 | ||
| 1898 | =head1 ERRORS DUE TO DIVISION BY ZERO OR LOGARITHM OF ZERO | |
| 1899 | ||
| 1900 | The division (/) and the following functions | |
| 1901 | ||
| 1902 | log ln log10 logn | |
| 1903 | tan sec csc cot | |
| 1904 | atan asec acsc acot | |
| 1905 | tanh sech csch coth | |
| 1906 | atanh asech acsch acoth | |
| 1907 | ||
| 1908 | cannot be computed for all arguments because that would mean dividing | |
| 1909 | by zero or taking logarithm of zero. These situations cause fatal | |
| 1910 | runtime errors looking like this | |
| 1911 | ||
| 1912 | cot(0): Division by zero. | |
| 1913 | (Because in the definition of cot(0), the divisor sin(0) is 0) | |
| 1914 | Died at ... | |
| 1915 | ||
| 1916 | or | |
| 1917 | ||
| 1918 | atanh(-1): Logarithm of zero. | |
| 1919 | Died at... | |
| 1920 | ||
| 1921 | For the C<csc>, C<cot>, C<asec>, C<acsc>, C<acot>, C<csch>, C<coth>, | |
| 1922 | C<asech>, C<acsch>, the argument cannot be C<0> (zero). For the | |
| 1923 | logarithmic functions and the C<atanh>, C<acoth>, the argument cannot | |
| 1924 | be C<1> (one). For the C<atanh>, C<acoth>, the argument cannot be | |
| 1925 | C<-1> (minus one). For the C<atan>, C<acot>, the argument cannot be | |
| 1926 | C<i> (the imaginary unit). For the C<atan>, C<acoth>, the argument | |
| 1927 | cannot be C<-i> (the negative imaginary unit). For the C<tan>, | |
| 1928 | C<sec>, C<tanh>, the argument cannot be I<pi/2 + k * pi>, where I<k> | |
| 1929 | is any integer. atan2(0, 0) is undefined, and if the complex arguments | |
| 1930 | are used for atan2(), a division by zero will happen if z1**2+z2**2 == 0. | |
| 1931 | ||
| 1932 | Note that because we are operating on approximations of real numbers, | |
| 1933 | these errors can happen when merely `too close' to the singularities | |
| 1934 | listed above. | |
| 1935 | ||
| 1936 | =head1 ERRORS DUE TO INDIGESTIBLE ARGUMENTS | |
| 1937 | ||
| 1938 | The C<make> and C<emake> accept both real and complex arguments. | |
| 1939 | When they cannot recognize the arguments they will die with error | |
| 1940 | messages like the following | |
| 1941 | ||
| 1942 | Math::Complex::make: Cannot take real part of ... | |
| 1943 | Math::Complex::make: Cannot take real part of ... | |
| 1944 | Math::Complex::emake: Cannot take rho of ... | |
| 1945 | Math::Complex::emake: Cannot take theta of ... | |
| 1946 | ||
| 1947 | =head1 BUGS | |
| 1948 | ||
| 1949 | Saying C<use Math::Complex;> exports many mathematical routines in the | |
| 1950 | caller environment and even overrides some (C<sqrt>, C<log>, C<atan2>). | |
| 1951 | This is construed as a feature by the Authors, actually... ;-) | |
| 1952 | ||
| 1953 | All routines expect to be given real or complex numbers. Don't attempt to | |
| 1954 | use BigFloat, since Perl has currently no rule to disambiguate a '+' | |
| 1955 | operation (for instance) between two overloaded entities. | |
| 1956 | ||
| 1957 | In Cray UNICOS there is some strange numerical instability that results | |
| 1958 | in root(), cos(), sin(), cosh(), sinh(), losing accuracy fast. Beware. | |
| 1959 | The bug may be in UNICOS math libs, in UNICOS C compiler, in Math::Complex. | |
| 1960 | Whatever it is, it does not manifest itself anywhere else where Perl runs. | |
| 1961 | ||
| 1962 | =head1 AUTHORS | |
| 1963 | ||
| 1964 | Daniel S. Lewart <F<d-lewart@uiuc.edu>> | |
| 1965 | ||
| 1966 | Original authors Raphael Manfredi <F<Raphael_Manfredi@pobox.com>> and | |
| 1967 | Jarkko Hietaniemi <F<jhi@iki.fi>> | |
| 1968 | ||
| 1969 | =cut | |
| 1970 | ||
| 1971 | 1; | |
| 1972 | ||
| 1973 | # eof |