Commit | Line | Data |
66730be0 |
1 | # |
2 | # Complex numbers and associated mathematical functions |
b42d0ec9 |
3 | # -- Raphael Manfredi Since Sep 1996 |
4 | # -- Jarkko Hietaniemi Since Mar 1997 |
5 | # -- Daniel S. Lewart Since Sep 1997 |
fb73857a |
6 | # |
a0d0e21e |
7 | |
8 | require Exporter; |
5aabfad6 |
9 | package Math::Complex; |
a0d0e21e |
10 | |
b42d0ec9 |
11 | use strict; |
fb73857a |
12 | |
b42d0ec9 |
13 | use vars qw($VERSION @ISA @EXPORT %EXPORT_TAGS); |
fb73857a |
14 | |
b42d0ec9 |
15 | my ( $i, $ip2, %logn ); |
0c721ce2 |
16 | |
2820d885 |
17 | $VERSION = sprintf("%s", q$Id: Complex.pm,v 1.26 1998/11/01 00:00:00 dsl Exp $ =~ /(\d+\.\d+)/); |
0c721ce2 |
18 | |
5aabfad6 |
19 | @ISA = qw(Exporter); |
20 | |
5aabfad6 |
21 | my @trig = qw( |
22 | pi |
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23 | tan |
5aabfad6 |
24 | csc cosec sec cot cotan |
25 | asin acos atan |
26 | acsc acosec asec acot acotan |
27 | sinh cosh tanh |
28 | csch cosech sech coth cotanh |
29 | asinh acosh atanh |
30 | acsch acosech asech acoth acotanh |
31 | ); |
32 | |
33 | @EXPORT = (qw( |
b42d0ec9 |
34 | i Re Im rho theta arg |
fb73857a |
35 | sqrt log ln |
5aabfad6 |
36 | log10 logn cbrt root |
37 | cplx cplxe |
38 | ), |
39 | @trig); |
40 | |
41 | %EXPORT_TAGS = ( |
42 | 'trig' => [@trig], |
66730be0 |
43 | ); |
a0d0e21e |
44 | |
a5f75d66 |
45 | use overload |
0c721ce2 |
46 | '+' => \&plus, |
47 | '-' => \&minus, |
48 | '*' => \&multiply, |
49 | '/' => \÷, |
66730be0 |
50 | '**' => \&power, |
51 | '<=>' => \&spaceship, |
52 | 'neg' => \&negate, |
0c721ce2 |
53 | '~' => \&conjugate, |
66730be0 |
54 | 'abs' => \&abs, |
55 | 'sqrt' => \&sqrt, |
56 | 'exp' => \&exp, |
57 | 'log' => \&log, |
58 | 'sin' => \&sin, |
59 | 'cos' => \&cos, |
0c721ce2 |
60 | 'tan' => \&tan, |
66730be0 |
61 | 'atan2' => \&atan2, |
62 | qw("" stringify); |
63 | |
64 | # |
b42d0ec9 |
65 | # Package "privates" |
66730be0 |
66 | # |
67 | |
b42d0ec9 |
68 | my $package = 'Math::Complex'; # Package name |
69 | my $display = 'cartesian'; # Default display format |
70 | my $eps = 1e-14; # Epsilon |
66730be0 |
71 | |
72 | # |
73 | # Object attributes (internal): |
74 | # cartesian [real, imaginary] -- cartesian form |
75 | # polar [rho, theta] -- polar form |
76 | # c_dirty cartesian form not up-to-date |
77 | # p_dirty polar form not up-to-date |
78 | # display display format (package's global when not set) |
79 | # |
80 | |
b42d0ec9 |
81 | # Die on bad *make() arguments. |
82 | |
83 | sub _cannot_make { |
84 | die "@{[(caller(1))[3]]}: Cannot take $_[0] of $_[1].\n"; |
85 | } |
86 | |
66730be0 |
87 | # |
88 | # ->make |
89 | # |
90 | # Create a new complex number (cartesian form) |
91 | # |
92 | sub make { |
93 | my $self = bless {}, shift; |
94 | my ($re, $im) = @_; |
b42d0ec9 |
95 | my $rre = ref $re; |
96 | if ( $rre ) { |
97 | if ( $rre eq ref $self ) { |
98 | $re = Re($re); |
99 | } else { |
100 | _cannot_make("real part", $rre); |
101 | } |
102 | } |
103 | my $rim = ref $im; |
104 | if ( $rim ) { |
105 | if ( $rim eq ref $self ) { |
106 | $im = Im($im); |
107 | } else { |
108 | _cannot_make("imaginary part", $rim); |
109 | } |
110 | } |
111 | $self->{'cartesian'} = [ $re, $im ]; |
66730be0 |
112 | $self->{c_dirty} = 0; |
113 | $self->{p_dirty} = 1; |
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114 | $self->display_format('cartesian'); |
66730be0 |
115 | return $self; |
116 | } |
117 | |
118 | # |
119 | # ->emake |
120 | # |
121 | # Create a new complex number (exponential form) |
122 | # |
123 | sub emake { |
124 | my $self = bless {}, shift; |
125 | my ($rho, $theta) = @_; |
b42d0ec9 |
126 | my $rrh = ref $rho; |
127 | if ( $rrh ) { |
128 | if ( $rrh eq ref $self ) { |
129 | $rho = rho($rho); |
130 | } else { |
131 | _cannot_make("rho", $rrh); |
132 | } |
133 | } |
134 | my $rth = ref $theta; |
135 | if ( $rth ) { |
136 | if ( $rth eq ref $self ) { |
137 | $theta = theta($theta); |
138 | } else { |
139 | _cannot_make("theta", $rth); |
140 | } |
141 | } |
fb73857a |
142 | if ($rho < 0) { |
143 | $rho = -$rho; |
144 | $theta = ($theta <= 0) ? $theta + pi() : $theta - pi(); |
145 | } |
146 | $self->{'polar'} = [$rho, $theta]; |
66730be0 |
147 | $self->{p_dirty} = 0; |
148 | $self->{c_dirty} = 1; |
b42d0ec9 |
149 | $self->display_format('polar'); |
66730be0 |
150 | return $self; |
151 | } |
152 | |
153 | sub new { &make } # For backward compatibility only. |
154 | |
155 | # |
156 | # cplx |
157 | # |
158 | # Creates a complex number from a (re, im) tuple. |
159 | # This avoids the burden of writing Math::Complex->make(re, im). |
160 | # |
161 | sub cplx { |
162 | my ($re, $im) = @_; |
0c721ce2 |
163 | return $package->make($re, defined $im ? $im : 0); |
66730be0 |
164 | } |
165 | |
166 | # |
167 | # cplxe |
168 | # |
169 | # Creates a complex number from a (rho, theta) tuple. |
170 | # This avoids the burden of writing Math::Complex->emake(rho, theta). |
171 | # |
172 | sub cplxe { |
173 | my ($rho, $theta) = @_; |
0c721ce2 |
174 | return $package->emake($rho, defined $theta ? $theta : 0); |
66730be0 |
175 | } |
176 | |
177 | # |
178 | # pi |
179 | # |
fb73857a |
180 | # The number defined as pi = 180 degrees |
66730be0 |
181 | # |
a8693bd3 |
182 | use constant pi => 4 * CORE::atan2(1, 1); |
5cd24f17 |
183 | |
184 | # |
fb73857a |
185 | # pit2 |
5cd24f17 |
186 | # |
fb73857a |
187 | # The full circle |
188 | # |
189 | use constant pit2 => 2 * pi; |
190 | |
5cd24f17 |
191 | # |
fb73857a |
192 | # pip2 |
193 | # |
194 | # The quarter circle |
195 | # |
196 | use constant pip2 => pi / 2; |
5cd24f17 |
197 | |
fb73857a |
198 | # |
d09ae4e6 |
199 | # deg1 |
200 | # |
201 | # One degree in radians, used in stringify_polar. |
202 | # |
203 | |
204 | use constant deg1 => pi / 180; |
205 | |
206 | # |
fb73857a |
207 | # uplog10 |
208 | # |
209 | # Used in log10(). |
210 | # |
a8693bd3 |
211 | use constant uplog10 => 1 / CORE::log(10); |
66730be0 |
212 | |
213 | # |
214 | # i |
215 | # |
216 | # The number defined as i*i = -1; |
217 | # |
218 | sub i () { |
5cd24f17 |
219 | return $i if ($i); |
220 | $i = bless {}; |
40da2db3 |
221 | $i->{'cartesian'} = [0, 1]; |
fb73857a |
222 | $i->{'polar'} = [1, pip2]; |
66730be0 |
223 | $i->{c_dirty} = 0; |
224 | $i->{p_dirty} = 0; |
225 | return $i; |
226 | } |
227 | |
228 | # |
229 | # Attribute access/set routines |
230 | # |
231 | |
0c721ce2 |
232 | sub cartesian {$_[0]->{c_dirty} ? |
233 | $_[0]->update_cartesian : $_[0]->{'cartesian'}} |
234 | sub polar {$_[0]->{p_dirty} ? |
235 | $_[0]->update_polar : $_[0]->{'polar'}} |
66730be0 |
236 | |
40da2db3 |
237 | sub set_cartesian { $_[0]->{p_dirty}++; $_[0]->{'cartesian'} = $_[1] } |
238 | sub set_polar { $_[0]->{c_dirty}++; $_[0]->{'polar'} = $_[1] } |
66730be0 |
239 | |
240 | # |
241 | # ->update_cartesian |
242 | # |
243 | # Recompute and return the cartesian form, given accurate polar form. |
244 | # |
245 | sub update_cartesian { |
246 | my $self = shift; |
40da2db3 |
247 | my ($r, $t) = @{$self->{'polar'}}; |
66730be0 |
248 | $self->{c_dirty} = 0; |
a8693bd3 |
249 | return $self->{'cartesian'} = [$r * CORE::cos($t), $r * CORE::sin($t)]; |
66730be0 |
250 | } |
251 | |
252 | # |
253 | # |
254 | # ->update_polar |
255 | # |
256 | # Recompute and return the polar form, given accurate cartesian form. |
257 | # |
258 | sub update_polar { |
259 | my $self = shift; |
40da2db3 |
260 | my ($x, $y) = @{$self->{'cartesian'}}; |
66730be0 |
261 | $self->{p_dirty} = 0; |
40da2db3 |
262 | return $self->{'polar'} = [0, 0] if $x == 0 && $y == 0; |
a8693bd3 |
263 | return $self->{'polar'} = [CORE::sqrt($x*$x + $y*$y), CORE::atan2($y, $x)]; |
66730be0 |
264 | } |
265 | |
266 | # |
267 | # (plus) |
268 | # |
269 | # Computes z1+z2. |
270 | # |
271 | sub plus { |
272 | my ($z1, $z2, $regular) = @_; |
273 | my ($re1, $im1) = @{$z1->cartesian}; |
0e505df1 |
274 | $z2 = cplx($z2) unless ref $z2; |
5cd24f17 |
275 | my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0); |
66730be0 |
276 | unless (defined $regular) { |
277 | $z1->set_cartesian([$re1 + $re2, $im1 + $im2]); |
278 | return $z1; |
279 | } |
280 | return (ref $z1)->make($re1 + $re2, $im1 + $im2); |
281 | } |
282 | |
283 | # |
284 | # (minus) |
285 | # |
286 | # Computes z1-z2. |
287 | # |
288 | sub minus { |
289 | my ($z1, $z2, $inverted) = @_; |
290 | my ($re1, $im1) = @{$z1->cartesian}; |
0e505df1 |
291 | $z2 = cplx($z2) unless ref $z2; |
292 | my ($re2, $im2) = @{$z2->cartesian}; |
66730be0 |
293 | unless (defined $inverted) { |
294 | $z1->set_cartesian([$re1 - $re2, $im1 - $im2]); |
295 | return $z1; |
296 | } |
297 | return $inverted ? |
298 | (ref $z1)->make($re2 - $re1, $im2 - $im1) : |
299 | (ref $z1)->make($re1 - $re2, $im1 - $im2); |
0e505df1 |
300 | |
66730be0 |
301 | } |
302 | |
303 | # |
304 | # (multiply) |
305 | # |
306 | # Computes z1*z2. |
307 | # |
308 | sub multiply { |
fb73857a |
309 | my ($z1, $z2, $regular) = @_; |
310 | if ($z1->{p_dirty} == 0 and ref $z2 and $z2->{p_dirty} == 0) { |
311 | # if both polar better use polar to avoid rounding errors |
312 | my ($r1, $t1) = @{$z1->polar}; |
313 | my ($r2, $t2) = @{$z2->polar}; |
314 | my $t = $t1 + $t2; |
315 | if ($t > pi()) { $t -= pit2 } |
316 | elsif ($t <= -pi()) { $t += pit2 } |
317 | unless (defined $regular) { |
318 | $z1->set_polar([$r1 * $r2, $t]); |
66730be0 |
319 | return $z1; |
fb73857a |
320 | } |
321 | return (ref $z1)->emake($r1 * $r2, $t); |
322 | } else { |
323 | my ($x1, $y1) = @{$z1->cartesian}; |
324 | if (ref $z2) { |
325 | my ($x2, $y2) = @{$z2->cartesian}; |
326 | return (ref $z1)->make($x1*$x2-$y1*$y2, $x1*$y2+$y1*$x2); |
327 | } else { |
328 | return (ref $z1)->make($x1*$z2, $y1*$z2); |
329 | } |
66730be0 |
330 | } |
66730be0 |
331 | } |
332 | |
333 | # |
0e505df1 |
334 | # _divbyzero |
0c721ce2 |
335 | # |
336 | # Die on division by zero. |
337 | # |
0e505df1 |
338 | sub _divbyzero { |
5cd24f17 |
339 | my $mess = "$_[0]: Division by zero.\n"; |
340 | |
341 | if (defined $_[1]) { |
342 | $mess .= "(Because in the definition of $_[0], the divisor "; |
343 | $mess .= "$_[1] " unless ($_[1] eq '0'); |
344 | $mess .= "is 0)\n"; |
345 | } |
346 | |
0c721ce2 |
347 | my @up = caller(1); |
fb73857a |
348 | |
5cd24f17 |
349 | $mess .= "Died at $up[1] line $up[2].\n"; |
350 | |
351 | die $mess; |
0c721ce2 |
352 | } |
353 | |
354 | # |
66730be0 |
355 | # (divide) |
356 | # |
357 | # Computes z1/z2. |
358 | # |
359 | sub divide { |
360 | my ($z1, $z2, $inverted) = @_; |
fb73857a |
361 | if ($z1->{p_dirty} == 0 and ref $z2 and $z2->{p_dirty} == 0) { |
362 | # if both polar better use polar to avoid rounding errors |
363 | my ($r1, $t1) = @{$z1->polar}; |
364 | my ($r2, $t2) = @{$z2->polar}; |
365 | my $t; |
366 | if ($inverted) { |
0e505df1 |
367 | _divbyzero "$z2/0" if ($r1 == 0); |
fb73857a |
368 | $t = $t2 - $t1; |
369 | if ($t > pi()) { $t -= pit2 } |
370 | elsif ($t <= -pi()) { $t += pit2 } |
371 | return (ref $z1)->emake($r2 / $r1, $t); |
372 | } else { |
0e505df1 |
373 | _divbyzero "$z1/0" if ($r2 == 0); |
fb73857a |
374 | $t = $t1 - $t2; |
375 | if ($t > pi()) { $t -= pit2 } |
376 | elsif ($t <= -pi()) { $t += pit2 } |
377 | return (ref $z1)->emake($r1 / $r2, $t); |
378 | } |
379 | } else { |
380 | my ($d, $x2, $y2); |
381 | if ($inverted) { |
382 | ($x2, $y2) = @{$z1->cartesian}; |
383 | $d = $x2*$x2 + $y2*$y2; |
384 | _divbyzero "$z2/0" if $d == 0; |
385 | return (ref $z1)->make(($x2*$z2)/$d, -($y2*$z2)/$d); |
386 | } else { |
387 | my ($x1, $y1) = @{$z1->cartesian}; |
388 | if (ref $z2) { |
389 | ($x2, $y2) = @{$z2->cartesian}; |
390 | $d = $x2*$x2 + $y2*$y2; |
391 | _divbyzero "$z1/0" if $d == 0; |
392 | my $u = ($x1*$x2 + $y1*$y2)/$d; |
393 | my $v = ($y1*$x2 - $x1*$y2)/$d; |
394 | return (ref $z1)->make($u, $v); |
395 | } else { |
396 | _divbyzero "$z1/0" if $z2 == 0; |
397 | return (ref $z1)->make($x1/$z2, $y1/$z2); |
398 | } |
399 | } |
0c721ce2 |
400 | } |
66730be0 |
401 | } |
402 | |
403 | # |
404 | # (power) |
405 | # |
406 | # Computes z1**z2 = exp(z2 * log z1)). |
407 | # |
408 | sub power { |
409 | my ($z1, $z2, $inverted) = @_; |
ace5de91 |
410 | if ($inverted) { |
2820d885 |
411 | return 1 if $z1 == 0 || $z2 == 1; |
412 | return 0 if $z2 == 0 && Re($z1) > 0; |
ace5de91 |
413 | } else { |
2820d885 |
414 | return 1 if $z2 == 0 || $z1 == 1; |
415 | return 0 if $z1 == 0 && Re($z2) > 0; |
ace5de91 |
416 | } |
2820d885 |
417 | my $w = $inverted ? CORE::exp($z1 * CORE::log($z2)) |
418 | : CORE::exp($z2 * CORE::log($z1)); |
d09ae4e6 |
419 | # If both arguments cartesian, return cartesian, else polar. |
420 | return $z1->{c_dirty} == 0 && |
421 | (not ref $z2 or $z2->{c_dirty} == 0) ? |
422 | cplx(@{$w->cartesian}) : $w; |
66730be0 |
423 | } |
424 | |
425 | # |
426 | # (spaceship) |
427 | # |
428 | # Computes z1 <=> z2. |
2820d885 |
429 | # Sorts on the real part first, then on the imaginary part. Thus 2-4i < 3+8i. |
66730be0 |
430 | # |
431 | sub spaceship { |
432 | my ($z1, $z2, $inverted) = @_; |
5cd24f17 |
433 | my ($re1, $im1) = ref $z1 ? @{$z1->cartesian} : ($z1, 0); |
434 | my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0); |
66730be0 |
435 | my $sgn = $inverted ? -1 : 1; |
436 | return $sgn * ($re1 <=> $re2) if $re1 != $re2; |
437 | return $sgn * ($im1 <=> $im2); |
438 | } |
439 | |
440 | # |
441 | # (negate) |
442 | # |
443 | # Computes -z. |
444 | # |
445 | sub negate { |
446 | my ($z) = @_; |
447 | if ($z->{c_dirty}) { |
448 | my ($r, $t) = @{$z->polar}; |
fb73857a |
449 | $t = ($t <= 0) ? $t + pi : $t - pi; |
450 | return (ref $z)->emake($r, $t); |
66730be0 |
451 | } |
452 | my ($re, $im) = @{$z->cartesian}; |
453 | return (ref $z)->make(-$re, -$im); |
454 | } |
455 | |
456 | # |
457 | # (conjugate) |
458 | # |
459 | # Compute complex's conjugate. |
460 | # |
461 | sub conjugate { |
462 | my ($z) = @_; |
463 | if ($z->{c_dirty}) { |
464 | my ($r, $t) = @{$z->polar}; |
465 | return (ref $z)->emake($r, -$t); |
466 | } |
467 | my ($re, $im) = @{$z->cartesian}; |
468 | return (ref $z)->make($re, -$im); |
469 | } |
470 | |
471 | # |
472 | # (abs) |
473 | # |
b42d0ec9 |
474 | # Compute or set complex's norm (rho). |
66730be0 |
475 | # |
476 | sub abs { |
b42d0ec9 |
477 | my ($z, $rho) = @_; |
478 | return $z unless ref $z; |
479 | if (defined $rho) { |
480 | $z->{'polar'} = [ $rho, ${$z->polar}[1] ]; |
481 | $z->{p_dirty} = 0; |
482 | $z->{c_dirty} = 1; |
483 | return $rho; |
484 | } else { |
485 | return ${$z->polar}[0]; |
486 | } |
487 | } |
488 | |
489 | sub _theta { |
490 | my $theta = $_[0]; |
491 | |
492 | if ($$theta > pi()) { $$theta -= pit2 } |
493 | elsif ($$theta <= -pi()) { $$theta += pit2 } |
66730be0 |
494 | } |
495 | |
496 | # |
497 | # arg |
498 | # |
b42d0ec9 |
499 | # Compute or set complex's argument (theta). |
66730be0 |
500 | # |
501 | sub arg { |
b42d0ec9 |
502 | my ($z, $theta) = @_; |
503 | return $z unless ref $z; |
504 | if (defined $theta) { |
505 | _theta(\$theta); |
506 | $z->{'polar'} = [ ${$z->polar}[0], $theta ]; |
507 | $z->{p_dirty} = 0; |
508 | $z->{c_dirty} = 1; |
509 | } else { |
510 | $theta = ${$z->polar}[1]; |
511 | _theta(\$theta); |
512 | } |
513 | return $theta; |
66730be0 |
514 | } |
515 | |
516 | # |
517 | # (sqrt) |
518 | # |
0c721ce2 |
519 | # Compute sqrt(z). |
66730be0 |
520 | # |
b42d0ec9 |
521 | # It is quite tempting to use wantarray here so that in list context |
522 | # sqrt() would return the two solutions. This, however, would |
523 | # break things like |
524 | # |
525 | # print "sqrt(z) = ", sqrt($z), "\n"; |
526 | # |
527 | # The two values would be printed side by side without no intervening |
528 | # whitespace, quite confusing. |
529 | # Therefore if you want the two solutions use the root(). |
530 | # |
66730be0 |
531 | sub sqrt { |
532 | my ($z) = @_; |
b42d0ec9 |
533 | my ($re, $im) = ref $z ? @{$z->cartesian} : ($z, 0); |
a8693bd3 |
534 | return $re < 0 ? cplx(0, CORE::sqrt(-$re)) : CORE::sqrt($re) if $im == 0; |
66730be0 |
535 | my ($r, $t) = @{$z->polar}; |
a8693bd3 |
536 | return (ref $z)->emake(CORE::sqrt($r), $t/2); |
66730be0 |
537 | } |
538 | |
539 | # |
540 | # cbrt |
541 | # |
0c721ce2 |
542 | # Compute cbrt(z) (cubic root). |
66730be0 |
543 | # |
b42d0ec9 |
544 | # Why are we not returning three values? The same answer as for sqrt(). |
545 | # |
66730be0 |
546 | sub cbrt { |
547 | my ($z) = @_; |
a8693bd3 |
548 | return $z < 0 ? -CORE::exp(CORE::log(-$z)/3) : ($z > 0 ? CORE::exp(CORE::log($z)/3): 0) |
fb73857a |
549 | unless ref $z; |
66730be0 |
550 | my ($r, $t) = @{$z->polar}; |
a8693bd3 |
551 | return (ref $z)->emake(CORE::exp(CORE::log($r)/3), $t/3); |
66730be0 |
552 | } |
553 | |
554 | # |
0e505df1 |
555 | # _rootbad |
556 | # |
557 | # Die on bad root. |
558 | # |
559 | sub _rootbad { |
560 | my $mess = "Root $_[0] not defined, root must be positive integer.\n"; |
561 | |
562 | my @up = caller(1); |
fb73857a |
563 | |
0e505df1 |
564 | $mess .= "Died at $up[1] line $up[2].\n"; |
565 | |
566 | die $mess; |
567 | } |
568 | |
569 | # |
66730be0 |
570 | # root |
571 | # |
572 | # Computes all nth root for z, returning an array whose size is n. |
573 | # `n' must be a positive integer. |
574 | # |
575 | # The roots are given by (for k = 0..n-1): |
576 | # |
577 | # z^(1/n) = r^(1/n) (cos ((t+2 k pi)/n) + i sin ((t+2 k pi)/n)) |
578 | # |
579 | sub root { |
580 | my ($z, $n) = @_; |
0e505df1 |
581 | _rootbad($n) if ($n < 1 or int($n) != $n); |
a8693bd3 |
582 | my ($r, $t) = ref $z ? @{$z->polar} : (CORE::abs($z), $z >= 0 ? 0 : pi); |
66730be0 |
583 | my @root; |
584 | my $k; |
fb73857a |
585 | my $theta_inc = pit2 / $n; |
66730be0 |
586 | my $rho = $r ** (1/$n); |
587 | my $theta; |
d09ae4e6 |
588 | my $cartesian = ref $z && $z->{c_dirty} == 0; |
66730be0 |
589 | for ($k = 0, $theta = $t / $n; $k < $n; $k++, $theta += $theta_inc) { |
d09ae4e6 |
590 | my $w = cplxe($rho, $theta); |
591 | # Yes, $cartesian is loop invariant. |
592 | push @root, $cartesian ? cplx(@{$w->cartesian}) : $w; |
a0d0e21e |
593 | } |
66730be0 |
594 | return @root; |
a0d0e21e |
595 | } |
596 | |
66730be0 |
597 | # |
598 | # Re |
599 | # |
b42d0ec9 |
600 | # Return or set Re(z). |
66730be0 |
601 | # |
a0d0e21e |
602 | sub Re { |
b42d0ec9 |
603 | my ($z, $Re) = @_; |
66730be0 |
604 | return $z unless ref $z; |
b42d0ec9 |
605 | if (defined $Re) { |
606 | $z->{'cartesian'} = [ $Re, ${$z->cartesian}[1] ]; |
607 | $z->{c_dirty} = 0; |
608 | $z->{p_dirty} = 1; |
609 | } else { |
610 | return ${$z->cartesian}[0]; |
611 | } |
a0d0e21e |
612 | } |
613 | |
66730be0 |
614 | # |
615 | # Im |
616 | # |
b42d0ec9 |
617 | # Return or set Im(z). |
66730be0 |
618 | # |
a0d0e21e |
619 | sub Im { |
b42d0ec9 |
620 | my ($z, $Im) = @_; |
621 | return $z unless ref $z; |
622 | if (defined $Im) { |
623 | $z->{'cartesian'} = [ ${$z->cartesian}[0], $Im ]; |
624 | $z->{c_dirty} = 0; |
625 | $z->{p_dirty} = 1; |
626 | } else { |
627 | return ${$z->cartesian}[1]; |
628 | } |
629 | } |
630 | |
631 | # |
632 | # rho |
633 | # |
634 | # Return or set rho(w). |
635 | # |
636 | sub rho { |
637 | Math::Complex::abs(@_); |
638 | } |
639 | |
640 | # |
641 | # theta |
642 | # |
643 | # Return or set theta(w). |
644 | # |
645 | sub theta { |
646 | Math::Complex::arg(@_); |
a0d0e21e |
647 | } |
648 | |
66730be0 |
649 | # |
650 | # (exp) |
651 | # |
652 | # Computes exp(z). |
653 | # |
654 | sub exp { |
655 | my ($z) = @_; |
656 | my ($x, $y) = @{$z->cartesian}; |
a8693bd3 |
657 | return (ref $z)->emake(CORE::exp($x), $y); |
66730be0 |
658 | } |
659 | |
660 | # |
8c03c583 |
661 | # _logofzero |
662 | # |
fb73857a |
663 | # Die on logarithm of zero. |
8c03c583 |
664 | # |
665 | sub _logofzero { |
666 | my $mess = "$_[0]: Logarithm of zero.\n"; |
667 | |
668 | if (defined $_[1]) { |
669 | $mess .= "(Because in the definition of $_[0], the argument "; |
670 | $mess .= "$_[1] " unless ($_[1] eq '0'); |
671 | $mess .= "is 0)\n"; |
672 | } |
673 | |
674 | my @up = caller(1); |
fb73857a |
675 | |
8c03c583 |
676 | $mess .= "Died at $up[1] line $up[2].\n"; |
677 | |
678 | die $mess; |
679 | } |
680 | |
681 | # |
66730be0 |
682 | # (log) |
683 | # |
684 | # Compute log(z). |
685 | # |
686 | sub log { |
687 | my ($z) = @_; |
fb73857a |
688 | unless (ref $z) { |
689 | _logofzero("log") if $z == 0; |
a8693bd3 |
690 | return $z > 0 ? CORE::log($z) : cplx(CORE::log(-$z), pi); |
fb73857a |
691 | } |
5cd24f17 |
692 | my ($r, $t) = @{$z->polar}; |
fb73857a |
693 | _logofzero("log") if $r == 0; |
694 | if ($t > pi()) { $t -= pit2 } |
695 | elsif ($t <= -pi()) { $t += pit2 } |
a8693bd3 |
696 | return (ref $z)->make(CORE::log($r), $t); |
66730be0 |
697 | } |
698 | |
699 | # |
0c721ce2 |
700 | # ln |
701 | # |
702 | # Alias for log(). |
703 | # |
704 | sub ln { Math::Complex::log(@_) } |
705 | |
706 | # |
66730be0 |
707 | # log10 |
708 | # |
709 | # Compute log10(z). |
710 | # |
5cd24f17 |
711 | |
66730be0 |
712 | sub log10 { |
fb73857a |
713 | return Math::Complex::log($_[0]) * uplog10; |
66730be0 |
714 | } |
715 | |
716 | # |
717 | # logn |
718 | # |
719 | # Compute logn(z,n) = log(z) / log(n) |
720 | # |
721 | sub logn { |
722 | my ($z, $n) = @_; |
0c721ce2 |
723 | $z = cplx($z, 0) unless ref $z; |
66730be0 |
724 | my $logn = $logn{$n}; |
a8693bd3 |
725 | $logn = $logn{$n} = CORE::log($n) unless defined $logn; # Cache log(n) |
726 | return CORE::log($z) / $logn; |
66730be0 |
727 | } |
728 | |
729 | # |
730 | # (cos) |
731 | # |
732 | # Compute cos(z) = (exp(iz) + exp(-iz))/2. |
733 | # |
734 | sub cos { |
735 | my ($z) = @_; |
736 | my ($x, $y) = @{$z->cartesian}; |
a8693bd3 |
737 | my $ey = CORE::exp($y); |
66730be0 |
738 | my $ey_1 = 1 / $ey; |
a8693bd3 |
739 | return (ref $z)->make(CORE::cos($x) * ($ey + $ey_1)/2, |
740 | CORE::sin($x) * ($ey_1 - $ey)/2); |
66730be0 |
741 | } |
742 | |
743 | # |
744 | # (sin) |
745 | # |
746 | # Compute sin(z) = (exp(iz) - exp(-iz))/2. |
747 | # |
748 | sub sin { |
749 | my ($z) = @_; |
750 | my ($x, $y) = @{$z->cartesian}; |
a8693bd3 |
751 | my $ey = CORE::exp($y); |
66730be0 |
752 | my $ey_1 = 1 / $ey; |
a8693bd3 |
753 | return (ref $z)->make(CORE::sin($x) * ($ey + $ey_1)/2, |
754 | CORE::cos($x) * ($ey - $ey_1)/2); |
66730be0 |
755 | } |
756 | |
757 | # |
758 | # tan |
759 | # |
760 | # Compute tan(z) = sin(z) / cos(z). |
761 | # |
762 | sub tan { |
763 | my ($z) = @_; |
a8693bd3 |
764 | my $cz = CORE::cos($z); |
765 | _divbyzero "tan($z)", "cos($z)" if (CORE::abs($cz) < $eps); |
766 | return CORE::sin($z) / $cz; |
66730be0 |
767 | } |
768 | |
769 | # |
0c721ce2 |
770 | # sec |
771 | # |
772 | # Computes the secant sec(z) = 1 / cos(z). |
773 | # |
774 | sub sec { |
775 | my ($z) = @_; |
a8693bd3 |
776 | my $cz = CORE::cos($z); |
0e505df1 |
777 | _divbyzero "sec($z)", "cos($z)" if ($cz == 0); |
0c721ce2 |
778 | return 1 / $cz; |
779 | } |
780 | |
781 | # |
782 | # csc |
783 | # |
784 | # Computes the cosecant csc(z) = 1 / sin(z). |
785 | # |
786 | sub csc { |
787 | my ($z) = @_; |
a8693bd3 |
788 | my $sz = CORE::sin($z); |
0e505df1 |
789 | _divbyzero "csc($z)", "sin($z)" if ($sz == 0); |
0c721ce2 |
790 | return 1 / $sz; |
791 | } |
792 | |
66730be0 |
793 | # |
0c721ce2 |
794 | # cosec |
66730be0 |
795 | # |
0c721ce2 |
796 | # Alias for csc(). |
797 | # |
798 | sub cosec { Math::Complex::csc(@_) } |
799 | |
800 | # |
801 | # cot |
802 | # |
fb73857a |
803 | # Computes cot(z) = cos(z) / sin(z). |
0c721ce2 |
804 | # |
805 | sub cot { |
66730be0 |
806 | my ($z) = @_; |
a8693bd3 |
807 | my $sz = CORE::sin($z); |
0e505df1 |
808 | _divbyzero "cot($z)", "sin($z)" if ($sz == 0); |
a8693bd3 |
809 | return CORE::cos($z) / $sz; |
66730be0 |
810 | } |
811 | |
812 | # |
0c721ce2 |
813 | # cotan |
814 | # |
815 | # Alias for cot(). |
816 | # |
817 | sub cotan { Math::Complex::cot(@_) } |
818 | |
819 | # |
66730be0 |
820 | # acos |
821 | # |
822 | # Computes the arc cosine acos(z) = -i log(z + sqrt(z*z-1)). |
823 | # |
824 | sub acos { |
fb73857a |
825 | my $z = $_[0]; |
a8693bd3 |
826 | return CORE::atan2(CORE::sqrt(1-$z*$z), $z) if (! ref $z) && CORE::abs($z) <= 1; |
fb73857a |
827 | my ($x, $y) = ref $z ? @{$z->cartesian} : ($z, 0); |
a8693bd3 |
828 | my $t1 = CORE::sqrt(($x+1)*($x+1) + $y*$y); |
829 | my $t2 = CORE::sqrt(($x-1)*($x-1) + $y*$y); |
fb73857a |
830 | my $alpha = ($t1 + $t2)/2; |
831 | my $beta = ($t1 - $t2)/2; |
832 | $alpha = 1 if $alpha < 1; |
833 | if ($beta > 1) { $beta = 1 } |
834 | elsif ($beta < -1) { $beta = -1 } |
a8693bd3 |
835 | my $u = CORE::atan2(CORE::sqrt(1-$beta*$beta), $beta); |
836 | my $v = CORE::log($alpha + CORE::sqrt($alpha*$alpha-1)); |
fb73857a |
837 | $v = -$v if $y > 0 || ($y == 0 && $x < -1); |
838 | return $package->make($u, $v); |
66730be0 |
839 | } |
840 | |
841 | # |
842 | # asin |
843 | # |
844 | # Computes the arc sine asin(z) = -i log(iz + sqrt(1-z*z)). |
845 | # |
846 | sub asin { |
fb73857a |
847 | my $z = $_[0]; |
a8693bd3 |
848 | return CORE::atan2($z, CORE::sqrt(1-$z*$z)) if (! ref $z) && CORE::abs($z) <= 1; |
fb73857a |
849 | my ($x, $y) = ref $z ? @{$z->cartesian} : ($z, 0); |
a8693bd3 |
850 | my $t1 = CORE::sqrt(($x+1)*($x+1) + $y*$y); |
851 | my $t2 = CORE::sqrt(($x-1)*($x-1) + $y*$y); |
fb73857a |
852 | my $alpha = ($t1 + $t2)/2; |
853 | my $beta = ($t1 - $t2)/2; |
854 | $alpha = 1 if $alpha < 1; |
855 | if ($beta > 1) { $beta = 1 } |
856 | elsif ($beta < -1) { $beta = -1 } |
a8693bd3 |
857 | my $u = CORE::atan2($beta, CORE::sqrt(1-$beta*$beta)); |
858 | my $v = -CORE::log($alpha + CORE::sqrt($alpha*$alpha-1)); |
fb73857a |
859 | $v = -$v if $y > 0 || ($y == 0 && $x < -1); |
860 | return $package->make($u, $v); |
66730be0 |
861 | } |
862 | |
863 | # |
864 | # atan |
865 | # |
0c721ce2 |
866 | # Computes the arc tangent atan(z) = i/2 log((i+z) / (i-z)). |
66730be0 |
867 | # |
868 | sub atan { |
869 | my ($z) = @_; |
a8693bd3 |
870 | return CORE::atan2($z, 1) unless ref $z; |
8c03c583 |
871 | _divbyzero "atan(i)" if ( $z == i); |
872 | _divbyzero "atan(-i)" if (-$z == i); |
a8693bd3 |
873 | my $log = CORE::log((i + $z) / (i - $z)); |
fb73857a |
874 | $ip2 = 0.5 * i unless defined $ip2; |
875 | return $ip2 * $log; |
a0d0e21e |
876 | } |
877 | |
66730be0 |
878 | # |
0c721ce2 |
879 | # asec |
880 | # |
881 | # Computes the arc secant asec(z) = acos(1 / z). |
882 | # |
883 | sub asec { |
884 | my ($z) = @_; |
0e505df1 |
885 | _divbyzero "asec($z)", $z if ($z == 0); |
fb73857a |
886 | return acos(1 / $z); |
0c721ce2 |
887 | } |
888 | |
889 | # |
5cd24f17 |
890 | # acsc |
0c721ce2 |
891 | # |
8c03c583 |
892 | # Computes the arc cosecant acsc(z) = asin(1 / z). |
0c721ce2 |
893 | # |
5cd24f17 |
894 | sub acsc { |
0c721ce2 |
895 | my ($z) = @_; |
0e505df1 |
896 | _divbyzero "acsc($z)", $z if ($z == 0); |
fb73857a |
897 | return asin(1 / $z); |
0c721ce2 |
898 | } |
899 | |
900 | # |
5cd24f17 |
901 | # acosec |
66730be0 |
902 | # |
5cd24f17 |
903 | # Alias for acsc(). |
0c721ce2 |
904 | # |
5cd24f17 |
905 | sub acosec { Math::Complex::acsc(@_) } |
0c721ce2 |
906 | |
66730be0 |
907 | # |
0c721ce2 |
908 | # acot |
909 | # |
8c03c583 |
910 | # Computes the arc cotangent acot(z) = atan(1 / z) |
0c721ce2 |
911 | # |
912 | sub acot { |
66730be0 |
913 | my ($z) = @_; |
a8693bd3 |
914 | _divbyzero "acot(0)" if (CORE::abs($z) < $eps); |
915 | return ($z >= 0) ? CORE::atan2(1, $z) : CORE::atan2(-1, -$z) unless ref $z; |
916 | _divbyzero "acot(i)" if (CORE::abs($z - i) < $eps); |
917 | _logofzero "acot(-i)" if (CORE::abs($z + i) < $eps); |
8c03c583 |
918 | return atan(1 / $z); |
66730be0 |
919 | } |
920 | |
921 | # |
0c721ce2 |
922 | # acotan |
923 | # |
924 | # Alias for acot(). |
925 | # |
926 | sub acotan { Math::Complex::acot(@_) } |
927 | |
928 | # |
66730be0 |
929 | # cosh |
930 | # |
931 | # Computes the hyperbolic cosine cosh(z) = (exp(z) + exp(-z))/2. |
932 | # |
933 | sub cosh { |
934 | my ($z) = @_; |
fb73857a |
935 | my $ex; |
0e505df1 |
936 | unless (ref $z) { |
a8693bd3 |
937 | $ex = CORE::exp($z); |
fb73857a |
938 | return ($ex + 1/$ex)/2; |
0e505df1 |
939 | } |
940 | my ($x, $y) = @{$z->cartesian}; |
a8693bd3 |
941 | $ex = CORE::exp($x); |
66730be0 |
942 | my $ex_1 = 1 / $ex; |
a8693bd3 |
943 | return (ref $z)->make(CORE::cos($y) * ($ex + $ex_1)/2, |
944 | CORE::sin($y) * ($ex - $ex_1)/2); |
66730be0 |
945 | } |
946 | |
947 | # |
948 | # sinh |
949 | # |
950 | # Computes the hyperbolic sine sinh(z) = (exp(z) - exp(-z))/2. |
951 | # |
952 | sub sinh { |
953 | my ($z) = @_; |
fb73857a |
954 | my $ex; |
0e505df1 |
955 | unless (ref $z) { |
a8693bd3 |
956 | $ex = CORE::exp($z); |
fb73857a |
957 | return ($ex - 1/$ex)/2; |
0e505df1 |
958 | } |
959 | my ($x, $y) = @{$z->cartesian}; |
a8693bd3 |
960 | $ex = CORE::exp($x); |
66730be0 |
961 | my $ex_1 = 1 / $ex; |
a8693bd3 |
962 | return (ref $z)->make(CORE::cos($y) * ($ex - $ex_1)/2, |
963 | CORE::sin($y) * ($ex + $ex_1)/2); |
66730be0 |
964 | } |
965 | |
966 | # |
967 | # tanh |
968 | # |
969 | # Computes the hyperbolic tangent tanh(z) = sinh(z) / cosh(z). |
970 | # |
971 | sub tanh { |
972 | my ($z) = @_; |
0c721ce2 |
973 | my $cz = cosh($z); |
0e505df1 |
974 | _divbyzero "tanh($z)", "cosh($z)" if ($cz == 0); |
0c721ce2 |
975 | return sinh($z) / $cz; |
66730be0 |
976 | } |
977 | |
978 | # |
0c721ce2 |
979 | # sech |
980 | # |
981 | # Computes the hyperbolic secant sech(z) = 1 / cosh(z). |
982 | # |
983 | sub sech { |
984 | my ($z) = @_; |
985 | my $cz = cosh($z); |
0e505df1 |
986 | _divbyzero "sech($z)", "cosh($z)" if ($cz == 0); |
0c721ce2 |
987 | return 1 / $cz; |
988 | } |
989 | |
990 | # |
991 | # csch |
992 | # |
993 | # Computes the hyperbolic cosecant csch(z) = 1 / sinh(z). |
66730be0 |
994 | # |
0c721ce2 |
995 | sub csch { |
996 | my ($z) = @_; |
997 | my $sz = sinh($z); |
0e505df1 |
998 | _divbyzero "csch($z)", "sinh($z)" if ($sz == 0); |
0c721ce2 |
999 | return 1 / $sz; |
1000 | } |
1001 | |
1002 | # |
1003 | # cosech |
1004 | # |
1005 | # Alias for csch(). |
1006 | # |
1007 | sub cosech { Math::Complex::csch(@_) } |
1008 | |
66730be0 |
1009 | # |
0c721ce2 |
1010 | # coth |
1011 | # |
1012 | # Computes the hyperbolic cotangent coth(z) = cosh(z) / sinh(z). |
1013 | # |
1014 | sub coth { |
66730be0 |
1015 | my ($z) = @_; |
0c721ce2 |
1016 | my $sz = sinh($z); |
0e505df1 |
1017 | _divbyzero "coth($z)", "sinh($z)" if ($sz == 0); |
0c721ce2 |
1018 | return cosh($z) / $sz; |
66730be0 |
1019 | } |
1020 | |
1021 | # |
0c721ce2 |
1022 | # cotanh |
1023 | # |
1024 | # Alias for coth(). |
1025 | # |
1026 | sub cotanh { Math::Complex::coth(@_) } |
1027 | |
1028 | # |
66730be0 |
1029 | # acosh |
1030 | # |
fb73857a |
1031 | # Computes the arc hyperbolic cosine acosh(z) = log(z + sqrt(z*z-1)). |
66730be0 |
1032 | # |
1033 | sub acosh { |
1034 | my ($z) = @_; |
fb73857a |
1035 | unless (ref $z) { |
a8693bd3 |
1036 | return CORE::log($z + CORE::sqrt($z*$z-1)) if $z >= 1; |
fb73857a |
1037 | $z = cplx($z, 0); |
1038 | } |
8c03c583 |
1039 | my ($re, $im) = @{$z->cartesian}; |
fb73857a |
1040 | if ($im == 0) { |
a8693bd3 |
1041 | return cplx(CORE::log($re + CORE::sqrt($re*$re - 1)), 0) if $re >= 1; |
1042 | return cplx(0, CORE::atan2(CORE::sqrt(1-$re*$re), $re)) if CORE::abs($re) <= 1; |
fb73857a |
1043 | } |
a8693bd3 |
1044 | return CORE::log($z + CORE::sqrt($z*$z - 1)); |
66730be0 |
1045 | } |
1046 | |
1047 | # |
1048 | # asinh |
1049 | # |
1050 | # Computes the arc hyperbolic sine asinh(z) = log(z + sqrt(z*z-1)) |
1051 | # |
1052 | sub asinh { |
1053 | my ($z) = @_; |
a8693bd3 |
1054 | return CORE::log($z + CORE::sqrt($z*$z + 1)); |
66730be0 |
1055 | } |
1056 | |
1057 | # |
1058 | # atanh |
1059 | # |
1060 | # Computes the arc hyperbolic tangent atanh(z) = 1/2 log((1+z) / (1-z)). |
1061 | # |
1062 | sub atanh { |
1063 | my ($z) = @_; |
fb73857a |
1064 | unless (ref $z) { |
a8693bd3 |
1065 | return CORE::log((1 + $z)/(1 - $z))/2 if CORE::abs($z) < 1; |
fb73857a |
1066 | $z = cplx($z, 0); |
1067 | } |
8c03c583 |
1068 | _divbyzero 'atanh(1)', "1 - $z" if ($z == 1); |
1069 | _logofzero 'atanh(-1)' if ($z == -1); |
a8693bd3 |
1070 | return 0.5 * CORE::log((1 + $z) / (1 - $z)); |
66730be0 |
1071 | } |
1072 | |
1073 | # |
0c721ce2 |
1074 | # asech |
1075 | # |
1076 | # Computes the hyperbolic arc secant asech(z) = acosh(1 / z). |
1077 | # |
1078 | sub asech { |
1079 | my ($z) = @_; |
0e505df1 |
1080 | _divbyzero 'asech(0)', $z if ($z == 0); |
0c721ce2 |
1081 | return acosh(1 / $z); |
1082 | } |
1083 | |
1084 | # |
1085 | # acsch |
66730be0 |
1086 | # |
0c721ce2 |
1087 | # Computes the hyperbolic arc cosecant acsch(z) = asinh(1 / z). |
66730be0 |
1088 | # |
0c721ce2 |
1089 | sub acsch { |
66730be0 |
1090 | my ($z) = @_; |
0e505df1 |
1091 | _divbyzero 'acsch(0)', $z if ($z == 0); |
0c721ce2 |
1092 | return asinh(1 / $z); |
1093 | } |
1094 | |
1095 | # |
1096 | # acosech |
1097 | # |
1098 | # Alias for acosh(). |
1099 | # |
1100 | sub acosech { Math::Complex::acsch(@_) } |
1101 | |
1102 | # |
1103 | # acoth |
1104 | # |
1105 | # Computes the arc hyperbolic cotangent acoth(z) = 1/2 log((1+z) / (z-1)). |
1106 | # |
1107 | sub acoth { |
1108 | my ($z) = @_; |
a8693bd3 |
1109 | _divbyzero 'acoth(0)' if (CORE::abs($z) < $eps); |
fb73857a |
1110 | unless (ref $z) { |
a8693bd3 |
1111 | return CORE::log(($z + 1)/($z - 1))/2 if CORE::abs($z) > 1; |
fb73857a |
1112 | $z = cplx($z, 0); |
1113 | } |
a8693bd3 |
1114 | _divbyzero 'acoth(1)', "$z - 1" if (CORE::abs($z - 1) < $eps); |
1115 | _logofzero 'acoth(-1)', "1 / $z" if (CORE::abs($z + 1) < $eps); |
1116 | return CORE::log((1 + $z) / ($z - 1)) / 2; |
66730be0 |
1117 | } |
1118 | |
1119 | # |
0c721ce2 |
1120 | # acotanh |
1121 | # |
1122 | # Alias for acot(). |
1123 | # |
1124 | sub acotanh { Math::Complex::acoth(@_) } |
1125 | |
1126 | # |
66730be0 |
1127 | # (atan2) |
1128 | # |
1129 | # Compute atan(z1/z2). |
1130 | # |
1131 | sub atan2 { |
1132 | my ($z1, $z2, $inverted) = @_; |
fb73857a |
1133 | my ($re1, $im1, $re2, $im2); |
1134 | if ($inverted) { |
1135 | ($re1, $im1) = ref $z2 ? @{$z2->cartesian} : ($z2, 0); |
1136 | ($re2, $im2) = @{$z1->cartesian}; |
66730be0 |
1137 | } else { |
fb73857a |
1138 | ($re1, $im1) = @{$z1->cartesian}; |
1139 | ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0); |
1140 | } |
1141 | if ($im2 == 0) { |
a8693bd3 |
1142 | return cplx(CORE::atan2($re1, $re2), 0) if $im1 == 0; |
fb73857a |
1143 | return cplx(($im1<=>0) * pip2, 0) if $re2 == 0; |
66730be0 |
1144 | } |
fb73857a |
1145 | my $w = atan($z1/$z2); |
1146 | my ($u, $v) = ref $w ? @{$w->cartesian} : ($w, 0); |
1147 | $u += pi if $re2 < 0; |
1148 | $u -= pit2 if $u > pi; |
1149 | return cplx($u, $v); |
66730be0 |
1150 | } |
1151 | |
1152 | # |
1153 | # display_format |
1154 | # ->display_format |
1155 | # |
1156 | # Set (fetch if no argument) display format for all complex numbers that |
fb73857a |
1157 | # don't happen to have overridden it via ->display_format |
66730be0 |
1158 | # |
1159 | # When called as a method, this actually sets the display format for |
1160 | # the current object. |
1161 | # |
1162 | # Valid object formats are 'c' and 'p' for cartesian and polar. The first |
1163 | # letter is used actually, so the type can be fully spelled out for clarity. |
1164 | # |
1165 | sub display_format { |
1166 | my $self = shift; |
1167 | my $format = undef; |
1168 | |
1169 | if (ref $self) { # Called as a method |
1170 | $format = shift; |
0c721ce2 |
1171 | } else { # Regular procedure call |
66730be0 |
1172 | $format = $self; |
1173 | undef $self; |
1174 | } |
1175 | |
1176 | if (defined $self) { |
1177 | return defined $self->{display} ? $self->{display} : $display |
1178 | unless defined $format; |
1179 | return $self->{display} = $format; |
1180 | } |
1181 | |
1182 | return $display unless defined $format; |
1183 | return $display = $format; |
1184 | } |
1185 | |
1186 | # |
1187 | # (stringify) |
1188 | # |
1189 | # Show nicely formatted complex number under its cartesian or polar form, |
1190 | # depending on the current display format: |
1191 | # |
1192 | # . If a specific display format has been recorded for this object, use it. |
1193 | # . Otherwise, use the generic current default for all complex numbers, |
1194 | # which is a package global variable. |
1195 | # |
a0d0e21e |
1196 | sub stringify { |
66730be0 |
1197 | my ($z) = shift; |
1198 | my $format; |
1199 | |
1200 | $format = $display; |
1201 | $format = $z->{display} if defined $z->{display}; |
1202 | |
1203 | return $z->stringify_polar if $format =~ /^p/i; |
1204 | return $z->stringify_cartesian; |
1205 | } |
1206 | |
1207 | # |
1208 | # ->stringify_cartesian |
1209 | # |
1210 | # Stringify as a cartesian representation 'a+bi'. |
1211 | # |
1212 | sub stringify_cartesian { |
1213 | my $z = shift; |
1214 | my ($x, $y) = @{$z->cartesian}; |
1215 | my ($re, $im); |
1216 | |
fb73857a |
1217 | $x = int($x + ($x < 0 ? -1 : 1) * $eps) |
a8693bd3 |
1218 | if int(CORE::abs($x)) != int(CORE::abs($x) + $eps); |
fb73857a |
1219 | $y = int($y + ($y < 0 ? -1 : 1) * $eps) |
a8693bd3 |
1220 | if int(CORE::abs($y)) != int(CORE::abs($y) + $eps); |
55497cff |
1221 | |
a8693bd3 |
1222 | $re = "$x" if CORE::abs($x) >= $eps; |
fb73857a |
1223 | if ($y == 1) { $im = 'i' } |
1224 | elsif ($y == -1) { $im = '-i' } |
a8693bd3 |
1225 | elsif (CORE::abs($y) >= $eps) { $im = $y . "i" } |
66730be0 |
1226 | |
0c721ce2 |
1227 | my $str = ''; |
66730be0 |
1228 | $str = $re if defined $re; |
1229 | $str .= "+$im" if defined $im; |
1230 | $str =~ s/\+-/-/; |
1231 | $str =~ s/^\+//; |
d09ae4e6 |
1232 | $str =~ s/([-+])1i/$1i/; # Not redundant with the above 1/-1 tests. |
66730be0 |
1233 | $str = '0' unless $str; |
1234 | |
1235 | return $str; |
1236 | } |
1237 | |
d09ae4e6 |
1238 | |
1239 | # Helper for stringify_polar, a Greatest Common Divisor with a memory. |
1240 | |
1241 | sub _gcd { |
1242 | my ($a, $b) = @_; |
1243 | |
1244 | use integer; |
1245 | |
1246 | # Loops forever if given negative inputs. |
1247 | |
1248 | if ($b and $a > $b) { return gcd($a % $b, $b) } |
1249 | elsif ($a and $b > $a) { return gcd($b % $a, $a) } |
1250 | else { return $a ? $a : $b } |
1251 | } |
1252 | |
1253 | my %gcd; |
1254 | |
1255 | sub gcd { |
1256 | my ($a, $b) = @_; |
1257 | |
1258 | my $id = "$a $b"; |
2820d885 |
1259 | |
d09ae4e6 |
1260 | unless (exists $gcd{$id}) { |
1261 | $gcd{$id} = _gcd($a, $b); |
1262 | $gcd{"$b $a"} = $gcd{$id}; |
1263 | } |
1264 | |
1265 | return $gcd{$id}; |
1266 | } |
1267 | |
66730be0 |
1268 | # |
1269 | # ->stringify_polar |
1270 | # |
1271 | # Stringify as a polar representation '[r,t]'. |
1272 | # |
1273 | sub stringify_polar { |
1274 | my $z = shift; |
1275 | my ($r, $t) = @{$z->polar}; |
1276 | my $theta; |
1277 | |
0c721ce2 |
1278 | return '[0,0]' if $r <= $eps; |
a0d0e21e |
1279 | |
fb73857a |
1280 | my $nt = $t / pit2; |
1281 | $nt = ($nt - int($nt)) * pit2; |
1282 | $nt += pit2 if $nt < 0; # Range [0, 2pi] |
a0d0e21e |
1283 | |
a8693bd3 |
1284 | if (CORE::abs($nt) <= $eps) { $theta = 0 } |
1285 | elsif (CORE::abs(pi-$nt) <= $eps) { $theta = 'pi' } |
66730be0 |
1286 | |
55497cff |
1287 | if (defined $theta) { |
0c721ce2 |
1288 | $r = int($r + ($r < 0 ? -1 : 1) * $eps) |
a8693bd3 |
1289 | if int(CORE::abs($r)) != int(CORE::abs($r) + $eps); |
0c721ce2 |
1290 | $theta = int($theta + ($theta < 0 ? -1 : 1) * $eps) |
1291 | if ($theta ne 'pi' and |
a8693bd3 |
1292 | int(CORE::abs($theta)) != int(CORE::abs($theta) + $eps)); |
55497cff |
1293 | return "\[$r,$theta\]"; |
1294 | } |
66730be0 |
1295 | |
1296 | # |
1297 | # Okay, number is not a real. Try to identify pi/n and friends... |
1298 | # |
1299 | |
fb73857a |
1300 | $nt -= pit2 if $nt > pi; |
fb73857a |
1301 | |
a8693bd3 |
1302 | if (CORE::abs($nt) >= deg1) { |
d09ae4e6 |
1303 | my ($n, $k, $kpi); |
1304 | |
1305 | for ($k = 1, $kpi = pi; $k < 10; $k++, $kpi += pi) { |
66730be0 |
1306 | $n = int($kpi / $nt + ($nt > 0 ? 1 : -1) * 0.5); |
a8693bd3 |
1307 | if (CORE::abs($kpi/$n - $nt) <= $eps) { |
1308 | $n = CORE::abs($n); |
d09ae4e6 |
1309 | my $gcd = gcd($k, $n); |
1310 | if ($gcd > 1) { |
1311 | $k /= $gcd; |
1312 | $n /= $gcd; |
1313 | } |
1314 | next if $n > 360; |
1315 | $theta = ($nt < 0 ? '-':''). |
1316 | ($k == 1 ? 'pi':"${k}pi"); |
1317 | $theta .= '/'.$n if $n > 1; |
1318 | last; |
66730be0 |
1319 | } |
d09ae4e6 |
1320 | } |
66730be0 |
1321 | } |
1322 | |
1323 | $theta = $nt unless defined $theta; |
1324 | |
0c721ce2 |
1325 | $r = int($r + ($r < 0 ? -1 : 1) * $eps) |
a8693bd3 |
1326 | if int(CORE::abs($r)) != int(CORE::abs($r) + $eps); |
0c721ce2 |
1327 | $theta = int($theta + ($theta < 0 ? -1 : 1) * $eps) |
1328 | if ($theta !~ m(^-?\d*pi/\d+$) and |
a8693bd3 |
1329 | int(CORE::abs($theta)) != int(CORE::abs($theta) + $eps)); |
55497cff |
1330 | |
66730be0 |
1331 | return "\[$r,$theta\]"; |
a0d0e21e |
1332 | } |
a5f75d66 |
1333 | |
1334 | 1; |
1335 | __END__ |
1336 | |
1337 | =head1 NAME |
1338 | |
66730be0 |
1339 | Math::Complex - complex numbers and associated mathematical functions |
a5f75d66 |
1340 | |
1341 | =head1 SYNOPSIS |
1342 | |
66730be0 |
1343 | use Math::Complex; |
fb73857a |
1344 | |
66730be0 |
1345 | $z = Math::Complex->make(5, 6); |
1346 | $t = 4 - 3*i + $z; |
1347 | $j = cplxe(1, 2*pi/3); |
a5f75d66 |
1348 | |
1349 | =head1 DESCRIPTION |
1350 | |
66730be0 |
1351 | This package lets you create and manipulate complex numbers. By default, |
1352 | I<Perl> limits itself to real numbers, but an extra C<use> statement brings |
1353 | full complex support, along with a full set of mathematical functions |
1354 | typically associated with and/or extended to complex numbers. |
1355 | |
1356 | If you wonder what complex numbers are, they were invented to be able to solve |
1357 | the following equation: |
1358 | |
1359 | x*x = -1 |
1360 | |
1361 | and by definition, the solution is noted I<i> (engineers use I<j> instead since |
1362 | I<i> usually denotes an intensity, but the name does not matter). The number |
1363 | I<i> is a pure I<imaginary> number. |
1364 | |
1365 | The arithmetics with pure imaginary numbers works just like you would expect |
1366 | it with real numbers... you just have to remember that |
1367 | |
1368 | i*i = -1 |
1369 | |
1370 | so you have: |
1371 | |
1372 | 5i + 7i = i * (5 + 7) = 12i |
1373 | 4i - 3i = i * (4 - 3) = i |
1374 | 4i * 2i = -8 |
1375 | 6i / 2i = 3 |
1376 | 1 / i = -i |
1377 | |
1378 | Complex numbers are numbers that have both a real part and an imaginary |
1379 | part, and are usually noted: |
1380 | |
1381 | a + bi |
1382 | |
1383 | where C<a> is the I<real> part and C<b> is the I<imaginary> part. The |
1384 | arithmetic with complex numbers is straightforward. You have to |
1385 | keep track of the real and the imaginary parts, but otherwise the |
1386 | rules used for real numbers just apply: |
1387 | |
1388 | (4 + 3i) + (5 - 2i) = (4 + 5) + i(3 - 2) = 9 + i |
1389 | (2 + i) * (4 - i) = 2*4 + 4i -2i -i*i = 8 + 2i + 1 = 9 + 2i |
1390 | |
1391 | A graphical representation of complex numbers is possible in a plane |
1392 | (also called the I<complex plane>, but it's really a 2D plane). |
1393 | The number |
1394 | |
1395 | z = a + bi |
1396 | |
1397 | is the point whose coordinates are (a, b). Actually, it would |
1398 | be the vector originating from (0, 0) to (a, b). It follows that the addition |
1399 | of two complex numbers is a vectorial addition. |
1400 | |
1401 | Since there is a bijection between a point in the 2D plane and a complex |
1402 | number (i.e. the mapping is unique and reciprocal), a complex number |
1403 | can also be uniquely identified with polar coordinates: |
1404 | |
1405 | [rho, theta] |
1406 | |
1407 | where C<rho> is the distance to the origin, and C<theta> the angle between |
1408 | the vector and the I<x> axis. There is a notation for this using the |
1409 | exponential form, which is: |
1410 | |
1411 | rho * exp(i * theta) |
1412 | |
1413 | where I<i> is the famous imaginary number introduced above. Conversion |
1414 | between this form and the cartesian form C<a + bi> is immediate: |
1415 | |
1416 | a = rho * cos(theta) |
1417 | b = rho * sin(theta) |
1418 | |
1419 | which is also expressed by this formula: |
1420 | |
fb73857a |
1421 | z = rho * exp(i * theta) = rho * (cos theta + i * sin theta) |
66730be0 |
1422 | |
1423 | In other words, it's the projection of the vector onto the I<x> and I<y> |
1424 | axes. Mathematicians call I<rho> the I<norm> or I<modulus> and I<theta> |
1425 | the I<argument> of the complex number. The I<norm> of C<z> will be |
1426 | noted C<abs(z)>. |
1427 | |
1428 | The polar notation (also known as the trigonometric |
1429 | representation) is much more handy for performing multiplications and |
1430 | divisions of complex numbers, whilst the cartesian notation is better |
fb73857a |
1431 | suited for additions and subtractions. Real numbers are on the I<x> |
1432 | axis, and therefore I<theta> is zero or I<pi>. |
66730be0 |
1433 | |
1434 | All the common operations that can be performed on a real number have |
1435 | been defined to work on complex numbers as well, and are merely |
1436 | I<extensions> of the operations defined on real numbers. This means |
1437 | they keep their natural meaning when there is no imaginary part, provided |
1438 | the number is within their definition set. |
1439 | |
1440 | For instance, the C<sqrt> routine which computes the square root of |
fb73857a |
1441 | its argument is only defined for non-negative real numbers and yields a |
1442 | non-negative real number (it is an application from B<R+> to B<R+>). |
66730be0 |
1443 | If we allow it to return a complex number, then it can be extended to |
1444 | negative real numbers to become an application from B<R> to B<C> (the |
1445 | set of complex numbers): |
1446 | |
1447 | sqrt(x) = x >= 0 ? sqrt(x) : sqrt(-x)*i |
1448 | |
1449 | It can also be extended to be an application from B<C> to B<C>, |
1450 | whilst its restriction to B<R> behaves as defined above by using |
1451 | the following definition: |
1452 | |
1453 | sqrt(z = [r,t]) = sqrt(r) * exp(i * t/2) |
1454 | |
fb73857a |
1455 | Indeed, a negative real number can be noted C<[x,pi]> (the modulus |
1456 | I<x> is always non-negative, so C<[x,pi]> is really C<-x>, a negative |
1457 | number) and the above definition states that |
66730be0 |
1458 | |
1459 | sqrt([x,pi]) = sqrt(x) * exp(i*pi/2) = [sqrt(x),pi/2] = sqrt(x)*i |
1460 | |
1461 | which is exactly what we had defined for negative real numbers above. |
b42d0ec9 |
1462 | The C<sqrt> returns only one of the solutions: if you want the both, |
1463 | use the C<root> function. |
a5f75d66 |
1464 | |
66730be0 |
1465 | All the common mathematical functions defined on real numbers that |
1466 | are extended to complex numbers share that same property of working |
1467 | I<as usual> when the imaginary part is zero (otherwise, it would not |
1468 | be called an extension, would it?). |
a5f75d66 |
1469 | |
66730be0 |
1470 | A I<new> operation possible on a complex number that is |
1471 | the identity for real numbers is called the I<conjugate>, and is noted |
1472 | with an horizontal bar above the number, or C<~z> here. |
a5f75d66 |
1473 | |
66730be0 |
1474 | z = a + bi |
1475 | ~z = a - bi |
a5f75d66 |
1476 | |
66730be0 |
1477 | Simple... Now look: |
a5f75d66 |
1478 | |
66730be0 |
1479 | z * ~z = (a + bi) * (a - bi) = a*a + b*b |
a5f75d66 |
1480 | |
66730be0 |
1481 | We saw that the norm of C<z> was noted C<abs(z)> and was defined as the |
1482 | distance to the origin, also known as: |
a5f75d66 |
1483 | |
66730be0 |
1484 | rho = abs(z) = sqrt(a*a + b*b) |
a5f75d66 |
1485 | |
66730be0 |
1486 | so |
1487 | |
1488 | z * ~z = abs(z) ** 2 |
1489 | |
1490 | If z is a pure real number (i.e. C<b == 0>), then the above yields: |
1491 | |
1492 | a * a = abs(a) ** 2 |
1493 | |
1494 | which is true (C<abs> has the regular meaning for real number, i.e. stands |
1495 | for the absolute value). This example explains why the norm of C<z> is |
1496 | noted C<abs(z)>: it extends the C<abs> function to complex numbers, yet |
1497 | is the regular C<abs> we know when the complex number actually has no |
1498 | imaginary part... This justifies I<a posteriori> our use of the C<abs> |
1499 | notation for the norm. |
1500 | |
1501 | =head1 OPERATIONS |
1502 | |
1503 | Given the following notations: |
1504 | |
1505 | z1 = a + bi = r1 * exp(i * t1) |
1506 | z2 = c + di = r2 * exp(i * t2) |
1507 | z = <any complex or real number> |
1508 | |
1509 | the following (overloaded) operations are supported on complex numbers: |
1510 | |
1511 | z1 + z2 = (a + c) + i(b + d) |
1512 | z1 - z2 = (a - c) + i(b - d) |
1513 | z1 * z2 = (r1 * r2) * exp(i * (t1 + t2)) |
1514 | z1 / z2 = (r1 / r2) * exp(i * (t1 - t2)) |
1515 | z1 ** z2 = exp(z2 * log z1) |
b42d0ec9 |
1516 | ~z = a - bi |
1517 | abs(z) = r1 = sqrt(a*a + b*b) |
1518 | sqrt(z) = sqrt(r1) * exp(i * t/2) |
1519 | exp(z) = exp(a) * exp(i * b) |
1520 | log(z) = log(r1) + i*t |
1521 | sin(z) = 1/2i (exp(i * z1) - exp(-i * z)) |
1522 | cos(z) = 1/2 (exp(i * z1) + exp(-i * z)) |
66730be0 |
1523 | atan2(z1, z2) = atan(z1/z2) |
1524 | |
1525 | The following extra operations are supported on both real and complex |
1526 | numbers: |
1527 | |
1528 | Re(z) = a |
1529 | Im(z) = b |
1530 | arg(z) = t |
b42d0ec9 |
1531 | abs(z) = r |
66730be0 |
1532 | |
1533 | cbrt(z) = z ** (1/3) |
1534 | log10(z) = log(z) / log(10) |
1535 | logn(z, n) = log(z) / log(n) |
1536 | |
1537 | tan(z) = sin(z) / cos(z) |
0c721ce2 |
1538 | |
5aabfad6 |
1539 | csc(z) = 1 / sin(z) |
1540 | sec(z) = 1 / cos(z) |
0c721ce2 |
1541 | cot(z) = 1 / tan(z) |
66730be0 |
1542 | |
1543 | asin(z) = -i * log(i*z + sqrt(1-z*z)) |
fb73857a |
1544 | acos(z) = -i * log(z + i*sqrt(1-z*z)) |
66730be0 |
1545 | atan(z) = i/2 * log((i+z) / (i-z)) |
0c721ce2 |
1546 | |
5aabfad6 |
1547 | acsc(z) = asin(1 / z) |
1548 | asec(z) = acos(1 / z) |
8c03c583 |
1549 | acot(z) = atan(1 / z) = -i/2 * log((i+z) / (z-i)) |
66730be0 |
1550 | |
1551 | sinh(z) = 1/2 (exp(z) - exp(-z)) |
1552 | cosh(z) = 1/2 (exp(z) + exp(-z)) |
0c721ce2 |
1553 | tanh(z) = sinh(z) / cosh(z) = (exp(z) - exp(-z)) / (exp(z) + exp(-z)) |
1554 | |
5aabfad6 |
1555 | csch(z) = 1 / sinh(z) |
1556 | sech(z) = 1 / cosh(z) |
0c721ce2 |
1557 | coth(z) = 1 / tanh(z) |
fb73857a |
1558 | |
66730be0 |
1559 | asinh(z) = log(z + sqrt(z*z+1)) |
1560 | acosh(z) = log(z + sqrt(z*z-1)) |
1561 | atanh(z) = 1/2 * log((1+z) / (1-z)) |
66730be0 |
1562 | |
5aabfad6 |
1563 | acsch(z) = asinh(1 / z) |
1564 | asech(z) = acosh(1 / z) |
0c721ce2 |
1565 | acoth(z) = atanh(1 / z) = 1/2 * log((1+z) / (z-1)) |
1566 | |
b42d0ec9 |
1567 | I<arg>, I<abs>, I<log>, I<csc>, I<cot>, I<acsc>, I<acot>, I<csch>, |
1568 | I<coth>, I<acosech>, I<acotanh>, have aliases I<rho>, I<theta>, I<ln>, |
1569 | I<cosec>, I<cotan>, I<acosec>, I<acotan>, I<cosech>, I<cotanh>, |
1570 | I<acosech>, I<acotanh>, respectively. C<Re>, C<Im>, C<arg>, C<abs>, |
1571 | C<rho>, and C<theta> can be used also also mutators. The C<cbrt> |
1572 | returns only one of the solutions: if you want all three, use the |
1573 | C<root> function. |
0c721ce2 |
1574 | |
1575 | The I<root> function is available to compute all the I<n> |
66730be0 |
1576 | roots of some complex, where I<n> is a strictly positive integer. |
1577 | There are exactly I<n> such roots, returned as a list. Getting the |
1578 | number mathematicians call C<j> such that: |
1579 | |
1580 | 1 + j + j*j = 0; |
1581 | |
1582 | is a simple matter of writing: |
1583 | |
1584 | $j = ((root(1, 3))[1]; |
1585 | |
1586 | The I<k>th root for C<z = [r,t]> is given by: |
1587 | |
1588 | (root(z, n))[k] = r**(1/n) * exp(i * (t + 2*k*pi)/n) |
1589 | |
f4837644 |
1590 | The I<spaceship> comparison operator, E<lt>=E<gt>, is also defined. In |
1591 | order to ensure its restriction to real numbers is conform to what you |
1592 | would expect, the comparison is run on the real part of the complex |
1593 | number first, and imaginary parts are compared only when the real |
1594 | parts match. |
66730be0 |
1595 | |
1596 | =head1 CREATION |
1597 | |
1598 | To create a complex number, use either: |
1599 | |
1600 | $z = Math::Complex->make(3, 4); |
1601 | $z = cplx(3, 4); |
1602 | |
1603 | if you know the cartesian form of the number, or |
1604 | |
1605 | $z = 3 + 4*i; |
1606 | |
fb73857a |
1607 | if you like. To create a number using the polar form, use either: |
66730be0 |
1608 | |
1609 | $z = Math::Complex->emake(5, pi/3); |
1610 | $x = cplxe(5, pi/3); |
1611 | |
0c721ce2 |
1612 | instead. The first argument is the modulus, the second is the angle |
fb73857a |
1613 | (in radians, the full circle is 2*pi). (Mnemonic: C<e> is used as a |
1614 | notation for complex numbers in the polar form). |
66730be0 |
1615 | |
1616 | It is possible to write: |
1617 | |
1618 | $x = cplxe(-3, pi/4); |
1619 | |
1620 | but that will be silently converted into C<[3,-3pi/4]>, since the modulus |
fb73857a |
1621 | must be non-negative (it represents the distance to the origin in the complex |
66730be0 |
1622 | plane). |
1623 | |
b42d0ec9 |
1624 | It is also possible to have a complex number as either argument of |
1625 | either the C<make> or C<emake>: the appropriate component of |
1626 | the argument will be used. |
1627 | |
1628 | $z1 = cplx(-2, 1); |
1629 | $z2 = cplx($z1, 4); |
1630 | |
66730be0 |
1631 | =head1 STRINGIFICATION |
1632 | |
1633 | When printed, a complex number is usually shown under its cartesian |
1634 | form I<a+bi>, but there are legitimate cases where the polar format |
1635 | I<[r,t]> is more appropriate. |
1636 | |
1637 | By calling the routine C<Math::Complex::display_format> and supplying either |
1638 | C<"polar"> or C<"cartesian">, you override the default display format, |
1639 | which is C<"cartesian">. Not supplying any argument returns the current |
1640 | setting. |
1641 | |
1642 | This default can be overridden on a per-number basis by calling the |
1643 | C<display_format> method instead. As before, not supplying any argument |
1644 | returns the current display format for this number. Otherwise whatever you |
1645 | specify will be the new display format for I<this> particular number. |
1646 | |
1647 | For instance: |
1648 | |
1649 | use Math::Complex; |
1650 | |
1651 | Math::Complex::display_format('polar'); |
1652 | $j = ((root(1, 3))[1]; |
1653 | print "j = $j\n"; # Prints "j = [1,2pi/3] |
1654 | $j->display_format('cartesian'); |
1655 | print "j = $j\n"; # Prints "j = -0.5+0.866025403784439i" |
1656 | |
1657 | The polar format attempts to emphasize arguments like I<k*pi/n> |
1658 | (where I<n> is a positive integer and I<k> an integer within [-9,+9]). |
1659 | |
1660 | =head1 USAGE |
1661 | |
1662 | Thanks to overloading, the handling of arithmetics with complex numbers |
1663 | is simple and almost transparent. |
1664 | |
1665 | Here are some examples: |
1666 | |
1667 | use Math::Complex; |
1668 | |
1669 | $j = cplxe(1, 2*pi/3); # $j ** 3 == 1 |
1670 | print "j = $j, j**3 = ", $j ** 3, "\n"; |
1671 | print "1 + j + j**2 = ", 1 + $j + $j**2, "\n"; |
1672 | |
1673 | $z = -16 + 0*i; # Force it to be a complex |
1674 | print "sqrt($z) = ", sqrt($z), "\n"; |
1675 | |
1676 | $k = exp(i * 2*pi/3); |
1677 | print "$j - $k = ", $j - $k, "\n"; |
a5f75d66 |
1678 | |
b42d0ec9 |
1679 | $z->Re(3); # Re, Im, arg, abs, |
1680 | $j->arg(2); # (the last two aka rho, theta) |
1681 | # can be used also as mutators. |
1682 | |
1683 | =head1 ERRORS DUE TO DIVISION BY ZERO OR LOGARITHM OF ZERO |
5aabfad6 |
1684 | |
1685 | The division (/) and the following functions |
1686 | |
b42d0ec9 |
1687 | log ln log10 logn |
2820d885 |
1688 | tan sec csc cot |
b42d0ec9 |
1689 | atan asec acsc acot |
1690 | tanh sech csch coth |
1691 | atanh asech acsch acoth |
5aabfad6 |
1692 | |
1693 | cannot be computed for all arguments because that would mean dividing |
8c03c583 |
1694 | by zero or taking logarithm of zero. These situations cause fatal |
1695 | runtime errors looking like this |
5aabfad6 |
1696 | |
1697 | cot(0): Division by zero. |
5cd24f17 |
1698 | (Because in the definition of cot(0), the divisor sin(0) is 0) |
5aabfad6 |
1699 | Died at ... |
1700 | |
8c03c583 |
1701 | or |
1702 | |
1703 | atanh(-1): Logarithm of zero. |
1704 | Died at... |
1705 | |
1706 | For the C<csc>, C<cot>, C<asec>, C<acsc>, C<acot>, C<csch>, C<coth>, |
b42d0ec9 |
1707 | C<asech>, C<acsch>, the argument cannot be C<0> (zero). For the the |
1708 | logarithmic functions and the C<atanh>, C<acoth>, the argument cannot |
1709 | be C<1> (one). For the C<atanh>, C<acoth>, the argument cannot be |
1710 | C<-1> (minus one). For the C<atan>, C<acot>, the argument cannot be |
1711 | C<i> (the imaginary unit). For the C<atan>, C<acoth>, the argument |
1712 | cannot be C<-i> (the negative imaginary unit). For the C<tan>, |
1713 | C<sec>, C<tanh>, the argument cannot be I<pi/2 + k * pi>, where I<k> |
1714 | is any integer. |
1715 | |
1716 | Note that because we are operating on approximations of real numbers, |
1717 | these errors can happen when merely `too close' to the singularities |
1718 | listed above. For example C<tan(2*atan2(1,1)+1e-15)> will die of |
1719 | division by zero. |
1720 | |
1721 | =head1 ERRORS DUE TO INDIGESTIBLE ARGUMENTS |
1722 | |
1723 | The C<make> and C<emake> accept both real and complex arguments. |
1724 | When they cannot recognize the arguments they will die with error |
1725 | messages like the following |
1726 | |
1727 | Math::Complex::make: Cannot take real part of ... |
1728 | Math::Complex::make: Cannot take real part of ... |
1729 | Math::Complex::emake: Cannot take rho of ... |
1730 | Math::Complex::emake: Cannot take theta of ... |
5cd24f17 |
1731 | |
a5f75d66 |
1732 | =head1 BUGS |
1733 | |
5cd24f17 |
1734 | Saying C<use Math::Complex;> exports many mathematical routines in the |
fb73857a |
1735 | caller environment and even overrides some (C<sqrt>, C<log>). |
1736 | This is construed as a feature by the Authors, actually... ;-) |
a5f75d66 |
1737 | |
66730be0 |
1738 | All routines expect to be given real or complex numbers. Don't attempt to |
1739 | use BigFloat, since Perl has currently no rule to disambiguate a '+' |
1740 | operation (for instance) between two overloaded entities. |
a5f75d66 |
1741 | |
d09ae4e6 |
1742 | In Cray UNICOS there is some strange numerical instability that results |
1743 | in root(), cos(), sin(), cosh(), sinh(), losing accuracy fast. Beware. |
1744 | The bug may be in UNICOS math libs, in UNICOS C compiler, in Math::Complex. |
1745 | Whatever it is, it does not manifest itself anywhere else where Perl runs. |
1746 | |
0c721ce2 |
1747 | =head1 AUTHORS |
a5f75d66 |
1748 | |
6e238990 |
1749 | Raphael Manfredi <F<Raphael_Manfredi@pobox.com>> and |
ace5de91 |
1750 | Jarkko Hietaniemi <F<jhi@iki.fi>>. |
5cd24f17 |
1751 | |
fb73857a |
1752 | Extensive patches by Daniel S. Lewart <F<d-lewart@uiuc.edu>>. |
1753 | |
5cd24f17 |
1754 | =cut |
1755 | |
b42d0ec9 |
1756 | 1; |
1757 | |
5cd24f17 |
1758 | # eof |