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 | |
17f410f9 |
11 | use 5.005_64; |
b42d0ec9 |
12 | use strict; |
fb73857a |
13 | |
17f410f9 |
14 | our($VERSION, @ISA, @EXPORT, %EXPORT_TAGS); |
fb73857a |
15 | |
1fa12f56 |
16 | my ( $i, %logn ); |
0c721ce2 |
17 | |
2820d885 |
18 | $VERSION = sprintf("%s", q$Id: Complex.pm,v 1.26 1998/11/01 00:00:00 dsl Exp $ =~ /(\d+\.\d+)/); |
0c721ce2 |
19 | |
5aabfad6 |
20 | @ISA = qw(Exporter); |
21 | |
5aabfad6 |
22 | my @trig = qw( |
23 | pi |
fb73857a |
24 | tan |
5aabfad6 |
25 | csc cosec sec cot cotan |
26 | asin acos atan |
27 | acsc acosec asec acot acotan |
28 | sinh cosh tanh |
29 | csch cosech sech coth cotanh |
30 | asinh acosh atanh |
31 | acsch acosech asech acoth acotanh |
32 | ); |
33 | |
34 | @EXPORT = (qw( |
b42d0ec9 |
35 | i Re Im rho theta arg |
fb73857a |
36 | sqrt log ln |
5aabfad6 |
37 | log10 logn cbrt root |
38 | cplx cplxe |
39 | ), |
40 | @trig); |
41 | |
42 | %EXPORT_TAGS = ( |
43 | 'trig' => [@trig], |
66730be0 |
44 | ); |
a0d0e21e |
45 | |
a5f75d66 |
46 | use overload |
0c721ce2 |
47 | '+' => \&plus, |
48 | '-' => \&minus, |
49 | '*' => \&multiply, |
50 | '/' => \÷, |
66730be0 |
51 | '**' => \&power, |
1fa12f56 |
52 | '==' => \&numeq, |
66730be0 |
53 | '<=>' => \&spaceship, |
54 | 'neg' => \&negate, |
0c721ce2 |
55 | '~' => \&conjugate, |
66730be0 |
56 | 'abs' => \&abs, |
57 | 'sqrt' => \&sqrt, |
58 | 'exp' => \&exp, |
59 | 'log' => \&log, |
60 | 'sin' => \&sin, |
61 | 'cos' => \&cos, |
0c721ce2 |
62 | 'tan' => \&tan, |
66730be0 |
63 | 'atan2' => \&atan2, |
64 | qw("" stringify); |
65 | |
66 | # |
b42d0ec9 |
67 | # Package "privates" |
66730be0 |
68 | # |
69 | |
16357284 |
70 | my $package = 'Math::Complex'; # Package name |
71 | my %DISPLAY_FORMAT = ('style' => 'cartesian', |
72 | 'polar_pretty_print' => 1); |
73 | my $eps = 1e-14; # Epsilon |
66730be0 |
74 | |
214f2c7f |
75 | my $Inf; |
3a188f30 |
76 | # Unicos gets a fatal runtime error without -h matherr=errno in ccflags |
77 | unless ($^O eq 'unicos') { |
78 | my $e = $!; |
214f2c7f |
79 | $Inf = CORE::exp(CORE::exp(30)); |
3a188f30 |
80 | $! = $e; # Clear ERANGE. |
81 | undef $Inf unless $Inf =~ /^inf$/; # Inf INF inf |
214f2c7f |
82 | } |
1fa12f56 |
83 | $Inf = "Inf" if !defined $Inf || !$Inf > 0; |
84 | |
66730be0 |
85 | # |
86 | # Object attributes (internal): |
87 | # cartesian [real, imaginary] -- cartesian form |
88 | # polar [rho, theta] -- polar form |
89 | # c_dirty cartesian form not up-to-date |
90 | # p_dirty polar form not up-to-date |
91 | # display display format (package's global when not set) |
92 | # |
93 | |
b42d0ec9 |
94 | # Die on bad *make() arguments. |
95 | |
96 | sub _cannot_make { |
97 | die "@{[(caller(1))[3]]}: Cannot take $_[0] of $_[1].\n"; |
98 | } |
99 | |
66730be0 |
100 | # |
101 | # ->make |
102 | # |
103 | # Create a new complex number (cartesian form) |
104 | # |
105 | sub make { |
106 | my $self = bless {}, shift; |
107 | my ($re, $im) = @_; |
b42d0ec9 |
108 | my $rre = ref $re; |
109 | if ( $rre ) { |
110 | if ( $rre eq ref $self ) { |
111 | $re = Re($re); |
112 | } else { |
113 | _cannot_make("real part", $rre); |
114 | } |
115 | } |
116 | my $rim = ref $im; |
117 | if ( $rim ) { |
118 | if ( $rim eq ref $self ) { |
119 | $im = Im($im); |
120 | } else { |
121 | _cannot_make("imaginary part", $rim); |
122 | } |
123 | } |
124 | $self->{'cartesian'} = [ $re, $im ]; |
66730be0 |
125 | $self->{c_dirty} = 0; |
126 | $self->{p_dirty} = 1; |
b42d0ec9 |
127 | $self->display_format('cartesian'); |
66730be0 |
128 | return $self; |
129 | } |
130 | |
131 | # |
132 | # ->emake |
133 | # |
134 | # Create a new complex number (exponential form) |
135 | # |
136 | sub emake { |
137 | my $self = bless {}, shift; |
138 | my ($rho, $theta) = @_; |
b42d0ec9 |
139 | my $rrh = ref $rho; |
140 | if ( $rrh ) { |
141 | if ( $rrh eq ref $self ) { |
142 | $rho = rho($rho); |
143 | } else { |
144 | _cannot_make("rho", $rrh); |
145 | } |
146 | } |
147 | my $rth = ref $theta; |
148 | if ( $rth ) { |
149 | if ( $rth eq ref $self ) { |
150 | $theta = theta($theta); |
151 | } else { |
152 | _cannot_make("theta", $rth); |
153 | } |
154 | } |
fb73857a |
155 | if ($rho < 0) { |
156 | $rho = -$rho; |
157 | $theta = ($theta <= 0) ? $theta + pi() : $theta - pi(); |
158 | } |
159 | $self->{'polar'} = [$rho, $theta]; |
66730be0 |
160 | $self->{p_dirty} = 0; |
161 | $self->{c_dirty} = 1; |
b42d0ec9 |
162 | $self->display_format('polar'); |
66730be0 |
163 | return $self; |
164 | } |
165 | |
166 | sub new { &make } # For backward compatibility only. |
167 | |
168 | # |
169 | # cplx |
170 | # |
171 | # Creates a complex number from a (re, im) tuple. |
172 | # This avoids the burden of writing Math::Complex->make(re, im). |
173 | # |
174 | sub cplx { |
175 | my ($re, $im) = @_; |
16357284 |
176 | return __PACKAGE__->make($re, defined $im ? $im : 0); |
66730be0 |
177 | } |
178 | |
179 | # |
180 | # cplxe |
181 | # |
182 | # Creates a complex number from a (rho, theta) tuple. |
183 | # This avoids the burden of writing Math::Complex->emake(rho, theta). |
184 | # |
185 | sub cplxe { |
186 | my ($rho, $theta) = @_; |
16357284 |
187 | return __PACKAGE__->emake($rho, defined $theta ? $theta : 0); |
66730be0 |
188 | } |
189 | |
190 | # |
191 | # pi |
192 | # |
fb73857a |
193 | # The number defined as pi = 180 degrees |
66730be0 |
194 | # |
6570f784 |
195 | sub pi () { 4 * CORE::atan2(1, 1) } |
5cd24f17 |
196 | |
197 | # |
fb73857a |
198 | # pit2 |
5cd24f17 |
199 | # |
fb73857a |
200 | # The full circle |
201 | # |
6570f784 |
202 | sub pit2 () { 2 * pi } |
fb73857a |
203 | |
5cd24f17 |
204 | # |
fb73857a |
205 | # pip2 |
206 | # |
207 | # The quarter circle |
208 | # |
6570f784 |
209 | sub pip2 () { pi / 2 } |
5cd24f17 |
210 | |
fb73857a |
211 | # |
d09ae4e6 |
212 | # deg1 |
213 | # |
214 | # One degree in radians, used in stringify_polar. |
215 | # |
216 | |
6570f784 |
217 | sub deg1 () { pi / 180 } |
d09ae4e6 |
218 | |
219 | # |
fb73857a |
220 | # uplog10 |
221 | # |
222 | # Used in log10(). |
223 | # |
6570f784 |
224 | sub uplog10 () { 1 / CORE::log(10) } |
66730be0 |
225 | |
226 | # |
227 | # i |
228 | # |
229 | # The number defined as i*i = -1; |
230 | # |
231 | sub i () { |
5cd24f17 |
232 | return $i if ($i); |
233 | $i = bless {}; |
40da2db3 |
234 | $i->{'cartesian'} = [0, 1]; |
fb73857a |
235 | $i->{'polar'} = [1, pip2]; |
66730be0 |
236 | $i->{c_dirty} = 0; |
237 | $i->{p_dirty} = 0; |
238 | return $i; |
239 | } |
240 | |
241 | # |
1fa12f56 |
242 | # ip2 |
243 | # |
244 | # Half of i. |
245 | # |
246 | sub ip2 () { i / 2 } |
247 | |
248 | # |
66730be0 |
249 | # Attribute access/set routines |
250 | # |
251 | |
0c721ce2 |
252 | sub cartesian {$_[0]->{c_dirty} ? |
253 | $_[0]->update_cartesian : $_[0]->{'cartesian'}} |
254 | sub polar {$_[0]->{p_dirty} ? |
255 | $_[0]->update_polar : $_[0]->{'polar'}} |
66730be0 |
256 | |
40da2db3 |
257 | sub set_cartesian { $_[0]->{p_dirty}++; $_[0]->{'cartesian'} = $_[1] } |
258 | sub set_polar { $_[0]->{c_dirty}++; $_[0]->{'polar'} = $_[1] } |
66730be0 |
259 | |
260 | # |
261 | # ->update_cartesian |
262 | # |
263 | # Recompute and return the cartesian form, given accurate polar form. |
264 | # |
265 | sub update_cartesian { |
266 | my $self = shift; |
40da2db3 |
267 | my ($r, $t) = @{$self->{'polar'}}; |
66730be0 |
268 | $self->{c_dirty} = 0; |
a8693bd3 |
269 | return $self->{'cartesian'} = [$r * CORE::cos($t), $r * CORE::sin($t)]; |
66730be0 |
270 | } |
271 | |
272 | # |
273 | # |
274 | # ->update_polar |
275 | # |
276 | # Recompute and return the polar form, given accurate cartesian form. |
277 | # |
278 | sub update_polar { |
279 | my $self = shift; |
40da2db3 |
280 | my ($x, $y) = @{$self->{'cartesian'}}; |
66730be0 |
281 | $self->{p_dirty} = 0; |
40da2db3 |
282 | return $self->{'polar'} = [0, 0] if $x == 0 && $y == 0; |
1fa12f56 |
283 | return $self->{'polar'} = [CORE::sqrt($x*$x + $y*$y), |
284 | CORE::atan2($y, $x)]; |
66730be0 |
285 | } |
286 | |
287 | # |
288 | # (plus) |
289 | # |
290 | # Computes z1+z2. |
291 | # |
292 | sub plus { |
293 | my ($z1, $z2, $regular) = @_; |
294 | my ($re1, $im1) = @{$z1->cartesian}; |
0e505df1 |
295 | $z2 = cplx($z2) unless ref $z2; |
5cd24f17 |
296 | my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0); |
66730be0 |
297 | unless (defined $regular) { |
298 | $z1->set_cartesian([$re1 + $re2, $im1 + $im2]); |
299 | return $z1; |
300 | } |
301 | return (ref $z1)->make($re1 + $re2, $im1 + $im2); |
302 | } |
303 | |
304 | # |
305 | # (minus) |
306 | # |
307 | # Computes z1-z2. |
308 | # |
309 | sub minus { |
310 | my ($z1, $z2, $inverted) = @_; |
311 | my ($re1, $im1) = @{$z1->cartesian}; |
0e505df1 |
312 | $z2 = cplx($z2) unless ref $z2; |
313 | my ($re2, $im2) = @{$z2->cartesian}; |
66730be0 |
314 | unless (defined $inverted) { |
315 | $z1->set_cartesian([$re1 - $re2, $im1 - $im2]); |
316 | return $z1; |
317 | } |
318 | return $inverted ? |
319 | (ref $z1)->make($re2 - $re1, $im2 - $im1) : |
320 | (ref $z1)->make($re1 - $re2, $im1 - $im2); |
0e505df1 |
321 | |
66730be0 |
322 | } |
323 | |
324 | # |
325 | # (multiply) |
326 | # |
327 | # Computes z1*z2. |
328 | # |
329 | sub multiply { |
fb73857a |
330 | my ($z1, $z2, $regular) = @_; |
331 | if ($z1->{p_dirty} == 0 and ref $z2 and $z2->{p_dirty} == 0) { |
332 | # if both polar better use polar to avoid rounding errors |
333 | my ($r1, $t1) = @{$z1->polar}; |
334 | my ($r2, $t2) = @{$z2->polar}; |
335 | my $t = $t1 + $t2; |
336 | if ($t > pi()) { $t -= pit2 } |
337 | elsif ($t <= -pi()) { $t += pit2 } |
338 | unless (defined $regular) { |
339 | $z1->set_polar([$r1 * $r2, $t]); |
66730be0 |
340 | return $z1; |
fb73857a |
341 | } |
342 | return (ref $z1)->emake($r1 * $r2, $t); |
343 | } else { |
344 | my ($x1, $y1) = @{$z1->cartesian}; |
345 | if (ref $z2) { |
346 | my ($x2, $y2) = @{$z2->cartesian}; |
347 | return (ref $z1)->make($x1*$x2-$y1*$y2, $x1*$y2+$y1*$x2); |
348 | } else { |
349 | return (ref $z1)->make($x1*$z2, $y1*$z2); |
350 | } |
66730be0 |
351 | } |
66730be0 |
352 | } |
353 | |
354 | # |
0e505df1 |
355 | # _divbyzero |
0c721ce2 |
356 | # |
357 | # Die on division by zero. |
358 | # |
0e505df1 |
359 | sub _divbyzero { |
5cd24f17 |
360 | my $mess = "$_[0]: Division by zero.\n"; |
361 | |
362 | if (defined $_[1]) { |
363 | $mess .= "(Because in the definition of $_[0], the divisor "; |
1fa12f56 |
364 | $mess .= "$_[1] " unless ("$_[1]" eq '0'); |
5cd24f17 |
365 | $mess .= "is 0)\n"; |
366 | } |
367 | |
0c721ce2 |
368 | my @up = caller(1); |
fb73857a |
369 | |
5cd24f17 |
370 | $mess .= "Died at $up[1] line $up[2].\n"; |
371 | |
372 | die $mess; |
0c721ce2 |
373 | } |
374 | |
375 | # |
66730be0 |
376 | # (divide) |
377 | # |
378 | # Computes z1/z2. |
379 | # |
380 | sub divide { |
381 | my ($z1, $z2, $inverted) = @_; |
fb73857a |
382 | if ($z1->{p_dirty} == 0 and ref $z2 and $z2->{p_dirty} == 0) { |
383 | # if both polar better use polar to avoid rounding errors |
384 | my ($r1, $t1) = @{$z1->polar}; |
385 | my ($r2, $t2) = @{$z2->polar}; |
386 | my $t; |
387 | if ($inverted) { |
0e505df1 |
388 | _divbyzero "$z2/0" if ($r1 == 0); |
fb73857a |
389 | $t = $t2 - $t1; |
390 | if ($t > pi()) { $t -= pit2 } |
391 | elsif ($t <= -pi()) { $t += pit2 } |
392 | return (ref $z1)->emake($r2 / $r1, $t); |
393 | } else { |
0e505df1 |
394 | _divbyzero "$z1/0" if ($r2 == 0); |
fb73857a |
395 | $t = $t1 - $t2; |
396 | if ($t > pi()) { $t -= pit2 } |
397 | elsif ($t <= -pi()) { $t += pit2 } |
398 | return (ref $z1)->emake($r1 / $r2, $t); |
399 | } |
400 | } else { |
401 | my ($d, $x2, $y2); |
402 | if ($inverted) { |
403 | ($x2, $y2) = @{$z1->cartesian}; |
404 | $d = $x2*$x2 + $y2*$y2; |
405 | _divbyzero "$z2/0" if $d == 0; |
406 | return (ref $z1)->make(($x2*$z2)/$d, -($y2*$z2)/$d); |
407 | } else { |
408 | my ($x1, $y1) = @{$z1->cartesian}; |
409 | if (ref $z2) { |
410 | ($x2, $y2) = @{$z2->cartesian}; |
411 | $d = $x2*$x2 + $y2*$y2; |
412 | _divbyzero "$z1/0" if $d == 0; |
413 | my $u = ($x1*$x2 + $y1*$y2)/$d; |
414 | my $v = ($y1*$x2 - $x1*$y2)/$d; |
415 | return (ref $z1)->make($u, $v); |
416 | } else { |
417 | _divbyzero "$z1/0" if $z2 == 0; |
418 | return (ref $z1)->make($x1/$z2, $y1/$z2); |
419 | } |
420 | } |
0c721ce2 |
421 | } |
66730be0 |
422 | } |
423 | |
424 | # |
425 | # (power) |
426 | # |
427 | # Computes z1**z2 = exp(z2 * log z1)). |
428 | # |
429 | sub power { |
430 | my ($z1, $z2, $inverted) = @_; |
ace5de91 |
431 | if ($inverted) { |
2820d885 |
432 | return 1 if $z1 == 0 || $z2 == 1; |
433 | return 0 if $z2 == 0 && Re($z1) > 0; |
ace5de91 |
434 | } else { |
2820d885 |
435 | return 1 if $z2 == 0 || $z1 == 1; |
436 | return 0 if $z1 == 0 && Re($z2) > 0; |
ace5de91 |
437 | } |
1fa12f56 |
438 | my $w = $inverted ? &exp($z1 * &log($z2)) |
439 | : &exp($z2 * &log($z1)); |
d09ae4e6 |
440 | # If both arguments cartesian, return cartesian, else polar. |
441 | return $z1->{c_dirty} == 0 && |
442 | (not ref $z2 or $z2->{c_dirty} == 0) ? |
443 | cplx(@{$w->cartesian}) : $w; |
66730be0 |
444 | } |
445 | |
446 | # |
447 | # (spaceship) |
448 | # |
449 | # Computes z1 <=> z2. |
2820d885 |
450 | # Sorts on the real part first, then on the imaginary part. Thus 2-4i < 3+8i. |
66730be0 |
451 | # |
452 | sub spaceship { |
453 | my ($z1, $z2, $inverted) = @_; |
5cd24f17 |
454 | my ($re1, $im1) = ref $z1 ? @{$z1->cartesian} : ($z1, 0); |
455 | my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0); |
66730be0 |
456 | my $sgn = $inverted ? -1 : 1; |
457 | return $sgn * ($re1 <=> $re2) if $re1 != $re2; |
458 | return $sgn * ($im1 <=> $im2); |
459 | } |
460 | |
461 | # |
1fa12f56 |
462 | # (numeq) |
463 | # |
464 | # Computes z1 == z2. |
465 | # |
466 | # (Required in addition to spaceship() because of NaNs.) |
467 | sub numeq { |
468 | my ($z1, $z2, $inverted) = @_; |
469 | my ($re1, $im1) = ref $z1 ? @{$z1->cartesian} : ($z1, 0); |
470 | my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0); |
471 | return $re1 == $re2 && $im1 == $im2 ? 1 : 0; |
472 | } |
473 | |
474 | # |
66730be0 |
475 | # (negate) |
476 | # |
477 | # Computes -z. |
478 | # |
479 | sub negate { |
480 | my ($z) = @_; |
481 | if ($z->{c_dirty}) { |
482 | my ($r, $t) = @{$z->polar}; |
fb73857a |
483 | $t = ($t <= 0) ? $t + pi : $t - pi; |
484 | return (ref $z)->emake($r, $t); |
66730be0 |
485 | } |
486 | my ($re, $im) = @{$z->cartesian}; |
487 | return (ref $z)->make(-$re, -$im); |
488 | } |
489 | |
490 | # |
491 | # (conjugate) |
492 | # |
493 | # Compute complex's conjugate. |
494 | # |
495 | sub conjugate { |
496 | my ($z) = @_; |
497 | if ($z->{c_dirty}) { |
498 | my ($r, $t) = @{$z->polar}; |
499 | return (ref $z)->emake($r, -$t); |
500 | } |
501 | my ($re, $im) = @{$z->cartesian}; |
502 | return (ref $z)->make($re, -$im); |
503 | } |
504 | |
505 | # |
506 | # (abs) |
507 | # |
b42d0ec9 |
508 | # Compute or set complex's norm (rho). |
66730be0 |
509 | # |
510 | sub abs { |
b42d0ec9 |
511 | my ($z, $rho) = @_; |
1fa12f56 |
512 | unless (ref $z) { |
513 | if (@_ == 2) { |
514 | $_[0] = $_[1]; |
515 | } else { |
516 | return CORE::abs($z); |
517 | } |
518 | } |
b42d0ec9 |
519 | if (defined $rho) { |
520 | $z->{'polar'} = [ $rho, ${$z->polar}[1] ]; |
521 | $z->{p_dirty} = 0; |
522 | $z->{c_dirty} = 1; |
523 | return $rho; |
524 | } else { |
525 | return ${$z->polar}[0]; |
526 | } |
527 | } |
528 | |
529 | sub _theta { |
530 | my $theta = $_[0]; |
531 | |
532 | if ($$theta > pi()) { $$theta -= pit2 } |
533 | elsif ($$theta <= -pi()) { $$theta += pit2 } |
66730be0 |
534 | } |
535 | |
536 | # |
537 | # arg |
538 | # |
b42d0ec9 |
539 | # Compute or set complex's argument (theta). |
66730be0 |
540 | # |
541 | sub arg { |
b42d0ec9 |
542 | my ($z, $theta) = @_; |
543 | return $z unless ref $z; |
544 | if (defined $theta) { |
545 | _theta(\$theta); |
546 | $z->{'polar'} = [ ${$z->polar}[0], $theta ]; |
547 | $z->{p_dirty} = 0; |
548 | $z->{c_dirty} = 1; |
549 | } else { |
550 | $theta = ${$z->polar}[1]; |
551 | _theta(\$theta); |
552 | } |
553 | return $theta; |
66730be0 |
554 | } |
555 | |
556 | # |
557 | # (sqrt) |
558 | # |
0c721ce2 |
559 | # Compute sqrt(z). |
66730be0 |
560 | # |
b42d0ec9 |
561 | # It is quite tempting to use wantarray here so that in list context |
562 | # sqrt() would return the two solutions. This, however, would |
563 | # break things like |
564 | # |
565 | # print "sqrt(z) = ", sqrt($z), "\n"; |
566 | # |
567 | # The two values would be printed side by side without no intervening |
568 | # whitespace, quite confusing. |
569 | # Therefore if you want the two solutions use the root(). |
570 | # |
66730be0 |
571 | sub sqrt { |
572 | my ($z) = @_; |
b42d0ec9 |
573 | my ($re, $im) = ref $z ? @{$z->cartesian} : ($z, 0); |
1fa12f56 |
574 | return $re < 0 ? cplx(0, CORE::sqrt(-$re)) : CORE::sqrt($re) |
575 | if $im == 0; |
66730be0 |
576 | my ($r, $t) = @{$z->polar}; |
a8693bd3 |
577 | return (ref $z)->emake(CORE::sqrt($r), $t/2); |
66730be0 |
578 | } |
579 | |
580 | # |
581 | # cbrt |
582 | # |
0c721ce2 |
583 | # Compute cbrt(z) (cubic root). |
66730be0 |
584 | # |
b42d0ec9 |
585 | # Why are we not returning three values? The same answer as for sqrt(). |
586 | # |
66730be0 |
587 | sub cbrt { |
588 | my ($z) = @_; |
1fa12f56 |
589 | return $z < 0 ? |
590 | -CORE::exp(CORE::log(-$z)/3) : |
591 | ($z > 0 ? CORE::exp(CORE::log($z)/3): 0) |
fb73857a |
592 | unless ref $z; |
66730be0 |
593 | my ($r, $t) = @{$z->polar}; |
1fa12f56 |
594 | return 0 if $r == 0; |
a8693bd3 |
595 | return (ref $z)->emake(CORE::exp(CORE::log($r)/3), $t/3); |
66730be0 |
596 | } |
597 | |
598 | # |
0e505df1 |
599 | # _rootbad |
600 | # |
601 | # Die on bad root. |
602 | # |
603 | sub _rootbad { |
1fa12f56 |
604 | my $mess = "Root $_[0] illegal, root rank must be positive integer.\n"; |
0e505df1 |
605 | |
606 | my @up = caller(1); |
fb73857a |
607 | |
0e505df1 |
608 | $mess .= "Died at $up[1] line $up[2].\n"; |
609 | |
610 | die $mess; |
611 | } |
612 | |
613 | # |
66730be0 |
614 | # root |
615 | # |
616 | # Computes all nth root for z, returning an array whose size is n. |
617 | # `n' must be a positive integer. |
618 | # |
619 | # The roots are given by (for k = 0..n-1): |
620 | # |
621 | # z^(1/n) = r^(1/n) (cos ((t+2 k pi)/n) + i sin ((t+2 k pi)/n)) |
622 | # |
623 | sub root { |
624 | my ($z, $n) = @_; |
0e505df1 |
625 | _rootbad($n) if ($n < 1 or int($n) != $n); |
1fa12f56 |
626 | my ($r, $t) = ref $z ? |
627 | @{$z->polar} : (CORE::abs($z), $z >= 0 ? 0 : pi); |
66730be0 |
628 | my @root; |
629 | my $k; |
fb73857a |
630 | my $theta_inc = pit2 / $n; |
66730be0 |
631 | my $rho = $r ** (1/$n); |
632 | my $theta; |
d09ae4e6 |
633 | my $cartesian = ref $z && $z->{c_dirty} == 0; |
66730be0 |
634 | for ($k = 0, $theta = $t / $n; $k < $n; $k++, $theta += $theta_inc) { |
d09ae4e6 |
635 | my $w = cplxe($rho, $theta); |
636 | # Yes, $cartesian is loop invariant. |
637 | push @root, $cartesian ? cplx(@{$w->cartesian}) : $w; |
a0d0e21e |
638 | } |
66730be0 |
639 | return @root; |
a0d0e21e |
640 | } |
641 | |
66730be0 |
642 | # |
643 | # Re |
644 | # |
b42d0ec9 |
645 | # Return or set Re(z). |
66730be0 |
646 | # |
a0d0e21e |
647 | sub Re { |
b42d0ec9 |
648 | my ($z, $Re) = @_; |
66730be0 |
649 | return $z unless ref $z; |
b42d0ec9 |
650 | if (defined $Re) { |
651 | $z->{'cartesian'} = [ $Re, ${$z->cartesian}[1] ]; |
652 | $z->{c_dirty} = 0; |
653 | $z->{p_dirty} = 1; |
654 | } else { |
655 | return ${$z->cartesian}[0]; |
656 | } |
a0d0e21e |
657 | } |
658 | |
66730be0 |
659 | # |
660 | # Im |
661 | # |
b42d0ec9 |
662 | # Return or set Im(z). |
66730be0 |
663 | # |
a0d0e21e |
664 | sub Im { |
b42d0ec9 |
665 | my ($z, $Im) = @_; |
666 | return $z unless ref $z; |
667 | if (defined $Im) { |
668 | $z->{'cartesian'} = [ ${$z->cartesian}[0], $Im ]; |
669 | $z->{c_dirty} = 0; |
670 | $z->{p_dirty} = 1; |
671 | } else { |
672 | return ${$z->cartesian}[1]; |
673 | } |
674 | } |
675 | |
676 | # |
677 | # rho |
678 | # |
679 | # Return or set rho(w). |
680 | # |
681 | sub rho { |
682 | Math::Complex::abs(@_); |
683 | } |
684 | |
685 | # |
686 | # theta |
687 | # |
688 | # Return or set theta(w). |
689 | # |
690 | sub theta { |
691 | Math::Complex::arg(@_); |
a0d0e21e |
692 | } |
693 | |
66730be0 |
694 | # |
695 | # (exp) |
696 | # |
697 | # Computes exp(z). |
698 | # |
699 | sub exp { |
700 | my ($z) = @_; |
701 | my ($x, $y) = @{$z->cartesian}; |
a8693bd3 |
702 | return (ref $z)->emake(CORE::exp($x), $y); |
66730be0 |
703 | } |
704 | |
705 | # |
8c03c583 |
706 | # _logofzero |
707 | # |
fb73857a |
708 | # Die on logarithm of zero. |
8c03c583 |
709 | # |
710 | sub _logofzero { |
711 | my $mess = "$_[0]: Logarithm of zero.\n"; |
712 | |
713 | if (defined $_[1]) { |
714 | $mess .= "(Because in the definition of $_[0], the argument "; |
715 | $mess .= "$_[1] " unless ($_[1] eq '0'); |
716 | $mess .= "is 0)\n"; |
717 | } |
718 | |
719 | my @up = caller(1); |
fb73857a |
720 | |
8c03c583 |
721 | $mess .= "Died at $up[1] line $up[2].\n"; |
722 | |
723 | die $mess; |
724 | } |
725 | |
726 | # |
66730be0 |
727 | # (log) |
728 | # |
729 | # Compute log(z). |
730 | # |
731 | sub log { |
732 | my ($z) = @_; |
fb73857a |
733 | unless (ref $z) { |
734 | _logofzero("log") if $z == 0; |
a8693bd3 |
735 | return $z > 0 ? CORE::log($z) : cplx(CORE::log(-$z), pi); |
fb73857a |
736 | } |
5cd24f17 |
737 | my ($r, $t) = @{$z->polar}; |
fb73857a |
738 | _logofzero("log") if $r == 0; |
739 | if ($t > pi()) { $t -= pit2 } |
740 | elsif ($t <= -pi()) { $t += pit2 } |
a8693bd3 |
741 | return (ref $z)->make(CORE::log($r), $t); |
66730be0 |
742 | } |
743 | |
744 | # |
0c721ce2 |
745 | # ln |
746 | # |
747 | # Alias for log(). |
748 | # |
749 | sub ln { Math::Complex::log(@_) } |
750 | |
751 | # |
66730be0 |
752 | # log10 |
753 | # |
754 | # Compute log10(z). |
755 | # |
5cd24f17 |
756 | |
66730be0 |
757 | sub log10 { |
fb73857a |
758 | return Math::Complex::log($_[0]) * uplog10; |
66730be0 |
759 | } |
760 | |
761 | # |
762 | # logn |
763 | # |
764 | # Compute logn(z,n) = log(z) / log(n) |
765 | # |
766 | sub logn { |
767 | my ($z, $n) = @_; |
0c721ce2 |
768 | $z = cplx($z, 0) unless ref $z; |
66730be0 |
769 | my $logn = $logn{$n}; |
a8693bd3 |
770 | $logn = $logn{$n} = CORE::log($n) unless defined $logn; # Cache log(n) |
1fa12f56 |
771 | return &log($z) / $logn; |
66730be0 |
772 | } |
773 | |
774 | # |
775 | # (cos) |
776 | # |
777 | # Compute cos(z) = (exp(iz) + exp(-iz))/2. |
778 | # |
779 | sub cos { |
780 | my ($z) = @_; |
1fa12f56 |
781 | return CORE::cos($z) unless ref $z; |
66730be0 |
782 | my ($x, $y) = @{$z->cartesian}; |
a8693bd3 |
783 | my $ey = CORE::exp($y); |
1fa12f56 |
784 | my $sx = CORE::sin($x); |
785 | my $cx = CORE::cos($x); |
786 | my $ey_1 = $ey ? 1 / $ey : $Inf; |
787 | return (ref $z)->make($cx * ($ey + $ey_1)/2, |
788 | $sx * ($ey_1 - $ey)/2); |
66730be0 |
789 | } |
790 | |
791 | # |
792 | # (sin) |
793 | # |
794 | # Compute sin(z) = (exp(iz) - exp(-iz))/2. |
795 | # |
796 | sub sin { |
797 | my ($z) = @_; |
1fa12f56 |
798 | return CORE::sin($z) unless ref $z; |
66730be0 |
799 | my ($x, $y) = @{$z->cartesian}; |
a8693bd3 |
800 | my $ey = CORE::exp($y); |
1fa12f56 |
801 | my $sx = CORE::sin($x); |
802 | my $cx = CORE::cos($x); |
803 | my $ey_1 = $ey ? 1 / $ey : $Inf; |
804 | return (ref $z)->make($sx * ($ey + $ey_1)/2, |
805 | $cx * ($ey - $ey_1)/2); |
66730be0 |
806 | } |
807 | |
808 | # |
809 | # tan |
810 | # |
811 | # Compute tan(z) = sin(z) / cos(z). |
812 | # |
813 | sub tan { |
814 | my ($z) = @_; |
1fa12f56 |
815 | my $cz = &cos($z); |
816 | _divbyzero "tan($z)", "cos($z)" if $cz == 0; |
817 | return &sin($z) / $cz; |
66730be0 |
818 | } |
819 | |
820 | # |
0c721ce2 |
821 | # sec |
822 | # |
823 | # Computes the secant sec(z) = 1 / cos(z). |
824 | # |
825 | sub sec { |
826 | my ($z) = @_; |
1fa12f56 |
827 | my $cz = &cos($z); |
0e505df1 |
828 | _divbyzero "sec($z)", "cos($z)" if ($cz == 0); |
0c721ce2 |
829 | return 1 / $cz; |
830 | } |
831 | |
832 | # |
833 | # csc |
834 | # |
835 | # Computes the cosecant csc(z) = 1 / sin(z). |
836 | # |
837 | sub csc { |
838 | my ($z) = @_; |
1fa12f56 |
839 | my $sz = &sin($z); |
0e505df1 |
840 | _divbyzero "csc($z)", "sin($z)" if ($sz == 0); |
0c721ce2 |
841 | return 1 / $sz; |
842 | } |
843 | |
66730be0 |
844 | # |
0c721ce2 |
845 | # cosec |
66730be0 |
846 | # |
0c721ce2 |
847 | # Alias for csc(). |
848 | # |
849 | sub cosec { Math::Complex::csc(@_) } |
850 | |
851 | # |
852 | # cot |
853 | # |
fb73857a |
854 | # Computes cot(z) = cos(z) / sin(z). |
0c721ce2 |
855 | # |
856 | sub cot { |
66730be0 |
857 | my ($z) = @_; |
1fa12f56 |
858 | my $sz = &sin($z); |
0e505df1 |
859 | _divbyzero "cot($z)", "sin($z)" if ($sz == 0); |
1fa12f56 |
860 | return &cos($z) / $sz; |
66730be0 |
861 | } |
862 | |
863 | # |
0c721ce2 |
864 | # cotan |
865 | # |
866 | # Alias for cot(). |
867 | # |
868 | sub cotan { Math::Complex::cot(@_) } |
869 | |
870 | # |
66730be0 |
871 | # acos |
872 | # |
873 | # Computes the arc cosine acos(z) = -i log(z + sqrt(z*z-1)). |
874 | # |
875 | sub acos { |
fb73857a |
876 | my $z = $_[0]; |
1fa12f56 |
877 | return CORE::atan2(CORE::sqrt(1-$z*$z), $z) |
878 | if (! ref $z) && CORE::abs($z) <= 1; |
fb73857a |
879 | my ($x, $y) = ref $z ? @{$z->cartesian} : ($z, 0); |
1fa12f56 |
880 | return 0 if $x == 1 && $y == 0; |
a8693bd3 |
881 | my $t1 = CORE::sqrt(($x+1)*($x+1) + $y*$y); |
882 | my $t2 = CORE::sqrt(($x-1)*($x-1) + $y*$y); |
fb73857a |
883 | my $alpha = ($t1 + $t2)/2; |
884 | my $beta = ($t1 - $t2)/2; |
885 | $alpha = 1 if $alpha < 1; |
886 | if ($beta > 1) { $beta = 1 } |
887 | elsif ($beta < -1) { $beta = -1 } |
a8693bd3 |
888 | my $u = CORE::atan2(CORE::sqrt(1-$beta*$beta), $beta); |
889 | my $v = CORE::log($alpha + CORE::sqrt($alpha*$alpha-1)); |
fb73857a |
890 | $v = -$v if $y > 0 || ($y == 0 && $x < -1); |
16357284 |
891 | return __PACKAGE__->make($u, $v); |
66730be0 |
892 | } |
893 | |
894 | # |
895 | # asin |
896 | # |
897 | # Computes the arc sine asin(z) = -i log(iz + sqrt(1-z*z)). |
898 | # |
899 | sub asin { |
fb73857a |
900 | my $z = $_[0]; |
1fa12f56 |
901 | return CORE::atan2($z, CORE::sqrt(1-$z*$z)) |
902 | if (! ref $z) && CORE::abs($z) <= 1; |
fb73857a |
903 | my ($x, $y) = ref $z ? @{$z->cartesian} : ($z, 0); |
1fa12f56 |
904 | return 0 if $x == 0 && $y == 0; |
a8693bd3 |
905 | my $t1 = CORE::sqrt(($x+1)*($x+1) + $y*$y); |
906 | my $t2 = CORE::sqrt(($x-1)*($x-1) + $y*$y); |
fb73857a |
907 | my $alpha = ($t1 + $t2)/2; |
908 | my $beta = ($t1 - $t2)/2; |
909 | $alpha = 1 if $alpha < 1; |
910 | if ($beta > 1) { $beta = 1 } |
911 | elsif ($beta < -1) { $beta = -1 } |
a8693bd3 |
912 | my $u = CORE::atan2($beta, CORE::sqrt(1-$beta*$beta)); |
913 | my $v = -CORE::log($alpha + CORE::sqrt($alpha*$alpha-1)); |
fb73857a |
914 | $v = -$v if $y > 0 || ($y == 0 && $x < -1); |
16357284 |
915 | return __PACKAGE__->make($u, $v); |
66730be0 |
916 | } |
917 | |
918 | # |
919 | # atan |
920 | # |
0c721ce2 |
921 | # Computes the arc tangent atan(z) = i/2 log((i+z) / (i-z)). |
66730be0 |
922 | # |
923 | sub atan { |
924 | my ($z) = @_; |
a8693bd3 |
925 | return CORE::atan2($z, 1) unless ref $z; |
1fa12f56 |
926 | my ($x, $y) = ref $z ? @{$z->cartesian} : ($z, 0); |
927 | return 0 if $x == 0 && $y == 0; |
8c03c583 |
928 | _divbyzero "atan(i)" if ( $z == i); |
1fa12f56 |
929 | _logofzero "atan(-i)" if (-$z == i); # -i is a bad file test... |
930 | my $log = &log((i + $z) / (i - $z)); |
931 | return ip2 * $log; |
a0d0e21e |
932 | } |
933 | |
66730be0 |
934 | # |
0c721ce2 |
935 | # asec |
936 | # |
937 | # Computes the arc secant asec(z) = acos(1 / z). |
938 | # |
939 | sub asec { |
940 | my ($z) = @_; |
0e505df1 |
941 | _divbyzero "asec($z)", $z if ($z == 0); |
fb73857a |
942 | return acos(1 / $z); |
0c721ce2 |
943 | } |
944 | |
945 | # |
5cd24f17 |
946 | # acsc |
0c721ce2 |
947 | # |
8c03c583 |
948 | # Computes the arc cosecant acsc(z) = asin(1 / z). |
0c721ce2 |
949 | # |
5cd24f17 |
950 | sub acsc { |
0c721ce2 |
951 | my ($z) = @_; |
0e505df1 |
952 | _divbyzero "acsc($z)", $z if ($z == 0); |
fb73857a |
953 | return asin(1 / $z); |
0c721ce2 |
954 | } |
955 | |
956 | # |
5cd24f17 |
957 | # acosec |
66730be0 |
958 | # |
5cd24f17 |
959 | # Alias for acsc(). |
0c721ce2 |
960 | # |
5cd24f17 |
961 | sub acosec { Math::Complex::acsc(@_) } |
0c721ce2 |
962 | |
66730be0 |
963 | # |
0c721ce2 |
964 | # acot |
965 | # |
8c03c583 |
966 | # Computes the arc cotangent acot(z) = atan(1 / z) |
0c721ce2 |
967 | # |
968 | sub acot { |
66730be0 |
969 | my ($z) = @_; |
1fa12f56 |
970 | _divbyzero "acot(0)" if $z == 0; |
971 | return ($z >= 0) ? CORE::atan2(1, $z) : CORE::atan2(-1, -$z) |
972 | unless ref $z; |
973 | _divbyzero "acot(i)" if ($z - i == 0); |
974 | _logofzero "acot(-i)" if ($z + i == 0); |
8c03c583 |
975 | return atan(1 / $z); |
66730be0 |
976 | } |
977 | |
978 | # |
0c721ce2 |
979 | # acotan |
980 | # |
981 | # Alias for acot(). |
982 | # |
983 | sub acotan { Math::Complex::acot(@_) } |
984 | |
985 | # |
66730be0 |
986 | # cosh |
987 | # |
988 | # Computes the hyperbolic cosine cosh(z) = (exp(z) + exp(-z))/2. |
989 | # |
990 | sub cosh { |
991 | my ($z) = @_; |
fb73857a |
992 | my $ex; |
0e505df1 |
993 | unless (ref $z) { |
a8693bd3 |
994 | $ex = CORE::exp($z); |
1fa12f56 |
995 | return $ex ? ($ex + 1/$ex)/2 : $Inf; |
0e505df1 |
996 | } |
997 | my ($x, $y) = @{$z->cartesian}; |
1fa12f56 |
998 | my $cy = CORE::cos($y); |
999 | my $sy = CORE::cos($y); |
a8693bd3 |
1000 | $ex = CORE::exp($x); |
1fa12f56 |
1001 | my $ex_1 = $ex ? 1 / $ex : $Inf; |
a8693bd3 |
1002 | return (ref $z)->make(CORE::cos($y) * ($ex + $ex_1)/2, |
1003 | CORE::sin($y) * ($ex - $ex_1)/2); |
66730be0 |
1004 | } |
1005 | |
1006 | # |
1007 | # sinh |
1008 | # |
1009 | # Computes the hyperbolic sine sinh(z) = (exp(z) - exp(-z))/2. |
1010 | # |
1011 | sub sinh { |
1012 | my ($z) = @_; |
fb73857a |
1013 | my $ex; |
0e505df1 |
1014 | unless (ref $z) { |
1fa12f56 |
1015 | return 0 if $z == 0; |
a8693bd3 |
1016 | $ex = CORE::exp($z); |
1fa12f56 |
1017 | return $ex ? ($ex - 1/$ex)/2 : "-$Inf"; |
0e505df1 |
1018 | } |
1019 | my ($x, $y) = @{$z->cartesian}; |
1fa12f56 |
1020 | my $cy = CORE::cos($y); |
1021 | my $sy = CORE::sin($y); |
a8693bd3 |
1022 | $ex = CORE::exp($x); |
1fa12f56 |
1023 | my $ex_1 = $ex ? 1 / $ex : $Inf; |
1024 | return (ref $z)->make($cy * ($ex - $ex_1)/2, |
1025 | $sy * ($ex + $ex_1)/2); |
66730be0 |
1026 | } |
1027 | |
1028 | # |
1029 | # tanh |
1030 | # |
1031 | # Computes the hyperbolic tangent tanh(z) = sinh(z) / cosh(z). |
1032 | # |
1033 | sub tanh { |
1034 | my ($z) = @_; |
0c721ce2 |
1035 | my $cz = cosh($z); |
0e505df1 |
1036 | _divbyzero "tanh($z)", "cosh($z)" if ($cz == 0); |
0c721ce2 |
1037 | return sinh($z) / $cz; |
66730be0 |
1038 | } |
1039 | |
1040 | # |
0c721ce2 |
1041 | # sech |
1042 | # |
1043 | # Computes the hyperbolic secant sech(z) = 1 / cosh(z). |
1044 | # |
1045 | sub sech { |
1046 | my ($z) = @_; |
1047 | my $cz = cosh($z); |
0e505df1 |
1048 | _divbyzero "sech($z)", "cosh($z)" if ($cz == 0); |
0c721ce2 |
1049 | return 1 / $cz; |
1050 | } |
1051 | |
1052 | # |
1053 | # csch |
1054 | # |
1055 | # Computes the hyperbolic cosecant csch(z) = 1 / sinh(z). |
66730be0 |
1056 | # |
0c721ce2 |
1057 | sub csch { |
1058 | my ($z) = @_; |
1059 | my $sz = sinh($z); |
0e505df1 |
1060 | _divbyzero "csch($z)", "sinh($z)" if ($sz == 0); |
0c721ce2 |
1061 | return 1 / $sz; |
1062 | } |
1063 | |
1064 | # |
1065 | # cosech |
1066 | # |
1067 | # Alias for csch(). |
1068 | # |
1069 | sub cosech { Math::Complex::csch(@_) } |
1070 | |
66730be0 |
1071 | # |
0c721ce2 |
1072 | # coth |
1073 | # |
1074 | # Computes the hyperbolic cotangent coth(z) = cosh(z) / sinh(z). |
1075 | # |
1076 | sub coth { |
66730be0 |
1077 | my ($z) = @_; |
0c721ce2 |
1078 | my $sz = sinh($z); |
1fa12f56 |
1079 | _divbyzero "coth($z)", "sinh($z)" if $sz == 0; |
0c721ce2 |
1080 | return cosh($z) / $sz; |
66730be0 |
1081 | } |
1082 | |
1083 | # |
0c721ce2 |
1084 | # cotanh |
1085 | # |
1086 | # Alias for coth(). |
1087 | # |
1088 | sub cotanh { Math::Complex::coth(@_) } |
1089 | |
1090 | # |
66730be0 |
1091 | # acosh |
1092 | # |
fb73857a |
1093 | # Computes the arc hyperbolic cosine acosh(z) = log(z + sqrt(z*z-1)). |
66730be0 |
1094 | # |
1095 | sub acosh { |
1096 | my ($z) = @_; |
fb73857a |
1097 | unless (ref $z) { |
fb73857a |
1098 | $z = cplx($z, 0); |
1099 | } |
8c03c583 |
1100 | my ($re, $im) = @{$z->cartesian}; |
fb73857a |
1101 | if ($im == 0) { |
1fa12f56 |
1102 | return CORE::log($re + CORE::sqrt($re*$re - 1)) |
1103 | if $re >= 1; |
1104 | return cplx(0, CORE::atan2(CORE::sqrt(1 - $re*$re), $re)) |
1105 | if CORE::abs($re) < 1; |
fb73857a |
1106 | } |
1fa12f56 |
1107 | my $s = &sqrt($z*$z - 1); |
1108 | my $t = $z + $s; |
1109 | $t = 1/(2*$s) if $t == 0 || $t && &abs(cosh(&log($t)) - $z) > $eps; |
1110 | return &log($t); |
66730be0 |
1111 | } |
1112 | |
1113 | # |
1114 | # asinh |
1115 | # |
1fa12f56 |
1116 | # Computes the arc hyperbolic sine asinh(z) = log(z + sqrt(z*z+1)) |
66730be0 |
1117 | # |
1118 | sub asinh { |
1119 | my ($z) = @_; |
1fa12f56 |
1120 | unless (ref $z) { |
1121 | my $t = $z + CORE::sqrt($z*$z + 1); |
1122 | return CORE::log($t) if $t; |
1123 | } |
1124 | my $s = &sqrt($z*$z + 1); |
1125 | my $t = $z + $s; |
1126 | # Try Taylor series if looking bad. |
1127 | $t = 1/(2*$s) if $t == 0 || $t && &abs(sinh(&log($t)) - $z) > $eps; |
1128 | return &log($t); |
66730be0 |
1129 | } |
1130 | |
1131 | # |
1132 | # atanh |
1133 | # |
1134 | # Computes the arc hyperbolic tangent atanh(z) = 1/2 log((1+z) / (1-z)). |
1135 | # |
1136 | sub atanh { |
1137 | my ($z) = @_; |
fb73857a |
1138 | unless (ref $z) { |
a8693bd3 |
1139 | return CORE::log((1 + $z)/(1 - $z))/2 if CORE::abs($z) < 1; |
fb73857a |
1140 | $z = cplx($z, 0); |
1141 | } |
1fa12f56 |
1142 | _divbyzero 'atanh(1)', "1 - $z" if (1 - $z == 0); |
1143 | _logofzero 'atanh(-1)' if (1 + $z == 0); |
1144 | return 0.5 * &log((1 + $z) / (1 - $z)); |
66730be0 |
1145 | } |
1146 | |
1147 | # |
0c721ce2 |
1148 | # asech |
1149 | # |
1150 | # Computes the hyperbolic arc secant asech(z) = acosh(1 / z). |
1151 | # |
1152 | sub asech { |
1153 | my ($z) = @_; |
1fa12f56 |
1154 | _divbyzero 'asech(0)', "$z" if ($z == 0); |
0c721ce2 |
1155 | return acosh(1 / $z); |
1156 | } |
1157 | |
1158 | # |
1159 | # acsch |
66730be0 |
1160 | # |
0c721ce2 |
1161 | # Computes the hyperbolic arc cosecant acsch(z) = asinh(1 / z). |
66730be0 |
1162 | # |
0c721ce2 |
1163 | sub acsch { |
66730be0 |
1164 | my ($z) = @_; |
0e505df1 |
1165 | _divbyzero 'acsch(0)', $z if ($z == 0); |
0c721ce2 |
1166 | return asinh(1 / $z); |
1167 | } |
1168 | |
1169 | # |
1170 | # acosech |
1171 | # |
1172 | # Alias for acosh(). |
1173 | # |
1174 | sub acosech { Math::Complex::acsch(@_) } |
1175 | |
1176 | # |
1177 | # acoth |
1178 | # |
1179 | # Computes the arc hyperbolic cotangent acoth(z) = 1/2 log((1+z) / (z-1)). |
1180 | # |
1181 | sub acoth { |
1182 | my ($z) = @_; |
1fa12f56 |
1183 | _divbyzero 'acoth(0)' if ($z == 0); |
fb73857a |
1184 | unless (ref $z) { |
a8693bd3 |
1185 | return CORE::log(($z + 1)/($z - 1))/2 if CORE::abs($z) > 1; |
fb73857a |
1186 | $z = cplx($z, 0); |
1187 | } |
1fa12f56 |
1188 | _divbyzero 'acoth(1)', "$z - 1" if ($z - 1 == 0); |
1189 | _logofzero 'acoth(-1)', "1 + $z" if (1 + $z == 0); |
1190 | return &log((1 + $z) / ($z - 1)) / 2; |
66730be0 |
1191 | } |
1192 | |
1193 | # |
0c721ce2 |
1194 | # acotanh |
1195 | # |
1196 | # Alias for acot(). |
1197 | # |
1198 | sub acotanh { Math::Complex::acoth(@_) } |
1199 | |
1200 | # |
66730be0 |
1201 | # (atan2) |
1202 | # |
1203 | # Compute atan(z1/z2). |
1204 | # |
1205 | sub atan2 { |
1206 | my ($z1, $z2, $inverted) = @_; |
fb73857a |
1207 | my ($re1, $im1, $re2, $im2); |
1208 | if ($inverted) { |
1209 | ($re1, $im1) = ref $z2 ? @{$z2->cartesian} : ($z2, 0); |
1210 | ($re2, $im2) = @{$z1->cartesian}; |
66730be0 |
1211 | } else { |
fb73857a |
1212 | ($re1, $im1) = @{$z1->cartesian}; |
1213 | ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0); |
1214 | } |
1215 | if ($im2 == 0) { |
1fa12f56 |
1216 | return CORE::atan2($re1, $re2) if $im1 == 0; |
1217 | return ($im1<=>0) * pip2 if $re2 == 0; |
66730be0 |
1218 | } |
fb73857a |
1219 | my $w = atan($z1/$z2); |
1220 | my ($u, $v) = ref $w ? @{$w->cartesian} : ($w, 0); |
1221 | $u += pi if $re2 < 0; |
1222 | $u -= pit2 if $u > pi; |
1223 | return cplx($u, $v); |
66730be0 |
1224 | } |
1225 | |
1226 | # |
1227 | # display_format |
1228 | # ->display_format |
1229 | # |
16357284 |
1230 | # Set (get if no argument) the display format for all complex numbers that |
fb73857a |
1231 | # don't happen to have overridden it via ->display_format |
66730be0 |
1232 | # |
16357284 |
1233 | # When called as an object method, this actually sets the display format for |
66730be0 |
1234 | # the current object. |
1235 | # |
1236 | # Valid object formats are 'c' and 'p' for cartesian and polar. The first |
1237 | # letter is used actually, so the type can be fully spelled out for clarity. |
1238 | # |
1239 | sub display_format { |
16357284 |
1240 | my $self = shift; |
1241 | my %display_format = %DISPLAY_FORMAT; |
66730be0 |
1242 | |
16357284 |
1243 | if (ref $self) { # Called as an object method |
1244 | if (exists $self->{display_format}) { |
1245 | my %obj = %{$self->{display_format}}; |
1246 | @display_format{keys %obj} = values %obj; |
1247 | } |
1248 | if (@_ == 1) { |
1249 | $display_format{style} = shift; |
1250 | } else { |
1251 | my %new = @_; |
1252 | @display_format{keys %new} = values %new; |
1253 | } |
1254 | } else { # Called as a class method |
1255 | if (@_ = 1) { |
1256 | $display_format{style} = $self; |
1257 | } else { |
1258 | my %new = @_; |
1259 | @display_format{keys %new} = values %new; |
1260 | } |
1261 | undef $self; |
66730be0 |
1262 | } |
1263 | |
1264 | if (defined $self) { |
16357284 |
1265 | $self->{display_format} = { %display_format }; |
1266 | return |
1267 | wantarray ? |
1268 | %{$self->{display_format}} : |
1269 | $self->{display_format}->{style}; |
66730be0 |
1270 | } |
1271 | |
16357284 |
1272 | %DISPLAY_FORMAT = %display_format; |
1273 | return |
1274 | wantarray ? |
1275 | %DISPLAY_FORMAT : |
1276 | $DISPLAY_FORMAT{style}; |
66730be0 |
1277 | } |
1278 | |
1279 | # |
1280 | # (stringify) |
1281 | # |
1282 | # Show nicely formatted complex number under its cartesian or polar form, |
1283 | # depending on the current display format: |
1284 | # |
1285 | # . If a specific display format has been recorded for this object, use it. |
1286 | # . Otherwise, use the generic current default for all complex numbers, |
1287 | # which is a package global variable. |
1288 | # |
a0d0e21e |
1289 | sub stringify { |
66730be0 |
1290 | my ($z) = shift; |
66730be0 |
1291 | |
16357284 |
1292 | my $style = $z->display_format; |
1293 | |
1294 | $style = $DISPLAY_FORMAT{style} unless defined $style; |
66730be0 |
1295 | |
16357284 |
1296 | return $z->stringify_polar if $style =~ /^p/i; |
66730be0 |
1297 | return $z->stringify_cartesian; |
1298 | } |
1299 | |
1300 | # |
1301 | # ->stringify_cartesian |
1302 | # |
1303 | # Stringify as a cartesian representation 'a+bi'. |
1304 | # |
1305 | sub stringify_cartesian { |
1306 | my $z = shift; |
1307 | my ($x, $y) = @{$z->cartesian}; |
1308 | my ($re, $im); |
1309 | |
16357284 |
1310 | my %format = $z->display_format; |
1311 | my $format = $format{format}; |
1312 | |
1fa12f56 |
1313 | if ($x) { |
1314 | if ($x =~ /^NaN[QS]?$/i) { |
1315 | $re = $x; |
1316 | } else { |
1317 | if ($x =~ /^-?$Inf$/oi) { |
1318 | $re = $x; |
1319 | } else { |
1320 | $re = defined $format ? sprintf($format, $x) : $x; |
1321 | } |
1322 | } |
1323 | } else { |
1324 | undef $re; |
1325 | } |
1326 | |
1327 | if ($y) { |
1328 | if ($y == 1) { $im = "" } |
1329 | elsif ($y == -1) { $im = "-" } |
1330 | elsif ($y =~ /^(NaN[QS]?)$/i) { |
1331 | $im = $y; |
1332 | } else { |
1333 | if ($y =~ /^-?$Inf$/oi) { |
1334 | $im = $y; |
1335 | } else { |
1336 | $im = defined $format ? sprintf($format, $y) : $y; |
1337 | } |
1338 | } |
1339 | $im .= "i"; |
1340 | } else { |
1341 | undef $im; |
16357284 |
1342 | } |
66730be0 |
1343 | |
1fa12f56 |
1344 | my $str = $re; |
1345 | |
16357284 |
1346 | if (defined $im) { |
1347 | if ($y < 0) { |
1348 | $str .= $im; |
1fa12f56 |
1349 | } elsif ($y > 0 || $im =~ /^NaN[QS]?i$/i) { |
16357284 |
1350 | $str .= "+" if defined $re; |
1351 | $str .= $im; |
1352 | } |
1fa12f56 |
1353 | } elsif (!defined $re) { |
1354 | $str = "0"; |
16357284 |
1355 | } |
66730be0 |
1356 | |
1357 | return $str; |
1358 | } |
1359 | |
d09ae4e6 |
1360 | |
66730be0 |
1361 | # |
1362 | # ->stringify_polar |
1363 | # |
1364 | # Stringify as a polar representation '[r,t]'. |
1365 | # |
1366 | sub stringify_polar { |
1367 | my $z = shift; |
1368 | my ($r, $t) = @{$z->polar}; |
1369 | my $theta; |
1370 | |
16357284 |
1371 | my %format = $z->display_format; |
1fa12f56 |
1372 | my $format = $format{format}; |
16357284 |
1373 | |
1fa12f56 |
1374 | if ($t =~ /^NaN[QS]?$/i || $t =~ /^-?$Inf$/oi) { |
1375 | $theta = $t; |
1376 | } elsif ($t == pi) { |
1377 | $theta = "pi"; |
1378 | } elsif ($r == 0 || $t == 0) { |
1379 | $theta = defined $format ? sprintf($format, $t) : $t; |
55497cff |
1380 | } |
66730be0 |
1381 | |
1fa12f56 |
1382 | return "[$r,$theta]" if defined $theta; |
1383 | |
66730be0 |
1384 | # |
1fa12f56 |
1385 | # Try to identify pi/n and friends. |
66730be0 |
1386 | # |
1387 | |
1fa12f56 |
1388 | $t -= int(CORE::abs($t) / pit2) * pit2; |
1389 | |
1390 | if ($format{polar_pretty_print}) { |
1391 | my ($a, $b); |
1392 | for $a (2, 3, 4, 6, 8, 12, 16, 24, 30, 32, 36, 48, 60, 64, 72) { |
1393 | $b = $t * $a / pi; |
1394 | if (int($b) == $b) { |
1395 | $b = $b < 0 ? "-" : "" if CORE::abs($b) == 1; |
1396 | $theta = "${b}pi/$a"; |
d09ae4e6 |
1397 | last; |
66730be0 |
1398 | } |
d09ae4e6 |
1399 | } |
66730be0 |
1400 | } |
1401 | |
16357284 |
1402 | if (defined $format) { |
1403 | $r = sprintf($format, $r); |
1fa12f56 |
1404 | $theta = sprintf($format, $theta) unless defined $theta; |
1405 | } else { |
1406 | $theta = $t unless defined $theta; |
16357284 |
1407 | } |
1408 | |
1fa12f56 |
1409 | return "[$r,$theta]"; |
a0d0e21e |
1410 | } |
a5f75d66 |
1411 | |
1412 | 1; |
1413 | __END__ |
1414 | |
1415 | =head1 NAME |
1416 | |
66730be0 |
1417 | Math::Complex - complex numbers and associated mathematical functions |
a5f75d66 |
1418 | |
1419 | =head1 SYNOPSIS |
1420 | |
66730be0 |
1421 | use Math::Complex; |
fb73857a |
1422 | |
66730be0 |
1423 | $z = Math::Complex->make(5, 6); |
1424 | $t = 4 - 3*i + $z; |
1425 | $j = cplxe(1, 2*pi/3); |
a5f75d66 |
1426 | |
1427 | =head1 DESCRIPTION |
1428 | |
66730be0 |
1429 | This package lets you create and manipulate complex numbers. By default, |
1430 | I<Perl> limits itself to real numbers, but an extra C<use> statement brings |
1431 | full complex support, along with a full set of mathematical functions |
1432 | typically associated with and/or extended to complex numbers. |
1433 | |
1434 | If you wonder what complex numbers are, they were invented to be able to solve |
1435 | the following equation: |
1436 | |
1437 | x*x = -1 |
1438 | |
1439 | and by definition, the solution is noted I<i> (engineers use I<j> instead since |
1440 | I<i> usually denotes an intensity, but the name does not matter). The number |
1441 | I<i> is a pure I<imaginary> number. |
1442 | |
1443 | The arithmetics with pure imaginary numbers works just like you would expect |
1444 | it with real numbers... you just have to remember that |
1445 | |
1446 | i*i = -1 |
1447 | |
1448 | so you have: |
1449 | |
1450 | 5i + 7i = i * (5 + 7) = 12i |
1451 | 4i - 3i = i * (4 - 3) = i |
1452 | 4i * 2i = -8 |
1453 | 6i / 2i = 3 |
1454 | 1 / i = -i |
1455 | |
1456 | Complex numbers are numbers that have both a real part and an imaginary |
1457 | part, and are usually noted: |
1458 | |
1459 | a + bi |
1460 | |
1461 | where C<a> is the I<real> part and C<b> is the I<imaginary> part. The |
1462 | arithmetic with complex numbers is straightforward. You have to |
1463 | keep track of the real and the imaginary parts, but otherwise the |
1464 | rules used for real numbers just apply: |
1465 | |
1466 | (4 + 3i) + (5 - 2i) = (4 + 5) + i(3 - 2) = 9 + i |
1467 | (2 + i) * (4 - i) = 2*4 + 4i -2i -i*i = 8 + 2i + 1 = 9 + 2i |
1468 | |
1469 | A graphical representation of complex numbers is possible in a plane |
1470 | (also called the I<complex plane>, but it's really a 2D plane). |
1471 | The number |
1472 | |
1473 | z = a + bi |
1474 | |
1475 | is the point whose coordinates are (a, b). Actually, it would |
1476 | be the vector originating from (0, 0) to (a, b). It follows that the addition |
1477 | of two complex numbers is a vectorial addition. |
1478 | |
1479 | Since there is a bijection between a point in the 2D plane and a complex |
1480 | number (i.e. the mapping is unique and reciprocal), a complex number |
1481 | can also be uniquely identified with polar coordinates: |
1482 | |
1483 | [rho, theta] |
1484 | |
1485 | where C<rho> is the distance to the origin, and C<theta> the angle between |
1486 | the vector and the I<x> axis. There is a notation for this using the |
1487 | exponential form, which is: |
1488 | |
1489 | rho * exp(i * theta) |
1490 | |
1491 | where I<i> is the famous imaginary number introduced above. Conversion |
1492 | between this form and the cartesian form C<a + bi> is immediate: |
1493 | |
1494 | a = rho * cos(theta) |
1495 | b = rho * sin(theta) |
1496 | |
1497 | which is also expressed by this formula: |
1498 | |
fb73857a |
1499 | z = rho * exp(i * theta) = rho * (cos theta + i * sin theta) |
66730be0 |
1500 | |
1501 | In other words, it's the projection of the vector onto the I<x> and I<y> |
1502 | axes. Mathematicians call I<rho> the I<norm> or I<modulus> and I<theta> |
1503 | the I<argument> of the complex number. The I<norm> of C<z> will be |
1504 | noted C<abs(z)>. |
1505 | |
1506 | The polar notation (also known as the trigonometric |
1507 | representation) is much more handy for performing multiplications and |
1508 | divisions of complex numbers, whilst the cartesian notation is better |
fb73857a |
1509 | suited for additions and subtractions. Real numbers are on the I<x> |
1510 | axis, and therefore I<theta> is zero or I<pi>. |
66730be0 |
1511 | |
1512 | All the common operations that can be performed on a real number have |
1513 | been defined to work on complex numbers as well, and are merely |
1514 | I<extensions> of the operations defined on real numbers. This means |
1515 | they keep their natural meaning when there is no imaginary part, provided |
1516 | the number is within their definition set. |
1517 | |
1518 | For instance, the C<sqrt> routine which computes the square root of |
fb73857a |
1519 | its argument is only defined for non-negative real numbers and yields a |
1520 | non-negative real number (it is an application from B<R+> to B<R+>). |
66730be0 |
1521 | If we allow it to return a complex number, then it can be extended to |
1522 | negative real numbers to become an application from B<R> to B<C> (the |
1523 | set of complex numbers): |
1524 | |
1525 | sqrt(x) = x >= 0 ? sqrt(x) : sqrt(-x)*i |
1526 | |
1527 | It can also be extended to be an application from B<C> to B<C>, |
1528 | whilst its restriction to B<R> behaves as defined above by using |
1529 | the following definition: |
1530 | |
1531 | sqrt(z = [r,t]) = sqrt(r) * exp(i * t/2) |
1532 | |
fb73857a |
1533 | Indeed, a negative real number can be noted C<[x,pi]> (the modulus |
1534 | I<x> is always non-negative, so C<[x,pi]> is really C<-x>, a negative |
1535 | number) and the above definition states that |
66730be0 |
1536 | |
1537 | sqrt([x,pi]) = sqrt(x) * exp(i*pi/2) = [sqrt(x),pi/2] = sqrt(x)*i |
1538 | |
1539 | which is exactly what we had defined for negative real numbers above. |
b42d0ec9 |
1540 | The C<sqrt> returns only one of the solutions: if you want the both, |
1541 | use the C<root> function. |
a5f75d66 |
1542 | |
66730be0 |
1543 | All the common mathematical functions defined on real numbers that |
1544 | are extended to complex numbers share that same property of working |
1545 | I<as usual> when the imaginary part is zero (otherwise, it would not |
1546 | be called an extension, would it?). |
a5f75d66 |
1547 | |
66730be0 |
1548 | A I<new> operation possible on a complex number that is |
1549 | the identity for real numbers is called the I<conjugate>, and is noted |
1550 | with an horizontal bar above the number, or C<~z> here. |
a5f75d66 |
1551 | |
66730be0 |
1552 | z = a + bi |
1553 | ~z = a - bi |
a5f75d66 |
1554 | |
66730be0 |
1555 | Simple... Now look: |
a5f75d66 |
1556 | |
66730be0 |
1557 | z * ~z = (a + bi) * (a - bi) = a*a + b*b |
a5f75d66 |
1558 | |
66730be0 |
1559 | We saw that the norm of C<z> was noted C<abs(z)> and was defined as the |
1560 | distance to the origin, also known as: |
a5f75d66 |
1561 | |
66730be0 |
1562 | rho = abs(z) = sqrt(a*a + b*b) |
a5f75d66 |
1563 | |
66730be0 |
1564 | so |
1565 | |
1566 | z * ~z = abs(z) ** 2 |
1567 | |
1568 | If z is a pure real number (i.e. C<b == 0>), then the above yields: |
1569 | |
1570 | a * a = abs(a) ** 2 |
1571 | |
1572 | which is true (C<abs> has the regular meaning for real number, i.e. stands |
1573 | for the absolute value). This example explains why the norm of C<z> is |
1574 | noted C<abs(z)>: it extends the C<abs> function to complex numbers, yet |
1575 | is the regular C<abs> we know when the complex number actually has no |
1576 | imaginary part... This justifies I<a posteriori> our use of the C<abs> |
1577 | notation for the norm. |
1578 | |
1579 | =head1 OPERATIONS |
1580 | |
1581 | Given the following notations: |
1582 | |
1583 | z1 = a + bi = r1 * exp(i * t1) |
1584 | z2 = c + di = r2 * exp(i * t2) |
1585 | z = <any complex or real number> |
1586 | |
1587 | the following (overloaded) operations are supported on complex numbers: |
1588 | |
1589 | z1 + z2 = (a + c) + i(b + d) |
1590 | z1 - z2 = (a - c) + i(b - d) |
1591 | z1 * z2 = (r1 * r2) * exp(i * (t1 + t2)) |
1592 | z1 / z2 = (r1 / r2) * exp(i * (t1 - t2)) |
1593 | z1 ** z2 = exp(z2 * log z1) |
b42d0ec9 |
1594 | ~z = a - bi |
1595 | abs(z) = r1 = sqrt(a*a + b*b) |
1596 | sqrt(z) = sqrt(r1) * exp(i * t/2) |
1597 | exp(z) = exp(a) * exp(i * b) |
1598 | log(z) = log(r1) + i*t |
1599 | sin(z) = 1/2i (exp(i * z1) - exp(-i * z)) |
1600 | cos(z) = 1/2 (exp(i * z1) + exp(-i * z)) |
66730be0 |
1601 | atan2(z1, z2) = atan(z1/z2) |
1602 | |
1603 | The following extra operations are supported on both real and complex |
1604 | numbers: |
1605 | |
1606 | Re(z) = a |
1607 | Im(z) = b |
1608 | arg(z) = t |
b42d0ec9 |
1609 | abs(z) = r |
66730be0 |
1610 | |
1611 | cbrt(z) = z ** (1/3) |
1612 | log10(z) = log(z) / log(10) |
1613 | logn(z, n) = log(z) / log(n) |
1614 | |
1615 | tan(z) = sin(z) / cos(z) |
0c721ce2 |
1616 | |
5aabfad6 |
1617 | csc(z) = 1 / sin(z) |
1618 | sec(z) = 1 / cos(z) |
0c721ce2 |
1619 | cot(z) = 1 / tan(z) |
66730be0 |
1620 | |
1621 | asin(z) = -i * log(i*z + sqrt(1-z*z)) |
fb73857a |
1622 | acos(z) = -i * log(z + i*sqrt(1-z*z)) |
66730be0 |
1623 | atan(z) = i/2 * log((i+z) / (i-z)) |
0c721ce2 |
1624 | |
5aabfad6 |
1625 | acsc(z) = asin(1 / z) |
1626 | asec(z) = acos(1 / z) |
8c03c583 |
1627 | acot(z) = atan(1 / z) = -i/2 * log((i+z) / (z-i)) |
66730be0 |
1628 | |
1629 | sinh(z) = 1/2 (exp(z) - exp(-z)) |
1630 | cosh(z) = 1/2 (exp(z) + exp(-z)) |
0c721ce2 |
1631 | tanh(z) = sinh(z) / cosh(z) = (exp(z) - exp(-z)) / (exp(z) + exp(-z)) |
1632 | |
5aabfad6 |
1633 | csch(z) = 1 / sinh(z) |
1634 | sech(z) = 1 / cosh(z) |
0c721ce2 |
1635 | coth(z) = 1 / tanh(z) |
fb73857a |
1636 | |
66730be0 |
1637 | asinh(z) = log(z + sqrt(z*z+1)) |
1638 | acosh(z) = log(z + sqrt(z*z-1)) |
1639 | atanh(z) = 1/2 * log((1+z) / (1-z)) |
66730be0 |
1640 | |
5aabfad6 |
1641 | acsch(z) = asinh(1 / z) |
1642 | asech(z) = acosh(1 / z) |
0c721ce2 |
1643 | acoth(z) = atanh(1 / z) = 1/2 * log((1+z) / (z-1)) |
1644 | |
b42d0ec9 |
1645 | I<arg>, I<abs>, I<log>, I<csc>, I<cot>, I<acsc>, I<acot>, I<csch>, |
1646 | I<coth>, I<acosech>, I<acotanh>, have aliases I<rho>, I<theta>, I<ln>, |
1647 | I<cosec>, I<cotan>, I<acosec>, I<acotan>, I<cosech>, I<cotanh>, |
1648 | I<acosech>, I<acotanh>, respectively. C<Re>, C<Im>, C<arg>, C<abs>, |
1649 | C<rho>, and C<theta> can be used also also mutators. The C<cbrt> |
1650 | returns only one of the solutions: if you want all three, use the |
1651 | C<root> function. |
0c721ce2 |
1652 | |
1653 | The I<root> function is available to compute all the I<n> |
66730be0 |
1654 | roots of some complex, where I<n> is a strictly positive integer. |
1655 | There are exactly I<n> such roots, returned as a list. Getting the |
1656 | number mathematicians call C<j> such that: |
1657 | |
1658 | 1 + j + j*j = 0; |
1659 | |
1660 | is a simple matter of writing: |
1661 | |
1662 | $j = ((root(1, 3))[1]; |
1663 | |
1664 | The I<k>th root for C<z = [r,t]> is given by: |
1665 | |
1666 | (root(z, n))[k] = r**(1/n) * exp(i * (t + 2*k*pi)/n) |
1667 | |
f4837644 |
1668 | The I<spaceship> comparison operator, E<lt>=E<gt>, is also defined. In |
1669 | order to ensure its restriction to real numbers is conform to what you |
1670 | would expect, the comparison is run on the real part of the complex |
1671 | number first, and imaginary parts are compared only when the real |
1672 | parts match. |
66730be0 |
1673 | |
1674 | =head1 CREATION |
1675 | |
1676 | To create a complex number, use either: |
1677 | |
1678 | $z = Math::Complex->make(3, 4); |
1679 | $z = cplx(3, 4); |
1680 | |
1681 | if you know the cartesian form of the number, or |
1682 | |
1683 | $z = 3 + 4*i; |
1684 | |
fb73857a |
1685 | if you like. To create a number using the polar form, use either: |
66730be0 |
1686 | |
1687 | $z = Math::Complex->emake(5, pi/3); |
1688 | $x = cplxe(5, pi/3); |
1689 | |
0c721ce2 |
1690 | instead. The first argument is the modulus, the second is the angle |
fb73857a |
1691 | (in radians, the full circle is 2*pi). (Mnemonic: C<e> is used as a |
1692 | notation for complex numbers in the polar form). |
66730be0 |
1693 | |
1694 | It is possible to write: |
1695 | |
1696 | $x = cplxe(-3, pi/4); |
1697 | |
16357284 |
1698 | but that will be silently converted into C<[3,-3pi/4]>, since the |
1699 | modulus must be non-negative (it represents the distance to the origin |
1700 | in the complex plane). |
66730be0 |
1701 | |
b42d0ec9 |
1702 | It is also possible to have a complex number as either argument of |
1703 | either the C<make> or C<emake>: the appropriate component of |
1704 | the argument will be used. |
1705 | |
1706 | $z1 = cplx(-2, 1); |
1707 | $z2 = cplx($z1, 4); |
1708 | |
66730be0 |
1709 | =head1 STRINGIFICATION |
1710 | |
1711 | When printed, a complex number is usually shown under its cartesian |
16357284 |
1712 | style I<a+bi>, but there are legitimate cases where the polar style |
66730be0 |
1713 | I<[r,t]> is more appropriate. |
1714 | |
16357284 |
1715 | By calling the class method C<Math::Complex::display_format> and |
1716 | supplying either C<"polar"> or C<"cartesian"> as an argument, you |
5287f86b |
1717 | override the default display style, which is C<"cartesian">. Not |
16357284 |
1718 | supplying any argument returns the current settings. |
66730be0 |
1719 | |
1720 | This default can be overridden on a per-number basis by calling the |
1721 | C<display_format> method instead. As before, not supplying any argument |
5287f86b |
1722 | returns the current display style for this number. Otherwise whatever you |
1723 | specify will be the new display style for I<this> particular number. |
66730be0 |
1724 | |
1725 | For instance: |
1726 | |
1727 | use Math::Complex; |
1728 | |
1729 | Math::Complex::display_format('polar'); |
16357284 |
1730 | $j = (root(1, 3))[1]; |
1731 | print "j = $j\n"; # Prints "j = [1,2pi/3]" |
66730be0 |
1732 | $j->display_format('cartesian'); |
1733 | print "j = $j\n"; # Prints "j = -0.5+0.866025403784439i" |
1734 | |
5287f86b |
1735 | The polar style attempts to emphasize arguments like I<k*pi/n> |
1736 | (where I<n> is a positive integer and I<k> an integer within [-9,+9]), |
1737 | this is called I<polar pretty-printing>. |
66730be0 |
1738 | |
16357284 |
1739 | =head2 CHANGED IN PERL 5.6 |
1740 | |
1741 | The C<display_format> class method and the corresponding |
1742 | C<display_format> object method can now be called using |
1743 | a parameter hash instead of just a one parameter. |
1744 | |
1745 | The old display format style, which can have values C<"cartesian"> or |
1746 | C<"polar">, can be changed using the C<"style"> parameter. (The one |
1747 | parameter calling convention also still works.) |
1748 | |
1749 | There are two new display parameters. |
1750 | |
1751 | The first one is C<"format">, which is a sprintf()-style format |
1752 | string to be used for both parts of the complex number(s). The |
1753 | default is C<undef>, which corresponds usually (this is somewhat |
1754 | system-dependent) to C<"%.15g">. You can revert to the default by |
1755 | setting the format string to C<undef>. |
1756 | |
1757 | # the $j from the above example |
1758 | |
1759 | $j->display_format('format' => '%.5f'); |
1760 | print "j = $j\n"; # Prints "j = -0.50000+0.86603i" |
1761 | $j->display_format('format' => '%.6f'); |
1762 | print "j = $j\n"; # Prints "j = -0.5+0.86603i" |
1763 | |
1764 | Notice that this affects also the return values of the |
1765 | C<display_format> methods: in list context the whole parameter hash |
1766 | will be returned, as opposed to only the style parameter value. If |
1767 | you want to know the whole truth for a complex number, you must call |
1768 | both the class method and the object method: |
1769 | |
5287f86b |
1770 | The second new display parameter is C<"polar_pretty_print">, which can |
1771 | be set to true or false, the default being true. See the previous |
1772 | section for what this means. |
16357284 |
1773 | |
66730be0 |
1774 | =head1 USAGE |
1775 | |
1776 | Thanks to overloading, the handling of arithmetics with complex numbers |
1777 | is simple and almost transparent. |
1778 | |
1779 | Here are some examples: |
1780 | |
1781 | use Math::Complex; |
1782 | |
1783 | $j = cplxe(1, 2*pi/3); # $j ** 3 == 1 |
1784 | print "j = $j, j**3 = ", $j ** 3, "\n"; |
1785 | print "1 + j + j**2 = ", 1 + $j + $j**2, "\n"; |
1786 | |
1787 | $z = -16 + 0*i; # Force it to be a complex |
1788 | print "sqrt($z) = ", sqrt($z), "\n"; |
1789 | |
1790 | $k = exp(i * 2*pi/3); |
1791 | print "$j - $k = ", $j - $k, "\n"; |
a5f75d66 |
1792 | |
b42d0ec9 |
1793 | $z->Re(3); # Re, Im, arg, abs, |
1794 | $j->arg(2); # (the last two aka rho, theta) |
1795 | # can be used also as mutators. |
1796 | |
1797 | =head1 ERRORS DUE TO DIVISION BY ZERO OR LOGARITHM OF ZERO |
5aabfad6 |
1798 | |
1799 | The division (/) and the following functions |
1800 | |
b42d0ec9 |
1801 | log ln log10 logn |
2820d885 |
1802 | tan sec csc cot |
b42d0ec9 |
1803 | atan asec acsc acot |
1804 | tanh sech csch coth |
1805 | atanh asech acsch acoth |
5aabfad6 |
1806 | |
1807 | cannot be computed for all arguments because that would mean dividing |
8c03c583 |
1808 | by zero or taking logarithm of zero. These situations cause fatal |
1809 | runtime errors looking like this |
5aabfad6 |
1810 | |
1811 | cot(0): Division by zero. |
5cd24f17 |
1812 | (Because in the definition of cot(0), the divisor sin(0) is 0) |
5aabfad6 |
1813 | Died at ... |
1814 | |
8c03c583 |
1815 | or |
1816 | |
1817 | atanh(-1): Logarithm of zero. |
1818 | Died at... |
1819 | |
1820 | For the C<csc>, C<cot>, C<asec>, C<acsc>, C<acot>, C<csch>, C<coth>, |
b42d0ec9 |
1821 | C<asech>, C<acsch>, the argument cannot be C<0> (zero). For the the |
1822 | logarithmic functions and the C<atanh>, C<acoth>, the argument cannot |
1823 | be C<1> (one). For the C<atanh>, C<acoth>, the argument cannot be |
1824 | C<-1> (minus one). For the C<atan>, C<acot>, the argument cannot be |
1825 | C<i> (the imaginary unit). For the C<atan>, C<acoth>, the argument |
1826 | cannot be C<-i> (the negative imaginary unit). For the C<tan>, |
1827 | C<sec>, C<tanh>, the argument cannot be I<pi/2 + k * pi>, where I<k> |
1828 | is any integer. |
1829 | |
1830 | Note that because we are operating on approximations of real numbers, |
1831 | these errors can happen when merely `too close' to the singularities |
1832 | listed above. For example C<tan(2*atan2(1,1)+1e-15)> will die of |
1833 | division by zero. |
1834 | |
1835 | =head1 ERRORS DUE TO INDIGESTIBLE ARGUMENTS |
1836 | |
1837 | The C<make> and C<emake> accept both real and complex arguments. |
1838 | When they cannot recognize the arguments they will die with error |
1839 | messages like the following |
1840 | |
1841 | Math::Complex::make: Cannot take real part of ... |
1842 | Math::Complex::make: Cannot take real part of ... |
1843 | Math::Complex::emake: Cannot take rho of ... |
1844 | Math::Complex::emake: Cannot take theta of ... |
5cd24f17 |
1845 | |
a5f75d66 |
1846 | =head1 BUGS |
1847 | |
5cd24f17 |
1848 | Saying C<use Math::Complex;> exports many mathematical routines in the |
fb73857a |
1849 | caller environment and even overrides some (C<sqrt>, C<log>). |
1850 | This is construed as a feature by the Authors, actually... ;-) |
a5f75d66 |
1851 | |
66730be0 |
1852 | All routines expect to be given real or complex numbers. Don't attempt to |
1853 | use BigFloat, since Perl has currently no rule to disambiguate a '+' |
1854 | operation (for instance) between two overloaded entities. |
a5f75d66 |
1855 | |
d09ae4e6 |
1856 | In Cray UNICOS there is some strange numerical instability that results |
1857 | in root(), cos(), sin(), cosh(), sinh(), losing accuracy fast. Beware. |
1858 | The bug may be in UNICOS math libs, in UNICOS C compiler, in Math::Complex. |
1859 | Whatever it is, it does not manifest itself anywhere else where Perl runs. |
1860 | |
0c721ce2 |
1861 | =head1 AUTHORS |
a5f75d66 |
1862 | |
6e238990 |
1863 | Raphael Manfredi <F<Raphael_Manfredi@pobox.com>> and |
ace5de91 |
1864 | Jarkko Hietaniemi <F<jhi@iki.fi>>. |
5cd24f17 |
1865 | |
fb73857a |
1866 | Extensive patches by Daniel S. Lewart <F<d-lewart@uiuc.edu>>. |
1867 | |
5cd24f17 |
1868 | =cut |
1869 | |
b42d0ec9 |
1870 | 1; |
1871 | |
5cd24f17 |
1872 | # eof |