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 | |
5aabfad6 |
8 | package Math::Complex; |
a0d0e21e |
9 | |
9fbe1b12 |
10 | $VERSION = "1.30"; |
11 | |
12 | our($VERSION, @ISA, @EXPORT, %EXPORT_TAGS, $Inf); |
13 | |
14 | BEGIN { |
15 | eval { require POSIX; import POSIX 'HUGE_VAL' }; |
16 | if (defined &HUGE_VAL) { |
17 | $Inf = sprintf "%g", &HUGE_VAL; |
18 | } else { |
19 | my $e = $!; |
20 | $Inf = CORE::exp(CORE::exp(30)); |
21 | $! = $e; # Clear ERANGE. |
22 | undef $Inf unless $Inf =~ /^inf$/; # Inf INF inf |
23 | } |
24 | $Inf = "Inf" if !defined $Inf || !($Inf > 0); |
25 | } |
fb73857a |
26 | |
9fbe1b12 |
27 | use strict; |
fb73857a |
28 | |
9fbe1b12 |
29 | my $i; |
30 | my %LOGN; |
0c721ce2 |
31 | |
9fbe1b12 |
32 | require Exporter; |
0c721ce2 |
33 | |
5aabfad6 |
34 | @ISA = qw(Exporter); |
35 | |
5aabfad6 |
36 | my @trig = qw( |
37 | pi |
fb73857a |
38 | tan |
5aabfad6 |
39 | csc cosec sec cot cotan |
40 | asin acos atan |
41 | acsc acosec asec acot acotan |
42 | sinh cosh tanh |
43 | csch cosech sech coth cotanh |
44 | asinh acosh atanh |
45 | acsch acosech asech acoth acotanh |
46 | ); |
47 | |
48 | @EXPORT = (qw( |
b42d0ec9 |
49 | i Re Im rho theta arg |
fb73857a |
50 | sqrt log ln |
5aabfad6 |
51 | log10 logn cbrt root |
52 | cplx cplxe |
53 | ), |
54 | @trig); |
55 | |
56 | %EXPORT_TAGS = ( |
57 | 'trig' => [@trig], |
66730be0 |
58 | ); |
a0d0e21e |
59 | |
a5f75d66 |
60 | use overload |
0c721ce2 |
61 | '+' => \&plus, |
62 | '-' => \&minus, |
63 | '*' => \&multiply, |
64 | '/' => \÷, |
66730be0 |
65 | '**' => \&power, |
1fa12f56 |
66 | '==' => \&numeq, |
66730be0 |
67 | '<=>' => \&spaceship, |
68 | 'neg' => \&negate, |
0c721ce2 |
69 | '~' => \&conjugate, |
66730be0 |
70 | 'abs' => \&abs, |
71 | 'sqrt' => \&sqrt, |
72 | 'exp' => \&exp, |
73 | 'log' => \&log, |
74 | 'sin' => \&sin, |
75 | 'cos' => \&cos, |
0c721ce2 |
76 | 'tan' => \&tan, |
66730be0 |
77 | 'atan2' => \&atan2, |
78 | qw("" stringify); |
79 | |
80 | # |
b42d0ec9 |
81 | # Package "privates" |
66730be0 |
82 | # |
83 | |
16357284 |
84 | my %DISPLAY_FORMAT = ('style' => 'cartesian', |
85 | 'polar_pretty_print' => 1); |
86 | my $eps = 1e-14; # Epsilon |
66730be0 |
87 | |
88 | # |
89 | # Object attributes (internal): |
90 | # cartesian [real, imaginary] -- cartesian form |
91 | # polar [rho, theta] -- polar form |
92 | # c_dirty cartesian form not up-to-date |
93 | # p_dirty polar form not up-to-date |
94 | # display display format (package's global when not set) |
95 | # |
96 | |
b42d0ec9 |
97 | # Die on bad *make() arguments. |
98 | |
99 | sub _cannot_make { |
100 | die "@{[(caller(1))[3]]}: Cannot take $_[0] of $_[1].\n"; |
101 | } |
102 | |
66730be0 |
103 | # |
104 | # ->make |
105 | # |
106 | # Create a new complex number (cartesian form) |
107 | # |
108 | sub make { |
109 | my $self = bless {}, shift; |
110 | my ($re, $im) = @_; |
b42d0ec9 |
111 | my $rre = ref $re; |
112 | if ( $rre ) { |
113 | if ( $rre eq ref $self ) { |
114 | $re = Re($re); |
115 | } else { |
116 | _cannot_make("real part", $rre); |
117 | } |
118 | } |
119 | my $rim = ref $im; |
120 | if ( $rim ) { |
121 | if ( $rim eq ref $self ) { |
122 | $im = Im($im); |
123 | } else { |
124 | _cannot_make("imaginary part", $rim); |
125 | } |
126 | } |
127 | $self->{'cartesian'} = [ $re, $im ]; |
66730be0 |
128 | $self->{c_dirty} = 0; |
129 | $self->{p_dirty} = 1; |
b42d0ec9 |
130 | $self->display_format('cartesian'); |
66730be0 |
131 | return $self; |
132 | } |
133 | |
134 | # |
135 | # ->emake |
136 | # |
137 | # Create a new complex number (exponential form) |
138 | # |
139 | sub emake { |
140 | my $self = bless {}, shift; |
141 | my ($rho, $theta) = @_; |
b42d0ec9 |
142 | my $rrh = ref $rho; |
143 | if ( $rrh ) { |
144 | if ( $rrh eq ref $self ) { |
145 | $rho = rho($rho); |
146 | } else { |
147 | _cannot_make("rho", $rrh); |
148 | } |
149 | } |
150 | my $rth = ref $theta; |
151 | if ( $rth ) { |
152 | if ( $rth eq ref $self ) { |
153 | $theta = theta($theta); |
154 | } else { |
155 | _cannot_make("theta", $rth); |
156 | } |
157 | } |
fb73857a |
158 | if ($rho < 0) { |
159 | $rho = -$rho; |
160 | $theta = ($theta <= 0) ? $theta + pi() : $theta - pi(); |
161 | } |
162 | $self->{'polar'} = [$rho, $theta]; |
66730be0 |
163 | $self->{p_dirty} = 0; |
164 | $self->{c_dirty} = 1; |
b42d0ec9 |
165 | $self->display_format('polar'); |
66730be0 |
166 | return $self; |
167 | } |
168 | |
169 | sub new { &make } # For backward compatibility only. |
170 | |
171 | # |
172 | # cplx |
173 | # |
174 | # Creates a complex number from a (re, im) tuple. |
175 | # This avoids the burden of writing Math::Complex->make(re, im). |
176 | # |
177 | sub cplx { |
178 | my ($re, $im) = @_; |
16357284 |
179 | return __PACKAGE__->make($re, defined $im ? $im : 0); |
66730be0 |
180 | } |
181 | |
182 | # |
183 | # cplxe |
184 | # |
185 | # Creates a complex number from a (rho, theta) tuple. |
186 | # This avoids the burden of writing Math::Complex->emake(rho, theta). |
187 | # |
188 | sub cplxe { |
189 | my ($rho, $theta) = @_; |
16357284 |
190 | return __PACKAGE__->emake($rho, defined $theta ? $theta : 0); |
66730be0 |
191 | } |
192 | |
193 | # |
194 | # pi |
195 | # |
fb73857a |
196 | # The number defined as pi = 180 degrees |
66730be0 |
197 | # |
6570f784 |
198 | sub pi () { 4 * CORE::atan2(1, 1) } |
5cd24f17 |
199 | |
200 | # |
fb73857a |
201 | # pit2 |
5cd24f17 |
202 | # |
fb73857a |
203 | # The full circle |
204 | # |
6570f784 |
205 | sub pit2 () { 2 * pi } |
fb73857a |
206 | |
5cd24f17 |
207 | # |
fb73857a |
208 | # pip2 |
209 | # |
210 | # The quarter circle |
211 | # |
6570f784 |
212 | sub pip2 () { pi / 2 } |
5cd24f17 |
213 | |
fb73857a |
214 | # |
d09ae4e6 |
215 | # deg1 |
216 | # |
217 | # One degree in radians, used in stringify_polar. |
218 | # |
219 | |
6570f784 |
220 | sub deg1 () { pi / 180 } |
d09ae4e6 |
221 | |
222 | # |
fb73857a |
223 | # uplog10 |
224 | # |
225 | # Used in log10(). |
226 | # |
6570f784 |
227 | sub uplog10 () { 1 / CORE::log(10) } |
66730be0 |
228 | |
229 | # |
230 | # i |
231 | # |
232 | # The number defined as i*i = -1; |
233 | # |
234 | sub i () { |
5cd24f17 |
235 | return $i if ($i); |
236 | $i = bless {}; |
40da2db3 |
237 | $i->{'cartesian'} = [0, 1]; |
fb73857a |
238 | $i->{'polar'} = [1, pip2]; |
66730be0 |
239 | $i->{c_dirty} = 0; |
240 | $i->{p_dirty} = 0; |
241 | return $i; |
242 | } |
243 | |
244 | # |
1fa12f56 |
245 | # ip2 |
246 | # |
247 | # Half of i. |
248 | # |
249 | sub ip2 () { i / 2 } |
250 | |
251 | # |
66730be0 |
252 | # Attribute access/set routines |
253 | # |
254 | |
0c721ce2 |
255 | sub cartesian {$_[0]->{c_dirty} ? |
256 | $_[0]->update_cartesian : $_[0]->{'cartesian'}} |
257 | sub polar {$_[0]->{p_dirty} ? |
258 | $_[0]->update_polar : $_[0]->{'polar'}} |
66730be0 |
259 | |
40da2db3 |
260 | sub set_cartesian { $_[0]->{p_dirty}++; $_[0]->{'cartesian'} = $_[1] } |
261 | sub set_polar { $_[0]->{c_dirty}++; $_[0]->{'polar'} = $_[1] } |
66730be0 |
262 | |
263 | # |
264 | # ->update_cartesian |
265 | # |
266 | # Recompute and return the cartesian form, given accurate polar form. |
267 | # |
268 | sub update_cartesian { |
269 | my $self = shift; |
40da2db3 |
270 | my ($r, $t) = @{$self->{'polar'}}; |
66730be0 |
271 | $self->{c_dirty} = 0; |
a8693bd3 |
272 | return $self->{'cartesian'} = [$r * CORE::cos($t), $r * CORE::sin($t)]; |
66730be0 |
273 | } |
274 | |
275 | # |
276 | # |
277 | # ->update_polar |
278 | # |
279 | # Recompute and return the polar form, given accurate cartesian form. |
280 | # |
281 | sub update_polar { |
282 | my $self = shift; |
40da2db3 |
283 | my ($x, $y) = @{$self->{'cartesian'}}; |
66730be0 |
284 | $self->{p_dirty} = 0; |
40da2db3 |
285 | return $self->{'polar'} = [0, 0] if $x == 0 && $y == 0; |
1fa12f56 |
286 | return $self->{'polar'} = [CORE::sqrt($x*$x + $y*$y), |
287 | CORE::atan2($y, $x)]; |
66730be0 |
288 | } |
289 | |
290 | # |
291 | # (plus) |
292 | # |
293 | # Computes z1+z2. |
294 | # |
295 | sub plus { |
296 | my ($z1, $z2, $regular) = @_; |
297 | my ($re1, $im1) = @{$z1->cartesian}; |
0e505df1 |
298 | $z2 = cplx($z2) unless ref $z2; |
5cd24f17 |
299 | my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0); |
66730be0 |
300 | unless (defined $regular) { |
301 | $z1->set_cartesian([$re1 + $re2, $im1 + $im2]); |
302 | return $z1; |
303 | } |
304 | return (ref $z1)->make($re1 + $re2, $im1 + $im2); |
305 | } |
306 | |
307 | # |
308 | # (minus) |
309 | # |
310 | # Computes z1-z2. |
311 | # |
312 | sub minus { |
313 | my ($z1, $z2, $inverted) = @_; |
314 | my ($re1, $im1) = @{$z1->cartesian}; |
0e505df1 |
315 | $z2 = cplx($z2) unless ref $z2; |
316 | my ($re2, $im2) = @{$z2->cartesian}; |
66730be0 |
317 | unless (defined $inverted) { |
318 | $z1->set_cartesian([$re1 - $re2, $im1 - $im2]); |
319 | return $z1; |
320 | } |
321 | return $inverted ? |
322 | (ref $z1)->make($re2 - $re1, $im2 - $im1) : |
323 | (ref $z1)->make($re1 - $re2, $im1 - $im2); |
0e505df1 |
324 | |
66730be0 |
325 | } |
326 | |
327 | # |
328 | # (multiply) |
329 | # |
330 | # Computes z1*z2. |
331 | # |
332 | sub multiply { |
fb73857a |
333 | my ($z1, $z2, $regular) = @_; |
334 | if ($z1->{p_dirty} == 0 and ref $z2 and $z2->{p_dirty} == 0) { |
335 | # if both polar better use polar to avoid rounding errors |
336 | my ($r1, $t1) = @{$z1->polar}; |
337 | my ($r2, $t2) = @{$z2->polar}; |
338 | my $t = $t1 + $t2; |
339 | if ($t > pi()) { $t -= pit2 } |
340 | elsif ($t <= -pi()) { $t += pit2 } |
341 | unless (defined $regular) { |
342 | $z1->set_polar([$r1 * $r2, $t]); |
66730be0 |
343 | return $z1; |
fb73857a |
344 | } |
345 | return (ref $z1)->emake($r1 * $r2, $t); |
346 | } else { |
347 | my ($x1, $y1) = @{$z1->cartesian}; |
348 | if (ref $z2) { |
349 | my ($x2, $y2) = @{$z2->cartesian}; |
350 | return (ref $z1)->make($x1*$x2-$y1*$y2, $x1*$y2+$y1*$x2); |
351 | } else { |
352 | return (ref $z1)->make($x1*$z2, $y1*$z2); |
353 | } |
66730be0 |
354 | } |
66730be0 |
355 | } |
356 | |
357 | # |
0e505df1 |
358 | # _divbyzero |
0c721ce2 |
359 | # |
360 | # Die on division by zero. |
361 | # |
0e505df1 |
362 | sub _divbyzero { |
5cd24f17 |
363 | my $mess = "$_[0]: Division by zero.\n"; |
364 | |
365 | if (defined $_[1]) { |
366 | $mess .= "(Because in the definition of $_[0], the divisor "; |
1fa12f56 |
367 | $mess .= "$_[1] " unless ("$_[1]" eq '0'); |
5cd24f17 |
368 | $mess .= "is 0)\n"; |
369 | } |
370 | |
0c721ce2 |
371 | my @up = caller(1); |
fb73857a |
372 | |
5cd24f17 |
373 | $mess .= "Died at $up[1] line $up[2].\n"; |
374 | |
375 | die $mess; |
0c721ce2 |
376 | } |
377 | |
378 | # |
66730be0 |
379 | # (divide) |
380 | # |
381 | # Computes z1/z2. |
382 | # |
383 | sub divide { |
384 | my ($z1, $z2, $inverted) = @_; |
fb73857a |
385 | if ($z1->{p_dirty} == 0 and ref $z2 and $z2->{p_dirty} == 0) { |
386 | # if both polar better use polar to avoid rounding errors |
387 | my ($r1, $t1) = @{$z1->polar}; |
388 | my ($r2, $t2) = @{$z2->polar}; |
389 | my $t; |
390 | if ($inverted) { |
0e505df1 |
391 | _divbyzero "$z2/0" if ($r1 == 0); |
fb73857a |
392 | $t = $t2 - $t1; |
393 | if ($t > pi()) { $t -= pit2 } |
394 | elsif ($t <= -pi()) { $t += pit2 } |
395 | return (ref $z1)->emake($r2 / $r1, $t); |
396 | } else { |
0e505df1 |
397 | _divbyzero "$z1/0" if ($r2 == 0); |
fb73857a |
398 | $t = $t1 - $t2; |
399 | if ($t > pi()) { $t -= pit2 } |
400 | elsif ($t <= -pi()) { $t += pit2 } |
401 | return (ref $z1)->emake($r1 / $r2, $t); |
402 | } |
403 | } else { |
404 | my ($d, $x2, $y2); |
405 | if ($inverted) { |
406 | ($x2, $y2) = @{$z1->cartesian}; |
407 | $d = $x2*$x2 + $y2*$y2; |
408 | _divbyzero "$z2/0" if $d == 0; |
409 | return (ref $z1)->make(($x2*$z2)/$d, -($y2*$z2)/$d); |
410 | } else { |
411 | my ($x1, $y1) = @{$z1->cartesian}; |
412 | if (ref $z2) { |
413 | ($x2, $y2) = @{$z2->cartesian}; |
414 | $d = $x2*$x2 + $y2*$y2; |
415 | _divbyzero "$z1/0" if $d == 0; |
416 | my $u = ($x1*$x2 + $y1*$y2)/$d; |
417 | my $v = ($y1*$x2 - $x1*$y2)/$d; |
418 | return (ref $z1)->make($u, $v); |
419 | } else { |
420 | _divbyzero "$z1/0" if $z2 == 0; |
421 | return (ref $z1)->make($x1/$z2, $y1/$z2); |
422 | } |
423 | } |
0c721ce2 |
424 | } |
66730be0 |
425 | } |
426 | |
427 | # |
428 | # (power) |
429 | # |
430 | # Computes z1**z2 = exp(z2 * log z1)). |
431 | # |
432 | sub power { |
433 | my ($z1, $z2, $inverted) = @_; |
ace5de91 |
434 | if ($inverted) { |
2820d885 |
435 | return 1 if $z1 == 0 || $z2 == 1; |
436 | return 0 if $z2 == 0 && Re($z1) > 0; |
ace5de91 |
437 | } else { |
2820d885 |
438 | return 1 if $z2 == 0 || $z1 == 1; |
439 | return 0 if $z1 == 0 && Re($z2) > 0; |
ace5de91 |
440 | } |
1fa12f56 |
441 | my $w = $inverted ? &exp($z1 * &log($z2)) |
442 | : &exp($z2 * &log($z1)); |
d09ae4e6 |
443 | # If both arguments cartesian, return cartesian, else polar. |
444 | return $z1->{c_dirty} == 0 && |
445 | (not ref $z2 or $z2->{c_dirty} == 0) ? |
446 | cplx(@{$w->cartesian}) : $w; |
66730be0 |
447 | } |
448 | |
449 | # |
450 | # (spaceship) |
451 | # |
452 | # Computes z1 <=> z2. |
2820d885 |
453 | # Sorts on the real part first, then on the imaginary part. Thus 2-4i < 3+8i. |
66730be0 |
454 | # |
455 | sub spaceship { |
456 | my ($z1, $z2, $inverted) = @_; |
5cd24f17 |
457 | my ($re1, $im1) = ref $z1 ? @{$z1->cartesian} : ($z1, 0); |
458 | my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0); |
66730be0 |
459 | my $sgn = $inverted ? -1 : 1; |
460 | return $sgn * ($re1 <=> $re2) if $re1 != $re2; |
461 | return $sgn * ($im1 <=> $im2); |
462 | } |
463 | |
464 | # |
1fa12f56 |
465 | # (numeq) |
466 | # |
467 | # Computes z1 == z2. |
468 | # |
469 | # (Required in addition to spaceship() because of NaNs.) |
470 | sub numeq { |
471 | my ($z1, $z2, $inverted) = @_; |
472 | my ($re1, $im1) = ref $z1 ? @{$z1->cartesian} : ($z1, 0); |
473 | my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0); |
474 | return $re1 == $re2 && $im1 == $im2 ? 1 : 0; |
475 | } |
476 | |
477 | # |
66730be0 |
478 | # (negate) |
479 | # |
480 | # Computes -z. |
481 | # |
482 | sub negate { |
483 | my ($z) = @_; |
484 | if ($z->{c_dirty}) { |
485 | my ($r, $t) = @{$z->polar}; |
fb73857a |
486 | $t = ($t <= 0) ? $t + pi : $t - pi; |
487 | return (ref $z)->emake($r, $t); |
66730be0 |
488 | } |
489 | my ($re, $im) = @{$z->cartesian}; |
490 | return (ref $z)->make(-$re, -$im); |
491 | } |
492 | |
493 | # |
494 | # (conjugate) |
495 | # |
496 | # Compute complex's conjugate. |
497 | # |
498 | sub conjugate { |
499 | my ($z) = @_; |
500 | if ($z->{c_dirty}) { |
501 | my ($r, $t) = @{$z->polar}; |
502 | return (ref $z)->emake($r, -$t); |
503 | } |
504 | my ($re, $im) = @{$z->cartesian}; |
505 | return (ref $z)->make($re, -$im); |
506 | } |
507 | |
508 | # |
509 | # (abs) |
510 | # |
b42d0ec9 |
511 | # Compute or set complex's norm (rho). |
66730be0 |
512 | # |
513 | sub abs { |
b42d0ec9 |
514 | my ($z, $rho) = @_; |
1fa12f56 |
515 | unless (ref $z) { |
516 | if (@_ == 2) { |
517 | $_[0] = $_[1]; |
518 | } else { |
519 | return CORE::abs($z); |
520 | } |
521 | } |
b42d0ec9 |
522 | if (defined $rho) { |
523 | $z->{'polar'} = [ $rho, ${$z->polar}[1] ]; |
524 | $z->{p_dirty} = 0; |
525 | $z->{c_dirty} = 1; |
526 | return $rho; |
527 | } else { |
528 | return ${$z->polar}[0]; |
529 | } |
530 | } |
531 | |
532 | sub _theta { |
533 | my $theta = $_[0]; |
534 | |
535 | if ($$theta > pi()) { $$theta -= pit2 } |
536 | elsif ($$theta <= -pi()) { $$theta += pit2 } |
66730be0 |
537 | } |
538 | |
539 | # |
540 | # arg |
541 | # |
b42d0ec9 |
542 | # Compute or set complex's argument (theta). |
66730be0 |
543 | # |
544 | sub arg { |
b42d0ec9 |
545 | my ($z, $theta) = @_; |
546 | return $z unless ref $z; |
547 | if (defined $theta) { |
548 | _theta(\$theta); |
549 | $z->{'polar'} = [ ${$z->polar}[0], $theta ]; |
550 | $z->{p_dirty} = 0; |
551 | $z->{c_dirty} = 1; |
552 | } else { |
553 | $theta = ${$z->polar}[1]; |
554 | _theta(\$theta); |
555 | } |
556 | return $theta; |
66730be0 |
557 | } |
558 | |
559 | # |
560 | # (sqrt) |
561 | # |
0c721ce2 |
562 | # Compute sqrt(z). |
66730be0 |
563 | # |
b42d0ec9 |
564 | # It is quite tempting to use wantarray here so that in list context |
565 | # sqrt() would return the two solutions. This, however, would |
566 | # break things like |
567 | # |
568 | # print "sqrt(z) = ", sqrt($z), "\n"; |
569 | # |
570 | # The two values would be printed side by side without no intervening |
571 | # whitespace, quite confusing. |
572 | # Therefore if you want the two solutions use the root(). |
573 | # |
66730be0 |
574 | sub sqrt { |
575 | my ($z) = @_; |
b42d0ec9 |
576 | my ($re, $im) = ref $z ? @{$z->cartesian} : ($z, 0); |
1fa12f56 |
577 | return $re < 0 ? cplx(0, CORE::sqrt(-$re)) : CORE::sqrt($re) |
578 | if $im == 0; |
66730be0 |
579 | my ($r, $t) = @{$z->polar}; |
a8693bd3 |
580 | return (ref $z)->emake(CORE::sqrt($r), $t/2); |
66730be0 |
581 | } |
582 | |
583 | # |
584 | # cbrt |
585 | # |
0c721ce2 |
586 | # Compute cbrt(z) (cubic root). |
66730be0 |
587 | # |
b42d0ec9 |
588 | # Why are we not returning three values? The same answer as for sqrt(). |
589 | # |
66730be0 |
590 | sub cbrt { |
591 | my ($z) = @_; |
1fa12f56 |
592 | return $z < 0 ? |
593 | -CORE::exp(CORE::log(-$z)/3) : |
594 | ($z > 0 ? CORE::exp(CORE::log($z)/3): 0) |
fb73857a |
595 | unless ref $z; |
66730be0 |
596 | my ($r, $t) = @{$z->polar}; |
1fa12f56 |
597 | return 0 if $r == 0; |
a8693bd3 |
598 | return (ref $z)->emake(CORE::exp(CORE::log($r)/3), $t/3); |
66730be0 |
599 | } |
600 | |
601 | # |
0e505df1 |
602 | # _rootbad |
603 | # |
604 | # Die on bad root. |
605 | # |
606 | sub _rootbad { |
1fa12f56 |
607 | my $mess = "Root $_[0] illegal, root rank must be positive integer.\n"; |
0e505df1 |
608 | |
609 | my @up = caller(1); |
fb73857a |
610 | |
0e505df1 |
611 | $mess .= "Died at $up[1] line $up[2].\n"; |
612 | |
613 | die $mess; |
614 | } |
615 | |
616 | # |
66730be0 |
617 | # root |
618 | # |
619 | # Computes all nth root for z, returning an array whose size is n. |
620 | # `n' must be a positive integer. |
621 | # |
622 | # The roots are given by (for k = 0..n-1): |
623 | # |
624 | # z^(1/n) = r^(1/n) (cos ((t+2 k pi)/n) + i sin ((t+2 k pi)/n)) |
625 | # |
626 | sub root { |
627 | my ($z, $n) = @_; |
0e505df1 |
628 | _rootbad($n) if ($n < 1 or int($n) != $n); |
1fa12f56 |
629 | my ($r, $t) = ref $z ? |
630 | @{$z->polar} : (CORE::abs($z), $z >= 0 ? 0 : pi); |
66730be0 |
631 | my @root; |
632 | my $k; |
fb73857a |
633 | my $theta_inc = pit2 / $n; |
66730be0 |
634 | my $rho = $r ** (1/$n); |
635 | my $theta; |
d09ae4e6 |
636 | my $cartesian = ref $z && $z->{c_dirty} == 0; |
66730be0 |
637 | for ($k = 0, $theta = $t / $n; $k < $n; $k++, $theta += $theta_inc) { |
d09ae4e6 |
638 | my $w = cplxe($rho, $theta); |
639 | # Yes, $cartesian is loop invariant. |
640 | push @root, $cartesian ? cplx(@{$w->cartesian}) : $w; |
a0d0e21e |
641 | } |
66730be0 |
642 | return @root; |
a0d0e21e |
643 | } |
644 | |
66730be0 |
645 | # |
646 | # Re |
647 | # |
b42d0ec9 |
648 | # Return or set Re(z). |
66730be0 |
649 | # |
a0d0e21e |
650 | sub Re { |
b42d0ec9 |
651 | my ($z, $Re) = @_; |
66730be0 |
652 | return $z unless ref $z; |
b42d0ec9 |
653 | if (defined $Re) { |
654 | $z->{'cartesian'} = [ $Re, ${$z->cartesian}[1] ]; |
655 | $z->{c_dirty} = 0; |
656 | $z->{p_dirty} = 1; |
657 | } else { |
658 | return ${$z->cartesian}[0]; |
659 | } |
a0d0e21e |
660 | } |
661 | |
66730be0 |
662 | # |
663 | # Im |
664 | # |
b42d0ec9 |
665 | # Return or set Im(z). |
66730be0 |
666 | # |
a0d0e21e |
667 | sub Im { |
b42d0ec9 |
668 | my ($z, $Im) = @_; |
669 | return $z unless ref $z; |
670 | if (defined $Im) { |
671 | $z->{'cartesian'} = [ ${$z->cartesian}[0], $Im ]; |
672 | $z->{c_dirty} = 0; |
673 | $z->{p_dirty} = 1; |
674 | } else { |
675 | return ${$z->cartesian}[1]; |
676 | } |
677 | } |
678 | |
679 | # |
680 | # rho |
681 | # |
682 | # Return or set rho(w). |
683 | # |
684 | sub rho { |
685 | Math::Complex::abs(@_); |
686 | } |
687 | |
688 | # |
689 | # theta |
690 | # |
691 | # Return or set theta(w). |
692 | # |
693 | sub theta { |
694 | Math::Complex::arg(@_); |
a0d0e21e |
695 | } |
696 | |
66730be0 |
697 | # |
698 | # (exp) |
699 | # |
700 | # Computes exp(z). |
701 | # |
702 | sub exp { |
703 | my ($z) = @_; |
704 | my ($x, $y) = @{$z->cartesian}; |
a8693bd3 |
705 | return (ref $z)->emake(CORE::exp($x), $y); |
66730be0 |
706 | } |
707 | |
708 | # |
8c03c583 |
709 | # _logofzero |
710 | # |
fb73857a |
711 | # Die on logarithm of zero. |
8c03c583 |
712 | # |
713 | sub _logofzero { |
714 | my $mess = "$_[0]: Logarithm of zero.\n"; |
715 | |
716 | if (defined $_[1]) { |
717 | $mess .= "(Because in the definition of $_[0], the argument "; |
718 | $mess .= "$_[1] " unless ($_[1] eq '0'); |
719 | $mess .= "is 0)\n"; |
720 | } |
721 | |
722 | my @up = caller(1); |
fb73857a |
723 | |
8c03c583 |
724 | $mess .= "Died at $up[1] line $up[2].\n"; |
725 | |
726 | die $mess; |
727 | } |
728 | |
729 | # |
66730be0 |
730 | # (log) |
731 | # |
732 | # Compute log(z). |
733 | # |
734 | sub log { |
735 | my ($z) = @_; |
fb73857a |
736 | unless (ref $z) { |
737 | _logofzero("log") if $z == 0; |
a8693bd3 |
738 | return $z > 0 ? CORE::log($z) : cplx(CORE::log(-$z), pi); |
fb73857a |
739 | } |
5cd24f17 |
740 | my ($r, $t) = @{$z->polar}; |
fb73857a |
741 | _logofzero("log") if $r == 0; |
742 | if ($t > pi()) { $t -= pit2 } |
743 | elsif ($t <= -pi()) { $t += pit2 } |
a8693bd3 |
744 | return (ref $z)->make(CORE::log($r), $t); |
66730be0 |
745 | } |
746 | |
747 | # |
0c721ce2 |
748 | # ln |
749 | # |
750 | # Alias for log(). |
751 | # |
752 | sub ln { Math::Complex::log(@_) } |
753 | |
754 | # |
66730be0 |
755 | # log10 |
756 | # |
757 | # Compute log10(z). |
758 | # |
5cd24f17 |
759 | |
66730be0 |
760 | sub log10 { |
fb73857a |
761 | return Math::Complex::log($_[0]) * uplog10; |
66730be0 |
762 | } |
763 | |
764 | # |
765 | # logn |
766 | # |
767 | # Compute logn(z,n) = log(z) / log(n) |
768 | # |
769 | sub logn { |
770 | my ($z, $n) = @_; |
0c721ce2 |
771 | $z = cplx($z, 0) unless ref $z; |
9fbe1b12 |
772 | my $logn = $LOGN{$n}; |
773 | $logn = $LOGN{$n} = CORE::log($n) unless defined $logn; # Cache log(n) |
1fa12f56 |
774 | return &log($z) / $logn; |
66730be0 |
775 | } |
776 | |
777 | # |
778 | # (cos) |
779 | # |
780 | # Compute cos(z) = (exp(iz) + exp(-iz))/2. |
781 | # |
782 | sub cos { |
783 | my ($z) = @_; |
1fa12f56 |
784 | return CORE::cos($z) unless ref $z; |
66730be0 |
785 | my ($x, $y) = @{$z->cartesian}; |
a8693bd3 |
786 | my $ey = CORE::exp($y); |
1fa12f56 |
787 | my $sx = CORE::sin($x); |
788 | my $cx = CORE::cos($x); |
789 | my $ey_1 = $ey ? 1 / $ey : $Inf; |
790 | return (ref $z)->make($cx * ($ey + $ey_1)/2, |
791 | $sx * ($ey_1 - $ey)/2); |
66730be0 |
792 | } |
793 | |
794 | # |
795 | # (sin) |
796 | # |
797 | # Compute sin(z) = (exp(iz) - exp(-iz))/2. |
798 | # |
799 | sub sin { |
800 | my ($z) = @_; |
1fa12f56 |
801 | return CORE::sin($z) unless ref $z; |
66730be0 |
802 | my ($x, $y) = @{$z->cartesian}; |
a8693bd3 |
803 | my $ey = CORE::exp($y); |
1fa12f56 |
804 | my $sx = CORE::sin($x); |
805 | my $cx = CORE::cos($x); |
806 | my $ey_1 = $ey ? 1 / $ey : $Inf; |
807 | return (ref $z)->make($sx * ($ey + $ey_1)/2, |
808 | $cx * ($ey - $ey_1)/2); |
66730be0 |
809 | } |
810 | |
811 | # |
812 | # tan |
813 | # |
814 | # Compute tan(z) = sin(z) / cos(z). |
815 | # |
816 | sub tan { |
817 | my ($z) = @_; |
1fa12f56 |
818 | my $cz = &cos($z); |
819 | _divbyzero "tan($z)", "cos($z)" if $cz == 0; |
820 | return &sin($z) / $cz; |
66730be0 |
821 | } |
822 | |
823 | # |
0c721ce2 |
824 | # sec |
825 | # |
826 | # Computes the secant sec(z) = 1 / cos(z). |
827 | # |
828 | sub sec { |
829 | my ($z) = @_; |
1fa12f56 |
830 | my $cz = &cos($z); |
0e505df1 |
831 | _divbyzero "sec($z)", "cos($z)" if ($cz == 0); |
0c721ce2 |
832 | return 1 / $cz; |
833 | } |
834 | |
835 | # |
836 | # csc |
837 | # |
838 | # Computes the cosecant csc(z) = 1 / sin(z). |
839 | # |
840 | sub csc { |
841 | my ($z) = @_; |
1fa12f56 |
842 | my $sz = &sin($z); |
0e505df1 |
843 | _divbyzero "csc($z)", "sin($z)" if ($sz == 0); |
0c721ce2 |
844 | return 1 / $sz; |
845 | } |
846 | |
66730be0 |
847 | # |
0c721ce2 |
848 | # cosec |
66730be0 |
849 | # |
0c721ce2 |
850 | # Alias for csc(). |
851 | # |
852 | sub cosec { Math::Complex::csc(@_) } |
853 | |
854 | # |
855 | # cot |
856 | # |
fb73857a |
857 | # Computes cot(z) = cos(z) / sin(z). |
0c721ce2 |
858 | # |
859 | sub cot { |
66730be0 |
860 | my ($z) = @_; |
1fa12f56 |
861 | my $sz = &sin($z); |
0e505df1 |
862 | _divbyzero "cot($z)", "sin($z)" if ($sz == 0); |
1fa12f56 |
863 | return &cos($z) / $sz; |
66730be0 |
864 | } |
865 | |
866 | # |
0c721ce2 |
867 | # cotan |
868 | # |
869 | # Alias for cot(). |
870 | # |
871 | sub cotan { Math::Complex::cot(@_) } |
872 | |
873 | # |
66730be0 |
874 | # acos |
875 | # |
876 | # Computes the arc cosine acos(z) = -i log(z + sqrt(z*z-1)). |
877 | # |
878 | sub acos { |
fb73857a |
879 | my $z = $_[0]; |
1fa12f56 |
880 | return CORE::atan2(CORE::sqrt(1-$z*$z), $z) |
881 | if (! ref $z) && CORE::abs($z) <= 1; |
fb73857a |
882 | my ($x, $y) = ref $z ? @{$z->cartesian} : ($z, 0); |
1fa12f56 |
883 | return 0 if $x == 1 && $y == 0; |
a8693bd3 |
884 | my $t1 = CORE::sqrt(($x+1)*($x+1) + $y*$y); |
885 | my $t2 = CORE::sqrt(($x-1)*($x-1) + $y*$y); |
fb73857a |
886 | my $alpha = ($t1 + $t2)/2; |
887 | my $beta = ($t1 - $t2)/2; |
888 | $alpha = 1 if $alpha < 1; |
889 | if ($beta > 1) { $beta = 1 } |
890 | elsif ($beta < -1) { $beta = -1 } |
a8693bd3 |
891 | my $u = CORE::atan2(CORE::sqrt(1-$beta*$beta), $beta); |
892 | my $v = CORE::log($alpha + CORE::sqrt($alpha*$alpha-1)); |
fb73857a |
893 | $v = -$v if $y > 0 || ($y == 0 && $x < -1); |
16357284 |
894 | return __PACKAGE__->make($u, $v); |
66730be0 |
895 | } |
896 | |
897 | # |
898 | # asin |
899 | # |
900 | # Computes the arc sine asin(z) = -i log(iz + sqrt(1-z*z)). |
901 | # |
902 | sub asin { |
fb73857a |
903 | my $z = $_[0]; |
1fa12f56 |
904 | return CORE::atan2($z, CORE::sqrt(1-$z*$z)) |
905 | if (! ref $z) && CORE::abs($z) <= 1; |
fb73857a |
906 | my ($x, $y) = ref $z ? @{$z->cartesian} : ($z, 0); |
1fa12f56 |
907 | return 0 if $x == 0 && $y == 0; |
a8693bd3 |
908 | my $t1 = CORE::sqrt(($x+1)*($x+1) + $y*$y); |
909 | my $t2 = CORE::sqrt(($x-1)*($x-1) + $y*$y); |
fb73857a |
910 | my $alpha = ($t1 + $t2)/2; |
911 | my $beta = ($t1 - $t2)/2; |
912 | $alpha = 1 if $alpha < 1; |
913 | if ($beta > 1) { $beta = 1 } |
914 | elsif ($beta < -1) { $beta = -1 } |
a8693bd3 |
915 | my $u = CORE::atan2($beta, CORE::sqrt(1-$beta*$beta)); |
916 | my $v = -CORE::log($alpha + CORE::sqrt($alpha*$alpha-1)); |
fb73857a |
917 | $v = -$v if $y > 0 || ($y == 0 && $x < -1); |
16357284 |
918 | return __PACKAGE__->make($u, $v); |
66730be0 |
919 | } |
920 | |
921 | # |
922 | # atan |
923 | # |
0c721ce2 |
924 | # Computes the arc tangent atan(z) = i/2 log((i+z) / (i-z)). |
66730be0 |
925 | # |
926 | sub atan { |
927 | my ($z) = @_; |
a8693bd3 |
928 | return CORE::atan2($z, 1) unless ref $z; |
1fa12f56 |
929 | my ($x, $y) = ref $z ? @{$z->cartesian} : ($z, 0); |
930 | return 0 if $x == 0 && $y == 0; |
8c03c583 |
931 | _divbyzero "atan(i)" if ( $z == i); |
1fa12f56 |
932 | _logofzero "atan(-i)" if (-$z == i); # -i is a bad file test... |
933 | my $log = &log((i + $z) / (i - $z)); |
934 | return ip2 * $log; |
a0d0e21e |
935 | } |
936 | |
66730be0 |
937 | # |
0c721ce2 |
938 | # asec |
939 | # |
940 | # Computes the arc secant asec(z) = acos(1 / z). |
941 | # |
942 | sub asec { |
943 | my ($z) = @_; |
0e505df1 |
944 | _divbyzero "asec($z)", $z if ($z == 0); |
fb73857a |
945 | return acos(1 / $z); |
0c721ce2 |
946 | } |
947 | |
948 | # |
5cd24f17 |
949 | # acsc |
0c721ce2 |
950 | # |
8c03c583 |
951 | # Computes the arc cosecant acsc(z) = asin(1 / z). |
0c721ce2 |
952 | # |
5cd24f17 |
953 | sub acsc { |
0c721ce2 |
954 | my ($z) = @_; |
0e505df1 |
955 | _divbyzero "acsc($z)", $z if ($z == 0); |
fb73857a |
956 | return asin(1 / $z); |
0c721ce2 |
957 | } |
958 | |
959 | # |
5cd24f17 |
960 | # acosec |
66730be0 |
961 | # |
5cd24f17 |
962 | # Alias for acsc(). |
0c721ce2 |
963 | # |
5cd24f17 |
964 | sub acosec { Math::Complex::acsc(@_) } |
0c721ce2 |
965 | |
66730be0 |
966 | # |
0c721ce2 |
967 | # acot |
968 | # |
8c03c583 |
969 | # Computes the arc cotangent acot(z) = atan(1 / z) |
0c721ce2 |
970 | # |
971 | sub acot { |
66730be0 |
972 | my ($z) = @_; |
1fa12f56 |
973 | _divbyzero "acot(0)" if $z == 0; |
974 | return ($z >= 0) ? CORE::atan2(1, $z) : CORE::atan2(-1, -$z) |
975 | unless ref $z; |
976 | _divbyzero "acot(i)" if ($z - i == 0); |
977 | _logofzero "acot(-i)" if ($z + i == 0); |
8c03c583 |
978 | return atan(1 / $z); |
66730be0 |
979 | } |
980 | |
981 | # |
0c721ce2 |
982 | # acotan |
983 | # |
984 | # Alias for acot(). |
985 | # |
986 | sub acotan { Math::Complex::acot(@_) } |
987 | |
988 | # |
66730be0 |
989 | # cosh |
990 | # |
991 | # Computes the hyperbolic cosine cosh(z) = (exp(z) + exp(-z))/2. |
992 | # |
993 | sub cosh { |
994 | my ($z) = @_; |
fb73857a |
995 | my $ex; |
0e505df1 |
996 | unless (ref $z) { |
a8693bd3 |
997 | $ex = CORE::exp($z); |
1fa12f56 |
998 | return $ex ? ($ex + 1/$ex)/2 : $Inf; |
0e505df1 |
999 | } |
1000 | my ($x, $y) = @{$z->cartesian}; |
1fa12f56 |
1001 | my $cy = CORE::cos($y); |
1002 | my $sy = CORE::cos($y); |
a8693bd3 |
1003 | $ex = CORE::exp($x); |
1fa12f56 |
1004 | my $ex_1 = $ex ? 1 / $ex : $Inf; |
a8693bd3 |
1005 | return (ref $z)->make(CORE::cos($y) * ($ex + $ex_1)/2, |
1006 | CORE::sin($y) * ($ex - $ex_1)/2); |
66730be0 |
1007 | } |
1008 | |
1009 | # |
1010 | # sinh |
1011 | # |
1012 | # Computes the hyperbolic sine sinh(z) = (exp(z) - exp(-z))/2. |
1013 | # |
1014 | sub sinh { |
1015 | my ($z) = @_; |
fb73857a |
1016 | my $ex; |
0e505df1 |
1017 | unless (ref $z) { |
1fa12f56 |
1018 | return 0 if $z == 0; |
a8693bd3 |
1019 | $ex = CORE::exp($z); |
1fa12f56 |
1020 | return $ex ? ($ex - 1/$ex)/2 : "-$Inf"; |
0e505df1 |
1021 | } |
1022 | my ($x, $y) = @{$z->cartesian}; |
1fa12f56 |
1023 | my $cy = CORE::cos($y); |
1024 | my $sy = CORE::sin($y); |
a8693bd3 |
1025 | $ex = CORE::exp($x); |
1fa12f56 |
1026 | my $ex_1 = $ex ? 1 / $ex : $Inf; |
1027 | return (ref $z)->make($cy * ($ex - $ex_1)/2, |
1028 | $sy * ($ex + $ex_1)/2); |
66730be0 |
1029 | } |
1030 | |
1031 | # |
1032 | # tanh |
1033 | # |
1034 | # Computes the hyperbolic tangent tanh(z) = sinh(z) / cosh(z). |
1035 | # |
1036 | sub tanh { |
1037 | my ($z) = @_; |
0c721ce2 |
1038 | my $cz = cosh($z); |
0e505df1 |
1039 | _divbyzero "tanh($z)", "cosh($z)" if ($cz == 0); |
0c721ce2 |
1040 | return sinh($z) / $cz; |
66730be0 |
1041 | } |
1042 | |
1043 | # |
0c721ce2 |
1044 | # sech |
1045 | # |
1046 | # Computes the hyperbolic secant sech(z) = 1 / cosh(z). |
1047 | # |
1048 | sub sech { |
1049 | my ($z) = @_; |
1050 | my $cz = cosh($z); |
0e505df1 |
1051 | _divbyzero "sech($z)", "cosh($z)" if ($cz == 0); |
0c721ce2 |
1052 | return 1 / $cz; |
1053 | } |
1054 | |
1055 | # |
1056 | # csch |
1057 | # |
1058 | # Computes the hyperbolic cosecant csch(z) = 1 / sinh(z). |
66730be0 |
1059 | # |
0c721ce2 |
1060 | sub csch { |
1061 | my ($z) = @_; |
1062 | my $sz = sinh($z); |
0e505df1 |
1063 | _divbyzero "csch($z)", "sinh($z)" if ($sz == 0); |
0c721ce2 |
1064 | return 1 / $sz; |
1065 | } |
1066 | |
1067 | # |
1068 | # cosech |
1069 | # |
1070 | # Alias for csch(). |
1071 | # |
1072 | sub cosech { Math::Complex::csch(@_) } |
1073 | |
66730be0 |
1074 | # |
0c721ce2 |
1075 | # coth |
1076 | # |
1077 | # Computes the hyperbolic cotangent coth(z) = cosh(z) / sinh(z). |
1078 | # |
1079 | sub coth { |
66730be0 |
1080 | my ($z) = @_; |
0c721ce2 |
1081 | my $sz = sinh($z); |
1fa12f56 |
1082 | _divbyzero "coth($z)", "sinh($z)" if $sz == 0; |
0c721ce2 |
1083 | return cosh($z) / $sz; |
66730be0 |
1084 | } |
1085 | |
1086 | # |
0c721ce2 |
1087 | # cotanh |
1088 | # |
1089 | # Alias for coth(). |
1090 | # |
1091 | sub cotanh { Math::Complex::coth(@_) } |
1092 | |
1093 | # |
66730be0 |
1094 | # acosh |
1095 | # |
fb73857a |
1096 | # Computes the arc hyperbolic cosine acosh(z) = log(z + sqrt(z*z-1)). |
66730be0 |
1097 | # |
1098 | sub acosh { |
1099 | my ($z) = @_; |
fb73857a |
1100 | unless (ref $z) { |
fb73857a |
1101 | $z = cplx($z, 0); |
1102 | } |
8c03c583 |
1103 | my ($re, $im) = @{$z->cartesian}; |
fb73857a |
1104 | if ($im == 0) { |
1fa12f56 |
1105 | return CORE::log($re + CORE::sqrt($re*$re - 1)) |
1106 | if $re >= 1; |
1107 | return cplx(0, CORE::atan2(CORE::sqrt(1 - $re*$re), $re)) |
1108 | if CORE::abs($re) < 1; |
fb73857a |
1109 | } |
1fa12f56 |
1110 | my $s = &sqrt($z*$z - 1); |
1111 | my $t = $z + $s; |
1112 | $t = 1/(2*$s) if $t == 0 || $t && &abs(cosh(&log($t)) - $z) > $eps; |
1113 | return &log($t); |
66730be0 |
1114 | } |
1115 | |
1116 | # |
1117 | # asinh |
1118 | # |
1fa12f56 |
1119 | # Computes the arc hyperbolic sine asinh(z) = log(z + sqrt(z*z+1)) |
66730be0 |
1120 | # |
1121 | sub asinh { |
1122 | my ($z) = @_; |
1fa12f56 |
1123 | unless (ref $z) { |
1124 | my $t = $z + CORE::sqrt($z*$z + 1); |
1125 | return CORE::log($t) if $t; |
1126 | } |
1127 | my $s = &sqrt($z*$z + 1); |
1128 | my $t = $z + $s; |
1129 | # Try Taylor series if looking bad. |
1130 | $t = 1/(2*$s) if $t == 0 || $t && &abs(sinh(&log($t)) - $z) > $eps; |
1131 | return &log($t); |
66730be0 |
1132 | } |
1133 | |
1134 | # |
1135 | # atanh |
1136 | # |
1137 | # Computes the arc hyperbolic tangent atanh(z) = 1/2 log((1+z) / (1-z)). |
1138 | # |
1139 | sub atanh { |
1140 | my ($z) = @_; |
fb73857a |
1141 | unless (ref $z) { |
a8693bd3 |
1142 | return CORE::log((1 + $z)/(1 - $z))/2 if CORE::abs($z) < 1; |
fb73857a |
1143 | $z = cplx($z, 0); |
1144 | } |
1fa12f56 |
1145 | _divbyzero 'atanh(1)', "1 - $z" if (1 - $z == 0); |
1146 | _logofzero 'atanh(-1)' if (1 + $z == 0); |
1147 | return 0.5 * &log((1 + $z) / (1 - $z)); |
66730be0 |
1148 | } |
1149 | |
1150 | # |
0c721ce2 |
1151 | # asech |
1152 | # |
1153 | # Computes the hyperbolic arc secant asech(z) = acosh(1 / z). |
1154 | # |
1155 | sub asech { |
1156 | my ($z) = @_; |
1fa12f56 |
1157 | _divbyzero 'asech(0)', "$z" if ($z == 0); |
0c721ce2 |
1158 | return acosh(1 / $z); |
1159 | } |
1160 | |
1161 | # |
1162 | # acsch |
66730be0 |
1163 | # |
0c721ce2 |
1164 | # Computes the hyperbolic arc cosecant acsch(z) = asinh(1 / z). |
66730be0 |
1165 | # |
0c721ce2 |
1166 | sub acsch { |
66730be0 |
1167 | my ($z) = @_; |
0e505df1 |
1168 | _divbyzero 'acsch(0)', $z if ($z == 0); |
0c721ce2 |
1169 | return asinh(1 / $z); |
1170 | } |
1171 | |
1172 | # |
1173 | # acosech |
1174 | # |
1175 | # Alias for acosh(). |
1176 | # |
1177 | sub acosech { Math::Complex::acsch(@_) } |
1178 | |
1179 | # |
1180 | # acoth |
1181 | # |
1182 | # Computes the arc hyperbolic cotangent acoth(z) = 1/2 log((1+z) / (z-1)). |
1183 | # |
1184 | sub acoth { |
1185 | my ($z) = @_; |
1fa12f56 |
1186 | _divbyzero 'acoth(0)' if ($z == 0); |
fb73857a |
1187 | unless (ref $z) { |
a8693bd3 |
1188 | return CORE::log(($z + 1)/($z - 1))/2 if CORE::abs($z) > 1; |
fb73857a |
1189 | $z = cplx($z, 0); |
1190 | } |
1fa12f56 |
1191 | _divbyzero 'acoth(1)', "$z - 1" if ($z - 1 == 0); |
1192 | _logofzero 'acoth(-1)', "1 + $z" if (1 + $z == 0); |
1193 | return &log((1 + $z) / ($z - 1)) / 2; |
66730be0 |
1194 | } |
1195 | |
1196 | # |
0c721ce2 |
1197 | # acotanh |
1198 | # |
1199 | # Alias for acot(). |
1200 | # |
1201 | sub acotanh { Math::Complex::acoth(@_) } |
1202 | |
1203 | # |
66730be0 |
1204 | # (atan2) |
1205 | # |
1206 | # Compute atan(z1/z2). |
1207 | # |
1208 | sub atan2 { |
1209 | my ($z1, $z2, $inverted) = @_; |
fb73857a |
1210 | my ($re1, $im1, $re2, $im2); |
1211 | if ($inverted) { |
1212 | ($re1, $im1) = ref $z2 ? @{$z2->cartesian} : ($z2, 0); |
1213 | ($re2, $im2) = @{$z1->cartesian}; |
66730be0 |
1214 | } else { |
fb73857a |
1215 | ($re1, $im1) = @{$z1->cartesian}; |
1216 | ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0); |
1217 | } |
1218 | if ($im2 == 0) { |
1fa12f56 |
1219 | return CORE::atan2($re1, $re2) if $im1 == 0; |
1220 | return ($im1<=>0) * pip2 if $re2 == 0; |
66730be0 |
1221 | } |
fb73857a |
1222 | my $w = atan($z1/$z2); |
1223 | my ($u, $v) = ref $w ? @{$w->cartesian} : ($w, 0); |
1224 | $u += pi if $re2 < 0; |
1225 | $u -= pit2 if $u > pi; |
1226 | return cplx($u, $v); |
66730be0 |
1227 | } |
1228 | |
1229 | # |
1230 | # display_format |
1231 | # ->display_format |
1232 | # |
16357284 |
1233 | # Set (get if no argument) the display format for all complex numbers that |
fb73857a |
1234 | # don't happen to have overridden it via ->display_format |
66730be0 |
1235 | # |
16357284 |
1236 | # When called as an object method, this actually sets the display format for |
66730be0 |
1237 | # the current object. |
1238 | # |
1239 | # Valid object formats are 'c' and 'p' for cartesian and polar. The first |
1240 | # letter is used actually, so the type can be fully spelled out for clarity. |
1241 | # |
1242 | sub display_format { |
16357284 |
1243 | my $self = shift; |
1244 | my %display_format = %DISPLAY_FORMAT; |
66730be0 |
1245 | |
16357284 |
1246 | if (ref $self) { # Called as an object method |
1247 | if (exists $self->{display_format}) { |
1248 | my %obj = %{$self->{display_format}}; |
1249 | @display_format{keys %obj} = values %obj; |
1250 | } |
1251 | if (@_ == 1) { |
1252 | $display_format{style} = shift; |
1253 | } else { |
1254 | my %new = @_; |
1255 | @display_format{keys %new} = values %new; |
1256 | } |
1257 | } else { # Called as a class method |
1258 | if (@_ = 1) { |
1259 | $display_format{style} = $self; |
1260 | } else { |
1261 | my %new = @_; |
1262 | @display_format{keys %new} = values %new; |
1263 | } |
1264 | undef $self; |
66730be0 |
1265 | } |
1266 | |
1267 | if (defined $self) { |
16357284 |
1268 | $self->{display_format} = { %display_format }; |
1269 | return |
1270 | wantarray ? |
1271 | %{$self->{display_format}} : |
1272 | $self->{display_format}->{style}; |
66730be0 |
1273 | } |
1274 | |
16357284 |
1275 | %DISPLAY_FORMAT = %display_format; |
1276 | return |
1277 | wantarray ? |
1278 | %DISPLAY_FORMAT : |
1279 | $DISPLAY_FORMAT{style}; |
66730be0 |
1280 | } |
1281 | |
1282 | # |
1283 | # (stringify) |
1284 | # |
1285 | # Show nicely formatted complex number under its cartesian or polar form, |
1286 | # depending on the current display format: |
1287 | # |
1288 | # . If a specific display format has been recorded for this object, use it. |
1289 | # . Otherwise, use the generic current default for all complex numbers, |
1290 | # which is a package global variable. |
1291 | # |
a0d0e21e |
1292 | sub stringify { |
66730be0 |
1293 | my ($z) = shift; |
66730be0 |
1294 | |
16357284 |
1295 | my $style = $z->display_format; |
1296 | |
1297 | $style = $DISPLAY_FORMAT{style} unless defined $style; |
66730be0 |
1298 | |
16357284 |
1299 | return $z->stringify_polar if $style =~ /^p/i; |
66730be0 |
1300 | return $z->stringify_cartesian; |
1301 | } |
1302 | |
1303 | # |
1304 | # ->stringify_cartesian |
1305 | # |
1306 | # Stringify as a cartesian representation 'a+bi'. |
1307 | # |
1308 | sub stringify_cartesian { |
1309 | my $z = shift; |
1310 | my ($x, $y) = @{$z->cartesian}; |
1311 | my ($re, $im); |
1312 | |
16357284 |
1313 | my %format = $z->display_format; |
1314 | my $format = $format{format}; |
1315 | |
1fa12f56 |
1316 | if ($x) { |
1317 | if ($x =~ /^NaN[QS]?$/i) { |
1318 | $re = $x; |
1319 | } else { |
1320 | if ($x =~ /^-?$Inf$/oi) { |
1321 | $re = $x; |
1322 | } else { |
1323 | $re = defined $format ? sprintf($format, $x) : $x; |
1324 | } |
1325 | } |
1326 | } else { |
1327 | undef $re; |
1328 | } |
1329 | |
1330 | if ($y) { |
1331 | if ($y == 1) { $im = "" } |
1332 | elsif ($y == -1) { $im = "-" } |
1333 | elsif ($y =~ /^(NaN[QS]?)$/i) { |
1334 | $im = $y; |
1335 | } else { |
1336 | if ($y =~ /^-?$Inf$/oi) { |
1337 | $im = $y; |
1338 | } else { |
1339 | $im = defined $format ? sprintf($format, $y) : $y; |
1340 | } |
1341 | } |
1342 | $im .= "i"; |
1343 | } else { |
1344 | undef $im; |
16357284 |
1345 | } |
66730be0 |
1346 | |
1fa12f56 |
1347 | my $str = $re; |
1348 | |
16357284 |
1349 | if (defined $im) { |
1350 | if ($y < 0) { |
1351 | $str .= $im; |
1fa12f56 |
1352 | } elsif ($y > 0 || $im =~ /^NaN[QS]?i$/i) { |
16357284 |
1353 | $str .= "+" if defined $re; |
1354 | $str .= $im; |
1355 | } |
1fa12f56 |
1356 | } elsif (!defined $re) { |
1357 | $str = "0"; |
16357284 |
1358 | } |
66730be0 |
1359 | |
1360 | return $str; |
1361 | } |
1362 | |
d09ae4e6 |
1363 | |
66730be0 |
1364 | # |
1365 | # ->stringify_polar |
1366 | # |
1367 | # Stringify as a polar representation '[r,t]'. |
1368 | # |
1369 | sub stringify_polar { |
1370 | my $z = shift; |
1371 | my ($r, $t) = @{$z->polar}; |
1372 | my $theta; |
1373 | |
16357284 |
1374 | my %format = $z->display_format; |
1fa12f56 |
1375 | my $format = $format{format}; |
16357284 |
1376 | |
1fa12f56 |
1377 | if ($t =~ /^NaN[QS]?$/i || $t =~ /^-?$Inf$/oi) { |
1378 | $theta = $t; |
1379 | } elsif ($t == pi) { |
1380 | $theta = "pi"; |
1381 | } elsif ($r == 0 || $t == 0) { |
1382 | $theta = defined $format ? sprintf($format, $t) : $t; |
55497cff |
1383 | } |
66730be0 |
1384 | |
1fa12f56 |
1385 | return "[$r,$theta]" if defined $theta; |
1386 | |
66730be0 |
1387 | # |
1fa12f56 |
1388 | # Try to identify pi/n and friends. |
66730be0 |
1389 | # |
1390 | |
1fa12f56 |
1391 | $t -= int(CORE::abs($t) / pit2) * pit2; |
1392 | |
1393 | if ($format{polar_pretty_print}) { |
1394 | my ($a, $b); |
1395 | for $a (2, 3, 4, 6, 8, 12, 16, 24, 30, 32, 36, 48, 60, 64, 72) { |
1396 | $b = $t * $a / pi; |
1397 | if (int($b) == $b) { |
1398 | $b = $b < 0 ? "-" : "" if CORE::abs($b) == 1; |
1399 | $theta = "${b}pi/$a"; |
d09ae4e6 |
1400 | last; |
66730be0 |
1401 | } |
d09ae4e6 |
1402 | } |
66730be0 |
1403 | } |
1404 | |
16357284 |
1405 | if (defined $format) { |
1406 | $r = sprintf($format, $r); |
1fa12f56 |
1407 | $theta = sprintf($format, $theta) unless defined $theta; |
1408 | } else { |
1409 | $theta = $t unless defined $theta; |
16357284 |
1410 | } |
1411 | |
1fa12f56 |
1412 | return "[$r,$theta]"; |
a0d0e21e |
1413 | } |
a5f75d66 |
1414 | |
1415 | 1; |
1416 | __END__ |
1417 | |
1418 | =head1 NAME |
1419 | |
66730be0 |
1420 | Math::Complex - complex numbers and associated mathematical functions |
a5f75d66 |
1421 | |
1422 | =head1 SYNOPSIS |
1423 | |
66730be0 |
1424 | use Math::Complex; |
fb73857a |
1425 | |
66730be0 |
1426 | $z = Math::Complex->make(5, 6); |
1427 | $t = 4 - 3*i + $z; |
1428 | $j = cplxe(1, 2*pi/3); |
a5f75d66 |
1429 | |
1430 | =head1 DESCRIPTION |
1431 | |
66730be0 |
1432 | This package lets you create and manipulate complex numbers. By default, |
1433 | I<Perl> limits itself to real numbers, but an extra C<use> statement brings |
1434 | full complex support, along with a full set of mathematical functions |
1435 | typically associated with and/or extended to complex numbers. |
1436 | |
1437 | If you wonder what complex numbers are, they were invented to be able to solve |
1438 | the following equation: |
1439 | |
1440 | x*x = -1 |
1441 | |
1442 | and by definition, the solution is noted I<i> (engineers use I<j> instead since |
1443 | I<i> usually denotes an intensity, but the name does not matter). The number |
1444 | I<i> is a pure I<imaginary> number. |
1445 | |
1446 | The arithmetics with pure imaginary numbers works just like you would expect |
1447 | it with real numbers... you just have to remember that |
1448 | |
1449 | i*i = -1 |
1450 | |
1451 | so you have: |
1452 | |
1453 | 5i + 7i = i * (5 + 7) = 12i |
1454 | 4i - 3i = i * (4 - 3) = i |
1455 | 4i * 2i = -8 |
1456 | 6i / 2i = 3 |
1457 | 1 / i = -i |
1458 | |
1459 | Complex numbers are numbers that have both a real part and an imaginary |
1460 | part, and are usually noted: |
1461 | |
1462 | a + bi |
1463 | |
1464 | where C<a> is the I<real> part and C<b> is the I<imaginary> part. The |
1465 | arithmetic with complex numbers is straightforward. You have to |
1466 | keep track of the real and the imaginary parts, but otherwise the |
1467 | rules used for real numbers just apply: |
1468 | |
1469 | (4 + 3i) + (5 - 2i) = (4 + 5) + i(3 - 2) = 9 + i |
1470 | (2 + i) * (4 - i) = 2*4 + 4i -2i -i*i = 8 + 2i + 1 = 9 + 2i |
1471 | |
1472 | A graphical representation of complex numbers is possible in a plane |
1473 | (also called the I<complex plane>, but it's really a 2D plane). |
1474 | The number |
1475 | |
1476 | z = a + bi |
1477 | |
1478 | is the point whose coordinates are (a, b). Actually, it would |
1479 | be the vector originating from (0, 0) to (a, b). It follows that the addition |
1480 | of two complex numbers is a vectorial addition. |
1481 | |
1482 | Since there is a bijection between a point in the 2D plane and a complex |
1483 | number (i.e. the mapping is unique and reciprocal), a complex number |
1484 | can also be uniquely identified with polar coordinates: |
1485 | |
1486 | [rho, theta] |
1487 | |
1488 | where C<rho> is the distance to the origin, and C<theta> the angle between |
1489 | the vector and the I<x> axis. There is a notation for this using the |
1490 | exponential form, which is: |
1491 | |
1492 | rho * exp(i * theta) |
1493 | |
1494 | where I<i> is the famous imaginary number introduced above. Conversion |
1495 | between this form and the cartesian form C<a + bi> is immediate: |
1496 | |
1497 | a = rho * cos(theta) |
1498 | b = rho * sin(theta) |
1499 | |
1500 | which is also expressed by this formula: |
1501 | |
fb73857a |
1502 | z = rho * exp(i * theta) = rho * (cos theta + i * sin theta) |
66730be0 |
1503 | |
1504 | In other words, it's the projection of the vector onto the I<x> and I<y> |
1505 | axes. Mathematicians call I<rho> the I<norm> or I<modulus> and I<theta> |
1506 | the I<argument> of the complex number. The I<norm> of C<z> will be |
1507 | noted C<abs(z)>. |
1508 | |
1509 | The polar notation (also known as the trigonometric |
1510 | representation) is much more handy for performing multiplications and |
1511 | divisions of complex numbers, whilst the cartesian notation is better |
fb73857a |
1512 | suited for additions and subtractions. Real numbers are on the I<x> |
1513 | axis, and therefore I<theta> is zero or I<pi>. |
66730be0 |
1514 | |
1515 | All the common operations that can be performed on a real number have |
1516 | been defined to work on complex numbers as well, and are merely |
1517 | I<extensions> of the operations defined on real numbers. This means |
1518 | they keep their natural meaning when there is no imaginary part, provided |
1519 | the number is within their definition set. |
1520 | |
1521 | For instance, the C<sqrt> routine which computes the square root of |
fb73857a |
1522 | its argument is only defined for non-negative real numbers and yields a |
1523 | non-negative real number (it is an application from B<R+> to B<R+>). |
66730be0 |
1524 | If we allow it to return a complex number, then it can be extended to |
1525 | negative real numbers to become an application from B<R> to B<C> (the |
1526 | set of complex numbers): |
1527 | |
1528 | sqrt(x) = x >= 0 ? sqrt(x) : sqrt(-x)*i |
1529 | |
1530 | It can also be extended to be an application from B<C> to B<C>, |
1531 | whilst its restriction to B<R> behaves as defined above by using |
1532 | the following definition: |
1533 | |
1534 | sqrt(z = [r,t]) = sqrt(r) * exp(i * t/2) |
1535 | |
fb73857a |
1536 | Indeed, a negative real number can be noted C<[x,pi]> (the modulus |
1537 | I<x> is always non-negative, so C<[x,pi]> is really C<-x>, a negative |
1538 | number) and the above definition states that |
66730be0 |
1539 | |
1540 | sqrt([x,pi]) = sqrt(x) * exp(i*pi/2) = [sqrt(x),pi/2] = sqrt(x)*i |
1541 | |
1542 | which is exactly what we had defined for negative real numbers above. |
b42d0ec9 |
1543 | The C<sqrt> returns only one of the solutions: if you want the both, |
1544 | use the C<root> function. |
a5f75d66 |
1545 | |
66730be0 |
1546 | All the common mathematical functions defined on real numbers that |
1547 | are extended to complex numbers share that same property of working |
1548 | I<as usual> when the imaginary part is zero (otherwise, it would not |
1549 | be called an extension, would it?). |
a5f75d66 |
1550 | |
66730be0 |
1551 | A I<new> operation possible on a complex number that is |
1552 | the identity for real numbers is called the I<conjugate>, and is noted |
1553 | with an horizontal bar above the number, or C<~z> here. |
a5f75d66 |
1554 | |
66730be0 |
1555 | z = a + bi |
1556 | ~z = a - bi |
a5f75d66 |
1557 | |
66730be0 |
1558 | Simple... Now look: |
a5f75d66 |
1559 | |
66730be0 |
1560 | z * ~z = (a + bi) * (a - bi) = a*a + b*b |
a5f75d66 |
1561 | |
66730be0 |
1562 | We saw that the norm of C<z> was noted C<abs(z)> and was defined as the |
1563 | distance to the origin, also known as: |
a5f75d66 |
1564 | |
66730be0 |
1565 | rho = abs(z) = sqrt(a*a + b*b) |
a5f75d66 |
1566 | |
66730be0 |
1567 | so |
1568 | |
1569 | z * ~z = abs(z) ** 2 |
1570 | |
1571 | If z is a pure real number (i.e. C<b == 0>), then the above yields: |
1572 | |
1573 | a * a = abs(a) ** 2 |
1574 | |
1575 | which is true (C<abs> has the regular meaning for real number, i.e. stands |
1576 | for the absolute value). This example explains why the norm of C<z> is |
1577 | noted C<abs(z)>: it extends the C<abs> function to complex numbers, yet |
1578 | is the regular C<abs> we know when the complex number actually has no |
1579 | imaginary part... This justifies I<a posteriori> our use of the C<abs> |
1580 | notation for the norm. |
1581 | |
1582 | =head1 OPERATIONS |
1583 | |
1584 | Given the following notations: |
1585 | |
1586 | z1 = a + bi = r1 * exp(i * t1) |
1587 | z2 = c + di = r2 * exp(i * t2) |
1588 | z = <any complex or real number> |
1589 | |
1590 | the following (overloaded) operations are supported on complex numbers: |
1591 | |
1592 | z1 + z2 = (a + c) + i(b + d) |
1593 | z1 - z2 = (a - c) + i(b - d) |
1594 | z1 * z2 = (r1 * r2) * exp(i * (t1 + t2)) |
1595 | z1 / z2 = (r1 / r2) * exp(i * (t1 - t2)) |
1596 | z1 ** z2 = exp(z2 * log z1) |
b42d0ec9 |
1597 | ~z = a - bi |
1598 | abs(z) = r1 = sqrt(a*a + b*b) |
1599 | sqrt(z) = sqrt(r1) * exp(i * t/2) |
1600 | exp(z) = exp(a) * exp(i * b) |
1601 | log(z) = log(r1) + i*t |
1602 | sin(z) = 1/2i (exp(i * z1) - exp(-i * z)) |
1603 | cos(z) = 1/2 (exp(i * z1) + exp(-i * z)) |
66730be0 |
1604 | atan2(z1, z2) = atan(z1/z2) |
1605 | |
1606 | The following extra operations are supported on both real and complex |
1607 | numbers: |
1608 | |
1609 | Re(z) = a |
1610 | Im(z) = b |
1611 | arg(z) = t |
b42d0ec9 |
1612 | abs(z) = r |
66730be0 |
1613 | |
1614 | cbrt(z) = z ** (1/3) |
1615 | log10(z) = log(z) / log(10) |
1616 | logn(z, n) = log(z) / log(n) |
1617 | |
1618 | tan(z) = sin(z) / cos(z) |
0c721ce2 |
1619 | |
5aabfad6 |
1620 | csc(z) = 1 / sin(z) |
1621 | sec(z) = 1 / cos(z) |
0c721ce2 |
1622 | cot(z) = 1 / tan(z) |
66730be0 |
1623 | |
1624 | asin(z) = -i * log(i*z + sqrt(1-z*z)) |
fb73857a |
1625 | acos(z) = -i * log(z + i*sqrt(1-z*z)) |
66730be0 |
1626 | atan(z) = i/2 * log((i+z) / (i-z)) |
0c721ce2 |
1627 | |
5aabfad6 |
1628 | acsc(z) = asin(1 / z) |
1629 | asec(z) = acos(1 / z) |
8c03c583 |
1630 | acot(z) = atan(1 / z) = -i/2 * log((i+z) / (z-i)) |
66730be0 |
1631 | |
1632 | sinh(z) = 1/2 (exp(z) - exp(-z)) |
1633 | cosh(z) = 1/2 (exp(z) + exp(-z)) |
0c721ce2 |
1634 | tanh(z) = sinh(z) / cosh(z) = (exp(z) - exp(-z)) / (exp(z) + exp(-z)) |
1635 | |
5aabfad6 |
1636 | csch(z) = 1 / sinh(z) |
1637 | sech(z) = 1 / cosh(z) |
0c721ce2 |
1638 | coth(z) = 1 / tanh(z) |
fb73857a |
1639 | |
66730be0 |
1640 | asinh(z) = log(z + sqrt(z*z+1)) |
1641 | acosh(z) = log(z + sqrt(z*z-1)) |
1642 | atanh(z) = 1/2 * log((1+z) / (1-z)) |
66730be0 |
1643 | |
5aabfad6 |
1644 | acsch(z) = asinh(1 / z) |
1645 | asech(z) = acosh(1 / z) |
0c721ce2 |
1646 | acoth(z) = atanh(1 / z) = 1/2 * log((1+z) / (z-1)) |
1647 | |
b42d0ec9 |
1648 | I<arg>, I<abs>, I<log>, I<csc>, I<cot>, I<acsc>, I<acot>, I<csch>, |
1649 | I<coth>, I<acosech>, I<acotanh>, have aliases I<rho>, I<theta>, I<ln>, |
1650 | I<cosec>, I<cotan>, I<acosec>, I<acotan>, I<cosech>, I<cotanh>, |
1651 | I<acosech>, I<acotanh>, respectively. C<Re>, C<Im>, C<arg>, C<abs>, |
1652 | C<rho>, and C<theta> can be used also also mutators. The C<cbrt> |
1653 | returns only one of the solutions: if you want all three, use the |
1654 | C<root> function. |
0c721ce2 |
1655 | |
1656 | The I<root> function is available to compute all the I<n> |
66730be0 |
1657 | roots of some complex, where I<n> is a strictly positive integer. |
1658 | There are exactly I<n> such roots, returned as a list. Getting the |
1659 | number mathematicians call C<j> such that: |
1660 | |
1661 | 1 + j + j*j = 0; |
1662 | |
1663 | is a simple matter of writing: |
1664 | |
1665 | $j = ((root(1, 3))[1]; |
1666 | |
1667 | The I<k>th root for C<z = [r,t]> is given by: |
1668 | |
1669 | (root(z, n))[k] = r**(1/n) * exp(i * (t + 2*k*pi)/n) |
1670 | |
f4837644 |
1671 | The I<spaceship> comparison operator, E<lt>=E<gt>, is also defined. In |
1672 | order to ensure its restriction to real numbers is conform to what you |
1673 | would expect, the comparison is run on the real part of the complex |
1674 | number first, and imaginary parts are compared only when the real |
1675 | parts match. |
66730be0 |
1676 | |
1677 | =head1 CREATION |
1678 | |
1679 | To create a complex number, use either: |
1680 | |
1681 | $z = Math::Complex->make(3, 4); |
1682 | $z = cplx(3, 4); |
1683 | |
1684 | if you know the cartesian form of the number, or |
1685 | |
1686 | $z = 3 + 4*i; |
1687 | |
fb73857a |
1688 | if you like. To create a number using the polar form, use either: |
66730be0 |
1689 | |
1690 | $z = Math::Complex->emake(5, pi/3); |
1691 | $x = cplxe(5, pi/3); |
1692 | |
0c721ce2 |
1693 | instead. The first argument is the modulus, the second is the angle |
fb73857a |
1694 | (in radians, the full circle is 2*pi). (Mnemonic: C<e> is used as a |
1695 | notation for complex numbers in the polar form). |
66730be0 |
1696 | |
1697 | It is possible to write: |
1698 | |
1699 | $x = cplxe(-3, pi/4); |
1700 | |
16357284 |
1701 | but that will be silently converted into C<[3,-3pi/4]>, since the |
1702 | modulus must be non-negative (it represents the distance to the origin |
1703 | in the complex plane). |
66730be0 |
1704 | |
b42d0ec9 |
1705 | It is also possible to have a complex number as either argument of |
1706 | either the C<make> or C<emake>: the appropriate component of |
1707 | the argument will be used. |
1708 | |
1709 | $z1 = cplx(-2, 1); |
1710 | $z2 = cplx($z1, 4); |
1711 | |
66730be0 |
1712 | =head1 STRINGIFICATION |
1713 | |
1714 | When printed, a complex number is usually shown under its cartesian |
16357284 |
1715 | style I<a+bi>, but there are legitimate cases where the polar style |
66730be0 |
1716 | I<[r,t]> is more appropriate. |
1717 | |
16357284 |
1718 | By calling the class method C<Math::Complex::display_format> and |
1719 | supplying either C<"polar"> or C<"cartesian"> as an argument, you |
5287f86b |
1720 | override the default display style, which is C<"cartesian">. Not |
16357284 |
1721 | supplying any argument returns the current settings. |
66730be0 |
1722 | |
1723 | This default can be overridden on a per-number basis by calling the |
1724 | C<display_format> method instead. As before, not supplying any argument |
5287f86b |
1725 | returns the current display style for this number. Otherwise whatever you |
1726 | specify will be the new display style for I<this> particular number. |
66730be0 |
1727 | |
1728 | For instance: |
1729 | |
1730 | use Math::Complex; |
1731 | |
1732 | Math::Complex::display_format('polar'); |
16357284 |
1733 | $j = (root(1, 3))[1]; |
1734 | print "j = $j\n"; # Prints "j = [1,2pi/3]" |
66730be0 |
1735 | $j->display_format('cartesian'); |
1736 | print "j = $j\n"; # Prints "j = -0.5+0.866025403784439i" |
1737 | |
5287f86b |
1738 | The polar style attempts to emphasize arguments like I<k*pi/n> |
1739 | (where I<n> is a positive integer and I<k> an integer within [-9,+9]), |
1740 | this is called I<polar pretty-printing>. |
66730be0 |
1741 | |
16357284 |
1742 | =head2 CHANGED IN PERL 5.6 |
1743 | |
1744 | The C<display_format> class method and the corresponding |
1745 | C<display_format> object method can now be called using |
1746 | a parameter hash instead of just a one parameter. |
1747 | |
1748 | The old display format style, which can have values C<"cartesian"> or |
1749 | C<"polar">, can be changed using the C<"style"> parameter. (The one |
1750 | parameter calling convention also still works.) |
1751 | |
1752 | There are two new display parameters. |
1753 | |
1754 | The first one is C<"format">, which is a sprintf()-style format |
1755 | string to be used for both parts of the complex number(s). The |
1756 | default is C<undef>, which corresponds usually (this is somewhat |
1757 | system-dependent) to C<"%.15g">. You can revert to the default by |
1758 | setting the format string to C<undef>. |
1759 | |
1760 | # the $j from the above example |
1761 | |
1762 | $j->display_format('format' => '%.5f'); |
1763 | print "j = $j\n"; # Prints "j = -0.50000+0.86603i" |
1764 | $j->display_format('format' => '%.6f'); |
1765 | print "j = $j\n"; # Prints "j = -0.5+0.86603i" |
1766 | |
1767 | Notice that this affects also the return values of the |
1768 | C<display_format> methods: in list context the whole parameter hash |
1769 | will be returned, as opposed to only the style parameter value. If |
1770 | you want to know the whole truth for a complex number, you must call |
1771 | both the class method and the object method: |
1772 | |
5287f86b |
1773 | The second new display parameter is C<"polar_pretty_print">, which can |
1774 | be set to true or false, the default being true. See the previous |
1775 | section for what this means. |
16357284 |
1776 | |
66730be0 |
1777 | =head1 USAGE |
1778 | |
1779 | Thanks to overloading, the handling of arithmetics with complex numbers |
1780 | is simple and almost transparent. |
1781 | |
1782 | Here are some examples: |
1783 | |
1784 | use Math::Complex; |
1785 | |
1786 | $j = cplxe(1, 2*pi/3); # $j ** 3 == 1 |
1787 | print "j = $j, j**3 = ", $j ** 3, "\n"; |
1788 | print "1 + j + j**2 = ", 1 + $j + $j**2, "\n"; |
1789 | |
1790 | $z = -16 + 0*i; # Force it to be a complex |
1791 | print "sqrt($z) = ", sqrt($z), "\n"; |
1792 | |
1793 | $k = exp(i * 2*pi/3); |
1794 | print "$j - $k = ", $j - $k, "\n"; |
a5f75d66 |
1795 | |
b42d0ec9 |
1796 | $z->Re(3); # Re, Im, arg, abs, |
1797 | $j->arg(2); # (the last two aka rho, theta) |
1798 | # can be used also as mutators. |
1799 | |
1800 | =head1 ERRORS DUE TO DIVISION BY ZERO OR LOGARITHM OF ZERO |
5aabfad6 |
1801 | |
1802 | The division (/) and the following functions |
1803 | |
b42d0ec9 |
1804 | log ln log10 logn |
2820d885 |
1805 | tan sec csc cot |
b42d0ec9 |
1806 | atan asec acsc acot |
1807 | tanh sech csch coth |
1808 | atanh asech acsch acoth |
5aabfad6 |
1809 | |
1810 | cannot be computed for all arguments because that would mean dividing |
8c03c583 |
1811 | by zero or taking logarithm of zero. These situations cause fatal |
1812 | runtime errors looking like this |
5aabfad6 |
1813 | |
1814 | cot(0): Division by zero. |
5cd24f17 |
1815 | (Because in the definition of cot(0), the divisor sin(0) is 0) |
5aabfad6 |
1816 | Died at ... |
1817 | |
8c03c583 |
1818 | or |
1819 | |
1820 | atanh(-1): Logarithm of zero. |
1821 | Died at... |
1822 | |
1823 | For the C<csc>, C<cot>, C<asec>, C<acsc>, C<acot>, C<csch>, C<coth>, |
b42d0ec9 |
1824 | C<asech>, C<acsch>, the argument cannot be C<0> (zero). For the the |
1825 | logarithmic functions and the C<atanh>, C<acoth>, the argument cannot |
1826 | be C<1> (one). For the C<atanh>, C<acoth>, the argument cannot be |
1827 | C<-1> (minus one). For the C<atan>, C<acot>, the argument cannot be |
1828 | C<i> (the imaginary unit). For the C<atan>, C<acoth>, the argument |
1829 | cannot be C<-i> (the negative imaginary unit). For the C<tan>, |
1830 | C<sec>, C<tanh>, the argument cannot be I<pi/2 + k * pi>, where I<k> |
1831 | is any integer. |
1832 | |
1833 | Note that because we are operating on approximations of real numbers, |
1834 | these errors can happen when merely `too close' to the singularities |
1835 | listed above. For example C<tan(2*atan2(1,1)+1e-15)> will die of |
1836 | division by zero. |
1837 | |
1838 | =head1 ERRORS DUE TO INDIGESTIBLE ARGUMENTS |
1839 | |
1840 | The C<make> and C<emake> accept both real and complex arguments. |
1841 | When they cannot recognize the arguments they will die with error |
1842 | messages like the following |
1843 | |
1844 | Math::Complex::make: Cannot take real part of ... |
1845 | Math::Complex::make: Cannot take real part of ... |
1846 | Math::Complex::emake: Cannot take rho of ... |
1847 | Math::Complex::emake: Cannot take theta of ... |
5cd24f17 |
1848 | |
a5f75d66 |
1849 | =head1 BUGS |
1850 | |
5cd24f17 |
1851 | Saying C<use Math::Complex;> exports many mathematical routines in the |
fb73857a |
1852 | caller environment and even overrides some (C<sqrt>, C<log>). |
1853 | This is construed as a feature by the Authors, actually... ;-) |
a5f75d66 |
1854 | |
66730be0 |
1855 | All routines expect to be given real or complex numbers. Don't attempt to |
1856 | use BigFloat, since Perl has currently no rule to disambiguate a '+' |
1857 | operation (for instance) between two overloaded entities. |
a5f75d66 |
1858 | |
d09ae4e6 |
1859 | In Cray UNICOS there is some strange numerical instability that results |
1860 | in root(), cos(), sin(), cosh(), sinh(), losing accuracy fast. Beware. |
1861 | The bug may be in UNICOS math libs, in UNICOS C compiler, in Math::Complex. |
1862 | Whatever it is, it does not manifest itself anywhere else where Perl runs. |
1863 | |
0c721ce2 |
1864 | =head1 AUTHORS |
a5f75d66 |
1865 | |
6e238990 |
1866 | Raphael Manfredi <F<Raphael_Manfredi@pobox.com>> and |
ace5de91 |
1867 | Jarkko Hietaniemi <F<jhi@iki.fi>>. |
5cd24f17 |
1868 | |
fb73857a |
1869 | Extensive patches by Daniel S. Lewart <F<d-lewart@uiuc.edu>>. |
1870 | |
5cd24f17 |
1871 | =cut |
1872 | |
b42d0ec9 |
1873 | 1; |
1874 | |
5cd24f17 |
1875 | # eof |