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 | our($VERSION, @ISA, @EXPORT, %EXPORT_TAGS, $Inf); |
11 | |
d6a466d7 |
12 | $VERSION = 1.32; |
476757f7 |
13 | |
9fbe1b12 |
14 | BEGIN { |
ffb4440d |
15 | unless ($^O eq 'unicosmk') { |
16 | my $e = $!; |
830ec763 |
17 | # We do want an arithmetic overflow, Inf INF inf Infinity:. |
18 | undef $Inf unless eval <<'EOE' and $Inf =~ /^inf(?:inity)?$/i; |
19 | local $SIG{FPE} = sub {die}; |
20 | my $t = CORE::exp 30; |
21 | $Inf = CORE::exp $t; |
22 | EOE |
23 | if (!defined $Inf) { # Try a different method |
24 | undef $Inf unless eval <<'EOE' and $Inf =~ /^inf(?:inity)?$/i; |
25 | local $SIG{FPE} = sub {die}; |
26 | my $t = 1; |
27 | $Inf = $t + "1e99999999999999999999999999999999"; |
28 | EOE |
29 | } |
ffb4440d |
30 | $! = $e; # Clear ERANGE. |
ffb4440d |
31 | } |
5240e574 |
32 | $Inf = "Inf" if !defined $Inf || !($Inf > 0); # Desperation. |
9fbe1b12 |
33 | } |
fb73857a |
34 | |
9fbe1b12 |
35 | use strict; |
fb73857a |
36 | |
9fbe1b12 |
37 | my $i; |
38 | my %LOGN; |
0c721ce2 |
39 | |
9fbe1b12 |
40 | require Exporter; |
0c721ce2 |
41 | |
5aabfad6 |
42 | @ISA = qw(Exporter); |
43 | |
5aabfad6 |
44 | my @trig = qw( |
45 | pi |
fb73857a |
46 | tan |
5aabfad6 |
47 | csc cosec sec cot cotan |
48 | asin acos atan |
49 | acsc acosec asec acot acotan |
50 | sinh cosh tanh |
51 | csch cosech sech coth cotanh |
52 | asinh acosh atanh |
53 | acsch acosech asech acoth acotanh |
54 | ); |
55 | |
56 | @EXPORT = (qw( |
b42d0ec9 |
57 | i Re Im rho theta arg |
fb73857a |
58 | sqrt log ln |
5aabfad6 |
59 | log10 logn cbrt root |
60 | cplx cplxe |
61 | ), |
62 | @trig); |
63 | |
64 | %EXPORT_TAGS = ( |
65 | 'trig' => [@trig], |
66730be0 |
66 | ); |
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67 | |
a5f75d66 |
68 | use overload |
0c721ce2 |
69 | '+' => \&plus, |
70 | '-' => \&minus, |
71 | '*' => \&multiply, |
72 | '/' => \÷, |
66730be0 |
73 | '**' => \&power, |
1fa12f56 |
74 | '==' => \&numeq, |
66730be0 |
75 | '<=>' => \&spaceship, |
76 | 'neg' => \&negate, |
0c721ce2 |
77 | '~' => \&conjugate, |
66730be0 |
78 | 'abs' => \&abs, |
79 | 'sqrt' => \&sqrt, |
80 | 'exp' => \&exp, |
81 | 'log' => \&log, |
82 | 'sin' => \&sin, |
83 | 'cos' => \&cos, |
0c721ce2 |
84 | 'tan' => \&tan, |
66730be0 |
85 | 'atan2' => \&atan2, |
86 | qw("" stringify); |
87 | |
88 | # |
b42d0ec9 |
89 | # Package "privates" |
66730be0 |
90 | # |
91 | |
16357284 |
92 | my %DISPLAY_FORMAT = ('style' => 'cartesian', |
93 | 'polar_pretty_print' => 1); |
94 | my $eps = 1e-14; # Epsilon |
66730be0 |
95 | |
96 | # |
97 | # Object attributes (internal): |
98 | # cartesian [real, imaginary] -- cartesian form |
99 | # polar [rho, theta] -- polar form |
100 | # c_dirty cartesian form not up-to-date |
101 | # p_dirty polar form not up-to-date |
102 | # display display format (package's global when not set) |
103 | # |
104 | |
b42d0ec9 |
105 | # Die on bad *make() arguments. |
106 | |
107 | sub _cannot_make { |
108 | die "@{[(caller(1))[3]]}: Cannot take $_[0] of $_[1].\n"; |
109 | } |
110 | |
66730be0 |
111 | # |
112 | # ->make |
113 | # |
114 | # Create a new complex number (cartesian form) |
115 | # |
116 | sub make { |
117 | my $self = bless {}, shift; |
118 | my ($re, $im) = @_; |
b42d0ec9 |
119 | my $rre = ref $re; |
120 | if ( $rre ) { |
121 | if ( $rre eq ref $self ) { |
122 | $re = Re($re); |
123 | } else { |
124 | _cannot_make("real part", $rre); |
125 | } |
126 | } |
127 | my $rim = ref $im; |
128 | if ( $rim ) { |
129 | if ( $rim eq ref $self ) { |
130 | $im = Im($im); |
131 | } else { |
132 | _cannot_make("imaginary part", $rim); |
133 | } |
134 | } |
135 | $self->{'cartesian'} = [ $re, $im ]; |
66730be0 |
136 | $self->{c_dirty} = 0; |
137 | $self->{p_dirty} = 1; |
b42d0ec9 |
138 | $self->display_format('cartesian'); |
66730be0 |
139 | return $self; |
140 | } |
141 | |
142 | # |
143 | # ->emake |
144 | # |
145 | # Create a new complex number (exponential form) |
146 | # |
147 | sub emake { |
148 | my $self = bless {}, shift; |
149 | my ($rho, $theta) = @_; |
b42d0ec9 |
150 | my $rrh = ref $rho; |
151 | if ( $rrh ) { |
152 | if ( $rrh eq ref $self ) { |
153 | $rho = rho($rho); |
154 | } else { |
155 | _cannot_make("rho", $rrh); |
156 | } |
157 | } |
158 | my $rth = ref $theta; |
159 | if ( $rth ) { |
160 | if ( $rth eq ref $self ) { |
161 | $theta = theta($theta); |
162 | } else { |
163 | _cannot_make("theta", $rth); |
164 | } |
165 | } |
fb73857a |
166 | if ($rho < 0) { |
167 | $rho = -$rho; |
168 | $theta = ($theta <= 0) ? $theta + pi() : $theta - pi(); |
169 | } |
170 | $self->{'polar'} = [$rho, $theta]; |
66730be0 |
171 | $self->{p_dirty} = 0; |
172 | $self->{c_dirty} = 1; |
b42d0ec9 |
173 | $self->display_format('polar'); |
66730be0 |
174 | return $self; |
175 | } |
176 | |
177 | sub new { &make } # For backward compatibility only. |
178 | |
179 | # |
180 | # cplx |
181 | # |
182 | # Creates a complex number from a (re, im) tuple. |
183 | # This avoids the burden of writing Math::Complex->make(re, im). |
184 | # |
185 | sub cplx { |
186 | my ($re, $im) = @_; |
16357284 |
187 | return __PACKAGE__->make($re, defined $im ? $im : 0); |
66730be0 |
188 | } |
189 | |
190 | # |
191 | # cplxe |
192 | # |
193 | # Creates a complex number from a (rho, theta) tuple. |
194 | # This avoids the burden of writing Math::Complex->emake(rho, theta). |
195 | # |
196 | sub cplxe { |
197 | my ($rho, $theta) = @_; |
16357284 |
198 | return __PACKAGE__->emake($rho, defined $theta ? $theta : 0); |
66730be0 |
199 | } |
200 | |
201 | # |
202 | # pi |
203 | # |
fb73857a |
204 | # The number defined as pi = 180 degrees |
66730be0 |
205 | # |
6570f784 |
206 | sub pi () { 4 * CORE::atan2(1, 1) } |
5cd24f17 |
207 | |
208 | # |
fb73857a |
209 | # pit2 |
5cd24f17 |
210 | # |
fb73857a |
211 | # The full circle |
212 | # |
6570f784 |
213 | sub pit2 () { 2 * pi } |
fb73857a |
214 | |
5cd24f17 |
215 | # |
fb73857a |
216 | # pip2 |
217 | # |
218 | # The quarter circle |
219 | # |
6570f784 |
220 | sub pip2 () { pi / 2 } |
5cd24f17 |
221 | |
fb73857a |
222 | # |
d09ae4e6 |
223 | # deg1 |
224 | # |
225 | # One degree in radians, used in stringify_polar. |
226 | # |
227 | |
6570f784 |
228 | sub deg1 () { pi / 180 } |
d09ae4e6 |
229 | |
230 | # |
fb73857a |
231 | # uplog10 |
232 | # |
233 | # Used in log10(). |
234 | # |
6570f784 |
235 | sub uplog10 () { 1 / CORE::log(10) } |
66730be0 |
236 | |
237 | # |
238 | # i |
239 | # |
240 | # The number defined as i*i = -1; |
241 | # |
242 | sub i () { |
5cd24f17 |
243 | return $i if ($i); |
244 | $i = bless {}; |
40da2db3 |
245 | $i->{'cartesian'} = [0, 1]; |
fb73857a |
246 | $i->{'polar'} = [1, pip2]; |
66730be0 |
247 | $i->{c_dirty} = 0; |
248 | $i->{p_dirty} = 0; |
249 | return $i; |
250 | } |
251 | |
252 | # |
1fa12f56 |
253 | # ip2 |
254 | # |
255 | # Half of i. |
256 | # |
257 | sub ip2 () { i / 2 } |
258 | |
259 | # |
66730be0 |
260 | # Attribute access/set routines |
261 | # |
262 | |
0c721ce2 |
263 | sub cartesian {$_[0]->{c_dirty} ? |
264 | $_[0]->update_cartesian : $_[0]->{'cartesian'}} |
265 | sub polar {$_[0]->{p_dirty} ? |
266 | $_[0]->update_polar : $_[0]->{'polar'}} |
66730be0 |
267 | |
40da2db3 |
268 | sub set_cartesian { $_[0]->{p_dirty}++; $_[0]->{'cartesian'} = $_[1] } |
269 | sub set_polar { $_[0]->{c_dirty}++; $_[0]->{'polar'} = $_[1] } |
66730be0 |
270 | |
271 | # |
272 | # ->update_cartesian |
273 | # |
274 | # Recompute and return the cartesian form, given accurate polar form. |
275 | # |
276 | sub update_cartesian { |
277 | my $self = shift; |
40da2db3 |
278 | my ($r, $t) = @{$self->{'polar'}}; |
66730be0 |
279 | $self->{c_dirty} = 0; |
a8693bd3 |
280 | return $self->{'cartesian'} = [$r * CORE::cos($t), $r * CORE::sin($t)]; |
66730be0 |
281 | } |
282 | |
283 | # |
284 | # |
285 | # ->update_polar |
286 | # |
287 | # Recompute and return the polar form, given accurate cartesian form. |
288 | # |
289 | sub update_polar { |
290 | my $self = shift; |
40da2db3 |
291 | my ($x, $y) = @{$self->{'cartesian'}}; |
66730be0 |
292 | $self->{p_dirty} = 0; |
40da2db3 |
293 | return $self->{'polar'} = [0, 0] if $x == 0 && $y == 0; |
1fa12f56 |
294 | return $self->{'polar'} = [CORE::sqrt($x*$x + $y*$y), |
295 | CORE::atan2($y, $x)]; |
66730be0 |
296 | } |
297 | |
298 | # |
299 | # (plus) |
300 | # |
301 | # Computes z1+z2. |
302 | # |
303 | sub plus { |
304 | my ($z1, $z2, $regular) = @_; |
305 | my ($re1, $im1) = @{$z1->cartesian}; |
0e505df1 |
306 | $z2 = cplx($z2) unless ref $z2; |
5cd24f17 |
307 | my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0); |
66730be0 |
308 | unless (defined $regular) { |
309 | $z1->set_cartesian([$re1 + $re2, $im1 + $im2]); |
310 | return $z1; |
311 | } |
312 | return (ref $z1)->make($re1 + $re2, $im1 + $im2); |
313 | } |
314 | |
315 | # |
316 | # (minus) |
317 | # |
318 | # Computes z1-z2. |
319 | # |
320 | sub minus { |
321 | my ($z1, $z2, $inverted) = @_; |
322 | my ($re1, $im1) = @{$z1->cartesian}; |
0e505df1 |
323 | $z2 = cplx($z2) unless ref $z2; |
324 | my ($re2, $im2) = @{$z2->cartesian}; |
66730be0 |
325 | unless (defined $inverted) { |
326 | $z1->set_cartesian([$re1 - $re2, $im1 - $im2]); |
327 | return $z1; |
328 | } |
329 | return $inverted ? |
330 | (ref $z1)->make($re2 - $re1, $im2 - $im1) : |
331 | (ref $z1)->make($re1 - $re2, $im1 - $im2); |
0e505df1 |
332 | |
66730be0 |
333 | } |
334 | |
335 | # |
336 | # (multiply) |
337 | # |
338 | # Computes z1*z2. |
339 | # |
340 | sub multiply { |
fb73857a |
341 | my ($z1, $z2, $regular) = @_; |
342 | if ($z1->{p_dirty} == 0 and ref $z2 and $z2->{p_dirty} == 0) { |
343 | # if both polar better use polar to avoid rounding errors |
344 | my ($r1, $t1) = @{$z1->polar}; |
345 | my ($r2, $t2) = @{$z2->polar}; |
346 | my $t = $t1 + $t2; |
347 | if ($t > pi()) { $t -= pit2 } |
348 | elsif ($t <= -pi()) { $t += pit2 } |
349 | unless (defined $regular) { |
350 | $z1->set_polar([$r1 * $r2, $t]); |
66730be0 |
351 | return $z1; |
fb73857a |
352 | } |
353 | return (ref $z1)->emake($r1 * $r2, $t); |
354 | } else { |
355 | my ($x1, $y1) = @{$z1->cartesian}; |
356 | if (ref $z2) { |
357 | my ($x2, $y2) = @{$z2->cartesian}; |
358 | return (ref $z1)->make($x1*$x2-$y1*$y2, $x1*$y2+$y1*$x2); |
359 | } else { |
360 | return (ref $z1)->make($x1*$z2, $y1*$z2); |
361 | } |
66730be0 |
362 | } |
66730be0 |
363 | } |
364 | |
365 | # |
0e505df1 |
366 | # _divbyzero |
0c721ce2 |
367 | # |
368 | # Die on division by zero. |
369 | # |
0e505df1 |
370 | sub _divbyzero { |
5cd24f17 |
371 | my $mess = "$_[0]: Division by zero.\n"; |
372 | |
373 | if (defined $_[1]) { |
374 | $mess .= "(Because in the definition of $_[0], the divisor "; |
1fa12f56 |
375 | $mess .= "$_[1] " unless ("$_[1]" eq '0'); |
5cd24f17 |
376 | $mess .= "is 0)\n"; |
377 | } |
378 | |
0c721ce2 |
379 | my @up = caller(1); |
fb73857a |
380 | |
5cd24f17 |
381 | $mess .= "Died at $up[1] line $up[2].\n"; |
382 | |
383 | die $mess; |
0c721ce2 |
384 | } |
385 | |
386 | # |
66730be0 |
387 | # (divide) |
388 | # |
389 | # Computes z1/z2. |
390 | # |
391 | sub divide { |
392 | my ($z1, $z2, $inverted) = @_; |
fb73857a |
393 | if ($z1->{p_dirty} == 0 and ref $z2 and $z2->{p_dirty} == 0) { |
394 | # if both polar better use polar to avoid rounding errors |
395 | my ($r1, $t1) = @{$z1->polar}; |
396 | my ($r2, $t2) = @{$z2->polar}; |
397 | my $t; |
398 | if ($inverted) { |
0e505df1 |
399 | _divbyzero "$z2/0" if ($r1 == 0); |
fb73857a |
400 | $t = $t2 - $t1; |
401 | if ($t > pi()) { $t -= pit2 } |
402 | elsif ($t <= -pi()) { $t += pit2 } |
403 | return (ref $z1)->emake($r2 / $r1, $t); |
404 | } else { |
0e505df1 |
405 | _divbyzero "$z1/0" if ($r2 == 0); |
fb73857a |
406 | $t = $t1 - $t2; |
407 | if ($t > pi()) { $t -= pit2 } |
408 | elsif ($t <= -pi()) { $t += pit2 } |
409 | return (ref $z1)->emake($r1 / $r2, $t); |
410 | } |
411 | } else { |
412 | my ($d, $x2, $y2); |
413 | if ($inverted) { |
414 | ($x2, $y2) = @{$z1->cartesian}; |
415 | $d = $x2*$x2 + $y2*$y2; |
416 | _divbyzero "$z2/0" if $d == 0; |
417 | return (ref $z1)->make(($x2*$z2)/$d, -($y2*$z2)/$d); |
418 | } else { |
419 | my ($x1, $y1) = @{$z1->cartesian}; |
420 | if (ref $z2) { |
421 | ($x2, $y2) = @{$z2->cartesian}; |
422 | $d = $x2*$x2 + $y2*$y2; |
423 | _divbyzero "$z1/0" if $d == 0; |
424 | my $u = ($x1*$x2 + $y1*$y2)/$d; |
425 | my $v = ($y1*$x2 - $x1*$y2)/$d; |
426 | return (ref $z1)->make($u, $v); |
427 | } else { |
428 | _divbyzero "$z1/0" if $z2 == 0; |
429 | return (ref $z1)->make($x1/$z2, $y1/$z2); |
430 | } |
431 | } |
0c721ce2 |
432 | } |
66730be0 |
433 | } |
434 | |
435 | # |
436 | # (power) |
437 | # |
438 | # Computes z1**z2 = exp(z2 * log z1)). |
439 | # |
440 | sub power { |
441 | my ($z1, $z2, $inverted) = @_; |
ace5de91 |
442 | if ($inverted) { |
2820d885 |
443 | return 1 if $z1 == 0 || $z2 == 1; |
444 | return 0 if $z2 == 0 && Re($z1) > 0; |
ace5de91 |
445 | } else { |
2820d885 |
446 | return 1 if $z2 == 0 || $z1 == 1; |
447 | return 0 if $z1 == 0 && Re($z2) > 0; |
ace5de91 |
448 | } |
1fa12f56 |
449 | my $w = $inverted ? &exp($z1 * &log($z2)) |
450 | : &exp($z2 * &log($z1)); |
d09ae4e6 |
451 | # If both arguments cartesian, return cartesian, else polar. |
452 | return $z1->{c_dirty} == 0 && |
453 | (not ref $z2 or $z2->{c_dirty} == 0) ? |
454 | cplx(@{$w->cartesian}) : $w; |
66730be0 |
455 | } |
456 | |
457 | # |
458 | # (spaceship) |
459 | # |
460 | # Computes z1 <=> z2. |
2820d885 |
461 | # Sorts on the real part first, then on the imaginary part. Thus 2-4i < 3+8i. |
66730be0 |
462 | # |
463 | sub spaceship { |
464 | my ($z1, $z2, $inverted) = @_; |
5cd24f17 |
465 | my ($re1, $im1) = ref $z1 ? @{$z1->cartesian} : ($z1, 0); |
466 | my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0); |
66730be0 |
467 | my $sgn = $inverted ? -1 : 1; |
468 | return $sgn * ($re1 <=> $re2) if $re1 != $re2; |
469 | return $sgn * ($im1 <=> $im2); |
470 | } |
471 | |
472 | # |
1fa12f56 |
473 | # (numeq) |
474 | # |
475 | # Computes z1 == z2. |
476 | # |
477 | # (Required in addition to spaceship() because of NaNs.) |
478 | sub numeq { |
479 | my ($z1, $z2, $inverted) = @_; |
480 | my ($re1, $im1) = ref $z1 ? @{$z1->cartesian} : ($z1, 0); |
481 | my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0); |
482 | return $re1 == $re2 && $im1 == $im2 ? 1 : 0; |
483 | } |
484 | |
485 | # |
66730be0 |
486 | # (negate) |
487 | # |
488 | # Computes -z. |
489 | # |
490 | sub negate { |
491 | my ($z) = @_; |
492 | if ($z->{c_dirty}) { |
493 | my ($r, $t) = @{$z->polar}; |
fb73857a |
494 | $t = ($t <= 0) ? $t + pi : $t - pi; |
495 | return (ref $z)->emake($r, $t); |
66730be0 |
496 | } |
497 | my ($re, $im) = @{$z->cartesian}; |
498 | return (ref $z)->make(-$re, -$im); |
499 | } |
500 | |
501 | # |
502 | # (conjugate) |
503 | # |
504 | # Compute complex's conjugate. |
505 | # |
506 | sub conjugate { |
507 | my ($z) = @_; |
508 | if ($z->{c_dirty}) { |
509 | my ($r, $t) = @{$z->polar}; |
510 | return (ref $z)->emake($r, -$t); |
511 | } |
512 | my ($re, $im) = @{$z->cartesian}; |
513 | return (ref $z)->make($re, -$im); |
514 | } |
515 | |
516 | # |
517 | # (abs) |
518 | # |
b42d0ec9 |
519 | # Compute or set complex's norm (rho). |
66730be0 |
520 | # |
521 | sub abs { |
b42d0ec9 |
522 | my ($z, $rho) = @_; |
1fa12f56 |
523 | unless (ref $z) { |
524 | if (@_ == 2) { |
525 | $_[0] = $_[1]; |
526 | } else { |
527 | return CORE::abs($z); |
528 | } |
529 | } |
b42d0ec9 |
530 | if (defined $rho) { |
531 | $z->{'polar'} = [ $rho, ${$z->polar}[1] ]; |
532 | $z->{p_dirty} = 0; |
533 | $z->{c_dirty} = 1; |
534 | return $rho; |
535 | } else { |
536 | return ${$z->polar}[0]; |
537 | } |
538 | } |
539 | |
540 | sub _theta { |
541 | my $theta = $_[0]; |
542 | |
543 | if ($$theta > pi()) { $$theta -= pit2 } |
544 | elsif ($$theta <= -pi()) { $$theta += pit2 } |
66730be0 |
545 | } |
546 | |
547 | # |
548 | # arg |
549 | # |
b42d0ec9 |
550 | # Compute or set complex's argument (theta). |
66730be0 |
551 | # |
552 | sub arg { |
b42d0ec9 |
553 | my ($z, $theta) = @_; |
554 | return $z unless ref $z; |
555 | if (defined $theta) { |
556 | _theta(\$theta); |
557 | $z->{'polar'} = [ ${$z->polar}[0], $theta ]; |
558 | $z->{p_dirty} = 0; |
559 | $z->{c_dirty} = 1; |
560 | } else { |
561 | $theta = ${$z->polar}[1]; |
562 | _theta(\$theta); |
563 | } |
564 | return $theta; |
66730be0 |
565 | } |
566 | |
567 | # |
568 | # (sqrt) |
569 | # |
0c721ce2 |
570 | # Compute sqrt(z). |
66730be0 |
571 | # |
b42d0ec9 |
572 | # It is quite tempting to use wantarray here so that in list context |
573 | # sqrt() would return the two solutions. This, however, would |
574 | # break things like |
575 | # |
576 | # print "sqrt(z) = ", sqrt($z), "\n"; |
577 | # |
578 | # The two values would be printed side by side without no intervening |
579 | # whitespace, quite confusing. |
580 | # Therefore if you want the two solutions use the root(). |
581 | # |
66730be0 |
582 | sub sqrt { |
583 | my ($z) = @_; |
b42d0ec9 |
584 | my ($re, $im) = ref $z ? @{$z->cartesian} : ($z, 0); |
1fa12f56 |
585 | return $re < 0 ? cplx(0, CORE::sqrt(-$re)) : CORE::sqrt($re) |
586 | if $im == 0; |
66730be0 |
587 | my ($r, $t) = @{$z->polar}; |
a8693bd3 |
588 | return (ref $z)->emake(CORE::sqrt($r), $t/2); |
66730be0 |
589 | } |
590 | |
591 | # |
592 | # cbrt |
593 | # |
0c721ce2 |
594 | # Compute cbrt(z) (cubic root). |
66730be0 |
595 | # |
b42d0ec9 |
596 | # Why are we not returning three values? The same answer as for sqrt(). |
597 | # |
66730be0 |
598 | sub cbrt { |
599 | my ($z) = @_; |
1fa12f56 |
600 | return $z < 0 ? |
601 | -CORE::exp(CORE::log(-$z)/3) : |
602 | ($z > 0 ? CORE::exp(CORE::log($z)/3): 0) |
fb73857a |
603 | unless ref $z; |
66730be0 |
604 | my ($r, $t) = @{$z->polar}; |
1fa12f56 |
605 | return 0 if $r == 0; |
a8693bd3 |
606 | return (ref $z)->emake(CORE::exp(CORE::log($r)/3), $t/3); |
66730be0 |
607 | } |
608 | |
609 | # |
0e505df1 |
610 | # _rootbad |
611 | # |
612 | # Die on bad root. |
613 | # |
614 | sub _rootbad { |
1fa12f56 |
615 | my $mess = "Root $_[0] illegal, root rank must be positive integer.\n"; |
0e505df1 |
616 | |
617 | my @up = caller(1); |
fb73857a |
618 | |
0e505df1 |
619 | $mess .= "Died at $up[1] line $up[2].\n"; |
620 | |
621 | die $mess; |
622 | } |
623 | |
624 | # |
66730be0 |
625 | # root |
626 | # |
627 | # Computes all nth root for z, returning an array whose size is n. |
628 | # `n' must be a positive integer. |
629 | # |
630 | # The roots are given by (for k = 0..n-1): |
631 | # |
632 | # z^(1/n) = r^(1/n) (cos ((t+2 k pi)/n) + i sin ((t+2 k pi)/n)) |
633 | # |
634 | sub root { |
635 | my ($z, $n) = @_; |
0e505df1 |
636 | _rootbad($n) if ($n < 1 or int($n) != $n); |
1fa12f56 |
637 | my ($r, $t) = ref $z ? |
638 | @{$z->polar} : (CORE::abs($z), $z >= 0 ? 0 : pi); |
66730be0 |
639 | my @root; |
640 | my $k; |
fb73857a |
641 | my $theta_inc = pit2 / $n; |
66730be0 |
642 | my $rho = $r ** (1/$n); |
643 | my $theta; |
d09ae4e6 |
644 | my $cartesian = ref $z && $z->{c_dirty} == 0; |
66730be0 |
645 | for ($k = 0, $theta = $t / $n; $k < $n; $k++, $theta += $theta_inc) { |
d09ae4e6 |
646 | my $w = cplxe($rho, $theta); |
647 | # Yes, $cartesian is loop invariant. |
648 | push @root, $cartesian ? cplx(@{$w->cartesian}) : $w; |
a0d0e21e |
649 | } |
66730be0 |
650 | return @root; |
a0d0e21e |
651 | } |
652 | |
66730be0 |
653 | # |
654 | # Re |
655 | # |
b42d0ec9 |
656 | # Return or set Re(z). |
66730be0 |
657 | # |
a0d0e21e |
658 | sub Re { |
b42d0ec9 |
659 | my ($z, $Re) = @_; |
66730be0 |
660 | return $z unless ref $z; |
b42d0ec9 |
661 | if (defined $Re) { |
662 | $z->{'cartesian'} = [ $Re, ${$z->cartesian}[1] ]; |
663 | $z->{c_dirty} = 0; |
664 | $z->{p_dirty} = 1; |
665 | } else { |
666 | return ${$z->cartesian}[0]; |
667 | } |
a0d0e21e |
668 | } |
669 | |
66730be0 |
670 | # |
671 | # Im |
672 | # |
b42d0ec9 |
673 | # Return or set Im(z). |
66730be0 |
674 | # |
a0d0e21e |
675 | sub Im { |
b42d0ec9 |
676 | my ($z, $Im) = @_; |
178326e7 |
677 | return 0 unless ref $z; |
b42d0ec9 |
678 | if (defined $Im) { |
679 | $z->{'cartesian'} = [ ${$z->cartesian}[0], $Im ]; |
680 | $z->{c_dirty} = 0; |
681 | $z->{p_dirty} = 1; |
682 | } else { |
683 | return ${$z->cartesian}[1]; |
684 | } |
685 | } |
686 | |
687 | # |
688 | # rho |
689 | # |
690 | # Return or set rho(w). |
691 | # |
692 | sub rho { |
693 | Math::Complex::abs(@_); |
694 | } |
695 | |
696 | # |
697 | # theta |
698 | # |
699 | # Return or set theta(w). |
700 | # |
701 | sub theta { |
702 | Math::Complex::arg(@_); |
a0d0e21e |
703 | } |
704 | |
66730be0 |
705 | # |
706 | # (exp) |
707 | # |
708 | # Computes exp(z). |
709 | # |
710 | sub exp { |
711 | my ($z) = @_; |
712 | my ($x, $y) = @{$z->cartesian}; |
a8693bd3 |
713 | return (ref $z)->emake(CORE::exp($x), $y); |
66730be0 |
714 | } |
715 | |
716 | # |
8c03c583 |
717 | # _logofzero |
718 | # |
fb73857a |
719 | # Die on logarithm of zero. |
8c03c583 |
720 | # |
721 | sub _logofzero { |
722 | my $mess = "$_[0]: Logarithm of zero.\n"; |
723 | |
724 | if (defined $_[1]) { |
725 | $mess .= "(Because in the definition of $_[0], the argument "; |
726 | $mess .= "$_[1] " unless ($_[1] eq '0'); |
727 | $mess .= "is 0)\n"; |
728 | } |
729 | |
730 | my @up = caller(1); |
fb73857a |
731 | |
8c03c583 |
732 | $mess .= "Died at $up[1] line $up[2].\n"; |
733 | |
734 | die $mess; |
735 | } |
736 | |
737 | # |
66730be0 |
738 | # (log) |
739 | # |
740 | # Compute log(z). |
741 | # |
742 | sub log { |
743 | my ($z) = @_; |
fb73857a |
744 | unless (ref $z) { |
745 | _logofzero("log") if $z == 0; |
a8693bd3 |
746 | return $z > 0 ? CORE::log($z) : cplx(CORE::log(-$z), pi); |
fb73857a |
747 | } |
5cd24f17 |
748 | my ($r, $t) = @{$z->polar}; |
fb73857a |
749 | _logofzero("log") if $r == 0; |
750 | if ($t > pi()) { $t -= pit2 } |
751 | elsif ($t <= -pi()) { $t += pit2 } |
a8693bd3 |
752 | return (ref $z)->make(CORE::log($r), $t); |
66730be0 |
753 | } |
754 | |
755 | # |
0c721ce2 |
756 | # ln |
757 | # |
758 | # Alias for log(). |
759 | # |
760 | sub ln { Math::Complex::log(@_) } |
761 | |
762 | # |
66730be0 |
763 | # log10 |
764 | # |
765 | # Compute log10(z). |
766 | # |
5cd24f17 |
767 | |
66730be0 |
768 | sub log10 { |
fb73857a |
769 | return Math::Complex::log($_[0]) * uplog10; |
66730be0 |
770 | } |
771 | |
772 | # |
773 | # logn |
774 | # |
775 | # Compute logn(z,n) = log(z) / log(n) |
776 | # |
777 | sub logn { |
778 | my ($z, $n) = @_; |
0c721ce2 |
779 | $z = cplx($z, 0) unless ref $z; |
9fbe1b12 |
780 | my $logn = $LOGN{$n}; |
781 | $logn = $LOGN{$n} = CORE::log($n) unless defined $logn; # Cache log(n) |
1fa12f56 |
782 | return &log($z) / $logn; |
66730be0 |
783 | } |
784 | |
785 | # |
786 | # (cos) |
787 | # |
788 | # Compute cos(z) = (exp(iz) + exp(-iz))/2. |
789 | # |
790 | sub cos { |
791 | my ($z) = @_; |
1fa12f56 |
792 | return CORE::cos($z) unless ref $z; |
66730be0 |
793 | my ($x, $y) = @{$z->cartesian}; |
a8693bd3 |
794 | my $ey = CORE::exp($y); |
1fa12f56 |
795 | my $sx = CORE::sin($x); |
796 | my $cx = CORE::cos($x); |
797 | my $ey_1 = $ey ? 1 / $ey : $Inf; |
798 | return (ref $z)->make($cx * ($ey + $ey_1)/2, |
799 | $sx * ($ey_1 - $ey)/2); |
66730be0 |
800 | } |
801 | |
802 | # |
803 | # (sin) |
804 | # |
805 | # Compute sin(z) = (exp(iz) - exp(-iz))/2. |
806 | # |
807 | sub sin { |
808 | my ($z) = @_; |
1fa12f56 |
809 | return CORE::sin($z) unless ref $z; |
66730be0 |
810 | my ($x, $y) = @{$z->cartesian}; |
a8693bd3 |
811 | my $ey = CORE::exp($y); |
1fa12f56 |
812 | my $sx = CORE::sin($x); |
813 | my $cx = CORE::cos($x); |
814 | my $ey_1 = $ey ? 1 / $ey : $Inf; |
815 | return (ref $z)->make($sx * ($ey + $ey_1)/2, |
816 | $cx * ($ey - $ey_1)/2); |
66730be0 |
817 | } |
818 | |
819 | # |
820 | # tan |
821 | # |
822 | # Compute tan(z) = sin(z) / cos(z). |
823 | # |
824 | sub tan { |
825 | my ($z) = @_; |
1fa12f56 |
826 | my $cz = &cos($z); |
827 | _divbyzero "tan($z)", "cos($z)" if $cz == 0; |
828 | return &sin($z) / $cz; |
66730be0 |
829 | } |
830 | |
831 | # |
0c721ce2 |
832 | # sec |
833 | # |
834 | # Computes the secant sec(z) = 1 / cos(z). |
835 | # |
836 | sub sec { |
837 | my ($z) = @_; |
1fa12f56 |
838 | my $cz = &cos($z); |
0e505df1 |
839 | _divbyzero "sec($z)", "cos($z)" if ($cz == 0); |
0c721ce2 |
840 | return 1 / $cz; |
841 | } |
842 | |
843 | # |
844 | # csc |
845 | # |
846 | # Computes the cosecant csc(z) = 1 / sin(z). |
847 | # |
848 | sub csc { |
849 | my ($z) = @_; |
1fa12f56 |
850 | my $sz = &sin($z); |
0e505df1 |
851 | _divbyzero "csc($z)", "sin($z)" if ($sz == 0); |
0c721ce2 |
852 | return 1 / $sz; |
853 | } |
854 | |
66730be0 |
855 | # |
0c721ce2 |
856 | # cosec |
66730be0 |
857 | # |
0c721ce2 |
858 | # Alias for csc(). |
859 | # |
860 | sub cosec { Math::Complex::csc(@_) } |
861 | |
862 | # |
863 | # cot |
864 | # |
fb73857a |
865 | # Computes cot(z) = cos(z) / sin(z). |
0c721ce2 |
866 | # |
867 | sub cot { |
66730be0 |
868 | my ($z) = @_; |
1fa12f56 |
869 | my $sz = &sin($z); |
0e505df1 |
870 | _divbyzero "cot($z)", "sin($z)" if ($sz == 0); |
1fa12f56 |
871 | return &cos($z) / $sz; |
66730be0 |
872 | } |
873 | |
874 | # |
0c721ce2 |
875 | # cotan |
876 | # |
877 | # Alias for cot(). |
878 | # |
879 | sub cotan { Math::Complex::cot(@_) } |
880 | |
881 | # |
66730be0 |
882 | # acos |
883 | # |
884 | # Computes the arc cosine acos(z) = -i log(z + sqrt(z*z-1)). |
885 | # |
886 | sub acos { |
fb73857a |
887 | my $z = $_[0]; |
1fa12f56 |
888 | return CORE::atan2(CORE::sqrt(1-$z*$z), $z) |
889 | if (! ref $z) && CORE::abs($z) <= 1; |
40b904b7 |
890 | $z = cplx($z, 0) unless ref $z; |
891 | my ($x, $y) = @{$z->cartesian}; |
1fa12f56 |
892 | return 0 if $x == 1 && $y == 0; |
a8693bd3 |
893 | my $t1 = CORE::sqrt(($x+1)*($x+1) + $y*$y); |
894 | my $t2 = CORE::sqrt(($x-1)*($x-1) + $y*$y); |
fb73857a |
895 | my $alpha = ($t1 + $t2)/2; |
896 | my $beta = ($t1 - $t2)/2; |
897 | $alpha = 1 if $alpha < 1; |
898 | if ($beta > 1) { $beta = 1 } |
899 | elsif ($beta < -1) { $beta = -1 } |
a8693bd3 |
900 | my $u = CORE::atan2(CORE::sqrt(1-$beta*$beta), $beta); |
901 | my $v = CORE::log($alpha + CORE::sqrt($alpha*$alpha-1)); |
fb73857a |
902 | $v = -$v if $y > 0 || ($y == 0 && $x < -1); |
40b904b7 |
903 | return (ref $z)->make($u, $v); |
66730be0 |
904 | } |
905 | |
906 | # |
907 | # asin |
908 | # |
909 | # Computes the arc sine asin(z) = -i log(iz + sqrt(1-z*z)). |
910 | # |
911 | sub asin { |
fb73857a |
912 | my $z = $_[0]; |
1fa12f56 |
913 | return CORE::atan2($z, CORE::sqrt(1-$z*$z)) |
914 | if (! ref $z) && CORE::abs($z) <= 1; |
40b904b7 |
915 | $z = cplx($z, 0) unless ref $z; |
916 | my ($x, $y) = @{$z->cartesian}; |
1fa12f56 |
917 | return 0 if $x == 0 && $y == 0; |
a8693bd3 |
918 | my $t1 = CORE::sqrt(($x+1)*($x+1) + $y*$y); |
919 | my $t2 = CORE::sqrt(($x-1)*($x-1) + $y*$y); |
fb73857a |
920 | my $alpha = ($t1 + $t2)/2; |
921 | my $beta = ($t1 - $t2)/2; |
922 | $alpha = 1 if $alpha < 1; |
923 | if ($beta > 1) { $beta = 1 } |
924 | elsif ($beta < -1) { $beta = -1 } |
a8693bd3 |
925 | my $u = CORE::atan2($beta, CORE::sqrt(1-$beta*$beta)); |
926 | my $v = -CORE::log($alpha + CORE::sqrt($alpha*$alpha-1)); |
fb73857a |
927 | $v = -$v if $y > 0 || ($y == 0 && $x < -1); |
40b904b7 |
928 | return (ref $z)->make($u, $v); |
66730be0 |
929 | } |
930 | |
931 | # |
932 | # atan |
933 | # |
0c721ce2 |
934 | # Computes the arc tangent atan(z) = i/2 log((i+z) / (i-z)). |
66730be0 |
935 | # |
936 | sub atan { |
937 | my ($z) = @_; |
a8693bd3 |
938 | return CORE::atan2($z, 1) unless ref $z; |
1fa12f56 |
939 | my ($x, $y) = ref $z ? @{$z->cartesian} : ($z, 0); |
940 | return 0 if $x == 0 && $y == 0; |
8c03c583 |
941 | _divbyzero "atan(i)" if ( $z == i); |
1fa12f56 |
942 | _logofzero "atan(-i)" if (-$z == i); # -i is a bad file test... |
943 | my $log = &log((i + $z) / (i - $z)); |
944 | return ip2 * $log; |
a0d0e21e |
945 | } |
946 | |
66730be0 |
947 | # |
0c721ce2 |
948 | # asec |
949 | # |
950 | # Computes the arc secant asec(z) = acos(1 / z). |
951 | # |
952 | sub asec { |
953 | my ($z) = @_; |
0e505df1 |
954 | _divbyzero "asec($z)", $z if ($z == 0); |
fb73857a |
955 | return acos(1 / $z); |
0c721ce2 |
956 | } |
957 | |
958 | # |
5cd24f17 |
959 | # acsc |
0c721ce2 |
960 | # |
8c03c583 |
961 | # Computes the arc cosecant acsc(z) = asin(1 / z). |
0c721ce2 |
962 | # |
5cd24f17 |
963 | sub acsc { |
0c721ce2 |
964 | my ($z) = @_; |
0e505df1 |
965 | _divbyzero "acsc($z)", $z if ($z == 0); |
fb73857a |
966 | return asin(1 / $z); |
0c721ce2 |
967 | } |
968 | |
969 | # |
5cd24f17 |
970 | # acosec |
66730be0 |
971 | # |
5cd24f17 |
972 | # Alias for acsc(). |
0c721ce2 |
973 | # |
5cd24f17 |
974 | sub acosec { Math::Complex::acsc(@_) } |
0c721ce2 |
975 | |
66730be0 |
976 | # |
0c721ce2 |
977 | # acot |
978 | # |
8c03c583 |
979 | # Computes the arc cotangent acot(z) = atan(1 / z) |
0c721ce2 |
980 | # |
981 | sub acot { |
66730be0 |
982 | my ($z) = @_; |
1fa12f56 |
983 | _divbyzero "acot(0)" if $z == 0; |
984 | return ($z >= 0) ? CORE::atan2(1, $z) : CORE::atan2(-1, -$z) |
985 | unless ref $z; |
986 | _divbyzero "acot(i)" if ($z - i == 0); |
987 | _logofzero "acot(-i)" if ($z + i == 0); |
8c03c583 |
988 | return atan(1 / $z); |
66730be0 |
989 | } |
990 | |
991 | # |
0c721ce2 |
992 | # acotan |
993 | # |
994 | # Alias for acot(). |
995 | # |
996 | sub acotan { Math::Complex::acot(@_) } |
997 | |
998 | # |
66730be0 |
999 | # cosh |
1000 | # |
1001 | # Computes the hyperbolic cosine cosh(z) = (exp(z) + exp(-z))/2. |
1002 | # |
1003 | sub cosh { |
1004 | my ($z) = @_; |
fb73857a |
1005 | my $ex; |
0e505df1 |
1006 | unless (ref $z) { |
a8693bd3 |
1007 | $ex = CORE::exp($z); |
1fa12f56 |
1008 | return $ex ? ($ex + 1/$ex)/2 : $Inf; |
0e505df1 |
1009 | } |
1010 | my ($x, $y) = @{$z->cartesian}; |
a8693bd3 |
1011 | $ex = CORE::exp($x); |
1fa12f56 |
1012 | my $ex_1 = $ex ? 1 / $ex : $Inf; |
a8693bd3 |
1013 | return (ref $z)->make(CORE::cos($y) * ($ex + $ex_1)/2, |
1014 | CORE::sin($y) * ($ex - $ex_1)/2); |
66730be0 |
1015 | } |
1016 | |
1017 | # |
1018 | # sinh |
1019 | # |
1020 | # Computes the hyperbolic sine sinh(z) = (exp(z) - exp(-z))/2. |
1021 | # |
1022 | sub sinh { |
1023 | my ($z) = @_; |
fb73857a |
1024 | my $ex; |
0e505df1 |
1025 | unless (ref $z) { |
1fa12f56 |
1026 | return 0 if $z == 0; |
a8693bd3 |
1027 | $ex = CORE::exp($z); |
1fa12f56 |
1028 | return $ex ? ($ex - 1/$ex)/2 : "-$Inf"; |
0e505df1 |
1029 | } |
1030 | my ($x, $y) = @{$z->cartesian}; |
1fa12f56 |
1031 | my $cy = CORE::cos($y); |
1032 | my $sy = CORE::sin($y); |
a8693bd3 |
1033 | $ex = CORE::exp($x); |
1fa12f56 |
1034 | my $ex_1 = $ex ? 1 / $ex : $Inf; |
5240e574 |
1035 | return (ref $z)->make(CORE::cos($y) * ($ex - $ex_1)/2, |
1036 | CORE::sin($y) * ($ex + $ex_1)/2); |
66730be0 |
1037 | } |
1038 | |
1039 | # |
1040 | # tanh |
1041 | # |
1042 | # Computes the hyperbolic tangent tanh(z) = sinh(z) / cosh(z). |
1043 | # |
1044 | sub tanh { |
1045 | my ($z) = @_; |
0c721ce2 |
1046 | my $cz = cosh($z); |
0e505df1 |
1047 | _divbyzero "tanh($z)", "cosh($z)" if ($cz == 0); |
0c721ce2 |
1048 | return sinh($z) / $cz; |
66730be0 |
1049 | } |
1050 | |
1051 | # |
0c721ce2 |
1052 | # sech |
1053 | # |
1054 | # Computes the hyperbolic secant sech(z) = 1 / cosh(z). |
1055 | # |
1056 | sub sech { |
1057 | my ($z) = @_; |
1058 | my $cz = cosh($z); |
0e505df1 |
1059 | _divbyzero "sech($z)", "cosh($z)" if ($cz == 0); |
0c721ce2 |
1060 | return 1 / $cz; |
1061 | } |
1062 | |
1063 | # |
1064 | # csch |
1065 | # |
1066 | # Computes the hyperbolic cosecant csch(z) = 1 / sinh(z). |
66730be0 |
1067 | # |
0c721ce2 |
1068 | sub csch { |
1069 | my ($z) = @_; |
1070 | my $sz = sinh($z); |
0e505df1 |
1071 | _divbyzero "csch($z)", "sinh($z)" if ($sz == 0); |
0c721ce2 |
1072 | return 1 / $sz; |
1073 | } |
1074 | |
1075 | # |
1076 | # cosech |
1077 | # |
1078 | # Alias for csch(). |
1079 | # |
1080 | sub cosech { Math::Complex::csch(@_) } |
1081 | |
66730be0 |
1082 | # |
0c721ce2 |
1083 | # coth |
1084 | # |
1085 | # Computes the hyperbolic cotangent coth(z) = cosh(z) / sinh(z). |
1086 | # |
1087 | sub coth { |
66730be0 |
1088 | my ($z) = @_; |
0c721ce2 |
1089 | my $sz = sinh($z); |
1fa12f56 |
1090 | _divbyzero "coth($z)", "sinh($z)" if $sz == 0; |
0c721ce2 |
1091 | return cosh($z) / $sz; |
66730be0 |
1092 | } |
1093 | |
1094 | # |
0c721ce2 |
1095 | # cotanh |
1096 | # |
1097 | # Alias for coth(). |
1098 | # |
1099 | sub cotanh { Math::Complex::coth(@_) } |
1100 | |
1101 | # |
66730be0 |
1102 | # acosh |
1103 | # |
fb73857a |
1104 | # Computes the arc hyperbolic cosine acosh(z) = log(z + sqrt(z*z-1)). |
66730be0 |
1105 | # |
1106 | sub acosh { |
1107 | my ($z) = @_; |
fb73857a |
1108 | unless (ref $z) { |
fb73857a |
1109 | $z = cplx($z, 0); |
1110 | } |
8c03c583 |
1111 | my ($re, $im) = @{$z->cartesian}; |
fb73857a |
1112 | if ($im == 0) { |
1fa12f56 |
1113 | return CORE::log($re + CORE::sqrt($re*$re - 1)) |
1114 | if $re >= 1; |
1115 | return cplx(0, CORE::atan2(CORE::sqrt(1 - $re*$re), $re)) |
1116 | if CORE::abs($re) < 1; |
fb73857a |
1117 | } |
9bc5fa8d |
1118 | my $t = &sqrt($z * $z - 1) + $z; |
40b904b7 |
1119 | # Try Taylor if looking bad (this usually means that |
1120 | # $z was large negative, therefore the sqrt is really |
1121 | # close to abs(z), summing that with z...) |
9bc5fa8d |
1122 | $t = 1/(2 * $z) - 1/(8 * $z**3) + 1/(16 * $z**5) - 5/(128 * $z**7) |
1123 | if $t == 0; |
1124 | my $u = &log($t); |
40b904b7 |
1125 | $u->Im(-$u->Im) if $re < 0 && $im == 0; |
9bc5fa8d |
1126 | return $re < 0 ? -$u : $u; |
66730be0 |
1127 | } |
1128 | |
1129 | # |
1130 | # asinh |
1131 | # |
1fa12f56 |
1132 | # Computes the arc hyperbolic sine asinh(z) = log(z + sqrt(z*z+1)) |
66730be0 |
1133 | # |
1134 | sub asinh { |
1135 | my ($z) = @_; |
1fa12f56 |
1136 | unless (ref $z) { |
1137 | my $t = $z + CORE::sqrt($z*$z + 1); |
1138 | return CORE::log($t) if $t; |
1139 | } |
9bc5fa8d |
1140 | my $t = &sqrt($z * $z + 1) + $z; |
40b904b7 |
1141 | # Try Taylor if looking bad (this usually means that |
1142 | # $z was large negative, therefore the sqrt is really |
1143 | # close to abs(z), summing that with z...) |
9bc5fa8d |
1144 | $t = 1/(2 * $z) - 1/(8 * $z**3) + 1/(16 * $z**5) - 5/(128 * $z**7) |
1145 | if $t == 0; |
1fa12f56 |
1146 | return &log($t); |
66730be0 |
1147 | } |
1148 | |
1149 | # |
1150 | # atanh |
1151 | # |
1152 | # Computes the arc hyperbolic tangent atanh(z) = 1/2 log((1+z) / (1-z)). |
1153 | # |
1154 | sub atanh { |
1155 | my ($z) = @_; |
fb73857a |
1156 | unless (ref $z) { |
a8693bd3 |
1157 | return CORE::log((1 + $z)/(1 - $z))/2 if CORE::abs($z) < 1; |
fb73857a |
1158 | $z = cplx($z, 0); |
1159 | } |
1fa12f56 |
1160 | _divbyzero 'atanh(1)', "1 - $z" if (1 - $z == 0); |
1161 | _logofzero 'atanh(-1)' if (1 + $z == 0); |
1162 | return 0.5 * &log((1 + $z) / (1 - $z)); |
66730be0 |
1163 | } |
1164 | |
1165 | # |
0c721ce2 |
1166 | # asech |
1167 | # |
1168 | # Computes the hyperbolic arc secant asech(z) = acosh(1 / z). |
1169 | # |
1170 | sub asech { |
1171 | my ($z) = @_; |
1fa12f56 |
1172 | _divbyzero 'asech(0)', "$z" if ($z == 0); |
0c721ce2 |
1173 | return acosh(1 / $z); |
1174 | } |
1175 | |
1176 | # |
1177 | # acsch |
66730be0 |
1178 | # |
0c721ce2 |
1179 | # Computes the hyperbolic arc cosecant acsch(z) = asinh(1 / z). |
66730be0 |
1180 | # |
0c721ce2 |
1181 | sub acsch { |
66730be0 |
1182 | my ($z) = @_; |
0e505df1 |
1183 | _divbyzero 'acsch(0)', $z if ($z == 0); |
0c721ce2 |
1184 | return asinh(1 / $z); |
1185 | } |
1186 | |
1187 | # |
1188 | # acosech |
1189 | # |
1190 | # Alias for acosh(). |
1191 | # |
1192 | sub acosech { Math::Complex::acsch(@_) } |
1193 | |
1194 | # |
1195 | # acoth |
1196 | # |
1197 | # Computes the arc hyperbolic cotangent acoth(z) = 1/2 log((1+z) / (z-1)). |
1198 | # |
1199 | sub acoth { |
1200 | my ($z) = @_; |
1fa12f56 |
1201 | _divbyzero 'acoth(0)' if ($z == 0); |
fb73857a |
1202 | unless (ref $z) { |
a8693bd3 |
1203 | return CORE::log(($z + 1)/($z - 1))/2 if CORE::abs($z) > 1; |
fb73857a |
1204 | $z = cplx($z, 0); |
1205 | } |
1fa12f56 |
1206 | _divbyzero 'acoth(1)', "$z - 1" if ($z - 1 == 0); |
1207 | _logofzero 'acoth(-1)', "1 + $z" if (1 + $z == 0); |
1208 | return &log((1 + $z) / ($z - 1)) / 2; |
66730be0 |
1209 | } |
1210 | |
1211 | # |
0c721ce2 |
1212 | # acotanh |
1213 | # |
1214 | # Alias for acot(). |
1215 | # |
1216 | sub acotanh { Math::Complex::acoth(@_) } |
1217 | |
1218 | # |
66730be0 |
1219 | # (atan2) |
1220 | # |
1221 | # Compute atan(z1/z2). |
1222 | # |
1223 | sub atan2 { |
1224 | my ($z1, $z2, $inverted) = @_; |
fb73857a |
1225 | my ($re1, $im1, $re2, $im2); |
1226 | if ($inverted) { |
1227 | ($re1, $im1) = ref $z2 ? @{$z2->cartesian} : ($z2, 0); |
1228 | ($re2, $im2) = @{$z1->cartesian}; |
66730be0 |
1229 | } else { |
fb73857a |
1230 | ($re1, $im1) = @{$z1->cartesian}; |
1231 | ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0); |
1232 | } |
1233 | if ($im2 == 0) { |
1fa12f56 |
1234 | return CORE::atan2($re1, $re2) if $im1 == 0; |
1235 | return ($im1<=>0) * pip2 if $re2 == 0; |
66730be0 |
1236 | } |
fb73857a |
1237 | my $w = atan($z1/$z2); |
1238 | my ($u, $v) = ref $w ? @{$w->cartesian} : ($w, 0); |
1239 | $u += pi if $re2 < 0; |
1240 | $u -= pit2 if $u > pi; |
1241 | return cplx($u, $v); |
66730be0 |
1242 | } |
1243 | |
1244 | # |
1245 | # display_format |
1246 | # ->display_format |
1247 | # |
16357284 |
1248 | # Set (get if no argument) the display format for all complex numbers that |
fb73857a |
1249 | # don't happen to have overridden it via ->display_format |
66730be0 |
1250 | # |
16357284 |
1251 | # When called as an object method, this actually sets the display format for |
66730be0 |
1252 | # the current object. |
1253 | # |
1254 | # Valid object formats are 'c' and 'p' for cartesian and polar. The first |
1255 | # letter is used actually, so the type can be fully spelled out for clarity. |
1256 | # |
1257 | sub display_format { |
16357284 |
1258 | my $self = shift; |
1259 | my %display_format = %DISPLAY_FORMAT; |
66730be0 |
1260 | |
16357284 |
1261 | if (ref $self) { # Called as an object method |
1262 | if (exists $self->{display_format}) { |
1263 | my %obj = %{$self->{display_format}}; |
1264 | @display_format{keys %obj} = values %obj; |
1265 | } |
476757f7 |
1266 | } |
1267 | if (@_ == 1) { |
1268 | $display_format{style} = shift; |
1269 | } else { |
1270 | my %new = @_; |
1271 | @display_format{keys %new} = values %new; |
66730be0 |
1272 | } |
1273 | |
476757f7 |
1274 | if (ref $self) { # Called as an object method |
16357284 |
1275 | $self->{display_format} = { %display_format }; |
1276 | return |
1277 | wantarray ? |
1278 | %{$self->{display_format}} : |
1279 | $self->{display_format}->{style}; |
66730be0 |
1280 | } |
1281 | |
476757f7 |
1282 | # Called as a class method |
16357284 |
1283 | %DISPLAY_FORMAT = %display_format; |
1284 | return |
1285 | wantarray ? |
1286 | %DISPLAY_FORMAT : |
1287 | $DISPLAY_FORMAT{style}; |
66730be0 |
1288 | } |
1289 | |
1290 | # |
1291 | # (stringify) |
1292 | # |
1293 | # Show nicely formatted complex number under its cartesian or polar form, |
1294 | # depending on the current display format: |
1295 | # |
1296 | # . If a specific display format has been recorded for this object, use it. |
1297 | # . Otherwise, use the generic current default for all complex numbers, |
1298 | # which is a package global variable. |
1299 | # |
a0d0e21e |
1300 | sub stringify { |
66730be0 |
1301 | my ($z) = shift; |
66730be0 |
1302 | |
16357284 |
1303 | my $style = $z->display_format; |
1304 | |
1305 | $style = $DISPLAY_FORMAT{style} unless defined $style; |
66730be0 |
1306 | |
16357284 |
1307 | return $z->stringify_polar if $style =~ /^p/i; |
66730be0 |
1308 | return $z->stringify_cartesian; |
1309 | } |
1310 | |
1311 | # |
1312 | # ->stringify_cartesian |
1313 | # |
1314 | # Stringify as a cartesian representation 'a+bi'. |
1315 | # |
1316 | sub stringify_cartesian { |
1317 | my $z = shift; |
1318 | my ($x, $y) = @{$z->cartesian}; |
1319 | my ($re, $im); |
1320 | |
16357284 |
1321 | my %format = $z->display_format; |
1322 | my $format = $format{format}; |
1323 | |
1fa12f56 |
1324 | if ($x) { |
1325 | if ($x =~ /^NaN[QS]?$/i) { |
1326 | $re = $x; |
1327 | } else { |
1328 | if ($x =~ /^-?$Inf$/oi) { |
1329 | $re = $x; |
1330 | } else { |
1331 | $re = defined $format ? sprintf($format, $x) : $x; |
1332 | } |
1333 | } |
1334 | } else { |
1335 | undef $re; |
1336 | } |
1337 | |
1338 | if ($y) { |
40b904b7 |
1339 | if ($y =~ /^(NaN[QS]?)$/i) { |
1fa12f56 |
1340 | $im = $y; |
1341 | } else { |
1342 | if ($y =~ /^-?$Inf$/oi) { |
1343 | $im = $y; |
1344 | } else { |
40b904b7 |
1345 | $im = |
1346 | defined $format ? |
1347 | sprintf($format, $y) : |
1348 | ($y == 1 ? "" : ($y == -1 ? "-" : $y)); |
1fa12f56 |
1349 | } |
1350 | } |
1351 | $im .= "i"; |
1352 | } else { |
1353 | undef $im; |
16357284 |
1354 | } |
66730be0 |
1355 | |
1fa12f56 |
1356 | my $str = $re; |
1357 | |
16357284 |
1358 | if (defined $im) { |
1359 | if ($y < 0) { |
1360 | $str .= $im; |
1fa12f56 |
1361 | } elsif ($y > 0 || $im =~ /^NaN[QS]?i$/i) { |
16357284 |
1362 | $str .= "+" if defined $re; |
1363 | $str .= $im; |
1364 | } |
1fa12f56 |
1365 | } elsif (!defined $re) { |
1366 | $str = "0"; |
16357284 |
1367 | } |
66730be0 |
1368 | |
1369 | return $str; |
1370 | } |
1371 | |
d09ae4e6 |
1372 | |
66730be0 |
1373 | # |
1374 | # ->stringify_polar |
1375 | # |
1376 | # Stringify as a polar representation '[r,t]'. |
1377 | # |
1378 | sub stringify_polar { |
1379 | my $z = shift; |
1380 | my ($r, $t) = @{$z->polar}; |
1381 | my $theta; |
1382 | |
16357284 |
1383 | my %format = $z->display_format; |
1fa12f56 |
1384 | my $format = $format{format}; |
16357284 |
1385 | |
1fa12f56 |
1386 | if ($t =~ /^NaN[QS]?$/i || $t =~ /^-?$Inf$/oi) { |
1387 | $theta = $t; |
1388 | } elsif ($t == pi) { |
1389 | $theta = "pi"; |
1390 | } elsif ($r == 0 || $t == 0) { |
1391 | $theta = defined $format ? sprintf($format, $t) : $t; |
55497cff |
1392 | } |
66730be0 |
1393 | |
1fa12f56 |
1394 | return "[$r,$theta]" if defined $theta; |
1395 | |
66730be0 |
1396 | # |
1fa12f56 |
1397 | # Try to identify pi/n and friends. |
66730be0 |
1398 | # |
1399 | |
1fa12f56 |
1400 | $t -= int(CORE::abs($t) / pit2) * pit2; |
1401 | |
e97e26fa |
1402 | if ($format{polar_pretty_print} && $t) { |
1fa12f56 |
1403 | my ($a, $b); |
9bc5fa8d |
1404 | for $a (2..9) { |
1fa12f56 |
1405 | $b = $t * $a / pi; |
e97e26fa |
1406 | if ($b =~ /^-?\d+$/) { |
1fa12f56 |
1407 | $b = $b < 0 ? "-" : "" if CORE::abs($b) == 1; |
1408 | $theta = "${b}pi/$a"; |
d09ae4e6 |
1409 | last; |
66730be0 |
1410 | } |
d09ae4e6 |
1411 | } |
66730be0 |
1412 | } |
1413 | |
16357284 |
1414 | if (defined $format) { |
1415 | $r = sprintf($format, $r); |
1fa12f56 |
1416 | $theta = sprintf($format, $theta) unless defined $theta; |
1417 | } else { |
1418 | $theta = $t unless defined $theta; |
16357284 |
1419 | } |
1420 | |
1fa12f56 |
1421 | return "[$r,$theta]"; |
a0d0e21e |
1422 | } |
a5f75d66 |
1423 | |
1424 | 1; |
1425 | __END__ |
1426 | |
1cf6bcb8 |
1427 | =pod |
1428 | |
a5f75d66 |
1429 | =head1 NAME |
1430 | |
66730be0 |
1431 | Math::Complex - complex numbers and associated mathematical functions |
a5f75d66 |
1432 | |
1433 | =head1 SYNOPSIS |
1434 | |
66730be0 |
1435 | use Math::Complex; |
fb73857a |
1436 | |
66730be0 |
1437 | $z = Math::Complex->make(5, 6); |
1438 | $t = 4 - 3*i + $z; |
1439 | $j = cplxe(1, 2*pi/3); |
a5f75d66 |
1440 | |
1441 | =head1 DESCRIPTION |
1442 | |
66730be0 |
1443 | This package lets you create and manipulate complex numbers. By default, |
1444 | I<Perl> limits itself to real numbers, but an extra C<use> statement brings |
1445 | full complex support, along with a full set of mathematical functions |
1446 | typically associated with and/or extended to complex numbers. |
1447 | |
1448 | If you wonder what complex numbers are, they were invented to be able to solve |
1449 | the following equation: |
1450 | |
1451 | x*x = -1 |
1452 | |
1453 | and by definition, the solution is noted I<i> (engineers use I<j> instead since |
1454 | I<i> usually denotes an intensity, but the name does not matter). The number |
1455 | I<i> is a pure I<imaginary> number. |
1456 | |
1457 | The arithmetics with pure imaginary numbers works just like you would expect |
1458 | it with real numbers... you just have to remember that |
1459 | |
1460 | i*i = -1 |
1461 | |
1462 | so you have: |
1463 | |
1464 | 5i + 7i = i * (5 + 7) = 12i |
1465 | 4i - 3i = i * (4 - 3) = i |
1466 | 4i * 2i = -8 |
1467 | 6i / 2i = 3 |
1468 | 1 / i = -i |
1469 | |
1470 | Complex numbers are numbers that have both a real part and an imaginary |
1471 | part, and are usually noted: |
1472 | |
1473 | a + bi |
1474 | |
1475 | where C<a> is the I<real> part and C<b> is the I<imaginary> part. The |
1476 | arithmetic with complex numbers is straightforward. You have to |
1477 | keep track of the real and the imaginary parts, but otherwise the |
1478 | rules used for real numbers just apply: |
1479 | |
1480 | (4 + 3i) + (5 - 2i) = (4 + 5) + i(3 - 2) = 9 + i |
1481 | (2 + i) * (4 - i) = 2*4 + 4i -2i -i*i = 8 + 2i + 1 = 9 + 2i |
1482 | |
1483 | A graphical representation of complex numbers is possible in a plane |
1484 | (also called the I<complex plane>, but it's really a 2D plane). |
1485 | The number |
1486 | |
1487 | z = a + bi |
1488 | |
1489 | is the point whose coordinates are (a, b). Actually, it would |
1490 | be the vector originating from (0, 0) to (a, b). It follows that the addition |
1491 | of two complex numbers is a vectorial addition. |
1492 | |
1493 | Since there is a bijection between a point in the 2D plane and a complex |
1494 | number (i.e. the mapping is unique and reciprocal), a complex number |
1495 | can also be uniquely identified with polar coordinates: |
1496 | |
1497 | [rho, theta] |
1498 | |
1499 | where C<rho> is the distance to the origin, and C<theta> the angle between |
1500 | the vector and the I<x> axis. There is a notation for this using the |
1501 | exponential form, which is: |
1502 | |
1503 | rho * exp(i * theta) |
1504 | |
1505 | where I<i> is the famous imaginary number introduced above. Conversion |
1506 | between this form and the cartesian form C<a + bi> is immediate: |
1507 | |
1508 | a = rho * cos(theta) |
1509 | b = rho * sin(theta) |
1510 | |
1511 | which is also expressed by this formula: |
1512 | |
fb73857a |
1513 | z = rho * exp(i * theta) = rho * (cos theta + i * sin theta) |
66730be0 |
1514 | |
1515 | In other words, it's the projection of the vector onto the I<x> and I<y> |
1516 | axes. Mathematicians call I<rho> the I<norm> or I<modulus> and I<theta> |
1517 | the I<argument> of the complex number. The I<norm> of C<z> will be |
1518 | noted C<abs(z)>. |
1519 | |
1520 | The polar notation (also known as the trigonometric |
1521 | representation) is much more handy for performing multiplications and |
1522 | divisions of complex numbers, whilst the cartesian notation is better |
fb73857a |
1523 | suited for additions and subtractions. Real numbers are on the I<x> |
1524 | axis, and therefore I<theta> is zero or I<pi>. |
66730be0 |
1525 | |
1526 | All the common operations that can be performed on a real number have |
1527 | been defined to work on complex numbers as well, and are merely |
1528 | I<extensions> of the operations defined on real numbers. This means |
1529 | they keep their natural meaning when there is no imaginary part, provided |
1530 | the number is within their definition set. |
1531 | |
1532 | For instance, the C<sqrt> routine which computes the square root of |
fb73857a |
1533 | its argument is only defined for non-negative real numbers and yields a |
1534 | non-negative real number (it is an application from B<R+> to B<R+>). |
66730be0 |
1535 | If we allow it to return a complex number, then it can be extended to |
1536 | negative real numbers to become an application from B<R> to B<C> (the |
1537 | set of complex numbers): |
1538 | |
1539 | sqrt(x) = x >= 0 ? sqrt(x) : sqrt(-x)*i |
1540 | |
1541 | It can also be extended to be an application from B<C> to B<C>, |
1542 | whilst its restriction to B<R> behaves as defined above by using |
1543 | the following definition: |
1544 | |
1545 | sqrt(z = [r,t]) = sqrt(r) * exp(i * t/2) |
1546 | |
fb73857a |
1547 | Indeed, a negative real number can be noted C<[x,pi]> (the modulus |
1548 | I<x> is always non-negative, so C<[x,pi]> is really C<-x>, a negative |
1549 | number) and the above definition states that |
66730be0 |
1550 | |
1551 | sqrt([x,pi]) = sqrt(x) * exp(i*pi/2) = [sqrt(x),pi/2] = sqrt(x)*i |
1552 | |
1553 | which is exactly what we had defined for negative real numbers above. |
b42d0ec9 |
1554 | The C<sqrt> returns only one of the solutions: if you want the both, |
1555 | use the C<root> function. |
a5f75d66 |
1556 | |
66730be0 |
1557 | All the common mathematical functions defined on real numbers that |
1558 | are extended to complex numbers share that same property of working |
1559 | I<as usual> when the imaginary part is zero (otherwise, it would not |
1560 | be called an extension, would it?). |
a5f75d66 |
1561 | |
66730be0 |
1562 | A I<new> operation possible on a complex number that is |
1563 | the identity for real numbers is called the I<conjugate>, and is noted |
d1be9408 |
1564 | with a horizontal bar above the number, or C<~z> here. |
a5f75d66 |
1565 | |
66730be0 |
1566 | z = a + bi |
1567 | ~z = a - bi |
a5f75d66 |
1568 | |
66730be0 |
1569 | Simple... Now look: |
a5f75d66 |
1570 | |
66730be0 |
1571 | z * ~z = (a + bi) * (a - bi) = a*a + b*b |
a5f75d66 |
1572 | |
66730be0 |
1573 | We saw that the norm of C<z> was noted C<abs(z)> and was defined as the |
1574 | distance to the origin, also known as: |
a5f75d66 |
1575 | |
66730be0 |
1576 | rho = abs(z) = sqrt(a*a + b*b) |
a5f75d66 |
1577 | |
66730be0 |
1578 | so |
1579 | |
1580 | z * ~z = abs(z) ** 2 |
1581 | |
1582 | If z is a pure real number (i.e. C<b == 0>), then the above yields: |
1583 | |
1584 | a * a = abs(a) ** 2 |
1585 | |
1586 | which is true (C<abs> has the regular meaning for real number, i.e. stands |
1587 | for the absolute value). This example explains why the norm of C<z> is |
1588 | noted C<abs(z)>: it extends the C<abs> function to complex numbers, yet |
1589 | is the regular C<abs> we know when the complex number actually has no |
1590 | imaginary part... This justifies I<a posteriori> our use of the C<abs> |
1591 | notation for the norm. |
1592 | |
1593 | =head1 OPERATIONS |
1594 | |
1595 | Given the following notations: |
1596 | |
1597 | z1 = a + bi = r1 * exp(i * t1) |
1598 | z2 = c + di = r2 * exp(i * t2) |
1599 | z = <any complex or real number> |
1600 | |
1601 | the following (overloaded) operations are supported on complex numbers: |
1602 | |
1603 | z1 + z2 = (a + c) + i(b + d) |
1604 | z1 - z2 = (a - c) + i(b - d) |
1605 | z1 * z2 = (r1 * r2) * exp(i * (t1 + t2)) |
1606 | z1 / z2 = (r1 / r2) * exp(i * (t1 - t2)) |
1607 | z1 ** z2 = exp(z2 * log z1) |
b42d0ec9 |
1608 | ~z = a - bi |
1609 | abs(z) = r1 = sqrt(a*a + b*b) |
1610 | sqrt(z) = sqrt(r1) * exp(i * t/2) |
1611 | exp(z) = exp(a) * exp(i * b) |
1612 | log(z) = log(r1) + i*t |
1613 | sin(z) = 1/2i (exp(i * z1) - exp(-i * z)) |
1614 | cos(z) = 1/2 (exp(i * z1) + exp(-i * z)) |
66730be0 |
1615 | atan2(z1, z2) = atan(z1/z2) |
1616 | |
1617 | The following extra operations are supported on both real and complex |
1618 | numbers: |
1619 | |
1620 | Re(z) = a |
1621 | Im(z) = b |
1622 | arg(z) = t |
b42d0ec9 |
1623 | abs(z) = r |
66730be0 |
1624 | |
1625 | cbrt(z) = z ** (1/3) |
1626 | log10(z) = log(z) / log(10) |
1627 | logn(z, n) = log(z) / log(n) |
1628 | |
1629 | tan(z) = sin(z) / cos(z) |
0c721ce2 |
1630 | |
5aabfad6 |
1631 | csc(z) = 1 / sin(z) |
1632 | sec(z) = 1 / cos(z) |
0c721ce2 |
1633 | cot(z) = 1 / tan(z) |
66730be0 |
1634 | |
1635 | asin(z) = -i * log(i*z + sqrt(1-z*z)) |
fb73857a |
1636 | acos(z) = -i * log(z + i*sqrt(1-z*z)) |
66730be0 |
1637 | atan(z) = i/2 * log((i+z) / (i-z)) |
0c721ce2 |
1638 | |
5aabfad6 |
1639 | acsc(z) = asin(1 / z) |
1640 | asec(z) = acos(1 / z) |
8c03c583 |
1641 | acot(z) = atan(1 / z) = -i/2 * log((i+z) / (z-i)) |
66730be0 |
1642 | |
1643 | sinh(z) = 1/2 (exp(z) - exp(-z)) |
1644 | cosh(z) = 1/2 (exp(z) + exp(-z)) |
0c721ce2 |
1645 | tanh(z) = sinh(z) / cosh(z) = (exp(z) - exp(-z)) / (exp(z) + exp(-z)) |
1646 | |
5aabfad6 |
1647 | csch(z) = 1 / sinh(z) |
1648 | sech(z) = 1 / cosh(z) |
0c721ce2 |
1649 | coth(z) = 1 / tanh(z) |
fb73857a |
1650 | |
66730be0 |
1651 | asinh(z) = log(z + sqrt(z*z+1)) |
1652 | acosh(z) = log(z + sqrt(z*z-1)) |
1653 | atanh(z) = 1/2 * log((1+z) / (1-z)) |
66730be0 |
1654 | |
5aabfad6 |
1655 | acsch(z) = asinh(1 / z) |
1656 | asech(z) = acosh(1 / z) |
0c721ce2 |
1657 | acoth(z) = atanh(1 / z) = 1/2 * log((1+z) / (z-1)) |
1658 | |
b42d0ec9 |
1659 | I<arg>, I<abs>, I<log>, I<csc>, I<cot>, I<acsc>, I<acot>, I<csch>, |
1660 | I<coth>, I<acosech>, I<acotanh>, have aliases I<rho>, I<theta>, I<ln>, |
1661 | I<cosec>, I<cotan>, I<acosec>, I<acotan>, I<cosech>, I<cotanh>, |
1662 | I<acosech>, I<acotanh>, respectively. C<Re>, C<Im>, C<arg>, C<abs>, |
d1be9408 |
1663 | C<rho>, and C<theta> can be used also as mutators. The C<cbrt> |
b42d0ec9 |
1664 | returns only one of the solutions: if you want all three, use the |
1665 | C<root> function. |
0c721ce2 |
1666 | |
1667 | The I<root> function is available to compute all the I<n> |
66730be0 |
1668 | roots of some complex, where I<n> is a strictly positive integer. |
1669 | There are exactly I<n> such roots, returned as a list. Getting the |
1670 | number mathematicians call C<j> such that: |
1671 | |
1672 | 1 + j + j*j = 0; |
1673 | |
1674 | is a simple matter of writing: |
1675 | |
1676 | $j = ((root(1, 3))[1]; |
1677 | |
1678 | The I<k>th root for C<z = [r,t]> is given by: |
1679 | |
1680 | (root(z, n))[k] = r**(1/n) * exp(i * (t + 2*k*pi)/n) |
1681 | |
f4837644 |
1682 | The I<spaceship> comparison operator, E<lt>=E<gt>, is also defined. In |
1683 | order to ensure its restriction to real numbers is conform to what you |
1684 | would expect, the comparison is run on the real part of the complex |
1685 | number first, and imaginary parts are compared only when the real |
1686 | parts match. |
66730be0 |
1687 | |
1688 | =head1 CREATION |
1689 | |
1690 | To create a complex number, use either: |
1691 | |
1692 | $z = Math::Complex->make(3, 4); |
1693 | $z = cplx(3, 4); |
1694 | |
1695 | if you know the cartesian form of the number, or |
1696 | |
1697 | $z = 3 + 4*i; |
1698 | |
fb73857a |
1699 | if you like. To create a number using the polar form, use either: |
66730be0 |
1700 | |
1701 | $z = Math::Complex->emake(5, pi/3); |
1702 | $x = cplxe(5, pi/3); |
1703 | |
0c721ce2 |
1704 | instead. The first argument is the modulus, the second is the angle |
fb73857a |
1705 | (in radians, the full circle is 2*pi). (Mnemonic: C<e> is used as a |
1706 | notation for complex numbers in the polar form). |
66730be0 |
1707 | |
1708 | It is possible to write: |
1709 | |
1710 | $x = cplxe(-3, pi/4); |
1711 | |
16357284 |
1712 | but that will be silently converted into C<[3,-3pi/4]>, since the |
1713 | modulus must be non-negative (it represents the distance to the origin |
1714 | in the complex plane). |
66730be0 |
1715 | |
b42d0ec9 |
1716 | It is also possible to have a complex number as either argument of |
1717 | either the C<make> or C<emake>: the appropriate component of |
1718 | the argument will be used. |
1719 | |
1720 | $z1 = cplx(-2, 1); |
1721 | $z2 = cplx($z1, 4); |
1722 | |
66730be0 |
1723 | =head1 STRINGIFICATION |
1724 | |
1725 | When printed, a complex number is usually shown under its cartesian |
16357284 |
1726 | style I<a+bi>, but there are legitimate cases where the polar style |
66730be0 |
1727 | I<[r,t]> is more appropriate. |
1728 | |
16357284 |
1729 | By calling the class method C<Math::Complex::display_format> and |
1730 | supplying either C<"polar"> or C<"cartesian"> as an argument, you |
5287f86b |
1731 | override the default display style, which is C<"cartesian">. Not |
16357284 |
1732 | supplying any argument returns the current settings. |
66730be0 |
1733 | |
1734 | This default can be overridden on a per-number basis by calling the |
1735 | C<display_format> method instead. As before, not supplying any argument |
5287f86b |
1736 | returns the current display style for this number. Otherwise whatever you |
1737 | specify will be the new display style for I<this> particular number. |
66730be0 |
1738 | |
1739 | For instance: |
1740 | |
1741 | use Math::Complex; |
1742 | |
1743 | Math::Complex::display_format('polar'); |
16357284 |
1744 | $j = (root(1, 3))[1]; |
1745 | print "j = $j\n"; # Prints "j = [1,2pi/3]" |
66730be0 |
1746 | $j->display_format('cartesian'); |
1747 | print "j = $j\n"; # Prints "j = -0.5+0.866025403784439i" |
1748 | |
5287f86b |
1749 | The polar style attempts to emphasize arguments like I<k*pi/n> |
9bc5fa8d |
1750 | (where I<n> is a positive integer and I<k> an integer within [-9, +9]), |
5287f86b |
1751 | this is called I<polar pretty-printing>. |
66730be0 |
1752 | |
16357284 |
1753 | =head2 CHANGED IN PERL 5.6 |
1754 | |
1755 | The C<display_format> class method and the corresponding |
1756 | C<display_format> object method can now be called using |
1757 | a parameter hash instead of just a one parameter. |
1758 | |
1759 | The old display format style, which can have values C<"cartesian"> or |
40b904b7 |
1760 | C<"polar">, can be changed using the C<"style"> parameter. |
1761 | |
1762 | $j->display_format(style => "polar"); |
1763 | |
1764 | The one parameter calling convention also still works. |
1765 | |
1766 | $j->display_format("polar"); |
16357284 |
1767 | |
1768 | There are two new display parameters. |
1769 | |
40b904b7 |
1770 | The first one is C<"format">, which is a sprintf()-style format string |
1771 | to be used for both numeric parts of the complex number(s). The is |
1772 | somewhat system-dependent but most often it corresponds to C<"%.15g">. |
1773 | You can revert to the default by setting the C<format> to C<undef>. |
16357284 |
1774 | |
1775 | # the $j from the above example |
1776 | |
1777 | $j->display_format('format' => '%.5f'); |
1778 | print "j = $j\n"; # Prints "j = -0.50000+0.86603i" |
40b904b7 |
1779 | $j->display_format('format' => undef); |
16357284 |
1780 | print "j = $j\n"; # Prints "j = -0.5+0.86603i" |
1781 | |
1782 | Notice that this affects also the return values of the |
1783 | C<display_format> methods: in list context the whole parameter hash |
40b904b7 |
1784 | will be returned, as opposed to only the style parameter value. |
1785 | This is a potential incompatibility with earlier versions if you |
1786 | have been calling the C<display_format> method in list context. |
16357284 |
1787 | |
5287f86b |
1788 | The second new display parameter is C<"polar_pretty_print">, which can |
1789 | be set to true or false, the default being true. See the previous |
1790 | section for what this means. |
16357284 |
1791 | |
66730be0 |
1792 | =head1 USAGE |
1793 | |
1794 | Thanks to overloading, the handling of arithmetics with complex numbers |
1795 | is simple and almost transparent. |
1796 | |
1797 | Here are some examples: |
1798 | |
1799 | use Math::Complex; |
1800 | |
1801 | $j = cplxe(1, 2*pi/3); # $j ** 3 == 1 |
1802 | print "j = $j, j**3 = ", $j ** 3, "\n"; |
1803 | print "1 + j + j**2 = ", 1 + $j + $j**2, "\n"; |
1804 | |
1805 | $z = -16 + 0*i; # Force it to be a complex |
1806 | print "sqrt($z) = ", sqrt($z), "\n"; |
1807 | |
1808 | $k = exp(i * 2*pi/3); |
1809 | print "$j - $k = ", $j - $k, "\n"; |
a5f75d66 |
1810 | |
b42d0ec9 |
1811 | $z->Re(3); # Re, Im, arg, abs, |
1812 | $j->arg(2); # (the last two aka rho, theta) |
1813 | # can be used also as mutators. |
1814 | |
1815 | =head1 ERRORS DUE TO DIVISION BY ZERO OR LOGARITHM OF ZERO |
5aabfad6 |
1816 | |
1817 | The division (/) and the following functions |
1818 | |
b42d0ec9 |
1819 | log ln log10 logn |
2820d885 |
1820 | tan sec csc cot |
b42d0ec9 |
1821 | atan asec acsc acot |
1822 | tanh sech csch coth |
1823 | atanh asech acsch acoth |
5aabfad6 |
1824 | |
1825 | cannot be computed for all arguments because that would mean dividing |
8c03c583 |
1826 | by zero or taking logarithm of zero. These situations cause fatal |
1827 | runtime errors looking like this |
5aabfad6 |
1828 | |
1829 | cot(0): Division by zero. |
5cd24f17 |
1830 | (Because in the definition of cot(0), the divisor sin(0) is 0) |
5aabfad6 |
1831 | Died at ... |
1832 | |
8c03c583 |
1833 | or |
1834 | |
1835 | atanh(-1): Logarithm of zero. |
1836 | Died at... |
1837 | |
1838 | For the C<csc>, C<cot>, C<asec>, C<acsc>, C<acot>, C<csch>, C<coth>, |
d1be9408 |
1839 | C<asech>, C<acsch>, the argument cannot be C<0> (zero). For the |
b42d0ec9 |
1840 | logarithmic functions and the C<atanh>, C<acoth>, the argument cannot |
1841 | be C<1> (one). For the C<atanh>, C<acoth>, the argument cannot be |
1842 | C<-1> (minus one). For the C<atan>, C<acot>, the argument cannot be |
1843 | C<i> (the imaginary unit). For the C<atan>, C<acoth>, the argument |
1844 | cannot be C<-i> (the negative imaginary unit). For the C<tan>, |
1845 | C<sec>, C<tanh>, the argument cannot be I<pi/2 + k * pi>, where I<k> |
1846 | is any integer. |
1847 | |
1848 | Note that because we are operating on approximations of real numbers, |
1849 | these errors can happen when merely `too close' to the singularities |
40b904b7 |
1850 | listed above. |
b42d0ec9 |
1851 | |
1852 | =head1 ERRORS DUE TO INDIGESTIBLE ARGUMENTS |
1853 | |
1854 | The C<make> and C<emake> accept both real and complex arguments. |
1855 | When they cannot recognize the arguments they will die with error |
1856 | messages like the following |
1857 | |
1858 | Math::Complex::make: Cannot take real part of ... |
1859 | Math::Complex::make: Cannot take real part of ... |
1860 | Math::Complex::emake: Cannot take rho of ... |
1861 | Math::Complex::emake: Cannot take theta of ... |
5cd24f17 |
1862 | |
a5f75d66 |
1863 | =head1 BUGS |
1864 | |
5cd24f17 |
1865 | Saying C<use Math::Complex;> exports many mathematical routines in the |
fb73857a |
1866 | caller environment and even overrides some (C<sqrt>, C<log>). |
1867 | This is construed as a feature by the Authors, actually... ;-) |
a5f75d66 |
1868 | |
66730be0 |
1869 | All routines expect to be given real or complex numbers. Don't attempt to |
1870 | use BigFloat, since Perl has currently no rule to disambiguate a '+' |
1871 | operation (for instance) between two overloaded entities. |
a5f75d66 |
1872 | |
d09ae4e6 |
1873 | In Cray UNICOS there is some strange numerical instability that results |
1874 | in root(), cos(), sin(), cosh(), sinh(), losing accuracy fast. Beware. |
1875 | The bug may be in UNICOS math libs, in UNICOS C compiler, in Math::Complex. |
1876 | Whatever it is, it does not manifest itself anywhere else where Perl runs. |
1877 | |
0c721ce2 |
1878 | =head1 AUTHORS |
a5f75d66 |
1879 | |
6e238990 |
1880 | Raphael Manfredi <F<Raphael_Manfredi@pobox.com>> and |
ace5de91 |
1881 | Jarkko Hietaniemi <F<jhi@iki.fi>>. |
5cd24f17 |
1882 | |
fb73857a |
1883 | Extensive patches by Daniel S. Lewart <F<d-lewart@uiuc.edu>>. |
1884 | |
5cd24f17 |
1885 | =cut |
1886 | |
b42d0ec9 |
1887 | 1; |
1888 | |
5cd24f17 |
1889 | # eof |