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 | |
e6c12c3f |
12 | $VERSION = 1.34; |
476757f7 |
13 | |
9fbe1b12 |
14 | BEGIN { |
ffb4440d |
15 | unless ($^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 | |
91cb744f |
40 | # Regular expression for floating point numbers. |
41 | my $gre = qr'\s*([\+\-]?(?:(?:(?:\d+(?:_\d+)*(?:\.\d*(?:_\d+)*)?|\.\d+(?:_\d+)*)(?:[eE][\+\-]?\d+(?:_\d+)*)?)))'; |
42 | |
9fbe1b12 |
43 | require Exporter; |
0c721ce2 |
44 | |
5aabfad6 |
45 | @ISA = qw(Exporter); |
46 | |
5aabfad6 |
47 | my @trig = qw( |
48 | pi |
fb73857a |
49 | tan |
5aabfad6 |
50 | csc cosec sec cot cotan |
51 | asin acos atan |
52 | acsc acosec asec acot acotan |
53 | sinh cosh tanh |
54 | csch cosech sech coth cotanh |
55 | asinh acosh atanh |
56 | acsch acosech asech acoth acotanh |
57 | ); |
58 | |
59 | @EXPORT = (qw( |
b42d0ec9 |
60 | i Re Im rho theta arg |
fb73857a |
61 | sqrt log ln |
5aabfad6 |
62 | log10 logn cbrt root |
63 | cplx cplxe |
64 | ), |
65 | @trig); |
66 | |
67 | %EXPORT_TAGS = ( |
68 | 'trig' => [@trig], |
66730be0 |
69 | ); |
a0d0e21e |
70 | |
a5f75d66 |
71 | use overload |
0c721ce2 |
72 | '+' => \&plus, |
73 | '-' => \&minus, |
74 | '*' => \&multiply, |
75 | '/' => \÷, |
66730be0 |
76 | '**' => \&power, |
1fa12f56 |
77 | '==' => \&numeq, |
66730be0 |
78 | '<=>' => \&spaceship, |
79 | 'neg' => \&negate, |
0c721ce2 |
80 | '~' => \&conjugate, |
66730be0 |
81 | 'abs' => \&abs, |
82 | 'sqrt' => \&sqrt, |
83 | 'exp' => \&exp, |
84 | 'log' => \&log, |
85 | 'sin' => \&sin, |
86 | 'cos' => \&cos, |
0c721ce2 |
87 | 'tan' => \&tan, |
66730be0 |
88 | 'atan2' => \&atan2, |
89 | qw("" stringify); |
90 | |
91 | # |
b42d0ec9 |
92 | # Package "privates" |
66730be0 |
93 | # |
94 | |
16357284 |
95 | my %DISPLAY_FORMAT = ('style' => 'cartesian', |
96 | 'polar_pretty_print' => 1); |
97 | my $eps = 1e-14; # Epsilon |
66730be0 |
98 | |
99 | # |
100 | # Object attributes (internal): |
101 | # cartesian [real, imaginary] -- cartesian form |
102 | # polar [rho, theta] -- polar form |
103 | # c_dirty cartesian form not up-to-date |
104 | # p_dirty polar form not up-to-date |
105 | # display display format (package's global when not set) |
106 | # |
107 | |
b42d0ec9 |
108 | # Die on bad *make() arguments. |
109 | |
110 | sub _cannot_make { |
111 | die "@{[(caller(1))[3]]}: Cannot take $_[0] of $_[1].\n"; |
112 | } |
113 | |
91cb744f |
114 | sub _remake { |
115 | my $arg = shift; |
116 | my ($made, $p, $q); |
117 | |
118 | if ($arg =~ /^(?:$gre)?$gre\s*i\s*$/) { |
119 | ($p, $q) = ($1 || 0, $2); |
120 | $made = 'cart'; |
121 | } elsif ($arg =~ /^\s*\[\s*$gre\s*(?:,\s*$gre\s*)?\]\s*$/) { |
122 | ($p, $q) = ($1, $2 || 0); |
123 | $made = 'exp'; |
124 | } |
125 | |
126 | if ($made) { |
127 | $p =~ s/^\+//; |
128 | $q =~ s/^\+//; |
129 | } |
130 | |
131 | return ($made, $p, $q); |
132 | } |
133 | |
66730be0 |
134 | # |
135 | # ->make |
136 | # |
137 | # Create a new complex number (cartesian form) |
138 | # |
139 | sub make { |
140 | my $self = bless {}, shift; |
141 | my ($re, $im) = @_; |
91cb744f |
142 | if (@_ == 1) { |
143 | my ($remade, $p, $q) = _remake($re); |
144 | if ($remade) { |
145 | if ($remade eq 'cart') { |
146 | ($re, $im) = ($p, $q); |
147 | } else { |
148 | return (ref $self)->emake($p, $q); |
149 | } |
150 | } |
151 | } |
b42d0ec9 |
152 | my $rre = ref $re; |
153 | if ( $rre ) { |
154 | if ( $rre eq ref $self ) { |
155 | $re = Re($re); |
156 | } else { |
157 | _cannot_make("real part", $rre); |
158 | } |
159 | } |
160 | my $rim = ref $im; |
161 | if ( $rim ) { |
162 | if ( $rim eq ref $self ) { |
163 | $im = Im($im); |
164 | } else { |
165 | _cannot_make("imaginary part", $rim); |
166 | } |
167 | } |
91cb744f |
168 | _cannot_make("real part", $re) unless $re =~ /^$gre$/; |
169 | $im ||= 0; |
170 | _cannot_make("imaginary part", $im) unless $im =~ /^$gre$/; |
b42d0ec9 |
171 | $self->{'cartesian'} = [ $re, $im ]; |
66730be0 |
172 | $self->{c_dirty} = 0; |
173 | $self->{p_dirty} = 1; |
b42d0ec9 |
174 | $self->display_format('cartesian'); |
66730be0 |
175 | return $self; |
176 | } |
177 | |
178 | # |
179 | # ->emake |
180 | # |
181 | # Create a new complex number (exponential form) |
182 | # |
183 | sub emake { |
184 | my $self = bless {}, shift; |
185 | my ($rho, $theta) = @_; |
91cb744f |
186 | if (@_ == 1) { |
187 | my ($remade, $p, $q) = _remake($rho); |
188 | if ($remade) { |
189 | if ($remade eq 'exp') { |
190 | ($rho, $theta) = ($p, $q); |
191 | } else { |
192 | return (ref $self)->make($p, $q); |
193 | } |
194 | } |
195 | } |
b42d0ec9 |
196 | my $rrh = ref $rho; |
197 | if ( $rrh ) { |
198 | if ( $rrh eq ref $self ) { |
199 | $rho = rho($rho); |
200 | } else { |
201 | _cannot_make("rho", $rrh); |
202 | } |
203 | } |
204 | my $rth = ref $theta; |
205 | if ( $rth ) { |
206 | if ( $rth eq ref $self ) { |
207 | $theta = theta($theta); |
208 | } else { |
209 | _cannot_make("theta", $rth); |
210 | } |
211 | } |
fb73857a |
212 | if ($rho < 0) { |
213 | $rho = -$rho; |
214 | $theta = ($theta <= 0) ? $theta + pi() : $theta - pi(); |
215 | } |
91cb744f |
216 | _cannot_make("rho", $rho) unless $rho =~ /^$gre$/; |
217 | $theta ||= 0; |
218 | _cannot_make("theta", $theta) unless $theta =~ /^$gre$/; |
fb73857a |
219 | $self->{'polar'} = [$rho, $theta]; |
66730be0 |
220 | $self->{p_dirty} = 0; |
221 | $self->{c_dirty} = 1; |
b42d0ec9 |
222 | $self->display_format('polar'); |
66730be0 |
223 | return $self; |
224 | } |
225 | |
226 | sub new { &make } # For backward compatibility only. |
227 | |
228 | # |
229 | # cplx |
230 | # |
231 | # Creates a complex number from a (re, im) tuple. |
232 | # This avoids the burden of writing Math::Complex->make(re, im). |
233 | # |
234 | sub cplx { |
91cb744f |
235 | return __PACKAGE__->make(@_); |
66730be0 |
236 | } |
237 | |
238 | # |
239 | # cplxe |
240 | # |
241 | # Creates a complex number from a (rho, theta) tuple. |
242 | # This avoids the burden of writing Math::Complex->emake(rho, theta). |
243 | # |
244 | sub cplxe { |
91cb744f |
245 | return __PACKAGE__->emake(@_); |
66730be0 |
246 | } |
247 | |
248 | # |
249 | # pi |
250 | # |
fb73857a |
251 | # The number defined as pi = 180 degrees |
66730be0 |
252 | # |
6570f784 |
253 | sub pi () { 4 * CORE::atan2(1, 1) } |
5cd24f17 |
254 | |
255 | # |
fb73857a |
256 | # pit2 |
5cd24f17 |
257 | # |
fb73857a |
258 | # The full circle |
259 | # |
6570f784 |
260 | sub pit2 () { 2 * pi } |
fb73857a |
261 | |
5cd24f17 |
262 | # |
fb73857a |
263 | # pip2 |
264 | # |
265 | # The quarter circle |
266 | # |
6570f784 |
267 | sub pip2 () { pi / 2 } |
5cd24f17 |
268 | |
fb73857a |
269 | # |
d09ae4e6 |
270 | # deg1 |
271 | # |
272 | # One degree in radians, used in stringify_polar. |
273 | # |
274 | |
6570f784 |
275 | sub deg1 () { pi / 180 } |
d09ae4e6 |
276 | |
277 | # |
fb73857a |
278 | # uplog10 |
279 | # |
280 | # Used in log10(). |
281 | # |
6570f784 |
282 | sub uplog10 () { 1 / CORE::log(10) } |
66730be0 |
283 | |
284 | # |
285 | # i |
286 | # |
287 | # The number defined as i*i = -1; |
288 | # |
289 | sub i () { |
5cd24f17 |
290 | return $i if ($i); |
291 | $i = bless {}; |
40da2db3 |
292 | $i->{'cartesian'} = [0, 1]; |
fb73857a |
293 | $i->{'polar'} = [1, pip2]; |
66730be0 |
294 | $i->{c_dirty} = 0; |
295 | $i->{p_dirty} = 0; |
296 | return $i; |
297 | } |
298 | |
299 | # |
1fa12f56 |
300 | # ip2 |
301 | # |
302 | # Half of i. |
303 | # |
304 | sub ip2 () { i / 2 } |
305 | |
306 | # |
66730be0 |
307 | # Attribute access/set routines |
308 | # |
309 | |
0c721ce2 |
310 | sub cartesian {$_[0]->{c_dirty} ? |
311 | $_[0]->update_cartesian : $_[0]->{'cartesian'}} |
312 | sub polar {$_[0]->{p_dirty} ? |
313 | $_[0]->update_polar : $_[0]->{'polar'}} |
66730be0 |
314 | |
40da2db3 |
315 | sub set_cartesian { $_[0]->{p_dirty}++; $_[0]->{'cartesian'} = $_[1] } |
316 | sub set_polar { $_[0]->{c_dirty}++; $_[0]->{'polar'} = $_[1] } |
66730be0 |
317 | |
318 | # |
319 | # ->update_cartesian |
320 | # |
321 | # Recompute and return the cartesian form, given accurate polar form. |
322 | # |
323 | sub update_cartesian { |
324 | my $self = shift; |
40da2db3 |
325 | my ($r, $t) = @{$self->{'polar'}}; |
66730be0 |
326 | $self->{c_dirty} = 0; |
a8693bd3 |
327 | return $self->{'cartesian'} = [$r * CORE::cos($t), $r * CORE::sin($t)]; |
66730be0 |
328 | } |
329 | |
330 | # |
331 | # |
332 | # ->update_polar |
333 | # |
334 | # Recompute and return the polar form, given accurate cartesian form. |
335 | # |
336 | sub update_polar { |
337 | my $self = shift; |
40da2db3 |
338 | my ($x, $y) = @{$self->{'cartesian'}}; |
66730be0 |
339 | $self->{p_dirty} = 0; |
40da2db3 |
340 | return $self->{'polar'} = [0, 0] if $x == 0 && $y == 0; |
1fa12f56 |
341 | return $self->{'polar'} = [CORE::sqrt($x*$x + $y*$y), |
342 | CORE::atan2($y, $x)]; |
66730be0 |
343 | } |
344 | |
345 | # |
346 | # (plus) |
347 | # |
348 | # Computes z1+z2. |
349 | # |
350 | sub plus { |
351 | my ($z1, $z2, $regular) = @_; |
352 | my ($re1, $im1) = @{$z1->cartesian}; |
0e505df1 |
353 | $z2 = cplx($z2) unless ref $z2; |
5cd24f17 |
354 | my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0); |
66730be0 |
355 | unless (defined $regular) { |
356 | $z1->set_cartesian([$re1 + $re2, $im1 + $im2]); |
357 | return $z1; |
358 | } |
359 | return (ref $z1)->make($re1 + $re2, $im1 + $im2); |
360 | } |
361 | |
362 | # |
363 | # (minus) |
364 | # |
365 | # Computes z1-z2. |
366 | # |
367 | sub minus { |
368 | my ($z1, $z2, $inverted) = @_; |
369 | my ($re1, $im1) = @{$z1->cartesian}; |
0e505df1 |
370 | $z2 = cplx($z2) unless ref $z2; |
371 | my ($re2, $im2) = @{$z2->cartesian}; |
66730be0 |
372 | unless (defined $inverted) { |
373 | $z1->set_cartesian([$re1 - $re2, $im1 - $im2]); |
374 | return $z1; |
375 | } |
376 | return $inverted ? |
377 | (ref $z1)->make($re2 - $re1, $im2 - $im1) : |
378 | (ref $z1)->make($re1 - $re2, $im1 - $im2); |
0e505df1 |
379 | |
66730be0 |
380 | } |
381 | |
382 | # |
383 | # (multiply) |
384 | # |
385 | # Computes z1*z2. |
386 | # |
387 | sub multiply { |
fb73857a |
388 | my ($z1, $z2, $regular) = @_; |
389 | if ($z1->{p_dirty} == 0 and ref $z2 and $z2->{p_dirty} == 0) { |
390 | # if both polar better use polar to avoid rounding errors |
391 | my ($r1, $t1) = @{$z1->polar}; |
392 | my ($r2, $t2) = @{$z2->polar}; |
393 | my $t = $t1 + $t2; |
394 | if ($t > pi()) { $t -= pit2 } |
395 | elsif ($t <= -pi()) { $t += pit2 } |
396 | unless (defined $regular) { |
397 | $z1->set_polar([$r1 * $r2, $t]); |
66730be0 |
398 | return $z1; |
fb73857a |
399 | } |
400 | return (ref $z1)->emake($r1 * $r2, $t); |
401 | } else { |
402 | my ($x1, $y1) = @{$z1->cartesian}; |
403 | if (ref $z2) { |
404 | my ($x2, $y2) = @{$z2->cartesian}; |
405 | return (ref $z1)->make($x1*$x2-$y1*$y2, $x1*$y2+$y1*$x2); |
406 | } else { |
407 | return (ref $z1)->make($x1*$z2, $y1*$z2); |
408 | } |
66730be0 |
409 | } |
66730be0 |
410 | } |
411 | |
412 | # |
0e505df1 |
413 | # _divbyzero |
0c721ce2 |
414 | # |
415 | # Die on division by zero. |
416 | # |
0e505df1 |
417 | sub _divbyzero { |
5cd24f17 |
418 | my $mess = "$_[0]: Division by zero.\n"; |
419 | |
420 | if (defined $_[1]) { |
421 | $mess .= "(Because in the definition of $_[0], the divisor "; |
1fa12f56 |
422 | $mess .= "$_[1] " unless ("$_[1]" eq '0'); |
5cd24f17 |
423 | $mess .= "is 0)\n"; |
424 | } |
425 | |
0c721ce2 |
426 | my @up = caller(1); |
fb73857a |
427 | |
5cd24f17 |
428 | $mess .= "Died at $up[1] line $up[2].\n"; |
429 | |
430 | die $mess; |
0c721ce2 |
431 | } |
432 | |
433 | # |
66730be0 |
434 | # (divide) |
435 | # |
436 | # Computes z1/z2. |
437 | # |
438 | sub divide { |
439 | my ($z1, $z2, $inverted) = @_; |
fb73857a |
440 | if ($z1->{p_dirty} == 0 and ref $z2 and $z2->{p_dirty} == 0) { |
441 | # if both polar better use polar to avoid rounding errors |
442 | my ($r1, $t1) = @{$z1->polar}; |
443 | my ($r2, $t2) = @{$z2->polar}; |
444 | my $t; |
445 | if ($inverted) { |
0e505df1 |
446 | _divbyzero "$z2/0" if ($r1 == 0); |
fb73857a |
447 | $t = $t2 - $t1; |
448 | if ($t > pi()) { $t -= pit2 } |
449 | elsif ($t <= -pi()) { $t += pit2 } |
450 | return (ref $z1)->emake($r2 / $r1, $t); |
451 | } else { |
0e505df1 |
452 | _divbyzero "$z1/0" if ($r2 == 0); |
fb73857a |
453 | $t = $t1 - $t2; |
454 | if ($t > pi()) { $t -= pit2 } |
455 | elsif ($t <= -pi()) { $t += pit2 } |
456 | return (ref $z1)->emake($r1 / $r2, $t); |
457 | } |
458 | } else { |
459 | my ($d, $x2, $y2); |
460 | if ($inverted) { |
461 | ($x2, $y2) = @{$z1->cartesian}; |
462 | $d = $x2*$x2 + $y2*$y2; |
463 | _divbyzero "$z2/0" if $d == 0; |
464 | return (ref $z1)->make(($x2*$z2)/$d, -($y2*$z2)/$d); |
465 | } else { |
466 | my ($x1, $y1) = @{$z1->cartesian}; |
467 | if (ref $z2) { |
468 | ($x2, $y2) = @{$z2->cartesian}; |
469 | $d = $x2*$x2 + $y2*$y2; |
470 | _divbyzero "$z1/0" if $d == 0; |
471 | my $u = ($x1*$x2 + $y1*$y2)/$d; |
472 | my $v = ($y1*$x2 - $x1*$y2)/$d; |
473 | return (ref $z1)->make($u, $v); |
474 | } else { |
475 | _divbyzero "$z1/0" if $z2 == 0; |
476 | return (ref $z1)->make($x1/$z2, $y1/$z2); |
477 | } |
478 | } |
0c721ce2 |
479 | } |
66730be0 |
480 | } |
481 | |
482 | # |
483 | # (power) |
484 | # |
485 | # Computes z1**z2 = exp(z2 * log z1)). |
486 | # |
487 | sub power { |
488 | my ($z1, $z2, $inverted) = @_; |
ace5de91 |
489 | if ($inverted) { |
2820d885 |
490 | return 1 if $z1 == 0 || $z2 == 1; |
491 | return 0 if $z2 == 0 && Re($z1) > 0; |
ace5de91 |
492 | } else { |
2820d885 |
493 | return 1 if $z2 == 0 || $z1 == 1; |
494 | return 0 if $z1 == 0 && Re($z2) > 0; |
ace5de91 |
495 | } |
1fa12f56 |
496 | my $w = $inverted ? &exp($z1 * &log($z2)) |
497 | : &exp($z2 * &log($z1)); |
d09ae4e6 |
498 | # If both arguments cartesian, return cartesian, else polar. |
499 | return $z1->{c_dirty} == 0 && |
500 | (not ref $z2 or $z2->{c_dirty} == 0) ? |
501 | cplx(@{$w->cartesian}) : $w; |
66730be0 |
502 | } |
503 | |
504 | # |
505 | # (spaceship) |
506 | # |
507 | # Computes z1 <=> z2. |
2820d885 |
508 | # Sorts on the real part first, then on the imaginary part. Thus 2-4i < 3+8i. |
66730be0 |
509 | # |
510 | sub spaceship { |
511 | my ($z1, $z2, $inverted) = @_; |
5cd24f17 |
512 | my ($re1, $im1) = ref $z1 ? @{$z1->cartesian} : ($z1, 0); |
513 | my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0); |
66730be0 |
514 | my $sgn = $inverted ? -1 : 1; |
515 | return $sgn * ($re1 <=> $re2) if $re1 != $re2; |
516 | return $sgn * ($im1 <=> $im2); |
517 | } |
518 | |
519 | # |
1fa12f56 |
520 | # (numeq) |
521 | # |
522 | # Computes z1 == z2. |
523 | # |
524 | # (Required in addition to spaceship() because of NaNs.) |
525 | sub numeq { |
526 | my ($z1, $z2, $inverted) = @_; |
527 | my ($re1, $im1) = ref $z1 ? @{$z1->cartesian} : ($z1, 0); |
528 | my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0); |
529 | return $re1 == $re2 && $im1 == $im2 ? 1 : 0; |
530 | } |
531 | |
532 | # |
66730be0 |
533 | # (negate) |
534 | # |
535 | # Computes -z. |
536 | # |
537 | sub negate { |
538 | my ($z) = @_; |
539 | if ($z->{c_dirty}) { |
540 | my ($r, $t) = @{$z->polar}; |
fb73857a |
541 | $t = ($t <= 0) ? $t + pi : $t - pi; |
542 | return (ref $z)->emake($r, $t); |
66730be0 |
543 | } |
544 | my ($re, $im) = @{$z->cartesian}; |
545 | return (ref $z)->make(-$re, -$im); |
546 | } |
547 | |
548 | # |
549 | # (conjugate) |
550 | # |
551 | # Compute complex's conjugate. |
552 | # |
553 | sub conjugate { |
554 | my ($z) = @_; |
555 | if ($z->{c_dirty}) { |
556 | my ($r, $t) = @{$z->polar}; |
557 | return (ref $z)->emake($r, -$t); |
558 | } |
559 | my ($re, $im) = @{$z->cartesian}; |
560 | return (ref $z)->make($re, -$im); |
561 | } |
562 | |
563 | # |
564 | # (abs) |
565 | # |
b42d0ec9 |
566 | # Compute or set complex's norm (rho). |
66730be0 |
567 | # |
568 | sub abs { |
b42d0ec9 |
569 | my ($z, $rho) = @_; |
1fa12f56 |
570 | unless (ref $z) { |
571 | if (@_ == 2) { |
572 | $_[0] = $_[1]; |
573 | } else { |
574 | return CORE::abs($z); |
575 | } |
576 | } |
b42d0ec9 |
577 | if (defined $rho) { |
578 | $z->{'polar'} = [ $rho, ${$z->polar}[1] ]; |
579 | $z->{p_dirty} = 0; |
580 | $z->{c_dirty} = 1; |
581 | return $rho; |
582 | } else { |
583 | return ${$z->polar}[0]; |
584 | } |
585 | } |
586 | |
587 | sub _theta { |
588 | my $theta = $_[0]; |
589 | |
590 | if ($$theta > pi()) { $$theta -= pit2 } |
591 | elsif ($$theta <= -pi()) { $$theta += pit2 } |
66730be0 |
592 | } |
593 | |
594 | # |
595 | # arg |
596 | # |
b42d0ec9 |
597 | # Compute or set complex's argument (theta). |
66730be0 |
598 | # |
599 | sub arg { |
b42d0ec9 |
600 | my ($z, $theta) = @_; |
601 | return $z unless ref $z; |
602 | if (defined $theta) { |
603 | _theta(\$theta); |
604 | $z->{'polar'} = [ ${$z->polar}[0], $theta ]; |
605 | $z->{p_dirty} = 0; |
606 | $z->{c_dirty} = 1; |
607 | } else { |
608 | $theta = ${$z->polar}[1]; |
609 | _theta(\$theta); |
610 | } |
611 | return $theta; |
66730be0 |
612 | } |
613 | |
614 | # |
615 | # (sqrt) |
616 | # |
0c721ce2 |
617 | # Compute sqrt(z). |
66730be0 |
618 | # |
b42d0ec9 |
619 | # It is quite tempting to use wantarray here so that in list context |
620 | # sqrt() would return the two solutions. This, however, would |
621 | # break things like |
622 | # |
623 | # print "sqrt(z) = ", sqrt($z), "\n"; |
624 | # |
625 | # The two values would be printed side by side without no intervening |
626 | # whitespace, quite confusing. |
627 | # Therefore if you want the two solutions use the root(). |
628 | # |
66730be0 |
629 | sub sqrt { |
630 | my ($z) = @_; |
b42d0ec9 |
631 | my ($re, $im) = ref $z ? @{$z->cartesian} : ($z, 0); |
1fa12f56 |
632 | return $re < 0 ? cplx(0, CORE::sqrt(-$re)) : CORE::sqrt($re) |
633 | if $im == 0; |
66730be0 |
634 | my ($r, $t) = @{$z->polar}; |
a8693bd3 |
635 | return (ref $z)->emake(CORE::sqrt($r), $t/2); |
66730be0 |
636 | } |
637 | |
638 | # |
639 | # cbrt |
640 | # |
0c721ce2 |
641 | # Compute cbrt(z) (cubic root). |
66730be0 |
642 | # |
b42d0ec9 |
643 | # Why are we not returning three values? The same answer as for sqrt(). |
644 | # |
66730be0 |
645 | sub cbrt { |
646 | my ($z) = @_; |
1fa12f56 |
647 | return $z < 0 ? |
648 | -CORE::exp(CORE::log(-$z)/3) : |
649 | ($z > 0 ? CORE::exp(CORE::log($z)/3): 0) |
fb73857a |
650 | unless ref $z; |
66730be0 |
651 | my ($r, $t) = @{$z->polar}; |
1fa12f56 |
652 | return 0 if $r == 0; |
a8693bd3 |
653 | return (ref $z)->emake(CORE::exp(CORE::log($r)/3), $t/3); |
66730be0 |
654 | } |
655 | |
656 | # |
0e505df1 |
657 | # _rootbad |
658 | # |
659 | # Die on bad root. |
660 | # |
661 | sub _rootbad { |
1fa12f56 |
662 | my $mess = "Root $_[0] illegal, root rank must be positive integer.\n"; |
0e505df1 |
663 | |
664 | my @up = caller(1); |
fb73857a |
665 | |
0e505df1 |
666 | $mess .= "Died at $up[1] line $up[2].\n"; |
667 | |
668 | die $mess; |
669 | } |
670 | |
671 | # |
66730be0 |
672 | # root |
673 | # |
674 | # Computes all nth root for z, returning an array whose size is n. |
675 | # `n' must be a positive integer. |
676 | # |
677 | # The roots are given by (for k = 0..n-1): |
678 | # |
679 | # z^(1/n) = r^(1/n) (cos ((t+2 k pi)/n) + i sin ((t+2 k pi)/n)) |
680 | # |
681 | sub root { |
682 | my ($z, $n) = @_; |
0e505df1 |
683 | _rootbad($n) if ($n < 1 or int($n) != $n); |
1fa12f56 |
684 | my ($r, $t) = ref $z ? |
685 | @{$z->polar} : (CORE::abs($z), $z >= 0 ? 0 : pi); |
66730be0 |
686 | my @root; |
687 | my $k; |
fb73857a |
688 | my $theta_inc = pit2 / $n; |
66730be0 |
689 | my $rho = $r ** (1/$n); |
690 | my $theta; |
d09ae4e6 |
691 | my $cartesian = ref $z && $z->{c_dirty} == 0; |
66730be0 |
692 | for ($k = 0, $theta = $t / $n; $k < $n; $k++, $theta += $theta_inc) { |
d09ae4e6 |
693 | my $w = cplxe($rho, $theta); |
694 | # Yes, $cartesian is loop invariant. |
695 | push @root, $cartesian ? cplx(@{$w->cartesian}) : $w; |
a0d0e21e |
696 | } |
66730be0 |
697 | return @root; |
a0d0e21e |
698 | } |
699 | |
66730be0 |
700 | # |
701 | # Re |
702 | # |
b42d0ec9 |
703 | # Return or set Re(z). |
66730be0 |
704 | # |
a0d0e21e |
705 | sub Re { |
b42d0ec9 |
706 | my ($z, $Re) = @_; |
66730be0 |
707 | return $z unless ref $z; |
b42d0ec9 |
708 | if (defined $Re) { |
709 | $z->{'cartesian'} = [ $Re, ${$z->cartesian}[1] ]; |
710 | $z->{c_dirty} = 0; |
711 | $z->{p_dirty} = 1; |
712 | } else { |
713 | return ${$z->cartesian}[0]; |
714 | } |
a0d0e21e |
715 | } |
716 | |
66730be0 |
717 | # |
718 | # Im |
719 | # |
b42d0ec9 |
720 | # Return or set Im(z). |
66730be0 |
721 | # |
a0d0e21e |
722 | sub Im { |
b42d0ec9 |
723 | my ($z, $Im) = @_; |
178326e7 |
724 | return 0 unless ref $z; |
b42d0ec9 |
725 | if (defined $Im) { |
726 | $z->{'cartesian'} = [ ${$z->cartesian}[0], $Im ]; |
727 | $z->{c_dirty} = 0; |
728 | $z->{p_dirty} = 1; |
729 | } else { |
730 | return ${$z->cartesian}[1]; |
731 | } |
732 | } |
733 | |
734 | # |
735 | # rho |
736 | # |
737 | # Return or set rho(w). |
738 | # |
739 | sub rho { |
740 | Math::Complex::abs(@_); |
741 | } |
742 | |
743 | # |
744 | # theta |
745 | # |
746 | # Return or set theta(w). |
747 | # |
748 | sub theta { |
749 | Math::Complex::arg(@_); |
a0d0e21e |
750 | } |
751 | |
66730be0 |
752 | # |
753 | # (exp) |
754 | # |
755 | # Computes exp(z). |
756 | # |
757 | sub exp { |
758 | my ($z) = @_; |
759 | my ($x, $y) = @{$z->cartesian}; |
a8693bd3 |
760 | return (ref $z)->emake(CORE::exp($x), $y); |
66730be0 |
761 | } |
762 | |
763 | # |
8c03c583 |
764 | # _logofzero |
765 | # |
fb73857a |
766 | # Die on logarithm of zero. |
8c03c583 |
767 | # |
768 | sub _logofzero { |
769 | my $mess = "$_[0]: Logarithm of zero.\n"; |
770 | |
771 | if (defined $_[1]) { |
772 | $mess .= "(Because in the definition of $_[0], the argument "; |
773 | $mess .= "$_[1] " unless ($_[1] eq '0'); |
774 | $mess .= "is 0)\n"; |
775 | } |
776 | |
777 | my @up = caller(1); |
fb73857a |
778 | |
8c03c583 |
779 | $mess .= "Died at $up[1] line $up[2].\n"; |
780 | |
781 | die $mess; |
782 | } |
783 | |
784 | # |
66730be0 |
785 | # (log) |
786 | # |
787 | # Compute log(z). |
788 | # |
789 | sub log { |
790 | my ($z) = @_; |
fb73857a |
791 | unless (ref $z) { |
792 | _logofzero("log") if $z == 0; |
a8693bd3 |
793 | return $z > 0 ? CORE::log($z) : cplx(CORE::log(-$z), pi); |
fb73857a |
794 | } |
5cd24f17 |
795 | my ($r, $t) = @{$z->polar}; |
fb73857a |
796 | _logofzero("log") if $r == 0; |
797 | if ($t > pi()) { $t -= pit2 } |
798 | elsif ($t <= -pi()) { $t += pit2 } |
a8693bd3 |
799 | return (ref $z)->make(CORE::log($r), $t); |
66730be0 |
800 | } |
801 | |
802 | # |
0c721ce2 |
803 | # ln |
804 | # |
805 | # Alias for log(). |
806 | # |
807 | sub ln { Math::Complex::log(@_) } |
808 | |
809 | # |
66730be0 |
810 | # log10 |
811 | # |
812 | # Compute log10(z). |
813 | # |
5cd24f17 |
814 | |
66730be0 |
815 | sub log10 { |
fb73857a |
816 | return Math::Complex::log($_[0]) * uplog10; |
66730be0 |
817 | } |
818 | |
819 | # |
820 | # logn |
821 | # |
822 | # Compute logn(z,n) = log(z) / log(n) |
823 | # |
824 | sub logn { |
825 | my ($z, $n) = @_; |
0c721ce2 |
826 | $z = cplx($z, 0) unless ref $z; |
9fbe1b12 |
827 | my $logn = $LOGN{$n}; |
828 | $logn = $LOGN{$n} = CORE::log($n) unless defined $logn; # Cache log(n) |
1fa12f56 |
829 | return &log($z) / $logn; |
66730be0 |
830 | } |
831 | |
832 | # |
833 | # (cos) |
834 | # |
835 | # Compute cos(z) = (exp(iz) + exp(-iz))/2. |
836 | # |
837 | sub cos { |
838 | my ($z) = @_; |
1fa12f56 |
839 | return CORE::cos($z) unless ref $z; |
66730be0 |
840 | my ($x, $y) = @{$z->cartesian}; |
a8693bd3 |
841 | my $ey = CORE::exp($y); |
1fa12f56 |
842 | my $sx = CORE::sin($x); |
843 | my $cx = CORE::cos($x); |
844 | my $ey_1 = $ey ? 1 / $ey : $Inf; |
845 | return (ref $z)->make($cx * ($ey + $ey_1)/2, |
846 | $sx * ($ey_1 - $ey)/2); |
66730be0 |
847 | } |
848 | |
849 | # |
850 | # (sin) |
851 | # |
852 | # Compute sin(z) = (exp(iz) - exp(-iz))/2. |
853 | # |
854 | sub sin { |
855 | my ($z) = @_; |
1fa12f56 |
856 | return CORE::sin($z) unless ref $z; |
66730be0 |
857 | my ($x, $y) = @{$z->cartesian}; |
a8693bd3 |
858 | my $ey = CORE::exp($y); |
1fa12f56 |
859 | my $sx = CORE::sin($x); |
860 | my $cx = CORE::cos($x); |
861 | my $ey_1 = $ey ? 1 / $ey : $Inf; |
862 | return (ref $z)->make($sx * ($ey + $ey_1)/2, |
863 | $cx * ($ey - $ey_1)/2); |
66730be0 |
864 | } |
865 | |
866 | # |
867 | # tan |
868 | # |
869 | # Compute tan(z) = sin(z) / cos(z). |
870 | # |
871 | sub tan { |
872 | my ($z) = @_; |
1fa12f56 |
873 | my $cz = &cos($z); |
874 | _divbyzero "tan($z)", "cos($z)" if $cz == 0; |
875 | return &sin($z) / $cz; |
66730be0 |
876 | } |
877 | |
878 | # |
0c721ce2 |
879 | # sec |
880 | # |
881 | # Computes the secant sec(z) = 1 / cos(z). |
882 | # |
883 | sub sec { |
884 | my ($z) = @_; |
1fa12f56 |
885 | my $cz = &cos($z); |
0e505df1 |
886 | _divbyzero "sec($z)", "cos($z)" if ($cz == 0); |
0c721ce2 |
887 | return 1 / $cz; |
888 | } |
889 | |
890 | # |
891 | # csc |
892 | # |
893 | # Computes the cosecant csc(z) = 1 / sin(z). |
894 | # |
895 | sub csc { |
896 | my ($z) = @_; |
1fa12f56 |
897 | my $sz = &sin($z); |
0e505df1 |
898 | _divbyzero "csc($z)", "sin($z)" if ($sz == 0); |
0c721ce2 |
899 | return 1 / $sz; |
900 | } |
901 | |
66730be0 |
902 | # |
0c721ce2 |
903 | # cosec |
66730be0 |
904 | # |
0c721ce2 |
905 | # Alias for csc(). |
906 | # |
907 | sub cosec { Math::Complex::csc(@_) } |
908 | |
909 | # |
910 | # cot |
911 | # |
fb73857a |
912 | # Computes cot(z) = cos(z) / sin(z). |
0c721ce2 |
913 | # |
914 | sub cot { |
66730be0 |
915 | my ($z) = @_; |
1fa12f56 |
916 | my $sz = &sin($z); |
0e505df1 |
917 | _divbyzero "cot($z)", "sin($z)" if ($sz == 0); |
1fa12f56 |
918 | return &cos($z) / $sz; |
66730be0 |
919 | } |
920 | |
921 | # |
0c721ce2 |
922 | # cotan |
923 | # |
924 | # Alias for cot(). |
925 | # |
926 | sub cotan { Math::Complex::cot(@_) } |
927 | |
928 | # |
66730be0 |
929 | # acos |
930 | # |
931 | # Computes the arc cosine acos(z) = -i log(z + sqrt(z*z-1)). |
932 | # |
933 | sub acos { |
fb73857a |
934 | my $z = $_[0]; |
1fa12f56 |
935 | return CORE::atan2(CORE::sqrt(1-$z*$z), $z) |
936 | if (! ref $z) && CORE::abs($z) <= 1; |
40b904b7 |
937 | $z = cplx($z, 0) unless ref $z; |
938 | my ($x, $y) = @{$z->cartesian}; |
1fa12f56 |
939 | return 0 if $x == 1 && $y == 0; |
a8693bd3 |
940 | my $t1 = CORE::sqrt(($x+1)*($x+1) + $y*$y); |
941 | my $t2 = CORE::sqrt(($x-1)*($x-1) + $y*$y); |
fb73857a |
942 | my $alpha = ($t1 + $t2)/2; |
943 | my $beta = ($t1 - $t2)/2; |
944 | $alpha = 1 if $alpha < 1; |
945 | if ($beta > 1) { $beta = 1 } |
946 | elsif ($beta < -1) { $beta = -1 } |
a8693bd3 |
947 | my $u = CORE::atan2(CORE::sqrt(1-$beta*$beta), $beta); |
948 | my $v = CORE::log($alpha + CORE::sqrt($alpha*$alpha-1)); |
fb73857a |
949 | $v = -$v if $y > 0 || ($y == 0 && $x < -1); |
40b904b7 |
950 | return (ref $z)->make($u, $v); |
66730be0 |
951 | } |
952 | |
953 | # |
954 | # asin |
955 | # |
956 | # Computes the arc sine asin(z) = -i log(iz + sqrt(1-z*z)). |
957 | # |
958 | sub asin { |
fb73857a |
959 | my $z = $_[0]; |
1fa12f56 |
960 | return CORE::atan2($z, CORE::sqrt(1-$z*$z)) |
961 | if (! ref $z) && CORE::abs($z) <= 1; |
40b904b7 |
962 | $z = cplx($z, 0) unless ref $z; |
963 | my ($x, $y) = @{$z->cartesian}; |
1fa12f56 |
964 | return 0 if $x == 0 && $y == 0; |
a8693bd3 |
965 | my $t1 = CORE::sqrt(($x+1)*($x+1) + $y*$y); |
966 | my $t2 = CORE::sqrt(($x-1)*($x-1) + $y*$y); |
fb73857a |
967 | my $alpha = ($t1 + $t2)/2; |
968 | my $beta = ($t1 - $t2)/2; |
969 | $alpha = 1 if $alpha < 1; |
970 | if ($beta > 1) { $beta = 1 } |
971 | elsif ($beta < -1) { $beta = -1 } |
a8693bd3 |
972 | my $u = CORE::atan2($beta, CORE::sqrt(1-$beta*$beta)); |
973 | my $v = -CORE::log($alpha + CORE::sqrt($alpha*$alpha-1)); |
fb73857a |
974 | $v = -$v if $y > 0 || ($y == 0 && $x < -1); |
40b904b7 |
975 | return (ref $z)->make($u, $v); |
66730be0 |
976 | } |
977 | |
978 | # |
979 | # atan |
980 | # |
0c721ce2 |
981 | # Computes the arc tangent atan(z) = i/2 log((i+z) / (i-z)). |
66730be0 |
982 | # |
983 | sub atan { |
984 | my ($z) = @_; |
a8693bd3 |
985 | return CORE::atan2($z, 1) unless ref $z; |
1fa12f56 |
986 | my ($x, $y) = ref $z ? @{$z->cartesian} : ($z, 0); |
987 | return 0 if $x == 0 && $y == 0; |
8c03c583 |
988 | _divbyzero "atan(i)" if ( $z == i); |
1fa12f56 |
989 | _logofzero "atan(-i)" if (-$z == i); # -i is a bad file test... |
990 | my $log = &log((i + $z) / (i - $z)); |
991 | return ip2 * $log; |
a0d0e21e |
992 | } |
993 | |
66730be0 |
994 | # |
0c721ce2 |
995 | # asec |
996 | # |
997 | # Computes the arc secant asec(z) = acos(1 / z). |
998 | # |
999 | sub asec { |
1000 | my ($z) = @_; |
0e505df1 |
1001 | _divbyzero "asec($z)", $z if ($z == 0); |
fb73857a |
1002 | return acos(1 / $z); |
0c721ce2 |
1003 | } |
1004 | |
1005 | # |
5cd24f17 |
1006 | # acsc |
0c721ce2 |
1007 | # |
8c03c583 |
1008 | # Computes the arc cosecant acsc(z) = asin(1 / z). |
0c721ce2 |
1009 | # |
5cd24f17 |
1010 | sub acsc { |
0c721ce2 |
1011 | my ($z) = @_; |
0e505df1 |
1012 | _divbyzero "acsc($z)", $z if ($z == 0); |
fb73857a |
1013 | return asin(1 / $z); |
0c721ce2 |
1014 | } |
1015 | |
1016 | # |
5cd24f17 |
1017 | # acosec |
66730be0 |
1018 | # |
5cd24f17 |
1019 | # Alias for acsc(). |
0c721ce2 |
1020 | # |
5cd24f17 |
1021 | sub acosec { Math::Complex::acsc(@_) } |
0c721ce2 |
1022 | |
66730be0 |
1023 | # |
0c721ce2 |
1024 | # acot |
1025 | # |
8c03c583 |
1026 | # Computes the arc cotangent acot(z) = atan(1 / z) |
0c721ce2 |
1027 | # |
1028 | sub acot { |
66730be0 |
1029 | my ($z) = @_; |
1fa12f56 |
1030 | _divbyzero "acot(0)" if $z == 0; |
1031 | return ($z >= 0) ? CORE::atan2(1, $z) : CORE::atan2(-1, -$z) |
1032 | unless ref $z; |
1033 | _divbyzero "acot(i)" if ($z - i == 0); |
1034 | _logofzero "acot(-i)" if ($z + i == 0); |
8c03c583 |
1035 | return atan(1 / $z); |
66730be0 |
1036 | } |
1037 | |
1038 | # |
0c721ce2 |
1039 | # acotan |
1040 | # |
1041 | # Alias for acot(). |
1042 | # |
1043 | sub acotan { Math::Complex::acot(@_) } |
1044 | |
1045 | # |
66730be0 |
1046 | # cosh |
1047 | # |
1048 | # Computes the hyperbolic cosine cosh(z) = (exp(z) + exp(-z))/2. |
1049 | # |
1050 | sub cosh { |
1051 | my ($z) = @_; |
fb73857a |
1052 | my $ex; |
0e505df1 |
1053 | unless (ref $z) { |
a8693bd3 |
1054 | $ex = CORE::exp($z); |
1fa12f56 |
1055 | return $ex ? ($ex + 1/$ex)/2 : $Inf; |
0e505df1 |
1056 | } |
1057 | my ($x, $y) = @{$z->cartesian}; |
a8693bd3 |
1058 | $ex = CORE::exp($x); |
1fa12f56 |
1059 | my $ex_1 = $ex ? 1 / $ex : $Inf; |
a8693bd3 |
1060 | return (ref $z)->make(CORE::cos($y) * ($ex + $ex_1)/2, |
1061 | CORE::sin($y) * ($ex - $ex_1)/2); |
66730be0 |
1062 | } |
1063 | |
1064 | # |
1065 | # sinh |
1066 | # |
1067 | # Computes the hyperbolic sine sinh(z) = (exp(z) - exp(-z))/2. |
1068 | # |
1069 | sub sinh { |
1070 | my ($z) = @_; |
fb73857a |
1071 | my $ex; |
0e505df1 |
1072 | unless (ref $z) { |
1fa12f56 |
1073 | return 0 if $z == 0; |
a8693bd3 |
1074 | $ex = CORE::exp($z); |
1fa12f56 |
1075 | return $ex ? ($ex - 1/$ex)/2 : "-$Inf"; |
0e505df1 |
1076 | } |
1077 | my ($x, $y) = @{$z->cartesian}; |
1fa12f56 |
1078 | my $cy = CORE::cos($y); |
1079 | my $sy = CORE::sin($y); |
a8693bd3 |
1080 | $ex = CORE::exp($x); |
1fa12f56 |
1081 | my $ex_1 = $ex ? 1 / $ex : $Inf; |
5240e574 |
1082 | return (ref $z)->make(CORE::cos($y) * ($ex - $ex_1)/2, |
1083 | CORE::sin($y) * ($ex + $ex_1)/2); |
66730be0 |
1084 | } |
1085 | |
1086 | # |
1087 | # tanh |
1088 | # |
1089 | # Computes the hyperbolic tangent tanh(z) = sinh(z) / cosh(z). |
1090 | # |
1091 | sub tanh { |
1092 | my ($z) = @_; |
0c721ce2 |
1093 | my $cz = cosh($z); |
0e505df1 |
1094 | _divbyzero "tanh($z)", "cosh($z)" if ($cz == 0); |
0c721ce2 |
1095 | return sinh($z) / $cz; |
66730be0 |
1096 | } |
1097 | |
1098 | # |
0c721ce2 |
1099 | # sech |
1100 | # |
1101 | # Computes the hyperbolic secant sech(z) = 1 / cosh(z). |
1102 | # |
1103 | sub sech { |
1104 | my ($z) = @_; |
1105 | my $cz = cosh($z); |
0e505df1 |
1106 | _divbyzero "sech($z)", "cosh($z)" if ($cz == 0); |
0c721ce2 |
1107 | return 1 / $cz; |
1108 | } |
1109 | |
1110 | # |
1111 | # csch |
1112 | # |
1113 | # Computes the hyperbolic cosecant csch(z) = 1 / sinh(z). |
66730be0 |
1114 | # |
0c721ce2 |
1115 | sub csch { |
1116 | my ($z) = @_; |
1117 | my $sz = sinh($z); |
0e505df1 |
1118 | _divbyzero "csch($z)", "sinh($z)" if ($sz == 0); |
0c721ce2 |
1119 | return 1 / $sz; |
1120 | } |
1121 | |
1122 | # |
1123 | # cosech |
1124 | # |
1125 | # Alias for csch(). |
1126 | # |
1127 | sub cosech { Math::Complex::csch(@_) } |
1128 | |
66730be0 |
1129 | # |
0c721ce2 |
1130 | # coth |
1131 | # |
1132 | # Computes the hyperbolic cotangent coth(z) = cosh(z) / sinh(z). |
1133 | # |
1134 | sub coth { |
66730be0 |
1135 | my ($z) = @_; |
0c721ce2 |
1136 | my $sz = sinh($z); |
1fa12f56 |
1137 | _divbyzero "coth($z)", "sinh($z)" if $sz == 0; |
0c721ce2 |
1138 | return cosh($z) / $sz; |
66730be0 |
1139 | } |
1140 | |
1141 | # |
0c721ce2 |
1142 | # cotanh |
1143 | # |
1144 | # Alias for coth(). |
1145 | # |
1146 | sub cotanh { Math::Complex::coth(@_) } |
1147 | |
1148 | # |
66730be0 |
1149 | # acosh |
1150 | # |
fb73857a |
1151 | # Computes the arc hyperbolic cosine acosh(z) = log(z + sqrt(z*z-1)). |
66730be0 |
1152 | # |
1153 | sub acosh { |
1154 | my ($z) = @_; |
fb73857a |
1155 | unless (ref $z) { |
fb73857a |
1156 | $z = cplx($z, 0); |
1157 | } |
8c03c583 |
1158 | my ($re, $im) = @{$z->cartesian}; |
fb73857a |
1159 | if ($im == 0) { |
1fa12f56 |
1160 | return CORE::log($re + CORE::sqrt($re*$re - 1)) |
1161 | if $re >= 1; |
1162 | return cplx(0, CORE::atan2(CORE::sqrt(1 - $re*$re), $re)) |
1163 | if CORE::abs($re) < 1; |
fb73857a |
1164 | } |
9bc5fa8d |
1165 | my $t = &sqrt($z * $z - 1) + $z; |
40b904b7 |
1166 | # Try Taylor if looking bad (this usually means that |
1167 | # $z was large negative, therefore the sqrt is really |
1168 | # close to abs(z), summing that with z...) |
9bc5fa8d |
1169 | $t = 1/(2 * $z) - 1/(8 * $z**3) + 1/(16 * $z**5) - 5/(128 * $z**7) |
1170 | if $t == 0; |
1171 | my $u = &log($t); |
40b904b7 |
1172 | $u->Im(-$u->Im) if $re < 0 && $im == 0; |
9bc5fa8d |
1173 | return $re < 0 ? -$u : $u; |
66730be0 |
1174 | } |
1175 | |
1176 | # |
1177 | # asinh |
1178 | # |
1fa12f56 |
1179 | # Computes the arc hyperbolic sine asinh(z) = log(z + sqrt(z*z+1)) |
66730be0 |
1180 | # |
1181 | sub asinh { |
1182 | my ($z) = @_; |
1fa12f56 |
1183 | unless (ref $z) { |
1184 | my $t = $z + CORE::sqrt($z*$z + 1); |
1185 | return CORE::log($t) if $t; |
1186 | } |
9bc5fa8d |
1187 | my $t = &sqrt($z * $z + 1) + $z; |
40b904b7 |
1188 | # Try Taylor if looking bad (this usually means that |
1189 | # $z was large negative, therefore the sqrt is really |
1190 | # close to abs(z), summing that with z...) |
9bc5fa8d |
1191 | $t = 1/(2 * $z) - 1/(8 * $z**3) + 1/(16 * $z**5) - 5/(128 * $z**7) |
1192 | if $t == 0; |
1fa12f56 |
1193 | return &log($t); |
66730be0 |
1194 | } |
1195 | |
1196 | # |
1197 | # atanh |
1198 | # |
1199 | # Computes the arc hyperbolic tangent atanh(z) = 1/2 log((1+z) / (1-z)). |
1200 | # |
1201 | sub atanh { |
1202 | my ($z) = @_; |
fb73857a |
1203 | unless (ref $z) { |
a8693bd3 |
1204 | return CORE::log((1 + $z)/(1 - $z))/2 if CORE::abs($z) < 1; |
fb73857a |
1205 | $z = cplx($z, 0); |
1206 | } |
1fa12f56 |
1207 | _divbyzero 'atanh(1)', "1 - $z" if (1 - $z == 0); |
1208 | _logofzero 'atanh(-1)' if (1 + $z == 0); |
1209 | return 0.5 * &log((1 + $z) / (1 - $z)); |
66730be0 |
1210 | } |
1211 | |
1212 | # |
0c721ce2 |
1213 | # asech |
1214 | # |
1215 | # Computes the hyperbolic arc secant asech(z) = acosh(1 / z). |
1216 | # |
1217 | sub asech { |
1218 | my ($z) = @_; |
1fa12f56 |
1219 | _divbyzero 'asech(0)', "$z" if ($z == 0); |
0c721ce2 |
1220 | return acosh(1 / $z); |
1221 | } |
1222 | |
1223 | # |
1224 | # acsch |
66730be0 |
1225 | # |
0c721ce2 |
1226 | # Computes the hyperbolic arc cosecant acsch(z) = asinh(1 / z). |
66730be0 |
1227 | # |
0c721ce2 |
1228 | sub acsch { |
66730be0 |
1229 | my ($z) = @_; |
0e505df1 |
1230 | _divbyzero 'acsch(0)', $z if ($z == 0); |
0c721ce2 |
1231 | return asinh(1 / $z); |
1232 | } |
1233 | |
1234 | # |
1235 | # acosech |
1236 | # |
1237 | # Alias for acosh(). |
1238 | # |
1239 | sub acosech { Math::Complex::acsch(@_) } |
1240 | |
1241 | # |
1242 | # acoth |
1243 | # |
1244 | # Computes the arc hyperbolic cotangent acoth(z) = 1/2 log((1+z) / (z-1)). |
1245 | # |
1246 | sub acoth { |
1247 | my ($z) = @_; |
1fa12f56 |
1248 | _divbyzero 'acoth(0)' if ($z == 0); |
fb73857a |
1249 | unless (ref $z) { |
a8693bd3 |
1250 | return CORE::log(($z + 1)/($z - 1))/2 if CORE::abs($z) > 1; |
fb73857a |
1251 | $z = cplx($z, 0); |
1252 | } |
1fa12f56 |
1253 | _divbyzero 'acoth(1)', "$z - 1" if ($z - 1 == 0); |
1254 | _logofzero 'acoth(-1)', "1 + $z" if (1 + $z == 0); |
1255 | return &log((1 + $z) / ($z - 1)) / 2; |
66730be0 |
1256 | } |
1257 | |
1258 | # |
0c721ce2 |
1259 | # acotanh |
1260 | # |
1261 | # Alias for acot(). |
1262 | # |
1263 | sub acotanh { Math::Complex::acoth(@_) } |
1264 | |
1265 | # |
66730be0 |
1266 | # (atan2) |
1267 | # |
1268 | # Compute atan(z1/z2). |
1269 | # |
1270 | sub atan2 { |
1271 | my ($z1, $z2, $inverted) = @_; |
fb73857a |
1272 | my ($re1, $im1, $re2, $im2); |
1273 | if ($inverted) { |
1274 | ($re1, $im1) = ref $z2 ? @{$z2->cartesian} : ($z2, 0); |
1275 | ($re2, $im2) = @{$z1->cartesian}; |
66730be0 |
1276 | } else { |
fb73857a |
1277 | ($re1, $im1) = @{$z1->cartesian}; |
1278 | ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0); |
1279 | } |
1280 | if ($im2 == 0) { |
1fa12f56 |
1281 | return CORE::atan2($re1, $re2) if $im1 == 0; |
1282 | return ($im1<=>0) * pip2 if $re2 == 0; |
66730be0 |
1283 | } |
fb73857a |
1284 | my $w = atan($z1/$z2); |
1285 | my ($u, $v) = ref $w ? @{$w->cartesian} : ($w, 0); |
1286 | $u += pi if $re2 < 0; |
1287 | $u -= pit2 if $u > pi; |
1288 | return cplx($u, $v); |
66730be0 |
1289 | } |
1290 | |
1291 | # |
1292 | # display_format |
1293 | # ->display_format |
1294 | # |
16357284 |
1295 | # Set (get if no argument) the display format for all complex numbers that |
fb73857a |
1296 | # don't happen to have overridden it via ->display_format |
66730be0 |
1297 | # |
16357284 |
1298 | # When called as an object method, this actually sets the display format for |
66730be0 |
1299 | # the current object. |
1300 | # |
1301 | # Valid object formats are 'c' and 'p' for cartesian and polar. The first |
1302 | # letter is used actually, so the type can be fully spelled out for clarity. |
1303 | # |
1304 | sub display_format { |
16357284 |
1305 | my $self = shift; |
1306 | my %display_format = %DISPLAY_FORMAT; |
66730be0 |
1307 | |
16357284 |
1308 | if (ref $self) { # Called as an object method |
1309 | if (exists $self->{display_format}) { |
1310 | my %obj = %{$self->{display_format}}; |
1311 | @display_format{keys %obj} = values %obj; |
1312 | } |
476757f7 |
1313 | } |
1314 | if (@_ == 1) { |
1315 | $display_format{style} = shift; |
1316 | } else { |
1317 | my %new = @_; |
1318 | @display_format{keys %new} = values %new; |
66730be0 |
1319 | } |
1320 | |
476757f7 |
1321 | if (ref $self) { # Called as an object method |
16357284 |
1322 | $self->{display_format} = { %display_format }; |
1323 | return |
1324 | wantarray ? |
1325 | %{$self->{display_format}} : |
1326 | $self->{display_format}->{style}; |
66730be0 |
1327 | } |
1328 | |
476757f7 |
1329 | # Called as a class method |
16357284 |
1330 | %DISPLAY_FORMAT = %display_format; |
1331 | return |
1332 | wantarray ? |
1333 | %DISPLAY_FORMAT : |
1334 | $DISPLAY_FORMAT{style}; |
66730be0 |
1335 | } |
1336 | |
1337 | # |
1338 | # (stringify) |
1339 | # |
1340 | # Show nicely formatted complex number under its cartesian or polar form, |
1341 | # depending on the current display format: |
1342 | # |
1343 | # . If a specific display format has been recorded for this object, use it. |
1344 | # . Otherwise, use the generic current default for all complex numbers, |
1345 | # which is a package global variable. |
1346 | # |
a0d0e21e |
1347 | sub stringify { |
66730be0 |
1348 | my ($z) = shift; |
66730be0 |
1349 | |
16357284 |
1350 | my $style = $z->display_format; |
1351 | |
1352 | $style = $DISPLAY_FORMAT{style} unless defined $style; |
66730be0 |
1353 | |
16357284 |
1354 | return $z->stringify_polar if $style =~ /^p/i; |
66730be0 |
1355 | return $z->stringify_cartesian; |
1356 | } |
1357 | |
1358 | # |
1359 | # ->stringify_cartesian |
1360 | # |
1361 | # Stringify as a cartesian representation 'a+bi'. |
1362 | # |
1363 | sub stringify_cartesian { |
1364 | my $z = shift; |
1365 | my ($x, $y) = @{$z->cartesian}; |
1366 | my ($re, $im); |
1367 | |
16357284 |
1368 | my %format = $z->display_format; |
1369 | my $format = $format{format}; |
1370 | |
1fa12f56 |
1371 | if ($x) { |
1372 | if ($x =~ /^NaN[QS]?$/i) { |
1373 | $re = $x; |
1374 | } else { |
1375 | if ($x =~ /^-?$Inf$/oi) { |
1376 | $re = $x; |
1377 | } else { |
1378 | $re = defined $format ? sprintf($format, $x) : $x; |
1379 | } |
1380 | } |
1381 | } else { |
1382 | undef $re; |
1383 | } |
1384 | |
1385 | if ($y) { |
40b904b7 |
1386 | if ($y =~ /^(NaN[QS]?)$/i) { |
1fa12f56 |
1387 | $im = $y; |
1388 | } else { |
1389 | if ($y =~ /^-?$Inf$/oi) { |
1390 | $im = $y; |
1391 | } else { |
40b904b7 |
1392 | $im = |
1393 | defined $format ? |
1394 | sprintf($format, $y) : |
1395 | ($y == 1 ? "" : ($y == -1 ? "-" : $y)); |
1fa12f56 |
1396 | } |
1397 | } |
1398 | $im .= "i"; |
1399 | } else { |
1400 | undef $im; |
16357284 |
1401 | } |
66730be0 |
1402 | |
1fa12f56 |
1403 | my $str = $re; |
1404 | |
16357284 |
1405 | if (defined $im) { |
1406 | if ($y < 0) { |
1407 | $str .= $im; |
1fa12f56 |
1408 | } elsif ($y > 0 || $im =~ /^NaN[QS]?i$/i) { |
16357284 |
1409 | $str .= "+" if defined $re; |
1410 | $str .= $im; |
1411 | } |
1fa12f56 |
1412 | } elsif (!defined $re) { |
1413 | $str = "0"; |
16357284 |
1414 | } |
66730be0 |
1415 | |
1416 | return $str; |
1417 | } |
1418 | |
d09ae4e6 |
1419 | |
66730be0 |
1420 | # |
1421 | # ->stringify_polar |
1422 | # |
1423 | # Stringify as a polar representation '[r,t]'. |
1424 | # |
1425 | sub stringify_polar { |
1426 | my $z = shift; |
1427 | my ($r, $t) = @{$z->polar}; |
1428 | my $theta; |
1429 | |
16357284 |
1430 | my %format = $z->display_format; |
1fa12f56 |
1431 | my $format = $format{format}; |
16357284 |
1432 | |
1fa12f56 |
1433 | if ($t =~ /^NaN[QS]?$/i || $t =~ /^-?$Inf$/oi) { |
1434 | $theta = $t; |
1435 | } elsif ($t == pi) { |
1436 | $theta = "pi"; |
1437 | } elsif ($r == 0 || $t == 0) { |
1438 | $theta = defined $format ? sprintf($format, $t) : $t; |
55497cff |
1439 | } |
66730be0 |
1440 | |
1fa12f56 |
1441 | return "[$r,$theta]" if defined $theta; |
1442 | |
66730be0 |
1443 | # |
1fa12f56 |
1444 | # Try to identify pi/n and friends. |
66730be0 |
1445 | # |
1446 | |
1fa12f56 |
1447 | $t -= int(CORE::abs($t) / pit2) * pit2; |
1448 | |
e97e26fa |
1449 | if ($format{polar_pretty_print} && $t) { |
1fa12f56 |
1450 | my ($a, $b); |
9bc5fa8d |
1451 | for $a (2..9) { |
1fa12f56 |
1452 | $b = $t * $a / pi; |
e97e26fa |
1453 | if ($b =~ /^-?\d+$/) { |
1fa12f56 |
1454 | $b = $b < 0 ? "-" : "" if CORE::abs($b) == 1; |
1455 | $theta = "${b}pi/$a"; |
d09ae4e6 |
1456 | last; |
66730be0 |
1457 | } |
d09ae4e6 |
1458 | } |
66730be0 |
1459 | } |
1460 | |
16357284 |
1461 | if (defined $format) { |
1462 | $r = sprintf($format, $r); |
1fa12f56 |
1463 | $theta = sprintf($format, $theta) unless defined $theta; |
1464 | } else { |
1465 | $theta = $t unless defined $theta; |
16357284 |
1466 | } |
1467 | |
1fa12f56 |
1468 | return "[$r,$theta]"; |
a0d0e21e |
1469 | } |
a5f75d66 |
1470 | |
1471 | 1; |
1472 | __END__ |
1473 | |
1cf6bcb8 |
1474 | =pod |
1475 | |
a5f75d66 |
1476 | =head1 NAME |
1477 | |
66730be0 |
1478 | Math::Complex - complex numbers and associated mathematical functions |
a5f75d66 |
1479 | |
1480 | =head1 SYNOPSIS |
1481 | |
66730be0 |
1482 | use Math::Complex; |
fb73857a |
1483 | |
66730be0 |
1484 | $z = Math::Complex->make(5, 6); |
1485 | $t = 4 - 3*i + $z; |
1486 | $j = cplxe(1, 2*pi/3); |
a5f75d66 |
1487 | |
1488 | =head1 DESCRIPTION |
1489 | |
66730be0 |
1490 | This package lets you create and manipulate complex numbers. By default, |
1491 | I<Perl> limits itself to real numbers, but an extra C<use> statement brings |
1492 | full complex support, along with a full set of mathematical functions |
1493 | typically associated with and/or extended to complex numbers. |
1494 | |
1495 | If you wonder what complex numbers are, they were invented to be able to solve |
1496 | the following equation: |
1497 | |
1498 | x*x = -1 |
1499 | |
1500 | and by definition, the solution is noted I<i> (engineers use I<j> instead since |
1501 | I<i> usually denotes an intensity, but the name does not matter). The number |
1502 | I<i> is a pure I<imaginary> number. |
1503 | |
1504 | The arithmetics with pure imaginary numbers works just like you would expect |
1505 | it with real numbers... you just have to remember that |
1506 | |
1507 | i*i = -1 |
1508 | |
1509 | so you have: |
1510 | |
1511 | 5i + 7i = i * (5 + 7) = 12i |
1512 | 4i - 3i = i * (4 - 3) = i |
1513 | 4i * 2i = -8 |
1514 | 6i / 2i = 3 |
1515 | 1 / i = -i |
1516 | |
1517 | Complex numbers are numbers that have both a real part and an imaginary |
1518 | part, and are usually noted: |
1519 | |
1520 | a + bi |
1521 | |
1522 | where C<a> is the I<real> part and C<b> is the I<imaginary> part. The |
1523 | arithmetic with complex numbers is straightforward. You have to |
1524 | keep track of the real and the imaginary parts, but otherwise the |
1525 | rules used for real numbers just apply: |
1526 | |
1527 | (4 + 3i) + (5 - 2i) = (4 + 5) + i(3 - 2) = 9 + i |
1528 | (2 + i) * (4 - i) = 2*4 + 4i -2i -i*i = 8 + 2i + 1 = 9 + 2i |
1529 | |
1530 | A graphical representation of complex numbers is possible in a plane |
1531 | (also called the I<complex plane>, but it's really a 2D plane). |
1532 | The number |
1533 | |
1534 | z = a + bi |
1535 | |
1536 | is the point whose coordinates are (a, b). Actually, it would |
1537 | be the vector originating from (0, 0) to (a, b). It follows that the addition |
1538 | of two complex numbers is a vectorial addition. |
1539 | |
1540 | Since there is a bijection between a point in the 2D plane and a complex |
1541 | number (i.e. the mapping is unique and reciprocal), a complex number |
1542 | can also be uniquely identified with polar coordinates: |
1543 | |
1544 | [rho, theta] |
1545 | |
1546 | where C<rho> is the distance to the origin, and C<theta> the angle between |
1547 | the vector and the I<x> axis. There is a notation for this using the |
1548 | exponential form, which is: |
1549 | |
1550 | rho * exp(i * theta) |
1551 | |
1552 | where I<i> is the famous imaginary number introduced above. Conversion |
1553 | between this form and the cartesian form C<a + bi> is immediate: |
1554 | |
1555 | a = rho * cos(theta) |
1556 | b = rho * sin(theta) |
1557 | |
1558 | which is also expressed by this formula: |
1559 | |
fb73857a |
1560 | z = rho * exp(i * theta) = rho * (cos theta + i * sin theta) |
66730be0 |
1561 | |
1562 | In other words, it's the projection of the vector onto the I<x> and I<y> |
1563 | axes. Mathematicians call I<rho> the I<norm> or I<modulus> and I<theta> |
1564 | the I<argument> of the complex number. The I<norm> of C<z> will be |
1565 | noted C<abs(z)>. |
1566 | |
1567 | The polar notation (also known as the trigonometric |
1568 | representation) is much more handy for performing multiplications and |
1569 | divisions of complex numbers, whilst the cartesian notation is better |
fb73857a |
1570 | suited for additions and subtractions. Real numbers are on the I<x> |
1571 | axis, and therefore I<theta> is zero or I<pi>. |
66730be0 |
1572 | |
1573 | All the common operations that can be performed on a real number have |
1574 | been defined to work on complex numbers as well, and are merely |
1575 | I<extensions> of the operations defined on real numbers. This means |
1576 | they keep their natural meaning when there is no imaginary part, provided |
1577 | the number is within their definition set. |
1578 | |
1579 | For instance, the C<sqrt> routine which computes the square root of |
fb73857a |
1580 | its argument is only defined for non-negative real numbers and yields a |
1581 | non-negative real number (it is an application from B<R+> to B<R+>). |
66730be0 |
1582 | If we allow it to return a complex number, then it can be extended to |
1583 | negative real numbers to become an application from B<R> to B<C> (the |
1584 | set of complex numbers): |
1585 | |
1586 | sqrt(x) = x >= 0 ? sqrt(x) : sqrt(-x)*i |
1587 | |
1588 | It can also be extended to be an application from B<C> to B<C>, |
1589 | whilst its restriction to B<R> behaves as defined above by using |
1590 | the following definition: |
1591 | |
1592 | sqrt(z = [r,t]) = sqrt(r) * exp(i * t/2) |
1593 | |
fb73857a |
1594 | Indeed, a negative real number can be noted C<[x,pi]> (the modulus |
1595 | I<x> is always non-negative, so C<[x,pi]> is really C<-x>, a negative |
1596 | number) and the above definition states that |
66730be0 |
1597 | |
1598 | sqrt([x,pi]) = sqrt(x) * exp(i*pi/2) = [sqrt(x),pi/2] = sqrt(x)*i |
1599 | |
1600 | which is exactly what we had defined for negative real numbers above. |
b42d0ec9 |
1601 | The C<sqrt> returns only one of the solutions: if you want the both, |
1602 | use the C<root> function. |
a5f75d66 |
1603 | |
66730be0 |
1604 | All the common mathematical functions defined on real numbers that |
1605 | are extended to complex numbers share that same property of working |
1606 | I<as usual> when the imaginary part is zero (otherwise, it would not |
1607 | be called an extension, would it?). |
a5f75d66 |
1608 | |
66730be0 |
1609 | A I<new> operation possible on a complex number that is |
1610 | the identity for real numbers is called the I<conjugate>, and is noted |
d1be9408 |
1611 | with a horizontal bar above the number, or C<~z> here. |
a5f75d66 |
1612 | |
66730be0 |
1613 | z = a + bi |
1614 | ~z = a - bi |
a5f75d66 |
1615 | |
66730be0 |
1616 | Simple... Now look: |
a5f75d66 |
1617 | |
66730be0 |
1618 | z * ~z = (a + bi) * (a - bi) = a*a + b*b |
a5f75d66 |
1619 | |
66730be0 |
1620 | We saw that the norm of C<z> was noted C<abs(z)> and was defined as the |
1621 | distance to the origin, also known as: |
a5f75d66 |
1622 | |
66730be0 |
1623 | rho = abs(z) = sqrt(a*a + b*b) |
a5f75d66 |
1624 | |
66730be0 |
1625 | so |
1626 | |
1627 | z * ~z = abs(z) ** 2 |
1628 | |
1629 | If z is a pure real number (i.e. C<b == 0>), then the above yields: |
1630 | |
1631 | a * a = abs(a) ** 2 |
1632 | |
1633 | which is true (C<abs> has the regular meaning for real number, i.e. stands |
1634 | for the absolute value). This example explains why the norm of C<z> is |
1635 | noted C<abs(z)>: it extends the C<abs> function to complex numbers, yet |
1636 | is the regular C<abs> we know when the complex number actually has no |
1637 | imaginary part... This justifies I<a posteriori> our use of the C<abs> |
1638 | notation for the norm. |
1639 | |
1640 | =head1 OPERATIONS |
1641 | |
1642 | Given the following notations: |
1643 | |
1644 | z1 = a + bi = r1 * exp(i * t1) |
1645 | z2 = c + di = r2 * exp(i * t2) |
1646 | z = <any complex or real number> |
1647 | |
1648 | the following (overloaded) operations are supported on complex numbers: |
1649 | |
1650 | z1 + z2 = (a + c) + i(b + d) |
1651 | z1 - z2 = (a - c) + i(b - d) |
1652 | z1 * z2 = (r1 * r2) * exp(i * (t1 + t2)) |
1653 | z1 / z2 = (r1 / r2) * exp(i * (t1 - t2)) |
1654 | z1 ** z2 = exp(z2 * log z1) |
b42d0ec9 |
1655 | ~z = a - bi |
1656 | abs(z) = r1 = sqrt(a*a + b*b) |
1657 | sqrt(z) = sqrt(r1) * exp(i * t/2) |
1658 | exp(z) = exp(a) * exp(i * b) |
1659 | log(z) = log(r1) + i*t |
1660 | sin(z) = 1/2i (exp(i * z1) - exp(-i * z)) |
1661 | cos(z) = 1/2 (exp(i * z1) + exp(-i * z)) |
66730be0 |
1662 | atan2(z1, z2) = atan(z1/z2) |
1663 | |
1664 | The following extra operations are supported on both real and complex |
1665 | numbers: |
1666 | |
1667 | Re(z) = a |
1668 | Im(z) = b |
1669 | arg(z) = t |
b42d0ec9 |
1670 | abs(z) = r |
66730be0 |
1671 | |
1672 | cbrt(z) = z ** (1/3) |
1673 | log10(z) = log(z) / log(10) |
1674 | logn(z, n) = log(z) / log(n) |
1675 | |
1676 | tan(z) = sin(z) / cos(z) |
0c721ce2 |
1677 | |
5aabfad6 |
1678 | csc(z) = 1 / sin(z) |
1679 | sec(z) = 1 / cos(z) |
0c721ce2 |
1680 | cot(z) = 1 / tan(z) |
66730be0 |
1681 | |
1682 | asin(z) = -i * log(i*z + sqrt(1-z*z)) |
fb73857a |
1683 | acos(z) = -i * log(z + i*sqrt(1-z*z)) |
66730be0 |
1684 | atan(z) = i/2 * log((i+z) / (i-z)) |
0c721ce2 |
1685 | |
5aabfad6 |
1686 | acsc(z) = asin(1 / z) |
1687 | asec(z) = acos(1 / z) |
8c03c583 |
1688 | acot(z) = atan(1 / z) = -i/2 * log((i+z) / (z-i)) |
66730be0 |
1689 | |
1690 | sinh(z) = 1/2 (exp(z) - exp(-z)) |
1691 | cosh(z) = 1/2 (exp(z) + exp(-z)) |
0c721ce2 |
1692 | tanh(z) = sinh(z) / cosh(z) = (exp(z) - exp(-z)) / (exp(z) + exp(-z)) |
1693 | |
5aabfad6 |
1694 | csch(z) = 1 / sinh(z) |
1695 | sech(z) = 1 / cosh(z) |
0c721ce2 |
1696 | coth(z) = 1 / tanh(z) |
fb73857a |
1697 | |
66730be0 |
1698 | asinh(z) = log(z + sqrt(z*z+1)) |
1699 | acosh(z) = log(z + sqrt(z*z-1)) |
1700 | atanh(z) = 1/2 * log((1+z) / (1-z)) |
66730be0 |
1701 | |
5aabfad6 |
1702 | acsch(z) = asinh(1 / z) |
1703 | asech(z) = acosh(1 / z) |
0c721ce2 |
1704 | acoth(z) = atanh(1 / z) = 1/2 * log((1+z) / (z-1)) |
1705 | |
b42d0ec9 |
1706 | I<arg>, I<abs>, I<log>, I<csc>, I<cot>, I<acsc>, I<acot>, I<csch>, |
1707 | I<coth>, I<acosech>, I<acotanh>, have aliases I<rho>, I<theta>, I<ln>, |
1708 | I<cosec>, I<cotan>, I<acosec>, I<acotan>, I<cosech>, I<cotanh>, |
1709 | I<acosech>, I<acotanh>, respectively. C<Re>, C<Im>, C<arg>, C<abs>, |
d1be9408 |
1710 | C<rho>, and C<theta> can be used also as mutators. The C<cbrt> |
b42d0ec9 |
1711 | returns only one of the solutions: if you want all three, use the |
1712 | C<root> function. |
0c721ce2 |
1713 | |
1714 | The I<root> function is available to compute all the I<n> |
66730be0 |
1715 | roots of some complex, where I<n> is a strictly positive integer. |
1716 | There are exactly I<n> such roots, returned as a list. Getting the |
1717 | number mathematicians call C<j> such that: |
1718 | |
1719 | 1 + j + j*j = 0; |
1720 | |
1721 | is a simple matter of writing: |
1722 | |
1723 | $j = ((root(1, 3))[1]; |
1724 | |
1725 | The I<k>th root for C<z = [r,t]> is given by: |
1726 | |
1727 | (root(z, n))[k] = r**(1/n) * exp(i * (t + 2*k*pi)/n) |
1728 | |
f4837644 |
1729 | The I<spaceship> comparison operator, E<lt>=E<gt>, is also defined. In |
1730 | order to ensure its restriction to real numbers is conform to what you |
1731 | would expect, the comparison is run on the real part of the complex |
1732 | number first, and imaginary parts are compared only when the real |
1733 | parts match. |
66730be0 |
1734 | |
1735 | =head1 CREATION |
1736 | |
1737 | To create a complex number, use either: |
1738 | |
1739 | $z = Math::Complex->make(3, 4); |
1740 | $z = cplx(3, 4); |
1741 | |
1742 | if you know the cartesian form of the number, or |
1743 | |
1744 | $z = 3 + 4*i; |
1745 | |
fb73857a |
1746 | if you like. To create a number using the polar form, use either: |
66730be0 |
1747 | |
1748 | $z = Math::Complex->emake(5, pi/3); |
1749 | $x = cplxe(5, pi/3); |
1750 | |
0c721ce2 |
1751 | instead. The first argument is the modulus, the second is the angle |
fb73857a |
1752 | (in radians, the full circle is 2*pi). (Mnemonic: C<e> is used as a |
1753 | notation for complex numbers in the polar form). |
66730be0 |
1754 | |
1755 | It is possible to write: |
1756 | |
1757 | $x = cplxe(-3, pi/4); |
1758 | |
16357284 |
1759 | but that will be silently converted into C<[3,-3pi/4]>, since the |
1760 | modulus must be non-negative (it represents the distance to the origin |
1761 | in the complex plane). |
66730be0 |
1762 | |
91cb744f |
1763 | It is also possible to have a complex number as either argument of the |
1764 | C<make>, C<emake>, C<cplx>, and C<cplxe>: the appropriate component of |
b42d0ec9 |
1765 | the argument will be used. |
1766 | |
1767 | $z1 = cplx(-2, 1); |
1768 | $z2 = cplx($z1, 4); |
1769 | |
91cb744f |
1770 | The C<new>, C<make>, C<emake>, C<cplx>, and C<cplxe> will also |
1771 | understand a single (string) argument of the forms |
1772 | |
1773 | 2-3i |
1774 | -3i |
1775 | [2,3] |
1776 | [2] |
1777 | |
1778 | in which case the appropriate cartesian and exponential components |
1779 | will be parsed from the string and used to create new complex numbers. |
1780 | The imaginary component and the theta, respectively, will default to zero. |
1781 | |
66730be0 |
1782 | =head1 STRINGIFICATION |
1783 | |
1784 | When printed, a complex number is usually shown under its cartesian |
16357284 |
1785 | style I<a+bi>, but there are legitimate cases where the polar style |
66730be0 |
1786 | I<[r,t]> is more appropriate. |
1787 | |
16357284 |
1788 | By calling the class method C<Math::Complex::display_format> and |
1789 | supplying either C<"polar"> or C<"cartesian"> as an argument, you |
5287f86b |
1790 | override the default display style, which is C<"cartesian">. Not |
16357284 |
1791 | supplying any argument returns the current settings. |
66730be0 |
1792 | |
1793 | This default can be overridden on a per-number basis by calling the |
1794 | C<display_format> method instead. As before, not supplying any argument |
5287f86b |
1795 | returns the current display style for this number. Otherwise whatever you |
1796 | specify will be the new display style for I<this> particular number. |
66730be0 |
1797 | |
1798 | For instance: |
1799 | |
1800 | use Math::Complex; |
1801 | |
1802 | Math::Complex::display_format('polar'); |
16357284 |
1803 | $j = (root(1, 3))[1]; |
1804 | print "j = $j\n"; # Prints "j = [1,2pi/3]" |
66730be0 |
1805 | $j->display_format('cartesian'); |
1806 | print "j = $j\n"; # Prints "j = -0.5+0.866025403784439i" |
1807 | |
5287f86b |
1808 | The polar style attempts to emphasize arguments like I<k*pi/n> |
9bc5fa8d |
1809 | (where I<n> is a positive integer and I<k> an integer within [-9, +9]), |
5287f86b |
1810 | this is called I<polar pretty-printing>. |
66730be0 |
1811 | |
16357284 |
1812 | =head2 CHANGED IN PERL 5.6 |
1813 | |
1814 | The C<display_format> class method and the corresponding |
1815 | C<display_format> object method can now be called using |
1816 | a parameter hash instead of just a one parameter. |
1817 | |
1818 | The old display format style, which can have values C<"cartesian"> or |
40b904b7 |
1819 | C<"polar">, can be changed using the C<"style"> parameter. |
1820 | |
1821 | $j->display_format(style => "polar"); |
1822 | |
1823 | The one parameter calling convention also still works. |
1824 | |
1825 | $j->display_format("polar"); |
16357284 |
1826 | |
1827 | There are two new display parameters. |
1828 | |
40b904b7 |
1829 | The first one is C<"format">, which is a sprintf()-style format string |
1830 | to be used for both numeric parts of the complex number(s). The is |
1831 | somewhat system-dependent but most often it corresponds to C<"%.15g">. |
1832 | You can revert to the default by setting the C<format> to C<undef>. |
16357284 |
1833 | |
1834 | # the $j from the above example |
1835 | |
1836 | $j->display_format('format' => '%.5f'); |
1837 | print "j = $j\n"; # Prints "j = -0.50000+0.86603i" |
40b904b7 |
1838 | $j->display_format('format' => undef); |
16357284 |
1839 | print "j = $j\n"; # Prints "j = -0.5+0.86603i" |
1840 | |
1841 | Notice that this affects also the return values of the |
1842 | C<display_format> methods: in list context the whole parameter hash |
40b904b7 |
1843 | will be returned, as opposed to only the style parameter value. |
1844 | This is a potential incompatibility with earlier versions if you |
1845 | have been calling the C<display_format> method in list context. |
16357284 |
1846 | |
5287f86b |
1847 | The second new display parameter is C<"polar_pretty_print">, which can |
1848 | be set to true or false, the default being true. See the previous |
1849 | section for what this means. |
16357284 |
1850 | |
66730be0 |
1851 | =head1 USAGE |
1852 | |
1853 | Thanks to overloading, the handling of arithmetics with complex numbers |
1854 | is simple and almost transparent. |
1855 | |
1856 | Here are some examples: |
1857 | |
1858 | use Math::Complex; |
1859 | |
1860 | $j = cplxe(1, 2*pi/3); # $j ** 3 == 1 |
1861 | print "j = $j, j**3 = ", $j ** 3, "\n"; |
1862 | print "1 + j + j**2 = ", 1 + $j + $j**2, "\n"; |
1863 | |
1864 | $z = -16 + 0*i; # Force it to be a complex |
1865 | print "sqrt($z) = ", sqrt($z), "\n"; |
1866 | |
1867 | $k = exp(i * 2*pi/3); |
1868 | print "$j - $k = ", $j - $k, "\n"; |
a5f75d66 |
1869 | |
b42d0ec9 |
1870 | $z->Re(3); # Re, Im, arg, abs, |
1871 | $j->arg(2); # (the last two aka rho, theta) |
1872 | # can be used also as mutators. |
1873 | |
1874 | =head1 ERRORS DUE TO DIVISION BY ZERO OR LOGARITHM OF ZERO |
5aabfad6 |
1875 | |
1876 | The division (/) and the following functions |
1877 | |
b42d0ec9 |
1878 | log ln log10 logn |
2820d885 |
1879 | tan sec csc cot |
b42d0ec9 |
1880 | atan asec acsc acot |
1881 | tanh sech csch coth |
1882 | atanh asech acsch acoth |
5aabfad6 |
1883 | |
1884 | cannot be computed for all arguments because that would mean dividing |
8c03c583 |
1885 | by zero or taking logarithm of zero. These situations cause fatal |
1886 | runtime errors looking like this |
5aabfad6 |
1887 | |
1888 | cot(0): Division by zero. |
5cd24f17 |
1889 | (Because in the definition of cot(0), the divisor sin(0) is 0) |
5aabfad6 |
1890 | Died at ... |
1891 | |
8c03c583 |
1892 | or |
1893 | |
1894 | atanh(-1): Logarithm of zero. |
1895 | Died at... |
1896 | |
1897 | For the C<csc>, C<cot>, C<asec>, C<acsc>, C<acot>, C<csch>, C<coth>, |
d1be9408 |
1898 | C<asech>, C<acsch>, the argument cannot be C<0> (zero). For the |
b42d0ec9 |
1899 | logarithmic functions and the C<atanh>, C<acoth>, the argument cannot |
1900 | be C<1> (one). For the C<atanh>, C<acoth>, the argument cannot be |
1901 | C<-1> (minus one). For the C<atan>, C<acot>, the argument cannot be |
1902 | C<i> (the imaginary unit). For the C<atan>, C<acoth>, the argument |
1903 | cannot be C<-i> (the negative imaginary unit). For the C<tan>, |
1904 | C<sec>, C<tanh>, the argument cannot be I<pi/2 + k * pi>, where I<k> |
1905 | is any integer. |
1906 | |
1907 | Note that because we are operating on approximations of real numbers, |
1908 | these errors can happen when merely `too close' to the singularities |
40b904b7 |
1909 | listed above. |
b42d0ec9 |
1910 | |
1911 | =head1 ERRORS DUE TO INDIGESTIBLE ARGUMENTS |
1912 | |
1913 | The C<make> and C<emake> accept both real and complex arguments. |
1914 | When they cannot recognize the arguments they will die with error |
1915 | messages like the following |
1916 | |
1917 | Math::Complex::make: Cannot take real part of ... |
1918 | Math::Complex::make: Cannot take real part of ... |
1919 | Math::Complex::emake: Cannot take rho of ... |
1920 | Math::Complex::emake: Cannot take theta of ... |
5cd24f17 |
1921 | |
a5f75d66 |
1922 | =head1 BUGS |
1923 | |
5cd24f17 |
1924 | Saying C<use Math::Complex;> exports many mathematical routines in the |
fb73857a |
1925 | caller environment and even overrides some (C<sqrt>, C<log>). |
1926 | This is construed as a feature by the Authors, actually... ;-) |
a5f75d66 |
1927 | |
66730be0 |
1928 | All routines expect to be given real or complex numbers. Don't attempt to |
1929 | use BigFloat, since Perl has currently no rule to disambiguate a '+' |
1930 | operation (for instance) between two overloaded entities. |
a5f75d66 |
1931 | |
d09ae4e6 |
1932 | In Cray UNICOS there is some strange numerical instability that results |
1933 | in root(), cos(), sin(), cosh(), sinh(), losing accuracy fast. Beware. |
1934 | The bug may be in UNICOS math libs, in UNICOS C compiler, in Math::Complex. |
1935 | Whatever it is, it does not manifest itself anywhere else where Perl runs. |
1936 | |
0c721ce2 |
1937 | =head1 AUTHORS |
a5f75d66 |
1938 | |
e6c12c3f |
1939 | Daniel S. Lewart <F<d-lewart@uiuc.edu>> |
5cd24f17 |
1940 | |
e6c12c3f |
1941 | Original authors Raphael Manfredi <F<Raphael_Manfredi@pobox.com>> and |
1942 | Jarkko Hietaniemi <F<jhi@iki.fi>> |
fb73857a |
1943 | |
5cd24f17 |
1944 | =cut |
1945 | |
b42d0ec9 |
1946 | 1; |
1947 | |
5cd24f17 |
1948 | # eof |