Commit | Line | Data |
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86530b38 AT |
1 | # |
2 | # Complex numbers and associated mathematical functions | |
3 | # -- Raphael Manfredi Since Sep 1996 | |
4 | # -- Jarkko Hietaniemi Since Mar 1997 | |
5 | # -- Daniel S. Lewart Since Sep 1997 | |
6 | # | |
7 | ||
8 | package Math::Complex; | |
9 | ||
10 | our($VERSION, @ISA, @EXPORT, %EXPORT_TAGS, $Inf); | |
11 | ||
12 | $VERSION = 1.34; | |
13 | ||
14 | BEGIN { | |
15 | unless ($^O eq 'unicosmk') { | |
16 | my $e = $!; | |
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 | } | |
30 | $! = $e; # Clear ERANGE. | |
31 | } | |
32 | $Inf = "Inf" if !defined $Inf || !($Inf > 0); # Desperation. | |
33 | } | |
34 | ||
35 | use strict; | |
36 | ||
37 | my $i; | |
38 | my %LOGN; | |
39 | ||
40 | # Regular expression for floating point numbers. | |
41 | my $gre = qr'\s*([\+\-]?(?:(?:(?:\d+(?:_\d+)*(?:\.\d*(?:_\d+)*)?|\.\d+(?:_\d+)*)(?:[eE][\+\-]?\d+(?:_\d+)*)?)))'; | |
42 | ||
43 | require Exporter; | |
44 | ||
45 | @ISA = qw(Exporter); | |
46 | ||
47 | my @trig = qw( | |
48 | pi | |
49 | tan | |
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( | |
60 | i Re Im rho theta arg | |
61 | sqrt log ln | |
62 | log10 logn cbrt root | |
63 | cplx cplxe | |
64 | ), | |
65 | @trig); | |
66 | ||
67 | %EXPORT_TAGS = ( | |
68 | 'trig' => [@trig], | |
69 | ); | |
70 | ||
71 | use overload | |
72 | '+' => \&plus, | |
73 | '-' => \&minus, | |
74 | '*' => \&multiply, | |
75 | '/' => \÷, | |
76 | '**' => \&power, | |
77 | '==' => \&numeq, | |
78 | '<=>' => \&spaceship, | |
79 | 'neg' => \&negate, | |
80 | '~' => \&conjugate, | |
81 | 'abs' => \&abs, | |
82 | 'sqrt' => \&sqrt, | |
83 | 'exp' => \&exp, | |
84 | 'log' => \&log, | |
85 | 'sin' => \&sin, | |
86 | 'cos' => \&cos, | |
87 | 'tan' => \&tan, | |
88 | 'atan2' => \&atan2, | |
89 | qw("" stringify); | |
90 | ||
91 | # | |
92 | # Package "privates" | |
93 | # | |
94 | ||
95 | my %DISPLAY_FORMAT = ('style' => 'cartesian', | |
96 | 'polar_pretty_print' => 1); | |
97 | my $eps = 1e-14; # Epsilon | |
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 | ||
108 | # Die on bad *make() arguments. | |
109 | ||
110 | sub _cannot_make { | |
111 | die "@{[(caller(1))[3]]}: Cannot take $_[0] of $_[1].\n"; | |
112 | } | |
113 | ||
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 | ||
134 | # | |
135 | # ->make | |
136 | # | |
137 | # Create a new complex number (cartesian form) | |
138 | # | |
139 | sub make { | |
140 | my $self = bless {}, shift; | |
141 | my ($re, $im) = @_; | |
142 | if (@_ == 1) { | |
143 | my ($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 | } | |
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 | } | |
168 | _cannot_make("real part", $re) unless $re =~ /^$gre$/; | |
169 | $im ||= 0; | |
170 | _cannot_make("imaginary part", $im) unless $im =~ /^$gre$/; | |
171 | $self->{'cartesian'} = [ $re, $im ]; | |
172 | $self->{c_dirty} = 0; | |
173 | $self->{p_dirty} = 1; | |
174 | $self->display_format('cartesian'); | |
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, $theta) = @_; | |
186 | if (@_ == 1) { | |
187 | my ($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 | } | |
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", $rth); | |
210 | } | |
211 | } | |
212 | if ($rho < 0) { | |
213 | $rho = -$rho; | |
214 | $theta = ($theta <= 0) ? $theta + pi() : $theta - pi(); | |
215 | } | |
216 | _cannot_make("rho", $rho) unless $rho =~ /^$gre$/; | |
217 | $theta ||= 0; | |
218 | _cannot_make("theta", $theta) unless $theta =~ /^$gre$/; | |
219 | $self->{'polar'} = [$rho, $theta]; | |
220 | $self->{p_dirty} = 0; | |
221 | $self->{c_dirty} = 1; | |
222 | $self->display_format('polar'); | |
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 { | |
235 | return __PACKAGE__->make(@_); | |
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 { | |
245 | return __PACKAGE__->emake(@_); | |
246 | } | |
247 | ||
248 | # | |
249 | # pi | |
250 | # | |
251 | # The number defined as pi = 180 degrees | |
252 | # | |
253 | sub pi () { 4 * CORE::atan2(1, 1) } | |
254 | ||
255 | # | |
256 | # pit2 | |
257 | # | |
258 | # The full circle | |
259 | # | |
260 | sub pit2 () { 2 * pi } | |
261 | ||
262 | # | |
263 | # pip2 | |
264 | # | |
265 | # The quarter circle | |
266 | # | |
267 | sub pip2 () { pi / 2 } | |
268 | ||
269 | # | |
270 | # deg1 | |
271 | # | |
272 | # One degree in radians, used in stringify_polar. | |
273 | # | |
274 | ||
275 | sub deg1 () { pi / 180 } | |
276 | ||
277 | # | |
278 | # uplog10 | |
279 | # | |
280 | # Used in log10(). | |
281 | # | |
282 | sub uplog10 () { 1 / CORE::log(10) } | |
283 | ||
284 | # | |
285 | # i | |
286 | # | |
287 | # The number defined as i*i = -1; | |
288 | # | |
289 | sub i () { | |
290 | return $i if ($i); | |
291 | $i = bless {}; | |
292 | $i->{'cartesian'} = [0, 1]; | |
293 | $i->{'polar'} = [1, pip2]; | |
294 | $i->{c_dirty} = 0; | |
295 | $i->{p_dirty} = 0; | |
296 | return $i; | |
297 | } | |
298 | ||
299 | # | |
300 | # ip2 | |
301 | # | |
302 | # Half of i. | |
303 | # | |
304 | sub ip2 () { i / 2 } | |
305 | ||
306 | # | |
307 | # Attribute access/set routines | |
308 | # | |
309 | ||
310 | sub cartesian {$_[0]->{c_dirty} ? | |
311 | $_[0]->update_cartesian : $_[0]->{'cartesian'}} | |
312 | sub polar {$_[0]->{p_dirty} ? | |
313 | $_[0]->update_polar : $_[0]->{'polar'}} | |
314 | ||
315 | sub set_cartesian { $_[0]->{p_dirty}++; $_[0]->{'cartesian'} = $_[1] } | |
316 | sub set_polar { $_[0]->{c_dirty}++; $_[0]->{'polar'} = $_[1] } | |
317 | ||
318 | # | |
319 | # ->update_cartesian | |
320 | # | |
321 | # Recompute and return the cartesian form, given accurate polar form. | |
322 | # | |
323 | sub update_cartesian { | |
324 | my $self = shift; | |
325 | my ($r, $t) = @{$self->{'polar'}}; | |
326 | $self->{c_dirty} = 0; | |
327 | return $self->{'cartesian'} = [$r * CORE::cos($t), $r * CORE::sin($t)]; | |
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 $self = shift; | |
338 | my ($x, $y) = @{$self->{'cartesian'}}; | |
339 | $self->{p_dirty} = 0; | |
340 | return $self->{'polar'} = [0, 0] if $x == 0 && $y == 0; | |
341 | return $self->{'polar'} = [CORE::sqrt($x*$x + $y*$y), | |
342 | CORE::atan2($y, $x)]; | |
343 | } | |
344 | ||
345 | # | |
346 | # (plus) | |
347 | # | |
348 | # Computes z1+z2. | |
349 | # | |
350 | sub plus { | |
351 | my ($z1, $z2, $regular) = @_; | |
352 | my ($re1, $im1) = @{$z1->cartesian}; | |
353 | $z2 = cplx($z2) unless ref $z2; | |
354 | my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0); | |
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}; | |
370 | $z2 = cplx($z2) unless ref $z2; | |
371 | my ($re2, $im2) = @{$z2->cartesian}; | |
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 - $im2); | |
379 | ||
380 | } | |
381 | ||
382 | # | |
383 | # (multiply) | |
384 | # | |
385 | # Computes z1*z2. | |
386 | # | |
387 | sub multiply { | |
388 | my ($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]); | |
398 | return $z1; | |
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 | } | |
409 | } | |
410 | } | |
411 | ||
412 | # | |
413 | # _divbyzero | |
414 | # | |
415 | # Die on division by zero. | |
416 | # | |
417 | sub _divbyzero { | |
418 | my $mess = "$_[0]: Division by zero.\n"; | |
419 | ||
420 | if (defined $_[1]) { | |
421 | $mess .= "(Because in the definition of $_[0], the divisor "; | |
422 | $mess .= "$_[1] " unless ("$_[1]" eq '0'); | |
423 | $mess .= "is 0)\n"; | |
424 | } | |
425 | ||
426 | my @up = caller(1); | |
427 | ||
428 | $mess .= "Died at $up[1] line $up[2].\n"; | |
429 | ||
430 | die $mess; | |
431 | } | |
432 | ||
433 | # | |
434 | # (divide) | |
435 | # | |
436 | # Computes z1/z2. | |
437 | # | |
438 | sub divide { | |
439 | my ($z1, $z2, $inverted) = @_; | |
440 | if ($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) { | |
446 | _divbyzero "$z2/0" if ($r1 == 0); | |
447 | $t = $t2 - $t1; | |
448 | if ($t > pi()) { $t -= pit2 } | |
449 | elsif ($t <= -pi()) { $t += pit2 } | |
450 | return (ref $z1)->emake($r2 / $r1, $t); | |
451 | } else { | |
452 | _divbyzero "$z1/0" if ($r2 == 0); | |
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 | } | |
479 | } | |
480 | } | |
481 | ||
482 | # | |
483 | # (power) | |
484 | # | |
485 | # Computes z1**z2 = exp(z2 * log z1)). | |
486 | # | |
487 | sub power { | |
488 | my ($z1, $z2, $inverted) = @_; | |
489 | if ($inverted) { | |
490 | return 1 if $z1 == 0 || $z2 == 1; | |
491 | return 0 if $z2 == 0 && Re($z1) > 0; | |
492 | } else { | |
493 | return 1 if $z2 == 0 || $z1 == 1; | |
494 | return 0 if $z1 == 0 && Re($z2) > 0; | |
495 | } | |
496 | my $w = $inverted ? &exp($z1 * &log($z2)) | |
497 | : &exp($z2 * &log($z1)); | |
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}) : $w; | |
502 | } | |
503 | ||
504 | # | |
505 | # (spaceship) | |
506 | # | |
507 | # Computes z1 <=> z2. | |
508 | # Sorts on the real part first, then on the imaginary part. Thus 2-4i < 3+8i. | |
509 | # | |
510 | sub spaceship { | |
511 | my ($z1, $z2, $inverted) = @_; | |
512 | my ($re1, $im1) = ref $z1 ? @{$z1->cartesian} : ($z1, 0); | |
513 | my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0); | |
514 | my $sgn = $inverted ? -1 : 1; | |
515 | return $sgn * ($re1 <=> $re2) if $re1 != $re2; | |
516 | return $sgn * ($im1 <=> $im2); | |
517 | } | |
518 | ||
519 | # | |
520 | # (numeq) | |
521 | # | |
522 | # Computes z1 == z2. | |
523 | # | |
524 | # (Required in addition to spaceship() because of NaNs.) | |
525 | sub numeq { | |
526 | my ($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 == $im2 ? 1 : 0; | |
530 | } | |
531 | ||
532 | # | |
533 | # (negate) | |
534 | # | |
535 | # Computes -z. | |
536 | # | |
537 | sub negate { | |
538 | my ($z) = @_; | |
539 | if ($z->{c_dirty}) { | |
540 | my ($r, $t) = @{$z->polar}; | |
541 | $t = ($t <= 0) ? $t + pi : $t - pi; | |
542 | return (ref $z)->emake($r, $t); | |
543 | } | |
544 | my ($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, -$im); | |
561 | } | |
562 | ||
563 | # | |
564 | # (abs) | |
565 | # | |
566 | # Compute or set complex's norm (rho). | |
567 | # | |
568 | sub abs { | |
569 | my ($z, $rho) = @_; | |
570 | unless (ref $z) { | |
571 | if (@_ == 2) { | |
572 | $_[0] = $_[1]; | |
573 | } else { | |
574 | return CORE::abs($z); | |
575 | } | |
576 | } | |
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 } | |
592 | } | |
593 | ||
594 | # | |
595 | # arg | |
596 | # | |
597 | # Compute or set complex's argument (theta). | |
598 | # | |
599 | sub arg { | |
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 $theta; | |
612 | } | |
613 | ||
614 | # | |
615 | # (sqrt) | |
616 | # | |
617 | # Compute sqrt(z). | |
618 | # | |
619 | # It is quite tempting to use wantarray here so that in list context | |
620 | # sqrt() would return the two solutions. This, however, would | |
621 | # break things like | |
622 | # | |
623 | # print "sqrt(z) = ", sqrt($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 | # | |
629 | sub sqrt { | |
630 | my ($z) = @_; | |
631 | my ($re, $im) = ref $z ? @{$z->cartesian} : ($z, 0); | |
632 | return $re < 0 ? cplx(0, CORE::sqrt(-$re)) : CORE::sqrt($re) | |
633 | if $im == 0; | |
634 | my ($r, $t) = @{$z->polar}; | |
635 | return (ref $z)->emake(CORE::sqrt($r), $t/2); | |
636 | } | |
637 | ||
638 | # | |
639 | # cbrt | |
640 | # | |
641 | # Compute cbrt(z) (cubic root). | |
642 | # | |
643 | # Why are we not returning three values? The same answer as for sqrt(). | |
644 | # | |
645 | sub cbrt { | |
646 | my ($z) = @_; | |
647 | return $z < 0 ? | |
648 | -CORE::exp(CORE::log(-$z)/3) : | |
649 | ($z > 0 ? CORE::exp(CORE::log($z)/3): 0) | |
650 | unless ref $z; | |
651 | my ($r, $t) = @{$z->polar}; | |
652 | return 0 if $r == 0; | |
653 | return (ref $z)->emake(CORE::exp(CORE::log($r)/3), $t/3); | |
654 | } | |
655 | ||
656 | # | |
657 | # _rootbad | |
658 | # | |
659 | # Die on bad root. | |
660 | # | |
661 | sub _rootbad { | |
662 | my $mess = "Root $_[0] illegal, root rank must be positive integer.\n"; | |
663 | ||
664 | my @up = caller(1); | |
665 | ||
666 | $mess .= "Died at $up[1] line $up[2].\n"; | |
667 | ||
668 | die $mess; | |
669 | } | |
670 | ||
671 | # | |
672 | # root | |
673 | # | |
674 | # Computes all nth root for z, returning an array whose size is n. | |
675 | # `n' must be a positive integer. | |
676 | # | |
677 | # The roots are given by (for k = 0..n-1): | |
678 | # | |
679 | # z^(1/n) = r^(1/n) (cos ((t+2 k pi)/n) + i sin ((t+2 k pi)/n)) | |
680 | # | |
681 | sub root { | |
682 | my ($z, $n) = @_; | |
683 | _rootbad($n) if ($n < 1 or int($n) != $n); | |
684 | my ($r, $t) = ref $z ? | |
685 | @{$z->polar} : (CORE::abs($z), $z >= 0 ? 0 : pi); | |
686 | my @root; | |
687 | my $k; | |
688 | my $theta_inc = pit2 / $n; | |
689 | my $rho = $r ** (1/$n); | |
690 | my $theta; | |
691 | my $cartesian = ref $z && $z->{c_dirty} == 0; | |
692 | for ($k = 0, $theta = $t / $n; $k < $n; $k++, $theta += $theta_inc) { | |
693 | my $w = cplxe($rho, $theta); | |
694 | # Yes, $cartesian is loop invariant. | |
695 | push @root, $cartesian ? cplx(@{$w->cartesian}) : $w; | |
696 | } | |
697 | return @root; | |
698 | } | |
699 | ||
700 | # | |
701 | # Re | |
702 | # | |
703 | # Return or set Re(z). | |
704 | # | |
705 | sub Re { | |
706 | my ($z, $Re) = @_; | |
707 | return $z unless ref $z; | |
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 | } | |
715 | } | |
716 | ||
717 | # | |
718 | # Im | |
719 | # | |
720 | # Return or set Im(z). | |
721 | # | |
722 | sub Im { | |
723 | my ($z, $Im) = @_; | |
724 | return 0 unless ref $z; | |
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(@_); | |
750 | } | |
751 | ||
752 | # | |
753 | # (exp) | |
754 | # | |
755 | # Computes exp(z). | |
756 | # | |
757 | sub exp { | |
758 | my ($z) = @_; | |
759 | my ($x, $y) = @{$z->cartesian}; | |
760 | return (ref $z)->emake(CORE::exp($x), $y); | |
761 | } | |
762 | ||
763 | # | |
764 | # _logofzero | |
765 | # | |
766 | # Die on logarithm of zero. | |
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); | |
778 | ||
779 | $mess .= "Died at $up[1] line $up[2].\n"; | |
780 | ||
781 | die $mess; | |
782 | } | |
783 | ||
784 | # | |
785 | # (log) | |
786 | # | |
787 | # Compute log(z). | |
788 | # | |
789 | sub log { | |
790 | my ($z) = @_; | |
791 | unless (ref $z) { | |
792 | _logofzero("log") if $z == 0; | |
793 | return $z > 0 ? CORE::log($z) : cplx(CORE::log(-$z), pi); | |
794 | } | |
795 | my ($r, $t) = @{$z->polar}; | |
796 | _logofzero("log") if $r == 0; | |
797 | if ($t > pi()) { $t -= pit2 } | |
798 | elsif ($t <= -pi()) { $t += pit2 } | |
799 | return (ref $z)->make(CORE::log($r), $t); | |
800 | } | |
801 | ||
802 | # | |
803 | # ln | |
804 | # | |
805 | # Alias for log(). | |
806 | # | |
807 | sub ln { Math::Complex::log(@_) } | |
808 | ||
809 | # | |
810 | # log10 | |
811 | # | |
812 | # Compute log10(z). | |
813 | # | |
814 | ||
815 | sub log10 { | |
816 | return Math::Complex::log($_[0]) * uplog10; | |
817 | } | |
818 | ||
819 | # | |
820 | # logn | |
821 | # | |
822 | # Compute logn(z,n) = log(z) / log(n) | |
823 | # | |
824 | sub logn { | |
825 | my ($z, $n) = @_; | |
826 | $z = cplx($z, 0) unless ref $z; | |
827 | my $logn = $LOGN{$n}; | |
828 | $logn = $LOGN{$n} = CORE::log($n) unless defined $logn; # Cache log(n) | |
829 | return &log($z) / $logn; | |
830 | } | |
831 | ||
832 | # | |
833 | # (cos) | |
834 | # | |
835 | # Compute cos(z) = (exp(iz) + exp(-iz))/2. | |
836 | # | |
837 | sub cos { | |
838 | my ($z) = @_; | |
839 | return CORE::cos($z) unless ref $z; | |
840 | my ($x, $y) = @{$z->cartesian}; | |
841 | my $ey = CORE::exp($y); | |
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 - $ey)/2); | |
847 | } | |
848 | ||
849 | # | |
850 | # (sin) | |
851 | # | |
852 | # Compute sin(z) = (exp(iz) - exp(-iz))/2. | |
853 | # | |
854 | sub sin { | |
855 | my ($z) = @_; | |
856 | return CORE::sin($z) unless ref $z; | |
857 | my ($x, $y) = @{$z->cartesian}; | |
858 | my $ey = CORE::exp($y); | |
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 - $ey_1)/2); | |
864 | } | |
865 | ||
866 | # | |
867 | # tan | |
868 | # | |
869 | # Compute tan(z) = sin(z) / cos(z). | |
870 | # | |
871 | sub tan { | |
872 | my ($z) = @_; | |
873 | my $cz = &cos($z); | |
874 | _divbyzero "tan($z)", "cos($z)" if $cz == 0; | |
875 | return &sin($z) / $cz; | |
876 | } | |
877 | ||
878 | # | |
879 | # sec | |
880 | # | |
881 | # Computes the secant sec(z) = 1 / cos(z). | |
882 | # | |
883 | sub sec { | |
884 | my ($z) = @_; | |
885 | my $cz = &cos($z); | |
886 | _divbyzero "sec($z)", "cos($z)" if ($cz == 0); | |
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) = @_; | |
897 | my $sz = &sin($z); | |
898 | _divbyzero "csc($z)", "sin($z)" if ($sz == 0); | |
899 | return 1 / $sz; | |
900 | } | |
901 | ||
902 | # | |
903 | # cosec | |
904 | # | |
905 | # Alias for csc(). | |
906 | # | |
907 | sub cosec { Math::Complex::csc(@_) } | |
908 | ||
909 | # | |
910 | # cot | |
911 | # | |
912 | # Computes cot(z) = cos(z) / sin(z). | |
913 | # | |
914 | sub cot { | |
915 | my ($z) = @_; | |
916 | my $sz = &sin($z); | |
917 | _divbyzero "cot($z)", "sin($z)" if ($sz == 0); | |
918 | return &cos($z) / $sz; | |
919 | } | |
920 | ||
921 | # | |
922 | # cotan | |
923 | # | |
924 | # Alias for cot(). | |
925 | # | |
926 | sub cotan { Math::Complex::cot(@_) } | |
927 | ||
928 | # | |
929 | # acos | |
930 | # | |
931 | # Computes the arc cosine acos(z) = -i log(z + sqrt(z*z-1)). | |
932 | # | |
933 | sub acos { | |
934 | my $z = $_[0]; | |
935 | return CORE::atan2(CORE::sqrt(1-$z*$z), $z) | |
936 | if (! ref $z) && CORE::abs($z) <= 1; | |
937 | $z = cplx($z, 0) unless ref $z; | |
938 | my ($x, $y) = @{$z->cartesian}; | |
939 | return 0 if $x == 1 && $y == 0; | |
940 | my $t1 = CORE::sqrt(($x+1)*($x+1) + $y*$y); | |
941 | my $t2 = CORE::sqrt(($x-1)*($x-1) + $y*$y); | |
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 } | |
947 | my $u = CORE::atan2(CORE::sqrt(1-$beta*$beta), $beta); | |
948 | my $v = CORE::log($alpha + CORE::sqrt($alpha*$alpha-1)); | |
949 | $v = -$v if $y > 0 || ($y == 0 && $x < -1); | |
950 | return (ref $z)->make($u, $v); | |
951 | } | |
952 | ||
953 | # | |
954 | # asin | |
955 | # | |
956 | # Computes the arc sine asin(z) = -i log(iz + sqrt(1-z*z)). | |
957 | # | |
958 | sub asin { | |
959 | my $z = $_[0]; | |
960 | return CORE::atan2($z, CORE::sqrt(1-$z*$z)) | |
961 | if (! ref $z) && CORE::abs($z) <= 1; | |
962 | $z = cplx($z, 0) unless ref $z; | |
963 | my ($x, $y) = @{$z->cartesian}; | |
964 | return 0 if $x == 0 && $y == 0; | |
965 | my $t1 = CORE::sqrt(($x+1)*($x+1) + $y*$y); | |
966 | my $t2 = CORE::sqrt(($x-1)*($x-1) + $y*$y); | |
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 } | |
972 | my $u = CORE::atan2($beta, CORE::sqrt(1-$beta*$beta)); | |
973 | my $v = -CORE::log($alpha + CORE::sqrt($alpha*$alpha-1)); | |
974 | $v = -$v if $y > 0 || ($y == 0 && $x < -1); | |
975 | return (ref $z)->make($u, $v); | |
976 | } | |
977 | ||
978 | # | |
979 | # atan | |
980 | # | |
981 | # Computes the arc tangent atan(z) = i/2 log((i+z) / (i-z)). | |
982 | # | |
983 | sub atan { | |
984 | my ($z) = @_; | |
985 | return CORE::atan2($z, 1) unless ref $z; | |
986 | my ($x, $y) = ref $z ? @{$z->cartesian} : ($z, 0); | |
987 | return 0 if $x == 0 && $y == 0; | |
988 | _divbyzero "atan(i)" if ( $z == i); | |
989 | _logofzero "atan(-i)" if (-$z == i); # -i is a bad file test... | |
990 | my $log = &log((i + $z) / (i - $z)); | |
991 | return ip2 * $log; | |
992 | } | |
993 | ||
994 | # | |
995 | # asec | |
996 | # | |
997 | # Computes the arc secant asec(z) = acos(1 / z). | |
998 | # | |
999 | sub asec { | |
1000 | my ($z) = @_; | |
1001 | _divbyzero "asec($z)", $z if ($z == 0); | |
1002 | return acos(1 / $z); | |
1003 | } | |
1004 | ||
1005 | # | |
1006 | # acsc | |
1007 | # | |
1008 | # Computes the arc cosecant acsc(z) = asin(1 / z). | |
1009 | # | |
1010 | sub acsc { | |
1011 | my ($z) = @_; | |
1012 | _divbyzero "acsc($z)", $z if ($z == 0); | |
1013 | return asin(1 / $z); | |
1014 | } | |
1015 | ||
1016 | # | |
1017 | # acosec | |
1018 | # | |
1019 | # Alias for acsc(). | |
1020 | # | |
1021 | sub acosec { Math::Complex::acsc(@_) } | |
1022 | ||
1023 | # | |
1024 | # acot | |
1025 | # | |
1026 | # Computes the arc cotangent acot(z) = atan(1 / z) | |
1027 | # | |
1028 | sub acot { | |
1029 | my ($z) = @_; | |
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); | |
1035 | return atan(1 / $z); | |
1036 | } | |
1037 | ||
1038 | # | |
1039 | # acotan | |
1040 | # | |
1041 | # Alias for acot(). | |
1042 | # | |
1043 | sub acotan { Math::Complex::acot(@_) } | |
1044 | ||
1045 | # | |
1046 | # cosh | |
1047 | # | |
1048 | # Computes the hyperbolic cosine cosh(z) = (exp(z) + exp(-z))/2. | |
1049 | # | |
1050 | sub cosh { | |
1051 | my ($z) = @_; | |
1052 | my $ex; | |
1053 | unless (ref $z) { | |
1054 | $ex = CORE::exp($z); | |
1055 | return $ex ? ($ex + 1/$ex)/2 : $Inf; | |
1056 | } | |
1057 | my ($x, $y) = @{$z->cartesian}; | |
1058 | $ex = CORE::exp($x); | |
1059 | my $ex_1 = $ex ? 1 / $ex : $Inf; | |
1060 | return (ref $z)->make(CORE::cos($y) * ($ex + $ex_1)/2, | |
1061 | CORE::sin($y) * ($ex - $ex_1)/2); | |
1062 | } | |
1063 | ||
1064 | # | |
1065 | # sinh | |
1066 | # | |
1067 | # Computes the hyperbolic sine sinh(z) = (exp(z) - exp(-z))/2. | |
1068 | # | |
1069 | sub sinh { | |
1070 | my ($z) = @_; | |
1071 | my $ex; | |
1072 | unless (ref $z) { | |
1073 | return 0 if $z == 0; | |
1074 | $ex = CORE::exp($z); | |
1075 | return $ex ? ($ex - 1/$ex)/2 : "-$Inf"; | |
1076 | } | |
1077 | my ($x, $y) = @{$z->cartesian}; | |
1078 | my $cy = CORE::cos($y); | |
1079 | my $sy = CORE::sin($y); | |
1080 | $ex = CORE::exp($x); | |
1081 | my $ex_1 = $ex ? 1 / $ex : $Inf; | |
1082 | return (ref $z)->make(CORE::cos($y) * ($ex - $ex_1)/2, | |
1083 | CORE::sin($y) * ($ex + $ex_1)/2); | |
1084 | } | |
1085 | ||
1086 | # | |
1087 | # tanh | |
1088 | # | |
1089 | # Computes the hyperbolic tangent tanh(z) = sinh(z) / cosh(z). | |
1090 | # | |
1091 | sub tanh { | |
1092 | my ($z) = @_; | |
1093 | my $cz = cosh($z); | |
1094 | _divbyzero "tanh($z)", "cosh($z)" if ($cz == 0); | |
1095 | return sinh($z) / $cz; | |
1096 | } | |
1097 | ||
1098 | # | |
1099 | # sech | |
1100 | # | |
1101 | # Computes the hyperbolic secant sech(z) = 1 / cosh(z). | |
1102 | # | |
1103 | sub sech { | |
1104 | my ($z) = @_; | |
1105 | my $cz = cosh($z); | |
1106 | _divbyzero "sech($z)", "cosh($z)" if ($cz == 0); | |
1107 | return 1 / $cz; | |
1108 | } | |
1109 | ||
1110 | # | |
1111 | # csch | |
1112 | # | |
1113 | # Computes the hyperbolic cosecant csch(z) = 1 / sinh(z). | |
1114 | # | |
1115 | sub csch { | |
1116 | my ($z) = @_; | |
1117 | my $sz = sinh($z); | |
1118 | _divbyzero "csch($z)", "sinh($z)" if ($sz == 0); | |
1119 | return 1 / $sz; | |
1120 | } | |
1121 | ||
1122 | # | |
1123 | # cosech | |
1124 | # | |
1125 | # Alias for csch(). | |
1126 | # | |
1127 | sub cosech { Math::Complex::csch(@_) } | |
1128 | ||
1129 | # | |
1130 | # coth | |
1131 | # | |
1132 | # Computes the hyperbolic cotangent coth(z) = cosh(z) / sinh(z). | |
1133 | # | |
1134 | sub coth { | |
1135 | my ($z) = @_; | |
1136 | my $sz = sinh($z); | |
1137 | _divbyzero "coth($z)", "sinh($z)" if $sz == 0; | |
1138 | return cosh($z) / $sz; | |
1139 | } | |
1140 | ||
1141 | # | |
1142 | # cotanh | |
1143 | # | |
1144 | # Alias for coth(). | |
1145 | # | |
1146 | sub cotanh { Math::Complex::coth(@_) } | |
1147 | ||
1148 | # | |
1149 | # acosh | |
1150 | # | |
1151 | # Computes the arc hyperbolic cosine acosh(z) = log(z + sqrt(z*z-1)). | |
1152 | # | |
1153 | sub acosh { | |
1154 | my ($z) = @_; | |
1155 | unless (ref $z) { | |
1156 | $z = cplx($z, 0); | |
1157 | } | |
1158 | my ($re, $im) = @{$z->cartesian}; | |
1159 | if ($im == 0) { | |
1160 | return CORE::log($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; | |
1164 | } | |
1165 | my $t = &sqrt($z * $z - 1) + $z; | |
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...) | |
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); | |
1172 | $u->Im(-$u->Im) if $re < 0 && $im == 0; | |
1173 | return $re < 0 ? -$u : $u; | |
1174 | } | |
1175 | ||
1176 | # | |
1177 | # asinh | |
1178 | # | |
1179 | # Computes the arc hyperbolic sine asinh(z) = log(z + sqrt(z*z+1)) | |
1180 | # | |
1181 | sub asinh { | |
1182 | my ($z) = @_; | |
1183 | unless (ref $z) { | |
1184 | my $t = $z + CORE::sqrt($z*$z + 1); | |
1185 | return CORE::log($t) if $t; | |
1186 | } | |
1187 | my $t = &sqrt($z * $z + 1) + $z; | |
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...) | |
1191 | $t = 1/(2 * $z) - 1/(8 * $z**3) + 1/(16 * $z**5) - 5/(128 * $z**7) | |
1192 | if $t == 0; | |
1193 | return &log($t); | |
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) = @_; | |
1203 | unless (ref $z) { | |
1204 | return CORE::log((1 + $z)/(1 - $z))/2 if CORE::abs($z) < 1; | |
1205 | $z = cplx($z, 0); | |
1206 | } | |
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 - $z)); | |
1210 | } | |
1211 | ||
1212 | # | |
1213 | # asech | |
1214 | # | |
1215 | # Computes the hyperbolic arc secant asech(z) = acosh(1 / z). | |
1216 | # | |
1217 | sub asech { | |
1218 | my ($z) = @_; | |
1219 | _divbyzero 'asech(0)', "$z" if ($z == 0); | |
1220 | return acosh(1 / $z); | |
1221 | } | |
1222 | ||
1223 | # | |
1224 | # acsch | |
1225 | # | |
1226 | # Computes the hyperbolic arc cosecant acsch(z) = asinh(1 / z). | |
1227 | # | |
1228 | sub acsch { | |
1229 | my ($z) = @_; | |
1230 | _divbyzero 'acsch(0)', $z if ($z == 0); | |
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) = @_; | |
1248 | _divbyzero 'acoth(0)' if ($z == 0); | |
1249 | unless (ref $z) { | |
1250 | return CORE::log(($z + 1)/($z - 1))/2 if CORE::abs($z) > 1; | |
1251 | $z = cplx($z, 0); | |
1252 | } | |
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; | |
1256 | } | |
1257 | ||
1258 | # | |
1259 | # acotanh | |
1260 | # | |
1261 | # Alias for acot(). | |
1262 | # | |
1263 | sub acotanh { Math::Complex::acoth(@_) } | |
1264 | ||
1265 | # | |
1266 | # (atan2) | |
1267 | # | |
1268 | # Compute atan(z1/z2). | |
1269 | # | |
1270 | sub atan2 { | |
1271 | my ($z1, $z2, $inverted) = @_; | |
1272 | my ($re1, $im1, $re2, $im2); | |
1273 | if ($inverted) { | |
1274 | ($re1, $im1) = ref $z2 ? @{$z2->cartesian} : ($z2, 0); | |
1275 | ($re2, $im2) = @{$z1->cartesian}; | |
1276 | } else { | |
1277 | ($re1, $im1) = @{$z1->cartesian}; | |
1278 | ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0); | |
1279 | } | |
1280 | if ($im2 == 0) { | |
1281 | return CORE::atan2($re1, $re2) if $im1 == 0; | |
1282 | return ($im1<=>0) * pip2 if $re2 == 0; | |
1283 | } | |
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); | |
1289 | } | |
1290 | ||
1291 | # | |
1292 | # display_format | |
1293 | # ->display_format | |
1294 | # | |
1295 | # Set (get if no argument) the display format for all complex numbers that | |
1296 | # don't happen to have overridden it via ->display_format | |
1297 | # | |
1298 | # When called as an object method, this actually sets the display format for | |
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 { | |
1305 | my $self = shift; | |
1306 | my %display_format = %DISPLAY_FORMAT; | |
1307 | ||
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 | } | |
1313 | } | |
1314 | if (@_ == 1) { | |
1315 | $display_format{style} = shift; | |
1316 | } else { | |
1317 | my %new = @_; | |
1318 | @display_format{keys %new} = values %new; | |
1319 | } | |
1320 | ||
1321 | if (ref $self) { # Called as an object method | |
1322 | $self->{display_format} = { %display_format }; | |
1323 | return | |
1324 | wantarray ? | |
1325 | %{$self->{display_format}} : | |
1326 | $self->{display_format}->{style}; | |
1327 | } | |
1328 | ||
1329 | # Called as a class method | |
1330 | %DISPLAY_FORMAT = %display_format; | |
1331 | return | |
1332 | wantarray ? | |
1333 | %DISPLAY_FORMAT : | |
1334 | $DISPLAY_FORMAT{style}; | |
1335 | } | |
1336 | ||
1337 | # | |
1338 | # (stringify) | |
1339 | # | |
1340 | # Show nicely formatted complex number under its cartesian or polar form, | |
1341 | # depending on the current display format: | |
1342 | # | |
1343 | # . If a specific display format has been recorded for this object, use it. | |
1344 | # . Otherwise, use the generic current default for all complex numbers, | |
1345 | # which is a package global variable. | |
1346 | # | |
1347 | sub stringify { | |
1348 | my ($z) = shift; | |
1349 | ||
1350 | my $style = $z->display_format; | |
1351 | ||
1352 | $style = $DISPLAY_FORMAT{style} unless defined $style; | |
1353 | ||
1354 | return $z->stringify_polar if $style =~ /^p/i; | |
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 | ||
1368 | my %format = $z->display_format; | |
1369 | my $format = $format{format}; | |
1370 | ||
1371 | if ($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) { | |
1386 | if ($y =~ /^(NaN[QS]?)$/i) { | |
1387 | $im = $y; | |
1388 | } else { | |
1389 | if ($y =~ /^-?$Inf$/oi) { | |
1390 | $im = $y; | |
1391 | } else { | |
1392 | $im = | |
1393 | defined $format ? | |
1394 | sprintf($format, $y) : | |
1395 | ($y == 1 ? "" : ($y == -1 ? "-" : $y)); | |
1396 | } | |
1397 | } | |
1398 | $im .= "i"; | |
1399 | } else { | |
1400 | undef $im; | |
1401 | } | |
1402 | ||
1403 | my $str = $re; | |
1404 | ||
1405 | if (defined $im) { | |
1406 | if ($y < 0) { | |
1407 | $str .= $im; | |
1408 | } elsif ($y > 0 || $im =~ /^NaN[QS]?i$/i) { | |
1409 | $str .= "+" if defined $re; | |
1410 | $str .= $im; | |
1411 | } | |
1412 | } elsif (!defined $re) { | |
1413 | $str = "0"; | |
1414 | } | |
1415 | ||
1416 | return $str; | |
1417 | } | |
1418 | ||
1419 | ||
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 | ||
1430 | my %format = $z->display_format; | |
1431 | my $format = $format{format}; | |
1432 | ||
1433 | if ($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) : $t; | |
1439 | } | |
1440 | ||
1441 | return "[$r,$theta]" if defined $theta; | |
1442 | ||
1443 | # | |
1444 | # Try to identify pi/n and friends. | |
1445 | # | |
1446 | ||
1447 | $t -= int(CORE::abs($t) / pit2) * pit2; | |
1448 | ||
1449 | if ($format{polar_pretty_print} && $t) { | |
1450 | my ($a, $b); | |
1451 | for $a (2..9) { | |
1452 | $b = $t * $a / pi; | |
1453 | if ($b =~ /^-?\d+$/) { | |
1454 | $b = $b < 0 ? "-" : "" if CORE::abs($b) == 1; | |
1455 | $theta = "${b}pi/$a"; | |
1456 | last; | |
1457 | } | |
1458 | } | |
1459 | } | |
1460 | ||
1461 | if (defined $format) { | |
1462 | $r = sprintf($format, $r); | |
1463 | $theta = sprintf($format, $theta) unless defined $theta; | |
1464 | } else { | |
1465 | $theta = $t unless defined $theta; | |
1466 | } | |
1467 | ||
1468 | return "[$r,$theta]"; | |
1469 | } | |
1470 | ||
1471 | 1; | |
1472 | __END__ | |
1473 | ||
1474 | =pod | |
1475 | ||
1476 | =head1 NAME | |
1477 | ||
1478 | Math::Complex - complex numbers and associated mathematical functions | |
1479 | ||
1480 | =head1 SYNOPSIS | |
1481 | ||
1482 | use Math::Complex; | |
1483 | ||
1484 | $z = Math::Complex->make(5, 6); | |
1485 | $t = 4 - 3*i + $z; | |
1486 | $j = cplxe(1, 2*pi/3); | |
1487 | ||
1488 | =head1 DESCRIPTION | |
1489 | ||
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 | ||
1560 | z = rho * exp(i * theta) = rho * (cos theta + i * sin theta) | |
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 | |
1570 | suited for additions and subtractions. Real numbers are on the I<x> | |
1571 | axis, and therefore I<theta> is zero or I<pi>. | |
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 | |
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+>). | |
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 | ||
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 | |
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. | |
1601 | The C<sqrt> returns only one of the solutions: if you want the both, | |
1602 | use the C<root> function. | |
1603 | ||
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?). | |
1608 | ||
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 | |
1611 | with a horizontal bar above the number, or C<~z> here. | |
1612 | ||
1613 | z = a + bi | |
1614 | ~z = a - bi | |
1615 | ||
1616 | Simple... Now look: | |
1617 | ||
1618 | z * ~z = (a + bi) * (a - bi) = a*a + b*b | |
1619 | ||
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: | |
1622 | ||
1623 | rho = abs(z) = sqrt(a*a + b*b) | |
1624 | ||
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) | |
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)) | |
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 | |
1670 | abs(z) = r | |
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) | |
1677 | ||
1678 | csc(z) = 1 / sin(z) | |
1679 | sec(z) = 1 / cos(z) | |
1680 | cot(z) = 1 / tan(z) | |
1681 | ||
1682 | asin(z) = -i * log(i*z + sqrt(1-z*z)) | |
1683 | acos(z) = -i * log(z + i*sqrt(1-z*z)) | |
1684 | atan(z) = i/2 * log((i+z) / (i-z)) | |
1685 | ||
1686 | acsc(z) = asin(1 / z) | |
1687 | asec(z) = acos(1 / z) | |
1688 | acot(z) = atan(1 / z) = -i/2 * log((i+z) / (z-i)) | |
1689 | ||
1690 | sinh(z) = 1/2 (exp(z) - exp(-z)) | |
1691 | cosh(z) = 1/2 (exp(z) + exp(-z)) | |
1692 | tanh(z) = sinh(z) / cosh(z) = (exp(z) - exp(-z)) / (exp(z) + exp(-z)) | |
1693 | ||
1694 | csch(z) = 1 / sinh(z) | |
1695 | sech(z) = 1 / cosh(z) | |
1696 | coth(z) = 1 / tanh(z) | |
1697 | ||
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)) | |
1701 | ||
1702 | acsch(z) = asinh(1 / z) | |
1703 | asech(z) = acosh(1 / z) | |
1704 | acoth(z) = atanh(1 / z) = 1/2 * log((1+z) / (z-1)) | |
1705 | ||
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>, | |
1710 | C<rho>, and C<theta> can be used also as mutators. The C<cbrt> | |
1711 | returns only one of the solutions: if you want all three, use the | |
1712 | C<root> function. | |
1713 | ||
1714 | The I<root> function is available to compute all the I<n> | |
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 | ||
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. | |
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 | ||
1746 | if you like. To create a number using the polar form, use either: | |
1747 | ||
1748 | $z = Math::Complex->emake(5, pi/3); | |
1749 | $x = cplxe(5, pi/3); | |
1750 | ||
1751 | instead. The first argument is the modulus, the second is the angle | |
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). | |
1754 | ||
1755 | It is possible to write: | |
1756 | ||
1757 | $x = cplxe(-3, pi/4); | |
1758 | ||
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). | |
1762 | ||
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 | |
1765 | the argument will be used. | |
1766 | ||
1767 | $z1 = cplx(-2, 1); | |
1768 | $z2 = cplx($z1, 4); | |
1769 | ||
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 | ||
1782 | =head1 STRINGIFICATION | |
1783 | ||
1784 | When printed, a complex number is usually shown under its cartesian | |
1785 | style I<a+bi>, but there are legitimate cases where the polar style | |
1786 | I<[r,t]> is more appropriate. | |
1787 | ||
1788 | By calling the class method C<Math::Complex::display_format> and | |
1789 | supplying either C<"polar"> or C<"cartesian"> as an argument, you | |
1790 | override the default display style, which is C<"cartesian">. Not | |
1791 | supplying any argument returns the current settings. | |
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 | |
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. | |
1797 | ||
1798 | For instance: | |
1799 | ||
1800 | use Math::Complex; | |
1801 | ||
1802 | Math::Complex::display_format('polar'); | |
1803 | $j = (root(1, 3))[1]; | |
1804 | print "j = $j\n"; # Prints "j = [1,2pi/3]" | |
1805 | $j->display_format('cartesian'); | |
1806 | print "j = $j\n"; # Prints "j = -0.5+0.866025403784439i" | |
1807 | ||
1808 | The polar style attempts to emphasize arguments like I<k*pi/n> | |
1809 | (where I<n> is a positive integer and I<k> an integer within [-9, +9]), | |
1810 | this is called I<polar pretty-printing>. | |
1811 | ||
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 | |
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"); | |
1826 | ||
1827 | There are two new display parameters. | |
1828 | ||
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>. | |
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" | |
1838 | $j->display_format('format' => undef); | |
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 | |
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. | |
1846 | ||
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. | |
1850 | ||
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"; | |
1869 | ||
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 | |
1875 | ||
1876 | The division (/) and the following functions | |
1877 | ||
1878 | log ln log10 logn | |
1879 | tan sec csc cot | |
1880 | atan asec acsc acot | |
1881 | tanh sech csch coth | |
1882 | atanh asech acsch acoth | |
1883 | ||
1884 | cannot be computed for all arguments because that would mean dividing | |
1885 | by zero or taking logarithm of zero. These situations cause fatal | |
1886 | runtime errors looking like this | |
1887 | ||
1888 | cot(0): Division by zero. | |
1889 | (Because in the definition of cot(0), the divisor sin(0) is 0) | |
1890 | Died at ... | |
1891 | ||
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>, | |
1898 | C<asech>, C<acsch>, the argument cannot be C<0> (zero). For the | |
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 | |
1909 | listed above. | |
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 ... | |
1921 | ||
1922 | =head1 BUGS | |
1923 | ||
1924 | Saying C<use Math::Complex;> exports many mathematical routines in the | |
1925 | caller environment and even overrides some (C<sqrt>, C<log>). | |
1926 | This is construed as a feature by the Authors, actually... ;-) | |
1927 | ||
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. | |
1931 | ||
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 | ||
1937 | =head1 AUTHORS | |
1938 | ||
1939 | Daniel S. Lewart <F<d-lewart@uiuc.edu>> | |
1940 | ||
1941 | Original authors Raphael Manfredi <F<Raphael_Manfredi@pobox.com>> and | |
1942 | Jarkko Hietaniemi <F<jhi@iki.fi>> | |
1943 | ||
1944 | =cut | |
1945 | ||
1946 | 1; | |
1947 | ||
1948 | # eof |