Commit | Line | Data |
---|---|---|
920dae64 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 | use vars qw($VERSION @ISA @EXPORT @EXPORT_OK %EXPORT_TAGS $Inf); | |
11 | ||
12 | $VERSION = 1.35; | |
13 | ||
14 | BEGIN { | |
15 | unless ($^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 | # These days we could use Scalar::Util::lln(), I guess. | |
42 | my $gre = qr'\s*([\+\-]?(?:(?:(?:\d+(?:_\d+)*(?:\.\d*(?:_\d+)*)?|\.\d+(?:_\d+)*)(?:[eE][\+\-]?\d+(?:_\d+)*)?))|inf)'i; | |
43 | ||
44 | require Exporter; | |
45 | ||
46 | @ISA = qw(Exporter); | |
47 | ||
48 | my @trig = qw( | |
49 | pi | |
50 | tan | |
51 | csc cosec sec cot cotan | |
52 | asin acos atan | |
53 | acsc acosec asec acot acotan | |
54 | sinh cosh tanh | |
55 | csch cosech sech coth cotanh | |
56 | asinh acosh atanh | |
57 | acsch acosech asech acoth acotanh | |
58 | ); | |
59 | ||
60 | @EXPORT = (qw( | |
61 | i Re Im rho theta arg | |
62 | sqrt log ln | |
63 | log10 logn cbrt root | |
64 | cplx cplxe | |
65 | atan2 | |
66 | ), | |
67 | @trig); | |
68 | ||
69 | @EXPORT_OK = qw(decplx); | |
70 | ||
71 | %EXPORT_TAGS = ( | |
72 | 'trig' => [@trig], | |
73 | ); | |
74 | ||
75 | use overload | |
76 | '+' => \&plus, | |
77 | '-' => \&minus, | |
78 | '*' => \&multiply, | |
79 | '/' => \÷, | |
80 | '**' => \&power, | |
81 | '==' => \&numeq, | |
82 | '<=>' => \&spaceship, | |
83 | 'neg' => \&negate, | |
84 | '~' => \&conjugate, | |
85 | 'abs' => \&abs, | |
86 | 'sqrt' => \&sqrt, | |
87 | 'exp' => \&exp, | |
88 | 'log' => \&log, | |
89 | 'sin' => \&sin, | |
90 | 'cos' => \&cos, | |
91 | 'tan' => \&tan, | |
92 | 'atan2' => \&atan2, | |
93 | qw("" stringify); | |
94 | ||
95 | # | |
96 | # Package "privates" | |
97 | # | |
98 | ||
99 | my %DISPLAY_FORMAT = ('style' => 'cartesian', | |
100 | 'polar_pretty_print' => 1); | |
101 | my $eps = 1e-14; # Epsilon | |
102 | ||
103 | # | |
104 | # Object attributes (internal): | |
105 | # cartesian [real, imaginary] -- cartesian form | |
106 | # polar [rho, theta] -- polar form | |
107 | # c_dirty cartesian form not up-to-date | |
108 | # p_dirty polar form not up-to-date | |
109 | # display display format (package's global when not set) | |
110 | # | |
111 | ||
112 | # Die on bad *make() arguments. | |
113 | ||
114 | sub _cannot_make { | |
115 | die "@{[(caller(1))[3]]}: Cannot take $_[0] of '$_[1]'.\n"; | |
116 | } | |
117 | ||
118 | sub _make { | |
119 | my $arg = shift; | |
120 | my ($p, $q); | |
121 | ||
122 | if ($arg =~ /^$gre$/) { | |
123 | ($p, $q) = ($1, 0); | |
124 | } elsif ($arg =~ /^(?:$gre)?$gre\s*i\s*$/) { | |
125 | ($p, $q) = ($1 || 0, $2); | |
126 | } elsif ($arg =~ /^\s*\(\s*$gre\s*(?:,\s*$gre\s*)?\)\s*$/) { | |
127 | ($p, $q) = ($1, $2 || 0); | |
128 | } | |
129 | ||
130 | if (defined $p) { | |
131 | $p =~ s/^\+//; | |
132 | $p =~ s/^(-?)inf$/"${1}9**9**9"/e; | |
133 | $q =~ s/^\+//; | |
134 | $q =~ s/^(-?)inf$/"${1}9**9**9"/e; | |
135 | } | |
136 | ||
137 | return ($p, $q); | |
138 | } | |
139 | ||
140 | sub _emake { | |
141 | my $arg = shift; | |
142 | my ($p, $q); | |
143 | ||
144 | if ($arg =~ /^\s*\[\s*$gre\s*(?:,\s*$gre\s*)?\]\s*$/) { | |
145 | ($p, $q) = ($1, $2 || 0); | |
146 | } elsif ($arg =~ m!^\s*\[\s*$gre\s*(?:,\s*([-+]?\d*\s*)?pi(?:/\s*(\d+))?\s*)?\]\s*$!) { | |
147 | ($p, $q) = ($1, ($2 eq '-' ? -1 : ($2 || 1)) * pi() / ($3 || 1)); | |
148 | } elsif ($arg =~ /^\s*\[\s*$gre\s*\]\s*$/) { | |
149 | ($p, $q) = ($1, 0); | |
150 | } elsif ($arg =~ /^\s*$gre\s*$/) { | |
151 | ($p, $q) = ($1, 0); | |
152 | } | |
153 | ||
154 | if (defined $p) { | |
155 | $p =~ s/^\+//; | |
156 | $q =~ s/^\+//; | |
157 | $p =~ s/^(-?)inf$/"${1}9**9**9"/e; | |
158 | $q =~ s/^(-?)inf$/"${1}9**9**9"/e; | |
159 | } | |
160 | ||
161 | return ($p, $q); | |
162 | } | |
163 | ||
164 | # | |
165 | # ->make | |
166 | # | |
167 | # Create a new complex number (cartesian form) | |
168 | # | |
169 | sub make { | |
170 | my $self = bless {}, shift; | |
171 | my ($re, $im); | |
172 | if (@_ == 0) { | |
173 | ($re, $im) = (0, 0); | |
174 | } elsif (@_ == 1) { | |
175 | return (ref $self)->emake($_[0]) | |
176 | if ($_[0] =~ /^\s*\[/); | |
177 | ($re, $im) = _make($_[0]); | |
178 | } elsif (@_ == 2) { | |
179 | ($re, $im) = @_; | |
180 | } | |
181 | if (defined $re) { | |
182 | _cannot_make("real part", $re) unless $re =~ /^$gre$/; | |
183 | } | |
184 | $im ||= 0; | |
185 | _cannot_make("imaginary part", $im) unless $im =~ /^$gre$/; | |
186 | $self->set_cartesian([$re, $im ]); | |
187 | $self->display_format('cartesian'); | |
188 | ||
189 | return $self; | |
190 | } | |
191 | ||
192 | # | |
193 | # ->emake | |
194 | # | |
195 | # Create a new complex number (exponential form) | |
196 | # | |
197 | sub emake { | |
198 | my $self = bless {}, shift; | |
199 | my ($rho, $theta); | |
200 | if (@_ == 0) { | |
201 | ($rho, $theta) = (0, 0); | |
202 | } elsif (@_ == 1) { | |
203 | return (ref $self)->make($_[0]) | |
204 | if ($_[0] =~ /^\s*\(/ || $_[0] =~ /i\s*$/); | |
205 | ($rho, $theta) = _emake($_[0]); | |
206 | } elsif (@_ == 2) { | |
207 | ($rho, $theta) = @_; | |
208 | } | |
209 | if (defined $rho && defined $theta) { | |
210 | if ($rho < 0) { | |
211 | $rho = -$rho; | |
212 | $theta = ($theta <= 0) ? $theta + pi() : $theta - pi(); | |
213 | } | |
214 | } | |
215 | if (defined $rho) { | |
216 | _cannot_make("rho", $rho) unless $rho =~ /^$gre$/; | |
217 | } | |
218 | $theta ||= 0; | |
219 | _cannot_make("theta", $theta) unless $theta =~ /^$gre$/; | |
220 | $self->set_polar([$rho, $theta]); | |
221 | $self->display_format('polar'); | |
222 | ||
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]->{c_dirty} = 0; | |
316 | $_[0]->{'cartesian'} = $_[1] } | |
317 | sub set_polar { $_[0]->{c_dirty}++; $_[0]->{p_dirty} = 0; | |
318 | $_[0]->{'polar'} = $_[1] } | |
319 | ||
320 | # | |
321 | # ->update_cartesian | |
322 | # | |
323 | # Recompute and return the cartesian form, given accurate polar form. | |
324 | # | |
325 | sub update_cartesian { | |
326 | my $self = shift; | |
327 | my ($r, $t) = @{$self->{'polar'}}; | |
328 | $self->{c_dirty} = 0; | |
329 | return $self->{'cartesian'} = [$r * CORE::cos($t), $r * CORE::sin($t)]; | |
330 | } | |
331 | ||
332 | # | |
333 | # | |
334 | # ->update_polar | |
335 | # | |
336 | # Recompute and return the polar form, given accurate cartesian form. | |
337 | # | |
338 | sub update_polar { | |
339 | my $self = shift; | |
340 | my ($x, $y) = @{$self->{'cartesian'}}; | |
341 | $self->{p_dirty} = 0; | |
342 | return $self->{'polar'} = [0, 0] if $x == 0 && $y == 0; | |
343 | return $self->{'polar'} = [CORE::sqrt($x*$x + $y*$y), | |
344 | CORE::atan2($y, $x)]; | |
345 | } | |
346 | ||
347 | # | |
348 | # (plus) | |
349 | # | |
350 | # Computes z1+z2. | |
351 | # | |
352 | sub plus { | |
353 | my ($z1, $z2, $regular) = @_; | |
354 | my ($re1, $im1) = @{$z1->cartesian}; | |
355 | $z2 = cplx($z2) unless ref $z2; | |
356 | my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0); | |
357 | unless (defined $regular) { | |
358 | $z1->set_cartesian([$re1 + $re2, $im1 + $im2]); | |
359 | return $z1; | |
360 | } | |
361 | return (ref $z1)->make($re1 + $re2, $im1 + $im2); | |
362 | } | |
363 | ||
364 | # | |
365 | # (minus) | |
366 | # | |
367 | # Computes z1-z2. | |
368 | # | |
369 | sub minus { | |
370 | my ($z1, $z2, $inverted) = @_; | |
371 | my ($re1, $im1) = @{$z1->cartesian}; | |
372 | $z2 = cplx($z2) unless ref $z2; | |
373 | my ($re2, $im2) = @{$z2->cartesian}; | |
374 | unless (defined $inverted) { | |
375 | $z1->set_cartesian([$re1 - $re2, $im1 - $im2]); | |
376 | return $z1; | |
377 | } | |
378 | return $inverted ? | |
379 | (ref $z1)->make($re2 - $re1, $im2 - $im1) : | |
380 | (ref $z1)->make($re1 - $re2, $im1 - $im2); | |
381 | ||
382 | } | |
383 | ||
384 | # | |
385 | # (multiply) | |
386 | # | |
387 | # Computes z1*z2. | |
388 | # | |
389 | sub multiply { | |
390 | my ($z1, $z2, $regular) = @_; | |
391 | if ($z1->{p_dirty} == 0 and ref $z2 and $z2->{p_dirty} == 0) { | |
392 | # if both polar better use polar to avoid rounding errors | |
393 | my ($r1, $t1) = @{$z1->polar}; | |
394 | my ($r2, $t2) = @{$z2->polar}; | |
395 | my $t = $t1 + $t2; | |
396 | if ($t > pi()) { $t -= pit2 } | |
397 | elsif ($t <= -pi()) { $t += pit2 } | |
398 | unless (defined $regular) { | |
399 | $z1->set_polar([$r1 * $r2, $t]); | |
400 | return $z1; | |
401 | } | |
402 | return (ref $z1)->emake($r1 * $r2, $t); | |
403 | } else { | |
404 | my ($x1, $y1) = @{$z1->cartesian}; | |
405 | if (ref $z2) { | |
406 | my ($x2, $y2) = @{$z2->cartesian}; | |
407 | return (ref $z1)->make($x1*$x2-$y1*$y2, $x1*$y2+$y1*$x2); | |
408 | } else { | |
409 | return (ref $z1)->make($x1*$z2, $y1*$z2); | |
410 | } | |
411 | } | |
412 | } | |
413 | ||
414 | # | |
415 | # _divbyzero | |
416 | # | |
417 | # Die on division by zero. | |
418 | # | |
419 | sub _divbyzero { | |
420 | my $mess = "$_[0]: Division by zero.\n"; | |
421 | ||
422 | if (defined $_[1]) { | |
423 | $mess .= "(Because in the definition of $_[0], the divisor "; | |
424 | $mess .= "$_[1] " unless ("$_[1]" eq '0'); | |
425 | $mess .= "is 0)\n"; | |
426 | } | |
427 | ||
428 | my @up = caller(1); | |
429 | ||
430 | $mess .= "Died at $up[1] line $up[2].\n"; | |
431 | ||
432 | die $mess; | |
433 | } | |
434 | ||
435 | # | |
436 | # (divide) | |
437 | # | |
438 | # Computes z1/z2. | |
439 | # | |
440 | sub divide { | |
441 | my ($z1, $z2, $inverted) = @_; | |
442 | if ($z1->{p_dirty} == 0 and ref $z2 and $z2->{p_dirty} == 0) { | |
443 | # if both polar better use polar to avoid rounding errors | |
444 | my ($r1, $t1) = @{$z1->polar}; | |
445 | my ($r2, $t2) = @{$z2->polar}; | |
446 | my $t; | |
447 | if ($inverted) { | |
448 | _divbyzero "$z2/0" if ($r1 == 0); | |
449 | $t = $t2 - $t1; | |
450 | if ($t > pi()) { $t -= pit2 } | |
451 | elsif ($t <= -pi()) { $t += pit2 } | |
452 | return (ref $z1)->emake($r2 / $r1, $t); | |
453 | } else { | |
454 | _divbyzero "$z1/0" if ($r2 == 0); | |
455 | $t = $t1 - $t2; | |
456 | if ($t > pi()) { $t -= pit2 } | |
457 | elsif ($t <= -pi()) { $t += pit2 } | |
458 | return (ref $z1)->emake($r1 / $r2, $t); | |
459 | } | |
460 | } else { | |
461 | my ($d, $x2, $y2); | |
462 | if ($inverted) { | |
463 | ($x2, $y2) = @{$z1->cartesian}; | |
464 | $d = $x2*$x2 + $y2*$y2; | |
465 | _divbyzero "$z2/0" if $d == 0; | |
466 | return (ref $z1)->make(($x2*$z2)/$d, -($y2*$z2)/$d); | |
467 | } else { | |
468 | my ($x1, $y1) = @{$z1->cartesian}; | |
469 | if (ref $z2) { | |
470 | ($x2, $y2) = @{$z2->cartesian}; | |
471 | $d = $x2*$x2 + $y2*$y2; | |
472 | _divbyzero "$z1/0" if $d == 0; | |
473 | my $u = ($x1*$x2 + $y1*$y2)/$d; | |
474 | my $v = ($y1*$x2 - $x1*$y2)/$d; | |
475 | return (ref $z1)->make($u, $v); | |
476 | } else { | |
477 | _divbyzero "$z1/0" if $z2 == 0; | |
478 | return (ref $z1)->make($x1/$z2, $y1/$z2); | |
479 | } | |
480 | } | |
481 | } | |
482 | } | |
483 | ||
484 | # | |
485 | # (power) | |
486 | # | |
487 | # Computes z1**z2 = exp(z2 * log z1)). | |
488 | # | |
489 | sub power { | |
490 | my ($z1, $z2, $inverted) = @_; | |
491 | if ($inverted) { | |
492 | return 1 if $z1 == 0 || $z2 == 1; | |
493 | return 0 if $z2 == 0 && Re($z1) > 0; | |
494 | } else { | |
495 | return 1 if $z2 == 0 || $z1 == 1; | |
496 | return 0 if $z1 == 0 && Re($z2) > 0; | |
497 | } | |
498 | my $w = $inverted ? &exp($z1 * &log($z2)) | |
499 | : &exp($z2 * &log($z1)); | |
500 | # If both arguments cartesian, return cartesian, else polar. | |
501 | return $z1->{c_dirty} == 0 && | |
502 | (not ref $z2 or $z2->{c_dirty} == 0) ? | |
503 | cplx(@{$w->cartesian}) : $w; | |
504 | } | |
505 | ||
506 | # | |
507 | # (spaceship) | |
508 | # | |
509 | # Computes z1 <=> z2. | |
510 | # Sorts on the real part first, then on the imaginary part. Thus 2-4i < 3+8i. | |
511 | # | |
512 | sub spaceship { | |
513 | my ($z1, $z2, $inverted) = @_; | |
514 | my ($re1, $im1) = ref $z1 ? @{$z1->cartesian} : ($z1, 0); | |
515 | my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0); | |
516 | my $sgn = $inverted ? -1 : 1; | |
517 | return $sgn * ($re1 <=> $re2) if $re1 != $re2; | |
518 | return $sgn * ($im1 <=> $im2); | |
519 | } | |
520 | ||
521 | # | |
522 | # (numeq) | |
523 | # | |
524 | # Computes z1 == z2. | |
525 | # | |
526 | # (Required in addition to spaceship() because of NaNs.) | |
527 | sub numeq { | |
528 | my ($z1, $z2, $inverted) = @_; | |
529 | my ($re1, $im1) = ref $z1 ? @{$z1->cartesian} : ($z1, 0); | |
530 | my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0); | |
531 | return $re1 == $re2 && $im1 == $im2 ? 1 : 0; | |
532 | } | |
533 | ||
534 | # | |
535 | # (negate) | |
536 | # | |
537 | # Computes -z. | |
538 | # | |
539 | sub negate { | |
540 | my ($z) = @_; | |
541 | if ($z->{c_dirty}) { | |
542 | my ($r, $t) = @{$z->polar}; | |
543 | $t = ($t <= 0) ? $t + pi : $t - pi; | |
544 | return (ref $z)->emake($r, $t); | |
545 | } | |
546 | my ($re, $im) = @{$z->cartesian}; | |
547 | return (ref $z)->make(-$re, -$im); | |
548 | } | |
549 | ||
550 | # | |
551 | # (conjugate) | |
552 | # | |
553 | # Compute complex's conjugate. | |
554 | # | |
555 | sub conjugate { | |
556 | my ($z) = @_; | |
557 | if ($z->{c_dirty}) { | |
558 | my ($r, $t) = @{$z->polar}; | |
559 | return (ref $z)->emake($r, -$t); | |
560 | } | |
561 | my ($re, $im) = @{$z->cartesian}; | |
562 | return (ref $z)->make($re, -$im); | |
563 | } | |
564 | ||
565 | # | |
566 | # (abs) | |
567 | # | |
568 | # Compute or set complex's norm (rho). | |
569 | # | |
570 | sub abs { | |
571 | my ($z, $rho) = @_; | |
572 | unless (ref $z) { | |
573 | if (@_ == 2) { | |
574 | $_[0] = $_[1]; | |
575 | } else { | |
576 | return CORE::abs($z); | |
577 | } | |
578 | } | |
579 | if (defined $rho) { | |
580 | $z->{'polar'} = [ $rho, ${$z->polar}[1] ]; | |
581 | $z->{p_dirty} = 0; | |
582 | $z->{c_dirty} = 1; | |
583 | return $rho; | |
584 | } else { | |
585 | return ${$z->polar}[0]; | |
586 | } | |
587 | } | |
588 | ||
589 | sub _theta { | |
590 | my $theta = $_[0]; | |
591 | ||
592 | if ($$theta > pi()) { $$theta -= pit2 } | |
593 | elsif ($$theta <= -pi()) { $$theta += pit2 } | |
594 | } | |
595 | ||
596 | # | |
597 | # arg | |
598 | # | |
599 | # Compute or set complex's argument (theta). | |
600 | # | |
601 | sub arg { | |
602 | my ($z, $theta) = @_; | |
603 | return $z unless ref $z; | |
604 | if (defined $theta) { | |
605 | _theta(\$theta); | |
606 | $z->{'polar'} = [ ${$z->polar}[0], $theta ]; | |
607 | $z->{p_dirty} = 0; | |
608 | $z->{c_dirty} = 1; | |
609 | } else { | |
610 | $theta = ${$z->polar}[1]; | |
611 | _theta(\$theta); | |
612 | } | |
613 | return $theta; | |
614 | } | |
615 | ||
616 | # | |
617 | # (sqrt) | |
618 | # | |
619 | # Compute sqrt(z). | |
620 | # | |
621 | # It is quite tempting to use wantarray here so that in list context | |
622 | # sqrt() would return the two solutions. This, however, would | |
623 | # break things like | |
624 | # | |
625 | # print "sqrt(z) = ", sqrt($z), "\n"; | |
626 | # | |
627 | # The two values would be printed side by side without no intervening | |
628 | # whitespace, quite confusing. | |
629 | # Therefore if you want the two solutions use the root(). | |
630 | # | |
631 | sub sqrt { | |
632 | my ($z) = @_; | |
633 | my ($re, $im) = ref $z ? @{$z->cartesian} : ($z, 0); | |
634 | return $re < 0 ? cplx(0, CORE::sqrt(-$re)) : CORE::sqrt($re) | |
635 | if $im == 0; | |
636 | my ($r, $t) = @{$z->polar}; | |
637 | return (ref $z)->emake(CORE::sqrt($r), $t/2); | |
638 | } | |
639 | ||
640 | # | |
641 | # cbrt | |
642 | # | |
643 | # Compute cbrt(z) (cubic root). | |
644 | # | |
645 | # Why are we not returning three values? The same answer as for sqrt(). | |
646 | # | |
647 | sub cbrt { | |
648 | my ($z) = @_; | |
649 | return $z < 0 ? | |
650 | -CORE::exp(CORE::log(-$z)/3) : | |
651 | ($z > 0 ? CORE::exp(CORE::log($z)/3): 0) | |
652 | unless ref $z; | |
653 | my ($r, $t) = @{$z->polar}; | |
654 | return 0 if $r == 0; | |
655 | return (ref $z)->emake(CORE::exp(CORE::log($r)/3), $t/3); | |
656 | } | |
657 | ||
658 | # | |
659 | # _rootbad | |
660 | # | |
661 | # Die on bad root. | |
662 | # | |
663 | sub _rootbad { | |
664 | my $mess = "Root '$_[0]' illegal, root rank must be positive integer.\n"; | |
665 | ||
666 | my @up = caller(1); | |
667 | ||
668 | $mess .= "Died at $up[1] line $up[2].\n"; | |
669 | ||
670 | die $mess; | |
671 | } | |
672 | ||
673 | # | |
674 | # root | |
675 | # | |
676 | # Computes all nth root for z, returning an array whose size is n. | |
677 | # `n' must be a positive integer. | |
678 | # | |
679 | # The roots are given by (for k = 0..n-1): | |
680 | # | |
681 | # z^(1/n) = r^(1/n) (cos ((t+2 k pi)/n) + i sin ((t+2 k pi)/n)) | |
682 | # | |
683 | sub root { | |
684 | my ($z, $n, $k) = @_; | |
685 | _rootbad($n) if ($n < 1 or int($n) != $n); | |
686 | my ($r, $t) = ref $z ? | |
687 | @{$z->polar} : (CORE::abs($z), $z >= 0 ? 0 : pi); | |
688 | my $theta_inc = pit2 / $n; | |
689 | my $rho = $r ** (1/$n); | |
690 | my $cartesian = ref $z && $z->{c_dirty} == 0; | |
691 | if (@_ == 2) { | |
692 | my @root; | |
693 | for (my $i = 0, my $theta = $t / $n; | |
694 | $i < $n; | |
695 | $i++, $theta += $theta_inc) { | |
696 | my $w = cplxe($rho, $theta); | |
697 | # Yes, $cartesian is loop invariant. | |
698 | push @root, $cartesian ? cplx(@{$w->cartesian}) : $w; | |
699 | } | |
700 | return @root; | |
701 | } elsif (@_ == 3) { | |
702 | my $w = cplxe($rho, $t / $n + $k * $theta_inc); | |
703 | return $cartesian ? cplx(@{$w->cartesian}) : $w; | |
704 | } | |
705 | } | |
706 | ||
707 | # | |
708 | # Re | |
709 | # | |
710 | # Return or set Re(z). | |
711 | # | |
712 | sub Re { | |
713 | my ($z, $Re) = @_; | |
714 | return $z unless ref $z; | |
715 | if (defined $Re) { | |
716 | $z->{'cartesian'} = [ $Re, ${$z->cartesian}[1] ]; | |
717 | $z->{c_dirty} = 0; | |
718 | $z->{p_dirty} = 1; | |
719 | } else { | |
720 | return ${$z->cartesian}[0]; | |
721 | } | |
722 | } | |
723 | ||
724 | # | |
725 | # Im | |
726 | # | |
727 | # Return or set Im(z). | |
728 | # | |
729 | sub Im { | |
730 | my ($z, $Im) = @_; | |
731 | return 0 unless ref $z; | |
732 | if (defined $Im) { | |
733 | $z->{'cartesian'} = [ ${$z->cartesian}[0], $Im ]; | |
734 | $z->{c_dirty} = 0; | |
735 | $z->{p_dirty} = 1; | |
736 | } else { | |
737 | return ${$z->cartesian}[1]; | |
738 | } | |
739 | } | |
740 | ||
741 | # | |
742 | # rho | |
743 | # | |
744 | # Return or set rho(w). | |
745 | # | |
746 | sub rho { | |
747 | Math::Complex::abs(@_); | |
748 | } | |
749 | ||
750 | # | |
751 | # theta | |
752 | # | |
753 | # Return or set theta(w). | |
754 | # | |
755 | sub theta { | |
756 | Math::Complex::arg(@_); | |
757 | } | |
758 | ||
759 | # | |
760 | # (exp) | |
761 | # | |
762 | # Computes exp(z). | |
763 | # | |
764 | sub exp { | |
765 | my ($z) = @_; | |
766 | my ($x, $y) = @{$z->cartesian}; | |
767 | return (ref $z)->emake(CORE::exp($x), $y); | |
768 | } | |
769 | ||
770 | # | |
771 | # _logofzero | |
772 | # | |
773 | # Die on logarithm of zero. | |
774 | # | |
775 | sub _logofzero { | |
776 | my $mess = "$_[0]: Logarithm of zero.\n"; | |
777 | ||
778 | if (defined $_[1]) { | |
779 | $mess .= "(Because in the definition of $_[0], the argument "; | |
780 | $mess .= "$_[1] " unless ($_[1] eq '0'); | |
781 | $mess .= "is 0)\n"; | |
782 | } | |
783 | ||
784 | my @up = caller(1); | |
785 | ||
786 | $mess .= "Died at $up[1] line $up[2].\n"; | |
787 | ||
788 | die $mess; | |
789 | } | |
790 | ||
791 | # | |
792 | # (log) | |
793 | # | |
794 | # Compute log(z). | |
795 | # | |
796 | sub log { | |
797 | my ($z) = @_; | |
798 | unless (ref $z) { | |
799 | _logofzero("log") if $z == 0; | |
800 | return $z > 0 ? CORE::log($z) : cplx(CORE::log(-$z), pi); | |
801 | } | |
802 | my ($r, $t) = @{$z->polar}; | |
803 | _logofzero("log") if $r == 0; | |
804 | if ($t > pi()) { $t -= pit2 } | |
805 | elsif ($t <= -pi()) { $t += pit2 } | |
806 | return (ref $z)->make(CORE::log($r), $t); | |
807 | } | |
808 | ||
809 | # | |
810 | # ln | |
811 | # | |
812 | # Alias for log(). | |
813 | # | |
814 | sub ln { Math::Complex::log(@_) } | |
815 | ||
816 | # | |
817 | # log10 | |
818 | # | |
819 | # Compute log10(z). | |
820 | # | |
821 | ||
822 | sub log10 { | |
823 | return Math::Complex::log($_[0]) * uplog10; | |
824 | } | |
825 | ||
826 | # | |
827 | # logn | |
828 | # | |
829 | # Compute logn(z,n) = log(z) / log(n) | |
830 | # | |
831 | sub logn { | |
832 | my ($z, $n) = @_; | |
833 | $z = cplx($z, 0) unless ref $z; | |
834 | my $logn = $LOGN{$n}; | |
835 | $logn = $LOGN{$n} = CORE::log($n) unless defined $logn; # Cache log(n) | |
836 | return &log($z) / $logn; | |
837 | } | |
838 | ||
839 | # | |
840 | # (cos) | |
841 | # | |
842 | # Compute cos(z) = (exp(iz) + exp(-iz))/2. | |
843 | # | |
844 | sub cos { | |
845 | my ($z) = @_; | |
846 | return CORE::cos($z) unless ref $z; | |
847 | my ($x, $y) = @{$z->cartesian}; | |
848 | my $ey = CORE::exp($y); | |
849 | my $sx = CORE::sin($x); | |
850 | my $cx = CORE::cos($x); | |
851 | my $ey_1 = $ey ? 1 / $ey : $Inf; | |
852 | return (ref $z)->make($cx * ($ey + $ey_1)/2, | |
853 | $sx * ($ey_1 - $ey)/2); | |
854 | } | |
855 | ||
856 | # | |
857 | # (sin) | |
858 | # | |
859 | # Compute sin(z) = (exp(iz) - exp(-iz))/2. | |
860 | # | |
861 | sub sin { | |
862 | my ($z) = @_; | |
863 | return CORE::sin($z) unless ref $z; | |
864 | my ($x, $y) = @{$z->cartesian}; | |
865 | my $ey = CORE::exp($y); | |
866 | my $sx = CORE::sin($x); | |
867 | my $cx = CORE::cos($x); | |
868 | my $ey_1 = $ey ? 1 / $ey : $Inf; | |
869 | return (ref $z)->make($sx * ($ey + $ey_1)/2, | |
870 | $cx * ($ey - $ey_1)/2); | |
871 | } | |
872 | ||
873 | # | |
874 | # tan | |
875 | # | |
876 | # Compute tan(z) = sin(z) / cos(z). | |
877 | # | |
878 | sub tan { | |
879 | my ($z) = @_; | |
880 | my $cz = &cos($z); | |
881 | _divbyzero "tan($z)", "cos($z)" if $cz == 0; | |
882 | return &sin($z) / $cz; | |
883 | } | |
884 | ||
885 | # | |
886 | # sec | |
887 | # | |
888 | # Computes the secant sec(z) = 1 / cos(z). | |
889 | # | |
890 | sub sec { | |
891 | my ($z) = @_; | |
892 | my $cz = &cos($z); | |
893 | _divbyzero "sec($z)", "cos($z)" if ($cz == 0); | |
894 | return 1 / $cz; | |
895 | } | |
896 | ||
897 | # | |
898 | # csc | |
899 | # | |
900 | # Computes the cosecant csc(z) = 1 / sin(z). | |
901 | # | |
902 | sub csc { | |
903 | my ($z) = @_; | |
904 | my $sz = &sin($z); | |
905 | _divbyzero "csc($z)", "sin($z)" if ($sz == 0); | |
906 | return 1 / $sz; | |
907 | } | |
908 | ||
909 | # | |
910 | # cosec | |
911 | # | |
912 | # Alias for csc(). | |
913 | # | |
914 | sub cosec { Math::Complex::csc(@_) } | |
915 | ||
916 | # | |
917 | # cot | |
918 | # | |
919 | # Computes cot(z) = cos(z) / sin(z). | |
920 | # | |
921 | sub cot { | |
922 | my ($z) = @_; | |
923 | my $sz = &sin($z); | |
924 | _divbyzero "cot($z)", "sin($z)" if ($sz == 0); | |
925 | return &cos($z) / $sz; | |
926 | } | |
927 | ||
928 | # | |
929 | # cotan | |
930 | # | |
931 | # Alias for cot(). | |
932 | # | |
933 | sub cotan { Math::Complex::cot(@_) } | |
934 | ||
935 | # | |
936 | # acos | |
937 | # | |
938 | # Computes the arc cosine acos(z) = -i log(z + sqrt(z*z-1)). | |
939 | # | |
940 | sub acos { | |
941 | my $z = $_[0]; | |
942 | return CORE::atan2(CORE::sqrt(1-$z*$z), $z) | |
943 | if (! ref $z) && CORE::abs($z) <= 1; | |
944 | $z = cplx($z, 0) unless ref $z; | |
945 | my ($x, $y) = @{$z->cartesian}; | |
946 | return 0 if $x == 1 && $y == 0; | |
947 | my $t1 = CORE::sqrt(($x+1)*($x+1) + $y*$y); | |
948 | my $t2 = CORE::sqrt(($x-1)*($x-1) + $y*$y); | |
949 | my $alpha = ($t1 + $t2)/2; | |
950 | my $beta = ($t1 - $t2)/2; | |
951 | $alpha = 1 if $alpha < 1; | |
952 | if ($beta > 1) { $beta = 1 } | |
953 | elsif ($beta < -1) { $beta = -1 } | |
954 | my $u = CORE::atan2(CORE::sqrt(1-$beta*$beta), $beta); | |
955 | my $v = CORE::log($alpha + CORE::sqrt($alpha*$alpha-1)); | |
956 | $v = -$v if $y > 0 || ($y == 0 && $x < -1); | |
957 | return (ref $z)->make($u, $v); | |
958 | } | |
959 | ||
960 | # | |
961 | # asin | |
962 | # | |
963 | # Computes the arc sine asin(z) = -i log(iz + sqrt(1-z*z)). | |
964 | # | |
965 | sub asin { | |
966 | my $z = $_[0]; | |
967 | return CORE::atan2($z, CORE::sqrt(1-$z*$z)) | |
968 | if (! ref $z) && CORE::abs($z) <= 1; | |
969 | $z = cplx($z, 0) unless ref $z; | |
970 | my ($x, $y) = @{$z->cartesian}; | |
971 | return 0 if $x == 0 && $y == 0; | |
972 | my $t1 = CORE::sqrt(($x+1)*($x+1) + $y*$y); | |
973 | my $t2 = CORE::sqrt(($x-1)*($x-1) + $y*$y); | |
974 | my $alpha = ($t1 + $t2)/2; | |
975 | my $beta = ($t1 - $t2)/2; | |
976 | $alpha = 1 if $alpha < 1; | |
977 | if ($beta > 1) { $beta = 1 } | |
978 | elsif ($beta < -1) { $beta = -1 } | |
979 | my $u = CORE::atan2($beta, CORE::sqrt(1-$beta*$beta)); | |
980 | my $v = -CORE::log($alpha + CORE::sqrt($alpha*$alpha-1)); | |
981 | $v = -$v if $y > 0 || ($y == 0 && $x < -1); | |
982 | return (ref $z)->make($u, $v); | |
983 | } | |
984 | ||
985 | # | |
986 | # atan | |
987 | # | |
988 | # Computes the arc tangent atan(z) = i/2 log((i+z) / (i-z)). | |
989 | # | |
990 | sub atan { | |
991 | my ($z) = @_; | |
992 | return CORE::atan2($z, 1) unless ref $z; | |
993 | my ($x, $y) = ref $z ? @{$z->cartesian} : ($z, 0); | |
994 | return 0 if $x == 0 && $y == 0; | |
995 | _divbyzero "atan(i)" if ( $z == i); | |
996 | _logofzero "atan(-i)" if (-$z == i); # -i is a bad file test... | |
997 | my $log = &log((i + $z) / (i - $z)); | |
998 | return ip2 * $log; | |
999 | } | |
1000 | ||
1001 | # | |
1002 | # asec | |
1003 | # | |
1004 | # Computes the arc secant asec(z) = acos(1 / z). | |
1005 | # | |
1006 | sub asec { | |
1007 | my ($z) = @_; | |
1008 | _divbyzero "asec($z)", $z if ($z == 0); | |
1009 | return acos(1 / $z); | |
1010 | } | |
1011 | ||
1012 | # | |
1013 | # acsc | |
1014 | # | |
1015 | # Computes the arc cosecant acsc(z) = asin(1 / z). | |
1016 | # | |
1017 | sub acsc { | |
1018 | my ($z) = @_; | |
1019 | _divbyzero "acsc($z)", $z if ($z == 0); | |
1020 | return asin(1 / $z); | |
1021 | } | |
1022 | ||
1023 | # | |
1024 | # acosec | |
1025 | # | |
1026 | # Alias for acsc(). | |
1027 | # | |
1028 | sub acosec { Math::Complex::acsc(@_) } | |
1029 | ||
1030 | # | |
1031 | # acot | |
1032 | # | |
1033 | # Computes the arc cotangent acot(z) = atan(1 / z) | |
1034 | # | |
1035 | sub acot { | |
1036 | my ($z) = @_; | |
1037 | _divbyzero "acot(0)" if $z == 0; | |
1038 | return ($z >= 0) ? CORE::atan2(1, $z) : CORE::atan2(-1, -$z) | |
1039 | unless ref $z; | |
1040 | _divbyzero "acot(i)" if ($z - i == 0); | |
1041 | _logofzero "acot(-i)" if ($z + i == 0); | |
1042 | return atan(1 / $z); | |
1043 | } | |
1044 | ||
1045 | # | |
1046 | # acotan | |
1047 | # | |
1048 | # Alias for acot(). | |
1049 | # | |
1050 | sub acotan { Math::Complex::acot(@_) } | |
1051 | ||
1052 | # | |
1053 | # cosh | |
1054 | # | |
1055 | # Computes the hyperbolic cosine cosh(z) = (exp(z) + exp(-z))/2. | |
1056 | # | |
1057 | sub cosh { | |
1058 | my ($z) = @_; | |
1059 | my $ex; | |
1060 | unless (ref $z) { | |
1061 | $ex = CORE::exp($z); | |
1062 | return $ex ? ($ex + 1/$ex)/2 : $Inf; | |
1063 | } | |
1064 | my ($x, $y) = @{$z->cartesian}; | |
1065 | $ex = CORE::exp($x); | |
1066 | my $ex_1 = $ex ? 1 / $ex : $Inf; | |
1067 | return (ref $z)->make(CORE::cos($y) * ($ex + $ex_1)/2, | |
1068 | CORE::sin($y) * ($ex - $ex_1)/2); | |
1069 | } | |
1070 | ||
1071 | # | |
1072 | # sinh | |
1073 | # | |
1074 | # Computes the hyperbolic sine sinh(z) = (exp(z) - exp(-z))/2. | |
1075 | # | |
1076 | sub sinh { | |
1077 | my ($z) = @_; | |
1078 | my $ex; | |
1079 | unless (ref $z) { | |
1080 | return 0 if $z == 0; | |
1081 | $ex = CORE::exp($z); | |
1082 | return $ex ? ($ex - 1/$ex)/2 : "-$Inf"; | |
1083 | } | |
1084 | my ($x, $y) = @{$z->cartesian}; | |
1085 | my $cy = CORE::cos($y); | |
1086 | my $sy = CORE::sin($y); | |
1087 | $ex = CORE::exp($x); | |
1088 | my $ex_1 = $ex ? 1 / $ex : $Inf; | |
1089 | return (ref $z)->make(CORE::cos($y) * ($ex - $ex_1)/2, | |
1090 | CORE::sin($y) * ($ex + $ex_1)/2); | |
1091 | } | |
1092 | ||
1093 | # | |
1094 | # tanh | |
1095 | # | |
1096 | # Computes the hyperbolic tangent tanh(z) = sinh(z) / cosh(z). | |
1097 | # | |
1098 | sub tanh { | |
1099 | my ($z) = @_; | |
1100 | my $cz = cosh($z); | |
1101 | _divbyzero "tanh($z)", "cosh($z)" if ($cz == 0); | |
1102 | return sinh($z) / $cz; | |
1103 | } | |
1104 | ||
1105 | # | |
1106 | # sech | |
1107 | # | |
1108 | # Computes the hyperbolic secant sech(z) = 1 / cosh(z). | |
1109 | # | |
1110 | sub sech { | |
1111 | my ($z) = @_; | |
1112 | my $cz = cosh($z); | |
1113 | _divbyzero "sech($z)", "cosh($z)" if ($cz == 0); | |
1114 | return 1 / $cz; | |
1115 | } | |
1116 | ||
1117 | # | |
1118 | # csch | |
1119 | # | |
1120 | # Computes the hyperbolic cosecant csch(z) = 1 / sinh(z). | |
1121 | # | |
1122 | sub csch { | |
1123 | my ($z) = @_; | |
1124 | my $sz = sinh($z); | |
1125 | _divbyzero "csch($z)", "sinh($z)" if ($sz == 0); | |
1126 | return 1 / $sz; | |
1127 | } | |
1128 | ||
1129 | # | |
1130 | # cosech | |
1131 | # | |
1132 | # Alias for csch(). | |
1133 | # | |
1134 | sub cosech { Math::Complex::csch(@_) } | |
1135 | ||
1136 | # | |
1137 | # coth | |
1138 | # | |
1139 | # Computes the hyperbolic cotangent coth(z) = cosh(z) / sinh(z). | |
1140 | # | |
1141 | sub coth { | |
1142 | my ($z) = @_; | |
1143 | my $sz = sinh($z); | |
1144 | _divbyzero "coth($z)", "sinh($z)" if $sz == 0; | |
1145 | return cosh($z) / $sz; | |
1146 | } | |
1147 | ||
1148 | # | |
1149 | # cotanh | |
1150 | # | |
1151 | # Alias for coth(). | |
1152 | # | |
1153 | sub cotanh { Math::Complex::coth(@_) } | |
1154 | ||
1155 | # | |
1156 | # acosh | |
1157 | # | |
1158 | # Computes the arc hyperbolic cosine acosh(z) = log(z + sqrt(z*z-1)). | |
1159 | # | |
1160 | sub acosh { | |
1161 | my ($z) = @_; | |
1162 | unless (ref $z) { | |
1163 | $z = cplx($z, 0); | |
1164 | } | |
1165 | my ($re, $im) = @{$z->cartesian}; | |
1166 | if ($im == 0) { | |
1167 | return CORE::log($re + CORE::sqrt($re*$re - 1)) | |
1168 | if $re >= 1; | |
1169 | return cplx(0, CORE::atan2(CORE::sqrt(1 - $re*$re), $re)) | |
1170 | if CORE::abs($re) < 1; | |
1171 | } | |
1172 | my $t = &sqrt($z * $z - 1) + $z; | |
1173 | # Try Taylor if looking bad (this usually means that | |
1174 | # $z was large negative, therefore the sqrt is really | |
1175 | # close to abs(z), summing that with z...) | |
1176 | $t = 1/(2 * $z) - 1/(8 * $z**3) + 1/(16 * $z**5) - 5/(128 * $z**7) | |
1177 | if $t == 0; | |
1178 | my $u = &log($t); | |
1179 | $u->Im(-$u->Im) if $re < 0 && $im == 0; | |
1180 | return $re < 0 ? -$u : $u; | |
1181 | } | |
1182 | ||
1183 | # | |
1184 | # asinh | |
1185 | # | |
1186 | # Computes the arc hyperbolic sine asinh(z) = log(z + sqrt(z*z+1)) | |
1187 | # | |
1188 | sub asinh { | |
1189 | my ($z) = @_; | |
1190 | unless (ref $z) { | |
1191 | my $t = $z + CORE::sqrt($z*$z + 1); | |
1192 | return CORE::log($t) if $t; | |
1193 | } | |
1194 | my $t = &sqrt($z * $z + 1) + $z; | |
1195 | # Try Taylor if looking bad (this usually means that | |
1196 | # $z was large negative, therefore the sqrt is really | |
1197 | # close to abs(z), summing that with z...) | |
1198 | $t = 1/(2 * $z) - 1/(8 * $z**3) + 1/(16 * $z**5) - 5/(128 * $z**7) | |
1199 | if $t == 0; | |
1200 | return &log($t); | |
1201 | } | |
1202 | ||
1203 | # | |
1204 | # atanh | |
1205 | # | |
1206 | # Computes the arc hyperbolic tangent atanh(z) = 1/2 log((1+z) / (1-z)). | |
1207 | # | |
1208 | sub atanh { | |
1209 | my ($z) = @_; | |
1210 | unless (ref $z) { | |
1211 | return CORE::log((1 + $z)/(1 - $z))/2 if CORE::abs($z) < 1; | |
1212 | $z = cplx($z, 0); | |
1213 | } | |
1214 | _divbyzero 'atanh(1)', "1 - $z" if (1 - $z == 0); | |
1215 | _logofzero 'atanh(-1)' if (1 + $z == 0); | |
1216 | return 0.5 * &log((1 + $z) / (1 - $z)); | |
1217 | } | |
1218 | ||
1219 | # | |
1220 | # asech | |
1221 | # | |
1222 | # Computes the hyperbolic arc secant asech(z) = acosh(1 / z). | |
1223 | # | |
1224 | sub asech { | |
1225 | my ($z) = @_; | |
1226 | _divbyzero 'asech(0)', "$z" if ($z == 0); | |
1227 | return acosh(1 / $z); | |
1228 | } | |
1229 | ||
1230 | # | |
1231 | # acsch | |
1232 | # | |
1233 | # Computes the hyperbolic arc cosecant acsch(z) = asinh(1 / z). | |
1234 | # | |
1235 | sub acsch { | |
1236 | my ($z) = @_; | |
1237 | _divbyzero 'acsch(0)', $z if ($z == 0); | |
1238 | return asinh(1 / $z); | |
1239 | } | |
1240 | ||
1241 | # | |
1242 | # acosech | |
1243 | # | |
1244 | # Alias for acosh(). | |
1245 | # | |
1246 | sub acosech { Math::Complex::acsch(@_) } | |
1247 | ||
1248 | # | |
1249 | # acoth | |
1250 | # | |
1251 | # Computes the arc hyperbolic cotangent acoth(z) = 1/2 log((1+z) / (z-1)). | |
1252 | # | |
1253 | sub acoth { | |
1254 | my ($z) = @_; | |
1255 | _divbyzero 'acoth(0)' if ($z == 0); | |
1256 | unless (ref $z) { | |
1257 | return CORE::log(($z + 1)/($z - 1))/2 if CORE::abs($z) > 1; | |
1258 | $z = cplx($z, 0); | |
1259 | } | |
1260 | _divbyzero 'acoth(1)', "$z - 1" if ($z - 1 == 0); | |
1261 | _logofzero 'acoth(-1)', "1 + $z" if (1 + $z == 0); | |
1262 | return &log((1 + $z) / ($z - 1)) / 2; | |
1263 | } | |
1264 | ||
1265 | # | |
1266 | # acotanh | |
1267 | # | |
1268 | # Alias for acot(). | |
1269 | # | |
1270 | sub acotanh { Math::Complex::acoth(@_) } | |
1271 | ||
1272 | # | |
1273 | # (atan2) | |
1274 | # | |
1275 | # Compute atan(z1/z2), minding the right quadrant. | |
1276 | # | |
1277 | sub atan2 { | |
1278 | my ($z1, $z2, $inverted) = @_; | |
1279 | my ($re1, $im1, $re2, $im2); | |
1280 | if ($inverted) { | |
1281 | ($re1, $im1) = ref $z2 ? @{$z2->cartesian} : ($z2, 0); | |
1282 | ($re2, $im2) = ref $z1 ? @{$z1->cartesian} : ($z1, 0); | |
1283 | } else { | |
1284 | ($re1, $im1) = ref $z1 ? @{$z1->cartesian} : ($z1, 0); | |
1285 | ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0); | |
1286 | } | |
1287 | if ($im1 || $im2) { | |
1288 | # In MATLAB the imaginary parts are ignored. | |
1289 | # warn "atan2: Imaginary parts ignored"; | |
1290 | # http://documents.wolfram.com/mathematica/functions/ArcTan | |
1291 | # NOTE: Mathematica ArcTan[x,y] while atan2(y,x) | |
1292 | my $s = $z1 * $z1 + $z2 * $z2; | |
1293 | _divbyzero("atan2") if $s == 0; | |
1294 | my $i = &i; | |
1295 | my $r = $z2 + $z1 * $i; | |
1296 | return -$i * &log($r / &sqrt( $s )); | |
1297 | } | |
1298 | return CORE::atan2($re1, $re2); | |
1299 | } | |
1300 | ||
1301 | # | |
1302 | # display_format | |
1303 | # ->display_format | |
1304 | # | |
1305 | # Set (get if no argument) the display format for all complex numbers that | |
1306 | # don't happen to have overridden it via ->display_format | |
1307 | # | |
1308 | # When called as an object method, this actually sets the display format for | |
1309 | # the current object. | |
1310 | # | |
1311 | # Valid object formats are 'c' and 'p' for cartesian and polar. The first | |
1312 | # letter is used actually, so the type can be fully spelled out for clarity. | |
1313 | # | |
1314 | sub display_format { | |
1315 | my $self = shift; | |
1316 | my %display_format = %DISPLAY_FORMAT; | |
1317 | ||
1318 | if (ref $self) { # Called as an object method | |
1319 | if (exists $self->{display_format}) { | |
1320 | my %obj = %{$self->{display_format}}; | |
1321 | @display_format{keys %obj} = values %obj; | |
1322 | } | |
1323 | } | |
1324 | if (@_ == 1) { | |
1325 | $display_format{style} = shift; | |
1326 | } else { | |
1327 | my %new = @_; | |
1328 | @display_format{keys %new} = values %new; | |
1329 | } | |
1330 | ||
1331 | if (ref $self) { # Called as an object method | |
1332 | $self->{display_format} = { %display_format }; | |
1333 | return | |
1334 | wantarray ? | |
1335 | %{$self->{display_format}} : | |
1336 | $self->{display_format}->{style}; | |
1337 | } | |
1338 | ||
1339 | # Called as a class method | |
1340 | %DISPLAY_FORMAT = %display_format; | |
1341 | return | |
1342 | wantarray ? | |
1343 | %DISPLAY_FORMAT : | |
1344 | $DISPLAY_FORMAT{style}; | |
1345 | } | |
1346 | ||
1347 | # | |
1348 | # (stringify) | |
1349 | # | |
1350 | # Show nicely formatted complex number under its cartesian or polar form, | |
1351 | # depending on the current display format: | |
1352 | # | |
1353 | # . If a specific display format has been recorded for this object, use it. | |
1354 | # . Otherwise, use the generic current default for all complex numbers, | |
1355 | # which is a package global variable. | |
1356 | # | |
1357 | sub stringify { | |
1358 | my ($z) = shift; | |
1359 | ||
1360 | my $style = $z->display_format; | |
1361 | ||
1362 | $style = $DISPLAY_FORMAT{style} unless defined $style; | |
1363 | ||
1364 | return $z->stringify_polar if $style =~ /^p/i; | |
1365 | return $z->stringify_cartesian; | |
1366 | } | |
1367 | ||
1368 | # | |
1369 | # ->stringify_cartesian | |
1370 | # | |
1371 | # Stringify as a cartesian representation 'a+bi'. | |
1372 | # | |
1373 | sub stringify_cartesian { | |
1374 | my $z = shift; | |
1375 | my ($x, $y) = @{$z->cartesian}; | |
1376 | my ($re, $im); | |
1377 | ||
1378 | my %format = $z->display_format; | |
1379 | my $format = $format{format}; | |
1380 | ||
1381 | if ($x) { | |
1382 | if ($x =~ /^NaN[QS]?$/i) { | |
1383 | $re = $x; | |
1384 | } else { | |
1385 | if ($x =~ /^-?$Inf$/oi) { | |
1386 | $re = $x; | |
1387 | } else { | |
1388 | $re = defined $format ? sprintf($format, $x) : $x; | |
1389 | } | |
1390 | } | |
1391 | } else { | |
1392 | undef $re; | |
1393 | } | |
1394 | ||
1395 | if ($y) { | |
1396 | if ($y =~ /^(NaN[QS]?)$/i) { | |
1397 | $im = $y; | |
1398 | } else { | |
1399 | if ($y =~ /^-?$Inf$/oi) { | |
1400 | $im = $y; | |
1401 | } else { | |
1402 | $im = | |
1403 | defined $format ? | |
1404 | sprintf($format, $y) : | |
1405 | ($y == 1 ? "" : ($y == -1 ? "-" : $y)); | |
1406 | } | |
1407 | } | |
1408 | $im .= "i"; | |
1409 | } else { | |
1410 | undef $im; | |
1411 | } | |
1412 | ||
1413 | my $str = $re; | |
1414 | ||
1415 | if (defined $im) { | |
1416 | if ($y < 0) { | |
1417 | $str .= $im; | |
1418 | } elsif ($y > 0 || $im =~ /^NaN[QS]?i$/i) { | |
1419 | $str .= "+" if defined $re; | |
1420 | $str .= $im; | |
1421 | } | |
1422 | } elsif (!defined $re) { | |
1423 | $str = "0"; | |
1424 | } | |
1425 | ||
1426 | return $str; | |
1427 | } | |
1428 | ||
1429 | ||
1430 | # | |
1431 | # ->stringify_polar | |
1432 | # | |
1433 | # Stringify as a polar representation '[r,t]'. | |
1434 | # | |
1435 | sub stringify_polar { | |
1436 | my $z = shift; | |
1437 | my ($r, $t) = @{$z->polar}; | |
1438 | my $theta; | |
1439 | ||
1440 | my %format = $z->display_format; | |
1441 | my $format = $format{format}; | |
1442 | ||
1443 | if ($t =~ /^NaN[QS]?$/i || $t =~ /^-?$Inf$/oi) { | |
1444 | $theta = $t; | |
1445 | } elsif ($t == pi) { | |
1446 | $theta = "pi"; | |
1447 | } elsif ($r == 0 || $t == 0) { | |
1448 | $theta = defined $format ? sprintf($format, $t) : $t; | |
1449 | } | |
1450 | ||
1451 | return "[$r,$theta]" if defined $theta; | |
1452 | ||
1453 | # | |
1454 | # Try to identify pi/n and friends. | |
1455 | # | |
1456 | ||
1457 | $t -= int(CORE::abs($t) / pit2) * pit2; | |
1458 | ||
1459 | if ($format{polar_pretty_print} && $t) { | |
1460 | my ($a, $b); | |
1461 | for $a (2..9) { | |
1462 | $b = $t * $a / pi; | |
1463 | if ($b =~ /^-?\d+$/) { | |
1464 | $b = $b < 0 ? "-" : "" if CORE::abs($b) == 1; | |
1465 | $theta = "${b}pi/$a"; | |
1466 | last; | |
1467 | } | |
1468 | } | |
1469 | } | |
1470 | ||
1471 | if (defined $format) { | |
1472 | $r = sprintf($format, $r); | |
1473 | $theta = sprintf($format, $theta) unless defined $theta; | |
1474 | } else { | |
1475 | $theta = $t unless defined $theta; | |
1476 | } | |
1477 | ||
1478 | return "[$r,$theta]"; | |
1479 | } | |
1480 | ||
1481 | 1; | |
1482 | __END__ | |
1483 | ||
1484 | =pod | |
1485 | ||
1486 | =head1 NAME | |
1487 | ||
1488 | Math::Complex - complex numbers and associated mathematical functions | |
1489 | ||
1490 | =head1 SYNOPSIS | |
1491 | ||
1492 | use Math::Complex; | |
1493 | ||
1494 | $z = Math::Complex->make(5, 6); | |
1495 | $t = 4 - 3*i + $z; | |
1496 | $j = cplxe(1, 2*pi/3); | |
1497 | ||
1498 | =head1 DESCRIPTION | |
1499 | ||
1500 | This package lets you create and manipulate complex numbers. By default, | |
1501 | I<Perl> limits itself to real numbers, but an extra C<use> statement brings | |
1502 | full complex support, along with a full set of mathematical functions | |
1503 | typically associated with and/or extended to complex numbers. | |
1504 | ||
1505 | If you wonder what complex numbers are, they were invented to be able to solve | |
1506 | the following equation: | |
1507 | ||
1508 | x*x = -1 | |
1509 | ||
1510 | and by definition, the solution is noted I<i> (engineers use I<j> instead since | |
1511 | I<i> usually denotes an intensity, but the name does not matter). The number | |
1512 | I<i> is a pure I<imaginary> number. | |
1513 | ||
1514 | The arithmetics with pure imaginary numbers works just like you would expect | |
1515 | it with real numbers... you just have to remember that | |
1516 | ||
1517 | i*i = -1 | |
1518 | ||
1519 | so you have: | |
1520 | ||
1521 | 5i + 7i = i * (5 + 7) = 12i | |
1522 | 4i - 3i = i * (4 - 3) = i | |
1523 | 4i * 2i = -8 | |
1524 | 6i / 2i = 3 | |
1525 | 1 / i = -i | |
1526 | ||
1527 | Complex numbers are numbers that have both a real part and an imaginary | |
1528 | part, and are usually noted: | |
1529 | ||
1530 | a + bi | |
1531 | ||
1532 | where C<a> is the I<real> part and C<b> is the I<imaginary> part. The | |
1533 | arithmetic with complex numbers is straightforward. You have to | |
1534 | keep track of the real and the imaginary parts, but otherwise the | |
1535 | rules used for real numbers just apply: | |
1536 | ||
1537 | (4 + 3i) + (5 - 2i) = (4 + 5) + i(3 - 2) = 9 + i | |
1538 | (2 + i) * (4 - i) = 2*4 + 4i -2i -i*i = 8 + 2i + 1 = 9 + 2i | |
1539 | ||
1540 | A graphical representation of complex numbers is possible in a plane | |
1541 | (also called the I<complex plane>, but it's really a 2D plane). | |
1542 | The number | |
1543 | ||
1544 | z = a + bi | |
1545 | ||
1546 | is the point whose coordinates are (a, b). Actually, it would | |
1547 | be the vector originating from (0, 0) to (a, b). It follows that the addition | |
1548 | of two complex numbers is a vectorial addition. | |
1549 | ||
1550 | Since there is a bijection between a point in the 2D plane and a complex | |
1551 | number (i.e. the mapping is unique and reciprocal), a complex number | |
1552 | can also be uniquely identified with polar coordinates: | |
1553 | ||
1554 | [rho, theta] | |
1555 | ||
1556 | where C<rho> is the distance to the origin, and C<theta> the angle between | |
1557 | the vector and the I<x> axis. There is a notation for this using the | |
1558 | exponential form, which is: | |
1559 | ||
1560 | rho * exp(i * theta) | |
1561 | ||
1562 | where I<i> is the famous imaginary number introduced above. Conversion | |
1563 | between this form and the cartesian form C<a + bi> is immediate: | |
1564 | ||
1565 | a = rho * cos(theta) | |
1566 | b = rho * sin(theta) | |
1567 | ||
1568 | which is also expressed by this formula: | |
1569 | ||
1570 | z = rho * exp(i * theta) = rho * (cos theta + i * sin theta) | |
1571 | ||
1572 | In other words, it's the projection of the vector onto the I<x> and I<y> | |
1573 | axes. Mathematicians call I<rho> the I<norm> or I<modulus> and I<theta> | |
1574 | the I<argument> of the complex number. The I<norm> of C<z> will be | |
1575 | noted C<abs(z)>. | |
1576 | ||
1577 | The polar notation (also known as the trigonometric | |
1578 | representation) is much more handy for performing multiplications and | |
1579 | divisions of complex numbers, whilst the cartesian notation is better | |
1580 | suited for additions and subtractions. Real numbers are on the I<x> | |
1581 | axis, and therefore I<theta> is zero or I<pi>. | |
1582 | ||
1583 | All the common operations that can be performed on a real number have | |
1584 | been defined to work on complex numbers as well, and are merely | |
1585 | I<extensions> of the operations defined on real numbers. This means | |
1586 | they keep their natural meaning when there is no imaginary part, provided | |
1587 | the number is within their definition set. | |
1588 | ||
1589 | For instance, the C<sqrt> routine which computes the square root of | |
1590 | its argument is only defined for non-negative real numbers and yields a | |
1591 | non-negative real number (it is an application from B<R+> to B<R+>). | |
1592 | If we allow it to return a complex number, then it can be extended to | |
1593 | negative real numbers to become an application from B<R> to B<C> (the | |
1594 | set of complex numbers): | |
1595 | ||
1596 | sqrt(x) = x >= 0 ? sqrt(x) : sqrt(-x)*i | |
1597 | ||
1598 | It can also be extended to be an application from B<C> to B<C>, | |
1599 | whilst its restriction to B<R> behaves as defined above by using | |
1600 | the following definition: | |
1601 | ||
1602 | sqrt(z = [r,t]) = sqrt(r) * exp(i * t/2) | |
1603 | ||
1604 | Indeed, a negative real number can be noted C<[x,pi]> (the modulus | |
1605 | I<x> is always non-negative, so C<[x,pi]> is really C<-x>, a negative | |
1606 | number) and the above definition states that | |
1607 | ||
1608 | sqrt([x,pi]) = sqrt(x) * exp(i*pi/2) = [sqrt(x),pi/2] = sqrt(x)*i | |
1609 | ||
1610 | which is exactly what we had defined for negative real numbers above. | |
1611 | The C<sqrt> returns only one of the solutions: if you want the both, | |
1612 | use the C<root> function. | |
1613 | ||
1614 | All the common mathematical functions defined on real numbers that | |
1615 | are extended to complex numbers share that same property of working | |
1616 | I<as usual> when the imaginary part is zero (otherwise, it would not | |
1617 | be called an extension, would it?). | |
1618 | ||
1619 | A I<new> operation possible on a complex number that is | |
1620 | the identity for real numbers is called the I<conjugate>, and is noted | |
1621 | with a horizontal bar above the number, or C<~z> here. | |
1622 | ||
1623 | z = a + bi | |
1624 | ~z = a - bi | |
1625 | ||
1626 | Simple... Now look: | |
1627 | ||
1628 | z * ~z = (a + bi) * (a - bi) = a*a + b*b | |
1629 | ||
1630 | We saw that the norm of C<z> was noted C<abs(z)> and was defined as the | |
1631 | distance to the origin, also known as: | |
1632 | ||
1633 | rho = abs(z) = sqrt(a*a + b*b) | |
1634 | ||
1635 | so | |
1636 | ||
1637 | z * ~z = abs(z) ** 2 | |
1638 | ||
1639 | If z is a pure real number (i.e. C<b == 0>), then the above yields: | |
1640 | ||
1641 | a * a = abs(a) ** 2 | |
1642 | ||
1643 | which is true (C<abs> has the regular meaning for real number, i.e. stands | |
1644 | for the absolute value). This example explains why the norm of C<z> is | |
1645 | noted C<abs(z)>: it extends the C<abs> function to complex numbers, yet | |
1646 | is the regular C<abs> we know when the complex number actually has no | |
1647 | imaginary part... This justifies I<a posteriori> our use of the C<abs> | |
1648 | notation for the norm. | |
1649 | ||
1650 | =head1 OPERATIONS | |
1651 | ||
1652 | Given the following notations: | |
1653 | ||
1654 | z1 = a + bi = r1 * exp(i * t1) | |
1655 | z2 = c + di = r2 * exp(i * t2) | |
1656 | z = <any complex or real number> | |
1657 | ||
1658 | the following (overloaded) operations are supported on complex numbers: | |
1659 | ||
1660 | z1 + z2 = (a + c) + i(b + d) | |
1661 | z1 - z2 = (a - c) + i(b - d) | |
1662 | z1 * z2 = (r1 * r2) * exp(i * (t1 + t2)) | |
1663 | z1 / z2 = (r1 / r2) * exp(i * (t1 - t2)) | |
1664 | z1 ** z2 = exp(z2 * log z1) | |
1665 | ~z = a - bi | |
1666 | abs(z) = r1 = sqrt(a*a + b*b) | |
1667 | sqrt(z) = sqrt(r1) * exp(i * t/2) | |
1668 | exp(z) = exp(a) * exp(i * b) | |
1669 | log(z) = log(r1) + i*t | |
1670 | sin(z) = 1/2i (exp(i * z1) - exp(-i * z)) | |
1671 | cos(z) = 1/2 (exp(i * z1) + exp(-i * z)) | |
1672 | atan2(y, x) = atan(y / x) # Minding the right quadrant, note the order. | |
1673 | ||
1674 | The definition used for complex arguments of atan2() is | |
1675 | ||
1676 | -i log((x + iy)/sqrt(x*x+y*y)) | |
1677 | ||
1678 | The following extra operations are supported on both real and complex | |
1679 | numbers: | |
1680 | ||
1681 | Re(z) = a | |
1682 | Im(z) = b | |
1683 | arg(z) = t | |
1684 | abs(z) = r | |
1685 | ||
1686 | cbrt(z) = z ** (1/3) | |
1687 | log10(z) = log(z) / log(10) | |
1688 | logn(z, n) = log(z) / log(n) | |
1689 | ||
1690 | tan(z) = sin(z) / cos(z) | |
1691 | ||
1692 | csc(z) = 1 / sin(z) | |
1693 | sec(z) = 1 / cos(z) | |
1694 | cot(z) = 1 / tan(z) | |
1695 | ||
1696 | asin(z) = -i * log(i*z + sqrt(1-z*z)) | |
1697 | acos(z) = -i * log(z + i*sqrt(1-z*z)) | |
1698 | atan(z) = i/2 * log((i+z) / (i-z)) | |
1699 | ||
1700 | acsc(z) = asin(1 / z) | |
1701 | asec(z) = acos(1 / z) | |
1702 | acot(z) = atan(1 / z) = -i/2 * log((i+z) / (z-i)) | |
1703 | ||
1704 | sinh(z) = 1/2 (exp(z) - exp(-z)) | |
1705 | cosh(z) = 1/2 (exp(z) + exp(-z)) | |
1706 | tanh(z) = sinh(z) / cosh(z) = (exp(z) - exp(-z)) / (exp(z) + exp(-z)) | |
1707 | ||
1708 | csch(z) = 1 / sinh(z) | |
1709 | sech(z) = 1 / cosh(z) | |
1710 | coth(z) = 1 / tanh(z) | |
1711 | ||
1712 | asinh(z) = log(z + sqrt(z*z+1)) | |
1713 | acosh(z) = log(z + sqrt(z*z-1)) | |
1714 | atanh(z) = 1/2 * log((1+z) / (1-z)) | |
1715 | ||
1716 | acsch(z) = asinh(1 / z) | |
1717 | asech(z) = acosh(1 / z) | |
1718 | acoth(z) = atanh(1 / z) = 1/2 * log((1+z) / (z-1)) | |
1719 | ||
1720 | I<arg>, I<abs>, I<log>, I<csc>, I<cot>, I<acsc>, I<acot>, I<csch>, | |
1721 | I<coth>, I<acosech>, I<acotanh>, have aliases I<rho>, I<theta>, I<ln>, | |
1722 | I<cosec>, I<cotan>, I<acosec>, I<acotan>, I<cosech>, I<cotanh>, | |
1723 | I<acosech>, I<acotanh>, respectively. C<Re>, C<Im>, C<arg>, C<abs>, | |
1724 | C<rho>, and C<theta> can be used also as mutators. The C<cbrt> | |
1725 | returns only one of the solutions: if you want all three, use the | |
1726 | C<root> function. | |
1727 | ||
1728 | The I<root> function is available to compute all the I<n> | |
1729 | roots of some complex, where I<n> is a strictly positive integer. | |
1730 | There are exactly I<n> such roots, returned as a list. Getting the | |
1731 | number mathematicians call C<j> such that: | |
1732 | ||
1733 | 1 + j + j*j = 0; | |
1734 | ||
1735 | is a simple matter of writing: | |
1736 | ||
1737 | $j = ((root(1, 3))[1]; | |
1738 | ||
1739 | The I<k>th root for C<z = [r,t]> is given by: | |
1740 | ||
1741 | (root(z, n))[k] = r**(1/n) * exp(i * (t + 2*k*pi)/n) | |
1742 | ||
1743 | You can return the I<k>th root directly by C<root(z, n, k)>, | |
1744 | indexing starting from I<zero> and ending at I<n - 1>. | |
1745 | ||
1746 | The I<spaceship> comparison operator, E<lt>=E<gt>, is also defined. In | |
1747 | order to ensure its restriction to real numbers is conform to what you | |
1748 | would expect, the comparison is run on the real part of the complex | |
1749 | number first, and imaginary parts are compared only when the real | |
1750 | parts match. | |
1751 | ||
1752 | =head1 CREATION | |
1753 | ||
1754 | To create a complex number, use either: | |
1755 | ||
1756 | $z = Math::Complex->make(3, 4); | |
1757 | $z = cplx(3, 4); | |
1758 | ||
1759 | if you know the cartesian form of the number, or | |
1760 | ||
1761 | $z = 3 + 4*i; | |
1762 | ||
1763 | if you like. To create a number using the polar form, use either: | |
1764 | ||
1765 | $z = Math::Complex->emake(5, pi/3); | |
1766 | $x = cplxe(5, pi/3); | |
1767 | ||
1768 | instead. The first argument is the modulus, the second is the angle | |
1769 | (in radians, the full circle is 2*pi). (Mnemonic: C<e> is used as a | |
1770 | notation for complex numbers in the polar form). | |
1771 | ||
1772 | It is possible to write: | |
1773 | ||
1774 | $x = cplxe(-3, pi/4); | |
1775 | ||
1776 | but that will be silently converted into C<[3,-3pi/4]>, since the | |
1777 | modulus must be non-negative (it represents the distance to the origin | |
1778 | in the complex plane). | |
1779 | ||
1780 | It is also possible to have a complex number as either argument of the | |
1781 | C<make>, C<emake>, C<cplx>, and C<cplxe>: the appropriate component of | |
1782 | the argument will be used. | |
1783 | ||
1784 | $z1 = cplx(-2, 1); | |
1785 | $z2 = cplx($z1, 4); | |
1786 | ||
1787 | The C<new>, C<make>, C<emake>, C<cplx>, and C<cplxe> will also | |
1788 | understand a single (string) argument of the forms | |
1789 | ||
1790 | 2-3i | |
1791 | -3i | |
1792 | [2,3] | |
1793 | [2,-3pi/4] | |
1794 | [2] | |
1795 | ||
1796 | in which case the appropriate cartesian and exponential components | |
1797 | will be parsed from the string and used to create new complex numbers. | |
1798 | The imaginary component and the theta, respectively, will default to zero. | |
1799 | ||
1800 | The C<new>, C<make>, C<emake>, C<cplx>, and C<cplxe> will also | |
1801 | understand the case of no arguments: this means plain zero or (0, 0). | |
1802 | ||
1803 | =head1 DISPLAYING | |
1804 | ||
1805 | When printed, a complex number is usually shown under its cartesian | |
1806 | style I<a+bi>, but there are legitimate cases where the polar style | |
1807 | I<[r,t]> is more appropriate. The process of converting the complex | |
1808 | number into a string that can be displayed is known as I<stringification>. | |
1809 | ||
1810 | By calling the class method C<Math::Complex::display_format> and | |
1811 | supplying either C<"polar"> or C<"cartesian"> as an argument, you | |
1812 | override the default display style, which is C<"cartesian">. Not | |
1813 | supplying any argument returns the current settings. | |
1814 | ||
1815 | This default can be overridden on a per-number basis by calling the | |
1816 | C<display_format> method instead. As before, not supplying any argument | |
1817 | returns the current display style for this number. Otherwise whatever you | |
1818 | specify will be the new display style for I<this> particular number. | |
1819 | ||
1820 | For instance: | |
1821 | ||
1822 | use Math::Complex; | |
1823 | ||
1824 | Math::Complex::display_format('polar'); | |
1825 | $j = (root(1, 3))[1]; | |
1826 | print "j = $j\n"; # Prints "j = [1,2pi/3]" | |
1827 | $j->display_format('cartesian'); | |
1828 | print "j = $j\n"; # Prints "j = -0.5+0.866025403784439i" | |
1829 | ||
1830 | The polar style attempts to emphasize arguments like I<k*pi/n> | |
1831 | (where I<n> is a positive integer and I<k> an integer within [-9, +9]), | |
1832 | this is called I<polar pretty-printing>. | |
1833 | ||
1834 | For the reverse of stringifying, see the C<make> and C<emake>. | |
1835 | ||
1836 | =head2 CHANGED IN PERL 5.6 | |
1837 | ||
1838 | The C<display_format> class method and the corresponding | |
1839 | C<display_format> object method can now be called using | |
1840 | a parameter hash instead of just a one parameter. | |
1841 | ||
1842 | The old display format style, which can have values C<"cartesian"> or | |
1843 | C<"polar">, can be changed using the C<"style"> parameter. | |
1844 | ||
1845 | $j->display_format(style => "polar"); | |
1846 | ||
1847 | The one parameter calling convention also still works. | |
1848 | ||
1849 | $j->display_format("polar"); | |
1850 | ||
1851 | There are two new display parameters. | |
1852 | ||
1853 | The first one is C<"format">, which is a sprintf()-style format string | |
1854 | to be used for both numeric parts of the complex number(s). The is | |
1855 | somewhat system-dependent but most often it corresponds to C<"%.15g">. | |
1856 | You can revert to the default by setting the C<format> to C<undef>. | |
1857 | ||
1858 | # the $j from the above example | |
1859 | ||
1860 | $j->display_format('format' => '%.5f'); | |
1861 | print "j = $j\n"; # Prints "j = -0.50000+0.86603i" | |
1862 | $j->display_format('format' => undef); | |
1863 | print "j = $j\n"; # Prints "j = -0.5+0.86603i" | |
1864 | ||
1865 | Notice that this affects also the return values of the | |
1866 | C<display_format> methods: in list context the whole parameter hash | |
1867 | will be returned, as opposed to only the style parameter value. | |
1868 | This is a potential incompatibility with earlier versions if you | |
1869 | have been calling the C<display_format> method in list context. | |
1870 | ||
1871 | The second new display parameter is C<"polar_pretty_print">, which can | |
1872 | be set to true or false, the default being true. See the previous | |
1873 | section for what this means. | |
1874 | ||
1875 | =head1 USAGE | |
1876 | ||
1877 | Thanks to overloading, the handling of arithmetics with complex numbers | |
1878 | is simple and almost transparent. | |
1879 | ||
1880 | Here are some examples: | |
1881 | ||
1882 | use Math::Complex; | |
1883 | ||
1884 | $j = cplxe(1, 2*pi/3); # $j ** 3 == 1 | |
1885 | print "j = $j, j**3 = ", $j ** 3, "\n"; | |
1886 | print "1 + j + j**2 = ", 1 + $j + $j**2, "\n"; | |
1887 | ||
1888 | $z = -16 + 0*i; # Force it to be a complex | |
1889 | print "sqrt($z) = ", sqrt($z), "\n"; | |
1890 | ||
1891 | $k = exp(i * 2*pi/3); | |
1892 | print "$j - $k = ", $j - $k, "\n"; | |
1893 | ||
1894 | $z->Re(3); # Re, Im, arg, abs, | |
1895 | $j->arg(2); # (the last two aka rho, theta) | |
1896 | # can be used also as mutators. | |
1897 | ||
1898 | =head1 ERRORS DUE TO DIVISION BY ZERO OR LOGARITHM OF ZERO | |
1899 | ||
1900 | The division (/) and the following functions | |
1901 | ||
1902 | log ln log10 logn | |
1903 | tan sec csc cot | |
1904 | atan asec acsc acot | |
1905 | tanh sech csch coth | |
1906 | atanh asech acsch acoth | |
1907 | ||
1908 | cannot be computed for all arguments because that would mean dividing | |
1909 | by zero or taking logarithm of zero. These situations cause fatal | |
1910 | runtime errors looking like this | |
1911 | ||
1912 | cot(0): Division by zero. | |
1913 | (Because in the definition of cot(0), the divisor sin(0) is 0) | |
1914 | Died at ... | |
1915 | ||
1916 | or | |
1917 | ||
1918 | atanh(-1): Logarithm of zero. | |
1919 | Died at... | |
1920 | ||
1921 | For the C<csc>, C<cot>, C<asec>, C<acsc>, C<acot>, C<csch>, C<coth>, | |
1922 | C<asech>, C<acsch>, the argument cannot be C<0> (zero). For the | |
1923 | logarithmic functions and the C<atanh>, C<acoth>, the argument cannot | |
1924 | be C<1> (one). For the C<atanh>, C<acoth>, the argument cannot be | |
1925 | C<-1> (minus one). For the C<atan>, C<acot>, the argument cannot be | |
1926 | C<i> (the imaginary unit). For the C<atan>, C<acoth>, the argument | |
1927 | cannot be C<-i> (the negative imaginary unit). For the C<tan>, | |
1928 | C<sec>, C<tanh>, the argument cannot be I<pi/2 + k * pi>, where I<k> | |
1929 | is any integer. atan2(0, 0) is undefined, and if the complex arguments | |
1930 | are used for atan2(), a division by zero will happen if z1**2+z2**2 == 0. | |
1931 | ||
1932 | Note that because we are operating on approximations of real numbers, | |
1933 | these errors can happen when merely `too close' to the singularities | |
1934 | listed above. | |
1935 | ||
1936 | =head1 ERRORS DUE TO INDIGESTIBLE ARGUMENTS | |
1937 | ||
1938 | The C<make> and C<emake> accept both real and complex arguments. | |
1939 | When they cannot recognize the arguments they will die with error | |
1940 | messages like the following | |
1941 | ||
1942 | Math::Complex::make: Cannot take real part of ... | |
1943 | Math::Complex::make: Cannot take real part of ... | |
1944 | Math::Complex::emake: Cannot take rho of ... | |
1945 | Math::Complex::emake: Cannot take theta of ... | |
1946 | ||
1947 | =head1 BUGS | |
1948 | ||
1949 | Saying C<use Math::Complex;> exports many mathematical routines in the | |
1950 | caller environment and even overrides some (C<sqrt>, C<log>, C<atan2>). | |
1951 | This is construed as a feature by the Authors, actually... ;-) | |
1952 | ||
1953 | All routines expect to be given real or complex numbers. Don't attempt to | |
1954 | use BigFloat, since Perl has currently no rule to disambiguate a '+' | |
1955 | operation (for instance) between two overloaded entities. | |
1956 | ||
1957 | In Cray UNICOS there is some strange numerical instability that results | |
1958 | in root(), cos(), sin(), cosh(), sinh(), losing accuracy fast. Beware. | |
1959 | The bug may be in UNICOS math libs, in UNICOS C compiler, in Math::Complex. | |
1960 | Whatever it is, it does not manifest itself anywhere else where Perl runs. | |
1961 | ||
1962 | =head1 AUTHORS | |
1963 | ||
1964 | Daniel S. Lewart <F<d-lewart@uiuc.edu>> | |
1965 | ||
1966 | Original authors Raphael Manfredi <F<Raphael_Manfredi@pobox.com>> and | |
1967 | Jarkko Hietaniemi <F<jhi@iki.fi>> | |
1968 | ||
1969 | =cut | |
1970 | ||
1971 | 1; | |
1972 | ||
1973 | # eof |