Commit | Line | Data |
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66730be0 RM |
1 | # |
2 | # Complex numbers and associated mathematical functions | |
b42d0ec9 JH |
3 | # -- Raphael Manfredi Since Sep 1996 |
4 | # -- Jarkko Hietaniemi Since Mar 1997 | |
5 | # -- Daniel S. Lewart Since Sep 1997 | |
fb73857a | 6 | # |
a0d0e21e | 7 | |
5aabfad6 | 8 | package Math::Complex; |
a0d0e21e | 9 | |
9fbe1b12 JH |
10 | $VERSION = "1.30"; |
11 | ||
12 | our($VERSION, @ISA, @EXPORT, %EXPORT_TAGS, $Inf); | |
13 | ||
14 | BEGIN { | |
806e78a9 JH |
15 | my $e = $!; |
16 | $Inf = CORE::exp(CORE::exp(30)); # We do want an arithmetic overflow. | |
17 | $! = $e; # Clear ERANGE. | |
5240e574 JH |
18 | undef $Inf unless $Inf =~ /^inf(?:inity)?$/i; # Inf INF inf Infinity |
19 | $Inf = "Inf" if !defined $Inf || !($Inf > 0); # Desperation. | |
9fbe1b12 | 20 | } |
fb73857a | 21 | |
9fbe1b12 | 22 | use strict; |
fb73857a | 23 | |
9fbe1b12 JH |
24 | my $i; |
25 | my %LOGN; | |
0c721ce2 | 26 | |
9fbe1b12 | 27 | require Exporter; |
0c721ce2 | 28 | |
5aabfad6 | 29 | @ISA = qw(Exporter); |
30 | ||
5aabfad6 | 31 | my @trig = qw( |
32 | pi | |
fb73857a | 33 | tan |
5aabfad6 | 34 | csc cosec sec cot cotan |
35 | asin acos atan | |
36 | acsc acosec asec acot acotan | |
37 | sinh cosh tanh | |
38 | csch cosech sech coth cotanh | |
39 | asinh acosh atanh | |
40 | acsch acosech asech acoth acotanh | |
41 | ); | |
42 | ||
43 | @EXPORT = (qw( | |
b42d0ec9 | 44 | i Re Im rho theta arg |
fb73857a | 45 | sqrt log ln |
5aabfad6 | 46 | log10 logn cbrt root |
47 | cplx cplxe | |
48 | ), | |
49 | @trig); | |
50 | ||
51 | %EXPORT_TAGS = ( | |
52 | 'trig' => [@trig], | |
66730be0 | 53 | ); |
a0d0e21e | 54 | |
a5f75d66 | 55 | use overload |
0c721ce2 JH |
56 | '+' => \&plus, |
57 | '-' => \&minus, | |
58 | '*' => \&multiply, | |
59 | '/' => \÷, | |
66730be0 | 60 | '**' => \&power, |
1fa12f56 | 61 | '==' => \&numeq, |
66730be0 RM |
62 | '<=>' => \&spaceship, |
63 | 'neg' => \&negate, | |
0c721ce2 | 64 | '~' => \&conjugate, |
66730be0 RM |
65 | 'abs' => \&abs, |
66 | 'sqrt' => \&sqrt, | |
67 | 'exp' => \&exp, | |
68 | 'log' => \&log, | |
69 | 'sin' => \&sin, | |
70 | 'cos' => \&cos, | |
0c721ce2 | 71 | 'tan' => \&tan, |
66730be0 RM |
72 | 'atan2' => \&atan2, |
73 | qw("" stringify); | |
74 | ||
75 | # | |
b42d0ec9 | 76 | # Package "privates" |
66730be0 RM |
77 | # |
78 | ||
16357284 JH |
79 | my %DISPLAY_FORMAT = ('style' => 'cartesian', |
80 | 'polar_pretty_print' => 1); | |
81 | my $eps = 1e-14; # Epsilon | |
66730be0 RM |
82 | |
83 | # | |
84 | # Object attributes (internal): | |
85 | # cartesian [real, imaginary] -- cartesian form | |
86 | # polar [rho, theta] -- polar form | |
87 | # c_dirty cartesian form not up-to-date | |
88 | # p_dirty polar form not up-to-date | |
89 | # display display format (package's global when not set) | |
90 | # | |
91 | ||
b42d0ec9 JH |
92 | # Die on bad *make() arguments. |
93 | ||
94 | sub _cannot_make { | |
95 | die "@{[(caller(1))[3]]}: Cannot take $_[0] of $_[1].\n"; | |
96 | } | |
97 | ||
66730be0 RM |
98 | # |
99 | # ->make | |
100 | # | |
101 | # Create a new complex number (cartesian form) | |
102 | # | |
103 | sub make { | |
104 | my $self = bless {}, shift; | |
105 | my ($re, $im) = @_; | |
b42d0ec9 JH |
106 | my $rre = ref $re; |
107 | if ( $rre ) { | |
108 | if ( $rre eq ref $self ) { | |
109 | $re = Re($re); | |
110 | } else { | |
111 | _cannot_make("real part", $rre); | |
112 | } | |
113 | } | |
114 | my $rim = ref $im; | |
115 | if ( $rim ) { | |
116 | if ( $rim eq ref $self ) { | |
117 | $im = Im($im); | |
118 | } else { | |
119 | _cannot_make("imaginary part", $rim); | |
120 | } | |
121 | } | |
122 | $self->{'cartesian'} = [ $re, $im ]; | |
66730be0 RM |
123 | $self->{c_dirty} = 0; |
124 | $self->{p_dirty} = 1; | |
b42d0ec9 | 125 | $self->display_format('cartesian'); |
66730be0 RM |
126 | return $self; |
127 | } | |
128 | ||
129 | # | |
130 | # ->emake | |
131 | # | |
132 | # Create a new complex number (exponential form) | |
133 | # | |
134 | sub emake { | |
135 | my $self = bless {}, shift; | |
136 | my ($rho, $theta) = @_; | |
b42d0ec9 JH |
137 | my $rrh = ref $rho; |
138 | if ( $rrh ) { | |
139 | if ( $rrh eq ref $self ) { | |
140 | $rho = rho($rho); | |
141 | } else { | |
142 | _cannot_make("rho", $rrh); | |
143 | } | |
144 | } | |
145 | my $rth = ref $theta; | |
146 | if ( $rth ) { | |
147 | if ( $rth eq ref $self ) { | |
148 | $theta = theta($theta); | |
149 | } else { | |
150 | _cannot_make("theta", $rth); | |
151 | } | |
152 | } | |
fb73857a | 153 | if ($rho < 0) { |
154 | $rho = -$rho; | |
155 | $theta = ($theta <= 0) ? $theta + pi() : $theta - pi(); | |
156 | } | |
157 | $self->{'polar'} = [$rho, $theta]; | |
66730be0 RM |
158 | $self->{p_dirty} = 0; |
159 | $self->{c_dirty} = 1; | |
b42d0ec9 | 160 | $self->display_format('polar'); |
66730be0 RM |
161 | return $self; |
162 | } | |
163 | ||
164 | sub new { &make } # For backward compatibility only. | |
165 | ||
166 | # | |
167 | # cplx | |
168 | # | |
169 | # Creates a complex number from a (re, im) tuple. | |
170 | # This avoids the burden of writing Math::Complex->make(re, im). | |
171 | # | |
172 | sub cplx { | |
173 | my ($re, $im) = @_; | |
16357284 | 174 | return __PACKAGE__->make($re, defined $im ? $im : 0); |
66730be0 RM |
175 | } |
176 | ||
177 | # | |
178 | # cplxe | |
179 | # | |
180 | # Creates a complex number from a (rho, theta) tuple. | |
181 | # This avoids the burden of writing Math::Complex->emake(rho, theta). | |
182 | # | |
183 | sub cplxe { | |
184 | my ($rho, $theta) = @_; | |
16357284 | 185 | return __PACKAGE__->emake($rho, defined $theta ? $theta : 0); |
66730be0 RM |
186 | } |
187 | ||
188 | # | |
189 | # pi | |
190 | # | |
fb73857a | 191 | # The number defined as pi = 180 degrees |
66730be0 | 192 | # |
6570f784 | 193 | sub pi () { 4 * CORE::atan2(1, 1) } |
5cd24f17 | 194 | |
195 | # | |
fb73857a | 196 | # pit2 |
5cd24f17 | 197 | # |
fb73857a | 198 | # The full circle |
199 | # | |
6570f784 | 200 | sub pit2 () { 2 * pi } |
fb73857a | 201 | |
5cd24f17 | 202 | # |
fb73857a | 203 | # pip2 |
204 | # | |
205 | # The quarter circle | |
206 | # | |
6570f784 | 207 | sub pip2 () { pi / 2 } |
5cd24f17 | 208 | |
fb73857a | 209 | # |
d09ae4e6 JH |
210 | # deg1 |
211 | # | |
212 | # One degree in radians, used in stringify_polar. | |
213 | # | |
214 | ||
6570f784 | 215 | sub deg1 () { pi / 180 } |
d09ae4e6 JH |
216 | |
217 | # | |
fb73857a | 218 | # uplog10 |
219 | # | |
220 | # Used in log10(). | |
221 | # | |
6570f784 | 222 | sub uplog10 () { 1 / CORE::log(10) } |
66730be0 RM |
223 | |
224 | # | |
225 | # i | |
226 | # | |
227 | # The number defined as i*i = -1; | |
228 | # | |
229 | sub i () { | |
5cd24f17 | 230 | return $i if ($i); |
231 | $i = bless {}; | |
40da2db3 | 232 | $i->{'cartesian'} = [0, 1]; |
fb73857a | 233 | $i->{'polar'} = [1, pip2]; |
66730be0 RM |
234 | $i->{c_dirty} = 0; |
235 | $i->{p_dirty} = 0; | |
236 | return $i; | |
237 | } | |
238 | ||
239 | # | |
1fa12f56 JH |
240 | # ip2 |
241 | # | |
242 | # Half of i. | |
243 | # | |
244 | sub ip2 () { i / 2 } | |
245 | ||
246 | # | |
66730be0 RM |
247 | # Attribute access/set routines |
248 | # | |
249 | ||
0c721ce2 JH |
250 | sub cartesian {$_[0]->{c_dirty} ? |
251 | $_[0]->update_cartesian : $_[0]->{'cartesian'}} | |
252 | sub polar {$_[0]->{p_dirty} ? | |
253 | $_[0]->update_polar : $_[0]->{'polar'}} | |
66730be0 | 254 | |
40da2db3 JH |
255 | sub set_cartesian { $_[0]->{p_dirty}++; $_[0]->{'cartesian'} = $_[1] } |
256 | sub set_polar { $_[0]->{c_dirty}++; $_[0]->{'polar'} = $_[1] } | |
66730be0 RM |
257 | |
258 | # | |
259 | # ->update_cartesian | |
260 | # | |
261 | # Recompute and return the cartesian form, given accurate polar form. | |
262 | # | |
263 | sub update_cartesian { | |
264 | my $self = shift; | |
40da2db3 | 265 | my ($r, $t) = @{$self->{'polar'}}; |
66730be0 | 266 | $self->{c_dirty} = 0; |
a8693bd3 | 267 | return $self->{'cartesian'} = [$r * CORE::cos($t), $r * CORE::sin($t)]; |
66730be0 RM |
268 | } |
269 | ||
270 | # | |
271 | # | |
272 | # ->update_polar | |
273 | # | |
274 | # Recompute and return the polar form, given accurate cartesian form. | |
275 | # | |
276 | sub update_polar { | |
277 | my $self = shift; | |
40da2db3 | 278 | my ($x, $y) = @{$self->{'cartesian'}}; |
66730be0 | 279 | $self->{p_dirty} = 0; |
40da2db3 | 280 | return $self->{'polar'} = [0, 0] if $x == 0 && $y == 0; |
1fa12f56 JH |
281 | return $self->{'polar'} = [CORE::sqrt($x*$x + $y*$y), |
282 | CORE::atan2($y, $x)]; | |
66730be0 RM |
283 | } |
284 | ||
285 | # | |
286 | # (plus) | |
287 | # | |
288 | # Computes z1+z2. | |
289 | # | |
290 | sub plus { | |
291 | my ($z1, $z2, $regular) = @_; | |
292 | my ($re1, $im1) = @{$z1->cartesian}; | |
0e505df1 | 293 | $z2 = cplx($z2) unless ref $z2; |
5cd24f17 | 294 | my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0); |
66730be0 RM |
295 | unless (defined $regular) { |
296 | $z1->set_cartesian([$re1 + $re2, $im1 + $im2]); | |
297 | return $z1; | |
298 | } | |
299 | return (ref $z1)->make($re1 + $re2, $im1 + $im2); | |
300 | } | |
301 | ||
302 | # | |
303 | # (minus) | |
304 | # | |
305 | # Computes z1-z2. | |
306 | # | |
307 | sub minus { | |
308 | my ($z1, $z2, $inverted) = @_; | |
309 | my ($re1, $im1) = @{$z1->cartesian}; | |
0e505df1 JH |
310 | $z2 = cplx($z2) unless ref $z2; |
311 | my ($re2, $im2) = @{$z2->cartesian}; | |
66730be0 RM |
312 | unless (defined $inverted) { |
313 | $z1->set_cartesian([$re1 - $re2, $im1 - $im2]); | |
314 | return $z1; | |
315 | } | |
316 | return $inverted ? | |
317 | (ref $z1)->make($re2 - $re1, $im2 - $im1) : | |
318 | (ref $z1)->make($re1 - $re2, $im1 - $im2); | |
0e505df1 | 319 | |
66730be0 RM |
320 | } |
321 | ||
322 | # | |
323 | # (multiply) | |
324 | # | |
325 | # Computes z1*z2. | |
326 | # | |
327 | sub multiply { | |
fb73857a | 328 | my ($z1, $z2, $regular) = @_; |
329 | if ($z1->{p_dirty} == 0 and ref $z2 and $z2->{p_dirty} == 0) { | |
330 | # if both polar better use polar to avoid rounding errors | |
331 | my ($r1, $t1) = @{$z1->polar}; | |
332 | my ($r2, $t2) = @{$z2->polar}; | |
333 | my $t = $t1 + $t2; | |
334 | if ($t > pi()) { $t -= pit2 } | |
335 | elsif ($t <= -pi()) { $t += pit2 } | |
336 | unless (defined $regular) { | |
337 | $z1->set_polar([$r1 * $r2, $t]); | |
66730be0 | 338 | return $z1; |
fb73857a | 339 | } |
340 | return (ref $z1)->emake($r1 * $r2, $t); | |
341 | } else { | |
342 | my ($x1, $y1) = @{$z1->cartesian}; | |
343 | if (ref $z2) { | |
344 | my ($x2, $y2) = @{$z2->cartesian}; | |
345 | return (ref $z1)->make($x1*$x2-$y1*$y2, $x1*$y2+$y1*$x2); | |
346 | } else { | |
347 | return (ref $z1)->make($x1*$z2, $y1*$z2); | |
348 | } | |
66730be0 | 349 | } |
66730be0 RM |
350 | } |
351 | ||
352 | # | |
0e505df1 | 353 | # _divbyzero |
0c721ce2 JH |
354 | # |
355 | # Die on division by zero. | |
356 | # | |
0e505df1 | 357 | sub _divbyzero { |
5cd24f17 | 358 | my $mess = "$_[0]: Division by zero.\n"; |
359 | ||
360 | if (defined $_[1]) { | |
361 | $mess .= "(Because in the definition of $_[0], the divisor "; | |
1fa12f56 | 362 | $mess .= "$_[1] " unless ("$_[1]" eq '0'); |
5cd24f17 | 363 | $mess .= "is 0)\n"; |
364 | } | |
365 | ||
0c721ce2 | 366 | my @up = caller(1); |
fb73857a | 367 | |
5cd24f17 | 368 | $mess .= "Died at $up[1] line $up[2].\n"; |
369 | ||
370 | die $mess; | |
0c721ce2 JH |
371 | } |
372 | ||
373 | # | |
66730be0 RM |
374 | # (divide) |
375 | # | |
376 | # Computes z1/z2. | |
377 | # | |
378 | sub divide { | |
379 | my ($z1, $z2, $inverted) = @_; | |
fb73857a | 380 | if ($z1->{p_dirty} == 0 and ref $z2 and $z2->{p_dirty} == 0) { |
381 | # if both polar better use polar to avoid rounding errors | |
382 | my ($r1, $t1) = @{$z1->polar}; | |
383 | my ($r2, $t2) = @{$z2->polar}; | |
384 | my $t; | |
385 | if ($inverted) { | |
0e505df1 | 386 | _divbyzero "$z2/0" if ($r1 == 0); |
fb73857a | 387 | $t = $t2 - $t1; |
388 | if ($t > pi()) { $t -= pit2 } | |
389 | elsif ($t <= -pi()) { $t += pit2 } | |
390 | return (ref $z1)->emake($r2 / $r1, $t); | |
391 | } else { | |
0e505df1 | 392 | _divbyzero "$z1/0" if ($r2 == 0); |
fb73857a | 393 | $t = $t1 - $t2; |
394 | if ($t > pi()) { $t -= pit2 } | |
395 | elsif ($t <= -pi()) { $t += pit2 } | |
396 | return (ref $z1)->emake($r1 / $r2, $t); | |
397 | } | |
398 | } else { | |
399 | my ($d, $x2, $y2); | |
400 | if ($inverted) { | |
401 | ($x2, $y2) = @{$z1->cartesian}; | |
402 | $d = $x2*$x2 + $y2*$y2; | |
403 | _divbyzero "$z2/0" if $d == 0; | |
404 | return (ref $z1)->make(($x2*$z2)/$d, -($y2*$z2)/$d); | |
405 | } else { | |
406 | my ($x1, $y1) = @{$z1->cartesian}; | |
407 | if (ref $z2) { | |
408 | ($x2, $y2) = @{$z2->cartesian}; | |
409 | $d = $x2*$x2 + $y2*$y2; | |
410 | _divbyzero "$z1/0" if $d == 0; | |
411 | my $u = ($x1*$x2 + $y1*$y2)/$d; | |
412 | my $v = ($y1*$x2 - $x1*$y2)/$d; | |
413 | return (ref $z1)->make($u, $v); | |
414 | } else { | |
415 | _divbyzero "$z1/0" if $z2 == 0; | |
416 | return (ref $z1)->make($x1/$z2, $y1/$z2); | |
417 | } | |
418 | } | |
0c721ce2 | 419 | } |
66730be0 RM |
420 | } |
421 | ||
422 | # | |
423 | # (power) | |
424 | # | |
425 | # Computes z1**z2 = exp(z2 * log z1)). | |
426 | # | |
427 | sub power { | |
428 | my ($z1, $z2, $inverted) = @_; | |
ace5de91 | 429 | if ($inverted) { |
2820d885 DL |
430 | return 1 if $z1 == 0 || $z2 == 1; |
431 | return 0 if $z2 == 0 && Re($z1) > 0; | |
ace5de91 | 432 | } else { |
2820d885 DL |
433 | return 1 if $z2 == 0 || $z1 == 1; |
434 | return 0 if $z1 == 0 && Re($z2) > 0; | |
ace5de91 | 435 | } |
1fa12f56 JH |
436 | my $w = $inverted ? &exp($z1 * &log($z2)) |
437 | : &exp($z2 * &log($z1)); | |
d09ae4e6 JH |
438 | # If both arguments cartesian, return cartesian, else polar. |
439 | return $z1->{c_dirty} == 0 && | |
440 | (not ref $z2 or $z2->{c_dirty} == 0) ? | |
441 | cplx(@{$w->cartesian}) : $w; | |
66730be0 RM |
442 | } |
443 | ||
444 | # | |
445 | # (spaceship) | |
446 | # | |
447 | # Computes z1 <=> z2. | |
2820d885 | 448 | # Sorts on the real part first, then on the imaginary part. Thus 2-4i < 3+8i. |
66730be0 RM |
449 | # |
450 | sub spaceship { | |
451 | my ($z1, $z2, $inverted) = @_; | |
5cd24f17 | 452 | my ($re1, $im1) = ref $z1 ? @{$z1->cartesian} : ($z1, 0); |
453 | my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0); | |
66730be0 RM |
454 | my $sgn = $inverted ? -1 : 1; |
455 | return $sgn * ($re1 <=> $re2) if $re1 != $re2; | |
456 | return $sgn * ($im1 <=> $im2); | |
457 | } | |
458 | ||
459 | # | |
1fa12f56 JH |
460 | # (numeq) |
461 | # | |
462 | # Computes z1 == z2. | |
463 | # | |
464 | # (Required in addition to spaceship() because of NaNs.) | |
465 | sub numeq { | |
466 | my ($z1, $z2, $inverted) = @_; | |
467 | my ($re1, $im1) = ref $z1 ? @{$z1->cartesian} : ($z1, 0); | |
468 | my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0); | |
469 | return $re1 == $re2 && $im1 == $im2 ? 1 : 0; | |
470 | } | |
471 | ||
472 | # | |
66730be0 RM |
473 | # (negate) |
474 | # | |
475 | # Computes -z. | |
476 | # | |
477 | sub negate { | |
478 | my ($z) = @_; | |
479 | if ($z->{c_dirty}) { | |
480 | my ($r, $t) = @{$z->polar}; | |
fb73857a | 481 | $t = ($t <= 0) ? $t + pi : $t - pi; |
482 | return (ref $z)->emake($r, $t); | |
66730be0 RM |
483 | } |
484 | my ($re, $im) = @{$z->cartesian}; | |
485 | return (ref $z)->make(-$re, -$im); | |
486 | } | |
487 | ||
488 | # | |
489 | # (conjugate) | |
490 | # | |
491 | # Compute complex's conjugate. | |
492 | # | |
493 | sub conjugate { | |
494 | my ($z) = @_; | |
495 | if ($z->{c_dirty}) { | |
496 | my ($r, $t) = @{$z->polar}; | |
497 | return (ref $z)->emake($r, -$t); | |
498 | } | |
499 | my ($re, $im) = @{$z->cartesian}; | |
500 | return (ref $z)->make($re, -$im); | |
501 | } | |
502 | ||
503 | # | |
504 | # (abs) | |
505 | # | |
b42d0ec9 | 506 | # Compute or set complex's norm (rho). |
66730be0 RM |
507 | # |
508 | sub abs { | |
b42d0ec9 | 509 | my ($z, $rho) = @_; |
1fa12f56 JH |
510 | unless (ref $z) { |
511 | if (@_ == 2) { | |
512 | $_[0] = $_[1]; | |
513 | } else { | |
514 | return CORE::abs($z); | |
515 | } | |
516 | } | |
b42d0ec9 JH |
517 | if (defined $rho) { |
518 | $z->{'polar'} = [ $rho, ${$z->polar}[1] ]; | |
519 | $z->{p_dirty} = 0; | |
520 | $z->{c_dirty} = 1; | |
521 | return $rho; | |
522 | } else { | |
523 | return ${$z->polar}[0]; | |
524 | } | |
525 | } | |
526 | ||
527 | sub _theta { | |
528 | my $theta = $_[0]; | |
529 | ||
530 | if ($$theta > pi()) { $$theta -= pit2 } | |
531 | elsif ($$theta <= -pi()) { $$theta += pit2 } | |
66730be0 RM |
532 | } |
533 | ||
534 | # | |
535 | # arg | |
536 | # | |
b42d0ec9 | 537 | # Compute or set complex's argument (theta). |
66730be0 RM |
538 | # |
539 | sub arg { | |
b42d0ec9 JH |
540 | my ($z, $theta) = @_; |
541 | return $z unless ref $z; | |
542 | if (defined $theta) { | |
543 | _theta(\$theta); | |
544 | $z->{'polar'} = [ ${$z->polar}[0], $theta ]; | |
545 | $z->{p_dirty} = 0; | |
546 | $z->{c_dirty} = 1; | |
547 | } else { | |
548 | $theta = ${$z->polar}[1]; | |
549 | _theta(\$theta); | |
550 | } | |
551 | return $theta; | |
66730be0 RM |
552 | } |
553 | ||
554 | # | |
555 | # (sqrt) | |
556 | # | |
0c721ce2 | 557 | # Compute sqrt(z). |
66730be0 | 558 | # |
b42d0ec9 JH |
559 | # It is quite tempting to use wantarray here so that in list context |
560 | # sqrt() would return the two solutions. This, however, would | |
561 | # break things like | |
562 | # | |
563 | # print "sqrt(z) = ", sqrt($z), "\n"; | |
564 | # | |
565 | # The two values would be printed side by side without no intervening | |
566 | # whitespace, quite confusing. | |
567 | # Therefore if you want the two solutions use the root(). | |
568 | # | |
66730be0 RM |
569 | sub sqrt { |
570 | my ($z) = @_; | |
b42d0ec9 | 571 | my ($re, $im) = ref $z ? @{$z->cartesian} : ($z, 0); |
1fa12f56 JH |
572 | return $re < 0 ? cplx(0, CORE::sqrt(-$re)) : CORE::sqrt($re) |
573 | if $im == 0; | |
66730be0 | 574 | my ($r, $t) = @{$z->polar}; |
a8693bd3 | 575 | return (ref $z)->emake(CORE::sqrt($r), $t/2); |
66730be0 RM |
576 | } |
577 | ||
578 | # | |
579 | # cbrt | |
580 | # | |
0c721ce2 | 581 | # Compute cbrt(z) (cubic root). |
66730be0 | 582 | # |
b42d0ec9 JH |
583 | # Why are we not returning three values? The same answer as for sqrt(). |
584 | # | |
66730be0 RM |
585 | sub cbrt { |
586 | my ($z) = @_; | |
1fa12f56 JH |
587 | return $z < 0 ? |
588 | -CORE::exp(CORE::log(-$z)/3) : | |
589 | ($z > 0 ? CORE::exp(CORE::log($z)/3): 0) | |
fb73857a | 590 | unless ref $z; |
66730be0 | 591 | my ($r, $t) = @{$z->polar}; |
1fa12f56 | 592 | return 0 if $r == 0; |
a8693bd3 | 593 | return (ref $z)->emake(CORE::exp(CORE::log($r)/3), $t/3); |
66730be0 RM |
594 | } |
595 | ||
596 | # | |
0e505df1 JH |
597 | # _rootbad |
598 | # | |
599 | # Die on bad root. | |
600 | # | |
601 | sub _rootbad { | |
1fa12f56 | 602 | my $mess = "Root $_[0] illegal, root rank must be positive integer.\n"; |
0e505df1 JH |
603 | |
604 | my @up = caller(1); | |
fb73857a | 605 | |
0e505df1 JH |
606 | $mess .= "Died at $up[1] line $up[2].\n"; |
607 | ||
608 | die $mess; | |
609 | } | |
610 | ||
611 | # | |
66730be0 RM |
612 | # root |
613 | # | |
614 | # Computes all nth root for z, returning an array whose size is n. | |
615 | # `n' must be a positive integer. | |
616 | # | |
617 | # The roots are given by (for k = 0..n-1): | |
618 | # | |
619 | # z^(1/n) = r^(1/n) (cos ((t+2 k pi)/n) + i sin ((t+2 k pi)/n)) | |
620 | # | |
621 | sub root { | |
622 | my ($z, $n) = @_; | |
0e505df1 | 623 | _rootbad($n) if ($n < 1 or int($n) != $n); |
1fa12f56 JH |
624 | my ($r, $t) = ref $z ? |
625 | @{$z->polar} : (CORE::abs($z), $z >= 0 ? 0 : pi); | |
66730be0 RM |
626 | my @root; |
627 | my $k; | |
fb73857a | 628 | my $theta_inc = pit2 / $n; |
66730be0 RM |
629 | my $rho = $r ** (1/$n); |
630 | my $theta; | |
d09ae4e6 | 631 | my $cartesian = ref $z && $z->{c_dirty} == 0; |
66730be0 | 632 | for ($k = 0, $theta = $t / $n; $k < $n; $k++, $theta += $theta_inc) { |
d09ae4e6 JH |
633 | my $w = cplxe($rho, $theta); |
634 | # Yes, $cartesian is loop invariant. | |
635 | push @root, $cartesian ? cplx(@{$w->cartesian}) : $w; | |
a0d0e21e | 636 | } |
66730be0 | 637 | return @root; |
a0d0e21e LW |
638 | } |
639 | ||
66730be0 RM |
640 | # |
641 | # Re | |
642 | # | |
b42d0ec9 | 643 | # Return or set Re(z). |
66730be0 | 644 | # |
a0d0e21e | 645 | sub Re { |
b42d0ec9 | 646 | my ($z, $Re) = @_; |
66730be0 | 647 | return $z unless ref $z; |
b42d0ec9 JH |
648 | if (defined $Re) { |
649 | $z->{'cartesian'} = [ $Re, ${$z->cartesian}[1] ]; | |
650 | $z->{c_dirty} = 0; | |
651 | $z->{p_dirty} = 1; | |
652 | } else { | |
653 | return ${$z->cartesian}[0]; | |
654 | } | |
a0d0e21e LW |
655 | } |
656 | ||
66730be0 RM |
657 | # |
658 | # Im | |
659 | # | |
b42d0ec9 | 660 | # Return or set Im(z). |
66730be0 | 661 | # |
a0d0e21e | 662 | sub Im { |
b42d0ec9 JH |
663 | my ($z, $Im) = @_; |
664 | return $z unless ref $z; | |
665 | if (defined $Im) { | |
666 | $z->{'cartesian'} = [ ${$z->cartesian}[0], $Im ]; | |
667 | $z->{c_dirty} = 0; | |
668 | $z->{p_dirty} = 1; | |
669 | } else { | |
670 | return ${$z->cartesian}[1]; | |
671 | } | |
672 | } | |
673 | ||
674 | # | |
675 | # rho | |
676 | # | |
677 | # Return or set rho(w). | |
678 | # | |
679 | sub rho { | |
680 | Math::Complex::abs(@_); | |
681 | } | |
682 | ||
683 | # | |
684 | # theta | |
685 | # | |
686 | # Return or set theta(w). | |
687 | # | |
688 | sub theta { | |
689 | Math::Complex::arg(@_); | |
a0d0e21e LW |
690 | } |
691 | ||
66730be0 RM |
692 | # |
693 | # (exp) | |
694 | # | |
695 | # Computes exp(z). | |
696 | # | |
697 | sub exp { | |
698 | my ($z) = @_; | |
699 | my ($x, $y) = @{$z->cartesian}; | |
a8693bd3 | 700 | return (ref $z)->emake(CORE::exp($x), $y); |
66730be0 RM |
701 | } |
702 | ||
703 | # | |
8c03c583 JH |
704 | # _logofzero |
705 | # | |
fb73857a | 706 | # Die on logarithm of zero. |
8c03c583 JH |
707 | # |
708 | sub _logofzero { | |
709 | my $mess = "$_[0]: Logarithm of zero.\n"; | |
710 | ||
711 | if (defined $_[1]) { | |
712 | $mess .= "(Because in the definition of $_[0], the argument "; | |
713 | $mess .= "$_[1] " unless ($_[1] eq '0'); | |
714 | $mess .= "is 0)\n"; | |
715 | } | |
716 | ||
717 | my @up = caller(1); | |
fb73857a | 718 | |
8c03c583 JH |
719 | $mess .= "Died at $up[1] line $up[2].\n"; |
720 | ||
721 | die $mess; | |
722 | } | |
723 | ||
724 | # | |
66730be0 RM |
725 | # (log) |
726 | # | |
727 | # Compute log(z). | |
728 | # | |
729 | sub log { | |
730 | my ($z) = @_; | |
fb73857a | 731 | unless (ref $z) { |
732 | _logofzero("log") if $z == 0; | |
a8693bd3 | 733 | return $z > 0 ? CORE::log($z) : cplx(CORE::log(-$z), pi); |
fb73857a | 734 | } |
5cd24f17 | 735 | my ($r, $t) = @{$z->polar}; |
fb73857a | 736 | _logofzero("log") if $r == 0; |
737 | if ($t > pi()) { $t -= pit2 } | |
738 | elsif ($t <= -pi()) { $t += pit2 } | |
a8693bd3 | 739 | return (ref $z)->make(CORE::log($r), $t); |
66730be0 RM |
740 | } |
741 | ||
742 | # | |
0c721ce2 JH |
743 | # ln |
744 | # | |
745 | # Alias for log(). | |
746 | # | |
747 | sub ln { Math::Complex::log(@_) } | |
748 | ||
749 | # | |
66730be0 RM |
750 | # log10 |
751 | # | |
752 | # Compute log10(z). | |
753 | # | |
5cd24f17 | 754 | |
66730be0 | 755 | sub log10 { |
fb73857a | 756 | return Math::Complex::log($_[0]) * uplog10; |
66730be0 RM |
757 | } |
758 | ||
759 | # | |
760 | # logn | |
761 | # | |
762 | # Compute logn(z,n) = log(z) / log(n) | |
763 | # | |
764 | sub logn { | |
765 | my ($z, $n) = @_; | |
0c721ce2 | 766 | $z = cplx($z, 0) unless ref $z; |
9fbe1b12 JH |
767 | my $logn = $LOGN{$n}; |
768 | $logn = $LOGN{$n} = CORE::log($n) unless defined $logn; # Cache log(n) | |
1fa12f56 | 769 | return &log($z) / $logn; |
66730be0 RM |
770 | } |
771 | ||
772 | # | |
773 | # (cos) | |
774 | # | |
775 | # Compute cos(z) = (exp(iz) + exp(-iz))/2. | |
776 | # | |
777 | sub cos { | |
778 | my ($z) = @_; | |
1fa12f56 | 779 | return CORE::cos($z) unless ref $z; |
66730be0 | 780 | my ($x, $y) = @{$z->cartesian}; |
a8693bd3 | 781 | my $ey = CORE::exp($y); |
1fa12f56 JH |
782 | my $sx = CORE::sin($x); |
783 | my $cx = CORE::cos($x); | |
784 | my $ey_1 = $ey ? 1 / $ey : $Inf; | |
785 | return (ref $z)->make($cx * ($ey + $ey_1)/2, | |
786 | $sx * ($ey_1 - $ey)/2); | |
66730be0 RM |
787 | } |
788 | ||
789 | # | |
790 | # (sin) | |
791 | # | |
792 | # Compute sin(z) = (exp(iz) - exp(-iz))/2. | |
793 | # | |
794 | sub sin { | |
795 | my ($z) = @_; | |
1fa12f56 | 796 | return CORE::sin($z) unless ref $z; |
66730be0 | 797 | my ($x, $y) = @{$z->cartesian}; |
a8693bd3 | 798 | my $ey = CORE::exp($y); |
1fa12f56 JH |
799 | my $sx = CORE::sin($x); |
800 | my $cx = CORE::cos($x); | |
801 | my $ey_1 = $ey ? 1 / $ey : $Inf; | |
802 | return (ref $z)->make($sx * ($ey + $ey_1)/2, | |
803 | $cx * ($ey - $ey_1)/2); | |
66730be0 RM |
804 | } |
805 | ||
806 | # | |
807 | # tan | |
808 | # | |
809 | # Compute tan(z) = sin(z) / cos(z). | |
810 | # | |
811 | sub tan { | |
812 | my ($z) = @_; | |
1fa12f56 JH |
813 | my $cz = &cos($z); |
814 | _divbyzero "tan($z)", "cos($z)" if $cz == 0; | |
815 | return &sin($z) / $cz; | |
66730be0 RM |
816 | } |
817 | ||
818 | # | |
0c721ce2 JH |
819 | # sec |
820 | # | |
821 | # Computes the secant sec(z) = 1 / cos(z). | |
822 | # | |
823 | sub sec { | |
824 | my ($z) = @_; | |
1fa12f56 | 825 | my $cz = &cos($z); |
0e505df1 | 826 | _divbyzero "sec($z)", "cos($z)" if ($cz == 0); |
0c721ce2 JH |
827 | return 1 / $cz; |
828 | } | |
829 | ||
830 | # | |
831 | # csc | |
832 | # | |
833 | # Computes the cosecant csc(z) = 1 / sin(z). | |
834 | # | |
835 | sub csc { | |
836 | my ($z) = @_; | |
1fa12f56 | 837 | my $sz = &sin($z); |
0e505df1 | 838 | _divbyzero "csc($z)", "sin($z)" if ($sz == 0); |
0c721ce2 JH |
839 | return 1 / $sz; |
840 | } | |
841 | ||
66730be0 | 842 | # |
0c721ce2 | 843 | # cosec |
66730be0 | 844 | # |
0c721ce2 JH |
845 | # Alias for csc(). |
846 | # | |
847 | sub cosec { Math::Complex::csc(@_) } | |
848 | ||
849 | # | |
850 | # cot | |
851 | # | |
fb73857a | 852 | # Computes cot(z) = cos(z) / sin(z). |
0c721ce2 JH |
853 | # |
854 | sub cot { | |
66730be0 | 855 | my ($z) = @_; |
1fa12f56 | 856 | my $sz = &sin($z); |
0e505df1 | 857 | _divbyzero "cot($z)", "sin($z)" if ($sz == 0); |
1fa12f56 | 858 | return &cos($z) / $sz; |
66730be0 RM |
859 | } |
860 | ||
861 | # | |
0c721ce2 JH |
862 | # cotan |
863 | # | |
864 | # Alias for cot(). | |
865 | # | |
866 | sub cotan { Math::Complex::cot(@_) } | |
867 | ||
868 | # | |
66730be0 RM |
869 | # acos |
870 | # | |
871 | # Computes the arc cosine acos(z) = -i log(z + sqrt(z*z-1)). | |
872 | # | |
873 | sub acos { | |
fb73857a | 874 | my $z = $_[0]; |
1fa12f56 JH |
875 | return CORE::atan2(CORE::sqrt(1-$z*$z), $z) |
876 | if (! ref $z) && CORE::abs($z) <= 1; | |
40b904b7 JH |
877 | $z = cplx($z, 0) unless ref $z; |
878 | my ($x, $y) = @{$z->cartesian}; | |
1fa12f56 | 879 | return 0 if $x == 1 && $y == 0; |
a8693bd3 NIS |
880 | my $t1 = CORE::sqrt(($x+1)*($x+1) + $y*$y); |
881 | my $t2 = CORE::sqrt(($x-1)*($x-1) + $y*$y); | |
fb73857a | 882 | my $alpha = ($t1 + $t2)/2; |
883 | my $beta = ($t1 - $t2)/2; | |
884 | $alpha = 1 if $alpha < 1; | |
885 | if ($beta > 1) { $beta = 1 } | |
886 | elsif ($beta < -1) { $beta = -1 } | |
a8693bd3 NIS |
887 | my $u = CORE::atan2(CORE::sqrt(1-$beta*$beta), $beta); |
888 | my $v = CORE::log($alpha + CORE::sqrt($alpha*$alpha-1)); | |
fb73857a | 889 | $v = -$v if $y > 0 || ($y == 0 && $x < -1); |
40b904b7 | 890 | return (ref $z)->make($u, $v); |
66730be0 RM |
891 | } |
892 | ||
893 | # | |
894 | # asin | |
895 | # | |
896 | # Computes the arc sine asin(z) = -i log(iz + sqrt(1-z*z)). | |
897 | # | |
898 | sub asin { | |
fb73857a | 899 | my $z = $_[0]; |
1fa12f56 JH |
900 | return CORE::atan2($z, CORE::sqrt(1-$z*$z)) |
901 | if (! ref $z) && CORE::abs($z) <= 1; | |
40b904b7 JH |
902 | $z = cplx($z, 0) unless ref $z; |
903 | my ($x, $y) = @{$z->cartesian}; | |
1fa12f56 | 904 | return 0 if $x == 0 && $y == 0; |
a8693bd3 NIS |
905 | my $t1 = CORE::sqrt(($x+1)*($x+1) + $y*$y); |
906 | my $t2 = CORE::sqrt(($x-1)*($x-1) + $y*$y); | |
fb73857a | 907 | my $alpha = ($t1 + $t2)/2; |
908 | my $beta = ($t1 - $t2)/2; | |
909 | $alpha = 1 if $alpha < 1; | |
910 | if ($beta > 1) { $beta = 1 } | |
911 | elsif ($beta < -1) { $beta = -1 } | |
a8693bd3 NIS |
912 | my $u = CORE::atan2($beta, CORE::sqrt(1-$beta*$beta)); |
913 | my $v = -CORE::log($alpha + CORE::sqrt($alpha*$alpha-1)); | |
fb73857a | 914 | $v = -$v if $y > 0 || ($y == 0 && $x < -1); |
40b904b7 | 915 | return (ref $z)->make($u, $v); |
66730be0 RM |
916 | } |
917 | ||
918 | # | |
919 | # atan | |
920 | # | |
0c721ce2 | 921 | # Computes the arc tangent atan(z) = i/2 log((i+z) / (i-z)). |
66730be0 RM |
922 | # |
923 | sub atan { | |
924 | my ($z) = @_; | |
a8693bd3 | 925 | return CORE::atan2($z, 1) unless ref $z; |
1fa12f56 JH |
926 | my ($x, $y) = ref $z ? @{$z->cartesian} : ($z, 0); |
927 | return 0 if $x == 0 && $y == 0; | |
8c03c583 | 928 | _divbyzero "atan(i)" if ( $z == i); |
1fa12f56 JH |
929 | _logofzero "atan(-i)" if (-$z == i); # -i is a bad file test... |
930 | my $log = &log((i + $z) / (i - $z)); | |
931 | return ip2 * $log; | |
a0d0e21e LW |
932 | } |
933 | ||
66730be0 | 934 | # |
0c721ce2 JH |
935 | # asec |
936 | # | |
937 | # Computes the arc secant asec(z) = acos(1 / z). | |
938 | # | |
939 | sub asec { | |
940 | my ($z) = @_; | |
0e505df1 | 941 | _divbyzero "asec($z)", $z if ($z == 0); |
fb73857a | 942 | return acos(1 / $z); |
0c721ce2 JH |
943 | } |
944 | ||
945 | # | |
5cd24f17 | 946 | # acsc |
0c721ce2 | 947 | # |
8c03c583 | 948 | # Computes the arc cosecant acsc(z) = asin(1 / z). |
0c721ce2 | 949 | # |
5cd24f17 | 950 | sub acsc { |
0c721ce2 | 951 | my ($z) = @_; |
0e505df1 | 952 | _divbyzero "acsc($z)", $z if ($z == 0); |
fb73857a | 953 | return asin(1 / $z); |
0c721ce2 JH |
954 | } |
955 | ||
956 | # | |
5cd24f17 | 957 | # acosec |
66730be0 | 958 | # |
5cd24f17 | 959 | # Alias for acsc(). |
0c721ce2 | 960 | # |
5cd24f17 | 961 | sub acosec { Math::Complex::acsc(@_) } |
0c721ce2 | 962 | |
66730be0 | 963 | # |
0c721ce2 JH |
964 | # acot |
965 | # | |
8c03c583 | 966 | # Computes the arc cotangent acot(z) = atan(1 / z) |
0c721ce2 JH |
967 | # |
968 | sub acot { | |
66730be0 | 969 | my ($z) = @_; |
1fa12f56 JH |
970 | _divbyzero "acot(0)" if $z == 0; |
971 | return ($z >= 0) ? CORE::atan2(1, $z) : CORE::atan2(-1, -$z) | |
972 | unless ref $z; | |
973 | _divbyzero "acot(i)" if ($z - i == 0); | |
974 | _logofzero "acot(-i)" if ($z + i == 0); | |
8c03c583 | 975 | return atan(1 / $z); |
66730be0 RM |
976 | } |
977 | ||
978 | # | |
0c721ce2 JH |
979 | # acotan |
980 | # | |
981 | # Alias for acot(). | |
982 | # | |
983 | sub acotan { Math::Complex::acot(@_) } | |
984 | ||
985 | # | |
66730be0 RM |
986 | # cosh |
987 | # | |
988 | # Computes the hyperbolic cosine cosh(z) = (exp(z) + exp(-z))/2. | |
989 | # | |
990 | sub cosh { | |
991 | my ($z) = @_; | |
fb73857a | 992 | my $ex; |
0e505df1 | 993 | unless (ref $z) { |
a8693bd3 | 994 | $ex = CORE::exp($z); |
1fa12f56 | 995 | return $ex ? ($ex + 1/$ex)/2 : $Inf; |
0e505df1 JH |
996 | } |
997 | my ($x, $y) = @{$z->cartesian}; | |
a8693bd3 | 998 | $ex = CORE::exp($x); |
1fa12f56 | 999 | my $ex_1 = $ex ? 1 / $ex : $Inf; |
a8693bd3 NIS |
1000 | return (ref $z)->make(CORE::cos($y) * ($ex + $ex_1)/2, |
1001 | CORE::sin($y) * ($ex - $ex_1)/2); | |
66730be0 RM |
1002 | } |
1003 | ||
1004 | # | |
1005 | # sinh | |
1006 | # | |
1007 | # Computes the hyperbolic sine sinh(z) = (exp(z) - exp(-z))/2. | |
1008 | # | |
1009 | sub sinh { | |
1010 | my ($z) = @_; | |
fb73857a | 1011 | my $ex; |
0e505df1 | 1012 | unless (ref $z) { |
1fa12f56 | 1013 | return 0 if $z == 0; |
a8693bd3 | 1014 | $ex = CORE::exp($z); |
1fa12f56 | 1015 | return $ex ? ($ex - 1/$ex)/2 : "-$Inf"; |
0e505df1 JH |
1016 | } |
1017 | my ($x, $y) = @{$z->cartesian}; | |
1fa12f56 JH |
1018 | my $cy = CORE::cos($y); |
1019 | my $sy = CORE::sin($y); | |
a8693bd3 | 1020 | $ex = CORE::exp($x); |
1fa12f56 | 1021 | my $ex_1 = $ex ? 1 / $ex : $Inf; |
5240e574 JH |
1022 | return (ref $z)->make(CORE::cos($y) * ($ex - $ex_1)/2, |
1023 | CORE::sin($y) * ($ex + $ex_1)/2); | |
66730be0 RM |
1024 | } |
1025 | ||
1026 | # | |
1027 | # tanh | |
1028 | # | |
1029 | # Computes the hyperbolic tangent tanh(z) = sinh(z) / cosh(z). | |
1030 | # | |
1031 | sub tanh { | |
1032 | my ($z) = @_; | |
0c721ce2 | 1033 | my $cz = cosh($z); |
0e505df1 | 1034 | _divbyzero "tanh($z)", "cosh($z)" if ($cz == 0); |
0c721ce2 | 1035 | return sinh($z) / $cz; |
66730be0 RM |
1036 | } |
1037 | ||
1038 | # | |
0c721ce2 JH |
1039 | # sech |
1040 | # | |
1041 | # Computes the hyperbolic secant sech(z) = 1 / cosh(z). | |
1042 | # | |
1043 | sub sech { | |
1044 | my ($z) = @_; | |
1045 | my $cz = cosh($z); | |
0e505df1 | 1046 | _divbyzero "sech($z)", "cosh($z)" if ($cz == 0); |
0c721ce2 JH |
1047 | return 1 / $cz; |
1048 | } | |
1049 | ||
1050 | # | |
1051 | # csch | |
1052 | # | |
1053 | # Computes the hyperbolic cosecant csch(z) = 1 / sinh(z). | |
66730be0 | 1054 | # |
0c721ce2 JH |
1055 | sub csch { |
1056 | my ($z) = @_; | |
1057 | my $sz = sinh($z); | |
0e505df1 | 1058 | _divbyzero "csch($z)", "sinh($z)" if ($sz == 0); |
0c721ce2 JH |
1059 | return 1 / $sz; |
1060 | } | |
1061 | ||
1062 | # | |
1063 | # cosech | |
1064 | # | |
1065 | # Alias for csch(). | |
1066 | # | |
1067 | sub cosech { Math::Complex::csch(@_) } | |
1068 | ||
66730be0 | 1069 | # |
0c721ce2 JH |
1070 | # coth |
1071 | # | |
1072 | # Computes the hyperbolic cotangent coth(z) = cosh(z) / sinh(z). | |
1073 | # | |
1074 | sub coth { | |
66730be0 | 1075 | my ($z) = @_; |
0c721ce2 | 1076 | my $sz = sinh($z); |
1fa12f56 | 1077 | _divbyzero "coth($z)", "sinh($z)" if $sz == 0; |
0c721ce2 | 1078 | return cosh($z) / $sz; |
66730be0 RM |
1079 | } |
1080 | ||
1081 | # | |
0c721ce2 JH |
1082 | # cotanh |
1083 | # | |
1084 | # Alias for coth(). | |
1085 | # | |
1086 | sub cotanh { Math::Complex::coth(@_) } | |
1087 | ||
1088 | # | |
66730be0 RM |
1089 | # acosh |
1090 | # | |
fb73857a | 1091 | # Computes the arc hyperbolic cosine acosh(z) = log(z + sqrt(z*z-1)). |
66730be0 RM |
1092 | # |
1093 | sub acosh { | |
1094 | my ($z) = @_; | |
fb73857a | 1095 | unless (ref $z) { |
fb73857a | 1096 | $z = cplx($z, 0); |
1097 | } | |
8c03c583 | 1098 | my ($re, $im) = @{$z->cartesian}; |
fb73857a | 1099 | if ($im == 0) { |
1fa12f56 JH |
1100 | return CORE::log($re + CORE::sqrt($re*$re - 1)) |
1101 | if $re >= 1; | |
1102 | return cplx(0, CORE::atan2(CORE::sqrt(1 - $re*$re), $re)) | |
1103 | if CORE::abs($re) < 1; | |
fb73857a | 1104 | } |
9bc5fa8d | 1105 | my $t = &sqrt($z * $z - 1) + $z; |
40b904b7 JH |
1106 | # Try Taylor if looking bad (this usually means that |
1107 | # $z was large negative, therefore the sqrt is really | |
1108 | # close to abs(z), summing that with z...) | |
9bc5fa8d JH |
1109 | $t = 1/(2 * $z) - 1/(8 * $z**3) + 1/(16 * $z**5) - 5/(128 * $z**7) |
1110 | if $t == 0; | |
1111 | my $u = &log($t); | |
40b904b7 | 1112 | $u->Im(-$u->Im) if $re < 0 && $im == 0; |
9bc5fa8d | 1113 | return $re < 0 ? -$u : $u; |
66730be0 RM |
1114 | } |
1115 | ||
1116 | # | |
1117 | # asinh | |
1118 | # | |
1fa12f56 | 1119 | # Computes the arc hyperbolic sine asinh(z) = log(z + sqrt(z*z+1)) |
66730be0 RM |
1120 | # |
1121 | sub asinh { | |
1122 | my ($z) = @_; | |
1fa12f56 JH |
1123 | unless (ref $z) { |
1124 | my $t = $z + CORE::sqrt($z*$z + 1); | |
1125 | return CORE::log($t) if $t; | |
1126 | } | |
9bc5fa8d | 1127 | my $t = &sqrt($z * $z + 1) + $z; |
40b904b7 JH |
1128 | # Try Taylor if looking bad (this usually means that |
1129 | # $z was large negative, therefore the sqrt is really | |
1130 | # close to abs(z), summing that with z...) | |
9bc5fa8d JH |
1131 | $t = 1/(2 * $z) - 1/(8 * $z**3) + 1/(16 * $z**5) - 5/(128 * $z**7) |
1132 | if $t == 0; | |
1fa12f56 | 1133 | return &log($t); |
66730be0 RM |
1134 | } |
1135 | ||
1136 | # | |
1137 | # atanh | |
1138 | # | |
1139 | # Computes the arc hyperbolic tangent atanh(z) = 1/2 log((1+z) / (1-z)). | |
1140 | # | |
1141 | sub atanh { | |
1142 | my ($z) = @_; | |
fb73857a | 1143 | unless (ref $z) { |
a8693bd3 | 1144 | return CORE::log((1 + $z)/(1 - $z))/2 if CORE::abs($z) < 1; |
fb73857a | 1145 | $z = cplx($z, 0); |
1146 | } | |
1fa12f56 JH |
1147 | _divbyzero 'atanh(1)', "1 - $z" if (1 - $z == 0); |
1148 | _logofzero 'atanh(-1)' if (1 + $z == 0); | |
1149 | return 0.5 * &log((1 + $z) / (1 - $z)); | |
66730be0 RM |
1150 | } |
1151 | ||
1152 | # | |
0c721ce2 JH |
1153 | # asech |
1154 | # | |
1155 | # Computes the hyperbolic arc secant asech(z) = acosh(1 / z). | |
1156 | # | |
1157 | sub asech { | |
1158 | my ($z) = @_; | |
1fa12f56 | 1159 | _divbyzero 'asech(0)', "$z" if ($z == 0); |
0c721ce2 JH |
1160 | return acosh(1 / $z); |
1161 | } | |
1162 | ||
1163 | # | |
1164 | # acsch | |
66730be0 | 1165 | # |
0c721ce2 | 1166 | # Computes the hyperbolic arc cosecant acsch(z) = asinh(1 / z). |
66730be0 | 1167 | # |
0c721ce2 | 1168 | sub acsch { |
66730be0 | 1169 | my ($z) = @_; |
0e505df1 | 1170 | _divbyzero 'acsch(0)', $z if ($z == 0); |
0c721ce2 JH |
1171 | return asinh(1 / $z); |
1172 | } | |
1173 | ||
1174 | # | |
1175 | # acosech | |
1176 | # | |
1177 | # Alias for acosh(). | |
1178 | # | |
1179 | sub acosech { Math::Complex::acsch(@_) } | |
1180 | ||
1181 | # | |
1182 | # acoth | |
1183 | # | |
1184 | # Computes the arc hyperbolic cotangent acoth(z) = 1/2 log((1+z) / (z-1)). | |
1185 | # | |
1186 | sub acoth { | |
1187 | my ($z) = @_; | |
1fa12f56 | 1188 | _divbyzero 'acoth(0)' if ($z == 0); |
fb73857a | 1189 | unless (ref $z) { |
a8693bd3 | 1190 | return CORE::log(($z + 1)/($z - 1))/2 if CORE::abs($z) > 1; |
fb73857a | 1191 | $z = cplx($z, 0); |
1192 | } | |
1fa12f56 JH |
1193 | _divbyzero 'acoth(1)', "$z - 1" if ($z - 1 == 0); |
1194 | _logofzero 'acoth(-1)', "1 + $z" if (1 + $z == 0); | |
1195 | return &log((1 + $z) / ($z - 1)) / 2; | |
66730be0 RM |
1196 | } |
1197 | ||
1198 | # | |
0c721ce2 JH |
1199 | # acotanh |
1200 | # | |
1201 | # Alias for acot(). | |
1202 | # | |
1203 | sub acotanh { Math::Complex::acoth(@_) } | |
1204 | ||
1205 | # | |
66730be0 RM |
1206 | # (atan2) |
1207 | # | |
1208 | # Compute atan(z1/z2). | |
1209 | # | |
1210 | sub atan2 { | |
1211 | my ($z1, $z2, $inverted) = @_; | |
fb73857a | 1212 | my ($re1, $im1, $re2, $im2); |
1213 | if ($inverted) { | |
1214 | ($re1, $im1) = ref $z2 ? @{$z2->cartesian} : ($z2, 0); | |
1215 | ($re2, $im2) = @{$z1->cartesian}; | |
66730be0 | 1216 | } else { |
fb73857a | 1217 | ($re1, $im1) = @{$z1->cartesian}; |
1218 | ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0); | |
1219 | } | |
1220 | if ($im2 == 0) { | |
1fa12f56 JH |
1221 | return CORE::atan2($re1, $re2) if $im1 == 0; |
1222 | return ($im1<=>0) * pip2 if $re2 == 0; | |
66730be0 | 1223 | } |
fb73857a | 1224 | my $w = atan($z1/$z2); |
1225 | my ($u, $v) = ref $w ? @{$w->cartesian} : ($w, 0); | |
1226 | $u += pi if $re2 < 0; | |
1227 | $u -= pit2 if $u > pi; | |
1228 | return cplx($u, $v); | |
66730be0 RM |
1229 | } |
1230 | ||
1231 | # | |
1232 | # display_format | |
1233 | # ->display_format | |
1234 | # | |
16357284 | 1235 | # Set (get if no argument) the display format for all complex numbers that |
fb73857a | 1236 | # don't happen to have overridden it via ->display_format |
66730be0 | 1237 | # |
16357284 | 1238 | # When called as an object method, this actually sets the display format for |
66730be0 RM |
1239 | # the current object. |
1240 | # | |
1241 | # Valid object formats are 'c' and 'p' for cartesian and polar. The first | |
1242 | # letter is used actually, so the type can be fully spelled out for clarity. | |
1243 | # | |
1244 | sub display_format { | |
16357284 JH |
1245 | my $self = shift; |
1246 | my %display_format = %DISPLAY_FORMAT; | |
66730be0 | 1247 | |
16357284 JH |
1248 | if (ref $self) { # Called as an object method |
1249 | if (exists $self->{display_format}) { | |
1250 | my %obj = %{$self->{display_format}}; | |
1251 | @display_format{keys %obj} = values %obj; | |
1252 | } | |
1253 | if (@_ == 1) { | |
1254 | $display_format{style} = shift; | |
1255 | } else { | |
1256 | my %new = @_; | |
1257 | @display_format{keys %new} = values %new; | |
1258 | } | |
1259 | } else { # Called as a class method | |
1260 | if (@_ = 1) { | |
1261 | $display_format{style} = $self; | |
1262 | } else { | |
1263 | my %new = @_; | |
1264 | @display_format{keys %new} = values %new; | |
1265 | } | |
1266 | undef $self; | |
66730be0 RM |
1267 | } |
1268 | ||
1269 | if (defined $self) { | |
16357284 JH |
1270 | $self->{display_format} = { %display_format }; |
1271 | return | |
1272 | wantarray ? | |
1273 | %{$self->{display_format}} : | |
1274 | $self->{display_format}->{style}; | |
66730be0 RM |
1275 | } |
1276 | ||
16357284 JH |
1277 | %DISPLAY_FORMAT = %display_format; |
1278 | return | |
1279 | wantarray ? | |
1280 | %DISPLAY_FORMAT : | |
1281 | $DISPLAY_FORMAT{style}; | |
66730be0 RM |
1282 | } |
1283 | ||
1284 | # | |
1285 | # (stringify) | |
1286 | # | |
1287 | # Show nicely formatted complex number under its cartesian or polar form, | |
1288 | # depending on the current display format: | |
1289 | # | |
1290 | # . If a specific display format has been recorded for this object, use it. | |
1291 | # . Otherwise, use the generic current default for all complex numbers, | |
1292 | # which is a package global variable. | |
1293 | # | |
a0d0e21e | 1294 | sub stringify { |
66730be0 | 1295 | my ($z) = shift; |
66730be0 | 1296 | |
16357284 JH |
1297 | my $style = $z->display_format; |
1298 | ||
1299 | $style = $DISPLAY_FORMAT{style} unless defined $style; | |
66730be0 | 1300 | |
16357284 | 1301 | return $z->stringify_polar if $style =~ /^p/i; |
66730be0 RM |
1302 | return $z->stringify_cartesian; |
1303 | } | |
1304 | ||
1305 | # | |
1306 | # ->stringify_cartesian | |
1307 | # | |
1308 | # Stringify as a cartesian representation 'a+bi'. | |
1309 | # | |
1310 | sub stringify_cartesian { | |
1311 | my $z = shift; | |
1312 | my ($x, $y) = @{$z->cartesian}; | |
1313 | my ($re, $im); | |
1314 | ||
16357284 JH |
1315 | my %format = $z->display_format; |
1316 | my $format = $format{format}; | |
1317 | ||
1fa12f56 JH |
1318 | if ($x) { |
1319 | if ($x =~ /^NaN[QS]?$/i) { | |
1320 | $re = $x; | |
1321 | } else { | |
1322 | if ($x =~ /^-?$Inf$/oi) { | |
1323 | $re = $x; | |
1324 | } else { | |
1325 | $re = defined $format ? sprintf($format, $x) : $x; | |
1326 | } | |
1327 | } | |
1328 | } else { | |
1329 | undef $re; | |
1330 | } | |
1331 | ||
1332 | if ($y) { | |
40b904b7 | 1333 | if ($y =~ /^(NaN[QS]?)$/i) { |
1fa12f56 JH |
1334 | $im = $y; |
1335 | } else { | |
1336 | if ($y =~ /^-?$Inf$/oi) { | |
1337 | $im = $y; | |
1338 | } else { | |
40b904b7 JH |
1339 | $im = |
1340 | defined $format ? | |
1341 | sprintf($format, $y) : | |
1342 | ($y == 1 ? "" : ($y == -1 ? "-" : $y)); | |
1fa12f56 JH |
1343 | } |
1344 | } | |
1345 | $im .= "i"; | |
1346 | } else { | |
1347 | undef $im; | |
16357284 | 1348 | } |
66730be0 | 1349 | |
1fa12f56 JH |
1350 | my $str = $re; |
1351 | ||
16357284 JH |
1352 | if (defined $im) { |
1353 | if ($y < 0) { | |
1354 | $str .= $im; | |
1fa12f56 | 1355 | } elsif ($y > 0 || $im =~ /^NaN[QS]?i$/i) { |
16357284 JH |
1356 | $str .= "+" if defined $re; |
1357 | $str .= $im; | |
1358 | } | |
1fa12f56 JH |
1359 | } elsif (!defined $re) { |
1360 | $str = "0"; | |
16357284 | 1361 | } |
66730be0 RM |
1362 | |
1363 | return $str; | |
1364 | } | |
1365 | ||
d09ae4e6 | 1366 | |
66730be0 RM |
1367 | # |
1368 | # ->stringify_polar | |
1369 | # | |
1370 | # Stringify as a polar representation '[r,t]'. | |
1371 | # | |
1372 | sub stringify_polar { | |
1373 | my $z = shift; | |
1374 | my ($r, $t) = @{$z->polar}; | |
1375 | my $theta; | |
1376 | ||
16357284 | 1377 | my %format = $z->display_format; |
1fa12f56 | 1378 | my $format = $format{format}; |
16357284 | 1379 | |
1fa12f56 JH |
1380 | if ($t =~ /^NaN[QS]?$/i || $t =~ /^-?$Inf$/oi) { |
1381 | $theta = $t; | |
1382 | } elsif ($t == pi) { | |
1383 | $theta = "pi"; | |
1384 | } elsif ($r == 0 || $t == 0) { | |
1385 | $theta = defined $format ? sprintf($format, $t) : $t; | |
55497cff | 1386 | } |
66730be0 | 1387 | |
1fa12f56 JH |
1388 | return "[$r,$theta]" if defined $theta; |
1389 | ||
66730be0 | 1390 | # |
1fa12f56 | 1391 | # Try to identify pi/n and friends. |
66730be0 RM |
1392 | # |
1393 | ||
1fa12f56 JH |
1394 | $t -= int(CORE::abs($t) / pit2) * pit2; |
1395 | ||
1396 | if ($format{polar_pretty_print}) { | |
1397 | my ($a, $b); | |
9bc5fa8d | 1398 | for $a (2..9) { |
1fa12f56 JH |
1399 | $b = $t * $a / pi; |
1400 | if (int($b) == $b) { | |
1401 | $b = $b < 0 ? "-" : "" if CORE::abs($b) == 1; | |
1402 | $theta = "${b}pi/$a"; | |
d09ae4e6 | 1403 | last; |
66730be0 | 1404 | } |
d09ae4e6 | 1405 | } |
66730be0 RM |
1406 | } |
1407 | ||
16357284 JH |
1408 | if (defined $format) { |
1409 | $r = sprintf($format, $r); | |
1fa12f56 JH |
1410 | $theta = sprintf($format, $theta) unless defined $theta; |
1411 | } else { | |
1412 | $theta = $t unless defined $theta; | |
16357284 JH |
1413 | } |
1414 | ||
1fa12f56 | 1415 | return "[$r,$theta]"; |
a0d0e21e | 1416 | } |
a5f75d66 AD |
1417 | |
1418 | 1; | |
1419 | __END__ | |
1420 | ||
1421 | =head1 NAME | |
1422 | ||
66730be0 | 1423 | Math::Complex - complex numbers and associated mathematical functions |
a5f75d66 AD |
1424 | |
1425 | =head1 SYNOPSIS | |
1426 | ||
66730be0 | 1427 | use Math::Complex; |
fb73857a | 1428 | |
66730be0 RM |
1429 | $z = Math::Complex->make(5, 6); |
1430 | $t = 4 - 3*i + $z; | |
1431 | $j = cplxe(1, 2*pi/3); | |
a5f75d66 AD |
1432 | |
1433 | =head1 DESCRIPTION | |
1434 | ||
66730be0 RM |
1435 | This package lets you create and manipulate complex numbers. By default, |
1436 | I<Perl> limits itself to real numbers, but an extra C<use> statement brings | |
1437 | full complex support, along with a full set of mathematical functions | |
1438 | typically associated with and/or extended to complex numbers. | |
1439 | ||
1440 | If you wonder what complex numbers are, they were invented to be able to solve | |
1441 | the following equation: | |
1442 | ||
1443 | x*x = -1 | |
1444 | ||
1445 | and by definition, the solution is noted I<i> (engineers use I<j> instead since | |
1446 | I<i> usually denotes an intensity, but the name does not matter). The number | |
1447 | I<i> is a pure I<imaginary> number. | |
1448 | ||
1449 | The arithmetics with pure imaginary numbers works just like you would expect | |
1450 | it with real numbers... you just have to remember that | |
1451 | ||
1452 | i*i = -1 | |
1453 | ||
1454 | so you have: | |
1455 | ||
1456 | 5i + 7i = i * (5 + 7) = 12i | |
1457 | 4i - 3i = i * (4 - 3) = i | |
1458 | 4i * 2i = -8 | |
1459 | 6i / 2i = 3 | |
1460 | 1 / i = -i | |
1461 | ||
1462 | Complex numbers are numbers that have both a real part and an imaginary | |
1463 | part, and are usually noted: | |
1464 | ||
1465 | a + bi | |
1466 | ||
1467 | where C<a> is the I<real> part and C<b> is the I<imaginary> part. The | |
1468 | arithmetic with complex numbers is straightforward. You have to | |
1469 | keep track of the real and the imaginary parts, but otherwise the | |
1470 | rules used for real numbers just apply: | |
1471 | ||
1472 | (4 + 3i) + (5 - 2i) = (4 + 5) + i(3 - 2) = 9 + i | |
1473 | (2 + i) * (4 - i) = 2*4 + 4i -2i -i*i = 8 + 2i + 1 = 9 + 2i | |
1474 | ||
1475 | A graphical representation of complex numbers is possible in a plane | |
1476 | (also called the I<complex plane>, but it's really a 2D plane). | |
1477 | The number | |
1478 | ||
1479 | z = a + bi | |
1480 | ||
1481 | is the point whose coordinates are (a, b). Actually, it would | |
1482 | be the vector originating from (0, 0) to (a, b). It follows that the addition | |
1483 | of two complex numbers is a vectorial addition. | |
1484 | ||
1485 | Since there is a bijection between a point in the 2D plane and a complex | |
1486 | number (i.e. the mapping is unique and reciprocal), a complex number | |
1487 | can also be uniquely identified with polar coordinates: | |
1488 | ||
1489 | [rho, theta] | |
1490 | ||
1491 | where C<rho> is the distance to the origin, and C<theta> the angle between | |
1492 | the vector and the I<x> axis. There is a notation for this using the | |
1493 | exponential form, which is: | |
1494 | ||
1495 | rho * exp(i * theta) | |
1496 | ||
1497 | where I<i> is the famous imaginary number introduced above. Conversion | |
1498 | between this form and the cartesian form C<a + bi> is immediate: | |
1499 | ||
1500 | a = rho * cos(theta) | |
1501 | b = rho * sin(theta) | |
1502 | ||
1503 | which is also expressed by this formula: | |
1504 | ||
fb73857a | 1505 | z = rho * exp(i * theta) = rho * (cos theta + i * sin theta) |
66730be0 RM |
1506 | |
1507 | In other words, it's the projection of the vector onto the I<x> and I<y> | |
1508 | axes. Mathematicians call I<rho> the I<norm> or I<modulus> and I<theta> | |
1509 | the I<argument> of the complex number. The I<norm> of C<z> will be | |
1510 | noted C<abs(z)>. | |
1511 | ||
1512 | The polar notation (also known as the trigonometric | |
1513 | representation) is much more handy for performing multiplications and | |
1514 | divisions of complex numbers, whilst the cartesian notation is better | |
fb73857a | 1515 | suited for additions and subtractions. Real numbers are on the I<x> |
1516 | axis, and therefore I<theta> is zero or I<pi>. | |
66730be0 RM |
1517 | |
1518 | All the common operations that can be performed on a real number have | |
1519 | been defined to work on complex numbers as well, and are merely | |
1520 | I<extensions> of the operations defined on real numbers. This means | |
1521 | they keep their natural meaning when there is no imaginary part, provided | |
1522 | the number is within their definition set. | |
1523 | ||
1524 | For instance, the C<sqrt> routine which computes the square root of | |
fb73857a | 1525 | its argument is only defined for non-negative real numbers and yields a |
1526 | non-negative real number (it is an application from B<R+> to B<R+>). | |
66730be0 RM |
1527 | If we allow it to return a complex number, then it can be extended to |
1528 | negative real numbers to become an application from B<R> to B<C> (the | |
1529 | set of complex numbers): | |
1530 | ||
1531 | sqrt(x) = x >= 0 ? sqrt(x) : sqrt(-x)*i | |
1532 | ||
1533 | It can also be extended to be an application from B<C> to B<C>, | |
1534 | whilst its restriction to B<R> behaves as defined above by using | |
1535 | the following definition: | |
1536 | ||
1537 | sqrt(z = [r,t]) = sqrt(r) * exp(i * t/2) | |
1538 | ||
fb73857a | 1539 | Indeed, a negative real number can be noted C<[x,pi]> (the modulus |
1540 | I<x> is always non-negative, so C<[x,pi]> is really C<-x>, a negative | |
1541 | number) and the above definition states that | |
66730be0 RM |
1542 | |
1543 | sqrt([x,pi]) = sqrt(x) * exp(i*pi/2) = [sqrt(x),pi/2] = sqrt(x)*i | |
1544 | ||
1545 | which is exactly what we had defined for negative real numbers above. | |
b42d0ec9 JH |
1546 | The C<sqrt> returns only one of the solutions: if you want the both, |
1547 | use the C<root> function. | |
a5f75d66 | 1548 | |
66730be0 RM |
1549 | All the common mathematical functions defined on real numbers that |
1550 | are extended to complex numbers share that same property of working | |
1551 | I<as usual> when the imaginary part is zero (otherwise, it would not | |
1552 | be called an extension, would it?). | |
a5f75d66 | 1553 | |
66730be0 RM |
1554 | A I<new> operation possible on a complex number that is |
1555 | the identity for real numbers is called the I<conjugate>, and is noted | |
1556 | with an horizontal bar above the number, or C<~z> here. | |
a5f75d66 | 1557 | |
66730be0 RM |
1558 | z = a + bi |
1559 | ~z = a - bi | |
a5f75d66 | 1560 | |
66730be0 | 1561 | Simple... Now look: |
a5f75d66 | 1562 | |
66730be0 | 1563 | z * ~z = (a + bi) * (a - bi) = a*a + b*b |
a5f75d66 | 1564 | |
66730be0 RM |
1565 | We saw that the norm of C<z> was noted C<abs(z)> and was defined as the |
1566 | distance to the origin, also known as: | |
a5f75d66 | 1567 | |
66730be0 | 1568 | rho = abs(z) = sqrt(a*a + b*b) |
a5f75d66 | 1569 | |
66730be0 RM |
1570 | so |
1571 | ||
1572 | z * ~z = abs(z) ** 2 | |
1573 | ||
1574 | If z is a pure real number (i.e. C<b == 0>), then the above yields: | |
1575 | ||
1576 | a * a = abs(a) ** 2 | |
1577 | ||
1578 | which is true (C<abs> has the regular meaning for real number, i.e. stands | |
1579 | for the absolute value). This example explains why the norm of C<z> is | |
1580 | noted C<abs(z)>: it extends the C<abs> function to complex numbers, yet | |
1581 | is the regular C<abs> we know when the complex number actually has no | |
1582 | imaginary part... This justifies I<a posteriori> our use of the C<abs> | |
1583 | notation for the norm. | |
1584 | ||
1585 | =head1 OPERATIONS | |
1586 | ||
1587 | Given the following notations: | |
1588 | ||
1589 | z1 = a + bi = r1 * exp(i * t1) | |
1590 | z2 = c + di = r2 * exp(i * t2) | |
1591 | z = <any complex or real number> | |
1592 | ||
1593 | the following (overloaded) operations are supported on complex numbers: | |
1594 | ||
1595 | z1 + z2 = (a + c) + i(b + d) | |
1596 | z1 - z2 = (a - c) + i(b - d) | |
1597 | z1 * z2 = (r1 * r2) * exp(i * (t1 + t2)) | |
1598 | z1 / z2 = (r1 / r2) * exp(i * (t1 - t2)) | |
1599 | z1 ** z2 = exp(z2 * log z1) | |
b42d0ec9 JH |
1600 | ~z = a - bi |
1601 | abs(z) = r1 = sqrt(a*a + b*b) | |
1602 | sqrt(z) = sqrt(r1) * exp(i * t/2) | |
1603 | exp(z) = exp(a) * exp(i * b) | |
1604 | log(z) = log(r1) + i*t | |
1605 | sin(z) = 1/2i (exp(i * z1) - exp(-i * z)) | |
1606 | cos(z) = 1/2 (exp(i * z1) + exp(-i * z)) | |
66730be0 RM |
1607 | atan2(z1, z2) = atan(z1/z2) |
1608 | ||
1609 | The following extra operations are supported on both real and complex | |
1610 | numbers: | |
1611 | ||
1612 | Re(z) = a | |
1613 | Im(z) = b | |
1614 | arg(z) = t | |
b42d0ec9 | 1615 | abs(z) = r |
66730be0 RM |
1616 | |
1617 | cbrt(z) = z ** (1/3) | |
1618 | log10(z) = log(z) / log(10) | |
1619 | logn(z, n) = log(z) / log(n) | |
1620 | ||
1621 | tan(z) = sin(z) / cos(z) | |
0c721ce2 | 1622 | |
5aabfad6 | 1623 | csc(z) = 1 / sin(z) |
1624 | sec(z) = 1 / cos(z) | |
0c721ce2 | 1625 | cot(z) = 1 / tan(z) |
66730be0 RM |
1626 | |
1627 | asin(z) = -i * log(i*z + sqrt(1-z*z)) | |
fb73857a | 1628 | acos(z) = -i * log(z + i*sqrt(1-z*z)) |
66730be0 | 1629 | atan(z) = i/2 * log((i+z) / (i-z)) |
0c721ce2 | 1630 | |
5aabfad6 | 1631 | acsc(z) = asin(1 / z) |
1632 | asec(z) = acos(1 / z) | |
8c03c583 | 1633 | acot(z) = atan(1 / z) = -i/2 * log((i+z) / (z-i)) |
66730be0 RM |
1634 | |
1635 | sinh(z) = 1/2 (exp(z) - exp(-z)) | |
1636 | cosh(z) = 1/2 (exp(z) + exp(-z)) | |
0c721ce2 JH |
1637 | tanh(z) = sinh(z) / cosh(z) = (exp(z) - exp(-z)) / (exp(z) + exp(-z)) |
1638 | ||
5aabfad6 | 1639 | csch(z) = 1 / sinh(z) |
1640 | sech(z) = 1 / cosh(z) | |
0c721ce2 | 1641 | coth(z) = 1 / tanh(z) |
fb73857a | 1642 | |
66730be0 RM |
1643 | asinh(z) = log(z + sqrt(z*z+1)) |
1644 | acosh(z) = log(z + sqrt(z*z-1)) | |
1645 | atanh(z) = 1/2 * log((1+z) / (1-z)) | |
66730be0 | 1646 | |
5aabfad6 | 1647 | acsch(z) = asinh(1 / z) |
1648 | asech(z) = acosh(1 / z) | |
0c721ce2 JH |
1649 | acoth(z) = atanh(1 / z) = 1/2 * log((1+z) / (z-1)) |
1650 | ||
b42d0ec9 JH |
1651 | I<arg>, I<abs>, I<log>, I<csc>, I<cot>, I<acsc>, I<acot>, I<csch>, |
1652 | I<coth>, I<acosech>, I<acotanh>, have aliases I<rho>, I<theta>, I<ln>, | |
1653 | I<cosec>, I<cotan>, I<acosec>, I<acotan>, I<cosech>, I<cotanh>, | |
1654 | I<acosech>, I<acotanh>, respectively. C<Re>, C<Im>, C<arg>, C<abs>, | |
1655 | C<rho>, and C<theta> can be used also also mutators. The C<cbrt> | |
1656 | returns only one of the solutions: if you want all three, use the | |
1657 | C<root> function. | |
0c721ce2 JH |
1658 | |
1659 | The I<root> function is available to compute all the I<n> | |
66730be0 RM |
1660 | roots of some complex, where I<n> is a strictly positive integer. |
1661 | There are exactly I<n> such roots, returned as a list. Getting the | |
1662 | number mathematicians call C<j> such that: | |
1663 | ||
1664 | 1 + j + j*j = 0; | |
1665 | ||
1666 | is a simple matter of writing: | |
1667 | ||
1668 | $j = ((root(1, 3))[1]; | |
1669 | ||
1670 | The I<k>th root for C<z = [r,t]> is given by: | |
1671 | ||
1672 | (root(z, n))[k] = r**(1/n) * exp(i * (t + 2*k*pi)/n) | |
1673 | ||
f4837644 JH |
1674 | The I<spaceship> comparison operator, E<lt>=E<gt>, is also defined. In |
1675 | order to ensure its restriction to real numbers is conform to what you | |
1676 | would expect, the comparison is run on the real part of the complex | |
1677 | number first, and imaginary parts are compared only when the real | |
1678 | parts match. | |
66730be0 RM |
1679 | |
1680 | =head1 CREATION | |
1681 | ||
1682 | To create a complex number, use either: | |
1683 | ||
1684 | $z = Math::Complex->make(3, 4); | |
1685 | $z = cplx(3, 4); | |
1686 | ||
1687 | if you know the cartesian form of the number, or | |
1688 | ||
1689 | $z = 3 + 4*i; | |
1690 | ||
fb73857a | 1691 | if you like. To create a number using the polar form, use either: |
66730be0 RM |
1692 | |
1693 | $z = Math::Complex->emake(5, pi/3); | |
1694 | $x = cplxe(5, pi/3); | |
1695 | ||
0c721ce2 | 1696 | instead. The first argument is the modulus, the second is the angle |
fb73857a | 1697 | (in radians, the full circle is 2*pi). (Mnemonic: C<e> is used as a |
1698 | notation for complex numbers in the polar form). | |
66730be0 RM |
1699 | |
1700 | It is possible to write: | |
1701 | ||
1702 | $x = cplxe(-3, pi/4); | |
1703 | ||
16357284 JH |
1704 | but that will be silently converted into C<[3,-3pi/4]>, since the |
1705 | modulus must be non-negative (it represents the distance to the origin | |
1706 | in the complex plane). | |
66730be0 | 1707 | |
b42d0ec9 JH |
1708 | It is also possible to have a complex number as either argument of |
1709 | either the C<make> or C<emake>: the appropriate component of | |
1710 | the argument will be used. | |
1711 | ||
1712 | $z1 = cplx(-2, 1); | |
1713 | $z2 = cplx($z1, 4); | |
1714 | ||
66730be0 RM |
1715 | =head1 STRINGIFICATION |
1716 | ||
1717 | When printed, a complex number is usually shown under its cartesian | |
16357284 | 1718 | style I<a+bi>, but there are legitimate cases where the polar style |
66730be0 RM |
1719 | I<[r,t]> is more appropriate. |
1720 | ||
16357284 JH |
1721 | By calling the class method C<Math::Complex::display_format> and |
1722 | supplying either C<"polar"> or C<"cartesian"> as an argument, you | |
5287f86b | 1723 | override the default display style, which is C<"cartesian">. Not |
16357284 | 1724 | supplying any argument returns the current settings. |
66730be0 RM |
1725 | |
1726 | This default can be overridden on a per-number basis by calling the | |
1727 | C<display_format> method instead. As before, not supplying any argument | |
5287f86b JH |
1728 | returns the current display style for this number. Otherwise whatever you |
1729 | specify will be the new display style for I<this> particular number. | |
66730be0 RM |
1730 | |
1731 | For instance: | |
1732 | ||
1733 | use Math::Complex; | |
1734 | ||
1735 | Math::Complex::display_format('polar'); | |
16357284 JH |
1736 | $j = (root(1, 3))[1]; |
1737 | print "j = $j\n"; # Prints "j = [1,2pi/3]" | |
66730be0 RM |
1738 | $j->display_format('cartesian'); |
1739 | print "j = $j\n"; # Prints "j = -0.5+0.866025403784439i" | |
1740 | ||
5287f86b | 1741 | The polar style attempts to emphasize arguments like I<k*pi/n> |
9bc5fa8d | 1742 | (where I<n> is a positive integer and I<k> an integer within [-9, +9]), |
5287f86b | 1743 | this is called I<polar pretty-printing>. |
66730be0 | 1744 | |
16357284 JH |
1745 | =head2 CHANGED IN PERL 5.6 |
1746 | ||
1747 | The C<display_format> class method and the corresponding | |
1748 | C<display_format> object method can now be called using | |
1749 | a parameter hash instead of just a one parameter. | |
1750 | ||
1751 | The old display format style, which can have values C<"cartesian"> or | |
40b904b7 JH |
1752 | C<"polar">, can be changed using the C<"style"> parameter. |
1753 | ||
1754 | $j->display_format(style => "polar"); | |
1755 | ||
1756 | The one parameter calling convention also still works. | |
1757 | ||
1758 | $j->display_format("polar"); | |
16357284 JH |
1759 | |
1760 | There are two new display parameters. | |
1761 | ||
40b904b7 JH |
1762 | The first one is C<"format">, which is a sprintf()-style format string |
1763 | to be used for both numeric parts of the complex number(s). The is | |
1764 | somewhat system-dependent but most often it corresponds to C<"%.15g">. | |
1765 | You can revert to the default by setting the C<format> to C<undef>. | |
16357284 JH |
1766 | |
1767 | # the $j from the above example | |
1768 | ||
1769 | $j->display_format('format' => '%.5f'); | |
1770 | print "j = $j\n"; # Prints "j = -0.50000+0.86603i" | |
40b904b7 | 1771 | $j->display_format('format' => undef); |
16357284 JH |
1772 | print "j = $j\n"; # Prints "j = -0.5+0.86603i" |
1773 | ||
1774 | Notice that this affects also the return values of the | |
1775 | C<display_format> methods: in list context the whole parameter hash | |
40b904b7 JH |
1776 | will be returned, as opposed to only the style parameter value. |
1777 | This is a potential incompatibility with earlier versions if you | |
1778 | have been calling the C<display_format> method in list context. | |
16357284 | 1779 | |
5287f86b JH |
1780 | The second new display parameter is C<"polar_pretty_print">, which can |
1781 | be set to true or false, the default being true. See the previous | |
1782 | section for what this means. | |
16357284 | 1783 | |
66730be0 RM |
1784 | =head1 USAGE |
1785 | ||
1786 | Thanks to overloading, the handling of arithmetics with complex numbers | |
1787 | is simple and almost transparent. | |
1788 | ||
1789 | Here are some examples: | |
1790 | ||
1791 | use Math::Complex; | |
1792 | ||
1793 | $j = cplxe(1, 2*pi/3); # $j ** 3 == 1 | |
1794 | print "j = $j, j**3 = ", $j ** 3, "\n"; | |
1795 | print "1 + j + j**2 = ", 1 + $j + $j**2, "\n"; | |
1796 | ||
1797 | $z = -16 + 0*i; # Force it to be a complex | |
1798 | print "sqrt($z) = ", sqrt($z), "\n"; | |
1799 | ||
1800 | $k = exp(i * 2*pi/3); | |
1801 | print "$j - $k = ", $j - $k, "\n"; | |
a5f75d66 | 1802 | |
b42d0ec9 JH |
1803 | $z->Re(3); # Re, Im, arg, abs, |
1804 | $j->arg(2); # (the last two aka rho, theta) | |
1805 | # can be used also as mutators. | |
1806 | ||
1807 | =head1 ERRORS DUE TO DIVISION BY ZERO OR LOGARITHM OF ZERO | |
5aabfad6 | 1808 | |
1809 | The division (/) and the following functions | |
1810 | ||
b42d0ec9 | 1811 | log ln log10 logn |
2820d885 | 1812 | tan sec csc cot |
b42d0ec9 JH |
1813 | atan asec acsc acot |
1814 | tanh sech csch coth | |
1815 | atanh asech acsch acoth | |
5aabfad6 | 1816 | |
1817 | cannot be computed for all arguments because that would mean dividing | |
8c03c583 JH |
1818 | by zero or taking logarithm of zero. These situations cause fatal |
1819 | runtime errors looking like this | |
5aabfad6 | 1820 | |
1821 | cot(0): Division by zero. | |
5cd24f17 | 1822 | (Because in the definition of cot(0), the divisor sin(0) is 0) |
5aabfad6 | 1823 | Died at ... |
1824 | ||
8c03c583 JH |
1825 | or |
1826 | ||
1827 | atanh(-1): Logarithm of zero. | |
1828 | Died at... | |
1829 | ||
1830 | For the C<csc>, C<cot>, C<asec>, C<acsc>, C<acot>, C<csch>, C<coth>, | |
b42d0ec9 JH |
1831 | C<asech>, C<acsch>, the argument cannot be C<0> (zero). For the the |
1832 | logarithmic functions and the C<atanh>, C<acoth>, the argument cannot | |
1833 | be C<1> (one). For the C<atanh>, C<acoth>, the argument cannot be | |
1834 | C<-1> (minus one). For the C<atan>, C<acot>, the argument cannot be | |
1835 | C<i> (the imaginary unit). For the C<atan>, C<acoth>, the argument | |
1836 | cannot be C<-i> (the negative imaginary unit). For the C<tan>, | |
1837 | C<sec>, C<tanh>, the argument cannot be I<pi/2 + k * pi>, where I<k> | |
1838 | is any integer. | |
1839 | ||
1840 | Note that because we are operating on approximations of real numbers, | |
1841 | these errors can happen when merely `too close' to the singularities | |
40b904b7 | 1842 | listed above. |
b42d0ec9 JH |
1843 | |
1844 | =head1 ERRORS DUE TO INDIGESTIBLE ARGUMENTS | |
1845 | ||
1846 | The C<make> and C<emake> accept both real and complex arguments. | |
1847 | When they cannot recognize the arguments they will die with error | |
1848 | messages like the following | |
1849 | ||
1850 | Math::Complex::make: Cannot take real part of ... | |
1851 | Math::Complex::make: Cannot take real part of ... | |
1852 | Math::Complex::emake: Cannot take rho of ... | |
1853 | Math::Complex::emake: Cannot take theta of ... | |
5cd24f17 | 1854 | |
a5f75d66 AD |
1855 | =head1 BUGS |
1856 | ||
5cd24f17 | 1857 | Saying C<use Math::Complex;> exports many mathematical routines in the |
fb73857a | 1858 | caller environment and even overrides some (C<sqrt>, C<log>). |
1859 | This is construed as a feature by the Authors, actually... ;-) | |
a5f75d66 | 1860 | |
66730be0 RM |
1861 | All routines expect to be given real or complex numbers. Don't attempt to |
1862 | use BigFloat, since Perl has currently no rule to disambiguate a '+' | |
1863 | operation (for instance) between two overloaded entities. | |
a5f75d66 | 1864 | |
d09ae4e6 JH |
1865 | In Cray UNICOS there is some strange numerical instability that results |
1866 | in root(), cos(), sin(), cosh(), sinh(), losing accuracy fast. Beware. | |
1867 | The bug may be in UNICOS math libs, in UNICOS C compiler, in Math::Complex. | |
1868 | Whatever it is, it does not manifest itself anywhere else where Perl runs. | |
1869 | ||
0c721ce2 | 1870 | =head1 AUTHORS |
a5f75d66 | 1871 | |
6e238990 | 1872 | Raphael Manfredi <F<Raphael_Manfredi@pobox.com>> and |
ace5de91 | 1873 | Jarkko Hietaniemi <F<jhi@iki.fi>>. |
5cd24f17 | 1874 | |
fb73857a | 1875 | Extensive patches by Daniel S. Lewart <F<d-lewart@uiuc.edu>>. |
1876 | ||
5cd24f17 | 1877 | =cut |
1878 | ||
b42d0ec9 JH |
1879 | 1; |
1880 | ||
5cd24f17 | 1881 | # eof |