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1 | # $RCSFile$ |
2 | # |
3 | # Complex numbers and associated mathematical functions |
4 | # -- Raphael Manfredi, Sept 1996 |
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5 | |
6 | require Exporter; |
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7 | package Math::Complex; @ISA = qw(Exporter); |
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8 | |
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9 | @EXPORT = qw( |
10 | pi i Re Im arg |
11 | log10 logn cbrt root |
12 | tan cotan asin acos atan acotan |
13 | sinh cosh tanh cotanh asinh acosh atanh acotanh |
14 | cplx cplxe |
15 | ); |
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16 | |
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17 | use overload |
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18 | '+' => \&plus, |
19 | '-' => \&minus, |
20 | '*' => \&multiply, |
21 | '/' => \÷, |
22 | '**' => \&power, |
23 | '<=>' => \&spaceship, |
24 | 'neg' => \&negate, |
25 | '~' => \&conjugate, |
26 | 'abs' => \&abs, |
27 | 'sqrt' => \&sqrt, |
28 | 'exp' => \&exp, |
29 | 'log' => \&log, |
30 | 'sin' => \&sin, |
31 | 'cos' => \&cos, |
32 | 'atan2' => \&atan2, |
33 | qw("" stringify); |
34 | |
35 | # |
36 | # Package globals |
37 | # |
38 | |
39 | $package = 'Math::Complex'; # Package name |
40 | $display = 'cartesian'; # Default display format |
41 | |
42 | # |
43 | # Object attributes (internal): |
44 | # cartesian [real, imaginary] -- cartesian form |
45 | # polar [rho, theta] -- polar form |
46 | # c_dirty cartesian form not up-to-date |
47 | # p_dirty polar form not up-to-date |
48 | # display display format (package's global when not set) |
49 | # |
50 | |
51 | # |
52 | # ->make |
53 | # |
54 | # Create a new complex number (cartesian form) |
55 | # |
56 | sub make { |
57 | my $self = bless {}, shift; |
58 | my ($re, $im) = @_; |
59 | $self->{cartesian} = [$re, $im]; |
60 | $self->{c_dirty} = 0; |
61 | $self->{p_dirty} = 1; |
62 | return $self; |
63 | } |
64 | |
65 | # |
66 | # ->emake |
67 | # |
68 | # Create a new complex number (exponential form) |
69 | # |
70 | sub emake { |
71 | my $self = bless {}, shift; |
72 | my ($rho, $theta) = @_; |
73 | $theta += pi() if $rho < 0; |
74 | $self->{polar} = [abs($rho), $theta]; |
75 | $self->{p_dirty} = 0; |
76 | $self->{c_dirty} = 1; |
77 | return $self; |
78 | } |
79 | |
80 | sub new { &make } # For backward compatibility only. |
81 | |
82 | # |
83 | # cplx |
84 | # |
85 | # Creates a complex number from a (re, im) tuple. |
86 | # This avoids the burden of writing Math::Complex->make(re, im). |
87 | # |
88 | sub cplx { |
89 | my ($re, $im) = @_; |
90 | return $package->make($re, $im); |
91 | } |
92 | |
93 | # |
94 | # cplxe |
95 | # |
96 | # Creates a complex number from a (rho, theta) tuple. |
97 | # This avoids the burden of writing Math::Complex->emake(rho, theta). |
98 | # |
99 | sub cplxe { |
100 | my ($rho, $theta) = @_; |
101 | return $package->emake($rho, $theta); |
102 | } |
103 | |
104 | # |
105 | # pi |
106 | # |
107 | # The number defined as 2 * pi = 360 degrees |
108 | # |
109 | sub pi () { |
110 | $pi = 4 * atan2(1, 1) unless $pi; |
111 | return $pi; |
112 | } |
113 | |
114 | # |
115 | # i |
116 | # |
117 | # The number defined as i*i = -1; |
118 | # |
119 | sub i () { |
120 | $i = bless {} unless $i; # There can be only one i |
121 | $i->{cartesian} = [0, 1]; |
122 | $i->{polar} = [1, pi/2]; |
123 | $i->{c_dirty} = 0; |
124 | $i->{p_dirty} = 0; |
125 | return $i; |
126 | } |
127 | |
128 | # |
129 | # Attribute access/set routines |
130 | # |
131 | |
132 | sub cartesian {$_[0]->{c_dirty} ? $_[0]->update_cartesian : $_[0]->{cartesian}} |
133 | sub polar {$_[0]->{p_dirty} ? $_[0]->update_polar : $_[0]->{polar}} |
134 | |
135 | sub set_cartesian { $_[0]->{p_dirty}++; $_[0]->{cartesian} = $_[1] } |
136 | sub set_polar { $_[0]->{c_dirty}++; $_[0]->{polar} = $_[1] } |
137 | |
138 | # |
139 | # ->update_cartesian |
140 | # |
141 | # Recompute and return the cartesian form, given accurate polar form. |
142 | # |
143 | sub update_cartesian { |
144 | my $self = shift; |
145 | my ($r, $t) = @{$self->{polar}}; |
146 | $self->{c_dirty} = 0; |
147 | return $self->{cartesian} = [$r * cos $t, $r * sin $t]; |
148 | } |
149 | |
150 | # |
151 | # |
152 | # ->update_polar |
153 | # |
154 | # Recompute and return the polar form, given accurate cartesian form. |
155 | # |
156 | sub update_polar { |
157 | my $self = shift; |
158 | my ($x, $y) = @{$self->{cartesian}}; |
159 | $self->{p_dirty} = 0; |
160 | return $self->{polar} = [0, 0] if $x == 0 && $y == 0; |
161 | return $self->{polar} = [sqrt($x*$x + $y*$y), atan2($y, $x)]; |
162 | } |
163 | |
164 | # |
165 | # (plus) |
166 | # |
167 | # Computes z1+z2. |
168 | # |
169 | sub plus { |
170 | my ($z1, $z2, $regular) = @_; |
171 | my ($re1, $im1) = @{$z1->cartesian}; |
172 | my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2); |
173 | unless (defined $regular) { |
174 | $z1->set_cartesian([$re1 + $re2, $im1 + $im2]); |
175 | return $z1; |
176 | } |
177 | return (ref $z1)->make($re1 + $re2, $im1 + $im2); |
178 | } |
179 | |
180 | # |
181 | # (minus) |
182 | # |
183 | # Computes z1-z2. |
184 | # |
185 | sub minus { |
186 | my ($z1, $z2, $inverted) = @_; |
187 | my ($re1, $im1) = @{$z1->cartesian}; |
188 | my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2); |
189 | unless (defined $inverted) { |
190 | $z1->set_cartesian([$re1 - $re2, $im1 - $im2]); |
191 | return $z1; |
192 | } |
193 | return $inverted ? |
194 | (ref $z1)->make($re2 - $re1, $im2 - $im1) : |
195 | (ref $z1)->make($re1 - $re2, $im1 - $im2); |
196 | } |
197 | |
198 | # |
199 | # (multiply) |
200 | # |
201 | # Computes z1*z2. |
202 | # |
203 | sub multiply { |
204 | my ($z1, $z2, $regular) = @_; |
205 | my ($r1, $t1) = @{$z1->polar}; |
206 | my ($r2, $t2) = ref $z2 ? @{$z2->polar} : (abs($z2), $z2 >= 0 ? 0 : pi); |
207 | unless (defined $regular) { |
208 | $z1->set_polar([$r1 * $r2, $t1 + $t2]); |
209 | return $z1; |
210 | } |
211 | return (ref $z1)->emake($r1 * $r2, $t1 + $t2); |
212 | } |
213 | |
214 | # |
215 | # (divide) |
216 | # |
217 | # Computes z1/z2. |
218 | # |
219 | sub divide { |
220 | my ($z1, $z2, $inverted) = @_; |
221 | my ($r1, $t1) = @{$z1->polar}; |
222 | my ($r2, $t2) = ref $z2 ? @{$z2->polar} : (abs($z2), $z2 >= 0 ? 0 : pi); |
223 | unless (defined $inverted) { |
224 | $z1->set_polar([$r1 / $r2, $t1 - $t2]); |
225 | return $z1; |
226 | } |
227 | return $inverted ? |
228 | (ref $z1)->emake($r2 / $r1, $t2 - $t1) : |
229 | (ref $z1)->emake($r1 / $r2, $t1 - $t2); |
230 | } |
231 | |
232 | # |
233 | # (power) |
234 | # |
235 | # Computes z1**z2 = exp(z2 * log z1)). |
236 | # |
237 | sub power { |
238 | my ($z1, $z2, $inverted) = @_; |
239 | return exp($z1 * log $z2) if defined $inverted && $inverted; |
240 | return exp($z2 * log $z1); |
241 | } |
242 | |
243 | # |
244 | # (spaceship) |
245 | # |
246 | # Computes z1 <=> z2. |
247 | # Sorts on the real part first, then on the imaginary part. Thus 2-4i > 3+8i. |
248 | # |
249 | sub spaceship { |
250 | my ($z1, $z2, $inverted) = @_; |
251 | my ($re1, $im1) = @{$z1->cartesian}; |
252 | my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2); |
253 | my $sgn = $inverted ? -1 : 1; |
254 | return $sgn * ($re1 <=> $re2) if $re1 != $re2; |
255 | return $sgn * ($im1 <=> $im2); |
256 | } |
257 | |
258 | # |
259 | # (negate) |
260 | # |
261 | # Computes -z. |
262 | # |
263 | sub negate { |
264 | my ($z) = @_; |
265 | if ($z->{c_dirty}) { |
266 | my ($r, $t) = @{$z->polar}; |
267 | return (ref $z)->emake($r, pi + $t); |
268 | } |
269 | my ($re, $im) = @{$z->cartesian}; |
270 | return (ref $z)->make(-$re, -$im); |
271 | } |
272 | |
273 | # |
274 | # (conjugate) |
275 | # |
276 | # Compute complex's conjugate. |
277 | # |
278 | sub conjugate { |
279 | my ($z) = @_; |
280 | if ($z->{c_dirty}) { |
281 | my ($r, $t) = @{$z->polar}; |
282 | return (ref $z)->emake($r, -$t); |
283 | } |
284 | my ($re, $im) = @{$z->cartesian}; |
285 | return (ref $z)->make($re, -$im); |
286 | } |
287 | |
288 | # |
289 | # (abs) |
290 | # |
291 | # Compute complex's norm (rho). |
292 | # |
293 | sub abs { |
294 | my ($z) = @_; |
295 | my ($r, $t) = @{$z->polar}; |
296 | return abs($r); |
297 | } |
298 | |
299 | # |
300 | # arg |
301 | # |
302 | # Compute complex's argument (theta). |
303 | # |
304 | sub arg { |
305 | my ($z) = @_; |
306 | return 0 unless ref $z; |
307 | my ($r, $t) = @{$z->polar}; |
308 | return $t; |
309 | } |
310 | |
311 | # |
312 | # (sqrt) |
313 | # |
314 | # Compute sqrt(z) (positive only). |
315 | # |
316 | sub sqrt { |
317 | my ($z) = @_; |
318 | my ($r, $t) = @{$z->polar}; |
319 | return (ref $z)->emake(sqrt($r), $t/2); |
320 | } |
321 | |
322 | # |
323 | # cbrt |
324 | # |
325 | # Compute cbrt(z) (cubic root, primary only). |
326 | # |
327 | sub cbrt { |
328 | my ($z) = @_; |
329 | return $z ** (1/3) unless ref $z; |
330 | my ($r, $t) = @{$z->polar}; |
331 | return (ref $z)->emake($r**(1/3), $t/3); |
332 | } |
333 | |
334 | # |
335 | # root |
336 | # |
337 | # Computes all nth root for z, returning an array whose size is n. |
338 | # `n' must be a positive integer. |
339 | # |
340 | # The roots are given by (for k = 0..n-1): |
341 | # |
342 | # z^(1/n) = r^(1/n) (cos ((t+2 k pi)/n) + i sin ((t+2 k pi)/n)) |
343 | # |
344 | sub root { |
345 | my ($z, $n) = @_; |
346 | $n = int($n + 0.5); |
347 | return undef unless $n > 0; |
348 | my ($r, $t) = ref $z ? @{$z->polar} : (abs($z), $z >= 0 ? 0 : pi); |
349 | my @root; |
350 | my $k; |
351 | my $theta_inc = 2 * pi / $n; |
352 | my $rho = $r ** (1/$n); |
353 | my $theta; |
354 | my $complex = ref($z) || $package; |
355 | for ($k = 0, $theta = $t / $n; $k < $n; $k++, $theta += $theta_inc) { |
356 | push(@root, $complex->emake($rho, $theta)); |
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357 | } |
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358 | return @root; |
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359 | } |
360 | |
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361 | # |
362 | # Re |
363 | # |
364 | # Return Re(z). |
365 | # |
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366 | sub Re { |
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367 | my ($z) = @_; |
368 | return $z unless ref $z; |
369 | my ($re, $im) = @{$z->cartesian}; |
370 | return $re; |
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371 | } |
372 | |
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373 | # |
374 | # Im |
375 | # |
376 | # Return Im(z). |
377 | # |
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378 | sub Im { |
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379 | my ($z) = @_; |
380 | return 0 unless ref $z; |
381 | my ($re, $im) = @{$z->cartesian}; |
382 | return $im; |
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383 | } |
384 | |
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385 | # |
386 | # (exp) |
387 | # |
388 | # Computes exp(z). |
389 | # |
390 | sub exp { |
391 | my ($z) = @_; |
392 | my ($x, $y) = @{$z->cartesian}; |
393 | return (ref $z)->emake(exp($x), $y); |
394 | } |
395 | |
396 | # |
397 | # (log) |
398 | # |
399 | # Compute log(z). |
400 | # |
401 | sub log { |
402 | my ($z) = @_; |
403 | my ($r, $t) = @{$z->polar}; |
404 | return (ref $z)->make(log($r), $t); |
405 | } |
406 | |
407 | # |
408 | # log10 |
409 | # |
410 | # Compute log10(z). |
411 | # |
412 | sub log10 { |
413 | my ($z) = @_; |
414 | $log10 = log(10) unless defined $log10; |
415 | return log($z) / $log10 unless ref $z; |
416 | my ($r, $t) = @{$z->polar}; |
417 | return (ref $z)->make(log($r) / $log10, $t / $log10); |
418 | } |
419 | |
420 | # |
421 | # logn |
422 | # |
423 | # Compute logn(z,n) = log(z) / log(n) |
424 | # |
425 | sub logn { |
426 | my ($z, $n) = @_; |
427 | my $logn = $logn{$n}; |
428 | $logn = $logn{$n} = log($n) unless defined $logn; # Cache log(n) |
429 | return log($z) / log($n); |
430 | } |
431 | |
432 | # |
433 | # (cos) |
434 | # |
435 | # Compute cos(z) = (exp(iz) + exp(-iz))/2. |
436 | # |
437 | sub cos { |
438 | my ($z) = @_; |
439 | my ($x, $y) = @{$z->cartesian}; |
440 | my $ey = exp($y); |
441 | my $ey_1 = 1 / $ey; |
442 | return (ref $z)->make(cos($x) * ($ey + $ey_1)/2, sin($x) * ($ey_1 - $ey)/2); |
443 | } |
444 | |
445 | # |
446 | # (sin) |
447 | # |
448 | # Compute sin(z) = (exp(iz) - exp(-iz))/2. |
449 | # |
450 | sub sin { |
451 | my ($z) = @_; |
452 | my ($x, $y) = @{$z->cartesian}; |
453 | my $ey = exp($y); |
454 | my $ey_1 = 1 / $ey; |
455 | return (ref $z)->make(sin($x) * ($ey + $ey_1)/2, cos($x) * ($ey - $ey_1)/2); |
456 | } |
457 | |
458 | # |
459 | # tan |
460 | # |
461 | # Compute tan(z) = sin(z) / cos(z). |
462 | # |
463 | sub tan { |
464 | my ($z) = @_; |
465 | return sin($z) / cos($z); |
466 | } |
467 | |
468 | # |
469 | # cotan |
470 | # |
471 | # Computes cotan(z) = 1 / tan(z). |
472 | # |
473 | sub cotan { |
474 | my ($z) = @_; |
475 | return cos($z) / sin($z); |
476 | } |
477 | |
478 | # |
479 | # acos |
480 | # |
481 | # Computes the arc cosine acos(z) = -i log(z + sqrt(z*z-1)). |
482 | # |
483 | sub acos { |
484 | my ($z) = @_; |
485 | my $cz = $z*$z - 1; |
486 | $cz = cplx($cz, 0) if !ref $cz && $cz < 0; # Force complex if <0 |
487 | return ~i * log($z + sqrt $cz); # ~i is -i |
488 | } |
489 | |
490 | # |
491 | # asin |
492 | # |
493 | # Computes the arc sine asin(z) = -i log(iz + sqrt(1-z*z)). |
494 | # |
495 | sub asin { |
496 | my ($z) = @_; |
497 | my $cz = 1 - $z*$z; |
498 | $cz = cplx($cz, 0) if !ref $cz && $cz < 0; # Force complex if <0 |
499 | return ~i * log(i * $z + sqrt $cz); # ~i is -i |
500 | } |
501 | |
502 | # |
503 | # atan |
504 | # |
505 | # Computes the arc tagent atan(z) = i/2 log((i+z) / (i-z)). |
506 | # |
507 | sub atan { |
508 | my ($z) = @_; |
509 | return i/2 * log((i + $z) / (i - $z)); |
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510 | } |
511 | |
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512 | # |
513 | # acotan |
514 | # |
515 | # Computes the arc cotangent acotan(z) = -i/2 log((i+z) / (z-i)) |
516 | # |
517 | sub acotan { |
518 | my ($z) = @_; |
519 | return i/-2 * log((i + $z) / ($z - i)); |
520 | } |
521 | |
522 | # |
523 | # cosh |
524 | # |
525 | # Computes the hyperbolic cosine cosh(z) = (exp(z) + exp(-z))/2. |
526 | # |
527 | sub cosh { |
528 | my ($z) = @_; |
529 | my ($x, $y) = ref $z ? @{$z->cartesian} : ($z); |
530 | my $ex = exp($x); |
531 | my $ex_1 = 1 / $ex; |
532 | return ($ex + $ex_1)/2 unless ref $z; |
533 | return (ref $z)->make(cos($y) * ($ex + $ex_1)/2, sin($y) * ($ex - $ex_1)/2); |
534 | } |
535 | |
536 | # |
537 | # sinh |
538 | # |
539 | # Computes the hyperbolic sine sinh(z) = (exp(z) - exp(-z))/2. |
540 | # |
541 | sub sinh { |
542 | my ($z) = @_; |
543 | my ($x, $y) = ref $z ? @{$z->cartesian} : ($z); |
544 | my $ex = exp($x); |
545 | my $ex_1 = 1 / $ex; |
546 | return ($ex - $ex_1)/2 unless ref $z; |
547 | return (ref $z)->make(cos($y) * ($ex - $ex_1)/2, sin($y) * ($ex + $ex_1)/2); |
548 | } |
549 | |
550 | # |
551 | # tanh |
552 | # |
553 | # Computes the hyperbolic tangent tanh(z) = sinh(z) / cosh(z). |
554 | # |
555 | sub tanh { |
556 | my ($z) = @_; |
557 | return sinh($z) / cosh($z); |
558 | } |
559 | |
560 | # |
561 | # cotanh |
562 | # |
563 | # Comptutes the hyperbolic cotangent cotanh(z) = cosh(z) / sinh(z). |
564 | # |
565 | sub cotanh { |
566 | my ($z) = @_; |
567 | return cosh($z) / sinh($z); |
568 | } |
569 | |
570 | # |
571 | # acosh |
572 | # |
573 | # Computes the arc hyperbolic cosine acosh(z) = log(z + sqrt(z*z-1)). |
574 | # |
575 | sub acosh { |
576 | my ($z) = @_; |
577 | my $cz = $z*$z - 1; |
578 | $cz = cplx($cz, 0) if !ref $cz && $cz < 0; # Force complex if <0 |
579 | return log($z + sqrt $cz); |
580 | } |
581 | |
582 | # |
583 | # asinh |
584 | # |
585 | # Computes the arc hyperbolic sine asinh(z) = log(z + sqrt(z*z-1)) |
586 | # |
587 | sub asinh { |
588 | my ($z) = @_; |
589 | my $cz = $z*$z + 1; # Already complex if <0 |
590 | return log($z + sqrt $cz); |
591 | } |
592 | |
593 | # |
594 | # atanh |
595 | # |
596 | # Computes the arc hyperbolic tangent atanh(z) = 1/2 log((1+z) / (1-z)). |
597 | # |
598 | sub atanh { |
599 | my ($z) = @_; |
600 | my $cz = (1 + $z) / (1 - $z); |
601 | $cz = cplx($cz, 0) if !ref $cz && $cz < 0; # Force complex if <0 |
602 | return log($cz) / 2; |
603 | } |
604 | |
605 | # |
606 | # acotanh |
607 | # |
608 | # Computes the arc hyperbolic cotangent acotanh(z) = 1/2 log((1+z) / (z-1)). |
609 | # |
610 | sub acotanh { |
611 | my ($z) = @_; |
612 | my $cz = (1 + $z) / ($z - 1); |
613 | $cz = cplx($cz, 0) if !ref $cz && $cz < 0; # Force complex if <0 |
614 | return log($cz) / 2; |
615 | } |
616 | |
617 | # |
618 | # (atan2) |
619 | # |
620 | # Compute atan(z1/z2). |
621 | # |
622 | sub atan2 { |
623 | my ($z1, $z2, $inverted) = @_; |
624 | my ($re1, $im1) = @{$z1->cartesian}; |
625 | my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2); |
626 | my $tan; |
627 | if (defined $inverted && $inverted) { # atan(z2/z1) |
628 | return pi * ($re2 > 0 ? 1 : -1) if $re1 == 0 && $im1 == 0; |
629 | $tan = $z2 / $z1; |
630 | } else { |
631 | return pi * ($re1 > 0 ? 1 : -1) if $re2 == 0 && $im2 == 0; |
632 | $tan = $z1 / $z2; |
633 | } |
634 | return atan($tan); |
635 | } |
636 | |
637 | # |
638 | # display_format |
639 | # ->display_format |
640 | # |
641 | # Set (fetch if no argument) display format for all complex numbers that |
642 | # don't happen to have overrriden it via ->display_format |
643 | # |
644 | # When called as a method, this actually sets the display format for |
645 | # the current object. |
646 | # |
647 | # Valid object formats are 'c' and 'p' for cartesian and polar. The first |
648 | # letter is used actually, so the type can be fully spelled out for clarity. |
649 | # |
650 | sub display_format { |
651 | my $self = shift; |
652 | my $format = undef; |
653 | |
654 | if (ref $self) { # Called as a method |
655 | $format = shift; |
656 | } else { # Regular procedure call |
657 | $format = $self; |
658 | undef $self; |
659 | } |
660 | |
661 | if (defined $self) { |
662 | return defined $self->{display} ? $self->{display} : $display |
663 | unless defined $format; |
664 | return $self->{display} = $format; |
665 | } |
666 | |
667 | return $display unless defined $format; |
668 | return $display = $format; |
669 | } |
670 | |
671 | # |
672 | # (stringify) |
673 | # |
674 | # Show nicely formatted complex number under its cartesian or polar form, |
675 | # depending on the current display format: |
676 | # |
677 | # . If a specific display format has been recorded for this object, use it. |
678 | # . Otherwise, use the generic current default for all complex numbers, |
679 | # which is a package global variable. |
680 | # |
a0d0e21e |
681 | sub stringify { |
66730be0 |
682 | my ($z) = shift; |
683 | my $format; |
684 | |
685 | $format = $display; |
686 | $format = $z->{display} if defined $z->{display}; |
687 | |
688 | return $z->stringify_polar if $format =~ /^p/i; |
689 | return $z->stringify_cartesian; |
690 | } |
691 | |
692 | # |
693 | # ->stringify_cartesian |
694 | # |
695 | # Stringify as a cartesian representation 'a+bi'. |
696 | # |
697 | sub stringify_cartesian { |
698 | my $z = shift; |
699 | my ($x, $y) = @{$z->cartesian}; |
700 | my ($re, $im); |
701 | |
702 | $re = "$x" if abs($x) >= 1e-14; |
703 | if ($y == 1) { $im = 'i' } |
704 | elsif ($y == -1) { $im = '-i' } |
705 | elsif (abs($y) >= 1e-14) { $im = "${y}i" } |
706 | |
707 | my $str; |
708 | $str = $re if defined $re; |
709 | $str .= "+$im" if defined $im; |
710 | $str =~ s/\+-/-/; |
711 | $str =~ s/^\+//; |
712 | $str = '0' unless $str; |
713 | |
714 | return $str; |
715 | } |
716 | |
717 | # |
718 | # ->stringify_polar |
719 | # |
720 | # Stringify as a polar representation '[r,t]'. |
721 | # |
722 | sub stringify_polar { |
723 | my $z = shift; |
724 | my ($r, $t) = @{$z->polar}; |
725 | my $theta; |
726 | |
727 | return '[0,0]' if $r <= 1e-14; |
a0d0e21e |
728 | |
66730be0 |
729 | my $tpi = 2 * pi; |
730 | my $nt = $t / $tpi; |
731 | $nt = ($nt - int($nt)) * $tpi; |
732 | $nt += $tpi if $nt < 0; # Range [0, 2pi] |
a0d0e21e |
733 | |
66730be0 |
734 | if (abs($nt) <= 1e-14) { $theta = 0 } |
735 | elsif (abs(pi-$nt) <= 1e-14) { $theta = 'pi' } |
736 | |
737 | return "\[$r,$theta\]" if defined $theta; |
738 | |
739 | # |
740 | # Okay, number is not a real. Try to identify pi/n and friends... |
741 | # |
742 | |
743 | $nt -= $tpi if $nt > pi; |
744 | my ($n, $k, $kpi); |
745 | |
746 | for ($k = 1, $kpi = pi; $k < 10; $k++, $kpi += pi) { |
747 | $n = int($kpi / $nt + ($nt > 0 ? 1 : -1) * 0.5); |
748 | if (abs($kpi/$n - $nt) <= 1e-14) { |
749 | $theta = ($nt < 0 ? '-':'').($k == 1 ? 'pi':"${k}pi").'/'.abs($n); |
750 | last; |
751 | } |
752 | } |
753 | |
754 | $theta = $nt unless defined $theta; |
755 | |
756 | return "\[$r,$theta\]"; |
a0d0e21e |
757 | } |
a5f75d66 |
758 | |
759 | 1; |
760 | __END__ |
761 | |
762 | =head1 NAME |
763 | |
66730be0 |
764 | Math::Complex - complex numbers and associated mathematical functions |
a5f75d66 |
765 | |
766 | =head1 SYNOPSIS |
767 | |
66730be0 |
768 | use Math::Complex; |
769 | $z = Math::Complex->make(5, 6); |
770 | $t = 4 - 3*i + $z; |
771 | $j = cplxe(1, 2*pi/3); |
a5f75d66 |
772 | |
773 | =head1 DESCRIPTION |
774 | |
66730be0 |
775 | This package lets you create and manipulate complex numbers. By default, |
776 | I<Perl> limits itself to real numbers, but an extra C<use> statement brings |
777 | full complex support, along with a full set of mathematical functions |
778 | typically associated with and/or extended to complex numbers. |
779 | |
780 | If you wonder what complex numbers are, they were invented to be able to solve |
781 | the following equation: |
782 | |
783 | x*x = -1 |
784 | |
785 | and by definition, the solution is noted I<i> (engineers use I<j> instead since |
786 | I<i> usually denotes an intensity, but the name does not matter). The number |
787 | I<i> is a pure I<imaginary> number. |
788 | |
789 | The arithmetics with pure imaginary numbers works just like you would expect |
790 | it with real numbers... you just have to remember that |
791 | |
792 | i*i = -1 |
793 | |
794 | so you have: |
795 | |
796 | 5i + 7i = i * (5 + 7) = 12i |
797 | 4i - 3i = i * (4 - 3) = i |
798 | 4i * 2i = -8 |
799 | 6i / 2i = 3 |
800 | 1 / i = -i |
801 | |
802 | Complex numbers are numbers that have both a real part and an imaginary |
803 | part, and are usually noted: |
804 | |
805 | a + bi |
806 | |
807 | where C<a> is the I<real> part and C<b> is the I<imaginary> part. The |
808 | arithmetic with complex numbers is straightforward. You have to |
809 | keep track of the real and the imaginary parts, but otherwise the |
810 | rules used for real numbers just apply: |
811 | |
812 | (4 + 3i) + (5 - 2i) = (4 + 5) + i(3 - 2) = 9 + i |
813 | (2 + i) * (4 - i) = 2*4 + 4i -2i -i*i = 8 + 2i + 1 = 9 + 2i |
814 | |
815 | A graphical representation of complex numbers is possible in a plane |
816 | (also called the I<complex plane>, but it's really a 2D plane). |
817 | The number |
818 | |
819 | z = a + bi |
820 | |
821 | is the point whose coordinates are (a, b). Actually, it would |
822 | be the vector originating from (0, 0) to (a, b). It follows that the addition |
823 | of two complex numbers is a vectorial addition. |
824 | |
825 | Since there is a bijection between a point in the 2D plane and a complex |
826 | number (i.e. the mapping is unique and reciprocal), a complex number |
827 | can also be uniquely identified with polar coordinates: |
828 | |
829 | [rho, theta] |
830 | |
831 | where C<rho> is the distance to the origin, and C<theta> the angle between |
832 | the vector and the I<x> axis. There is a notation for this using the |
833 | exponential form, which is: |
834 | |
835 | rho * exp(i * theta) |
836 | |
837 | where I<i> is the famous imaginary number introduced above. Conversion |
838 | between this form and the cartesian form C<a + bi> is immediate: |
839 | |
840 | a = rho * cos(theta) |
841 | b = rho * sin(theta) |
842 | |
843 | which is also expressed by this formula: |
844 | |
845 | z = rho * exp(i * theta) = rho * (cos theta + i * sin theta) |
846 | |
847 | In other words, it's the projection of the vector onto the I<x> and I<y> |
848 | axes. Mathematicians call I<rho> the I<norm> or I<modulus> and I<theta> |
849 | the I<argument> of the complex number. The I<norm> of C<z> will be |
850 | noted C<abs(z)>. |
851 | |
852 | The polar notation (also known as the trigonometric |
853 | representation) is much more handy for performing multiplications and |
854 | divisions of complex numbers, whilst the cartesian notation is better |
855 | suited for additions and substractions. Real numbers are on the I<x> |
856 | axis, and therefore I<theta> is zero. |
857 | |
858 | All the common operations that can be performed on a real number have |
859 | been defined to work on complex numbers as well, and are merely |
860 | I<extensions> of the operations defined on real numbers. This means |
861 | they keep their natural meaning when there is no imaginary part, provided |
862 | the number is within their definition set. |
863 | |
864 | For instance, the C<sqrt> routine which computes the square root of |
865 | its argument is only defined for positive real numbers and yields a |
866 | positive real number (it is an application from B<R+> to B<R+>). |
867 | If we allow it to return a complex number, then it can be extended to |
868 | negative real numbers to become an application from B<R> to B<C> (the |
869 | set of complex numbers): |
870 | |
871 | sqrt(x) = x >= 0 ? sqrt(x) : sqrt(-x)*i |
872 | |
873 | It can also be extended to be an application from B<C> to B<C>, |
874 | whilst its restriction to B<R> behaves as defined above by using |
875 | the following definition: |
876 | |
877 | sqrt(z = [r,t]) = sqrt(r) * exp(i * t/2) |
878 | |
879 | Indeed, a negative real number can be noted C<[x,pi]> |
880 | (the modulus I<x> is always positive, so C<[x,pi]> is really C<-x>, a |
881 | negative number) |
882 | and the above definition states that |
883 | |
884 | sqrt([x,pi]) = sqrt(x) * exp(i*pi/2) = [sqrt(x),pi/2] = sqrt(x)*i |
885 | |
886 | which is exactly what we had defined for negative real numbers above. |
a5f75d66 |
887 | |
66730be0 |
888 | All the common mathematical functions defined on real numbers that |
889 | are extended to complex numbers share that same property of working |
890 | I<as usual> when the imaginary part is zero (otherwise, it would not |
891 | be called an extension, would it?). |
a5f75d66 |
892 | |
66730be0 |
893 | A I<new> operation possible on a complex number that is |
894 | the identity for real numbers is called the I<conjugate>, and is noted |
895 | with an horizontal bar above the number, or C<~z> here. |
a5f75d66 |
896 | |
66730be0 |
897 | z = a + bi |
898 | ~z = a - bi |
a5f75d66 |
899 | |
66730be0 |
900 | Simple... Now look: |
a5f75d66 |
901 | |
66730be0 |
902 | z * ~z = (a + bi) * (a - bi) = a*a + b*b |
a5f75d66 |
903 | |
66730be0 |
904 | We saw that the norm of C<z> was noted C<abs(z)> and was defined as the |
905 | distance to the origin, also known as: |
a5f75d66 |
906 | |
66730be0 |
907 | rho = abs(z) = sqrt(a*a + b*b) |
a5f75d66 |
908 | |
66730be0 |
909 | so |
910 | |
911 | z * ~z = abs(z) ** 2 |
912 | |
913 | If z is a pure real number (i.e. C<b == 0>), then the above yields: |
914 | |
915 | a * a = abs(a) ** 2 |
916 | |
917 | which is true (C<abs> has the regular meaning for real number, i.e. stands |
918 | for the absolute value). This example explains why the norm of C<z> is |
919 | noted C<abs(z)>: it extends the C<abs> function to complex numbers, yet |
920 | is the regular C<abs> we know when the complex number actually has no |
921 | imaginary part... This justifies I<a posteriori> our use of the C<abs> |
922 | notation for the norm. |
923 | |
924 | =head1 OPERATIONS |
925 | |
926 | Given the following notations: |
927 | |
928 | z1 = a + bi = r1 * exp(i * t1) |
929 | z2 = c + di = r2 * exp(i * t2) |
930 | z = <any complex or real number> |
931 | |
932 | the following (overloaded) operations are supported on complex numbers: |
933 | |
934 | z1 + z2 = (a + c) + i(b + d) |
935 | z1 - z2 = (a - c) + i(b - d) |
936 | z1 * z2 = (r1 * r2) * exp(i * (t1 + t2)) |
937 | z1 / z2 = (r1 / r2) * exp(i * (t1 - t2)) |
938 | z1 ** z2 = exp(z2 * log z1) |
939 | ~z1 = a - bi |
940 | abs(z1) = r1 = sqrt(a*a + b*b) |
941 | sqrt(z1) = sqrt(r1) * exp(i * t1/2) |
942 | exp(z1) = exp(a) * exp(i * b) |
943 | log(z1) = log(r1) + i*t1 |
944 | sin(z1) = 1/2i (exp(i * z1) - exp(-i * z1)) |
945 | cos(z1) = 1/2 (exp(i * z1) + exp(-i * z1)) |
946 | abs(z1) = r1 |
947 | atan2(z1, z2) = atan(z1/z2) |
948 | |
949 | The following extra operations are supported on both real and complex |
950 | numbers: |
951 | |
952 | Re(z) = a |
953 | Im(z) = b |
954 | arg(z) = t |
955 | |
956 | cbrt(z) = z ** (1/3) |
957 | log10(z) = log(z) / log(10) |
958 | logn(z, n) = log(z) / log(n) |
959 | |
960 | tan(z) = sin(z) / cos(z) |
961 | cotan(z) = 1 / tan(z) |
962 | |
963 | asin(z) = -i * log(i*z + sqrt(1-z*z)) |
964 | acos(z) = -i * log(z + sqrt(z*z-1)) |
965 | atan(z) = i/2 * log((i+z) / (i-z)) |
966 | acotan(z) = -i/2 * log((i+z) / (z-i)) |
967 | |
968 | sinh(z) = 1/2 (exp(z) - exp(-z)) |
969 | cosh(z) = 1/2 (exp(z) + exp(-z)) |
970 | tanh(z) = sinh(z) / cosh(z) |
971 | cotanh(z) = 1 / tanh(z) |
972 | |
973 | asinh(z) = log(z + sqrt(z*z+1)) |
974 | acosh(z) = log(z + sqrt(z*z-1)) |
975 | atanh(z) = 1/2 * log((1+z) / (1-z)) |
976 | acotanh(z) = 1/2 * log((1+z) / (z-1)) |
977 | |
978 | The I<root> function is available to compute all the I<n>th |
979 | roots of some complex, where I<n> is a strictly positive integer. |
980 | There are exactly I<n> such roots, returned as a list. Getting the |
981 | number mathematicians call C<j> such that: |
982 | |
983 | 1 + j + j*j = 0; |
984 | |
985 | is a simple matter of writing: |
986 | |
987 | $j = ((root(1, 3))[1]; |
988 | |
989 | The I<k>th root for C<z = [r,t]> is given by: |
990 | |
991 | (root(z, n))[k] = r**(1/n) * exp(i * (t + 2*k*pi)/n) |
992 | |
993 | The I<spaceshift> operation is also defined. In order to ensure its |
994 | restriction to real numbers is conform to what you would expect, the |
995 | comparison is run on the real part of the complex number first, |
996 | and imaginary parts are compared only when the real parts match. |
997 | |
998 | =head1 CREATION |
999 | |
1000 | To create a complex number, use either: |
1001 | |
1002 | $z = Math::Complex->make(3, 4); |
1003 | $z = cplx(3, 4); |
1004 | |
1005 | if you know the cartesian form of the number, or |
1006 | |
1007 | $z = 3 + 4*i; |
1008 | |
1009 | if you like. To create a number using the trigonometric form, use either: |
1010 | |
1011 | $z = Math::Complex->emake(5, pi/3); |
1012 | $x = cplxe(5, pi/3); |
1013 | |
1014 | instead. The first argument is the modulus, the second is the angle (in radians). |
1015 | (Mnmemonic: C<e> is used as a notation for complex numbers in the trigonometric |
1016 | form). |
1017 | |
1018 | It is possible to write: |
1019 | |
1020 | $x = cplxe(-3, pi/4); |
1021 | |
1022 | but that will be silently converted into C<[3,-3pi/4]>, since the modulus |
1023 | must be positive (it represents the distance to the origin in the complex |
1024 | plane). |
1025 | |
1026 | =head1 STRINGIFICATION |
1027 | |
1028 | When printed, a complex number is usually shown under its cartesian |
1029 | form I<a+bi>, but there are legitimate cases where the polar format |
1030 | I<[r,t]> is more appropriate. |
1031 | |
1032 | By calling the routine C<Math::Complex::display_format> and supplying either |
1033 | C<"polar"> or C<"cartesian">, you override the default display format, |
1034 | which is C<"cartesian">. Not supplying any argument returns the current |
1035 | setting. |
1036 | |
1037 | This default can be overridden on a per-number basis by calling the |
1038 | C<display_format> method instead. As before, not supplying any argument |
1039 | returns the current display format for this number. Otherwise whatever you |
1040 | specify will be the new display format for I<this> particular number. |
1041 | |
1042 | For instance: |
1043 | |
1044 | use Math::Complex; |
1045 | |
1046 | Math::Complex::display_format('polar'); |
1047 | $j = ((root(1, 3))[1]; |
1048 | print "j = $j\n"; # Prints "j = [1,2pi/3] |
1049 | $j->display_format('cartesian'); |
1050 | print "j = $j\n"; # Prints "j = -0.5+0.866025403784439i" |
1051 | |
1052 | The polar format attempts to emphasize arguments like I<k*pi/n> |
1053 | (where I<n> is a positive integer and I<k> an integer within [-9,+9]). |
1054 | |
1055 | =head1 USAGE |
1056 | |
1057 | Thanks to overloading, the handling of arithmetics with complex numbers |
1058 | is simple and almost transparent. |
1059 | |
1060 | Here are some examples: |
1061 | |
1062 | use Math::Complex; |
1063 | |
1064 | $j = cplxe(1, 2*pi/3); # $j ** 3 == 1 |
1065 | print "j = $j, j**3 = ", $j ** 3, "\n"; |
1066 | print "1 + j + j**2 = ", 1 + $j + $j**2, "\n"; |
1067 | |
1068 | $z = -16 + 0*i; # Force it to be a complex |
1069 | print "sqrt($z) = ", sqrt($z), "\n"; |
1070 | |
1071 | $k = exp(i * 2*pi/3); |
1072 | print "$j - $k = ", $j - $k, "\n"; |
a5f75d66 |
1073 | |
1074 | =head1 BUGS |
1075 | |
66730be0 |
1076 | Saying C<use Math::Complex;> exports many mathematical routines in the caller |
1077 | environment. This is construed as a feature by the Author, actually... ;-) |
1078 | |
1079 | The code is not optimized for speed, although we try to use the cartesian |
1080 | form for addition-like operators and the trigonometric form for all |
1081 | multiplication-like operators. |
1082 | |
1083 | The arg() routine does not ensure the angle is within the range [-pi,+pi] |
1084 | (a side effect caused by multiplication and division using the trigonometric |
1085 | representation). |
a5f75d66 |
1086 | |
66730be0 |
1087 | All routines expect to be given real or complex numbers. Don't attempt to |
1088 | use BigFloat, since Perl has currently no rule to disambiguate a '+' |
1089 | operation (for instance) between two overloaded entities. |
a5f75d66 |
1090 | |
66730be0 |
1091 | =head1 AUTHOR |
a5f75d66 |
1092 | |
66730be0 |
1093 | Raphael Manfredi <F<Raphael_Manfredi@grenoble.hp.com>> |