Commit | Line | Data |
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1 | # $RCSFile$ |
2 | # |
3 | # Complex numbers and associated mathematical functions |
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4 | # -- Raphael Manfredi, September 1996 |
5 | # -- Jarkko Hietaniemi, March 1997 |
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6 | |
7 | require Exporter; |
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8 | package Math::Complex; @ISA = qw(Exporter); |
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9 | |
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10 | use strict; |
11 | |
12 | use vars qw(@EXPORT $package $display |
13 | $pi $i $ilog10 $logn %logn); |
14 | |
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15 | @EXPORT = qw( |
16 | pi i Re Im arg |
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17 | sqrt exp log ln |
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18 | log10 logn cbrt root |
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19 | tan |
20 | cosec csc sec cotan cot |
21 | asin acos atan |
22 | acosec acsc asec acotan acot |
23 | sinh cosh tanh |
24 | cosech csch sech cotanh coth |
25 | asinh acosh atanh |
26 | acosech acsch asech acotanh acoth |
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27 | cplx cplxe |
28 | ); |
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29 | |
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30 | use overload |
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31 | '+' => \&plus, |
32 | '-' => \&minus, |
33 | '*' => \&multiply, |
34 | '/' => \÷, |
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35 | '**' => \&power, |
36 | '<=>' => \&spaceship, |
37 | 'neg' => \&negate, |
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38 | '~' => \&conjugate, |
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39 | 'abs' => \&abs, |
40 | 'sqrt' => \&sqrt, |
41 | 'exp' => \&exp, |
42 | 'log' => \&log, |
43 | 'sin' => \&sin, |
44 | 'cos' => \&cos, |
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45 | 'tan' => \&tan, |
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46 | 'atan2' => \&atan2, |
47 | qw("" stringify); |
48 | |
49 | # |
50 | # Package globals |
51 | # |
52 | |
53 | $package = 'Math::Complex'; # Package name |
54 | $display = 'cartesian'; # Default display format |
55 | |
56 | # |
57 | # Object attributes (internal): |
58 | # cartesian [real, imaginary] -- cartesian form |
59 | # polar [rho, theta] -- polar form |
60 | # c_dirty cartesian form not up-to-date |
61 | # p_dirty polar form not up-to-date |
62 | # display display format (package's global when not set) |
63 | # |
64 | |
65 | # |
66 | # ->make |
67 | # |
68 | # Create a new complex number (cartesian form) |
69 | # |
70 | sub make { |
71 | my $self = bless {}, shift; |
72 | my ($re, $im) = @_; |
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73 | $self->{'cartesian'} = [$re, $im]; |
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74 | $self->{c_dirty} = 0; |
75 | $self->{p_dirty} = 1; |
76 | return $self; |
77 | } |
78 | |
79 | # |
80 | # ->emake |
81 | # |
82 | # Create a new complex number (exponential form) |
83 | # |
84 | sub emake { |
85 | my $self = bless {}, shift; |
86 | my ($rho, $theta) = @_; |
87 | $theta += pi() if $rho < 0; |
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88 | $self->{'polar'} = [abs($rho), $theta]; |
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89 | $self->{p_dirty} = 0; |
90 | $self->{c_dirty} = 1; |
91 | return $self; |
92 | } |
93 | |
94 | sub new { &make } # For backward compatibility only. |
95 | |
96 | # |
97 | # cplx |
98 | # |
99 | # Creates a complex number from a (re, im) tuple. |
100 | # This avoids the burden of writing Math::Complex->make(re, im). |
101 | # |
102 | sub cplx { |
103 | my ($re, $im) = @_; |
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104 | return $package->make($re, defined $im ? $im : 0); |
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105 | } |
106 | |
107 | # |
108 | # cplxe |
109 | # |
110 | # Creates a complex number from a (rho, theta) tuple. |
111 | # This avoids the burden of writing Math::Complex->emake(rho, theta). |
112 | # |
113 | sub cplxe { |
114 | my ($rho, $theta) = @_; |
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115 | return $package->emake($rho, defined $theta ? $theta : 0); |
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116 | } |
117 | |
118 | # |
119 | # pi |
120 | # |
121 | # The number defined as 2 * pi = 360 degrees |
122 | # |
123 | sub pi () { |
124 | $pi = 4 * atan2(1, 1) unless $pi; |
125 | return $pi; |
126 | } |
127 | |
128 | # |
129 | # i |
130 | # |
131 | # The number defined as i*i = -1; |
132 | # |
133 | sub i () { |
134 | $i = bless {} unless $i; # There can be only one i |
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135 | $i->{'cartesian'} = [0, 1]; |
136 | $i->{'polar'} = [1, pi/2]; |
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137 | $i->{c_dirty} = 0; |
138 | $i->{p_dirty} = 0; |
139 | return $i; |
140 | } |
141 | |
142 | # |
143 | # Attribute access/set routines |
144 | # |
145 | |
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146 | sub cartesian {$_[0]->{c_dirty} ? |
147 | $_[0]->update_cartesian : $_[0]->{'cartesian'}} |
148 | sub polar {$_[0]->{p_dirty} ? |
149 | $_[0]->update_polar : $_[0]->{'polar'}} |
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150 | |
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151 | sub set_cartesian { $_[0]->{p_dirty}++; $_[0]->{'cartesian'} = $_[1] } |
152 | sub set_polar { $_[0]->{c_dirty}++; $_[0]->{'polar'} = $_[1] } |
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153 | |
154 | # |
155 | # ->update_cartesian |
156 | # |
157 | # Recompute and return the cartesian form, given accurate polar form. |
158 | # |
159 | sub update_cartesian { |
160 | my $self = shift; |
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161 | my ($r, $t) = @{$self->{'polar'}}; |
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162 | $self->{c_dirty} = 0; |
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163 | return $self->{'cartesian'} = [$r * cos $t, $r * sin $t]; |
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164 | } |
165 | |
166 | # |
167 | # |
168 | # ->update_polar |
169 | # |
170 | # Recompute and return the polar form, given accurate cartesian form. |
171 | # |
172 | sub update_polar { |
173 | my $self = shift; |
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174 | my ($x, $y) = @{$self->{'cartesian'}}; |
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175 | $self->{p_dirty} = 0; |
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176 | return $self->{'polar'} = [0, 0] if $x == 0 && $y == 0; |
177 | return $self->{'polar'} = [sqrt($x*$x + $y*$y), atan2($y, $x)]; |
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178 | } |
179 | |
180 | # |
181 | # (plus) |
182 | # |
183 | # Computes z1+z2. |
184 | # |
185 | sub plus { |
186 | my ($z1, $z2, $regular) = @_; |
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187 | $z2 = cplx($z2, 0) unless ref $z2; |
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188 | my ($re1, $im1) = @{$z1->cartesian}; |
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189 | my ($re2, $im2) = @{$z2->cartesian}; |
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190 | unless (defined $regular) { |
191 | $z1->set_cartesian([$re1 + $re2, $im1 + $im2]); |
192 | return $z1; |
193 | } |
194 | return (ref $z1)->make($re1 + $re2, $im1 + $im2); |
195 | } |
196 | |
197 | # |
198 | # (minus) |
199 | # |
200 | # Computes z1-z2. |
201 | # |
202 | sub minus { |
203 | my ($z1, $z2, $inverted) = @_; |
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204 | $z2 = cplx($z2, 0) unless ref $z2; |
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205 | my ($re1, $im1) = @{$z1->cartesian}; |
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206 | my ($re2, $im2) = @{$z2->cartesian}; |
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207 | unless (defined $inverted) { |
208 | $z1->set_cartesian([$re1 - $re2, $im1 - $im2]); |
209 | return $z1; |
210 | } |
211 | return $inverted ? |
212 | (ref $z1)->make($re2 - $re1, $im2 - $im1) : |
213 | (ref $z1)->make($re1 - $re2, $im1 - $im2); |
214 | } |
215 | |
216 | # |
217 | # (multiply) |
218 | # |
219 | # Computes z1*z2. |
220 | # |
221 | sub multiply { |
222 | my ($z1, $z2, $regular) = @_; |
223 | my ($r1, $t1) = @{$z1->polar}; |
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224 | my ($r2, $t2) = ref $z2 ? |
225 | @{$z2->polar} : (abs($z2), $z2 >= 0 ? 0 : pi); |
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226 | unless (defined $regular) { |
227 | $z1->set_polar([$r1 * $r2, $t1 + $t2]); |
228 | return $z1; |
229 | } |
230 | return (ref $z1)->emake($r1 * $r2, $t1 + $t2); |
231 | } |
232 | |
233 | # |
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234 | # divbyzero |
235 | # |
236 | # Die on division by zero. |
237 | # |
238 | sub divbyzero { |
239 | warn $package . '::' . "$_[0]: Division by zero.\n"; |
240 | warn "(Because in the definition of $_[0], $_[1] is 0)\n" |
241 | if (defined $_[1]); |
242 | my @up = caller(1); |
243 | my $dmess = "Died at $up[1] line $up[2].\n"; |
244 | die $dmess; |
245 | } |
246 | |
247 | # |
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248 | # (divide) |
249 | # |
250 | # Computes z1/z2. |
251 | # |
252 | sub divide { |
253 | my ($z1, $z2, $inverted) = @_; |
254 | my ($r1, $t1) = @{$z1->polar}; |
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255 | my ($r2, $t2) = ref $z2 ? |
256 | @{$z2->polar} : (abs($z2), $z2 >= 0 ? 0 : pi); |
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257 | unless (defined $inverted) { |
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258 | divbyzero "$z1/0" if ($r2 == 0); |
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259 | $z1->set_polar([$r1 / $r2, $t1 - $t2]); |
260 | return $z1; |
261 | } |
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262 | if ($inverted) { |
263 | divbyzero "$z2/0" if ($r1 == 0); |
264 | return (ref $z1)->emake($r2 / $r1, $t2 - $t1); |
265 | } else { |
266 | divbyzero "$z1/0" if ($r2 == 0); |
267 | return (ref $z1)->emake($r1 / $r2, $t1 - $t2); |
268 | } |
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269 | } |
270 | |
271 | # |
272 | # (power) |
273 | # |
274 | # Computes z1**z2 = exp(z2 * log z1)). |
275 | # |
276 | sub power { |
277 | my ($z1, $z2, $inverted) = @_; |
278 | return exp($z1 * log $z2) if defined $inverted && $inverted; |
279 | return exp($z2 * log $z1); |
280 | } |
281 | |
282 | # |
283 | # (spaceship) |
284 | # |
285 | # Computes z1 <=> z2. |
286 | # Sorts on the real part first, then on the imaginary part. Thus 2-4i > 3+8i. |
287 | # |
288 | sub spaceship { |
289 | my ($z1, $z2, $inverted) = @_; |
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290 | $z2 = cplx($z2, 0) unless ref $z2; |
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291 | my ($re1, $im1) = @{$z1->cartesian}; |
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292 | my ($re2, $im2) = @{$z2->cartesian}; |
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293 | my $sgn = $inverted ? -1 : 1; |
294 | return $sgn * ($re1 <=> $re2) if $re1 != $re2; |
295 | return $sgn * ($im1 <=> $im2); |
296 | } |
297 | |
298 | # |
299 | # (negate) |
300 | # |
301 | # Computes -z. |
302 | # |
303 | sub negate { |
304 | my ($z) = @_; |
305 | if ($z->{c_dirty}) { |
306 | my ($r, $t) = @{$z->polar}; |
307 | return (ref $z)->emake($r, pi + $t); |
308 | } |
309 | my ($re, $im) = @{$z->cartesian}; |
310 | return (ref $z)->make(-$re, -$im); |
311 | } |
312 | |
313 | # |
314 | # (conjugate) |
315 | # |
316 | # Compute complex's conjugate. |
317 | # |
318 | sub conjugate { |
319 | my ($z) = @_; |
320 | if ($z->{c_dirty}) { |
321 | my ($r, $t) = @{$z->polar}; |
322 | return (ref $z)->emake($r, -$t); |
323 | } |
324 | my ($re, $im) = @{$z->cartesian}; |
325 | return (ref $z)->make($re, -$im); |
326 | } |
327 | |
328 | # |
329 | # (abs) |
330 | # |
331 | # Compute complex's norm (rho). |
332 | # |
333 | sub abs { |
334 | my ($z) = @_; |
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335 | return abs($z) unless ref $z; |
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336 | my ($r, $t) = @{$z->polar}; |
337 | return abs($r); |
338 | } |
339 | |
340 | # |
341 | # arg |
342 | # |
343 | # Compute complex's argument (theta). |
344 | # |
345 | sub arg { |
346 | my ($z) = @_; |
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347 | return ($z < 0 ? pi : 0) unless ref $z; |
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348 | my ($r, $t) = @{$z->polar}; |
349 | return $t; |
350 | } |
351 | |
352 | # |
353 | # (sqrt) |
354 | # |
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355 | # Compute sqrt(z). |
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356 | # |
357 | sub sqrt { |
358 | my ($z) = @_; |
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359 | $z = cplx($z, 0) unless ref $z; |
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360 | my ($r, $t) = @{$z->polar}; |
361 | return (ref $z)->emake(sqrt($r), $t/2); |
362 | } |
363 | |
364 | # |
365 | # cbrt |
366 | # |
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367 | # Compute cbrt(z) (cubic root). |
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368 | # |
369 | sub cbrt { |
370 | my ($z) = @_; |
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371 | return cplx($z, 0) ** (1/3) unless ref $z; |
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372 | my ($r, $t) = @{$z->polar}; |
373 | return (ref $z)->emake($r**(1/3), $t/3); |
374 | } |
375 | |
376 | # |
377 | # root |
378 | # |
379 | # Computes all nth root for z, returning an array whose size is n. |
380 | # `n' must be a positive integer. |
381 | # |
382 | # The roots are given by (for k = 0..n-1): |
383 | # |
384 | # z^(1/n) = r^(1/n) (cos ((t+2 k pi)/n) + i sin ((t+2 k pi)/n)) |
385 | # |
386 | sub root { |
387 | my ($z, $n) = @_; |
388 | $n = int($n + 0.5); |
389 | return undef unless $n > 0; |
390 | my ($r, $t) = ref $z ? @{$z->polar} : (abs($z), $z >= 0 ? 0 : pi); |
391 | my @root; |
392 | my $k; |
393 | my $theta_inc = 2 * pi / $n; |
394 | my $rho = $r ** (1/$n); |
395 | my $theta; |
396 | my $complex = ref($z) || $package; |
397 | for ($k = 0, $theta = $t / $n; $k < $n; $k++, $theta += $theta_inc) { |
398 | push(@root, $complex->emake($rho, $theta)); |
a0d0e21e |
399 | } |
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400 | return @root; |
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401 | } |
402 | |
66730be0 |
403 | # |
404 | # Re |
405 | # |
406 | # Return Re(z). |
407 | # |
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408 | sub Re { |
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409 | my ($z) = @_; |
410 | return $z unless ref $z; |
411 | my ($re, $im) = @{$z->cartesian}; |
412 | return $re; |
a0d0e21e |
413 | } |
414 | |
66730be0 |
415 | # |
416 | # Im |
417 | # |
418 | # Return Im(z). |
419 | # |
a0d0e21e |
420 | sub Im { |
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421 | my ($z) = @_; |
422 | return 0 unless ref $z; |
423 | my ($re, $im) = @{$z->cartesian}; |
424 | return $im; |
a0d0e21e |
425 | } |
426 | |
66730be0 |
427 | # |
428 | # (exp) |
429 | # |
430 | # Computes exp(z). |
431 | # |
432 | sub exp { |
433 | my ($z) = @_; |
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434 | $z = cplx($z, 0) unless ref $z; |
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435 | my ($x, $y) = @{$z->cartesian}; |
436 | return (ref $z)->emake(exp($x), $y); |
437 | } |
438 | |
439 | # |
440 | # (log) |
441 | # |
442 | # Compute log(z). |
443 | # |
444 | sub log { |
445 | my ($z) = @_; |
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446 | $z = cplx($z, 0) unless ref $z; |
66730be0 |
447 | my ($r, $t) = @{$z->polar}; |
0c721ce2 |
448 | my ($x, $y) = @{$z->cartesian}; |
449 | $t -= 2 * pi if ($t > pi() and $x < 0); |
450 | $t += 2 * pi if ($t < -pi() and $x < 0); |
66730be0 |
451 | return (ref $z)->make(log($r), $t); |
452 | } |
453 | |
454 | # |
0c721ce2 |
455 | # ln |
456 | # |
457 | # Alias for log(). |
458 | # |
459 | sub ln { Math::Complex::log(@_) } |
460 | |
461 | # |
66730be0 |
462 | # log10 |
463 | # |
464 | # Compute log10(z). |
465 | # |
466 | sub log10 { |
467 | my ($z) = @_; |
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468 | my $ilog10 = 1 / log(10) unless defined $ilog10; |
469 | return log(cplx($z, 0)) * $ilog10 unless ref $z; |
66730be0 |
470 | my ($r, $t) = @{$z->polar}; |
0c721ce2 |
471 | return (ref $z)->make(log($r) * $ilog10, $t * $ilog10); |
66730be0 |
472 | } |
473 | |
474 | # |
475 | # logn |
476 | # |
477 | # Compute logn(z,n) = log(z) / log(n) |
478 | # |
479 | sub logn { |
480 | my ($z, $n) = @_; |
0c721ce2 |
481 | $z = cplx($z, 0) unless ref $z; |
66730be0 |
482 | my $logn = $logn{$n}; |
483 | $logn = $logn{$n} = log($n) unless defined $logn; # Cache log(n) |
0c721ce2 |
484 | return log($z) / $logn; |
66730be0 |
485 | } |
486 | |
487 | # |
488 | # (cos) |
489 | # |
490 | # Compute cos(z) = (exp(iz) + exp(-iz))/2. |
491 | # |
492 | sub cos { |
493 | my ($z) = @_; |
494 | my ($x, $y) = @{$z->cartesian}; |
495 | my $ey = exp($y); |
496 | my $ey_1 = 1 / $ey; |
0c721ce2 |
497 | return (ref $z)->make(cos($x) * ($ey + $ey_1)/2, |
498 | sin($x) * ($ey_1 - $ey)/2); |
66730be0 |
499 | } |
500 | |
501 | # |
502 | # (sin) |
503 | # |
504 | # Compute sin(z) = (exp(iz) - exp(-iz))/2. |
505 | # |
506 | sub sin { |
507 | my ($z) = @_; |
508 | my ($x, $y) = @{$z->cartesian}; |
509 | my $ey = exp($y); |
510 | my $ey_1 = 1 / $ey; |
0c721ce2 |
511 | return (ref $z)->make(sin($x) * ($ey + $ey_1)/2, |
512 | cos($x) * ($ey - $ey_1)/2); |
66730be0 |
513 | } |
514 | |
515 | # |
516 | # tan |
517 | # |
518 | # Compute tan(z) = sin(z) / cos(z). |
519 | # |
520 | sub tan { |
521 | my ($z) = @_; |
0c721ce2 |
522 | my $cz = cos($z); |
523 | divbyzero "tan($z)", "cos($z)" if ($cz == 0); |
524 | return sin($z) / $cz; |
66730be0 |
525 | } |
526 | |
527 | # |
0c721ce2 |
528 | # sec |
529 | # |
530 | # Computes the secant sec(z) = 1 / cos(z). |
531 | # |
532 | sub sec { |
533 | my ($z) = @_; |
534 | my $cz = cos($z); |
535 | divbyzero "sec($z)", "cos($z)" if ($cz == 0); |
536 | return 1 / $cz; |
537 | } |
538 | |
539 | # |
540 | # csc |
541 | # |
542 | # Computes the cosecant csc(z) = 1 / sin(z). |
543 | # |
544 | sub csc { |
545 | my ($z) = @_; |
546 | my $sz = sin($z); |
547 | divbyzero "csc($z)", "sin($z)" if ($sz == 0); |
548 | return 1 / $sz; |
549 | } |
550 | |
66730be0 |
551 | # |
0c721ce2 |
552 | # cosec |
66730be0 |
553 | # |
0c721ce2 |
554 | # Alias for csc(). |
555 | # |
556 | sub cosec { Math::Complex::csc(@_) } |
557 | |
558 | # |
559 | # cot |
560 | # |
561 | # Computes cot(z) = 1 / tan(z). |
562 | # |
563 | sub cot { |
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564 | my ($z) = @_; |
0c721ce2 |
565 | my $sz = sin($z); |
566 | divbyzero "cot($z)", "sin($z)" if ($sz == 0); |
567 | return cos($z) / $sz; |
66730be0 |
568 | } |
569 | |
570 | # |
0c721ce2 |
571 | # cotan |
572 | # |
573 | # Alias for cot(). |
574 | # |
575 | sub cotan { Math::Complex::cot(@_) } |
576 | |
577 | # |
66730be0 |
578 | # acos |
579 | # |
580 | # Computes the arc cosine acos(z) = -i log(z + sqrt(z*z-1)). |
581 | # |
582 | sub acos { |
583 | my ($z) = @_; |
0c721ce2 |
584 | $z = cplx($z, 0) unless ref $z; |
585 | return ~i * log($z + (Re($z) * Im($z) > 0 ? 1 : -1) * sqrt($z*$z - 1)); |
66730be0 |
586 | } |
587 | |
588 | # |
589 | # asin |
590 | # |
591 | # Computes the arc sine asin(z) = -i log(iz + sqrt(1-z*z)). |
592 | # |
593 | sub asin { |
594 | my ($z) = @_; |
0c721ce2 |
595 | $z = cplx($z, 0) unless ref $z; |
596 | return ~i * log(i * $z + sqrt(1 - $z*$z)); |
66730be0 |
597 | } |
598 | |
599 | # |
600 | # atan |
601 | # |
0c721ce2 |
602 | # Computes the arc tangent atan(z) = i/2 log((i+z) / (i-z)). |
66730be0 |
603 | # |
604 | sub atan { |
605 | my ($z) = @_; |
0c721ce2 |
606 | divbyzero "atan($z)", "i - $z" if ($z == i); |
607 | return i/2*log((i + $z) / (i - $z)); |
a0d0e21e |
608 | } |
609 | |
66730be0 |
610 | # |
0c721ce2 |
611 | # asec |
612 | # |
613 | # Computes the arc secant asec(z) = acos(1 / z). |
614 | # |
615 | sub asec { |
616 | my ($z) = @_; |
617 | return acos(1 / $z); |
618 | } |
619 | |
620 | # |
621 | # acosec |
622 | # |
623 | # Computes the arc cosecant sec(z) = asin(1 / z). |
624 | # |
625 | sub acosec { |
626 | my ($z) = @_; |
627 | return asin(1 / $z); |
628 | } |
629 | |
630 | # |
631 | # acsc |
66730be0 |
632 | # |
0c721ce2 |
633 | # Alias for acosec(). |
634 | # |
635 | sub acsc { Math::Complex::acosec(@_) } |
636 | |
66730be0 |
637 | # |
0c721ce2 |
638 | # acot |
639 | # |
640 | # Computes the arc cotangent acot(z) = -i/2 log((i+z) / (z-i)) |
641 | # |
642 | sub acot { |
66730be0 |
643 | my ($z) = @_; |
0c721ce2 |
644 | divbyzero "acot($z)", "$z - i" if ($z == i); |
66730be0 |
645 | return i/-2 * log((i + $z) / ($z - i)); |
646 | } |
647 | |
648 | # |
0c721ce2 |
649 | # acotan |
650 | # |
651 | # Alias for acot(). |
652 | # |
653 | sub acotan { Math::Complex::acot(@_) } |
654 | |
655 | # |
66730be0 |
656 | # cosh |
657 | # |
658 | # Computes the hyperbolic cosine cosh(z) = (exp(z) + exp(-z))/2. |
659 | # |
660 | sub cosh { |
661 | my ($z) = @_; |
0c721ce2 |
662 | $z = cplx($z, 0) unless ref $z; |
663 | my ($x, $y) = @{$z->cartesian}; |
66730be0 |
664 | my $ex = exp($x); |
665 | my $ex_1 = 1 / $ex; |
666 | return ($ex + $ex_1)/2 unless ref $z; |
0c721ce2 |
667 | return (ref $z)->make(cos($y) * ($ex + $ex_1)/2, |
668 | sin($y) * ($ex - $ex_1)/2); |
66730be0 |
669 | } |
670 | |
671 | # |
672 | # sinh |
673 | # |
674 | # Computes the hyperbolic sine sinh(z) = (exp(z) - exp(-z))/2. |
675 | # |
676 | sub sinh { |
677 | my ($z) = @_; |
0c721ce2 |
678 | $z = cplx($z, 0) unless ref $z; |
679 | my ($x, $y) = @{$z->cartesian}; |
66730be0 |
680 | my $ex = exp($x); |
681 | my $ex_1 = 1 / $ex; |
682 | return ($ex - $ex_1)/2 unless ref $z; |
0c721ce2 |
683 | return (ref $z)->make(cos($y) * ($ex - $ex_1)/2, |
684 | sin($y) * ($ex + $ex_1)/2); |
66730be0 |
685 | } |
686 | |
687 | # |
688 | # tanh |
689 | # |
690 | # Computes the hyperbolic tangent tanh(z) = sinh(z) / cosh(z). |
691 | # |
692 | sub tanh { |
693 | my ($z) = @_; |
0c721ce2 |
694 | my $cz = cosh($z); |
695 | divbyzero "tanh($z)", "cosh($z)" if ($cz == 0); |
696 | return sinh($z) / $cz; |
66730be0 |
697 | } |
698 | |
699 | # |
0c721ce2 |
700 | # sech |
701 | # |
702 | # Computes the hyperbolic secant sech(z) = 1 / cosh(z). |
703 | # |
704 | sub sech { |
705 | my ($z) = @_; |
706 | my $cz = cosh($z); |
707 | divbyzero "sech($z)", "cosh($z)" if ($cz == 0); |
708 | return 1 / $cz; |
709 | } |
710 | |
711 | # |
712 | # csch |
713 | # |
714 | # Computes the hyperbolic cosecant csch(z) = 1 / sinh(z). |
66730be0 |
715 | # |
0c721ce2 |
716 | sub csch { |
717 | my ($z) = @_; |
718 | my $sz = sinh($z); |
719 | divbyzero "csch($z)", "sinh($z)" if ($sz == 0); |
720 | return 1 / $sz; |
721 | } |
722 | |
723 | # |
724 | # cosech |
725 | # |
726 | # Alias for csch(). |
727 | # |
728 | sub cosech { Math::Complex::csch(@_) } |
729 | |
66730be0 |
730 | # |
0c721ce2 |
731 | # coth |
732 | # |
733 | # Computes the hyperbolic cotangent coth(z) = cosh(z) / sinh(z). |
734 | # |
735 | sub coth { |
66730be0 |
736 | my ($z) = @_; |
0c721ce2 |
737 | my $sz = sinh($z); |
738 | divbyzero "coth($z)", "sinh($z)" if ($sz == 0); |
739 | return cosh($z) / $sz; |
66730be0 |
740 | } |
741 | |
742 | # |
0c721ce2 |
743 | # cotanh |
744 | # |
745 | # Alias for coth(). |
746 | # |
747 | sub cotanh { Math::Complex::coth(@_) } |
748 | |
749 | # |
66730be0 |
750 | # acosh |
751 | # |
752 | # Computes the arc hyperbolic cosine acosh(z) = log(z + sqrt(z*z-1)). |
753 | # |
754 | sub acosh { |
755 | my ($z) = @_; |
0c721ce2 |
756 | $z = cplx($z, 0) unless ref $z; # asinh(-2) |
757 | return log($z + sqrt($z*$z - 1)); |
66730be0 |
758 | } |
759 | |
760 | # |
761 | # asinh |
762 | # |
763 | # Computes the arc hyperbolic sine asinh(z) = log(z + sqrt(z*z-1)) |
764 | # |
765 | sub asinh { |
766 | my ($z) = @_; |
0c721ce2 |
767 | $z = cplx($z, 0) unless ref $z; # asinh(-2) |
768 | return log($z + sqrt($z*$z + 1)); |
66730be0 |
769 | } |
770 | |
771 | # |
772 | # atanh |
773 | # |
774 | # Computes the arc hyperbolic tangent atanh(z) = 1/2 log((1+z) / (1-z)). |
775 | # |
776 | sub atanh { |
777 | my ($z) = @_; |
0c721ce2 |
778 | $z = cplx($z, 0) unless ref $z; # atanh(-2) |
779 | divbyzero 'atanh(1)', "1 - $z" if ($z == 1); |
66730be0 |
780 | my $cz = (1 + $z) / (1 - $z); |
66730be0 |
781 | return log($cz) / 2; |
782 | } |
783 | |
784 | # |
0c721ce2 |
785 | # asech |
786 | # |
787 | # Computes the hyperbolic arc secant asech(z) = acosh(1 / z). |
788 | # |
789 | sub asech { |
790 | my ($z) = @_; |
791 | divbyzero 'asech(0)', $z if ($z == 0); |
792 | return acosh(1 / $z); |
793 | } |
794 | |
795 | # |
796 | # acsch |
66730be0 |
797 | # |
0c721ce2 |
798 | # Computes the hyperbolic arc cosecant acsch(z) = asinh(1 / z). |
66730be0 |
799 | # |
0c721ce2 |
800 | sub acsch { |
66730be0 |
801 | my ($z) = @_; |
0c721ce2 |
802 | divbyzero 'acsch(0)', $z if ($z == 0); |
803 | return asinh(1 / $z); |
804 | } |
805 | |
806 | # |
807 | # acosech |
808 | # |
809 | # Alias for acosh(). |
810 | # |
811 | sub acosech { Math::Complex::acsch(@_) } |
812 | |
813 | # |
814 | # acoth |
815 | # |
816 | # Computes the arc hyperbolic cotangent acoth(z) = 1/2 log((1+z) / (z-1)). |
817 | # |
818 | sub acoth { |
819 | my ($z) = @_; |
820 | $z = cplx($z, 0) unless ref $z; # acoth(-2) |
821 | divbyzero 'acoth(1)', "$z - 1" if ($z == 1); |
66730be0 |
822 | my $cz = (1 + $z) / ($z - 1); |
66730be0 |
823 | return log($cz) / 2; |
824 | } |
825 | |
826 | # |
0c721ce2 |
827 | # acotanh |
828 | # |
829 | # Alias for acot(). |
830 | # |
831 | sub acotanh { Math::Complex::acoth(@_) } |
832 | |
833 | # |
66730be0 |
834 | # (atan2) |
835 | # |
836 | # Compute atan(z1/z2). |
837 | # |
838 | sub atan2 { |
839 | my ($z1, $z2, $inverted) = @_; |
840 | my ($re1, $im1) = @{$z1->cartesian}; |
0c721ce2 |
841 | my ($re2, $im2) = @{$z2->cartesian}; |
66730be0 |
842 | my $tan; |
843 | if (defined $inverted && $inverted) { # atan(z2/z1) |
844 | return pi * ($re2 > 0 ? 1 : -1) if $re1 == 0 && $im1 == 0; |
845 | $tan = $z2 / $z1; |
846 | } else { |
847 | return pi * ($re1 > 0 ? 1 : -1) if $re2 == 0 && $im2 == 0; |
848 | $tan = $z1 / $z2; |
849 | } |
850 | return atan($tan); |
851 | } |
852 | |
853 | # |
854 | # display_format |
855 | # ->display_format |
856 | # |
857 | # Set (fetch if no argument) display format for all complex numbers that |
858 | # don't happen to have overrriden it via ->display_format |
859 | # |
860 | # When called as a method, this actually sets the display format for |
861 | # the current object. |
862 | # |
863 | # Valid object formats are 'c' and 'p' for cartesian and polar. The first |
864 | # letter is used actually, so the type can be fully spelled out for clarity. |
865 | # |
866 | sub display_format { |
867 | my $self = shift; |
868 | my $format = undef; |
869 | |
870 | if (ref $self) { # Called as a method |
871 | $format = shift; |
0c721ce2 |
872 | } else { # Regular procedure call |
66730be0 |
873 | $format = $self; |
874 | undef $self; |
875 | } |
876 | |
877 | if (defined $self) { |
878 | return defined $self->{display} ? $self->{display} : $display |
879 | unless defined $format; |
880 | return $self->{display} = $format; |
881 | } |
882 | |
883 | return $display unless defined $format; |
884 | return $display = $format; |
885 | } |
886 | |
887 | # |
888 | # (stringify) |
889 | # |
890 | # Show nicely formatted complex number under its cartesian or polar form, |
891 | # depending on the current display format: |
892 | # |
893 | # . If a specific display format has been recorded for this object, use it. |
894 | # . Otherwise, use the generic current default for all complex numbers, |
895 | # which is a package global variable. |
896 | # |
a0d0e21e |
897 | sub stringify { |
66730be0 |
898 | my ($z) = shift; |
899 | my $format; |
900 | |
901 | $format = $display; |
902 | $format = $z->{display} if defined $z->{display}; |
903 | |
904 | return $z->stringify_polar if $format =~ /^p/i; |
905 | return $z->stringify_cartesian; |
906 | } |
907 | |
908 | # |
909 | # ->stringify_cartesian |
910 | # |
911 | # Stringify as a cartesian representation 'a+bi'. |
912 | # |
913 | sub stringify_cartesian { |
914 | my $z = shift; |
915 | my ($x, $y) = @{$z->cartesian}; |
916 | my ($re, $im); |
917 | |
55497cff |
918 | $x = int($x + ($x < 0 ? -1 : 1) * 1e-14) |
919 | if int(abs($x)) != int(abs($x) + 1e-14); |
920 | $y = int($y + ($y < 0 ? -1 : 1) * 1e-14) |
921 | if int(abs($y)) != int(abs($y) + 1e-14); |
922 | |
66730be0 |
923 | $re = "$x" if abs($x) >= 1e-14; |
924 | if ($y == 1) { $im = 'i' } |
925 | elsif ($y == -1) { $im = '-i' } |
40da2db3 |
926 | elsif (abs($y) >= 1e-14) { $im = $y . "i" } |
66730be0 |
927 | |
0c721ce2 |
928 | my $str = ''; |
66730be0 |
929 | $str = $re if defined $re; |
930 | $str .= "+$im" if defined $im; |
931 | $str =~ s/\+-/-/; |
932 | $str =~ s/^\+//; |
933 | $str = '0' unless $str; |
934 | |
935 | return $str; |
936 | } |
937 | |
938 | # |
939 | # ->stringify_polar |
940 | # |
941 | # Stringify as a polar representation '[r,t]'. |
942 | # |
943 | sub stringify_polar { |
944 | my $z = shift; |
945 | my ($r, $t) = @{$z->polar}; |
946 | my $theta; |
0c721ce2 |
947 | my $eps = 1e-14; |
66730be0 |
948 | |
0c721ce2 |
949 | return '[0,0]' if $r <= $eps; |
a0d0e21e |
950 | |
66730be0 |
951 | my $tpi = 2 * pi; |
952 | my $nt = $t / $tpi; |
953 | $nt = ($nt - int($nt)) * $tpi; |
954 | $nt += $tpi if $nt < 0; # Range [0, 2pi] |
a0d0e21e |
955 | |
0c721ce2 |
956 | if (abs($nt) <= $eps) { $theta = 0 } |
957 | elsif (abs(pi-$nt) <= $eps) { $theta = 'pi' } |
66730be0 |
958 | |
55497cff |
959 | if (defined $theta) { |
0c721ce2 |
960 | $r = int($r + ($r < 0 ? -1 : 1) * $eps) |
961 | if int(abs($r)) != int(abs($r) + $eps); |
962 | $theta = int($theta + ($theta < 0 ? -1 : 1) * $eps) |
963 | if ($theta ne 'pi' and |
964 | int(abs($theta)) != int(abs($theta) + $eps)); |
55497cff |
965 | return "\[$r,$theta\]"; |
966 | } |
66730be0 |
967 | |
968 | # |
969 | # Okay, number is not a real. Try to identify pi/n and friends... |
970 | # |
971 | |
972 | $nt -= $tpi if $nt > pi; |
973 | my ($n, $k, $kpi); |
974 | |
975 | for ($k = 1, $kpi = pi; $k < 10; $k++, $kpi += pi) { |
976 | $n = int($kpi / $nt + ($nt > 0 ? 1 : -1) * 0.5); |
0c721ce2 |
977 | if (abs($kpi/$n - $nt) <= $eps) { |
978 | $theta = ($nt < 0 ? '-':''). |
979 | ($k == 1 ? 'pi':"${k}pi").'/'.abs($n); |
66730be0 |
980 | last; |
981 | } |
982 | } |
983 | |
984 | $theta = $nt unless defined $theta; |
985 | |
0c721ce2 |
986 | $r = int($r + ($r < 0 ? -1 : 1) * $eps) |
987 | if int(abs($r)) != int(abs($r) + $eps); |
988 | $theta = int($theta + ($theta < 0 ? -1 : 1) * $eps) |
989 | if ($theta !~ m(^-?\d*pi/\d+$) and |
990 | int(abs($theta)) != int(abs($theta) + $eps)); |
55497cff |
991 | |
66730be0 |
992 | return "\[$r,$theta\]"; |
a0d0e21e |
993 | } |
a5f75d66 |
994 | |
995 | 1; |
996 | __END__ |
997 | |
998 | =head1 NAME |
999 | |
66730be0 |
1000 | Math::Complex - complex numbers and associated mathematical functions |
a5f75d66 |
1001 | |
1002 | =head1 SYNOPSIS |
1003 | |
66730be0 |
1004 | use Math::Complex; |
1005 | $z = Math::Complex->make(5, 6); |
1006 | $t = 4 - 3*i + $z; |
1007 | $j = cplxe(1, 2*pi/3); |
a5f75d66 |
1008 | |
1009 | =head1 DESCRIPTION |
1010 | |
66730be0 |
1011 | This package lets you create and manipulate complex numbers. By default, |
1012 | I<Perl> limits itself to real numbers, but an extra C<use> statement brings |
1013 | full complex support, along with a full set of mathematical functions |
1014 | typically associated with and/or extended to complex numbers. |
1015 | |
1016 | If you wonder what complex numbers are, they were invented to be able to solve |
1017 | the following equation: |
1018 | |
1019 | x*x = -1 |
1020 | |
1021 | and by definition, the solution is noted I<i> (engineers use I<j> instead since |
1022 | I<i> usually denotes an intensity, but the name does not matter). The number |
1023 | I<i> is a pure I<imaginary> number. |
1024 | |
1025 | The arithmetics with pure imaginary numbers works just like you would expect |
1026 | it with real numbers... you just have to remember that |
1027 | |
1028 | i*i = -1 |
1029 | |
1030 | so you have: |
1031 | |
1032 | 5i + 7i = i * (5 + 7) = 12i |
1033 | 4i - 3i = i * (4 - 3) = i |
1034 | 4i * 2i = -8 |
1035 | 6i / 2i = 3 |
1036 | 1 / i = -i |
1037 | |
1038 | Complex numbers are numbers that have both a real part and an imaginary |
1039 | part, and are usually noted: |
1040 | |
1041 | a + bi |
1042 | |
1043 | where C<a> is the I<real> part and C<b> is the I<imaginary> part. The |
1044 | arithmetic with complex numbers is straightforward. You have to |
1045 | keep track of the real and the imaginary parts, but otherwise the |
1046 | rules used for real numbers just apply: |
1047 | |
1048 | (4 + 3i) + (5 - 2i) = (4 + 5) + i(3 - 2) = 9 + i |
1049 | (2 + i) * (4 - i) = 2*4 + 4i -2i -i*i = 8 + 2i + 1 = 9 + 2i |
1050 | |
1051 | A graphical representation of complex numbers is possible in a plane |
1052 | (also called the I<complex plane>, but it's really a 2D plane). |
1053 | The number |
1054 | |
1055 | z = a + bi |
1056 | |
1057 | is the point whose coordinates are (a, b). Actually, it would |
1058 | be the vector originating from (0, 0) to (a, b). It follows that the addition |
1059 | of two complex numbers is a vectorial addition. |
1060 | |
1061 | Since there is a bijection between a point in the 2D plane and a complex |
1062 | number (i.e. the mapping is unique and reciprocal), a complex number |
1063 | can also be uniquely identified with polar coordinates: |
1064 | |
1065 | [rho, theta] |
1066 | |
1067 | where C<rho> is the distance to the origin, and C<theta> the angle between |
1068 | the vector and the I<x> axis. There is a notation for this using the |
1069 | exponential form, which is: |
1070 | |
1071 | rho * exp(i * theta) |
1072 | |
1073 | where I<i> is the famous imaginary number introduced above. Conversion |
1074 | between this form and the cartesian form C<a + bi> is immediate: |
1075 | |
1076 | a = rho * cos(theta) |
1077 | b = rho * sin(theta) |
1078 | |
1079 | which is also expressed by this formula: |
1080 | |
1081 | z = rho * exp(i * theta) = rho * (cos theta + i * sin theta) |
1082 | |
1083 | In other words, it's the projection of the vector onto the I<x> and I<y> |
1084 | axes. Mathematicians call I<rho> the I<norm> or I<modulus> and I<theta> |
1085 | the I<argument> of the complex number. The I<norm> of C<z> will be |
1086 | noted C<abs(z)>. |
1087 | |
1088 | The polar notation (also known as the trigonometric |
1089 | representation) is much more handy for performing multiplications and |
1090 | divisions of complex numbers, whilst the cartesian notation is better |
1091 | suited for additions and substractions. Real numbers are on the I<x> |
1092 | axis, and therefore I<theta> is zero. |
1093 | |
1094 | All the common operations that can be performed on a real number have |
1095 | been defined to work on complex numbers as well, and are merely |
1096 | I<extensions> of the operations defined on real numbers. This means |
1097 | they keep their natural meaning when there is no imaginary part, provided |
1098 | the number is within their definition set. |
1099 | |
1100 | For instance, the C<sqrt> routine which computes the square root of |
1101 | its argument is only defined for positive real numbers and yields a |
1102 | positive real number (it is an application from B<R+> to B<R+>). |
1103 | If we allow it to return a complex number, then it can be extended to |
1104 | negative real numbers to become an application from B<R> to B<C> (the |
1105 | set of complex numbers): |
1106 | |
1107 | sqrt(x) = x >= 0 ? sqrt(x) : sqrt(-x)*i |
1108 | |
1109 | It can also be extended to be an application from B<C> to B<C>, |
1110 | whilst its restriction to B<R> behaves as defined above by using |
1111 | the following definition: |
1112 | |
1113 | sqrt(z = [r,t]) = sqrt(r) * exp(i * t/2) |
1114 | |
1115 | Indeed, a negative real number can be noted C<[x,pi]> |
1116 | (the modulus I<x> is always positive, so C<[x,pi]> is really C<-x>, a |
1117 | negative number) |
1118 | and the above definition states that |
1119 | |
1120 | sqrt([x,pi]) = sqrt(x) * exp(i*pi/2) = [sqrt(x),pi/2] = sqrt(x)*i |
1121 | |
1122 | which is exactly what we had defined for negative real numbers above. |
a5f75d66 |
1123 | |
66730be0 |
1124 | All the common mathematical functions defined on real numbers that |
1125 | are extended to complex numbers share that same property of working |
1126 | I<as usual> when the imaginary part is zero (otherwise, it would not |
1127 | be called an extension, would it?). |
a5f75d66 |
1128 | |
66730be0 |
1129 | A I<new> operation possible on a complex number that is |
1130 | the identity for real numbers is called the I<conjugate>, and is noted |
1131 | with an horizontal bar above the number, or C<~z> here. |
a5f75d66 |
1132 | |
66730be0 |
1133 | z = a + bi |
1134 | ~z = a - bi |
a5f75d66 |
1135 | |
66730be0 |
1136 | Simple... Now look: |
a5f75d66 |
1137 | |
66730be0 |
1138 | z * ~z = (a + bi) * (a - bi) = a*a + b*b |
a5f75d66 |
1139 | |
66730be0 |
1140 | We saw that the norm of C<z> was noted C<abs(z)> and was defined as the |
1141 | distance to the origin, also known as: |
a5f75d66 |
1142 | |
66730be0 |
1143 | rho = abs(z) = sqrt(a*a + b*b) |
a5f75d66 |
1144 | |
66730be0 |
1145 | so |
1146 | |
1147 | z * ~z = abs(z) ** 2 |
1148 | |
1149 | If z is a pure real number (i.e. C<b == 0>), then the above yields: |
1150 | |
1151 | a * a = abs(a) ** 2 |
1152 | |
1153 | which is true (C<abs> has the regular meaning for real number, i.e. stands |
1154 | for the absolute value). This example explains why the norm of C<z> is |
1155 | noted C<abs(z)>: it extends the C<abs> function to complex numbers, yet |
1156 | is the regular C<abs> we know when the complex number actually has no |
1157 | imaginary part... This justifies I<a posteriori> our use of the C<abs> |
1158 | notation for the norm. |
1159 | |
1160 | =head1 OPERATIONS |
1161 | |
1162 | Given the following notations: |
1163 | |
1164 | z1 = a + bi = r1 * exp(i * t1) |
1165 | z2 = c + di = r2 * exp(i * t2) |
1166 | z = <any complex or real number> |
1167 | |
1168 | the following (overloaded) operations are supported on complex numbers: |
1169 | |
1170 | z1 + z2 = (a + c) + i(b + d) |
1171 | z1 - z2 = (a - c) + i(b - d) |
1172 | z1 * z2 = (r1 * r2) * exp(i * (t1 + t2)) |
1173 | z1 / z2 = (r1 / r2) * exp(i * (t1 - t2)) |
1174 | z1 ** z2 = exp(z2 * log z1) |
1175 | ~z1 = a - bi |
1176 | abs(z1) = r1 = sqrt(a*a + b*b) |
1177 | sqrt(z1) = sqrt(r1) * exp(i * t1/2) |
1178 | exp(z1) = exp(a) * exp(i * b) |
1179 | log(z1) = log(r1) + i*t1 |
1180 | sin(z1) = 1/2i (exp(i * z1) - exp(-i * z1)) |
1181 | cos(z1) = 1/2 (exp(i * z1) + exp(-i * z1)) |
1182 | abs(z1) = r1 |
1183 | atan2(z1, z2) = atan(z1/z2) |
1184 | |
1185 | The following extra operations are supported on both real and complex |
1186 | numbers: |
1187 | |
1188 | Re(z) = a |
1189 | Im(z) = b |
1190 | arg(z) = t |
1191 | |
1192 | cbrt(z) = z ** (1/3) |
1193 | log10(z) = log(z) / log(10) |
1194 | logn(z, n) = log(z) / log(n) |
1195 | |
1196 | tan(z) = sin(z) / cos(z) |
0c721ce2 |
1197 | |
1198 | csc(z) = 1 / sin(z) |
1199 | sec(z) = 1 / cos(z) |
1200 | cot(z) = 1 / tan(z) |
66730be0 |
1201 | |
1202 | asin(z) = -i * log(i*z + sqrt(1-z*z)) |
1203 | acos(z) = -i * log(z + sqrt(z*z-1)) |
1204 | atan(z) = i/2 * log((i+z) / (i-z)) |
0c721ce2 |
1205 | |
1206 | acsc(z) = asin(1 / z) |
1207 | asec(z) = acos(1 / z) |
1208 | acot(z) = -i/2 * log((i+z) / (z-i)) |
66730be0 |
1209 | |
1210 | sinh(z) = 1/2 (exp(z) - exp(-z)) |
1211 | cosh(z) = 1/2 (exp(z) + exp(-z)) |
0c721ce2 |
1212 | tanh(z) = sinh(z) / cosh(z) = (exp(z) - exp(-z)) / (exp(z) + exp(-z)) |
1213 | |
1214 | csch(z) = 1 / sinh(z) |
1215 | sech(z) = 1 / cosh(z) |
1216 | coth(z) = 1 / tanh(z) |
66730be0 |
1217 | |
1218 | asinh(z) = log(z + sqrt(z*z+1)) |
1219 | acosh(z) = log(z + sqrt(z*z-1)) |
1220 | atanh(z) = 1/2 * log((1+z) / (1-z)) |
66730be0 |
1221 | |
0c721ce2 |
1222 | acsch(z) = asinh(1 / z) |
1223 | asech(z) = acosh(1 / z) |
1224 | acoth(z) = atanh(1 / z) = 1/2 * log((1+z) / (z-1)) |
1225 | |
1226 | I<log>, I<csc>, I<cot>, I<acsc>, I<acot>, I<csch>, I<coth>, |
1227 | I<acosech>, I<acotanh>, have aliases I<ln>, I<cosec>, I<cotan>, |
1228 | I<acosec>, I<acotan>, I<cosech>, I<cotanh>, I<acosech>, I<acotanh>, |
1229 | respectively. |
1230 | |
1231 | The I<root> function is available to compute all the I<n> |
66730be0 |
1232 | roots of some complex, where I<n> is a strictly positive integer. |
1233 | There are exactly I<n> such roots, returned as a list. Getting the |
1234 | number mathematicians call C<j> such that: |
1235 | |
1236 | 1 + j + j*j = 0; |
1237 | |
1238 | is a simple matter of writing: |
1239 | |
1240 | $j = ((root(1, 3))[1]; |
1241 | |
1242 | The I<k>th root for C<z = [r,t]> is given by: |
1243 | |
1244 | (root(z, n))[k] = r**(1/n) * exp(i * (t + 2*k*pi)/n) |
1245 | |
0c721ce2 |
1246 | The I<spaceship> comparison operator is also defined. In order to |
1247 | ensure its restriction to real numbers is conform to what you would |
1248 | expect, the comparison is run on the real part of the complex number |
1249 | first, and imaginary parts are compared only when the real parts |
1250 | match. |
66730be0 |
1251 | |
1252 | =head1 CREATION |
1253 | |
1254 | To create a complex number, use either: |
1255 | |
1256 | $z = Math::Complex->make(3, 4); |
1257 | $z = cplx(3, 4); |
1258 | |
1259 | if you know the cartesian form of the number, or |
1260 | |
1261 | $z = 3 + 4*i; |
1262 | |
1263 | if you like. To create a number using the trigonometric form, use either: |
1264 | |
1265 | $z = Math::Complex->emake(5, pi/3); |
1266 | $x = cplxe(5, pi/3); |
1267 | |
0c721ce2 |
1268 | instead. The first argument is the modulus, the second is the angle |
1269 | (in radians, the full circle is 2*pi). (Mnmemonic: C<e> is used as a |
1270 | notation for complex numbers in the trigonometric form). |
66730be0 |
1271 | |
1272 | It is possible to write: |
1273 | |
1274 | $x = cplxe(-3, pi/4); |
1275 | |
1276 | but that will be silently converted into C<[3,-3pi/4]>, since the modulus |
1277 | must be positive (it represents the distance to the origin in the complex |
1278 | plane). |
1279 | |
1280 | =head1 STRINGIFICATION |
1281 | |
1282 | When printed, a complex number is usually shown under its cartesian |
1283 | form I<a+bi>, but there are legitimate cases where the polar format |
1284 | I<[r,t]> is more appropriate. |
1285 | |
1286 | By calling the routine C<Math::Complex::display_format> and supplying either |
1287 | C<"polar"> or C<"cartesian">, you override the default display format, |
1288 | which is C<"cartesian">. Not supplying any argument returns the current |
1289 | setting. |
1290 | |
1291 | This default can be overridden on a per-number basis by calling the |
1292 | C<display_format> method instead. As before, not supplying any argument |
1293 | returns the current display format for this number. Otherwise whatever you |
1294 | specify will be the new display format for I<this> particular number. |
1295 | |
1296 | For instance: |
1297 | |
1298 | use Math::Complex; |
1299 | |
1300 | Math::Complex::display_format('polar'); |
1301 | $j = ((root(1, 3))[1]; |
1302 | print "j = $j\n"; # Prints "j = [1,2pi/3] |
1303 | $j->display_format('cartesian'); |
1304 | print "j = $j\n"; # Prints "j = -0.5+0.866025403784439i" |
1305 | |
1306 | The polar format attempts to emphasize arguments like I<k*pi/n> |
1307 | (where I<n> is a positive integer and I<k> an integer within [-9,+9]). |
1308 | |
1309 | =head1 USAGE |
1310 | |
1311 | Thanks to overloading, the handling of arithmetics with complex numbers |
1312 | is simple and almost transparent. |
1313 | |
1314 | Here are some examples: |
1315 | |
1316 | use Math::Complex; |
1317 | |
1318 | $j = cplxe(1, 2*pi/3); # $j ** 3 == 1 |
1319 | print "j = $j, j**3 = ", $j ** 3, "\n"; |
1320 | print "1 + j + j**2 = ", 1 + $j + $j**2, "\n"; |
1321 | |
1322 | $z = -16 + 0*i; # Force it to be a complex |
1323 | print "sqrt($z) = ", sqrt($z), "\n"; |
1324 | |
1325 | $k = exp(i * 2*pi/3); |
1326 | print "$j - $k = ", $j - $k, "\n"; |
a5f75d66 |
1327 | |
1328 | =head1 BUGS |
1329 | |
66730be0 |
1330 | Saying C<use Math::Complex;> exports many mathematical routines in the caller |
1331 | environment. This is construed as a feature by the Author, actually... ;-) |
1332 | |
1333 | The code is not optimized for speed, although we try to use the cartesian |
1334 | form for addition-like operators and the trigonometric form for all |
1335 | multiplication-like operators. |
1336 | |
1337 | The arg() routine does not ensure the angle is within the range [-pi,+pi] |
1338 | (a side effect caused by multiplication and division using the trigonometric |
1339 | representation). |
a5f75d66 |
1340 | |
66730be0 |
1341 | All routines expect to be given real or complex numbers. Don't attempt to |
1342 | use BigFloat, since Perl has currently no rule to disambiguate a '+' |
1343 | operation (for instance) between two overloaded entities. |
a5f75d66 |
1344 | |
0c721ce2 |
1345 | =head1 AUTHORS |
a5f75d66 |
1346 | |
0c721ce2 |
1347 | Raphael Manfredi <F<Raphael_Manfredi@grenoble.hp.com>> |
1348 | Jarkko Hietaniemi <F<jhi@iki.fi>> |