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LCMathComplex.pm
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Sun Nov 10 23:52:07 2019 UTC (4 years, 5 months ago) by
raeburn
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CVS tags:
version_2_12_X,
version_2_11_X,
version_2_11_4_uiuc,
version_2_11_4_msu,
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HEAD
- Update to Math::Complex rev. 1.59_01 to eliminate warnings in perl > 5.26
"Undefined value assigned to typeglob" then modify (and rename as:
LONCAPA::LCMathComplex) to eliminate use of Config.pm and Scalar::Util.pm,
when run in perl Safe container).
1: #
2: # Complex numbers and associated mathematical functions
3: # -- Raphael Manfredi Since Sep 1996
4: # -- Jarkko Hietaniemi Since Mar 1997
5: # -- Daniel S. Lewart Since Sep 1997
6: #
7: # -- Stuart Raeburn: renamed package (rev. 1.55) as LCMathComplex Oct 2013
8: # renamed package (rev. 1.59_01) as LCMathComplex Nov 2019
9: # with minor changes to allow use in Safe Space
10: #
11:
12: package LONCAPA::LCMathComplex;
13:
14: { use 5.006; }
15: use strict;
16:
17: our $VERSION = 1.59_01;
18:
19: our ($Inf, $ExpInf);
20: our ($vax_float, $has_inf, $has_nan);
21:
22: BEGIN {
23: $vax_float = (pack("d",1) =~ /^[\x80\x10]\x40/);
24: $has_inf = !$vax_float;
25: $has_nan = !$vax_float;
26:
27: unless ($has_inf) {
28: # For example in vax, there is no Inf,
29: # and just mentioning the DBL_MAX (1.70141183460469229e+38)
30: # causes SIGFPE.
31:
32: # These are pretty useless without a real infinity,
33: # but setting them makes for less warnings about their
34: # undefined values.
35: $Inf = "Inf";
36: $ExpInf = "Inf";
37: return;
38: }
39:
40: my %DBL_MAX = # These are IEEE 754 maxima.
41: (
42: 4 => '1.70141183460469229e+38',
43: 8 => '1.7976931348623157e+308',
44: # AFAICT the 10, 12, and 16-byte long doubles
45: # all have the same maximum.
46: 10 => '1.1897314953572317650857593266280070162E+4932',
47: 12 => '1.1897314953572317650857593266280070162E+4932',
48: 16 => '1.1897314953572317650857593266280070162E+4932',
49: );
50:
51: my $nvsize = 8;
52: die "Math::Complex: Could not figure out nvsize\n"
53: unless defined $nvsize;
54: die "LONCAPA::LCMathComplex: Cannot not figure out max nv (nvsize = $nvsize)\n"
55: unless defined $DBL_MAX{$nvsize};
56: my $DBL_MAX = eval $DBL_MAX{$nvsize};
57: die "LONCAPA::LCMathComplex: Could not figure out max nv (nvsize = $nvsize)\n"
58: unless defined $DBL_MAX;
59: my $BIGGER_THAN_THIS = 1e30; # Must find something bigger than this.
60: if ($^O eq 'unicosmk') {
61: $Inf = $DBL_MAX;
62: } else {
63: local $SIG{FPE} = sub { };
64: local $!;
65: # We do want an arithmetic overflow, Inf INF inf Infinity.
66: for my $t (
67: 'exp(99999)', # Enough even with 128-bit long doubles.
68: 'inf',
69: 'Inf',
70: 'INF',
71: 'infinity',
72: 'Infinity',
73: 'INFINITY',
74: '1e99999',
75: ) {
76: local $^W = 0;
77: my $i = eval "$t+1.0";
78: if (defined $i && $i > $BIGGER_THAN_THIS) {
79: $Inf = $i;
80: last;
81: }
82: }
83: $Inf = $DBL_MAX unless defined $Inf; # Oh well, close enough.
84: die "LONCAPA::LCMathComplex: Could not get Infinity"
85: unless $Inf > $BIGGER_THAN_THIS;
86: $ExpInf = eval 'exp(99999)';
87: }
88: # print "# On this machine, Inf = '$Inf'\n";
89: }
90:
91: use warnings;
92: no warnings 'syntax'; # To avoid the (_) warnings.
93:
94: my $i;
95: my %LOGN;
96:
97: # Regular expression for floating point numbers.
98: # These days we could use Scalar::Util::lln(), I guess.
99: my $gre = qr'\s*([\+\-]?(?:(?:(?:\d+(?:_\d+)*(?:\.\d*(?:_\d+)*)?|\.\d+(?:_\d+)*)(?:[eE][\+\-]?\d+(?:_\d+)*)?))|inf)'i;
100:
101: require Exporter;
102:
103: our @ISA = qw(Exporter);
104:
105: my @trig = qw(
106: pi
107: tan
108: csc cosec sec cot cotan
109: asin acos atan
110: acsc acosec asec acot acotan
111: sinh cosh tanh
112: csch cosech sech coth cotanh
113: asinh acosh atanh
114: acsch acosech asech acoth acotanh
115: );
116:
117: our @EXPORT = (qw(
118: i Re Im rho theta arg
119: sqrt log ln
120: log10 logn cbrt root
121: cplx cplxe
122: atan2
123: ),
124: @trig);
125:
126: my @pi = qw(pi pi2 pi4 pip2 pip4 Inf);
127:
128: our @EXPORT_OK = @pi;
129:
130: our %EXPORT_TAGS = (
131: 'trig' => [@trig],
132: 'pi' => [@pi],
133: );
134:
135: use overload
136: '=' => \&_copy,
137: '+=' => \&_plus,
138: '+' => \&_plus,
139: '-=' => \&_minus,
140: '-' => \&_minus,
141: '*=' => \&_multiply,
142: '*' => \&_multiply,
143: '/=' => \&_divide,
144: '/' => \&_divide,
145: '**=' => \&_power,
146: '**' => \&_power,
147: '==' => \&_numeq,
148: '<=>' => \&_spaceship,
149: 'neg' => \&_negate,
150: '~' => \&_conjugate,
151: 'abs' => \&abs,
152: 'sqrt' => \&sqrt,
153: 'exp' => \&exp,
154: 'log' => \&log,
155: 'sin' => \&sin,
156: 'cos' => \&cos,
157: 'atan2' => \&atan2,
158: '""' => \&_stringify;
159:
160: #
161: # Package "privates"
162: #
163:
164: my %DISPLAY_FORMAT = ('style' => 'cartesian',
165: 'polar_pretty_print' => 1);
166: my $eps = 1e-14; # Epsilon
167:
168: #
169: # Object attributes (internal):
170: # cartesian [real, imaginary] -- cartesian form
171: # polar [rho, theta] -- polar form
172: # c_dirty cartesian form not up-to-date
173: # p_dirty polar form not up-to-date
174: # display display format (package's global when not set)
175: #
176:
177: # Die on bad *make() arguments.
178:
179: sub _cannot_make {
180: die "@{[(caller(1))[3]]}: Cannot take $_[0] of '$_[1]'.\n";
181: }
182:
183: sub _make {
184: my $arg = shift;
185: my ($p, $q);
186:
187: if ($arg =~ /^$gre$/) {
188: ($p, $q) = ($1, 0);
189: } elsif ($arg =~ /^(?:$gre)?$gre\s*i\s*$/) {
190: ($p, $q) = ($1 || 0, $2);
191: } elsif ($arg =~ /^\s*\(\s*$gre\s*(?:,\s*$gre\s*)?\)\s*$/) {
192: ($p, $q) = ($1, $2 || 0);
193: }
194:
195: if (defined $p) {
196: $p =~ s/^\+//;
197: $p =~ s/^(-?)inf$/"${1}9**9**9"/e if $has_inf;
198: $q =~ s/^\+//;
199: $q =~ s/^(-?)inf$/"${1}9**9**9"/e if $has_inf;
200: }
201:
202: return ($p, $q);
203: }
204:
205: sub _emake {
206: my $arg = shift;
207: my ($p, $q);
208:
209: if ($arg =~ /^\s*\[\s*$gre\s*(?:,\s*$gre\s*)?\]\s*$/) {
210: ($p, $q) = ($1, $2 || 0);
211: } elsif ($arg =~ m!^\s*\[\s*$gre\s*(?:,\s*([-+]?\d*\s*)?pi(?:/\s*(\d+))?\s*)?\]\s*$!) {
212: ($p, $q) = ($1, ($2 eq '-' ? -1 : ($2 || 1)) * pi() / ($3 || 1));
213: } elsif ($arg =~ /^\s*\[\s*$gre\s*\]\s*$/) {
214: ($p, $q) = ($1, 0);
215: } elsif ($arg =~ /^\s*$gre\s*$/) {
216: ($p, $q) = ($1, 0);
217: }
218:
219: if (defined $p) {
220: $p =~ s/^\+//;
221: $q =~ s/^\+//;
222: $p =~ s/^(-?)inf$/"${1}9**9**9"/e if $has_inf;
223: $q =~ s/^(-?)inf$/"${1}9**9**9"/e if $has_inf;
224: }
225:
226: return ($p, $q);
227: }
228:
229: sub _copy {
230: my $self = shift;
231: my $clone = {%$self};
232: if ($self->{'cartesian'}) {
233: $clone->{'cartesian'} = [@{$self->{'cartesian'}}];
234: }
235: if ($self->{'polar'}) {
236: $clone->{'polar'} = [@{$self->{'polar'}}];
237: }
238: bless $clone,__PACKAGE__;
239: return $clone;
240: }
241:
242: #
243: # ->make
244: #
245: # Create a new complex number (cartesian form)
246: #
247: sub make {
248: my $self = bless {}, shift;
249: my ($re, $im);
250: if (@_ == 0) {
251: ($re, $im) = (0, 0);
252: } elsif (@_ == 1) {
253: return (ref $self)->emake($_[0])
254: if ($_[0] =~ /^\s*\[/);
255: ($re, $im) = _make($_[0]);
256: } elsif (@_ == 2) {
257: ($re, $im) = @_;
258: }
259: if (defined $re) {
260: _cannot_make("real part", $re) unless $re =~ /^$gre$/;
261: }
262: $im ||= 0;
263: _cannot_make("imaginary part", $im) unless $im =~ /^$gre$/;
264: $self->_set_cartesian([$re, $im ]);
265: $self->display_format('cartesian');
266:
267: return $self;
268: }
269:
270: #
271: # ->emake
272: #
273: # Create a new complex number (exponential form)
274: #
275: sub emake {
276: my $self = bless {}, shift;
277: my ($rho, $theta);
278: if (@_ == 0) {
279: ($rho, $theta) = (0, 0);
280: } elsif (@_ == 1) {
281: return (ref $self)->make($_[0])
282: if ($_[0] =~ /^\s*\(/ || $_[0] =~ /i\s*$/);
283: ($rho, $theta) = _emake($_[0]);
284: } elsif (@_ == 2) {
285: ($rho, $theta) = @_;
286: }
287: if (defined $rho && defined $theta) {
288: if ($rho < 0) {
289: $rho = -$rho;
290: $theta = ($theta <= 0) ? $theta + pi() : $theta - pi();
291: }
292: }
293: if (defined $rho) {
294: _cannot_make("rho", $rho) unless $rho =~ /^$gre$/;
295: }
296: $theta ||= 0;
297: _cannot_make("theta", $theta) unless $theta =~ /^$gre$/;
298: $self->_set_polar([$rho, $theta]);
299: $self->display_format('polar');
300:
301: return $self;
302: }
303:
304: sub new { &make } # For backward compatibility only.
305:
306: #
307: # cplx
308: #
309: # Creates a complex number from a (re, im) tuple.
310: # This avoids the burden of writing LONCAPA::LCMathComplex->make(re, im).
311: #
312: sub cplx {
313: return __PACKAGE__->make(@_);
314: }
315:
316: #
317: # cplxe
318: #
319: # Creates a complex number from a (rho, theta) tuple.
320: # This avoids the burden of writing LONCAPA::LCMathComplex->emake(rho, theta).
321: #
322: sub cplxe {
323: return __PACKAGE__->emake(@_);
324: }
325:
326: #
327: # pi
328: #
329: # The number defined as pi = 180 degrees
330: #
331: sub pi () { 4 * CORE::atan2(1, 1) }
332:
333: #
334: # pi2
335: #
336: # The full circle
337: #
338: sub pi2 () { 2 * pi }
339:
340: #
341: # pi4
342: #
343: # The full circle twice.
344: #
345: sub pi4 () { 4 * pi }
346:
347: #
348: # pip2
349: #
350: # The quarter circle
351: #
352: sub pip2 () { pi / 2 }
353:
354: #
355: # pip4
356: #
357: # The eighth circle.
358: #
359: sub pip4 () { pi / 4 }
360:
361: #
362: # _uplog10
363: #
364: # Used in log10().
365: #
366: sub _uplog10 () { 1 / CORE::log(10) }
367:
368: #
369: # i
370: #
371: # The number defined as i*i = -1;
372: #
373: sub i () {
374: return $i if ($i);
375: $i = bless {};
376: $i->{'cartesian'} = [0, 1];
377: $i->{'polar'} = [1, pip2];
378: $i->{c_dirty} = 0;
379: $i->{p_dirty} = 0;
380: return $i;
381: }
382:
383: #
384: # _ip2
385: #
386: # Half of i.
387: #
388: sub _ip2 () { i / 2 }
389:
390: #
391: # Attribute access/set routines
392: #
393:
394: sub _cartesian {$_[0]->{c_dirty} ?
395: $_[0]->_update_cartesian : $_[0]->{'cartesian'}}
396: sub _polar {$_[0]->{p_dirty} ?
397: $_[0]->_update_polar : $_[0]->{'polar'}}
398:
399: sub _set_cartesian { $_[0]->{p_dirty}++; $_[0]->{c_dirty} = 0;
400: $_[0]->{'cartesian'} = $_[1] }
401: sub _set_polar { $_[0]->{c_dirty}++; $_[0]->{p_dirty} = 0;
402: $_[0]->{'polar'} = $_[1] }
403:
404: #
405: # ->_update_cartesian
406: #
407: # Recompute and return the cartesian form, given accurate polar form.
408: #
409: sub _update_cartesian {
410: my $self = shift;
411: my ($r, $t) = @{$self->{'polar'}};
412: $self->{c_dirty} = 0;
413: return $self->{'cartesian'} = [$r * CORE::cos($t), $r * CORE::sin($t)];
414: }
415:
416: #
417: #
418: # ->_update_polar
419: #
420: # Recompute and return the polar form, given accurate cartesian form.
421: #
422: sub _update_polar {
423: my $self = shift;
424: my ($x, $y) = @{$self->{'cartesian'}};
425: $self->{p_dirty} = 0;
426: return $self->{'polar'} = [0, 0] if $x == 0 && $y == 0;
427: return $self->{'polar'} = [CORE::sqrt($x*$x + $y*$y),
428: CORE::atan2($y, $x)];
429: }
430:
431: #
432: # (_plus)
433: #
434: # Computes z1+z2.
435: #
436: sub _plus {
437: my ($z1, $z2, $regular) = @_;
438: my ($re1, $im1) = @{$z1->_cartesian};
439: $z2 = cplx($z2) unless ref $z2;
440: my ($re2, $im2) = ref $z2 ? @{$z2->_cartesian} : ($z2, 0);
441: unless (defined $regular) {
442: $z1->_set_cartesian([$re1 + $re2, $im1 + $im2]);
443: return $z1;
444: }
445: return (ref $z1)->make($re1 + $re2, $im1 + $im2);
446: }
447:
448: #
449: # (_minus)
450: #
451: # Computes z1-z2.
452: #
453: sub _minus {
454: my ($z1, $z2, $inverted) = @_;
455: my ($re1, $im1) = @{$z1->_cartesian};
456: $z2 = cplx($z2) unless ref $z2;
457: my ($re2, $im2) = @{$z2->_cartesian};
458: unless (defined $inverted) {
459: $z1->_set_cartesian([$re1 - $re2, $im1 - $im2]);
460: return $z1;
461: }
462: return $inverted ?
463: (ref $z1)->make($re2 - $re1, $im2 - $im1) :
464: (ref $z1)->make($re1 - $re2, $im1 - $im2);
465:
466: }
467:
468: #
469: # (_multiply)
470: #
471: # Computes z1*z2.
472: #
473: sub _multiply {
474: my ($z1, $z2, $regular) = @_;
475: if ($z1->{p_dirty} == 0 and ref $z2 and $z2->{p_dirty} == 0) {
476: # if both polar better use polar to avoid rounding errors
477: my ($r1, $t1) = @{$z1->_polar};
478: my ($r2, $t2) = @{$z2->_polar};
479: my $t = $t1 + $t2;
480: if ($t > pi()) { $t -= pi2 }
481: elsif ($t <= -pi()) { $t += pi2 }
482: unless (defined $regular) {
483: $z1->_set_polar([$r1 * $r2, $t]);
484: return $z1;
485: }
486: return (ref $z1)->emake($r1 * $r2, $t);
487: } else {
488: my ($x1, $y1) = @{$z1->_cartesian};
489: if (ref $z2) {
490: my ($x2, $y2) = @{$z2->_cartesian};
491: return (ref $z1)->make($x1*$x2-$y1*$y2, $x1*$y2+$y1*$x2);
492: } else {
493: return (ref $z1)->make($x1*$z2, $y1*$z2);
494: }
495: }
496: }
497:
498: #
499: # _divbyzero
500: #
501: # Die on division by zero.
502: #
503: sub _divbyzero {
504: my $mess = "$_[0]: Division by zero.\n";
505:
506: if (defined $_[1]) {
507: $mess .= "(Because in the definition of $_[0], the divisor ";
508: $mess .= "$_[1] " unless ("$_[1]" eq '0');
509: $mess .= "is 0)\n";
510: }
511:
512: my @up = caller(1);
513:
514: $mess .= "Died at $up[1] line $up[2].\n";
515:
516: die $mess;
517: }
518:
519: #
520: # (_divide)
521: #
522: # Computes z1/z2.
523: #
524: sub _divide {
525: my ($z1, $z2, $inverted) = @_;
526: if ($z1->{p_dirty} == 0 and ref $z2 and $z2->{p_dirty} == 0) {
527: # if both polar better use polar to avoid rounding errors
528: my ($r1, $t1) = @{$z1->_polar};
529: my ($r2, $t2) = @{$z2->_polar};
530: my $t;
531: if ($inverted) {
532: _divbyzero "$z2/0" if ($r1 == 0);
533: $t = $t2 - $t1;
534: if ($t > pi()) { $t -= pi2 }
535: elsif ($t <= -pi()) { $t += pi2 }
536: return (ref $z1)->emake($r2 / $r1, $t);
537: } else {
538: _divbyzero "$z1/0" if ($r2 == 0);
539: $t = $t1 - $t2;
540: if ($t > pi()) { $t -= pi2 }
541: elsif ($t <= -pi()) { $t += pi2 }
542: return (ref $z1)->emake($r1 / $r2, $t);
543: }
544: } else {
545: my ($d, $x2, $y2);
546: if ($inverted) {
547: ($x2, $y2) = @{$z1->_cartesian};
548: $d = $x2*$x2 + $y2*$y2;
549: _divbyzero "$z2/0" if $d == 0;
550: return (ref $z1)->make(($x2*$z2)/$d, -($y2*$z2)/$d);
551: } else {
552: my ($x1, $y1) = @{$z1->_cartesian};
553: if (ref $z2) {
554: ($x2, $y2) = @{$z2->_cartesian};
555: $d = $x2*$x2 + $y2*$y2;
556: _divbyzero "$z1/0" if $d == 0;
557: my $u = ($x1*$x2 + $y1*$y2)/$d;
558: my $v = ($y1*$x2 - $x1*$y2)/$d;
559: return (ref $z1)->make($u, $v);
560: } else {
561: _divbyzero "$z1/0" if $z2 == 0;
562: return (ref $z1)->make($x1/$z2, $y1/$z2);
563: }
564: }
565: }
566: }
567:
568: #
569: # (_power)
570: #
571: # Computes z1**z2 = exp(z2 * log z1)).
572: #
573: sub _power {
574: my ($z1, $z2, $inverted) = @_;
575: if ($inverted) {
576: return 1 if $z1 == 0 || $z2 == 1;
577: return 0 if $z2 == 0 && Re($z1) > 0;
578: } else {
579: return 1 if $z2 == 0 || $z1 == 1;
580: return 0 if $z1 == 0 && Re($z2) > 0;
581: }
582: my $w = $inverted ? &exp($z1 * &log($z2))
583: : &exp($z2 * &log($z1));
584: # If both arguments cartesian, return cartesian, else polar.
585: return $z1->{c_dirty} == 0 &&
586: (not ref $z2 or $z2->{c_dirty} == 0) ?
587: cplx(@{$w->_cartesian}) : $w;
588: }
589:
590: #
591: # (_spaceship)
592: #
593: # Computes z1 <=> z2.
594: # Sorts on the real part first, then on the imaginary part. Thus 2-4i < 3+8i.
595: #
596: sub _spaceship {
597: my ($z1, $z2, $inverted) = @_;
598: my ($re1, $im1) = ref $z1 ? @{$z1->_cartesian} : ($z1, 0);
599: my ($re2, $im2) = ref $z2 ? @{$z2->_cartesian} : ($z2, 0);
600: my $sgn = $inverted ? -1 : 1;
601: return $sgn * ($re1 <=> $re2) if $re1 != $re2;
602: return $sgn * ($im1 <=> $im2);
603: }
604:
605: #
606: # (_numeq)
607: #
608: # Computes z1 == z2.
609: #
610: # (Required in addition to _spaceship() because of NaNs.)
611: sub _numeq {
612: my ($z1, $z2, $inverted) = @_;
613: my ($re1, $im1) = ref $z1 ? @{$z1->_cartesian} : ($z1, 0);
614: my ($re2, $im2) = ref $z2 ? @{$z2->_cartesian} : ($z2, 0);
615: return $re1 == $re2 && $im1 == $im2 ? 1 : 0;
616: }
617:
618: #
619: # (_negate)
620: #
621: # Computes -z.
622: #
623: sub _negate {
624: my ($z) = @_;
625: if ($z->{c_dirty}) {
626: my ($r, $t) = @{$z->_polar};
627: $t = ($t <= 0) ? $t + pi : $t - pi;
628: return (ref $z)->emake($r, $t);
629: }
630: my ($re, $im) = @{$z->_cartesian};
631: return (ref $z)->make(-$re, -$im);
632: }
633:
634: #
635: # (_conjugate)
636: #
637: # Compute complex's _conjugate.
638: #
639: sub _conjugate {
640: my ($z) = @_;
641: if ($z->{c_dirty}) {
642: my ($r, $t) = @{$z->_polar};
643: return (ref $z)->emake($r, -$t);
644: }
645: my ($re, $im) = @{$z->_cartesian};
646: return (ref $z)->make($re, -$im);
647: }
648:
649: #
650: # (abs)
651: #
652: # Compute or set complex's norm (rho).
653: #
654: sub abs {
655: my ($z, $rho) = @_ ? @_ : $_;
656: unless (ref $z) {
657: if (@_ == 2) {
658: $_[0] = $_[1];
659: } else {
660: return CORE::abs($z);
661: }
662: }
663: if (defined $rho) {
664: $z->{'polar'} = [ $rho, ${$z->_polar}[1] ];
665: $z->{p_dirty} = 0;
666: $z->{c_dirty} = 1;
667: return $rho;
668: } else {
669: return ${$z->_polar}[0];
670: }
671: }
672:
673: sub _theta {
674: my $theta = $_[0];
675:
676: if ($$theta > pi()) { $$theta -= pi2 }
677: elsif ($$theta <= -pi()) { $$theta += pi2 }
678: }
679:
680: #
681: # arg
682: #
683: # Compute or set complex's argument (theta).
684: #
685: sub arg {
686: my ($z, $theta) = @_;
687: return $z unless ref $z;
688: if (defined $theta) {
689: _theta(\$theta);
690: $z->{'polar'} = [ ${$z->_polar}[0], $theta ];
691: $z->{p_dirty} = 0;
692: $z->{c_dirty} = 1;
693: } else {
694: $theta = ${$z->_polar}[1];
695: _theta(\$theta);
696: }
697: return $theta;
698: }
699:
700: #
701: # (sqrt)
702: #
703: # Compute sqrt(z).
704: #
705: # It is quite tempting to use wantarray here so that in list context
706: # sqrt() would return the two solutions. This, however, would
707: # break things like
708: #
709: # print "sqrt(z) = ", sqrt($z), "\n";
710: #
711: # The two values would be printed side by side without no intervening
712: # whitespace, quite confusing.
713: # Therefore if you want the two solutions use the root().
714: #
715: sub sqrt {
716: my ($z) = @_ ? $_[0] : $_;
717: my ($re, $im) = ref $z ? @{$z->_cartesian} : ($z, 0);
718: return $re < 0 ? cplx(0, CORE::sqrt(-$re)) : CORE::sqrt($re)
719: if $im == 0;
720: my ($r, $t) = @{$z->_polar};
721: return (ref $z)->emake(CORE::sqrt($r), $t/2);
722: }
723:
724: #
725: # cbrt
726: #
727: # Compute cbrt(z) (cubic root).
728: #
729: # Why are we not returning three values? The same answer as for sqrt().
730: #
731: sub cbrt {
732: my ($z) = @_;
733: return $z < 0 ?
734: -CORE::exp(CORE::log(-$z)/3) :
735: ($z > 0 ? CORE::exp(CORE::log($z)/3): 0)
736: unless ref $z;
737: my ($r, $t) = @{$z->_polar};
738: return 0 if $r == 0;
739: return (ref $z)->emake(CORE::exp(CORE::log($r)/3), $t/3);
740: }
741:
742: #
743: # _rootbad
744: #
745: # Die on bad root.
746: #
747: sub _rootbad {
748: my $mess = "Root '$_[0]' illegal, root rank must be positive integer.\n";
749:
750: my @up = caller(1);
751:
752: $mess .= "Died at $up[1] line $up[2].\n";
753:
754: die $mess;
755: }
756:
757: #
758: # root
759: #
760: # Computes all nth root for z, returning an array whose size is n.
761: # `n' must be a positive integer.
762: #
763: # The roots are given by (for k = 0..n-1):
764: #
765: # z^(1/n) = r^(1/n) (cos ((t+2 k pi)/n) + i sin ((t+2 k pi)/n))
766: #
767: sub root {
768: my ($z, $n, $k) = @_;
769: _rootbad($n) if ($n < 1 or int($n) != $n);
770: my ($r, $t) = ref $z ?
771: @{$z->_polar} : (CORE::abs($z), $z >= 0 ? 0 : pi);
772: my $theta_inc = pi2 / $n;
773: my $rho = $r ** (1/$n);
774: my $cartesian = ref $z && $z->{c_dirty} == 0;
775: if (@_ == 2) {
776: my @root;
777: for (my $i = 0, my $theta = $t / $n;
778: $i < $n;
779: $i++, $theta += $theta_inc) {
780: my $w = cplxe($rho, $theta);
781: # Yes, $cartesian is loop invariant.
782: push @root, $cartesian ? cplx(@{$w->_cartesian}) : $w;
783: }
784: return @root;
785: } elsif (@_ == 3) {
786: my $w = cplxe($rho, $t / $n + $k * $theta_inc);
787: return $cartesian ? cplx(@{$w->_cartesian}) : $w;
788: }
789: }
790:
791: #
792: # Re
793: #
794: # Return or set Re(z).
795: #
796: sub Re {
797: my ($z, $Re) = @_;
798: return $z unless ref $z;
799: if (defined $Re) {
800: $z->{'cartesian'} = [ $Re, ${$z->_cartesian}[1] ];
801: $z->{c_dirty} = 0;
802: $z->{p_dirty} = 1;
803: } else {
804: return ${$z->_cartesian}[0];
805: }
806: }
807:
808: #
809: # Im
810: #
811: # Return or set Im(z).
812: #
813: sub Im {
814: my ($z, $Im) = @_;
815: return 0 unless ref $z;
816: if (defined $Im) {
817: $z->{'cartesian'} = [ ${$z->_cartesian}[0], $Im ];
818: $z->{c_dirty} = 0;
819: $z->{p_dirty} = 1;
820: } else {
821: return ${$z->_cartesian}[1];
822: }
823: }
824:
825: #
826: # rho
827: #
828: # Return or set rho(w).
829: #
830: sub rho {
831: LONCAPA::LCMathComplex::abs(@_);
832: }
833:
834: #
835: # theta
836: #
837: # Return or set theta(w).
838: #
839: sub theta {
840: LONCAPA::LCMathComplex::arg(@_);
841: }
842:
843: #
844: # (exp)
845: #
846: # Computes exp(z).
847: #
848: sub exp {
849: my ($z) = @_ ? @_ : $_;
850: return CORE::exp($z) unless ref $z;
851: my ($x, $y) = @{$z->_cartesian};
852: return (ref $z)->emake(CORE::exp($x), $y);
853: }
854:
855: #
856: # _logofzero
857: #
858: # Die on logarithm of zero.
859: #
860: sub _logofzero {
861: my $mess = "$_[0]: Logarithm of zero.\n";
862:
863: if (defined $_[1]) {
864: $mess .= "(Because in the definition of $_[0], the argument ";
865: $mess .= "$_[1] " unless ($_[1] eq '0');
866: $mess .= "is 0)\n";
867: }
868:
869: my @up = caller(1);
870:
871: $mess .= "Died at $up[1] line $up[2].\n";
872:
873: die $mess;
874: }
875:
876: #
877: # (log)
878: #
879: # Compute log(z).
880: #
881: sub log {
882: my ($z) = @_ ? @_ : $_;
883: unless (ref $z) {
884: _logofzero("log") if $z == 0;
885: return $z > 0 ? CORE::log($z) : cplx(CORE::log(-$z), pi);
886: }
887: my ($r, $t) = @{$z->_polar};
888: _logofzero("log") if $r == 0;
889: if ($t > pi()) { $t -= pi2 }
890: elsif ($t <= -pi()) { $t += pi2 }
891: return (ref $z)->make(CORE::log($r), $t);
892: }
893:
894: #
895: # ln
896: #
897: # Alias for log().
898: #
899: sub ln { LONCAPA::LCMathComplex::log(@_) }
900:
901: #
902: # log10
903: #
904: # Compute log10(z).
905: #
906:
907: sub log10 {
908: return LONCAPA::LCMathComplex::log($_[0]) * _uplog10;
909: }
910:
911: #
912: # logn
913: #
914: # Compute logn(z,n) = log(z) / log(n)
915: #
916: sub logn {
917: my ($z, $n) = @_;
918: $z = cplx($z, 0) unless ref $z;
919: my $logn = $LOGN{$n};
920: $logn = $LOGN{$n} = CORE::log($n) unless defined $logn; # Cache log(n)
921: return &log($z) / $logn;
922: }
923:
924: #
925: # (cos)
926: #
927: # Compute cos(z) = (exp(iz) + exp(-iz))/2.
928: #
929: sub cos {
930: my ($z) = @_ ? @_ : $_;
931: return CORE::cos($z) unless ref $z;
932: my ($x, $y) = @{$z->_cartesian};
933: my $ey = CORE::exp($y);
934: my $sx = CORE::sin($x);
935: my $cx = CORE::cos($x);
936: my $ey_1 = $ey ? 1 / $ey : Inf();
937: return (ref $z)->make($cx * ($ey + $ey_1)/2,
938: $sx * ($ey_1 - $ey)/2);
939: }
940:
941: #
942: # (sin)
943: #
944: # Compute sin(z) = (exp(iz) - exp(-iz))/2.
945: #
946: sub sin {
947: my ($z) = @_ ? @_ : $_;
948: return CORE::sin($z) unless ref $z;
949: my ($x, $y) = @{$z->_cartesian};
950: my $ey = CORE::exp($y);
951: my $sx = CORE::sin($x);
952: my $cx = CORE::cos($x);
953: my $ey_1 = $ey ? 1 / $ey : Inf();
954: return (ref $z)->make($sx * ($ey + $ey_1)/2,
955: $cx * ($ey - $ey_1)/2);
956: }
957:
958: #
959: # tan
960: #
961: # Compute tan(z) = sin(z) / cos(z).
962: #
963: sub tan {
964: my ($z) = @_;
965: my $cz = &cos($z);
966: _divbyzero "tan($z)", "cos($z)" if $cz == 0;
967: return &sin($z) / $cz;
968: }
969:
970: #
971: # sec
972: #
973: # Computes the secant sec(z) = 1 / cos(z).
974: #
975: sub sec {
976: my ($z) = @_;
977: my $cz = &cos($z);
978: _divbyzero "sec($z)", "cos($z)" if ($cz == 0);
979: return 1 / $cz;
980: }
981:
982: #
983: # csc
984: #
985: # Computes the cosecant csc(z) = 1 / sin(z).
986: #
987: sub csc {
988: my ($z) = @_;
989: my $sz = &sin($z);
990: _divbyzero "csc($z)", "sin($z)" if ($sz == 0);
991: return 1 / $sz;
992: }
993:
994: #
995: # cosec
996: #
997: # Alias for csc().
998: #
999: sub cosec { LONCAPA::LCMathComplex::csc(@_) }
1000:
1001: #
1002: # cot
1003: #
1004: # Computes cot(z) = cos(z) / sin(z).
1005: #
1006: sub cot {
1007: my ($z) = @_;
1008: my $sz = &sin($z);
1009: _divbyzero "cot($z)", "sin($z)" if ($sz == 0);
1010: return &cos($z) / $sz;
1011: }
1012:
1013: #
1014: # cotan
1015: #
1016: # Alias for cot().
1017: #
1018: sub cotan { LONCAPA::LCMathComplex::cot(@_) }
1019:
1020: #
1021: # acos
1022: #
1023: # Computes the arc cosine acos(z) = -i log(z + sqrt(z*z-1)).
1024: #
1025: sub acos {
1026: my $z = $_[0];
1027: return CORE::atan2(CORE::sqrt(1-$z*$z), $z)
1028: if (! ref $z) && CORE::abs($z) <= 1;
1029: $z = cplx($z, 0) unless ref $z;
1030: my ($x, $y) = @{$z->_cartesian};
1031: return 0 if $x == 1 && $y == 0;
1032: my $t1 = CORE::sqrt(($x+1)*($x+1) + $y*$y);
1033: my $t2 = CORE::sqrt(($x-1)*($x-1) + $y*$y);
1034: my $alpha = ($t1 + $t2)/2;
1035: my $beta = ($t1 - $t2)/2;
1036: $alpha = 1 if $alpha < 1;
1037: if ($beta > 1) { $beta = 1 }
1038: elsif ($beta < -1) { $beta = -1 }
1039: my $u = CORE::atan2(CORE::sqrt(1-$beta*$beta), $beta);
1040: my $v = CORE::log($alpha + CORE::sqrt($alpha*$alpha-1));
1041: $v = -$v if $y > 0 || ($y == 0 && $x < -1);
1042: return (ref $z)->make($u, $v);
1043: }
1044:
1045: #
1046: # asin
1047: #
1048: # Computes the arc sine asin(z) = -i log(iz + sqrt(1-z*z)).
1049: #
1050: sub asin {
1051: my $z = $_[0];
1052: return CORE::atan2($z, CORE::sqrt(1-$z*$z))
1053: if (! ref $z) && CORE::abs($z) <= 1;
1054: $z = cplx($z, 0) unless ref $z;
1055: my ($x, $y) = @{$z->_cartesian};
1056: return 0 if $x == 0 && $y == 0;
1057: my $t1 = CORE::sqrt(($x+1)*($x+1) + $y*$y);
1058: my $t2 = CORE::sqrt(($x-1)*($x-1) + $y*$y);
1059: my $alpha = ($t1 + $t2)/2;
1060: my $beta = ($t1 - $t2)/2;
1061: $alpha = 1 if $alpha < 1;
1062: if ($beta > 1) { $beta = 1 }
1063: elsif ($beta < -1) { $beta = -1 }
1064: my $u = CORE::atan2($beta, CORE::sqrt(1-$beta*$beta));
1065: my $v = -CORE::log($alpha + CORE::sqrt($alpha*$alpha-1));
1066: $v = -$v if $y > 0 || ($y == 0 && $x < -1);
1067: return (ref $z)->make($u, $v);
1068: }
1069:
1070: #
1071: # atan
1072: #
1073: # Computes the arc tangent atan(z) = i/2 log((i+z) / (i-z)).
1074: #
1075: sub atan {
1076: my ($z) = @_;
1077: return CORE::atan2($z, 1) unless ref $z;
1078: my ($x, $y) = ref $z ? @{$z->_cartesian} : ($z, 0);
1079: return 0 if $x == 0 && $y == 0;
1080: _divbyzero "atan(i)" if ( $z == i);
1081: _logofzero "atan(-i)" if (-$z == i); # -i is a bad file test...
1082: my $log = &log((i + $z) / (i - $z));
1083: return _ip2 * $log;
1084: }
1085:
1086: #
1087: # asec
1088: #
1089: # Computes the arc secant asec(z) = acos(1 / z).
1090: #
1091: sub asec {
1092: my ($z) = @_;
1093: _divbyzero "asec($z)", $z if ($z == 0);
1094: return acos(1 / $z);
1095: }
1096:
1097: #
1098: # acsc
1099: #
1100: # Computes the arc cosecant acsc(z) = asin(1 / z).
1101: #
1102: sub acsc {
1103: my ($z) = @_;
1104: _divbyzero "acsc($z)", $z if ($z == 0);
1105: return asin(1 / $z);
1106: }
1107:
1108: #
1109: # acosec
1110: #
1111: # Alias for acsc().
1112: #
1113: sub acosec { LONCAPA::LCMathComplex::acsc(@_) }
1114:
1115: #
1116: # acot
1117: #
1118: # Computes the arc cotangent acot(z) = atan(1 / z)
1119: #
1120: sub acot {
1121: my ($z) = @_;
1122: _divbyzero "acot(0)" if $z == 0;
1123: return ($z >= 0) ? CORE::atan2(1, $z) : CORE::atan2(-1, -$z)
1124: unless ref $z;
1125: _divbyzero "acot(i)" if ($z - i == 0);
1126: _logofzero "acot(-i)" if ($z + i == 0);
1127: return atan(1 / $z);
1128: }
1129:
1130: #
1131: # acotan
1132: #
1133: # Alias for acot().
1134: #
1135: sub acotan { LONCAPA::LCMathComplex::acot(@_) }
1136:
1137: #
1138: # cosh
1139: #
1140: # Computes the hyperbolic cosine cosh(z) = (exp(z) + exp(-z))/2.
1141: #
1142: sub cosh {
1143: my ($z) = @_;
1144: my $ex;
1145: unless (ref $z) {
1146: $ex = CORE::exp($z);
1147: return $ex ? ($ex == $ExpInf ? Inf() : ($ex + 1/$ex)/2) : Inf();
1148: }
1149: my ($x, $y) = @{$z->_cartesian};
1150: $ex = CORE::exp($x);
1151: my $ex_1 = $ex ? 1 / $ex : Inf();
1152: return (ref $z)->make(CORE::cos($y) * ($ex + $ex_1)/2,
1153: CORE::sin($y) * ($ex - $ex_1)/2);
1154: }
1155:
1156: #
1157: # sinh
1158: #
1159: # Computes the hyperbolic sine sinh(z) = (exp(z) - exp(-z))/2.
1160: #
1161: sub sinh {
1162: my ($z) = @_;
1163: my $ex;
1164: unless (ref $z) {
1165: return 0 if $z == 0;
1166: $ex = CORE::exp($z);
1167: return $ex ? ($ex == $ExpInf ? Inf() : ($ex - 1/$ex)/2) : -Inf();
1168: }
1169: my ($x, $y) = @{$z->_cartesian};
1170: my $cy = CORE::cos($y);
1171: my $sy = CORE::sin($y);
1172: $ex = CORE::exp($x);
1173: my $ex_1 = $ex ? 1 / $ex : Inf();
1174: return (ref $z)->make(CORE::cos($y) * ($ex - $ex_1)/2,
1175: CORE::sin($y) * ($ex + $ex_1)/2);
1176: }
1177:
1178: #
1179: # tanh
1180: #
1181: # Computes the hyperbolic tangent tanh(z) = sinh(z) / cosh(z).
1182: #
1183: sub tanh {
1184: my ($z) = @_;
1185: my $cz = cosh($z);
1186: _divbyzero "tanh($z)", "cosh($z)" if ($cz == 0);
1187: my $sz = sinh($z);
1188: return 1 if $cz == $sz;
1189: return -1 if $cz == -$sz;
1190: return $sz / $cz;
1191: }
1192:
1193: #
1194: # sech
1195: #
1196: # Computes the hyperbolic secant sech(z) = 1 / cosh(z).
1197: #
1198: sub sech {
1199: my ($z) = @_;
1200: my $cz = cosh($z);
1201: _divbyzero "sech($z)", "cosh($z)" if ($cz == 0);
1202: return 1 / $cz;
1203: }
1204:
1205: #
1206: # csch
1207: #
1208: # Computes the hyperbolic cosecant csch(z) = 1 / sinh(z).
1209: #
1210: sub csch {
1211: my ($z) = @_;
1212: my $sz = sinh($z);
1213: _divbyzero "csch($z)", "sinh($z)" if ($sz == 0);
1214: return 1 / $sz;
1215: }
1216:
1217: #
1218: # cosech
1219: #
1220: # Alias for csch().
1221: #
1222: sub cosech { LONCAPA::LCMathComplex::csch(@_) }
1223:
1224: #
1225: # coth
1226: #
1227: # Computes the hyperbolic cotangent coth(z) = cosh(z) / sinh(z).
1228: #
1229: sub coth {
1230: my ($z) = @_;
1231: my $sz = sinh($z);
1232: _divbyzero "coth($z)", "sinh($z)" if $sz == 0;
1233: my $cz = cosh($z);
1234: return 1 if $cz == $sz;
1235: return -1 if $cz == -$sz;
1236: return $cz / $sz;
1237: }
1238:
1239: #
1240: # cotanh
1241: #
1242: # Alias for coth().
1243: #
1244: sub cotanh { LONCAPA::LCMathComplex::coth(@_) }
1245:
1246: #
1247: # acosh
1248: #
1249: # Computes the area/inverse hyperbolic cosine acosh(z) = log(z + sqrt(z*z-1)).
1250: #
1251: sub acosh {
1252: my ($z) = @_;
1253: unless (ref $z) {
1254: $z = cplx($z, 0);
1255: }
1256: my ($re, $im) = @{$z->_cartesian};
1257: if ($im == 0) {
1258: return CORE::log($re + CORE::sqrt($re*$re - 1))
1259: if $re >= 1;
1260: return cplx(0, CORE::atan2(CORE::sqrt(1 - $re*$re), $re))
1261: if CORE::abs($re) < 1;
1262: }
1263: my $t = &sqrt($z * $z - 1) + $z;
1264: # Try Taylor if looking bad (this usually means that
1265: # $z was large negative, therefore the sqrt is really
1266: # close to abs(z), summing that with z...)
1267: $t = 1/(2 * $z) - 1/(8 * $z**3) + 1/(16 * $z**5) - 5/(128 * $z**7)
1268: if $t == 0;
1269: my $u = &log($t);
1270: $u->Im(-$u->Im) if $re < 0 && $im == 0;
1271: return $re < 0 ? -$u : $u;
1272: }
1273:
1274: #
1275: # asinh
1276: #
1277: # Computes the area/inverse hyperbolic sine asinh(z) = log(z + sqrt(z*z+1))
1278: #
1279: sub asinh {
1280: my ($z) = @_;
1281: unless (ref $z) {
1282: my $t = $z + CORE::sqrt($z*$z + 1);
1283: return CORE::log($t) if $t;
1284: }
1285: my $t = &sqrt($z * $z + 1) + $z;
1286: # Try Taylor if looking bad (this usually means that
1287: # $z was large negative, therefore the sqrt is really
1288: # close to abs(z), summing that with z...)
1289: $t = 1/(2 * $z) - 1/(8 * $z**3) + 1/(16 * $z**5) - 5/(128 * $z**7)
1290: if $t == 0;
1291: return &log($t);
1292: }
1293:
1294: #
1295: # atanh
1296: #
1297: # Computes the area/inverse hyperbolic tangent atanh(z) = 1/2 log((1+z) / (1-z)).
1298: #
1299: sub atanh {
1300: my ($z) = @_;
1301: unless (ref $z) {
1302: return CORE::log((1 + $z)/(1 - $z))/2 if CORE::abs($z) < 1;
1303: $z = cplx($z, 0);
1304: }
1305: _divbyzero 'atanh(1)', "1 - $z" if (1 - $z == 0);
1306: _logofzero 'atanh(-1)' if (1 + $z == 0);
1307: return 0.5 * &log((1 + $z) / (1 - $z));
1308: }
1309:
1310: #
1311: # asech
1312: #
1313: # Computes the area/inverse hyperbolic secant asech(z) = acosh(1 / z).
1314: #
1315: sub asech {
1316: my ($z) = @_;
1317: _divbyzero 'asech(0)', "$z" if ($z == 0);
1318: return acosh(1 / $z);
1319: }
1320:
1321: #
1322: # acsch
1323: #
1324: # Computes the area/inverse hyperbolic cosecant acsch(z) = asinh(1 / z).
1325: #
1326: sub acsch {
1327: my ($z) = @_;
1328: _divbyzero 'acsch(0)', $z if ($z == 0);
1329: return asinh(1 / $z);
1330: }
1331:
1332: #
1333: # acosech
1334: #
1335: # Alias for acosh().
1336: #
1337: sub acosech { LONCAPA::LCMathComplex::acsch(@_) }
1338:
1339: #
1340: # acoth
1341: #
1342: # Computes the area/inverse hyperbolic cotangent acoth(z) = 1/2 log((1+z) / (z-1)).
1343: #
1344: sub acoth {
1345: my ($z) = @_;
1346: _divbyzero 'acoth(0)' if ($z == 0);
1347: unless (ref $z) {
1348: return CORE::log(($z + 1)/($z - 1))/2 if CORE::abs($z) > 1;
1349: $z = cplx($z, 0);
1350: }
1351: _divbyzero 'acoth(1)', "$z - 1" if ($z - 1 == 0);
1352: _logofzero 'acoth(-1)', "1 + $z" if (1 + $z == 0);
1353: return &log((1 + $z) / ($z - 1)) / 2;
1354: }
1355:
1356: #
1357: # acotanh
1358: #
1359: # Alias for acot().
1360: #
1361: sub acotanh { LONCAPA::LCMathComplex::acoth(@_) }
1362:
1363: #
1364: # (atan2)
1365: #
1366: # Compute atan(z1/z2), minding the right quadrant.
1367: #
1368: sub atan2 {
1369: my ($z1, $z2, $inverted) = @_;
1370: my ($re1, $im1, $re2, $im2);
1371: if ($inverted) {
1372: ($re1, $im1) = ref $z2 ? @{$z2->_cartesian} : ($z2, 0);
1373: ($re2, $im2) = ref $z1 ? @{$z1->_cartesian} : ($z1, 0);
1374: } else {
1375: ($re1, $im1) = ref $z1 ? @{$z1->_cartesian} : ($z1, 0);
1376: ($re2, $im2) = ref $z2 ? @{$z2->_cartesian} : ($z2, 0);
1377: }
1378: if ($im1 || $im2) {
1379: # In MATLAB the imaginary parts are ignored.
1380: # warn "atan2: Imaginary parts ignored";
1381: # http://documents.wolfram.com/mathematica/functions/ArcTan
1382: # NOTE: Mathematica ArcTan[x,y] while atan2(y,x)
1383: my $s = $z1 * $z1 + $z2 * $z2;
1384: _divbyzero("atan2") if $s == 0;
1385: my $i = &i;
1386: my $r = $z2 + $z1 * $i;
1387: return -$i * &log($r / &sqrt( $s ));
1388: }
1389: return CORE::atan2($re1, $re2);
1390: }
1391:
1392: #
1393: # display_format
1394: # ->display_format
1395: #
1396: # Set (get if no argument) the display format for all complex numbers that
1397: # don't happen to have overridden it via ->display_format
1398: #
1399: # When called as an object method, this actually sets the display format for
1400: # the current object.
1401: #
1402: # Valid object formats are 'c' and 'p' for cartesian and polar. The first
1403: # letter is used actually, so the type can be fully spelled out for clarity.
1404: #
1405: sub display_format {
1406: my $self = shift;
1407: my %display_format = %DISPLAY_FORMAT;
1408:
1409: if (ref $self) { # Called as an object method
1410: if (exists $self->{display_format}) {
1411: my %obj = %{$self->{display_format}};
1412: @display_format{keys %obj} = values %obj;
1413: }
1414: }
1415: if (@_ == 1) {
1416: $display_format{style} = shift;
1417: } else {
1418: my %new = @_;
1419: @display_format{keys %new} = values %new;
1420: }
1421:
1422: if (ref $self) { # Called as an object method
1423: $self->{display_format} = { %display_format };
1424: return
1425: wantarray ?
1426: %{$self->{display_format}} :
1427: $self->{display_format}->{style};
1428: }
1429:
1430: # Called as a class method
1431: %DISPLAY_FORMAT = %display_format;
1432: return
1433: wantarray ?
1434: %DISPLAY_FORMAT :
1435: $DISPLAY_FORMAT{style};
1436: }
1437:
1438: #
1439: # (_stringify)
1440: #
1441: # Show nicely formatted complex number under its cartesian or polar form,
1442: # depending on the current display format:
1443: #
1444: # . If a specific display format has been recorded for this object, use it.
1445: # . Otherwise, use the generic current default for all complex numbers,
1446: # which is a package global variable.
1447: #
1448: sub _stringify {
1449: my ($z) = shift;
1450:
1451: my $style = $z->display_format;
1452:
1453: $style = $DISPLAY_FORMAT{style} unless defined $style;
1454:
1455: return $z->_stringify_polar if $style =~ /^p/i;
1456: return $z->_stringify_cartesian;
1457: }
1458:
1459: #
1460: # ->_stringify_cartesian
1461: #
1462: # Stringify as a cartesian representation 'a+bi'.
1463: #
1464: sub _stringify_cartesian {
1465: my $z = shift;
1466: my ($x, $y) = @{$z->_cartesian};
1467: my ($re, $im);
1468:
1469: my %format = $z->display_format;
1470: my $format = $format{format};
1471:
1472: if ($x) {
1473: if ($x =~ /^NaN[QS]?$/i) {
1474: $re = $x;
1475: } else {
1476: if ($x =~ /^-?\Q$Inf\E$/oi) {
1477: $re = $x;
1478: } else {
1479: $re = defined $format ? sprintf($format, $x) : $x;
1480: }
1481: }
1482: } else {
1483: undef $re;
1484: }
1485:
1486: if ($y) {
1487: if ($y =~ /^(NaN[QS]?)$/i) {
1488: $im = $y;
1489: } else {
1490: if ($y =~ /^-?\Q$Inf\E$/oi) {
1491: $im = $y;
1492: } else {
1493: $im =
1494: defined $format ?
1495: sprintf($format, $y) :
1496: ($y == 1 ? "" : ($y == -1 ? "-" : $y));
1497: }
1498: }
1499: $im .= "i";
1500: } else {
1501: undef $im;
1502: }
1503:
1504: my $str = $re;
1505:
1506: if (defined $im) {
1507: if ($y < 0) {
1508: $str .= $im;
1509: } elsif ($y > 0 || $im =~ /^NaN[QS]?i$/i) {
1510: $str .= "+" if defined $re;
1511: $str .= $im;
1512: }
1513: } elsif (!defined $re) {
1514: $str = "0";
1515: }
1516:
1517: return $str;
1518: }
1519:
1520:
1521: #
1522: # ->_stringify_polar
1523: #
1524: # Stringify as a polar representation '[r,t]'.
1525: #
1526: sub _stringify_polar {
1527: my $z = shift;
1528: my ($r, $t) = @{$z->_polar};
1529: my $theta;
1530:
1531: my %format = $z->display_format;
1532: my $format = $format{format};
1533:
1534: if ($t =~ /^NaN[QS]?$/i || $t =~ /^-?\Q$Inf\E$/oi) {
1535: $theta = $t;
1536: } elsif ($t == pi) {
1537: $theta = "pi";
1538: } elsif ($r == 0 || $t == 0) {
1539: $theta = defined $format ? sprintf($format, $t) : $t;
1540: }
1541:
1542: return "[$r,$theta]" if defined $theta;
1543:
1544: #
1545: # Try to identify pi/n and friends.
1546: #
1547:
1548: $t -= int(CORE::abs($t) / pi2) * pi2;
1549:
1550: if ($format{polar_pretty_print} && $t) {
1551: my ($a, $b);
1552: for $a (2..9) {
1553: $b = $t * $a / pi;
1554: if ($b =~ /^-?\d+$/) {
1555: $b = $b < 0 ? "-" : "" if CORE::abs($b) == 1;
1556: $theta = "${b}pi/$a";
1557: last;
1558: }
1559: }
1560: }
1561:
1562: if (defined $format) {
1563: $r = sprintf($format, $r);
1564: $theta = sprintf($format, $t) unless defined $theta;
1565: } else {
1566: $theta = $t unless defined $theta;
1567: }
1568:
1569: return "[$r,$theta]";
1570: }
1571:
1572: sub Inf {
1573: return $Inf;
1574: }
1575:
1576: 1;
1577: __END__
1578:
1579: =pod
1580:
1581: =head1 NAME
1582:
1583: LONCAPA::LCMathComplex - complex numbers and associated mathematical functions
1584:
1585: =head1 SYNOPSIS
1586:
1587: use LONCAPA::LCMathComplex;
1588:
1589: $z = LONCAPA::LCMathComplex->make(5, 6);
1590: $t = 4 - 3*i + $z;
1591: $j = cplxe(1, 2*pi/3);
1592:
1593: =head1 DESCRIPTION
1594:
1595: Derived from Math::Complex.
1596:
1597: Modified for use in Safe Space in LON-CAPA by removing the dependency on
1598: Config.pm introduced in rev. 1.51 (which contains calls that are disallowed
1599: in Safe Space). In addition, Scalar::Util::set_prototype() is not used for
1600: abs(), cos(), exp(), log(), sin(), and sqrt(), to avoid warnings in logs:
1601: of type: "Use of uninitialized value" (Config.pm line 62).
1602:
1603: In this LON-CAPA-specific version, the following code changes were made.
1604:
1605: 15,16d17
1606: < use Config;
1607: <
1608: 49,51c50
1609: < my $nvsize = $Config{nvsize} ||
1610: < ($Config{uselongdouble} && $Config{longdblsize}) ||
1611: < $Config{doublesize};
1612: ---
1613: > my $nvsize = 8;
1614:
1615: 91,92d89
1616: < use Scalar::Util qw(set_prototype);
1617: <
1618: 96,109d92
1619: < BEGIN {
1620: < # For certain functions that we override, in 5.10 or better
1621: < # we can set a smarter prototype that will handle the lexical $_
1622: < # (also a 5.10+ feature).
1623: < if ($] >= 5.010000) {
1624: < set_prototype \&abs, '_';
1625: < set_prototype \&cos, '_';
1626: < set_prototype \&exp, '_';
1627: < set_prototype \&log, '_';
1628: < set_prototype \&sin, '_';
1629: < set_prototype \&sqrt, '_';
1630: < }
1631: < }
1632:
1633: Note: the value assigned to $nvsize is 8 by default.
1634:
1635: Whenever ./UPDATE is run to install or update LON-CAPA, the code which
1636: sets $nvsize in the standard Math::Complex script will be run in
1637: LCMathComplex_check.piml and the value of $nvsize will be set to the
1638: appropriate value: 4, 8, 10, 12 or 16.
1639:
1640: In addition all instances referring to Math::Complex were changed to
1641: refer to LONCAPA::LCMathComplex instead.
1642:
1643: This package lets you create and manipulate complex numbers. By default,
1644: I<Perl> limits itself to real numbers, but an extra C<use> statement brings
1645: full complex support, along with a full set of mathematical functions
1646: typically associated with and/or extended to complex numbers.
1647:
1648: If you wonder what complex numbers are, they were invented to be able to solve
1649: the following equation:
1650:
1651: x*x = -1
1652:
1653: and by definition, the solution is noted I<i> (engineers use I<j> instead since
1654: I<i> usually denotes an intensity, but the name does not matter). The number
1655: I<i> is a pure I<imaginary> number.
1656:
1657: The arithmetics with pure imaginary numbers works just like you would expect
1658: it with real numbers... you just have to remember that
1659:
1660: i*i = -1
1661:
1662: so you have:
1663:
1664: 5i + 7i = i * (5 + 7) = 12i
1665: 4i - 3i = i * (4 - 3) = i
1666: 4i * 2i = -8
1667: 6i / 2i = 3
1668: 1 / i = -i
1669:
1670: Complex numbers are numbers that have both a real part and an imaginary
1671: part, and are usually noted:
1672:
1673: a + bi
1674:
1675: where C<a> is the I<real> part and C<b> is the I<imaginary> part. The
1676: arithmetic with complex numbers is straightforward. You have to
1677: keep track of the real and the imaginary parts, but otherwise the
1678: rules used for real numbers just apply:
1679:
1680: (4 + 3i) + (5 - 2i) = (4 + 5) + i(3 - 2) = 9 + i
1681: (2 + i) * (4 - i) = 2*4 + 4i -2i -i*i = 8 + 2i + 1 = 9 + 2i
1682:
1683: A graphical representation of complex numbers is possible in a plane
1684: (also called the I<complex plane>, but it's really a 2D plane).
1685: The number
1686:
1687: z = a + bi
1688:
1689: is the point whose coordinates are (a, b). Actually, it would
1690: be the vector originating from (0, 0) to (a, b). It follows that the addition
1691: of two complex numbers is a vectorial addition.
1692:
1693: Since there is a bijection between a point in the 2D plane and a complex
1694: number (i.e. the mapping is unique and reciprocal), a complex number
1695: can also be uniquely identified with polar coordinates:
1696:
1697: [rho, theta]
1698:
1699: where C<rho> is the distance to the origin, and C<theta> the angle between
1700: the vector and the I<x> axis. There is a notation for this using the
1701: exponential form, which is:
1702:
1703: rho * exp(i * theta)
1704:
1705: where I<i> is the famous imaginary number introduced above. Conversion
1706: between this form and the cartesian form C<a + bi> is immediate:
1707:
1708: a = rho * cos(theta)
1709: b = rho * sin(theta)
1710:
1711: which is also expressed by this formula:
1712:
1713: z = rho * exp(i * theta) = rho * (cos theta + i * sin theta)
1714:
1715: In other words, it's the projection of the vector onto the I<x> and I<y>
1716: axes. Mathematicians call I<rho> the I<norm> or I<modulus> and I<theta>
1717: the I<argument> of the complex number. The I<norm> of C<z> is
1718: marked here as C<abs(z)>.
1719:
1720: The polar notation (also known as the trigonometric representation) is
1721: much more handy for performing multiplications and divisions of
1722: complex numbers, whilst the cartesian notation is better suited for
1723: additions and subtractions. Real numbers are on the I<x> axis, and
1724: therefore I<y> or I<theta> is zero or I<pi>.
1725:
1726: All the common operations that can be performed on a real number have
1727: been defined to work on complex numbers as well, and are merely
1728: I<extensions> of the operations defined on real numbers. This means
1729: they keep their natural meaning when there is no imaginary part, provided
1730: the number is within their definition set.
1731:
1732: For instance, the C<sqrt> routine which computes the square root of
1733: its argument is only defined for non-negative real numbers and yields a
1734: non-negative real number (it is an application from B<R+> to B<R+>).
1735: If we allow it to return a complex number, then it can be extended to
1736: negative real numbers to become an application from B<R> to B<C> (the
1737: set of complex numbers):
1738:
1739: sqrt(x) = x >= 0 ? sqrt(x) : sqrt(-x)*i
1740:
1741: It can also be extended to be an application from B<C> to B<C>,
1742: whilst its restriction to B<R> behaves as defined above by using
1743: the following definition:
1744:
1745: sqrt(z = [r,t]) = sqrt(r) * exp(i * t/2)
1746:
1747: Indeed, a negative real number can be noted C<[x,pi]> (the modulus
1748: I<x> is always non-negative, so C<[x,pi]> is really C<-x>, a negative
1749: number) and the above definition states that
1750:
1751: sqrt([x,pi]) = sqrt(x) * exp(i*pi/2) = [sqrt(x),pi/2] = sqrt(x)*i
1752:
1753: which is exactly what we had defined for negative real numbers above.
1754: The C<sqrt> returns only one of the solutions: if you want the both,
1755: use the C<root> function.
1756:
1757: All the common mathematical functions defined on real numbers that
1758: are extended to complex numbers share that same property of working
1759: I<as usual> when the imaginary part is zero (otherwise, it would not
1760: be called an extension, would it?).
1761:
1762: A I<new> operation possible on a complex number that is
1763: the identity for real numbers is called the I<conjugate>, and is noted
1764: with a horizontal bar above the number, or C<~z> here.
1765:
1766: z = a + bi
1767: ~z = a - bi
1768:
1769: Simple... Now look:
1770:
1771: z * ~z = (a + bi) * (a - bi) = a*a + b*b
1772:
1773: We saw that the norm of C<z> was noted C<abs(z)> and was defined as the
1774: distance to the origin, also known as:
1775:
1776: rho = abs(z) = sqrt(a*a + b*b)
1777:
1778: so
1779:
1780: z * ~z = abs(z) ** 2
1781:
1782: If z is a pure real number (i.e. C<b == 0>), then the above yields:
1783:
1784: a * a = abs(a) ** 2
1785:
1786: which is true (C<abs> has the regular meaning for real number, i.e. stands
1787: for the absolute value). This example explains why the norm of C<z> is
1788: noted C<abs(z)>: it extends the C<abs> function to complex numbers, yet
1789: is the regular C<abs> we know when the complex number actually has no
1790: imaginary part... This justifies I<a posteriori> our use of the C<abs>
1791: notation for the norm.
1792:
1793: =head1 OPERATIONS
1794:
1795: Given the following notations:
1796:
1797: z1 = a + bi = r1 * exp(i * t1)
1798: z2 = c + di = r2 * exp(i * t2)
1799: z = <any complex or real number>
1800:
1801: the following (overloaded) operations are supported on complex numbers:
1802:
1803: z1 + z2 = (a + c) + i(b + d)
1804: z1 - z2 = (a - c) + i(b - d)
1805: z1 * z2 = (r1 * r2) * exp(i * (t1 + t2))
1806: z1 / z2 = (r1 / r2) * exp(i * (t1 - t2))
1807: z1 ** z2 = exp(z2 * log z1)
1808: ~z = a - bi
1809: abs(z) = r1 = sqrt(a*a + b*b)
1810: sqrt(z) = sqrt(r1) * exp(i * t/2)
1811: exp(z) = exp(a) * exp(i * b)
1812: log(z) = log(r1) + i*t
1813: sin(z) = 1/2i (exp(i * z1) - exp(-i * z))
1814: cos(z) = 1/2 (exp(i * z1) + exp(-i * z))
1815: atan2(y, x) = atan(y / x) # Minding the right quadrant, note the order.
1816:
1817: The definition used for complex arguments of atan2() is
1818:
1819: -i log((x + iy)/sqrt(x*x+y*y))
1820:
1821: Note that atan2(0, 0) is not well-defined.
1822:
1823: The following extra operations are supported on both real and complex
1824: numbers:
1825:
1826: Re(z) = a
1827: Im(z) = b
1828: arg(z) = t
1829: abs(z) = r
1830:
1831: cbrt(z) = z ** (1/3)
1832: log10(z) = log(z) / log(10)
1833: logn(z, n) = log(z) / log(n)
1834:
1835: tan(z) = sin(z) / cos(z)
1836:
1837: csc(z) = 1 / sin(z)
1838: sec(z) = 1 / cos(z)
1839: cot(z) = 1 / tan(z)
1840:
1841: asin(z) = -i * log(i*z + sqrt(1-z*z))
1842: acos(z) = -i * log(z + i*sqrt(1-z*z))
1843: atan(z) = i/2 * log((i+z) / (i-z))
1844:
1845: acsc(z) = asin(1 / z)
1846: asec(z) = acos(1 / z)
1847: acot(z) = atan(1 / z) = -i/2 * log((i+z) / (z-i))
1848:
1849: sinh(z) = 1/2 (exp(z) - exp(-z))
1850: cosh(z) = 1/2 (exp(z) + exp(-z))
1851: tanh(z) = sinh(z) / cosh(z) = (exp(z) - exp(-z)) / (exp(z) + exp(-z))
1852:
1853: csch(z) = 1 / sinh(z)
1854: sech(z) = 1 / cosh(z)
1855: coth(z) = 1 / tanh(z)
1856:
1857: asinh(z) = log(z + sqrt(z*z+1))
1858: acosh(z) = log(z + sqrt(z*z-1))
1859: atanh(z) = 1/2 * log((1+z) / (1-z))
1860:
1861: acsch(z) = asinh(1 / z)
1862: asech(z) = acosh(1 / z)
1863: acoth(z) = atanh(1 / z) = 1/2 * log((1+z) / (z-1))
1864:
1865: I<arg>, I<abs>, I<log>, I<csc>, I<cot>, I<acsc>, I<acot>, I<csch>,
1866: I<coth>, I<acosech>, I<acotanh>, have aliases I<rho>, I<theta>, I<ln>,
1867: I<cosec>, I<cotan>, I<acosec>, I<acotan>, I<cosech>, I<cotanh>,
1868: I<acosech>, I<acotanh>, respectively. C<Re>, C<Im>, C<arg>, C<abs>,
1869: C<rho>, and C<theta> can be used also as mutators. The C<cbrt>
1870: returns only one of the solutions: if you want all three, use the
1871: C<root> function.
1872:
1873: The I<root> function is available to compute all the I<n>
1874: roots of some complex, where I<n> is a strictly positive integer.
1875: There are exactly I<n> such roots, returned as a list. Getting the
1876: number mathematicians call C<j> such that:
1877:
1878: 1 + j + j*j = 0;
1879:
1880: is a simple matter of writing:
1881:
1882: $j = ((root(1, 3))[1];
1883:
1884: The I<k>th root for C<z = [r,t]> is given by:
1885:
1886: (root(z, n))[k] = r**(1/n) * exp(i * (t + 2*k*pi)/n)
1887:
1888: You can return the I<k>th root directly by C<root(z, n, k)>,
1889: indexing starting from I<zero> and ending at I<n - 1>.
1890:
1891: The I<spaceship> numeric comparison operator, E<lt>=E<gt>, is also
1892: defined. In order to ensure its restriction to real numbers is conform
1893: to what you would expect, the comparison is run on the real part of
1894: the complex number first, and imaginary parts are compared only when
1895: the real parts match.
1896:
1897: =head1 CREATION
1898:
1899: To create a complex number, use either:
1900:
1901: $z = LONCAPA::LCMathComplex->make(3, 4);
1902: $z = cplx(3, 4);
1903:
1904: if you know the cartesian form of the number, or
1905:
1906: $z = 3 + 4*i;
1907:
1908: if you like. To create a number using the polar form, use either:
1909:
1910: $z = LONCAPA::LCMathComplex->emake(5, pi/3);
1911: $x = cplxe(5, pi/3);
1912:
1913: instead. The first argument is the modulus, the second is the angle
1914: (in radians, the full circle is 2*pi). (Mnemonic: C<e> is used as a
1915: notation for complex numbers in the polar form).
1916:
1917: It is possible to write:
1918:
1919: $x = cplxe(-3, pi/4);
1920:
1921: but that will be silently converted into C<[3,-3pi/4]>, since the
1922: modulus must be non-negative (it represents the distance to the origin
1923: in the complex plane).
1924:
1925: It is also possible to have a complex number as either argument of the
1926: C<make>, C<emake>, C<cplx>, and C<cplxe>: the appropriate component of
1927: the argument will be used.
1928:
1929: $z1 = cplx(-2, 1);
1930: $z2 = cplx($z1, 4);
1931:
1932: The C<new>, C<make>, C<emake>, C<cplx>, and C<cplxe> will also
1933: understand a single (string) argument of the forms
1934:
1935: 2-3i
1936: -3i
1937: [2,3]
1938: [2,-3pi/4]
1939: [2]
1940:
1941: in which case the appropriate cartesian and exponential components
1942: will be parsed from the string and used to create new complex numbers.
1943: The imaginary component and the theta, respectively, will default to zero.
1944:
1945: The C<new>, C<make>, C<emake>, C<cplx>, and C<cplxe> will also
1946: understand the case of no arguments: this means plain zero or (0, 0).
1947:
1948: =head1 DISPLAYING
1949:
1950: When printed, a complex number is usually shown under its cartesian
1951: style I<a+bi>, but there are legitimate cases where the polar style
1952: I<[r,t]> is more appropriate. The process of converting the complex
1953: number into a string that can be displayed is known as I<stringification>.
1954:
1955: By calling the class method C<LONCAPA::LCMathComplex::display_format> and
1956: supplying either C<"polar"> or C<"cartesian"> as an argument, you
1957: override the default display style, which is C<"cartesian">. Not
1958: supplying any argument returns the current settings.
1959:
1960: This default can be overridden on a per-number basis by calling the
1961: C<display_format> method instead. As before, not supplying any argument
1962: returns the current display style for this number. Otherwise whatever you
1963: specify will be the new display style for I<this> particular number.
1964:
1965: For instance:
1966:
1967: use LONCAPA::LCMathComplex;
1968:
1969: LONCAPA::LCMathComplex::display_format('polar');
1970: $j = (root(1, 3))[1];
1971: print "j = $j\n"; # Prints "j = [1,2pi/3]"
1972: $j->display_format('cartesian');
1973: print "j = $j\n"; # Prints "j = -0.5+0.866025403784439i"
1974:
1975: The polar style attempts to emphasize arguments like I<k*pi/n>
1976: (where I<n> is a positive integer and I<k> an integer within [-9, +9]),
1977: this is called I<polar pretty-printing>.
1978:
1979: For the reverse of stringifying, see the C<make> and C<emake>.
1980:
1981: =head2 CHANGED IN PERL 5.6
1982:
1983: The C<display_format> class method and the corresponding
1984: C<display_format> object method can now be called using
1985: a parameter hash instead of just a one parameter.
1986:
1987: The old display format style, which can have values C<"cartesian"> or
1988: C<"polar">, can be changed using the C<"style"> parameter.
1989:
1990: $j->display_format(style => "polar");
1991:
1992: The one parameter calling convention also still works.
1993:
1994: $j->display_format("polar");
1995:
1996: There are two new display parameters.
1997:
1998: The first one is C<"format">, which is a sprintf()-style format string
1999: to be used for both numeric parts of the complex number(s). The is
2000: somewhat system-dependent but most often it corresponds to C<"%.15g">.
2001: You can revert to the default by setting the C<format> to C<undef>.
2002:
2003: # the $j from the above example
2004:
2005: $j->display_format('format' => '%.5f');
2006: print "j = $j\n"; # Prints "j = -0.50000+0.86603i"
2007: $j->display_format('format' => undef);
2008: print "j = $j\n"; # Prints "j = -0.5+0.86603i"
2009:
2010: Notice that this affects also the return values of the
2011: C<display_format> methods: in list context the whole parameter hash
2012: will be returned, as opposed to only the style parameter value.
2013: This is a potential incompatibility with earlier versions if you
2014: have been calling the C<display_format> method in list context.
2015:
2016: The second new display parameter is C<"polar_pretty_print">, which can
2017: be set to true or false, the default being true. See the previous
2018: section for what this means.
2019:
2020: =head1 USAGE
2021:
2022: Thanks to overloading, the handling of arithmetics with complex numbers
2023: is simple and almost transparent.
2024:
2025: Here are some examples:
2026:
2027: use LONCAPA::LCMathComplex;
2028:
2029: $j = cplxe(1, 2*pi/3); # $j ** 3 == 1
2030: print "j = $j, j**3 = ", $j ** 3, "\n";
2031: print "1 + j + j**2 = ", 1 + $j + $j**2, "\n";
2032:
2033: $z = -16 + 0*i; # Force it to be a complex
2034: print "sqrt($z) = ", sqrt($z), "\n";
2035:
2036: $k = exp(i * 2*pi/3);
2037: print "$j - $k = ", $j - $k, "\n";
2038:
2039: $z->Re(3); # Re, Im, arg, abs,
2040: $j->arg(2); # (the last two aka rho, theta)
2041: # can be used also as mutators.
2042:
2043: =head1 CONSTANTS
2044:
2045: =head2 PI
2046:
2047: The constant C<pi> and some handy multiples of it (pi2, pi4,
2048: and pip2 (pi/2) and pip4 (pi/4)) are also available if separately
2049: exported:
2050:
2051: use LONCAPA::LCMathComplex ':pi';
2052: $third_of_circle = pi2 / 3;
2053:
2054: =head2 Inf
2055:
2056: The floating point infinity can be exported as a subroutine Inf():
2057:
2058: use LONCAPA::LCMathComplex qw(Inf sinh);
2059: my $AlsoInf = Inf() + 42;
2060: my $AnotherInf = sinh(1e42);
2061: print "$AlsoInf is $AnotherInf\n" if $AlsoInf == $AnotherInf;
2062:
2063: Note that the stringified form of infinity varies between platforms:
2064: it can be for example any of
2065:
2066: inf
2067: infinity
2068: INF
2069: 1.#INF
2070:
2071: or it can be something else.
2072:
2073: Also note that in some platforms trying to use the infinity in
2074: arithmetic operations may result in Perl crashing because using
2075: an infinity causes SIGFPE or its moral equivalent to be sent.
2076: The way to ignore this is
2077:
2078: local $SIG{FPE} = sub { };
2079:
2080: =head1 ERRORS DUE TO DIVISION BY ZERO OR LOGARITHM OF ZERO
2081:
2082: The division (/) and the following functions
2083:
2084: log ln log10 logn
2085: tan sec csc cot
2086: atan asec acsc acot
2087: tanh sech csch coth
2088: atanh asech acsch acoth
2089:
2090: cannot be computed for all arguments because that would mean dividing
2091: by zero or taking logarithm of zero. These situations cause fatal
2092: runtime errors looking like this
2093:
2094: cot(0): Division by zero.
2095: (Because in the definition of cot(0), the divisor sin(0) is 0)
2096: Died at ...
2097:
2098: or
2099:
2100: atanh(-1): Logarithm of zero.
2101: Died at...
2102:
2103: For the C<csc>, C<cot>, C<asec>, C<acsc>, C<acot>, C<csch>, C<coth>,
2104: C<asech>, C<acsch>, the argument cannot be C<0> (zero). For the
2105: logarithmic functions and the C<atanh>, C<acoth>, the argument cannot
2106: be C<1> (one). For the C<atanh>, C<acoth>, the argument cannot be
2107: C<-1> (minus one). For the C<atan>, C<acot>, the argument cannot be
2108: C<i> (the imaginary unit). For the C<atan>, C<acoth>, the argument
2109: cannot be C<-i> (the negative imaginary unit). For the C<tan>,
2110: C<sec>, C<tanh>, the argument cannot be I<pi/2 + k * pi>, where I<k>
2111: is any integer. atan2(0, 0) is undefined, and if the complex arguments
2112: are used for atan2(), a division by zero will happen if z1**2+z2**2 == 0.
2113:
2114: Note that because we are operating on approximations of real numbers,
2115: these errors can happen when merely `too close' to the singularities
2116: listed above.
2117:
2118: =head1 ERRORS DUE TO INDIGESTIBLE ARGUMENTS
2119:
2120: The C<make> and C<emake> accept both real and complex arguments.
2121: When they cannot recognize the arguments they will die with error
2122: messages like the following
2123:
2124: LONCAPA::LCMathComplex::make: Cannot take real part of ...
2125: LONCAPA::LCMathComplex::make: Cannot take real part of ...
2126: LONCAPA::LCMathComplex::emake: Cannot take rho of ...
2127: LONCAPA::LCMathComplex::emake: Cannot take theta of ...
2128:
2129: =head1 BUGS
2130:
2131: Saying C<use LONCAPA::LCMathComplex;> exports many mathematical routines in the
2132: caller environment and even overrides some (C<sqrt>, C<log>, C<atan2>).
2133: This is construed as a feature by the Authors, actually... ;-)
2134:
2135: All routines expect to be given real or complex numbers. Don't attempt to
2136: use BigFloat, since Perl has currently no rule to disambiguate a '+'
2137: operation (for instance) between two overloaded entities.
2138:
2139: In Cray UNICOS there is some strange numerical instability that results
2140: in root(), cos(), sin(), cosh(), sinh(), losing accuracy fast. Beware.
2141: The bug may be in UNICOS math libs, in UNICOS C compiler, in LONCAPA::LCMathComplex.
2142: Whatever it is, it does not manifest itself anywhere else where Perl runs.
2143:
2144: =head1 SEE ALSO
2145:
2146: L<Math::Trig>
2147:
2148: =head1 AUTHORS
2149:
2150: Daniel S. Lewart <F<lewart!at!uiuc.edu>>,
2151: Jarkko Hietaniemi <F<jhi!at!iki.fi>>,
2152: Raphael Manfredi <F<Raphael_Manfredi!at!pobox.com>>,
2153: Zefram <zefram@fysh.org>
2154:
2155: =head1 LICENSE
2156:
2157: This library is free software; you can redistribute it and/or modify
2158: it under the same terms as Perl itself.
2159:
2160: =cut
2161:
2162: 1;
2163:
2164: # eof
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