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
10 our($VERSION, @ISA, @EXPORT, %EXPORT_TAGS, $Inf);
15 unless ($^O eq 'unicosmk') {
17 # We do want an arithmetic overflow.
18 eval '$Inf = CORE::exp(CORE::exp(30))';
19 $! = $e; # Clear ERANGE.
20 undef $Inf unless $Inf =~ /^inf(?:inity)?$/i; # Inf INF inf Infinity
22 $Inf = "Inf" if !defined $Inf || !($Inf > 0); # Desperation.
37 csc cosec sec cot cotan
39 acsc acosec asec acot acotan
41 csch cosech sech coth cotanh
43 acsch acosech asech acoth acotanh
82 my %DISPLAY_FORMAT = ('style' => 'cartesian',
83 'polar_pretty_print' => 1);
84 my $eps = 1e-14; # Epsilon
87 # Object attributes (internal):
88 # cartesian [real, imaginary] -- cartesian form
89 # polar [rho, theta] -- polar form
90 # c_dirty cartesian form not up-to-date
91 # p_dirty polar form not up-to-date
92 # display display format (package's global when not set)
95 # Die on bad *make() arguments.
98 die "@{[(caller(1))[3]]}: Cannot take $_[0] of $_[1].\n";
104 # Create a new complex number (cartesian form)
107 my $self = bless {}, shift;
111 if ( $rre eq ref $self ) {
114 _cannot_make("real part", $rre);
119 if ( $rim eq ref $self ) {
122 _cannot_make("imaginary part", $rim);
125 $self->{'cartesian'} = [ $re, $im ];
126 $self->{c_dirty} = 0;
127 $self->{p_dirty} = 1;
128 $self->display_format('cartesian');
135 # Create a new complex number (exponential form)
138 my $self = bless {}, shift;
139 my ($rho, $theta) = @_;
142 if ( $rrh eq ref $self ) {
145 _cannot_make("rho", $rrh);
148 my $rth = ref $theta;
150 if ( $rth eq ref $self ) {
151 $theta = theta($theta);
153 _cannot_make("theta", $rth);
158 $theta = ($theta <= 0) ? $theta + pi() : $theta - pi();
160 $self->{'polar'} = [$rho, $theta];
161 $self->{p_dirty} = 0;
162 $self->{c_dirty} = 1;
163 $self->display_format('polar');
167 sub new { &make } # For backward compatibility only.
172 # Creates a complex number from a (re, im) tuple.
173 # This avoids the burden of writing Math::Complex->make(re, im).
177 return __PACKAGE__->make($re, defined $im ? $im : 0);
183 # Creates a complex number from a (rho, theta) tuple.
184 # This avoids the burden of writing Math::Complex->emake(rho, theta).
187 my ($rho, $theta) = @_;
188 return __PACKAGE__->emake($rho, defined $theta ? $theta : 0);
194 # The number defined as pi = 180 degrees
196 sub pi () { 4 * CORE::atan2(1, 1) }
203 sub pit2 () { 2 * pi }
210 sub pip2 () { pi / 2 }
215 # One degree in radians, used in stringify_polar.
218 sub deg1 () { pi / 180 }
225 sub uplog10 () { 1 / CORE::log(10) }
230 # The number defined as i*i = -1;
235 $i->{'cartesian'} = [0, 1];
236 $i->{'polar'} = [1, pip2];
250 # Attribute access/set routines
253 sub cartesian {$_[0]->{c_dirty} ?
254 $_[0]->update_cartesian : $_[0]->{'cartesian'}}
255 sub polar {$_[0]->{p_dirty} ?
256 $_[0]->update_polar : $_[0]->{'polar'}}
258 sub set_cartesian { $_[0]->{p_dirty}++; $_[0]->{'cartesian'} = $_[1] }
259 sub set_polar { $_[0]->{c_dirty}++; $_[0]->{'polar'} = $_[1] }
264 # Recompute and return the cartesian form, given accurate polar form.
266 sub update_cartesian {
268 my ($r, $t) = @{$self->{'polar'}};
269 $self->{c_dirty} = 0;
270 return $self->{'cartesian'} = [$r * CORE::cos($t), $r * CORE::sin($t)];
277 # Recompute and return the polar form, given accurate cartesian form.
281 my ($x, $y) = @{$self->{'cartesian'}};
282 $self->{p_dirty} = 0;
283 return $self->{'polar'} = [0, 0] if $x == 0 && $y == 0;
284 return $self->{'polar'} = [CORE::sqrt($x*$x + $y*$y),
285 CORE::atan2($y, $x)];
294 my ($z1, $z2, $regular) = @_;
295 my ($re1, $im1) = @{$z1->cartesian};
296 $z2 = cplx($z2) unless ref $z2;
297 my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0);
298 unless (defined $regular) {
299 $z1->set_cartesian([$re1 + $re2, $im1 + $im2]);
302 return (ref $z1)->make($re1 + $re2, $im1 + $im2);
311 my ($z1, $z2, $inverted) = @_;
312 my ($re1, $im1) = @{$z1->cartesian};
313 $z2 = cplx($z2) unless ref $z2;
314 my ($re2, $im2) = @{$z2->cartesian};
315 unless (defined $inverted) {
316 $z1->set_cartesian([$re1 - $re2, $im1 - $im2]);
320 (ref $z1)->make($re2 - $re1, $im2 - $im1) :
321 (ref $z1)->make($re1 - $re2, $im1 - $im2);
331 my ($z1, $z2, $regular) = @_;
332 if ($z1->{p_dirty} == 0 and ref $z2 and $z2->{p_dirty} == 0) {
333 # if both polar better use polar to avoid rounding errors
334 my ($r1, $t1) = @{$z1->polar};
335 my ($r2, $t2) = @{$z2->polar};
337 if ($t > pi()) { $t -= pit2 }
338 elsif ($t <= -pi()) { $t += pit2 }
339 unless (defined $regular) {
340 $z1->set_polar([$r1 * $r2, $t]);
343 return (ref $z1)->emake($r1 * $r2, $t);
345 my ($x1, $y1) = @{$z1->cartesian};
347 my ($x2, $y2) = @{$z2->cartesian};
348 return (ref $z1)->make($x1*$x2-$y1*$y2, $x1*$y2+$y1*$x2);
350 return (ref $z1)->make($x1*$z2, $y1*$z2);
358 # Die on division by zero.
361 my $mess = "$_[0]: Division by zero.\n";
364 $mess .= "(Because in the definition of $_[0], the divisor ";
365 $mess .= "$_[1] " unless ("$_[1]" eq '0');
371 $mess .= "Died at $up[1] line $up[2].\n";
382 my ($z1, $z2, $inverted) = @_;
383 if ($z1->{p_dirty} == 0 and ref $z2 and $z2->{p_dirty} == 0) {
384 # if both polar better use polar to avoid rounding errors
385 my ($r1, $t1) = @{$z1->polar};
386 my ($r2, $t2) = @{$z2->polar};
389 _divbyzero "$z2/0" if ($r1 == 0);
391 if ($t > pi()) { $t -= pit2 }
392 elsif ($t <= -pi()) { $t += pit2 }
393 return (ref $z1)->emake($r2 / $r1, $t);
395 _divbyzero "$z1/0" if ($r2 == 0);
397 if ($t > pi()) { $t -= pit2 }
398 elsif ($t <= -pi()) { $t += pit2 }
399 return (ref $z1)->emake($r1 / $r2, $t);
404 ($x2, $y2) = @{$z1->cartesian};
405 $d = $x2*$x2 + $y2*$y2;
406 _divbyzero "$z2/0" if $d == 0;
407 return (ref $z1)->make(($x2*$z2)/$d, -($y2*$z2)/$d);
409 my ($x1, $y1) = @{$z1->cartesian};
411 ($x2, $y2) = @{$z2->cartesian};
412 $d = $x2*$x2 + $y2*$y2;
413 _divbyzero "$z1/0" if $d == 0;
414 my $u = ($x1*$x2 + $y1*$y2)/$d;
415 my $v = ($y1*$x2 - $x1*$y2)/$d;
416 return (ref $z1)->make($u, $v);
418 _divbyzero "$z1/0" if $z2 == 0;
419 return (ref $z1)->make($x1/$z2, $y1/$z2);
428 # Computes z1**z2 = exp(z2 * log z1)).
431 my ($z1, $z2, $inverted) = @_;
433 return 1 if $z1 == 0 || $z2 == 1;
434 return 0 if $z2 == 0 && Re($z1) > 0;
436 return 1 if $z2 == 0 || $z1 == 1;
437 return 0 if $z1 == 0 && Re($z2) > 0;
439 my $w = $inverted ? &exp($z1 * &log($z2))
440 : &exp($z2 * &log($z1));
441 # If both arguments cartesian, return cartesian, else polar.
442 return $z1->{c_dirty} == 0 &&
443 (not ref $z2 or $z2->{c_dirty} == 0) ?
444 cplx(@{$w->cartesian}) : $w;
450 # Computes z1 <=> z2.
451 # Sorts on the real part first, then on the imaginary part. Thus 2-4i < 3+8i.
454 my ($z1, $z2, $inverted) = @_;
455 my ($re1, $im1) = ref $z1 ? @{$z1->cartesian} : ($z1, 0);
456 my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0);
457 my $sgn = $inverted ? -1 : 1;
458 return $sgn * ($re1 <=> $re2) if $re1 != $re2;
459 return $sgn * ($im1 <=> $im2);
467 # (Required in addition to spaceship() because of NaNs.)
469 my ($z1, $z2, $inverted) = @_;
470 my ($re1, $im1) = ref $z1 ? @{$z1->cartesian} : ($z1, 0);
471 my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0);
472 return $re1 == $re2 && $im1 == $im2 ? 1 : 0;
483 my ($r, $t) = @{$z->polar};
484 $t = ($t <= 0) ? $t + pi : $t - pi;
485 return (ref $z)->emake($r, $t);
487 my ($re, $im) = @{$z->cartesian};
488 return (ref $z)->make(-$re, -$im);
494 # Compute complex's conjugate.
499 my ($r, $t) = @{$z->polar};
500 return (ref $z)->emake($r, -$t);
502 my ($re, $im) = @{$z->cartesian};
503 return (ref $z)->make($re, -$im);
509 # Compute or set complex's norm (rho).
517 return CORE::abs($z);
521 $z->{'polar'} = [ $rho, ${$z->polar}[1] ];
526 return ${$z->polar}[0];
533 if ($$theta > pi()) { $$theta -= pit2 }
534 elsif ($$theta <= -pi()) { $$theta += pit2 }
540 # Compute or set complex's argument (theta).
543 my ($z, $theta) = @_;
544 return $z unless ref $z;
545 if (defined $theta) {
547 $z->{'polar'} = [ ${$z->polar}[0], $theta ];
551 $theta = ${$z->polar}[1];
562 # It is quite tempting to use wantarray here so that in list context
563 # sqrt() would return the two solutions. This, however, would
566 # print "sqrt(z) = ", sqrt($z), "\n";
568 # The two values would be printed side by side without no intervening
569 # whitespace, quite confusing.
570 # Therefore if you want the two solutions use the root().
574 my ($re, $im) = ref $z ? @{$z->cartesian} : ($z, 0);
575 return $re < 0 ? cplx(0, CORE::sqrt(-$re)) : CORE::sqrt($re)
577 my ($r, $t) = @{$z->polar};
578 return (ref $z)->emake(CORE::sqrt($r), $t/2);
584 # Compute cbrt(z) (cubic root).
586 # Why are we not returning three values? The same answer as for sqrt().
591 -CORE::exp(CORE::log(-$z)/3) :
592 ($z > 0 ? CORE::exp(CORE::log($z)/3): 0)
594 my ($r, $t) = @{$z->polar};
596 return (ref $z)->emake(CORE::exp(CORE::log($r)/3), $t/3);
605 my $mess = "Root $_[0] illegal, root rank must be positive integer.\n";
609 $mess .= "Died at $up[1] line $up[2].\n";
617 # Computes all nth root for z, returning an array whose size is n.
618 # `n' must be a positive integer.
620 # The roots are given by (for k = 0..n-1):
622 # z^(1/n) = r^(1/n) (cos ((t+2 k pi)/n) + i sin ((t+2 k pi)/n))
626 _rootbad($n) if ($n < 1 or int($n) != $n);
627 my ($r, $t) = ref $z ?
628 @{$z->polar} : (CORE::abs($z), $z >= 0 ? 0 : pi);
631 my $theta_inc = pit2 / $n;
632 my $rho = $r ** (1/$n);
634 my $cartesian = ref $z && $z->{c_dirty} == 0;
635 for ($k = 0, $theta = $t / $n; $k < $n; $k++, $theta += $theta_inc) {
636 my $w = cplxe($rho, $theta);
637 # Yes, $cartesian is loop invariant.
638 push @root, $cartesian ? cplx(@{$w->cartesian}) : $w;
646 # Return or set Re(z).
650 return $z unless ref $z;
652 $z->{'cartesian'} = [ $Re, ${$z->cartesian}[1] ];
656 return ${$z->cartesian}[0];
663 # Return or set Im(z).
667 return $z unless ref $z;
669 $z->{'cartesian'} = [ ${$z->cartesian}[0], $Im ];
673 return ${$z->cartesian}[1];
680 # Return or set rho(w).
683 Math::Complex::abs(@_);
689 # Return or set theta(w).
692 Math::Complex::arg(@_);
702 my ($x, $y) = @{$z->cartesian};
703 return (ref $z)->emake(CORE::exp($x), $y);
709 # Die on logarithm of zero.
712 my $mess = "$_[0]: Logarithm of zero.\n";
715 $mess .= "(Because in the definition of $_[0], the argument ";
716 $mess .= "$_[1] " unless ($_[1] eq '0');
722 $mess .= "Died at $up[1] line $up[2].\n";
735 _logofzero("log") if $z == 0;
736 return $z > 0 ? CORE::log($z) : cplx(CORE::log(-$z), pi);
738 my ($r, $t) = @{$z->polar};
739 _logofzero("log") if $r == 0;
740 if ($t > pi()) { $t -= pit2 }
741 elsif ($t <= -pi()) { $t += pit2 }
742 return (ref $z)->make(CORE::log($r), $t);
750 sub ln { Math::Complex::log(@_) }
759 return Math::Complex::log($_[0]) * uplog10;
765 # Compute logn(z,n) = log(z) / log(n)
769 $z = cplx($z, 0) unless ref $z;
770 my $logn = $LOGN{$n};
771 $logn = $LOGN{$n} = CORE::log($n) unless defined $logn; # Cache log(n)
772 return &log($z) / $logn;
778 # Compute cos(z) = (exp(iz) + exp(-iz))/2.
782 return CORE::cos($z) unless ref $z;
783 my ($x, $y) = @{$z->cartesian};
784 my $ey = CORE::exp($y);
785 my $sx = CORE::sin($x);
786 my $cx = CORE::cos($x);
787 my $ey_1 = $ey ? 1 / $ey : $Inf;
788 return (ref $z)->make($cx * ($ey + $ey_1)/2,
789 $sx * ($ey_1 - $ey)/2);
795 # Compute sin(z) = (exp(iz) - exp(-iz))/2.
799 return CORE::sin($z) unless ref $z;
800 my ($x, $y) = @{$z->cartesian};
801 my $ey = CORE::exp($y);
802 my $sx = CORE::sin($x);
803 my $cx = CORE::cos($x);
804 my $ey_1 = $ey ? 1 / $ey : $Inf;
805 return (ref $z)->make($sx * ($ey + $ey_1)/2,
806 $cx * ($ey - $ey_1)/2);
812 # Compute tan(z) = sin(z) / cos(z).
817 _divbyzero "tan($z)", "cos($z)" if $cz == 0;
818 return &sin($z) / $cz;
824 # Computes the secant sec(z) = 1 / cos(z).
829 _divbyzero "sec($z)", "cos($z)" if ($cz == 0);
836 # Computes the cosecant csc(z) = 1 / sin(z).
841 _divbyzero "csc($z)", "sin($z)" if ($sz == 0);
850 sub cosec { Math::Complex::csc(@_) }
855 # Computes cot(z) = cos(z) / sin(z).
860 _divbyzero "cot($z)", "sin($z)" if ($sz == 0);
861 return &cos($z) / $sz;
869 sub cotan { Math::Complex::cot(@_) }
874 # Computes the arc cosine acos(z) = -i log(z + sqrt(z*z-1)).
878 return CORE::atan2(CORE::sqrt(1-$z*$z), $z)
879 if (! ref $z) && CORE::abs($z) <= 1;
880 $z = cplx($z, 0) unless ref $z;
881 my ($x, $y) = @{$z->cartesian};
882 return 0 if $x == 1 && $y == 0;
883 my $t1 = CORE::sqrt(($x+1)*($x+1) + $y*$y);
884 my $t2 = CORE::sqrt(($x-1)*($x-1) + $y*$y);
885 my $alpha = ($t1 + $t2)/2;
886 my $beta = ($t1 - $t2)/2;
887 $alpha = 1 if $alpha < 1;
888 if ($beta > 1) { $beta = 1 }
889 elsif ($beta < -1) { $beta = -1 }
890 my $u = CORE::atan2(CORE::sqrt(1-$beta*$beta), $beta);
891 my $v = CORE::log($alpha + CORE::sqrt($alpha*$alpha-1));
892 $v = -$v if $y > 0 || ($y == 0 && $x < -1);
893 return (ref $z)->make($u, $v);
899 # Computes the arc sine asin(z) = -i log(iz + sqrt(1-z*z)).
903 return CORE::atan2($z, CORE::sqrt(1-$z*$z))
904 if (! ref $z) && CORE::abs($z) <= 1;
905 $z = cplx($z, 0) unless ref $z;
906 my ($x, $y) = @{$z->cartesian};
907 return 0 if $x == 0 && $y == 0;
908 my $t1 = CORE::sqrt(($x+1)*($x+1) + $y*$y);
909 my $t2 = CORE::sqrt(($x-1)*($x-1) + $y*$y);
910 my $alpha = ($t1 + $t2)/2;
911 my $beta = ($t1 - $t2)/2;
912 $alpha = 1 if $alpha < 1;
913 if ($beta > 1) { $beta = 1 }
914 elsif ($beta < -1) { $beta = -1 }
915 my $u = CORE::atan2($beta, CORE::sqrt(1-$beta*$beta));
916 my $v = -CORE::log($alpha + CORE::sqrt($alpha*$alpha-1));
917 $v = -$v if $y > 0 || ($y == 0 && $x < -1);
918 return (ref $z)->make($u, $v);
924 # Computes the arc tangent atan(z) = i/2 log((i+z) / (i-z)).
928 return CORE::atan2($z, 1) unless ref $z;
929 my ($x, $y) = ref $z ? @{$z->cartesian} : ($z, 0);
930 return 0 if $x == 0 && $y == 0;
931 _divbyzero "atan(i)" if ( $z == i);
932 _logofzero "atan(-i)" if (-$z == i); # -i is a bad file test...
933 my $log = &log((i + $z) / (i - $z));
940 # Computes the arc secant asec(z) = acos(1 / z).
944 _divbyzero "asec($z)", $z if ($z == 0);
951 # Computes the arc cosecant acsc(z) = asin(1 / z).
955 _divbyzero "acsc($z)", $z if ($z == 0);
964 sub acosec { Math::Complex::acsc(@_) }
969 # Computes the arc cotangent acot(z) = atan(1 / z)
973 _divbyzero "acot(0)" if $z == 0;
974 return ($z >= 0) ? CORE::atan2(1, $z) : CORE::atan2(-1, -$z)
976 _divbyzero "acot(i)" if ($z - i == 0);
977 _logofzero "acot(-i)" if ($z + i == 0);
986 sub acotan { Math::Complex::acot(@_) }
991 # Computes the hyperbolic cosine cosh(z) = (exp(z) + exp(-z))/2.
998 return $ex ? ($ex + 1/$ex)/2 : $Inf;
1000 my ($x, $y) = @{$z->cartesian};
1001 $ex = CORE::exp($x);
1002 my $ex_1 = $ex ? 1 / $ex : $Inf;
1003 return (ref $z)->make(CORE::cos($y) * ($ex + $ex_1)/2,
1004 CORE::sin($y) * ($ex - $ex_1)/2);
1010 # Computes the hyperbolic sine sinh(z) = (exp(z) - exp(-z))/2.
1016 return 0 if $z == 0;
1017 $ex = CORE::exp($z);
1018 return $ex ? ($ex - 1/$ex)/2 : "-$Inf";
1020 my ($x, $y) = @{$z->cartesian};
1021 my $cy = CORE::cos($y);
1022 my $sy = CORE::sin($y);
1023 $ex = CORE::exp($x);
1024 my $ex_1 = $ex ? 1 / $ex : $Inf;
1025 return (ref $z)->make(CORE::cos($y) * ($ex - $ex_1)/2,
1026 CORE::sin($y) * ($ex + $ex_1)/2);
1032 # Computes the hyperbolic tangent tanh(z) = sinh(z) / cosh(z).
1037 _divbyzero "tanh($z)", "cosh($z)" if ($cz == 0);
1038 return sinh($z) / $cz;
1044 # Computes the hyperbolic secant sech(z) = 1 / cosh(z).
1049 _divbyzero "sech($z)", "cosh($z)" if ($cz == 0);
1056 # Computes the hyperbolic cosecant csch(z) = 1 / sinh(z).
1061 _divbyzero "csch($z)", "sinh($z)" if ($sz == 0);
1070 sub cosech { Math::Complex::csch(@_) }
1075 # Computes the hyperbolic cotangent coth(z) = cosh(z) / sinh(z).
1080 _divbyzero "coth($z)", "sinh($z)" if $sz == 0;
1081 return cosh($z) / $sz;
1089 sub cotanh { Math::Complex::coth(@_) }
1094 # Computes the arc hyperbolic cosine acosh(z) = log(z + sqrt(z*z-1)).
1101 my ($re, $im) = @{$z->cartesian};
1103 return CORE::log($re + CORE::sqrt($re*$re - 1))
1105 return cplx(0, CORE::atan2(CORE::sqrt(1 - $re*$re), $re))
1106 if CORE::abs($re) < 1;
1108 my $t = &sqrt($z * $z - 1) + $z;
1109 # Try Taylor if looking bad (this usually means that
1110 # $z was large negative, therefore the sqrt is really
1111 # close to abs(z), summing that with z...)
1112 $t = 1/(2 * $z) - 1/(8 * $z**3) + 1/(16 * $z**5) - 5/(128 * $z**7)
1115 $u->Im(-$u->Im) if $re < 0 && $im == 0;
1116 return $re < 0 ? -$u : $u;
1122 # Computes the arc hyperbolic sine asinh(z) = log(z + sqrt(z*z+1))
1127 my $t = $z + CORE::sqrt($z*$z + 1);
1128 return CORE::log($t) if $t;
1130 my $t = &sqrt($z * $z + 1) + $z;
1131 # Try Taylor if looking bad (this usually means that
1132 # $z was large negative, therefore the sqrt is really
1133 # close to abs(z), summing that with z...)
1134 $t = 1/(2 * $z) - 1/(8 * $z**3) + 1/(16 * $z**5) - 5/(128 * $z**7)
1142 # Computes the arc hyperbolic tangent atanh(z) = 1/2 log((1+z) / (1-z)).
1147 return CORE::log((1 + $z)/(1 - $z))/2 if CORE::abs($z) < 1;
1150 _divbyzero 'atanh(1)', "1 - $z" if (1 - $z == 0);
1151 _logofzero 'atanh(-1)' if (1 + $z == 0);
1152 return 0.5 * &log((1 + $z) / (1 - $z));
1158 # Computes the hyperbolic arc secant asech(z) = acosh(1 / z).
1162 _divbyzero 'asech(0)', "$z" if ($z == 0);
1163 return acosh(1 / $z);
1169 # Computes the hyperbolic arc cosecant acsch(z) = asinh(1 / z).
1173 _divbyzero 'acsch(0)', $z if ($z == 0);
1174 return asinh(1 / $z);
1180 # Alias for acosh().
1182 sub acosech { Math::Complex::acsch(@_) }
1187 # Computes the arc hyperbolic cotangent acoth(z) = 1/2 log((1+z) / (z-1)).
1191 _divbyzero 'acoth(0)' if ($z == 0);
1193 return CORE::log(($z + 1)/($z - 1))/2 if CORE::abs($z) > 1;
1196 _divbyzero 'acoth(1)', "$z - 1" if ($z - 1 == 0);
1197 _logofzero 'acoth(-1)', "1 + $z" if (1 + $z == 0);
1198 return &log((1 + $z) / ($z - 1)) / 2;
1206 sub acotanh { Math::Complex::acoth(@_) }
1211 # Compute atan(z1/z2).
1214 my ($z1, $z2, $inverted) = @_;
1215 my ($re1, $im1, $re2, $im2);
1217 ($re1, $im1) = ref $z2 ? @{$z2->cartesian} : ($z2, 0);
1218 ($re2, $im2) = @{$z1->cartesian};
1220 ($re1, $im1) = @{$z1->cartesian};
1221 ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0);
1224 return CORE::atan2($re1, $re2) if $im1 == 0;
1225 return ($im1<=>0) * pip2 if $re2 == 0;
1227 my $w = atan($z1/$z2);
1228 my ($u, $v) = ref $w ? @{$w->cartesian} : ($w, 0);
1229 $u += pi if $re2 < 0;
1230 $u -= pit2 if $u > pi;
1231 return cplx($u, $v);
1238 # Set (get if no argument) the display format for all complex numbers that
1239 # don't happen to have overridden it via ->display_format
1241 # When called as an object method, this actually sets the display format for
1242 # the current object.
1244 # Valid object formats are 'c' and 'p' for cartesian and polar. The first
1245 # letter is used actually, so the type can be fully spelled out for clarity.
1247 sub display_format {
1249 my %display_format = %DISPLAY_FORMAT;
1251 if (ref $self) { # Called as an object method
1252 if (exists $self->{display_format}) {
1253 my %obj = %{$self->{display_format}};
1254 @display_format{keys %obj} = values %obj;
1258 $display_format{style} = shift;
1261 @display_format{keys %new} = values %new;
1264 if (ref $self) { # Called as an object method
1265 $self->{display_format} = { %display_format };
1268 %{$self->{display_format}} :
1269 $self->{display_format}->{style};
1272 # Called as a class method
1273 %DISPLAY_FORMAT = %display_format;
1277 $DISPLAY_FORMAT{style};
1283 # Show nicely formatted complex number under its cartesian or polar form,
1284 # depending on the current display format:
1286 # . If a specific display format has been recorded for this object, use it.
1287 # . Otherwise, use the generic current default for all complex numbers,
1288 # which is a package global variable.
1293 my $style = $z->display_format;
1295 $style = $DISPLAY_FORMAT{style} unless defined $style;
1297 return $z->stringify_polar if $style =~ /^p/i;
1298 return $z->stringify_cartesian;
1302 # ->stringify_cartesian
1304 # Stringify as a cartesian representation 'a+bi'.
1306 sub stringify_cartesian {
1308 my ($x, $y) = @{$z->cartesian};
1311 my %format = $z->display_format;
1312 my $format = $format{format};
1315 if ($x =~ /^NaN[QS]?$/i) {
1318 if ($x =~ /^-?$Inf$/oi) {
1321 $re = defined $format ? sprintf($format, $x) : $x;
1329 if ($y =~ /^(NaN[QS]?)$/i) {
1332 if ($y =~ /^-?$Inf$/oi) {
1337 sprintf($format, $y) :
1338 ($y == 1 ? "" : ($y == -1 ? "-" : $y));
1351 } elsif ($y > 0 || $im =~ /^NaN[QS]?i$/i) {
1352 $str .= "+" if defined $re;
1355 } elsif (!defined $re) {
1366 # Stringify as a polar representation '[r,t]'.
1368 sub stringify_polar {
1370 my ($r, $t) = @{$z->polar};
1373 my %format = $z->display_format;
1374 my $format = $format{format};
1376 if ($t =~ /^NaN[QS]?$/i || $t =~ /^-?$Inf$/oi) {
1378 } elsif ($t == pi) {
1380 } elsif ($r == 0 || $t == 0) {
1381 $theta = defined $format ? sprintf($format, $t) : $t;
1384 return "[$r,$theta]" if defined $theta;
1387 # Try to identify pi/n and friends.
1390 $t -= int(CORE::abs($t) / pit2) * pit2;
1392 if ($format{polar_pretty_print} && $t) {
1396 if ($b =~ /^-?\d+$/) {
1397 $b = $b < 0 ? "-" : "" if CORE::abs($b) == 1;
1398 $theta = "${b}pi/$a";
1404 if (defined $format) {
1405 $r = sprintf($format, $r);
1406 $theta = sprintf($format, $theta) unless defined $theta;
1408 $theta = $t unless defined $theta;
1411 return "[$r,$theta]";
1419 Math::Complex - complex numbers and associated mathematical functions
1425 $z = Math::Complex->make(5, 6);
1427 $j = cplxe(1, 2*pi/3);
1431 This package lets you create and manipulate complex numbers. By default,
1432 I<Perl> limits itself to real numbers, but an extra C<use> statement brings
1433 full complex support, along with a full set of mathematical functions
1434 typically associated with and/or extended to complex numbers.
1436 If you wonder what complex numbers are, they were invented to be able to solve
1437 the following equation:
1441 and by definition, the solution is noted I<i> (engineers use I<j> instead since
1442 I<i> usually denotes an intensity, but the name does not matter). The number
1443 I<i> is a pure I<imaginary> number.
1445 The arithmetics with pure imaginary numbers works just like you would expect
1446 it with real numbers... you just have to remember that
1452 5i + 7i = i * (5 + 7) = 12i
1453 4i - 3i = i * (4 - 3) = i
1458 Complex numbers are numbers that have both a real part and an imaginary
1459 part, and are usually noted:
1463 where C<a> is the I<real> part and C<b> is the I<imaginary> part. The
1464 arithmetic with complex numbers is straightforward. You have to
1465 keep track of the real and the imaginary parts, but otherwise the
1466 rules used for real numbers just apply:
1468 (4 + 3i) + (5 - 2i) = (4 + 5) + i(3 - 2) = 9 + i
1469 (2 + i) * (4 - i) = 2*4 + 4i -2i -i*i = 8 + 2i + 1 = 9 + 2i
1471 A graphical representation of complex numbers is possible in a plane
1472 (also called the I<complex plane>, but it's really a 2D plane).
1477 is the point whose coordinates are (a, b). Actually, it would
1478 be the vector originating from (0, 0) to (a, b). It follows that the addition
1479 of two complex numbers is a vectorial addition.
1481 Since there is a bijection between a point in the 2D plane and a complex
1482 number (i.e. the mapping is unique and reciprocal), a complex number
1483 can also be uniquely identified with polar coordinates:
1487 where C<rho> is the distance to the origin, and C<theta> the angle between
1488 the vector and the I<x> axis. There is a notation for this using the
1489 exponential form, which is:
1491 rho * exp(i * theta)
1493 where I<i> is the famous imaginary number introduced above. Conversion
1494 between this form and the cartesian form C<a + bi> is immediate:
1496 a = rho * cos(theta)
1497 b = rho * sin(theta)
1499 which is also expressed by this formula:
1501 z = rho * exp(i * theta) = rho * (cos theta + i * sin theta)
1503 In other words, it's the projection of the vector onto the I<x> and I<y>
1504 axes. Mathematicians call I<rho> the I<norm> or I<modulus> and I<theta>
1505 the I<argument> of the complex number. The I<norm> of C<z> will be
1508 The polar notation (also known as the trigonometric
1509 representation) is much more handy for performing multiplications and
1510 divisions of complex numbers, whilst the cartesian notation is better
1511 suited for additions and subtractions. Real numbers are on the I<x>
1512 axis, and therefore I<theta> is zero or I<pi>.
1514 All the common operations that can be performed on a real number have
1515 been defined to work on complex numbers as well, and are merely
1516 I<extensions> of the operations defined on real numbers. This means
1517 they keep their natural meaning when there is no imaginary part, provided
1518 the number is within their definition set.
1520 For instance, the C<sqrt> routine which computes the square root of
1521 its argument is only defined for non-negative real numbers and yields a
1522 non-negative real number (it is an application from B<R+> to B<R+>).
1523 If we allow it to return a complex number, then it can be extended to
1524 negative real numbers to become an application from B<R> to B<C> (the
1525 set of complex numbers):
1527 sqrt(x) = x >= 0 ? sqrt(x) : sqrt(-x)*i
1529 It can also be extended to be an application from B<C> to B<C>,
1530 whilst its restriction to B<R> behaves as defined above by using
1531 the following definition:
1533 sqrt(z = [r,t]) = sqrt(r) * exp(i * t/2)
1535 Indeed, a negative real number can be noted C<[x,pi]> (the modulus
1536 I<x> is always non-negative, so C<[x,pi]> is really C<-x>, a negative
1537 number) and the above definition states that
1539 sqrt([x,pi]) = sqrt(x) * exp(i*pi/2) = [sqrt(x),pi/2] = sqrt(x)*i
1541 which is exactly what we had defined for negative real numbers above.
1542 The C<sqrt> returns only one of the solutions: if you want the both,
1543 use the C<root> function.
1545 All the common mathematical functions defined on real numbers that
1546 are extended to complex numbers share that same property of working
1547 I<as usual> when the imaginary part is zero (otherwise, it would not
1548 be called an extension, would it?).
1550 A I<new> operation possible on a complex number that is
1551 the identity for real numbers is called the I<conjugate>, and is noted
1552 with an horizontal bar above the number, or C<~z> here.
1559 z * ~z = (a + bi) * (a - bi) = a*a + b*b
1561 We saw that the norm of C<z> was noted C<abs(z)> and was defined as the
1562 distance to the origin, also known as:
1564 rho = abs(z) = sqrt(a*a + b*b)
1568 z * ~z = abs(z) ** 2
1570 If z is a pure real number (i.e. C<b == 0>), then the above yields:
1574 which is true (C<abs> has the regular meaning for real number, i.e. stands
1575 for the absolute value). This example explains why the norm of C<z> is
1576 noted C<abs(z)>: it extends the C<abs> function to complex numbers, yet
1577 is the regular C<abs> we know when the complex number actually has no
1578 imaginary part... This justifies I<a posteriori> our use of the C<abs>
1579 notation for the norm.
1583 Given the following notations:
1585 z1 = a + bi = r1 * exp(i * t1)
1586 z2 = c + di = r2 * exp(i * t2)
1587 z = <any complex or real number>
1589 the following (overloaded) operations are supported on complex numbers:
1591 z1 + z2 = (a + c) + i(b + d)
1592 z1 - z2 = (a - c) + i(b - d)
1593 z1 * z2 = (r1 * r2) * exp(i * (t1 + t2))
1594 z1 / z2 = (r1 / r2) * exp(i * (t1 - t2))
1595 z1 ** z2 = exp(z2 * log z1)
1597 abs(z) = r1 = sqrt(a*a + b*b)
1598 sqrt(z) = sqrt(r1) * exp(i * t/2)
1599 exp(z) = exp(a) * exp(i * b)
1600 log(z) = log(r1) + i*t
1601 sin(z) = 1/2i (exp(i * z1) - exp(-i * z))
1602 cos(z) = 1/2 (exp(i * z1) + exp(-i * z))
1603 atan2(z1, z2) = atan(z1/z2)
1605 The following extra operations are supported on both real and complex
1613 cbrt(z) = z ** (1/3)
1614 log10(z) = log(z) / log(10)
1615 logn(z, n) = log(z) / log(n)
1617 tan(z) = sin(z) / cos(z)
1623 asin(z) = -i * log(i*z + sqrt(1-z*z))
1624 acos(z) = -i * log(z + i*sqrt(1-z*z))
1625 atan(z) = i/2 * log((i+z) / (i-z))
1627 acsc(z) = asin(1 / z)
1628 asec(z) = acos(1 / z)
1629 acot(z) = atan(1 / z) = -i/2 * log((i+z) / (z-i))
1631 sinh(z) = 1/2 (exp(z) - exp(-z))
1632 cosh(z) = 1/2 (exp(z) + exp(-z))
1633 tanh(z) = sinh(z) / cosh(z) = (exp(z) - exp(-z)) / (exp(z) + exp(-z))
1635 csch(z) = 1 / sinh(z)
1636 sech(z) = 1 / cosh(z)
1637 coth(z) = 1 / tanh(z)
1639 asinh(z) = log(z + sqrt(z*z+1))
1640 acosh(z) = log(z + sqrt(z*z-1))
1641 atanh(z) = 1/2 * log((1+z) / (1-z))
1643 acsch(z) = asinh(1 / z)
1644 asech(z) = acosh(1 / z)
1645 acoth(z) = atanh(1 / z) = 1/2 * log((1+z) / (z-1))
1647 I<arg>, I<abs>, I<log>, I<csc>, I<cot>, I<acsc>, I<acot>, I<csch>,
1648 I<coth>, I<acosech>, I<acotanh>, have aliases I<rho>, I<theta>, I<ln>,
1649 I<cosec>, I<cotan>, I<acosec>, I<acotan>, I<cosech>, I<cotanh>,
1650 I<acosech>, I<acotanh>, respectively. C<Re>, C<Im>, C<arg>, C<abs>,
1651 C<rho>, and C<theta> can be used also also mutators. The C<cbrt>
1652 returns only one of the solutions: if you want all three, use the
1655 The I<root> function is available to compute all the I<n>
1656 roots of some complex, where I<n> is a strictly positive integer.
1657 There are exactly I<n> such roots, returned as a list. Getting the
1658 number mathematicians call C<j> such that:
1662 is a simple matter of writing:
1664 $j = ((root(1, 3))[1];
1666 The I<k>th root for C<z = [r,t]> is given by:
1668 (root(z, n))[k] = r**(1/n) * exp(i * (t + 2*k*pi)/n)
1670 The I<spaceship> comparison operator, E<lt>=E<gt>, is also defined. In
1671 order to ensure its restriction to real numbers is conform to what you
1672 would expect, the comparison is run on the real part of the complex
1673 number first, and imaginary parts are compared only when the real
1678 To create a complex number, use either:
1680 $z = Math::Complex->make(3, 4);
1683 if you know the cartesian form of the number, or
1687 if you like. To create a number using the polar form, use either:
1689 $z = Math::Complex->emake(5, pi/3);
1690 $x = cplxe(5, pi/3);
1692 instead. The first argument is the modulus, the second is the angle
1693 (in radians, the full circle is 2*pi). (Mnemonic: C<e> is used as a
1694 notation for complex numbers in the polar form).
1696 It is possible to write:
1698 $x = cplxe(-3, pi/4);
1700 but that will be silently converted into C<[3,-3pi/4]>, since the
1701 modulus must be non-negative (it represents the distance to the origin
1702 in the complex plane).
1704 It is also possible to have a complex number as either argument of
1705 either the C<make> or C<emake>: the appropriate component of
1706 the argument will be used.
1711 =head1 STRINGIFICATION
1713 When printed, a complex number is usually shown under its cartesian
1714 style I<a+bi>, but there are legitimate cases where the polar style
1715 I<[r,t]> is more appropriate.
1717 By calling the class method C<Math::Complex::display_format> and
1718 supplying either C<"polar"> or C<"cartesian"> as an argument, you
1719 override the default display style, which is C<"cartesian">. Not
1720 supplying any argument returns the current settings.
1722 This default can be overridden on a per-number basis by calling the
1723 C<display_format> method instead. As before, not supplying any argument
1724 returns the current display style for this number. Otherwise whatever you
1725 specify will be the new display style for I<this> particular number.
1731 Math::Complex::display_format('polar');
1732 $j = (root(1, 3))[1];
1733 print "j = $j\n"; # Prints "j = [1,2pi/3]"
1734 $j->display_format('cartesian');
1735 print "j = $j\n"; # Prints "j = -0.5+0.866025403784439i"
1737 The polar style attempts to emphasize arguments like I<k*pi/n>
1738 (where I<n> is a positive integer and I<k> an integer within [-9, +9]),
1739 this is called I<polar pretty-printing>.
1741 =head2 CHANGED IN PERL 5.6
1743 The C<display_format> class method and the corresponding
1744 C<display_format> object method can now be called using
1745 a parameter hash instead of just a one parameter.
1747 The old display format style, which can have values C<"cartesian"> or
1748 C<"polar">, can be changed using the C<"style"> parameter.
1750 $j->display_format(style => "polar");
1752 The one parameter calling convention also still works.
1754 $j->display_format("polar");
1756 There are two new display parameters.
1758 The first one is C<"format">, which is a sprintf()-style format string
1759 to be used for both numeric parts of the complex number(s). The is
1760 somewhat system-dependent but most often it corresponds to C<"%.15g">.
1761 You can revert to the default by setting the C<format> to C<undef>.
1763 # the $j from the above example
1765 $j->display_format('format' => '%.5f');
1766 print "j = $j\n"; # Prints "j = -0.50000+0.86603i"
1767 $j->display_format('format' => undef);
1768 print "j = $j\n"; # Prints "j = -0.5+0.86603i"
1770 Notice that this affects also the return values of the
1771 C<display_format> methods: in list context the whole parameter hash
1772 will be returned, as opposed to only the style parameter value.
1773 This is a potential incompatibility with earlier versions if you
1774 have been calling the C<display_format> method in list context.
1776 The second new display parameter is C<"polar_pretty_print">, which can
1777 be set to true or false, the default being true. See the previous
1778 section for what this means.
1782 Thanks to overloading, the handling of arithmetics with complex numbers
1783 is simple and almost transparent.
1785 Here are some examples:
1789 $j = cplxe(1, 2*pi/3); # $j ** 3 == 1
1790 print "j = $j, j**3 = ", $j ** 3, "\n";
1791 print "1 + j + j**2 = ", 1 + $j + $j**2, "\n";
1793 $z = -16 + 0*i; # Force it to be a complex
1794 print "sqrt($z) = ", sqrt($z), "\n";
1796 $k = exp(i * 2*pi/3);
1797 print "$j - $k = ", $j - $k, "\n";
1799 $z->Re(3); # Re, Im, arg, abs,
1800 $j->arg(2); # (the last two aka rho, theta)
1801 # can be used also as mutators.
1803 =head1 ERRORS DUE TO DIVISION BY ZERO OR LOGARITHM OF ZERO
1805 The division (/) and the following functions
1811 atanh asech acsch acoth
1813 cannot be computed for all arguments because that would mean dividing
1814 by zero or taking logarithm of zero. These situations cause fatal
1815 runtime errors looking like this
1817 cot(0): Division by zero.
1818 (Because in the definition of cot(0), the divisor sin(0) is 0)
1823 atanh(-1): Logarithm of zero.
1826 For the C<csc>, C<cot>, C<asec>, C<acsc>, C<acot>, C<csch>, C<coth>,
1827 C<asech>, C<acsch>, the argument cannot be C<0> (zero). For the the
1828 logarithmic functions and the C<atanh>, C<acoth>, the argument cannot
1829 be C<1> (one). For the C<atanh>, C<acoth>, the argument cannot be
1830 C<-1> (minus one). For the C<atan>, C<acot>, the argument cannot be
1831 C<i> (the imaginary unit). For the C<atan>, C<acoth>, the argument
1832 cannot be C<-i> (the negative imaginary unit). For the C<tan>,
1833 C<sec>, C<tanh>, the argument cannot be I<pi/2 + k * pi>, where I<k>
1836 Note that because we are operating on approximations of real numbers,
1837 these errors can happen when merely `too close' to the singularities
1840 =head1 ERRORS DUE TO INDIGESTIBLE ARGUMENTS
1842 The C<make> and C<emake> accept both real and complex arguments.
1843 When they cannot recognize the arguments they will die with error
1844 messages like the following
1846 Math::Complex::make: Cannot take real part of ...
1847 Math::Complex::make: Cannot take real part of ...
1848 Math::Complex::emake: Cannot take rho of ...
1849 Math::Complex::emake: Cannot take theta of ...
1853 Saying C<use Math::Complex;> exports many mathematical routines in the
1854 caller environment and even overrides some (C<sqrt>, C<log>).
1855 This is construed as a feature by the Authors, actually... ;-)
1857 All routines expect to be given real or complex numbers. Don't attempt to
1858 use BigFloat, since Perl has currently no rule to disambiguate a '+'
1859 operation (for instance) between two overloaded entities.
1861 In Cray UNICOS there is some strange numerical instability that results
1862 in root(), cos(), sin(), cosh(), sinh(), losing accuracy fast. Beware.
1863 The bug may be in UNICOS math libs, in UNICOS C compiler, in Math::Complex.
1864 Whatever it is, it does not manifest itself anywhere else where Perl runs.
1868 Raphael Manfredi <F<Raphael_Manfredi@pobox.com>> and
1869 Jarkko Hietaniemi <F<jhi@iki.fi>>.
1871 Extensive patches by Daniel S. Lewart <F<d-lewart@uiuc.edu>>.