-# $RCSFile$
#
# Complex numbers and associated mathematical functions
-# -- Raphael Manfredi, September 1996
-# -- Jarkko Hietaniemi, March-April 1997
+# -- Raphael Manfredi Since Sep 1996
+# -- Jarkko Hietaniemi Since Mar 1997
+# -- Daniel S. Lewart Since Sep 1997
+#
require Exporter;
package Math::Complex;
use strict;
-use vars qw($VERSION @ISA
- @EXPORT %EXPORT_TAGS
- $package $display
- $i $logn %logn);
+use vars qw($VERSION @ISA @EXPORT %EXPORT_TAGS);
-@ISA = qw(Exporter);
+my ( $i, $ip2, %logn );
+
+$VERSION = sprintf("%s", q$Id: Complex.pm,v 1.25 1998/02/05 16:07:37 jhi Exp $ =~ /(\d+\.\d+)/);
-$VERSION = 1.01;
+@ISA = qw(Exporter);
my @trig = qw(
pi
- sin cos tan
+ tan
csc cosec sec cot cotan
asin acos atan
acsc acosec asec acot acotan
);
@EXPORT = (qw(
- i Re Im arg
- sqrt exp log ln
+ i Re Im rho theta arg
+ sqrt log ln
log10 logn cbrt root
cplx cplxe
),
qw("" stringify);
#
-# Package globals
+# Package "privates"
#
-$package = 'Math::Complex'; # Package name
-$display = 'cartesian'; # Default display format
+my $package = 'Math::Complex'; # Package name
+my $display = 'cartesian'; # Default display format
+my $eps = 1e-14; # Epsilon
#
# Object attributes (internal):
# display display format (package's global when not set)
#
+# Die on bad *make() arguments.
+
+sub _cannot_make {
+ die "@{[(caller(1))[3]]}: Cannot take $_[0] of $_[1].\n";
+}
+
#
# ->make
#
sub make {
my $self = bless {}, shift;
my ($re, $im) = @_;
- $self->{'cartesian'} = [$re, $im];
+ my $rre = ref $re;
+ if ( $rre ) {
+ if ( $rre eq ref $self ) {
+ $re = Re($re);
+ } else {
+ _cannot_make("real part", $rre);
+ }
+ }
+ my $rim = ref $im;
+ if ( $rim ) {
+ if ( $rim eq ref $self ) {
+ $im = Im($im);
+ } else {
+ _cannot_make("imaginary part", $rim);
+ }
+ }
+ $self->{'cartesian'} = [ $re, $im ];
$self->{c_dirty} = 0;
$self->{p_dirty} = 1;
+ $self->display_format('cartesian');
return $self;
}
sub emake {
my $self = bless {}, shift;
my ($rho, $theta) = @_;
- $theta += pi() if $rho < 0;
- $self->{'polar'} = [abs($rho), $theta];
+ my $rrh = ref $rho;
+ if ( $rrh ) {
+ if ( $rrh eq ref $self ) {
+ $rho = rho($rho);
+ } else {
+ _cannot_make("rho", $rrh);
+ }
+ }
+ my $rth = ref $theta;
+ if ( $rth ) {
+ if ( $rth eq ref $self ) {
+ $theta = theta($theta);
+ } else {
+ _cannot_make("theta", $rth);
+ }
+ }
+ if ($rho < 0) {
+ $rho = -$rho;
+ $theta = ($theta <= 0) ? $theta + pi() : $theta - pi();
+ }
+ $self->{'polar'} = [$rho, $theta];
$self->{p_dirty} = 0;
$self->{c_dirty} = 1;
+ $self->display_format('polar');
return $self;
}
#
# pi
#
-# The number defined as 2 * pi = 360 degrees
+# The number defined as pi = 180 degrees
#
-
use constant pi => 4 * atan2(1, 1);
#
-# log2inv
+# pit2
#
-# Used in log10().
+# The full circle
+#
+use constant pit2 => 2 * pi;
+
+#
+# pip2
+#
+# The quarter circle
#
+use constant pip2 => pi / 2;
-use constant log10inv => 1 / log(10);
+#
+# deg1
+#
+# One degree in radians, used in stringify_polar.
+#
+
+use constant deg1 => pi / 180;
+
+#
+# uplog10
+#
+# Used in log10().
+#
+use constant uplog10 => 1 / log(10);
#
# i
return $i if ($i);
$i = bless {};
$i->{'cartesian'} = [0, 1];
- $i->{'polar'} = [1, pi/2];
+ $i->{'polar'} = [1, pip2];
$i->{c_dirty} = 0;
$i->{p_dirty} = 0;
return $i;
sub plus {
my ($z1, $z2, $regular) = @_;
my ($re1, $im1) = @{$z1->cartesian};
+ $z2 = cplx($z2) unless ref $z2;
my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0);
unless (defined $regular) {
$z1->set_cartesian([$re1 + $re2, $im1 + $im2]);
sub minus {
my ($z1, $z2, $inverted) = @_;
my ($re1, $im1) = @{$z1->cartesian};
- my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0);
+ $z2 = cplx($z2) unless ref $z2;
+ my ($re2, $im2) = @{$z2->cartesian};
unless (defined $inverted) {
$z1->set_cartesian([$re1 - $re2, $im1 - $im2]);
return $z1;
return $inverted ?
(ref $z1)->make($re2 - $re1, $im2 - $im1) :
(ref $z1)->make($re1 - $re2, $im1 - $im2);
+
}
#
# Computes z1*z2.
#
sub multiply {
- my ($z1, $z2, $regular) = @_;
- my ($r1, $t1) = @{$z1->polar};
- my ($r2, $t2) = ref $z2 ?
- @{$z2->polar} : (abs($z2), $z2 >= 0 ? 0 : pi);
- unless (defined $regular) {
- $z1->set_polar([$r1 * $r2, $t1 + $t2]);
+ my ($z1, $z2, $regular) = @_;
+ if ($z1->{p_dirty} == 0 and ref $z2 and $z2->{p_dirty} == 0) {
+ # if both polar better use polar to avoid rounding errors
+ my ($r1, $t1) = @{$z1->polar};
+ my ($r2, $t2) = @{$z2->polar};
+ my $t = $t1 + $t2;
+ if ($t > pi()) { $t -= pit2 }
+ elsif ($t <= -pi()) { $t += pit2 }
+ unless (defined $regular) {
+ $z1->set_polar([$r1 * $r2, $t]);
return $z1;
+ }
+ return (ref $z1)->emake($r1 * $r2, $t);
+ } else {
+ my ($x1, $y1) = @{$z1->cartesian};
+ if (ref $z2) {
+ my ($x2, $y2) = @{$z2->cartesian};
+ return (ref $z1)->make($x1*$x2-$y1*$y2, $x1*$y2+$y1*$x2);
+ } else {
+ return (ref $z1)->make($x1*$z2, $y1*$z2);
+ }
}
- return (ref $z1)->emake($r1 * $r2, $t1 + $t2);
}
#
-# divbyzero
+# _divbyzero
#
# Die on division by zero.
#
-sub divbyzero {
+sub _divbyzero {
my $mess = "$_[0]: Division by zero.\n";
if (defined $_[1]) {
}
my @up = caller(1);
-
+
$mess .= "Died at $up[1] line $up[2].\n";
die $mess;
#
sub divide {
my ($z1, $z2, $inverted) = @_;
- my ($r1, $t1) = @{$z1->polar};
- my ($r2, $t2) = ref $z2 ?
- @{$z2->polar} : (abs($z2), $z2 >= 0 ? 0 : pi);
- unless (defined $inverted) {
- divbyzero "$z1/0" if ($r2 == 0);
- $z1->set_polar([$r1 / $r2, $t1 - $t2]);
- return $z1;
- }
- if ($inverted) {
- divbyzero "$z2/0" if ($r1 == 0);
- return (ref $z1)->emake($r2 / $r1, $t2 - $t1);
+ if ($z1->{p_dirty} == 0 and ref $z2 and $z2->{p_dirty} == 0) {
+ # if both polar better use polar to avoid rounding errors
+ my ($r1, $t1) = @{$z1->polar};
+ my ($r2, $t2) = @{$z2->polar};
+ my $t;
+ if ($inverted) {
+ _divbyzero "$z2/0" if ($r1 == 0);
+ $t = $t2 - $t1;
+ if ($t > pi()) { $t -= pit2 }
+ elsif ($t <= -pi()) { $t += pit2 }
+ return (ref $z1)->emake($r2 / $r1, $t);
+ } else {
+ _divbyzero "$z1/0" if ($r2 == 0);
+ $t = $t1 - $t2;
+ if ($t > pi()) { $t -= pit2 }
+ elsif ($t <= -pi()) { $t += pit2 }
+ return (ref $z1)->emake($r1 / $r2, $t);
+ }
} else {
- divbyzero "$z1/0" if ($r2 == 0);
- return (ref $z1)->emake($r1 / $r2, $t1 - $t2);
+ my ($d, $x2, $y2);
+ if ($inverted) {
+ ($x2, $y2) = @{$z1->cartesian};
+ $d = $x2*$x2 + $y2*$y2;
+ _divbyzero "$z2/0" if $d == 0;
+ return (ref $z1)->make(($x2*$z2)/$d, -($y2*$z2)/$d);
+ } else {
+ my ($x1, $y1) = @{$z1->cartesian};
+ if (ref $z2) {
+ ($x2, $y2) = @{$z2->cartesian};
+ $d = $x2*$x2 + $y2*$y2;
+ _divbyzero "$z1/0" if $d == 0;
+ my $u = ($x1*$x2 + $y1*$y2)/$d;
+ my $v = ($y1*$x2 - $x1*$y2)/$d;
+ return (ref $z1)->make($u, $v);
+ } else {
+ _divbyzero "$z1/0" if $z2 == 0;
+ return (ref $z1)->make($x1/$z2, $y1/$z2);
+ }
+ }
}
}
#
+# _zerotozero
+#
+# Die on zero raised to the zeroth.
+#
+sub _zerotozero {
+ my $mess = "The zero raised to the zeroth power is not defined.\n";
+
+ my @up = caller(1);
+
+ $mess .= "Died at $up[1] line $up[2].\n";
+
+ die $mess;
+}
+
+#
# (power)
#
# Computes z1**z2 = exp(z2 * log z1)).
#
sub power {
my ($z1, $z2, $inverted) = @_;
- return exp($z1 * log $z2) if defined $inverted && $inverted;
- return exp($z2 * log $z1);
+ my $z1z = $z1 == 0;
+ my $z2z = $z2 == 0;
+ _zerotozero if ($z1z and $z2z);
+ if ($inverted) {
+ return 0 if ($z2z);
+ return 1 if ($z1z or $z2 == 1);
+ } else {
+ return 0 if ($z1z);
+ return 1 if ($z2z or $z1 == 1);
+ }
+ my $w = $inverted ? exp($z1 * log $z2) : exp($z2 * log $z1);
+ # If both arguments cartesian, return cartesian, else polar.
+ return $z1->{c_dirty} == 0 &&
+ (not ref $z2 or $z2->{c_dirty} == 0) ?
+ cplx(@{$w->cartesian}) : $w;
}
#
my ($z) = @_;
if ($z->{c_dirty}) {
my ($r, $t) = @{$z->polar};
- return (ref $z)->emake($r, pi + $t);
+ $t = ($t <= 0) ? $t + pi : $t - pi;
+ return (ref $z)->emake($r, $t);
}
my ($re, $im) = @{$z->cartesian};
return (ref $z)->make(-$re, -$im);
#
# (abs)
#
-# Compute complex's norm (rho).
+# Compute or set complex's norm (rho).
#
sub abs {
- my ($z) = @_;
- return abs($z) unless ref $z;
- my ($r, $t) = @{$z->polar};
- return abs($r);
+ my ($z, $rho) = @_;
+ return $z unless ref $z;
+ if (defined $rho) {
+ $z->{'polar'} = [ $rho, ${$z->polar}[1] ];
+ $z->{p_dirty} = 0;
+ $z->{c_dirty} = 1;
+ return $rho;
+ } else {
+ return ${$z->polar}[0];
+ }
+}
+
+sub _theta {
+ my $theta = $_[0];
+
+ if ($$theta > pi()) { $$theta -= pit2 }
+ elsif ($$theta <= -pi()) { $$theta += pit2 }
}
#
# arg
#
-# Compute complex's argument (theta).
+# Compute or set complex's argument (theta).
#
sub arg {
- my ($z) = @_;
- return ($z < 0 ? pi : 0) unless ref $z;
- my ($r, $t) = @{$z->polar};
- return $t;
+ my ($z, $theta) = @_;
+ return $z unless ref $z;
+ if (defined $theta) {
+ _theta(\$theta);
+ $z->{'polar'} = [ ${$z->polar}[0], $theta ];
+ $z->{p_dirty} = 0;
+ $z->{c_dirty} = 1;
+ } else {
+ $theta = ${$z->polar}[1];
+ _theta(\$theta);
+ }
+ return $theta;
}
#
#
# Compute sqrt(z).
#
+# It is quite tempting to use wantarray here so that in list context
+# sqrt() would return the two solutions. This, however, would
+# break things like
+#
+# print "sqrt(z) = ", sqrt($z), "\n";
+#
+# The two values would be printed side by side without no intervening
+# whitespace, quite confusing.
+# Therefore if you want the two solutions use the root().
+#
sub sqrt {
my ($z) = @_;
- $z = cplx($z, 0) unless ref $z;
+ my ($re, $im) = ref $z ? @{$z->cartesian} : ($z, 0);
+ return $re < 0 ? cplx(0, sqrt(-$re)) : sqrt($re) if $im == 0;
my ($r, $t) = @{$z->polar};
return (ref $z)->emake(sqrt($r), $t/2);
}
#
# Compute cbrt(z) (cubic root).
#
+# Why are we not returning three values? The same answer as for sqrt().
+#
sub cbrt {
my ($z) = @_;
- return cplx($z, 0) ** (1/3) unless ref $z;
+ return $z < 0 ? -exp(log(-$z)/3) : ($z > 0 ? exp(log($z)/3): 0)
+ unless ref $z;
my ($r, $t) = @{$z->polar};
- return (ref $z)->emake($r**(1/3), $t/3);
+ return (ref $z)->emake(exp(log($r)/3), $t/3);
+}
+
+#
+# _rootbad
+#
+# Die on bad root.
+#
+sub _rootbad {
+ my $mess = "Root $_[0] not defined, root must be positive integer.\n";
+
+ my @up = caller(1);
+
+ $mess .= "Died at $up[1] line $up[2].\n";
+
+ die $mess;
}
#
#
sub root {
my ($z, $n) = @_;
- $n = int($n + 0.5);
- return undef unless $n > 0;
+ _rootbad($n) if ($n < 1 or int($n) != $n);
my ($r, $t) = ref $z ? @{$z->polar} : (abs($z), $z >= 0 ? 0 : pi);
my @root;
my $k;
- my $theta_inc = 2 * pi / $n;
+ my $theta_inc = pit2 / $n;
my $rho = $r ** (1/$n);
my $theta;
- my $complex = ref($z) || $package;
+ my $cartesian = ref $z && $z->{c_dirty} == 0;
for ($k = 0, $theta = $t / $n; $k < $n; $k++, $theta += $theta_inc) {
- push(@root, $complex->emake($rho, $theta));
+ my $w = cplxe($rho, $theta);
+ # Yes, $cartesian is loop invariant.
+ push @root, $cartesian ? cplx(@{$w->cartesian}) : $w;
}
return @root;
}
#
# Re
#
-# Return Re(z).
+# Return or set Re(z).
#
sub Re {
- my ($z) = @_;
+ my ($z, $Re) = @_;
return $z unless ref $z;
- my ($re, $im) = @{$z->cartesian};
- return $re;
+ if (defined $Re) {
+ $z->{'cartesian'} = [ $Re, ${$z->cartesian}[1] ];
+ $z->{c_dirty} = 0;
+ $z->{p_dirty} = 1;
+ } else {
+ return ${$z->cartesian}[0];
+ }
}
#
# Im
#
-# Return Im(z).
+# Return or set Im(z).
#
sub Im {
- my ($z) = @_;
- return 0 unless ref $z;
- my ($re, $im) = @{$z->cartesian};
- return $im;
+ my ($z, $Im) = @_;
+ return $z unless ref $z;
+ if (defined $Im) {
+ $z->{'cartesian'} = [ ${$z->cartesian}[0], $Im ];
+ $z->{c_dirty} = 0;
+ $z->{p_dirty} = 1;
+ } else {
+ return ${$z->cartesian}[1];
+ }
+}
+
+#
+# rho
+#
+# Return or set rho(w).
+#
+sub rho {
+ Math::Complex::abs(@_);
+}
+
+#
+# theta
+#
+# Return or set theta(w).
+#
+sub theta {
+ Math::Complex::arg(@_);
}
#
#
sub exp {
my ($z) = @_;
- $z = cplx($z, 0) unless ref $z;
my ($x, $y) = @{$z->cartesian};
return (ref $z)->emake(exp($x), $y);
}
#
+# _logofzero
+#
+# Die on logarithm of zero.
+#
+sub _logofzero {
+ my $mess = "$_[0]: Logarithm of zero.\n";
+
+ if (defined $_[1]) {
+ $mess .= "(Because in the definition of $_[0], the argument ";
+ $mess .= "$_[1] " unless ($_[1] eq '0');
+ $mess .= "is 0)\n";
+ }
+
+ my @up = caller(1);
+
+ $mess .= "Died at $up[1] line $up[2].\n";
+
+ die $mess;
+}
+
+#
# (log)
#
# Compute log(z).
#
sub log {
my ($z) = @_;
- $z = cplx($z, 0) unless ref $z;
- my ($x, $y) = @{$z->cartesian};
+ unless (ref $z) {
+ _logofzero("log") if $z == 0;
+ return $z > 0 ? log($z) : cplx(log(-$z), pi);
+ }
my ($r, $t) = @{$z->polar};
- $t -= 2 * pi if ($t > pi() and $x < 0);
- $t += 2 * pi if ($t < -pi() and $x < 0);
+ _logofzero("log") if $r == 0;
+ if ($t > pi()) { $t -= pit2 }
+ elsif ($t <= -pi()) { $t += pit2 }
return (ref $z)->make(log($r), $t);
}
#
sub log10 {
- my ($z) = @_;
-
- return log(cplx($z, 0)) * log10inv unless ref $z;
- my ($r, $t) = @{$z->polar};
- return (ref $z)->make(log($r) * log10inv, $t * log10inv);
+ return Math::Complex::log($_[0]) * uplog10;
}
#
#
sub cos {
my ($z) = @_;
- $z = cplx($z, 0) unless ref $z;
my ($x, $y) = @{$z->cartesian};
my $ey = exp($y);
my $ey_1 = 1 / $ey;
#
sub sin {
my ($z) = @_;
- $z = cplx($z, 0) unless ref $z;
my ($x, $y) = @{$z->cartesian};
my $ey = exp($y);
my $ey_1 = 1 / $ey;
sub tan {
my ($z) = @_;
my $cz = cos($z);
- divbyzero "tan($z)", "cos($z)" if ($cz == 0);
+ _divbyzero "tan($z)", "cos($z)" if (abs($cz) < $eps);
return sin($z) / $cz;
}
sub sec {
my ($z) = @_;
my $cz = cos($z);
- divbyzero "sec($z)", "cos($z)" if ($cz == 0);
+ _divbyzero "sec($z)", "cos($z)" if ($cz == 0);
return 1 / $cz;
}
sub csc {
my ($z) = @_;
my $sz = sin($z);
- divbyzero "csc($z)", "sin($z)" if ($sz == 0);
+ _divbyzero "csc($z)", "sin($z)" if ($sz == 0);
return 1 / $sz;
}
#
# cot
#
-# Computes cot(z) = 1 / tan(z).
+# Computes cot(z) = cos(z) / sin(z).
#
sub cot {
my ($z) = @_;
my $sz = sin($z);
- divbyzero "cot($z)", "sin($z)" if ($sz == 0);
+ _divbyzero "cot($z)", "sin($z)" if ($sz == 0);
return cos($z) / $sz;
}
# Computes the arc cosine acos(z) = -i log(z + sqrt(z*z-1)).
#
sub acos {
- my ($z) = @_;
- $z = cplx($z, 0) unless ref $z;
- return ~i * log($z + (Re($z) * Im($z) > 0 ? 1 : -1) * sqrt($z*$z - 1));
+ my $z = $_[0];
+ return atan2(sqrt(1-$z*$z), $z) if (! ref $z) && abs($z) <= 1;
+ my ($x, $y) = ref $z ? @{$z->cartesian} : ($z, 0);
+ my $t1 = sqrt(($x+1)*($x+1) + $y*$y);
+ my $t2 = sqrt(($x-1)*($x-1) + $y*$y);
+ my $alpha = ($t1 + $t2)/2;
+ my $beta = ($t1 - $t2)/2;
+ $alpha = 1 if $alpha < 1;
+ if ($beta > 1) { $beta = 1 }
+ elsif ($beta < -1) { $beta = -1 }
+ my $u = atan2(sqrt(1-$beta*$beta), $beta);
+ my $v = log($alpha + sqrt($alpha*$alpha-1));
+ $v = -$v if $y > 0 || ($y == 0 && $x < -1);
+ return $package->make($u, $v);
}
#
# Computes the arc sine asin(z) = -i log(iz + sqrt(1-z*z)).
#
sub asin {
- my ($z) = @_;
- $z = cplx($z, 0) unless ref $z;
- return ~i * log(i * $z + sqrt(1 - $z*$z));
+ my $z = $_[0];
+ return atan2($z, sqrt(1-$z*$z)) if (! ref $z) && abs($z) <= 1;
+ my ($x, $y) = ref $z ? @{$z->cartesian} : ($z, 0);
+ my $t1 = sqrt(($x+1)*($x+1) + $y*$y);
+ my $t2 = sqrt(($x-1)*($x-1) + $y*$y);
+ my $alpha = ($t1 + $t2)/2;
+ my $beta = ($t1 - $t2)/2;
+ $alpha = 1 if $alpha < 1;
+ if ($beta > 1) { $beta = 1 }
+ elsif ($beta < -1) { $beta = -1 }
+ my $u = atan2($beta, sqrt(1-$beta*$beta));
+ my $v = -log($alpha + sqrt($alpha*$alpha-1));
+ $v = -$v if $y > 0 || ($y == 0 && $x < -1);
+ return $package->make($u, $v);
}
#
#
sub atan {
my ($z) = @_;
- $z = cplx($z, 0) unless ref $z;
- divbyzero "atan($z)", "i - $z" if ($z == i);
- return i/2*log((i + $z) / (i - $z));
+ return atan2($z, 1) unless ref $z;
+ _divbyzero "atan(i)" if ( $z == i);
+ _divbyzero "atan(-i)" if (-$z == i);
+ my $log = log((i + $z) / (i - $z));
+ $ip2 = 0.5 * i unless defined $ip2;
+ return $ip2 * $log;
}
#
#
sub asec {
my ($z) = @_;
- divbyzero "asec($z)", $z if ($z == 0);
+ _divbyzero "asec($z)", $z if ($z == 0);
return acos(1 / $z);
}
#
# acsc
#
-# Computes the arc cosecant sec(z) = asin(1 / z).
+# Computes the arc cosecant acsc(z) = asin(1 / z).
#
sub acsc {
my ($z) = @_;
- divbyzero "acsc($z)", $z if ($z == 0);
+ _divbyzero "acsc($z)", $z if ($z == 0);
return asin(1 / $z);
}
#
# acot
#
-# Computes the arc cotangent acot(z) = -i/2 log((i+z) / (z-i))
+# Computes the arc cotangent acot(z) = atan(1 / z)
#
sub acot {
my ($z) = @_;
- $z = cplx($z, 0) unless ref $z;
- divbyzero "acot($z)", "$z - i" if ($z == i);
- return i/-2 * log((i + $z) / ($z - i));
+ _divbyzero "acot(0)" if (abs($z) < $eps);
+ return ($z >= 0) ? atan2(1, $z) : atan2(-1, -$z) unless ref $z;
+ _divbyzero "acot(i)" if (abs($z - i) < $eps);
+ _logofzero "acot(-i)" if (abs($z + i) < $eps);
+ return atan(1 / $z);
}
#
#
sub cosh {
my ($z) = @_;
- my ($x, $y) = ref $z ? @{$z->cartesian} : ($z, 0);
- my $ex = exp($x);
+ my $ex;
+ unless (ref $z) {
+ $ex = exp($z);
+ return ($ex + 1/$ex)/2;
+ }
+ my ($x, $y) = @{$z->cartesian};
+ $ex = exp($x);
my $ex_1 = 1 / $ex;
- return ($ex + $ex_1)/2 unless ref $z;
return (ref $z)->make(cos($y) * ($ex + $ex_1)/2,
sin($y) * ($ex - $ex_1)/2);
}
#
sub sinh {
my ($z) = @_;
- my ($x, $y) = ref $z ? @{$z->cartesian} : ($z, 0);
- my $ex = exp($x);
+ my $ex;
+ unless (ref $z) {
+ $ex = exp($z);
+ return ($ex - 1/$ex)/2;
+ }
+ my ($x, $y) = @{$z->cartesian};
+ $ex = exp($x);
my $ex_1 = 1 / $ex;
- return ($ex - $ex_1)/2 unless ref $z;
return (ref $z)->make(cos($y) * ($ex - $ex_1)/2,
sin($y) * ($ex + $ex_1)/2);
}
sub tanh {
my ($z) = @_;
my $cz = cosh($z);
- divbyzero "tanh($z)", "cosh($z)" if ($cz == 0);
+ _divbyzero "tanh($z)", "cosh($z)" if ($cz == 0);
return sinh($z) / $cz;
}
sub sech {
my ($z) = @_;
my $cz = cosh($z);
- divbyzero "sech($z)", "cosh($z)" if ($cz == 0);
+ _divbyzero "sech($z)", "cosh($z)" if ($cz == 0);
return 1 / $cz;
}
sub csch {
my ($z) = @_;
my $sz = sinh($z);
- divbyzero "csch($z)", "sinh($z)" if ($sz == 0);
+ _divbyzero "csch($z)", "sinh($z)" if ($sz == 0);
return 1 / $sz;
}
sub coth {
my ($z) = @_;
my $sz = sinh($z);
- divbyzero "coth($z)", "sinh($z)" if ($sz == 0);
+ _divbyzero "coth($z)", "sinh($z)" if ($sz == 0);
return cosh($z) / $sz;
}
#
sub acosh {
my ($z) = @_;
- $z = cplx($z, 0) unless ref $z;
+ unless (ref $z) {
+ return log($z + sqrt($z*$z-1)) if $z >= 1;
+ $z = cplx($z, 0);
+ }
+ my ($re, $im) = @{$z->cartesian};
+ if ($im == 0) {
+ return cplx(log($re + sqrt($re*$re - 1)), 0) if $re >= 1;
+ return cplx(0, atan2(sqrt(1-$re*$re), $re)) if abs($re) <= 1;
+ }
return log($z + sqrt($z*$z - 1));
}
#
sub asinh {
my ($z) = @_;
- $z = cplx($z, 0) unless ref $z;
return log($z + sqrt($z*$z + 1));
}
#
sub atanh {
my ($z) = @_;
- divbyzero 'atanh(1)', "1 - $z" if ($z == 1);
- $z = cplx($z, 0) unless ref $z;
- my $cz = (1 + $z) / (1 - $z);
- return log($cz) / 2;
+ unless (ref $z) {
+ return log((1 + $z)/(1 - $z))/2 if abs($z) < 1;
+ $z = cplx($z, 0);
+ }
+ _divbyzero 'atanh(1)', "1 - $z" if ($z == 1);
+ _logofzero 'atanh(-1)' if ($z == -1);
+ return 0.5 * log((1 + $z) / (1 - $z));
}
#
#
sub asech {
my ($z) = @_;
- divbyzero 'asech(0)', $z if ($z == 0);
+ _divbyzero 'asech(0)', $z if ($z == 0);
return acosh(1 / $z);
}
#
sub acsch {
my ($z) = @_;
- divbyzero 'acsch(0)', $z if ($z == 0);
+ _divbyzero 'acsch(0)', $z if ($z == 0);
return asinh(1 / $z);
}
#
sub acoth {
my ($z) = @_;
- divbyzero 'acoth(1)', "$z - 1" if ($z == 1);
- $z = cplx($z, 0) unless ref $z;
- my $cz = (1 + $z) / ($z - 1);
- return log($cz) / 2;
+ _divbyzero 'acoth(0)' if (abs($z) < $eps);
+ unless (ref $z) {
+ return log(($z + 1)/($z - 1))/2 if abs($z) > 1;
+ $z = cplx($z, 0);
+ }
+ _divbyzero 'acoth(1)', "$z - 1" if (abs($z - 1) < $eps);
+ _logofzero 'acoth(-1)', "1 / $z" if (abs($z + 1) < $eps);
+ return log((1 + $z) / ($z - 1)) / 2;
}
#
#
sub atan2 {
my ($z1, $z2, $inverted) = @_;
- my ($re1, $im1) = ref $z1 ? @{$z1->cartesian} : ($z1, 0);
- my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0);
- my $tan;
- if (defined $inverted && $inverted) { # atan(z2/z1)
- return pi * ($re2 > 0 ? 1 : -1) if $re1 == 0 && $im1 == 0;
- $tan = $z2 / $z1;
+ my ($re1, $im1, $re2, $im2);
+ if ($inverted) {
+ ($re1, $im1) = ref $z2 ? @{$z2->cartesian} : ($z2, 0);
+ ($re2, $im2) = @{$z1->cartesian};
} else {
- return pi * ($re1 > 0 ? 1 : -1) if $re2 == 0 && $im2 == 0;
- $tan = $z1 / $z2;
+ ($re1, $im1) = @{$z1->cartesian};
+ ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0);
}
- return atan($tan);
+ if ($im2 == 0) {
+ return cplx(atan2($re1, $re2), 0) if $im1 == 0;
+ return cplx(($im1<=>0) * pip2, 0) if $re2 == 0;
+ }
+ my $w = atan($z1/$z2);
+ my ($u, $v) = ref $w ? @{$w->cartesian} : ($w, 0);
+ $u += pi if $re2 < 0;
+ $u -= pit2 if $u > pi;
+ return cplx($u, $v);
}
#
# ->display_format
#
# Set (fetch if no argument) display format for all complex numbers that
-# don't happen to have overrriden it via ->display_format
+# don't happen to have overridden it via ->display_format
#
# When called as a method, this actually sets the display format for
# the current object.
my ($x, $y) = @{$z->cartesian};
my ($re, $im);
- $x = int($x + ($x < 0 ? -1 : 1) * 1e-14)
- if int(abs($x)) != int(abs($x) + 1e-14);
- $y = int($y + ($y < 0 ? -1 : 1) * 1e-14)
- if int(abs($y)) != int(abs($y) + 1e-14);
+ $x = int($x + ($x < 0 ? -1 : 1) * $eps)
+ if int(abs($x)) != int(abs($x) + $eps);
+ $y = int($y + ($y < 0 ? -1 : 1) * $eps)
+ if int(abs($y)) != int(abs($y) + $eps);
- $re = "$x" if abs($x) >= 1e-14;
- if ($y == 1) { $im = 'i' }
- elsif ($y == -1) { $im = '-i' }
- elsif (abs($y) >= 1e-14) { $im = $y . "i" }
+ $re = "$x" if abs($x) >= $eps;
+ if ($y == 1) { $im = 'i' }
+ elsif ($y == -1) { $im = '-i' }
+ elsif (abs($y) >= $eps) { $im = $y . "i" }
my $str = '';
$str = $re if defined $re;
$str .= "+$im" if defined $im;
$str =~ s/\+-/-/;
$str =~ s/^\+//;
+ $str =~ s/([-+])1i/$1i/; # Not redundant with the above 1/-1 tests.
$str = '0' unless $str;
return $str;
}
+
+# Helper for stringify_polar, a Greatest Common Divisor with a memory.
+
+sub _gcd {
+ my ($a, $b) = @_;
+
+ use integer;
+
+ # Loops forever if given negative inputs.
+
+ if ($b and $a > $b) { return gcd($a % $b, $b) }
+ elsif ($a and $b > $a) { return gcd($b % $a, $a) }
+ else { return $a ? $a : $b }
+}
+
+my %gcd;
+
+sub gcd {
+ my ($a, $b) = @_;
+
+ my $id = "$a $b";
+
+ unless (exists $gcd{$id}) {
+ $gcd{$id} = _gcd($a, $b);
+ $gcd{"$b $a"} = $gcd{$id};
+ }
+
+ return $gcd{$id};
+}
+
#
# ->stringify_polar
#
my $z = shift;
my ($r, $t) = @{$z->polar};
my $theta;
- my $eps = 1e-14;
return '[0,0]' if $r <= $eps;
- my $tpi = 2 * pi;
- my $nt = $t / $tpi;
- $nt = ($nt - int($nt)) * $tpi;
- $nt += $tpi if $nt < 0; # Range [0, 2pi]
+ my $nt = $t / pit2;
+ $nt = ($nt - int($nt)) * pit2;
+ $nt += pit2 if $nt < 0; # Range [0, 2pi]
if (abs($nt) <= $eps) { $theta = 0 }
elsif (abs(pi-$nt) <= $eps) { $theta = 'pi' }
# Okay, number is not a real. Try to identify pi/n and friends...
#
- $nt -= $tpi if $nt > pi;
- my ($n, $k, $kpi);
-
- for ($k = 1, $kpi = pi; $k < 10; $k++, $kpi += pi) {
+ $nt -= pit2 if $nt > pi;
+
+ if (abs($nt) >= deg1) {
+ my ($n, $k, $kpi);
+
+ for ($k = 1, $kpi = pi; $k < 10; $k++, $kpi += pi) {
$n = int($kpi / $nt + ($nt > 0 ? 1 : -1) * 0.5);
if (abs($kpi/$n - $nt) <= $eps) {
- $theta = ($nt < 0 ? '-':'').
- ($k == 1 ? 'pi':"${k}pi").'/'.abs($n);
- last;
+ $n = abs $n;
+ my $gcd = gcd($k, $n);
+ if ($gcd > 1) {
+ $k /= $gcd;
+ $n /= $gcd;
+ }
+ next if $n > 360;
+ $theta = ($nt < 0 ? '-':'').
+ ($k == 1 ? 'pi':"${k}pi");
+ $theta .= '/'.$n if $n > 1;
+ last;
}
+ }
}
$theta = $nt unless defined $theta;
=head1 SYNOPSIS
use Math::Complex;
-
+
$z = Math::Complex->make(5, 6);
$t = 4 - 3*i + $z;
$j = cplxe(1, 2*pi/3);
which is also expressed by this formula:
- z = rho * exp(i * theta) = rho * (cos theta + i * sin theta)
+ z = rho * exp(i * theta) = rho * (cos theta + i * sin theta)
In other words, it's the projection of the vector onto the I<x> and I<y>
axes. Mathematicians call I<rho> the I<norm> or I<modulus> and I<theta>
The polar notation (also known as the trigonometric
representation) is much more handy for performing multiplications and
divisions of complex numbers, whilst the cartesian notation is better
-suited for additions and substractions. Real numbers are on the I<x>
-axis, and therefore I<theta> is zero.
+suited for additions and subtractions. Real numbers are on the I<x>
+axis, and therefore I<theta> is zero or I<pi>.
All the common operations that can be performed on a real number have
been defined to work on complex numbers as well, and are merely
the number is within their definition set.
For instance, the C<sqrt> routine which computes the square root of
-its argument is only defined for positive real numbers and yields a
-positive real number (it is an application from B<R+> to B<R+>).
+its argument is only defined for non-negative real numbers and yields a
+non-negative real number (it is an application from B<R+> to B<R+>).
If we allow it to return a complex number, then it can be extended to
negative real numbers to become an application from B<R> to B<C> (the
set of complex numbers):
sqrt(z = [r,t]) = sqrt(r) * exp(i * t/2)
-Indeed, a negative real number can be noted C<[x,pi]>
-(the modulus I<x> is always positive, so C<[x,pi]> is really C<-x>, a
-negative number)
-and the above definition states that
+Indeed, a negative real number can be noted C<[x,pi]> (the modulus
+I<x> is always non-negative, so C<[x,pi]> is really C<-x>, a negative
+number) and the above definition states that
sqrt([x,pi]) = sqrt(x) * exp(i*pi/2) = [sqrt(x),pi/2] = sqrt(x)*i
which is exactly what we had defined for negative real numbers above.
+The C<sqrt> returns only one of the solutions: if you want the both,
+use the C<root> function.
All the common mathematical functions defined on real numbers that
are extended to complex numbers share that same property of working
z1 * z2 = (r1 * r2) * exp(i * (t1 + t2))
z1 / z2 = (r1 / r2) * exp(i * (t1 - t2))
z1 ** z2 = exp(z2 * log z1)
- ~z1 = a - bi
- abs(z1) = r1 = sqrt(a*a + b*b)
- sqrt(z1) = sqrt(r1) * exp(i * t1/2)
- exp(z1) = exp(a) * exp(i * b)
- log(z1) = log(r1) + i*t1
- sin(z1) = 1/2i (exp(i * z1) - exp(-i * z1))
- cos(z1) = 1/2 (exp(i * z1) + exp(-i * z1))
- abs(z1) = r1
+ ~z = a - bi
+ abs(z) = r1 = sqrt(a*a + b*b)
+ sqrt(z) = sqrt(r1) * exp(i * t/2)
+ exp(z) = exp(a) * exp(i * b)
+ log(z) = log(r1) + i*t
+ sin(z) = 1/2i (exp(i * z1) - exp(-i * z))
+ cos(z) = 1/2 (exp(i * z1) + exp(-i * z))
atan2(z1, z2) = atan(z1/z2)
The following extra operations are supported on both real and complex
Re(z) = a
Im(z) = b
arg(z) = t
+ abs(z) = r
cbrt(z) = z ** (1/3)
log10(z) = log(z) / log(10)
cot(z) = 1 / tan(z)
asin(z) = -i * log(i*z + sqrt(1-z*z))
- acos(z) = -i * log(z + sqrt(z*z-1))
+ acos(z) = -i * log(z + i*sqrt(1-z*z))
atan(z) = i/2 * log((i+z) / (i-z))
acsc(z) = asin(1 / z)
asec(z) = acos(1 / z)
- acot(z) = -i/2 * log((i+z) / (z-i))
+ acot(z) = atan(1 / z) = -i/2 * log((i+z) / (z-i))
sinh(z) = 1/2 (exp(z) - exp(-z))
cosh(z) = 1/2 (exp(z) + exp(-z))
csch(z) = 1 / sinh(z)
sech(z) = 1 / cosh(z)
coth(z) = 1 / tanh(z)
-
+
asinh(z) = log(z + sqrt(z*z+1))
acosh(z) = log(z + sqrt(z*z-1))
atanh(z) = 1/2 * log((1+z) / (1-z))
asech(z) = acosh(1 / z)
acoth(z) = atanh(1 / z) = 1/2 * log((1+z) / (z-1))
-I<log>, I<csc>, I<cot>, I<acsc>, I<acot>, I<csch>, I<coth>,
-I<acosech>, I<acotanh>, have aliases I<ln>, I<cosec>, I<cotan>,
-I<acosec>, I<acotan>, I<cosech>, I<cotanh>, I<acosech>, I<acotanh>,
-respectively.
+I<arg>, I<abs>, I<log>, I<csc>, I<cot>, I<acsc>, I<acot>, I<csch>,
+I<coth>, I<acosech>, I<acotanh>, have aliases I<rho>, I<theta>, I<ln>,
+I<cosec>, I<cotan>, I<acosec>, I<acotan>, I<cosech>, I<cotanh>,
+I<acosech>, I<acotanh>, respectively. C<Re>, C<Im>, C<arg>, C<abs>,
+C<rho>, and C<theta> can be used also also mutators. The C<cbrt>
+returns only one of the solutions: if you want all three, use the
+C<root> function.
The I<root> function is available to compute all the I<n>
roots of some complex, where I<n> is a strictly positive integer.
(root(z, n))[k] = r**(1/n) * exp(i * (t + 2*k*pi)/n)
-The I<spaceship> comparison operator is also defined. In order to
-ensure its restriction to real numbers is conform to what you would
-expect, the comparison is run on the real part of the complex number
-first, and imaginary parts are compared only when the real parts
-match.
+The I<spaceship> comparison operator, E<lt>=E<gt>, is also defined. In
+order to ensure its restriction to real numbers is conform to what you
+would expect, the comparison is run on the real part of the complex
+number first, and imaginary parts are compared only when the real
+parts match.
=head1 CREATION
$z = 3 + 4*i;
-if you like. To create a number using the trigonometric form, use either:
+if you like. To create a number using the polar form, use either:
$z = Math::Complex->emake(5, pi/3);
$x = cplxe(5, pi/3);
instead. The first argument is the modulus, the second is the angle
-(in radians, the full circle is 2*pi). (Mnmemonic: C<e> is used as a
-notation for complex numbers in the trigonometric form).
+(in radians, the full circle is 2*pi). (Mnemonic: C<e> is used as a
+notation for complex numbers in the polar form).
It is possible to write:
$x = cplxe(-3, pi/4);
but that will be silently converted into C<[3,-3pi/4]>, since the modulus
-must be positive (it represents the distance to the origin in the complex
+must be non-negative (it represents the distance to the origin in the complex
plane).
+It is also possible to have a complex number as either argument of
+either the C<make> or C<emake>: the appropriate component of
+the argument will be used.
+
+ $z1 = cplx(-2, 1);
+ $z2 = cplx($z1, 4);
+
=head1 STRINGIFICATION
When printed, a complex number is usually shown under its cartesian
$k = exp(i * 2*pi/3);
print "$j - $k = ", $j - $k, "\n";
-=head1 ERRORS DUE TO DIVISION BY ZERO
+ $z->Re(3); # Re, Im, arg, abs,
+ $j->arg(2); # (the last two aka rho, theta)
+ # can be used also as mutators.
+
+=head1 ERRORS DUE TO DIVISION BY ZERO OR LOGARITHM OF ZERO
The division (/) and the following functions
- tan
- sec
- csc
- cot
- asec
- acsc
- atan
- acot
- tanh
- sech
- csch
- coth
- atanh
- asech
- acsch
- acoth
+ log ln log10 logn
+ tan sec csc cot
+ atan asec acsc acot
+ tanh sech csch coth
+ atanh asech acsch acoth
cannot be computed for all arguments because that would mean dividing
-by zero. These situations cause fatal runtime errors looking like this
+by zero or taking logarithm of zero. These situations cause fatal
+runtime errors looking like this
cot(0): Division by zero.
(Because in the definition of cot(0), the divisor sin(0) is 0)
Died at ...
-For the C<csc>, C<cot>, C<asec>, C<acsc>, C<csch>, C<coth>, C<asech>,
-C<acsch>, the argument cannot be C<0> (zero). For the C<atanh>,
-C<acoth>, the argument cannot be C<1> (one). For the C<atan>, C<acot>,
-the argument cannot be C<i> (the imaginary unit). For the C<tan>,
-C<sec>, C<tanh>, C<sech>, the argument cannot be I<pi/2 + k * pi>, where
-I<k> is any integer.
+or
-=head1 BUGS
+ atanh(-1): Logarithm of zero.
+ Died at...
-Saying C<use Math::Complex;> exports many mathematical routines in the
-caller environment and even overrides some (C<sin>, C<cos>, C<sqrt>,
-C<log>, C<exp>). This is construed as a feature by the Authors,
-actually... ;-)
+For the C<csc>, C<cot>, C<asec>, C<acsc>, C<acot>, C<csch>, C<coth>,
+C<asech>, C<acsch>, the argument cannot be C<0> (zero). For the the
+logarithmic functions and the C<atanh>, C<acoth>, the argument cannot
+be C<1> (one). For the C<atanh>, C<acoth>, the argument cannot be
+C<-1> (minus one). For the C<atan>, C<acot>, the argument cannot be
+C<i> (the imaginary unit). For the C<atan>, C<acoth>, the argument
+cannot be C<-i> (the negative imaginary unit). For the C<tan>,
+C<sec>, C<tanh>, the argument cannot be I<pi/2 + k * pi>, where I<k>
+is any integer.
+
+Note that because we are operating on approximations of real numbers,
+these errors can happen when merely `too close' to the singularities
+listed above. For example C<tan(2*atan2(1,1)+1e-15)> will die of
+division by zero.
+
+=head1 ERRORS DUE TO INDIGESTIBLE ARGUMENTS
-The code is not optimized for speed, although we try to use the cartesian
-form for addition-like operators and the trigonometric form for all
-multiplication-like operators.
+The C<make> and C<emake> accept both real and complex arguments.
+When they cannot recognize the arguments they will die with error
+messages like the following
-The arg() routine does not ensure the angle is within the range [-pi,+pi]
-(a side effect caused by multiplication and division using the trigonometric
-representation).
+ Math::Complex::make: Cannot take real part of ...
+ Math::Complex::make: Cannot take real part of ...
+ Math::Complex::emake: Cannot take rho of ...
+ Math::Complex::emake: Cannot take theta of ...
+
+=head1 BUGS
+
+Saying C<use Math::Complex;> exports many mathematical routines in the
+caller environment and even overrides some (C<sqrt>, C<log>).
+This is construed as a feature by the Authors, actually... ;-)
All routines expect to be given real or complex numbers. Don't attempt to
use BigFloat, since Perl has currently no rule to disambiguate a '+'
operation (for instance) between two overloaded entities.
+In Cray UNICOS there is some strange numerical instability that results
+in root(), cos(), sin(), cosh(), sinh(), losing accuracy fast. Beware.
+The bug may be in UNICOS math libs, in UNICOS C compiler, in Math::Complex.
+Whatever it is, it does not manifest itself anywhere else where Perl runs.
+
=head1 AUTHORS
- Raphael Manfredi <F<Raphael_Manfredi@grenoble.hp.com>>
- Jarkko Hietaniemi <F<jhi@iki.fi>>
+Raphael Manfredi <F<Raphael_Manfredi@grenoble.hp.com>> and
+Jarkko Hietaniemi <F<jhi@iki.fi>>.
+
+Extensive patches by Daniel S. Lewart <F<d-lewart@uiuc.edu>>.
=cut
+1;
+
# eof