-package Math::Complex;
+# $RCSFile$
+#
+# Complex numbers and associated mathematical functions
+# -- Raphael Manfredi, Sept 1996
require Exporter;
+package Math::Complex; @ISA = qw(Exporter);
-@ISA = ('Exporter');
-
-# just to make use happy
+@EXPORT = qw(
+ pi i Re Im arg
+ log10 logn cbrt root
+ tan cotan asin acos atan acotan
+ sinh cosh tanh cotanh asinh acosh atanh acotanh
+ cplx cplxe
+);
use overload
- '+' => sub { my($x1,$y1,$x2,$y2) = (@{$_[0]},@{$_[1]});
- bless [ $x1+$x2, $y1+$y2];
- },
-
- '-' => sub { my($x1,$y1,$x2,$y2) = (@{$_[0]},@{$_[1]});
- bless [ $x1-$x2, $y1-$y2];
- },
-
- '*' => sub { my($x1,$y1,$x2,$y2) = (@{$_[0]},@{$_[1]});
- bless [ $x1*$x2-$y1*$y2,$x1*$y2+$x2*$y1];
- },
-
- '/' => sub { my($x1,$y1,$x2,$y2) = (@{$_[0]},@{$_[1]});
- my $q = $x2*$x2+$y2*$y2;
- bless [($x1*$x2+$y1*$y2)/$q, ($y1*$x2-$y2*$x1)/$q];
- },
-
- 'neg' => sub { my($x,$y) = @{$_[0]}; bless [ -$x, -$y];
- },
-
- '~' => sub { my($x,$y) = @{$_[0]}; bless [ $x, -$y];
- },
-
- 'abs' => sub { my($x,$y) = @{$_[0]}; sqrt $x*$x+$y*$y;
- },
-
- 'cos' => sub { my($x,$y) = @{$_[0]};
- my ($ab,$c,$s) = (exp $y, cos $x, sin $x);
- my $abr = 1/(2*$ab); $ab /= 2;
- bless [ ($abr+$ab)*$c, ($abr-$ab)*$s];
- },
-
- 'sin' => sub { my($x,$y) = @{$_[0]};
- my ($ab,$c,$s) = (exp $y, cos $x, sin $x);
- my $abr = 1/(2*$ab); $ab /= 2;
- bless [ (-$abr-$ab)*$s, ($abr-$ab)*$c];
- },
-
- 'exp' => sub { my($x,$y) = @{$_[0]};
- my ($ab,$c,$s) = (exp $x, cos $y, sin $y);
- bless [ $ab*$c, $ab*$s ];
- },
-
- 'sqrt' => sub {
- my($zr,$zi) = @{$_[0]};
- my ($x, $y, $r, $w);
- my $c = new Math::Complex (0,0);
- if (($zr == 0) && ($zi == 0)) {
- # nothing, $c already set
+ '+' => \&plus,
+ '-' => \&minus,
+ '*' => \&multiply,
+ '/' => \÷,
+ '**' => \&power,
+ '<=>' => \&spaceship,
+ 'neg' => \&negate,
+ '~' => \&conjugate,
+ 'abs' => \&abs,
+ 'sqrt' => \&sqrt,
+ 'exp' => \&exp,
+ 'log' => \&log,
+ 'sin' => \&sin,
+ 'cos' => \&cos,
+ 'atan2' => \&atan2,
+ qw("" stringify);
+
+#
+# Package globals
+#
+
+$package = 'Math::Complex'; # Package name
+$display = 'cartesian'; # Default display format
+
+#
+# Object attributes (internal):
+# cartesian [real, imaginary] -- cartesian form
+# polar [rho, theta] -- polar form
+# c_dirty cartesian form not up-to-date
+# p_dirty polar form not up-to-date
+# display display format (package's global when not set)
+#
+
+#
+# ->make
+#
+# Create a new complex number (cartesian form)
+#
+sub make {
+ my $self = bless {}, shift;
+ my ($re, $im) = @_;
+ $self->{cartesian} = [$re, $im];
+ $self->{c_dirty} = 0;
+ $self->{p_dirty} = 1;
+ return $self;
+}
+
+#
+# ->emake
+#
+# Create a new complex number (exponential form)
+#
+sub emake {
+ my $self = bless {}, shift;
+ my ($rho, $theta) = @_;
+ $theta += pi() if $rho < 0;
+ $self->{polar} = [abs($rho), $theta];
+ $self->{p_dirty} = 0;
+ $self->{c_dirty} = 1;
+ return $self;
+}
+
+sub new { &make } # For backward compatibility only.
+
+#
+# cplx
+#
+# Creates a complex number from a (re, im) tuple.
+# This avoids the burden of writing Math::Complex->make(re, im).
+#
+sub cplx {
+ my ($re, $im) = @_;
+ return $package->make($re, $im);
+}
+
+#
+# cplxe
+#
+# Creates a complex number from a (rho, theta) tuple.
+# This avoids the burden of writing Math::Complex->emake(rho, theta).
+#
+sub cplxe {
+ my ($rho, $theta) = @_;
+ return $package->emake($rho, $theta);
+}
+
+#
+# pi
+#
+# The number defined as 2 * pi = 360 degrees
+#
+sub pi () {
+ $pi = 4 * atan2(1, 1) unless $pi;
+ return $pi;
+}
+
+#
+# i
+#
+# The number defined as i*i = -1;
+#
+sub i () {
+ $i = bless {} unless $i; # There can be only one i
+ $i->{cartesian} = [0, 1];
+ $i->{polar} = [1, pi/2];
+ $i->{c_dirty} = 0;
+ $i->{p_dirty} = 0;
+ return $i;
+}
+
+#
+# Attribute access/set routines
+#
+
+sub cartesian {$_[0]->{c_dirty} ? $_[0]->update_cartesian : $_[0]->{cartesian}}
+sub polar {$_[0]->{p_dirty} ? $_[0]->update_polar : $_[0]->{polar}}
+
+sub set_cartesian { $_[0]->{p_dirty}++; $_[0]->{cartesian} = $_[1] }
+sub set_polar { $_[0]->{c_dirty}++; $_[0]->{polar} = $_[1] }
+
+#
+# ->update_cartesian
+#
+# Recompute and return the cartesian form, given accurate polar form.
+#
+sub update_cartesian {
+ my $self = shift;
+ my ($r, $t) = @{$self->{polar}};
+ $self->{c_dirty} = 0;
+ return $self->{cartesian} = [$r * cos $t, $r * sin $t];
+}
+
+#
+#
+# ->update_polar
+#
+# Recompute and return the polar form, given accurate cartesian form.
+#
+sub update_polar {
+ my $self = shift;
+ my ($x, $y) = @{$self->{cartesian}};
+ $self->{p_dirty} = 0;
+ return $self->{polar} = [0, 0] if $x == 0 && $y == 0;
+ return $self->{polar} = [sqrt($x*$x + $y*$y), atan2($y, $x)];
+}
+
+#
+# (plus)
+#
+# Computes z1+z2.
+#
+sub plus {
+ my ($z1, $z2, $regular) = @_;
+ my ($re1, $im1) = @{$z1->cartesian};
+ my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2);
+ unless (defined $regular) {
+ $z1->set_cartesian([$re1 + $re2, $im1 + $im2]);
+ return $z1;
+ }
+ return (ref $z1)->make($re1 + $re2, $im1 + $im2);
+}
+
+#
+# (minus)
+#
+# Computes z1-z2.
+#
+sub minus {
+ my ($z1, $z2, $inverted) = @_;
+ my ($re1, $im1) = @{$z1->cartesian};
+ my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2);
+ 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 - $im2);
+}
+
+#
+# (multiply)
+#
+# Computes z1*z2.
+#
+sub multiply {
+ my ($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]);
+ return $z1;
+ }
+ return (ref $z1)->emake($r1 * $r2, $t1 + $t2);
+}
+
+#
+# (divide)
+#
+# Computes z1/z2.
+#
+sub divide {
+ my ($z1, $z2, $inverted) = @_;
+ my ($r1, $t1) = @{$z1->polar};
+ my ($r2, $t2) = ref $z2 ? @{$z2->polar} : (abs($z2), $z2 >= 0 ? 0 : pi);
+ unless (defined $inverted) {
+ $z1->set_polar([$r1 / $r2, $t1 - $t2]);
+ return $z1;
+ }
+ return $inverted ?
+ (ref $z1)->emake($r2 / $r1, $t2 - $t1) :
+ (ref $z1)->emake($r1 / $r2, $t1 - $t2);
+}
+
+#
+# (power)
+#
+# Computes z1**z2 = exp(z2 * log z1)).
+#
+sub power {
+ my ($z1, $z2, $inverted) = @_;
+ return exp($z1 * log $z2) if defined $inverted && $inverted;
+ return exp($z2 * log $z1);
+}
+
+#
+# (spaceship)
+#
+# Computes z1 <=> z2.
+# Sorts on the real part first, then on the imaginary part. Thus 2-4i > 3+8i.
+#
+sub spaceship {
+ my ($z1, $z2, $inverted) = @_;
+ my ($re1, $im1) = @{$z1->cartesian};
+ my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2);
+ my $sgn = $inverted ? -1 : 1;
+ return $sgn * ($re1 <=> $re2) if $re1 != $re2;
+ return $sgn * ($im1 <=> $im2);
+}
+
+#
+# (negate)
+#
+# Computes -z.
+#
+sub negate {
+ my ($z) = @_;
+ if ($z->{c_dirty}) {
+ my ($r, $t) = @{$z->polar};
+ return (ref $z)->emake($r, pi + $t);
+ }
+ my ($re, $im) = @{$z->cartesian};
+ return (ref $z)->make(-$re, -$im);
+}
+
+#
+# (conjugate)
+#
+# Compute complex's conjugate.
+#
+sub conjugate {
+ my ($z) = @_;
+ if ($z->{c_dirty}) {
+ my ($r, $t) = @{$z->polar};
+ return (ref $z)->emake($r, -$t);
+ }
+ my ($re, $im) = @{$z->cartesian};
+ return (ref $z)->make($re, -$im);
+}
+
+#
+# (abs)
+#
+# Compute complex's norm (rho).
+#
+sub abs {
+ my ($z) = @_;
+ my ($r, $t) = @{$z->polar};
+ return abs($r);
+}
+
+#
+# arg
+#
+# Compute complex's argument (theta).
+#
+sub arg {
+ my ($z) = @_;
+ return 0 unless ref $z;
+ my ($r, $t) = @{$z->polar};
+ return $t;
+}
+
+#
+# (sqrt)
+#
+# Compute sqrt(z) (positive only).
+#
+sub sqrt {
+ my ($z) = @_;
+ my ($r, $t) = @{$z->polar};
+ return (ref $z)->emake(sqrt($r), $t/2);
+}
+
+#
+# cbrt
+#
+# Compute cbrt(z) (cubic root, primary only).
+#
+sub cbrt {
+ my ($z) = @_;
+ return $z ** (1/3) unless ref $z;
+ my ($r, $t) = @{$z->polar};
+ return (ref $z)->emake($r**(1/3), $t/3);
+}
+
+#
+# root
+#
+# Computes all nth root for z, returning an array whose size is n.
+# `n' must be a positive integer.
+#
+# The roots are given by (for k = 0..n-1):
+#
+# z^(1/n) = r^(1/n) (cos ((t+2 k pi)/n) + i sin ((t+2 k pi)/n))
+#
+sub root {
+ my ($z, $n) = @_;
+ $n = int($n + 0.5);
+ return undef unless $n > 0;
+ my ($r, $t) = ref $z ? @{$z->polar} : (abs($z), $z >= 0 ? 0 : pi);
+ my @root;
+ my $k;
+ my $theta_inc = 2 * pi / $n;
+ my $rho = $r ** (1/$n);
+ my $theta;
+ my $complex = ref($z) || $package;
+ for ($k = 0, $theta = $t / $n; $k < $n; $k++, $theta += $theta_inc) {
+ push(@root, $complex->emake($rho, $theta));
}
- else {
- $x = abs($zr);
- $y = abs($zi);
- if ($x >= $y) {
- $r = $y/$x;
- $w = sqrt(0.5 * $x * (1.0+sqrt(1.0+$r*$r)));
- }
- else {
- $r = $x/$y;
- $w = sqrt(0.5 * ($x + $y*sqrt(1.0+$r*$r)));
- }
- if ( $zr >= 0) {
- @$c = ($w, $zi/(2 * $w) );
- }
- else {
- $c->[1] = ($zi >= 0) ? $w : -$w;
- $c->[0] = $zi/(2.0* $c->[1]);
- }
- }
- return $c;
- },
-
- qw("" stringify)
-;
-
-sub new {
- my $class = shift;
- my @C = @_;
- bless \@C, $class;
+ return @root;
}
+#
+# Re
+#
+# Return Re(z).
+#
sub Re {
- my($x,$y) = @{$_[0]};
- $x;
+ my ($z) = @_;
+ return $z unless ref $z;
+ my ($re, $im) = @{$z->cartesian};
+ return $re;
}
+#
+# Im
+#
+# Return Im(z).
+#
sub Im {
- my($x,$y) = @{$_[0]};
- $y;
+ my ($z) = @_;
+ return 0 unless ref $z;
+ my ($re, $im) = @{$z->cartesian};
+ return $im;
}
-sub arg {
- my($x,$y) = @{$_[0]};
- atan2($y,$x);
+#
+# (exp)
+#
+# Computes exp(z).
+#
+sub exp {
+ my ($z) = @_;
+ my ($x, $y) = @{$z->cartesian};
+ return (ref $z)->emake(exp($x), $y);
+}
+
+#
+# (log)
+#
+# Compute log(z).
+#
+sub log {
+ my ($z) = @_;
+ my ($r, $t) = @{$z->polar};
+ return (ref $z)->make(log($r), $t);
+}
+
+#
+# log10
+#
+# Compute log10(z).
+#
+sub log10 {
+ my ($z) = @_;
+ $log10 = log(10) unless defined $log10;
+ return log($z) / $log10 unless ref $z;
+ my ($r, $t) = @{$z->polar};
+ return (ref $z)->make(log($r) / $log10, $t / $log10);
+}
+
+#
+# logn
+#
+# Compute logn(z,n) = log(z) / log(n)
+#
+sub logn {
+ my ($z, $n) = @_;
+ my $logn = $logn{$n};
+ $logn = $logn{$n} = log($n) unless defined $logn; # Cache log(n)
+ return log($z) / log($n);
+}
+
+#
+# (cos)
+#
+# Compute cos(z) = (exp(iz) + exp(-iz))/2.
+#
+sub cos {
+ my ($z) = @_;
+ my ($x, $y) = @{$z->cartesian};
+ my $ey = exp($y);
+ my $ey_1 = 1 / $ey;
+ return (ref $z)->make(cos($x) * ($ey + $ey_1)/2, sin($x) * ($ey_1 - $ey)/2);
+}
+
+#
+# (sin)
+#
+# Compute sin(z) = (exp(iz) - exp(-iz))/2.
+#
+sub sin {
+ my ($z) = @_;
+ my ($x, $y) = @{$z->cartesian};
+ my $ey = exp($y);
+ my $ey_1 = 1 / $ey;
+ return (ref $z)->make(sin($x) * ($ey + $ey_1)/2, cos($x) * ($ey - $ey_1)/2);
+}
+
+#
+# tan
+#
+# Compute tan(z) = sin(z) / cos(z).
+#
+sub tan {
+ my ($z) = @_;
+ return sin($z) / cos($z);
+}
+
+#
+# cotan
+#
+# Computes cotan(z) = 1 / tan(z).
+#
+sub cotan {
+ my ($z) = @_;
+ return cos($z) / sin($z);
+}
+
+#
+# acos
+#
+# Computes the arc cosine acos(z) = -i log(z + sqrt(z*z-1)).
+#
+sub acos {
+ my ($z) = @_;
+ my $cz = $z*$z - 1;
+ $cz = cplx($cz, 0) if !ref $cz && $cz < 0; # Force complex if <0
+ return ~i * log($z + sqrt $cz); # ~i is -i
+}
+
+#
+# asin
+#
+# Computes the arc sine asin(z) = -i log(iz + sqrt(1-z*z)).
+#
+sub asin {
+ my ($z) = @_;
+ my $cz = 1 - $z*$z;
+ $cz = cplx($cz, 0) if !ref $cz && $cz < 0; # Force complex if <0
+ return ~i * log(i * $z + sqrt $cz); # ~i is -i
+}
+
+#
+# atan
+#
+# Computes the arc tagent atan(z) = i/2 log((i+z) / (i-z)).
+#
+sub atan {
+ my ($z) = @_;
+ return i/2 * log((i + $z) / (i - $z));
}
+#
+# acotan
+#
+# Computes the arc cotangent acotan(z) = -i/2 log((i+z) / (z-i))
+#
+sub acotan {
+ my ($z) = @_;
+ return i/-2 * log((i + $z) / ($z - i));
+}
+
+#
+# cosh
+#
+# Computes the hyperbolic cosine cosh(z) = (exp(z) + exp(-z))/2.
+#
+sub cosh {
+ my ($z) = @_;
+ my ($x, $y) = ref $z ? @{$z->cartesian} : ($z);
+ my $ex = exp($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 - $ex_1)/2);
+}
+
+#
+# sinh
+#
+# Computes the hyperbolic sine sinh(z) = (exp(z) - exp(-z))/2.
+#
+sub sinh {
+ my ($z) = @_;
+ my ($x, $y) = ref $z ? @{$z->cartesian} : ($z);
+ my $ex = exp($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 + $ex_1)/2);
+}
+
+#
+# tanh
+#
+# Computes the hyperbolic tangent tanh(z) = sinh(z) / cosh(z).
+#
+sub tanh {
+ my ($z) = @_;
+ return sinh($z) / cosh($z);
+}
+
+#
+# cotanh
+#
+# Comptutes the hyperbolic cotangent cotanh(z) = cosh(z) / sinh(z).
+#
+sub cotanh {
+ my ($z) = @_;
+ return cosh($z) / sinh($z);
+}
+
+#
+# acosh
+#
+# Computes the arc hyperbolic cosine acosh(z) = log(z + sqrt(z*z-1)).
+#
+sub acosh {
+ my ($z) = @_;
+ my $cz = $z*$z - 1;
+ $cz = cplx($cz, 0) if !ref $cz && $cz < 0; # Force complex if <0
+ return log($z + sqrt $cz);
+}
+
+#
+# asinh
+#
+# Computes the arc hyperbolic sine asinh(z) = log(z + sqrt(z*z-1))
+#
+sub asinh {
+ my ($z) = @_;
+ my $cz = $z*$z + 1; # Already complex if <0
+ return log($z + sqrt $cz);
+}
+
+#
+# atanh
+#
+# Computes the arc hyperbolic tangent atanh(z) = 1/2 log((1+z) / (1-z)).
+#
+sub atanh {
+ my ($z) = @_;
+ my $cz = (1 + $z) / (1 - $z);
+ $cz = cplx($cz, 0) if !ref $cz && $cz < 0; # Force complex if <0
+ return log($cz) / 2;
+}
+
+#
+# acotanh
+#
+# Computes the arc hyperbolic cotangent acotanh(z) = 1/2 log((1+z) / (z-1)).
+#
+sub acotanh {
+ my ($z) = @_;
+ my $cz = (1 + $z) / ($z - 1);
+ $cz = cplx($cz, 0) if !ref $cz && $cz < 0; # Force complex if <0
+ return log($cz) / 2;
+}
+
+#
+# (atan2)
+#
+# Compute atan(z1/z2).
+#
+sub atan2 {
+ my ($z1, $z2, $inverted) = @_;
+ my ($re1, $im1) = @{$z1->cartesian};
+ my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2);
+ my $tan;
+ if (defined $inverted && $inverted) { # atan(z2/z1)
+ return pi * ($re2 > 0 ? 1 : -1) if $re1 == 0 && $im1 == 0;
+ $tan = $z2 / $z1;
+ } else {
+ return pi * ($re1 > 0 ? 1 : -1) if $re2 == 0 && $im2 == 0;
+ $tan = $z1 / $z2;
+ }
+ return atan($tan);
+}
+
+#
+# display_format
+# ->display_format
+#
+# Set (fetch if no argument) display format for all complex numbers that
+# don't happen to have overrriden it via ->display_format
+#
+# When called as a method, this actually sets the display format for
+# the current object.
+#
+# Valid object formats are 'c' and 'p' for cartesian and polar. The first
+# letter is used actually, so the type can be fully spelled out for clarity.
+#
+sub display_format {
+ my $self = shift;
+ my $format = undef;
+
+ if (ref $self) { # Called as a method
+ $format = shift;
+ } else { # Regular procedure call
+ $format = $self;
+ undef $self;
+ }
+
+ if (defined $self) {
+ return defined $self->{display} ? $self->{display} : $display
+ unless defined $format;
+ return $self->{display} = $format;
+ }
+
+ return $display unless defined $format;
+ return $display = $format;
+}
+
+#
+# (stringify)
+#
+# Show nicely formatted complex number under its cartesian or polar form,
+# depending on the current display format:
+#
+# . If a specific display format has been recorded for this object, use it.
+# . Otherwise, use the generic current default for all complex numbers,
+# which is a package global variable.
+#
sub stringify {
- my($x,$y) = @{$_[0]};
- my($re,$im);
+ my ($z) = shift;
+ my $format;
+
+ $format = $display;
+ $format = $z->{display} if defined $z->{display};
+
+ return $z->stringify_polar if $format =~ /^p/i;
+ return $z->stringify_cartesian;
+}
+
+#
+# ->stringify_cartesian
+#
+# Stringify as a cartesian representation 'a+bi'.
+#
+sub stringify_cartesian {
+ my $z = shift;
+ my ($x, $y) = @{$z->cartesian};
+ my ($re, $im);
+
+ $re = "$x" if abs($x) >= 1e-14;
+ if ($y == 1) { $im = 'i' }
+ elsif ($y == -1) { $im = '-i' }
+ elsif (abs($y) >= 1e-14) { $im = "${y}i" }
+
+ my $str;
+ $str = $re if defined $re;
+ $str .= "+$im" if defined $im;
+ $str =~ s/\+-/-/;
+ $str =~ s/^\+//;
+ $str = '0' unless $str;
+
+ return $str;
+}
+
+#
+# ->stringify_polar
+#
+# Stringify as a polar representation '[r,t]'.
+#
+sub stringify_polar {
+ my $z = shift;
+ my ($r, $t) = @{$z->polar};
+ my $theta;
+
+ return '[0,0]' if $r <= 1e-14;
- $re = $x if ($x);
- if ($y == 1) {$im = 'i';}
- elsif ($y == -1){$im = '-i';}
- elsif ($y) {$im = $y . 'i'; }
+ my $tpi = 2 * pi;
+ my $nt = $t / $tpi;
+ $nt = ($nt - int($nt)) * $tpi;
+ $nt += $tpi if $nt < 0; # Range [0, 2pi]
- local $_ = $re.'+'.$im;
- s/\+-/-/;
- s/^\+//;
- s/[\+-]$//;
- $_ = 0 if ($_ eq '');
- return $_;
+ if (abs($nt) <= 1e-14) { $theta = 0 }
+ elsif (abs(pi-$nt) <= 1e-14) { $theta = 'pi' }
+
+ return "\[$r,$theta\]" if defined $theta;
+
+ #
+ # 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) {
+ $n = int($kpi / $nt + ($nt > 0 ? 1 : -1) * 0.5);
+ if (abs($kpi/$n - $nt) <= 1e-14) {
+ $theta = ($nt < 0 ? '-':'').($k == 1 ? 'pi':"${k}pi").'/'.abs($n);
+ last;
+ }
+ }
+
+ $theta = $nt unless defined $theta;
+
+ return "\[$r,$theta\]";
}
1;
=head1 NAME
-Math::Complex - complex numbers package
+Math::Complex - complex numbers and associated mathematical functions
=head1 SYNOPSIS
- use Math::Complex;
- $i = new Math::Complex;
+ use Math::Complex;
+ $z = Math::Complex->make(5, 6);
+ $t = 4 - 3*i + $z;
+ $j = cplxe(1, 2*pi/3);
=head1 DESCRIPTION
-Complex numbers declared as
+This package lets you create and manipulate complex numbers. By default,
+I<Perl> limits itself to real numbers, but an extra C<use> statement brings
+full complex support, along with a full set of mathematical functions
+typically associated with and/or extended to complex numbers.
+
+If you wonder what complex numbers are, they were invented to be able to solve
+the following equation:
+
+ x*x = -1
+
+and by definition, the solution is noted I<i> (engineers use I<j> instead since
+I<i> usually denotes an intensity, but the name does not matter). The number
+I<i> is a pure I<imaginary> number.
+
+The arithmetics with pure imaginary numbers works just like you would expect
+it with real numbers... you just have to remember that
+
+ i*i = -1
+
+so you have:
+
+ 5i + 7i = i * (5 + 7) = 12i
+ 4i - 3i = i * (4 - 3) = i
+ 4i * 2i = -8
+ 6i / 2i = 3
+ 1 / i = -i
+
+Complex numbers are numbers that have both a real part and an imaginary
+part, and are usually noted:
+
+ a + bi
+
+where C<a> is the I<real> part and C<b> is the I<imaginary> part. The
+arithmetic with complex numbers is straightforward. You have to
+keep track of the real and the imaginary parts, but otherwise the
+rules used for real numbers just apply:
+
+ (4 + 3i) + (5 - 2i) = (4 + 5) + i(3 - 2) = 9 + i
+ (2 + i) * (4 - i) = 2*4 + 4i -2i -i*i = 8 + 2i + 1 = 9 + 2i
+
+A graphical representation of complex numbers is possible in a plane
+(also called the I<complex plane>, but it's really a 2D plane).
+The number
+
+ z = a + bi
+
+is the point whose coordinates are (a, b). Actually, it would
+be the vector originating from (0, 0) to (a, b). It follows that the addition
+of two complex numbers is a vectorial addition.
+
+Since there is a bijection between a point in the 2D plane and a complex
+number (i.e. the mapping is unique and reciprocal), a complex number
+can also be uniquely identified with polar coordinates:
+
+ [rho, theta]
+
+where C<rho> is the distance to the origin, and C<theta> the angle between
+the vector and the I<x> axis. There is a notation for this using the
+exponential form, which is:
+
+ rho * exp(i * theta)
+
+where I<i> is the famous imaginary number introduced above. Conversion
+between this form and the cartesian form C<a + bi> is immediate:
+
+ a = rho * cos(theta)
+ b = rho * sin(theta)
+
+which is also expressed by this formula:
+
+ 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 I<argument> of the complex number. The I<norm> of C<z> will be
+noted C<abs(z)>.
+
+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.
+
+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
+I<extensions> of the operations defined on real numbers. This means
+they keep their natural meaning when there is no imaginary part, provided
+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+>).
+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(x) = x >= 0 ? sqrt(x) : sqrt(-x)*i
+
+It can also be extended to be an application from B<C> to B<C>,
+whilst its restriction to B<R> behaves as defined above by using
+the following definition:
+
+ 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
+
+ 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.
- $i = Math::Complex->new(1,1);
+All the common mathematical functions defined on real numbers that
+are extended to complex numbers share that same property of working
+I<as usual> when the imaginary part is zero (otherwise, it would not
+be called an extension, would it?).
-can be manipulated with overloaded math operators. The operators
+A I<new> operation possible on a complex number that is
+the identity for real numbers is called the I<conjugate>, and is noted
+with an horizontal bar above the number, or C<~z> here.
- + - * / neg ~ abs cos sin exp sqrt
+ z = a + bi
+ ~z = a - bi
-are supported as well as
+Simple... Now look:
- "" (stringify)
+ z * ~z = (a + bi) * (a - bi) = a*a + b*b
-The methods
+We saw that the norm of C<z> was noted C<abs(z)> and was defined as the
+distance to the origin, also known as:
- Re Im arg
+ rho = abs(z) = sqrt(a*a + b*b)
-are also provided.
+so
+
+ z * ~z = abs(z) ** 2
+
+If z is a pure real number (i.e. C<b == 0>), then the above yields:
+
+ a * a = abs(a) ** 2
+
+which is true (C<abs> has the regular meaning for real number, i.e. stands
+for the absolute value). This example explains why the norm of C<z> is
+noted C<abs(z)>: it extends the C<abs> function to complex numbers, yet
+is the regular C<abs> we know when the complex number actually has no
+imaginary part... This justifies I<a posteriori> our use of the C<abs>
+notation for the norm.
+
+=head1 OPERATIONS
+
+Given the following notations:
+
+ z1 = a + bi = r1 * exp(i * t1)
+ z2 = c + di = r2 * exp(i * t2)
+ z = <any complex or real number>
+
+the following (overloaded) operations are supported on complex numbers:
+
+ z1 + z2 = (a + c) + i(b + d)
+ z1 - z2 = (a - c) + i(b - d)
+ 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
+ atan2(z1, z2) = atan(z1/z2)
+
+The following extra operations are supported on both real and complex
+numbers:
+
+ Re(z) = a
+ Im(z) = b
+ arg(z) = t
+
+ cbrt(z) = z ** (1/3)
+ log10(z) = log(z) / log(10)
+ logn(z, n) = log(z) / log(n)
+
+ tan(z) = sin(z) / cos(z)
+ cotan(z) = 1 / tan(z)
+
+ asin(z) = -i * log(i*z + sqrt(1-z*z))
+ acos(z) = -i * log(z + sqrt(z*z-1))
+ atan(z) = i/2 * log((i+z) / (i-z))
+ acotan(z) = -i/2 * log((i+z) / (z-i))
+
+ sinh(z) = 1/2 (exp(z) - exp(-z))
+ cosh(z) = 1/2 (exp(z) + exp(-z))
+ tanh(z) = sinh(z) / cosh(z)
+ cotanh(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))
+ acotanh(z) = 1/2 * log((1+z) / (z-1))
+
+The I<root> function is available to compute all the I<n>th
+roots of some complex, where I<n> is a strictly positive integer.
+There are exactly I<n> such roots, returned as a list. Getting the
+number mathematicians call C<j> such that:
+
+ 1 + j + j*j = 0;
+
+is a simple matter of writing:
+
+ $j = ((root(1, 3))[1];
+
+The I<k>th root for C<z = [r,t]> is given by:
+
+ (root(z, n))[k] = r**(1/n) * exp(i * (t + 2*k*pi)/n)
+
+The I<spaceshift> operation 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
+
+To create a complex number, use either:
+
+ $z = Math::Complex->make(3, 4);
+ $z = cplx(3, 4);
+
+if you know the cartesian form of the number, or
+
+ $z = 3 + 4*i;
+
+if you like. To create a number using the trigonometric 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).
+(Mnmemonic: C<e> is used as a notation for complex numbers in the trigonometric
+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
+plane).
+
+=head1 STRINGIFICATION
+
+When printed, a complex number is usually shown under its cartesian
+form I<a+bi>, but there are legitimate cases where the polar format
+I<[r,t]> is more appropriate.
+
+By calling the routine C<Math::Complex::display_format> and supplying either
+C<"polar"> or C<"cartesian">, you override the default display format,
+which is C<"cartesian">. Not supplying any argument returns the current
+setting.
+
+This default can be overridden on a per-number basis by calling the
+C<display_format> method instead. As before, not supplying any argument
+returns the current display format for this number. Otherwise whatever you
+specify will be the new display format for I<this> particular number.
+
+For instance:
+
+ use Math::Complex;
+
+ Math::Complex::display_format('polar');
+ $j = ((root(1, 3))[1];
+ print "j = $j\n"; # Prints "j = [1,2pi/3]
+ $j->display_format('cartesian');
+ print "j = $j\n"; # Prints "j = -0.5+0.866025403784439i"
+
+The polar format attempts to emphasize arguments like I<k*pi/n>
+(where I<n> is a positive integer and I<k> an integer within [-9,+9]).
+
+=head1 USAGE
+
+Thanks to overloading, the handling of arithmetics with complex numbers
+is simple and almost transparent.
+
+Here are some examples:
+
+ use Math::Complex;
+
+ $j = cplxe(1, 2*pi/3); # $j ** 3 == 1
+ print "j = $j, j**3 = ", $j ** 3, "\n";
+ print "1 + j + j**2 = ", 1 + $j + $j**2, "\n";
+
+ $z = -16 + 0*i; # Force it to be a complex
+ print "sqrt($z) = ", sqrt($z), "\n";
+
+ $k = exp(i * 2*pi/3);
+ print "$j - $k = ", $j - $k, "\n";
=head1 BUGS
-sqrt() should return two roots, but only returns one.
+Saying C<use Math::Complex;> exports many mathematical routines in the caller
+environment. This is construed as a feature by the Author, actually... ;-)
+
+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 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).
-=head1 AUTHORS
+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.
-Dave Nadler, Tom Christiansen, Tim Bunce, Larry Wall.
+=head1 AUTHOR
-=cut
+Raphael Manfredi <F<Raphael_Manfredi@grenoble.hp.com>>
projects / p5sagit/p5-mst-13.2.git / commitdiff