1776 lines
40 KiB
Perl
1776 lines
40 KiB
Perl
#
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# Complex numbers and associated mathematical functions
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# -- Raphael Manfredi Since Sep 1996
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# -- Jarkko Hietaniemi Since Mar 1997
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# -- Daniel S. Lewart Since Sep 1997
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#
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require Exporter;
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package Math::Complex;
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use strict;
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use vars qw($VERSION @ISA @EXPORT %EXPORT_TAGS);
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my ( $i, $ip2, %logn );
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$VERSION = sprintf("%s", q$Id: Complex.pm,v 1.25 1998/02/05 16:07:37 jhi Exp $ =~ /(\d+\.\d+)/);
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@ISA = qw(Exporter);
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my @trig = qw(
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pi
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tan
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csc cosec sec cot cotan
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asin acos atan
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acsc acosec asec acot acotan
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sinh cosh tanh
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csch cosech sech coth cotanh
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asinh acosh atanh
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acsch acosech asech acoth acotanh
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);
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@EXPORT = (qw(
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i Re Im rho theta arg
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sqrt log ln
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log10 logn cbrt root
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cplx cplxe
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),
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@trig);
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%EXPORT_TAGS = (
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'trig' => [@trig],
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);
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use overload
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'+' => \&plus,
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'-' => \&minus,
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'*' => \&multiply,
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'/' => \÷,
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'**' => \&power,
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'<=>' => \&spaceship,
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'neg' => \&negate,
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'~' => \&conjugate,
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'abs' => \&abs,
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'sqrt' => \&sqrt,
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'exp' => \&exp,
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'log' => \&log,
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'sin' => \&sin,
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'cos' => \&cos,
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'tan' => \&tan,
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'atan2' => \&atan2,
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qw("" stringify);
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#
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# Package "privates"
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#
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my $package = 'Math::Complex'; # Package name
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my $display = 'cartesian'; # Default display format
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my $eps = 1e-14; # Epsilon
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#
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# Object attributes (internal):
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# cartesian [real, imaginary] -- cartesian form
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# polar [rho, theta] -- polar form
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# c_dirty cartesian form not up-to-date
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# p_dirty polar form not up-to-date
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# display display format (package's global when not set)
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#
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# Die on bad *make() arguments.
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sub _cannot_make {
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die "@{[(caller(1))[3]]}: Cannot take $_[0] of $_[1].\n";
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}
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#
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# ->make
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#
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# Create a new complex number (cartesian form)
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#
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sub make {
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my $self = bless {}, shift;
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my ($re, $im) = @_;
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my $rre = ref $re;
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if ( $rre ) {
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if ( $rre eq ref $self ) {
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$re = Re($re);
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} else {
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_cannot_make("real part", $rre);
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}
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}
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my $rim = ref $im;
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if ( $rim ) {
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if ( $rim eq ref $self ) {
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$im = Im($im);
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} else {
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_cannot_make("imaginary part", $rim);
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}
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}
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$self->{'cartesian'} = [ $re, $im ];
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$self->{c_dirty} = 0;
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$self->{p_dirty} = 1;
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$self->display_format('cartesian');
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return $self;
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}
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#
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# ->emake
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#
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# Create a new complex number (exponential form)
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#
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sub emake {
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my $self = bless {}, shift;
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my ($rho, $theta) = @_;
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my $rrh = ref $rho;
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if ( $rrh ) {
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if ( $rrh eq ref $self ) {
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$rho = rho($rho);
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} else {
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_cannot_make("rho", $rrh);
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}
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}
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my $rth = ref $theta;
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if ( $rth ) {
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if ( $rth eq ref $self ) {
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$theta = theta($theta);
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} else {
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_cannot_make("theta", $rth);
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}
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}
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if ($rho < 0) {
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$rho = -$rho;
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$theta = ($theta <= 0) ? $theta + pi() : $theta - pi();
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}
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$self->{'polar'} = [$rho, $theta];
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$self->{p_dirty} = 0;
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$self->{c_dirty} = 1;
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$self->display_format('polar');
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return $self;
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}
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sub new { &make } # For backward compatibility only.
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#
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# cplx
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#
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# Creates a complex number from a (re, im) tuple.
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# This avoids the burden of writing Math::Complex->make(re, im).
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#
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sub cplx {
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my ($re, $im) = @_;
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return $package->make($re, defined $im ? $im : 0);
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}
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#
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# cplxe
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#
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# Creates a complex number from a (rho, theta) tuple.
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# This avoids the burden of writing Math::Complex->emake(rho, theta).
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#
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sub cplxe {
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my ($rho, $theta) = @_;
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return $package->emake($rho, defined $theta ? $theta : 0);
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}
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#
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# pi
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#
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# The number defined as pi = 180 degrees
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#
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use constant pi => 4 * CORE::atan2(1, 1);
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#
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# pit2
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#
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# The full circle
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#
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use constant pit2 => 2 * pi;
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#
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# pip2
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#
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# The quarter circle
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#
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use constant pip2 => pi / 2;
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#
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# deg1
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#
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# One degree in radians, used in stringify_polar.
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#
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use constant deg1 => pi / 180;
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#
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# uplog10
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#
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# Used in log10().
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#
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use constant uplog10 => 1 / CORE::log(10);
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#
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# i
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#
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# The number defined as i*i = -1;
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#
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sub i () {
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return $i if ($i);
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$i = bless {};
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$i->{'cartesian'} = [0, 1];
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$i->{'polar'} = [1, pip2];
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$i->{c_dirty} = 0;
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$i->{p_dirty} = 0;
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return $i;
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}
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#
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# Attribute access/set routines
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#
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sub cartesian {$_[0]->{c_dirty} ?
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$_[0]->update_cartesian : $_[0]->{'cartesian'}}
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sub polar {$_[0]->{p_dirty} ?
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$_[0]->update_polar : $_[0]->{'polar'}}
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sub set_cartesian { $_[0]->{p_dirty}++; $_[0]->{'cartesian'} = $_[1] }
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sub set_polar { $_[0]->{c_dirty}++; $_[0]->{'polar'} = $_[1] }
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#
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# ->update_cartesian
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#
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# Recompute and return the cartesian form, given accurate polar form.
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#
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sub update_cartesian {
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my $self = shift;
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my ($r, $t) = @{$self->{'polar'}};
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$self->{c_dirty} = 0;
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return $self->{'cartesian'} = [$r * CORE::cos($t), $r * CORE::sin($t)];
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}
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#
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#
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# ->update_polar
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#
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# Recompute and return the polar form, given accurate cartesian form.
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#
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sub update_polar {
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my $self = shift;
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my ($x, $y) = @{$self->{'cartesian'}};
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$self->{p_dirty} = 0;
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return $self->{'polar'} = [0, 0] if $x == 0 && $y == 0;
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return $self->{'polar'} = [CORE::sqrt($x*$x + $y*$y), CORE::atan2($y, $x)];
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}
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#
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# (plus)
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#
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# Computes z1+z2.
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#
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sub plus {
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my ($z1, $z2, $regular) = @_;
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my ($re1, $im1) = @{$z1->cartesian};
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$z2 = cplx($z2) unless ref $z2;
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my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0);
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unless (defined $regular) {
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$z1->set_cartesian([$re1 + $re2, $im1 + $im2]);
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return $z1;
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}
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return (ref $z1)->make($re1 + $re2, $im1 + $im2);
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}
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#
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# (minus)
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#
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# Computes z1-z2.
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#
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sub minus {
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my ($z1, $z2, $inverted) = @_;
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my ($re1, $im1) = @{$z1->cartesian};
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$z2 = cplx($z2) unless ref $z2;
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my ($re2, $im2) = @{$z2->cartesian};
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unless (defined $inverted) {
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$z1->set_cartesian([$re1 - $re2, $im1 - $im2]);
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return $z1;
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}
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return $inverted ?
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(ref $z1)->make($re2 - $re1, $im2 - $im1) :
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(ref $z1)->make($re1 - $re2, $im1 - $im2);
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}
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#
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# (multiply)
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#
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# Computes z1*z2.
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#
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sub multiply {
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my ($z1, $z2, $regular) = @_;
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if ($z1->{p_dirty} == 0 and ref $z2 and $z2->{p_dirty} == 0) {
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# if both polar better use polar to avoid rounding errors
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my ($r1, $t1) = @{$z1->polar};
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my ($r2, $t2) = @{$z2->polar};
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my $t = $t1 + $t2;
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if ($t > pi()) { $t -= pit2 }
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elsif ($t <= -pi()) { $t += pit2 }
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unless (defined $regular) {
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$z1->set_polar([$r1 * $r2, $t]);
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return $z1;
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}
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return (ref $z1)->emake($r1 * $r2, $t);
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} else {
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my ($x1, $y1) = @{$z1->cartesian};
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if (ref $z2) {
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my ($x2, $y2) = @{$z2->cartesian};
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return (ref $z1)->make($x1*$x2-$y1*$y2, $x1*$y2+$y1*$x2);
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} else {
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return (ref $z1)->make($x1*$z2, $y1*$z2);
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}
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}
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}
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#
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# _divbyzero
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#
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# Die on division by zero.
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#
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sub _divbyzero {
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my $mess = "$_[0]: Division by zero.\n";
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if (defined $_[1]) {
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$mess .= "(Because in the definition of $_[0], the divisor ";
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$mess .= "$_[1] " unless ($_[1] eq '0');
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$mess .= "is 0)\n";
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}
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my @up = caller(1);
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$mess .= "Died at $up[1] line $up[2].\n";
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die $mess;
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}
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#
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# (divide)
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#
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# Computes z1/z2.
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#
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sub divide {
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my ($z1, $z2, $inverted) = @_;
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if ($z1->{p_dirty} == 0 and ref $z2 and $z2->{p_dirty} == 0) {
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# if both polar better use polar to avoid rounding errors
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my ($r1, $t1) = @{$z1->polar};
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my ($r2, $t2) = @{$z2->polar};
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my $t;
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if ($inverted) {
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_divbyzero "$z2/0" if ($r1 == 0);
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$t = $t2 - $t1;
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if ($t > pi()) { $t -= pit2 }
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elsif ($t <= -pi()) { $t += pit2 }
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return (ref $z1)->emake($r2 / $r1, $t);
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} else {
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_divbyzero "$z1/0" if ($r2 == 0);
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$t = $t1 - $t2;
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if ($t > pi()) { $t -= pit2 }
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elsif ($t <= -pi()) { $t += pit2 }
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return (ref $z1)->emake($r1 / $r2, $t);
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}
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} else {
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my ($d, $x2, $y2);
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if ($inverted) {
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($x2, $y2) = @{$z1->cartesian};
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$d = $x2*$x2 + $y2*$y2;
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_divbyzero "$z2/0" if $d == 0;
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return (ref $z1)->make(($x2*$z2)/$d, -($y2*$z2)/$d);
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} else {
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my ($x1, $y1) = @{$z1->cartesian};
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if (ref $z2) {
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($x2, $y2) = @{$z2->cartesian};
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$d = $x2*$x2 + $y2*$y2;
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_divbyzero "$z1/0" if $d == 0;
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my $u = ($x1*$x2 + $y1*$y2)/$d;
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my $v = ($y1*$x2 - $x1*$y2)/$d;
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return (ref $z1)->make($u, $v);
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} else {
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_divbyzero "$z1/0" if $z2 == 0;
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return (ref $z1)->make($x1/$z2, $y1/$z2);
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}
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}
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}
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}
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#
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# _zerotozero
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#
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# Die on zero raised to the zeroth.
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#
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sub _zerotozero {
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my $mess = "The zero raised to the zeroth power is not defined.\n";
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my @up = caller(1);
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$mess .= "Died at $up[1] line $up[2].\n";
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die $mess;
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}
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#
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# (power)
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#
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# Computes z1**z2 = exp(z2 * log z1)).
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#
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sub power {
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my ($z1, $z2, $inverted) = @_;
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my $z1z = $z1 == 0;
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my $z2z = $z2 == 0;
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_zerotozero if ($z1z and $z2z);
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if ($inverted) {
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return 0 if ($z2z);
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return 1 if ($z1z or $z2 == 1);
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} else {
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return 0 if ($z1z);
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return 1 if ($z2z or $z1 == 1);
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}
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my $w = $inverted ? CORE::exp($z1 * CORE::log($z2)) : CORE::exp($z2 * CORE::log($z1));
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# If both arguments cartesian, return cartesian, else polar.
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return $z1->{c_dirty} == 0 &&
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(not ref $z2 or $z2->{c_dirty} == 0) ?
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cplx(@{$w->cartesian}) : $w;
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}
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#
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# (spaceship)
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#
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# Computes z1 <=> z2.
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# Sorts on the real part first, then on the imaginary part. Thus 2-4i > 3+8i.
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#
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sub spaceship {
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my ($z1, $z2, $inverted) = @_;
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my ($re1, $im1) = ref $z1 ? @{$z1->cartesian} : ($z1, 0);
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my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0);
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my $sgn = $inverted ? -1 : 1;
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return $sgn * ($re1 <=> $re2) if $re1 != $re2;
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return $sgn * ($im1 <=> $im2);
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}
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#
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# (negate)
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#
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# Computes -z.
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#
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sub negate {
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my ($z) = @_;
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if ($z->{c_dirty}) {
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my ($r, $t) = @{$z->polar};
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$t = ($t <= 0) ? $t + pi : $t - pi;
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return (ref $z)->emake($r, $t);
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}
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my ($re, $im) = @{$z->cartesian};
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return (ref $z)->make(-$re, -$im);
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}
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#
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# (conjugate)
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#
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# Compute complex's conjugate.
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#
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sub conjugate {
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my ($z) = @_;
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if ($z->{c_dirty}) {
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my ($r, $t) = @{$z->polar};
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return (ref $z)->emake($r, -$t);
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}
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my ($re, $im) = @{$z->cartesian};
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return (ref $z)->make($re, -$im);
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}
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#
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# (abs)
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#
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# Compute or set complex's norm (rho).
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#
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sub abs {
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my ($z, $rho) = @_;
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return $z unless ref $z;
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if (defined $rho) {
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$z->{'polar'} = [ $rho, ${$z->polar}[1] ];
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$z->{p_dirty} = 0;
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$z->{c_dirty} = 1;
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return $rho;
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} else {
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return ${$z->polar}[0];
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}
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}
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sub _theta {
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my $theta = $_[0];
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if ($$theta > pi()) { $$theta -= pit2 }
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elsif ($$theta <= -pi()) { $$theta += pit2 }
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}
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#
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# arg
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#
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# Compute or set complex's argument (theta).
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#
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sub arg {
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my ($z, $theta) = @_;
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return $z unless ref $z;
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if (defined $theta) {
|
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_theta(\$theta);
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$z->{'polar'} = [ ${$z->polar}[0], $theta ];
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$z->{p_dirty} = 0;
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$z->{c_dirty} = 1;
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} else {
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$theta = ${$z->polar}[1];
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_theta(\$theta);
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}
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return $theta;
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}
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#
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|
# (sqrt)
|
|
#
|
|
# Compute sqrt(z).
|
|
#
|
|
# It is quite tempting to use wantarray here so that in list context
|
|
# sqrt() would return the two solutions. This, however, would
|
|
# break things like
|
|
#
|
|
# print "sqrt(z) = ", sqrt($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 ($z) = @_;
|
|
my ($re, $im) = ref $z ? @{$z->cartesian} : ($z, 0);
|
|
return $re < 0 ? cplx(0, CORE::sqrt(-$re)) : CORE::sqrt($re) if $im == 0;
|
|
my ($r, $t) = @{$z->polar};
|
|
return (ref $z)->emake(CORE::sqrt($r), $t/2);
|
|
}
|
|
|
|
#
|
|
# cbrt
|
|
#
|
|
# Compute cbrt(z) (cubic root).
|
|
#
|
|
# Why are we not returning three values? The same answer as for sqrt().
|
|
#
|
|
sub cbrt {
|
|
my ($z) = @_;
|
|
return $z < 0 ? -CORE::exp(CORE::log(-$z)/3) : ($z > 0 ? CORE::exp(CORE::log($z)/3): 0)
|
|
unless ref $z;
|
|
my ($r, $t) = @{$z->polar};
|
|
return (ref $z)->emake(CORE::exp(CORE::log($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 $mess;
|
|
}
|
|
|
|
#
|
|
# root
|
|
#
|
|
# Computes all nth root for z, returning an array whose size is n.
|
|
# `n' must be a positive integer.
|
|
#
|
|
# The roots are given by (for k = 0..n-1):
|
|
#
|
|
# z^(1/n) = r^(1/n) (cos ((t+2 k pi)/n) + i sin ((t+2 k pi)/n))
|
|
#
|
|
sub root {
|
|
my ($z, $n) = @_;
|
|
_rootbad($n) if ($n < 1 or int($n) != $n);
|
|
my ($r, $t) = ref $z ? @{$z->polar} : (CORE::abs($z), $z >= 0 ? 0 : pi);
|
|
my @root;
|
|
my $k;
|
|
my $theta_inc = pit2 / $n;
|
|
my $rho = $r ** (1/$n);
|
|
my $theta;
|
|
my $cartesian = ref $z && $z->{c_dirty} == 0;
|
|
for ($k = 0, $theta = $t / $n; $k < $n; $k++, $theta += $theta_inc) {
|
|
my $w = cplxe($rho, $theta);
|
|
# Yes, $cartesian is loop invariant.
|
|
push @root, $cartesian ? cplx(@{$w->cartesian}) : $w;
|
|
}
|
|
return @root;
|
|
}
|
|
|
|
#
|
|
# Re
|
|
#
|
|
# Return or set Re(z).
|
|
#
|
|
sub Re {
|
|
my ($z, $Re) = @_;
|
|
return $z unless ref $z;
|
|
if (defined $Re) {
|
|
$z->{'cartesian'} = [ $Re, ${$z->cartesian}[1] ];
|
|
$z->{c_dirty} = 0;
|
|
$z->{p_dirty} = 1;
|
|
} else {
|
|
return ${$z->cartesian}[0];
|
|
}
|
|
}
|
|
|
|
#
|
|
# Im
|
|
#
|
|
# Return or set Im(z).
|
|
#
|
|
sub 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(@_);
|
|
}
|
|
|
|
#
|
|
# (exp)
|
|
#
|
|
# Computes exp(z).
|
|
#
|
|
sub exp {
|
|
my ($z) = @_;
|
|
my ($x, $y) = @{$z->cartesian};
|
|
return (ref $z)->emake(CORE::exp($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) = @_;
|
|
unless (ref $z) {
|
|
_logofzero("log") if $z == 0;
|
|
return $z > 0 ? CORE::log($z) : cplx(CORE::log(-$z), pi);
|
|
}
|
|
my ($r, $t) = @{$z->polar};
|
|
_logofzero("log") if $r == 0;
|
|
if ($t > pi()) { $t -= pit2 }
|
|
elsif ($t <= -pi()) { $t += pit2 }
|
|
return (ref $z)->make(CORE::log($r), $t);
|
|
}
|
|
|
|
#
|
|
# ln
|
|
#
|
|
# Alias for log().
|
|
#
|
|
sub ln { Math::Complex::log(@_) }
|
|
|
|
#
|
|
# log10
|
|
#
|
|
# Compute log10(z).
|
|
#
|
|
|
|
sub log10 {
|
|
return Math::Complex::log($_[0]) * uplog10;
|
|
}
|
|
|
|
#
|
|
# logn
|
|
#
|
|
# Compute logn(z,n) = log(z) / log(n)
|
|
#
|
|
sub logn {
|
|
my ($z, $n) = @_;
|
|
$z = cplx($z, 0) unless ref $z;
|
|
my $logn = $logn{$n};
|
|
$logn = $logn{$n} = CORE::log($n) unless defined $logn; # Cache log(n)
|
|
return CORE::log($z) / $logn;
|
|
}
|
|
|
|
#
|
|
# (cos)
|
|
#
|
|
# Compute cos(z) = (exp(iz) + exp(-iz))/2.
|
|
#
|
|
sub cos {
|
|
my ($z) = @_;
|
|
my ($x, $y) = @{$z->cartesian};
|
|
my $ey = CORE::exp($y);
|
|
my $ey_1 = 1 / $ey;
|
|
return (ref $z)->make(CORE::cos($x) * ($ey + $ey_1)/2,
|
|
CORE::sin($x) * ($ey_1 - $ey)/2);
|
|
}
|
|
|
|
#
|
|
# (sin)
|
|
#
|
|
# Compute sin(z) = (exp(iz) - exp(-iz))/2.
|
|
#
|
|
sub sin {
|
|
my ($z) = @_;
|
|
my ($x, $y) = @{$z->cartesian};
|
|
my $ey = CORE::exp($y);
|
|
my $ey_1 = 1 / $ey;
|
|
return (ref $z)->make(CORE::sin($x) * ($ey + $ey_1)/2,
|
|
CORE::cos($x) * ($ey - $ey_1)/2);
|
|
}
|
|
|
|
#
|
|
# tan
|
|
#
|
|
# Compute tan(z) = sin(z) / cos(z).
|
|
#
|
|
sub tan {
|
|
my ($z) = @_;
|
|
my $cz = CORE::cos($z);
|
|
_divbyzero "tan($z)", "cos($z)" if (CORE::abs($cz) < $eps);
|
|
return CORE::sin($z) / $cz;
|
|
}
|
|
|
|
#
|
|
# sec
|
|
#
|
|
# Computes the secant sec(z) = 1 / cos(z).
|
|
#
|
|
sub sec {
|
|
my ($z) = @_;
|
|
my $cz = CORE::cos($z);
|
|
_divbyzero "sec($z)", "cos($z)" if ($cz == 0);
|
|
return 1 / $cz;
|
|
}
|
|
|
|
#
|
|
# csc
|
|
#
|
|
# Computes the cosecant csc(z) = 1 / sin(z).
|
|
#
|
|
sub csc {
|
|
my ($z) = @_;
|
|
my $sz = CORE::sin($z);
|
|
_divbyzero "csc($z)", "sin($z)" if ($sz == 0);
|
|
return 1 / $sz;
|
|
}
|
|
|
|
#
|
|
# cosec
|
|
#
|
|
# Alias for csc().
|
|
#
|
|
sub cosec { Math::Complex::csc(@_) }
|
|
|
|
#
|
|
# cot
|
|
#
|
|
# Computes cot(z) = cos(z) / sin(z).
|
|
#
|
|
sub cot {
|
|
my ($z) = @_;
|
|
my $sz = CORE::sin($z);
|
|
_divbyzero "cot($z)", "sin($z)" if ($sz == 0);
|
|
return CORE::cos($z) / $sz;
|
|
}
|
|
|
|
#
|
|
# cotan
|
|
#
|
|
# Alias for cot().
|
|
#
|
|
sub cotan { Math::Complex::cot(@_) }
|
|
|
|
#
|
|
# acos
|
|
#
|
|
# Computes the arc cosine acos(z) = -i log(z + sqrt(z*z-1)).
|
|
#
|
|
sub acos {
|
|
my $z = $_[0];
|
|
return CORE::atan2(CORE::sqrt(1-$z*$z), $z) if (! ref $z) && CORE::abs($z) <= 1;
|
|
my ($x, $y) = ref $z ? @{$z->cartesian} : ($z, 0);
|
|
my $t1 = CORE::sqrt(($x+1)*($x+1) + $y*$y);
|
|
my $t2 = CORE::sqrt(($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 $u = CORE::atan2(CORE::sqrt(1-$beta*$beta), $beta);
|
|
my $v = CORE::log($alpha + CORE::sqrt($alpha*$alpha-1));
|
|
$v = -$v if $y > 0 || ($y == 0 && $x < -1);
|
|
return $package->make($u, $v);
|
|
}
|
|
|
|
#
|
|
# asin
|
|
#
|
|
# Computes the arc sine asin(z) = -i log(iz + sqrt(1-z*z)).
|
|
#
|
|
sub asin {
|
|
my $z = $_[0];
|
|
return CORE::atan2($z, CORE::sqrt(1-$z*$z)) if (! ref $z) && CORE::abs($z) <= 1;
|
|
my ($x, $y) = ref $z ? @{$z->cartesian} : ($z, 0);
|
|
my $t1 = CORE::sqrt(($x+1)*($x+1) + $y*$y);
|
|
my $t2 = CORE::sqrt(($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 $u = CORE::atan2($beta, CORE::sqrt(1-$beta*$beta));
|
|
my $v = -CORE::log($alpha + CORE::sqrt($alpha*$alpha-1));
|
|
$v = -$v if $y > 0 || ($y == 0 && $x < -1);
|
|
return $package->make($u, $v);
|
|
}
|
|
|
|
#
|
|
# atan
|
|
#
|
|
# Computes the arc tangent atan(z) = i/2 log((i+z) / (i-z)).
|
|
#
|
|
sub atan {
|
|
my ($z) = @_;
|
|
return CORE::atan2($z, 1) unless ref $z;
|
|
_divbyzero "atan(i)" if ( $z == i);
|
|
_divbyzero "atan(-i)" if (-$z == i);
|
|
my $log = CORE::log((i + $z) / (i - $z));
|
|
$ip2 = 0.5 * i unless defined $ip2;
|
|
return $ip2 * $log;
|
|
}
|
|
|
|
#
|
|
# asec
|
|
#
|
|
# Computes the arc secant asec(z) = acos(1 / z).
|
|
#
|
|
sub asec {
|
|
my ($z) = @_;
|
|
_divbyzero "asec($z)", $z if ($z == 0);
|
|
return acos(1 / $z);
|
|
}
|
|
|
|
#
|
|
# acsc
|
|
#
|
|
# Computes the arc cosecant acsc(z) = asin(1 / z).
|
|
#
|
|
sub acsc {
|
|
my ($z) = @_;
|
|
_divbyzero "acsc($z)", $z if ($z == 0);
|
|
return asin(1 / $z);
|
|
}
|
|
|
|
#
|
|
# acosec
|
|
#
|
|
# Alias for acsc().
|
|
#
|
|
sub acosec { Math::Complex::acsc(@_) }
|
|
|
|
#
|
|
# acot
|
|
#
|
|
# Computes the arc cotangent acot(z) = atan(1 / z)
|
|
#
|
|
sub acot {
|
|
my ($z) = @_;
|
|
_divbyzero "acot(0)" if (CORE::abs($z) < $eps);
|
|
return ($z >= 0) ? CORE::atan2(1, $z) : CORE::atan2(-1, -$z) unless ref $z;
|
|
_divbyzero "acot(i)" if (CORE::abs($z - i) < $eps);
|
|
_logofzero "acot(-i)" if (CORE::abs($z + i) < $eps);
|
|
return atan(1 / $z);
|
|
}
|
|
|
|
#
|
|
# acotan
|
|
#
|
|
# Alias for acot().
|
|
#
|
|
sub acotan { Math::Complex::acot(@_) }
|
|
|
|
#
|
|
# cosh
|
|
#
|
|
# Computes the hyperbolic cosine cosh(z) = (exp(z) + exp(-z))/2.
|
|
#
|
|
sub cosh {
|
|
my ($z) = @_;
|
|
my $ex;
|
|
unless (ref $z) {
|
|
$ex = CORE::exp($z);
|
|
return ($ex + 1/$ex)/2;
|
|
}
|
|
my ($x, $y) = @{$z->cartesian};
|
|
$ex = CORE::exp($x);
|
|
my $ex_1 = 1 / $ex;
|
|
return (ref $z)->make(CORE::cos($y) * ($ex + $ex_1)/2,
|
|
CORE::sin($y) * ($ex - $ex_1)/2);
|
|
}
|
|
|
|
#
|
|
# sinh
|
|
#
|
|
# Computes the hyperbolic sine sinh(z) = (exp(z) - exp(-z))/2.
|
|
#
|
|
sub sinh {
|
|
my ($z) = @_;
|
|
my $ex;
|
|
unless (ref $z) {
|
|
$ex = CORE::exp($z);
|
|
return ($ex - 1/$ex)/2;
|
|
}
|
|
my ($x, $y) = @{$z->cartesian};
|
|
$ex = CORE::exp($x);
|
|
my $ex_1 = 1 / $ex;
|
|
return (ref $z)->make(CORE::cos($y) * ($ex - $ex_1)/2,
|
|
CORE::sin($y) * ($ex + $ex_1)/2);
|
|
}
|
|
|
|
#
|
|
# tanh
|
|
#
|
|
# Computes the hyperbolic tangent tanh(z) = sinh(z) / cosh(z).
|
|
#
|
|
sub tanh {
|
|
my ($z) = @_;
|
|
my $cz = cosh($z);
|
|
_divbyzero "tanh($z)", "cosh($z)" if ($cz == 0);
|
|
return sinh($z) / $cz;
|
|
}
|
|
|
|
#
|
|
# sech
|
|
#
|
|
# Computes the hyperbolic secant sech(z) = 1 / cosh(z).
|
|
#
|
|
sub sech {
|
|
my ($z) = @_;
|
|
my $cz = cosh($z);
|
|
_divbyzero "sech($z)", "cosh($z)" if ($cz == 0);
|
|
return 1 / $cz;
|
|
}
|
|
|
|
#
|
|
# csch
|
|
#
|
|
# Computes the hyperbolic cosecant csch(z) = 1 / sinh(z).
|
|
#
|
|
sub csch {
|
|
my ($z) = @_;
|
|
my $sz = sinh($z);
|
|
_divbyzero "csch($z)", "sinh($z)" if ($sz == 0);
|
|
return 1 / $sz;
|
|
}
|
|
|
|
#
|
|
# cosech
|
|
#
|
|
# Alias for csch().
|
|
#
|
|
sub cosech { Math::Complex::csch(@_) }
|
|
|
|
#
|
|
# coth
|
|
#
|
|
# Computes the hyperbolic cotangent coth(z) = cosh(z) / sinh(z).
|
|
#
|
|
sub coth {
|
|
my ($z) = @_;
|
|
my $sz = sinh($z);
|
|
_divbyzero "coth($z)", "sinh($z)" if ($sz == 0);
|
|
return cosh($z) / $sz;
|
|
}
|
|
|
|
#
|
|
# cotanh
|
|
#
|
|
# Alias for coth().
|
|
#
|
|
sub cotanh { Math::Complex::coth(@_) }
|
|
|
|
#
|
|
# acosh
|
|
#
|
|
# Computes the arc hyperbolic cosine acosh(z) = log(z + sqrt(z*z-1)).
|
|
#
|
|
sub acosh {
|
|
my ($z) = @_;
|
|
unless (ref $z) {
|
|
return CORE::log($z + CORE::sqrt($z*$z-1)) if $z >= 1;
|
|
$z = cplx($z, 0);
|
|
}
|
|
my ($re, $im) = @{$z->cartesian};
|
|
if ($im == 0) {
|
|
return cplx(CORE::log($re + CORE::sqrt($re*$re - 1)), 0) if $re >= 1;
|
|
return cplx(0, CORE::atan2(CORE::sqrt(1-$re*$re), $re)) if CORE::abs($re) <= 1;
|
|
}
|
|
return CORE::log($z + CORE::sqrt($z*$z - 1));
|
|
}
|
|
|
|
#
|
|
# asinh
|
|
#
|
|
# Computes the arc hyperbolic sine asinh(z) = log(z + sqrt(z*z-1))
|
|
#
|
|
sub asinh {
|
|
my ($z) = @_;
|
|
return CORE::log($z + CORE::sqrt($z*$z + 1));
|
|
}
|
|
|
|
#
|
|
# atanh
|
|
#
|
|
# Computes the arc hyperbolic tangent atanh(z) = 1/2 log((1+z) / (1-z)).
|
|
#
|
|
sub atanh {
|
|
my ($z) = @_;
|
|
unless (ref $z) {
|
|
return CORE::log((1 + $z)/(1 - $z))/2 if CORE::abs($z) < 1;
|
|
$z = cplx($z, 0);
|
|
}
|
|
_divbyzero 'atanh(1)', "1 - $z" if ($z == 1);
|
|
_logofzero 'atanh(-1)' if ($z == -1);
|
|
return 0.5 * CORE::log((1 + $z) / (1 - $z));
|
|
}
|
|
|
|
#
|
|
# asech
|
|
#
|
|
# Computes the hyperbolic arc secant asech(z) = acosh(1 / z).
|
|
#
|
|
sub asech {
|
|
my ($z) = @_;
|
|
_divbyzero 'asech(0)', $z if ($z == 0);
|
|
return acosh(1 / $z);
|
|
}
|
|
|
|
#
|
|
# acsch
|
|
#
|
|
# Computes the hyperbolic arc cosecant acsch(z) = asinh(1 / z).
|
|
#
|
|
sub acsch {
|
|
my ($z) = @_;
|
|
_divbyzero 'acsch(0)', $z if ($z == 0);
|
|
return asinh(1 / $z);
|
|
}
|
|
|
|
#
|
|
# acosech
|
|
#
|
|
# Alias for acosh().
|
|
#
|
|
sub acosech { Math::Complex::acsch(@_) }
|
|
|
|
#
|
|
# acoth
|
|
#
|
|
# Computes the arc hyperbolic cotangent acoth(z) = 1/2 log((1+z) / (z-1)).
|
|
#
|
|
sub acoth {
|
|
my ($z) = @_;
|
|
_divbyzero 'acoth(0)' if (CORE::abs($z) < $eps);
|
|
unless (ref $z) {
|
|
return CORE::log(($z + 1)/($z - 1))/2 if CORE::abs($z) > 1;
|
|
$z = cplx($z, 0);
|
|
}
|
|
_divbyzero 'acoth(1)', "$z - 1" if (CORE::abs($z - 1) < $eps);
|
|
_logofzero 'acoth(-1)', "1 / $z" if (CORE::abs($z + 1) < $eps);
|
|
return CORE::log((1 + $z) / ($z - 1)) / 2;
|
|
}
|
|
|
|
#
|
|
# acotanh
|
|
#
|
|
# Alias for acot().
|
|
#
|
|
sub acotanh { Math::Complex::acoth(@_) }
|
|
|
|
#
|
|
# (atan2)
|
|
#
|
|
# Compute atan(z1/z2).
|
|
#
|
|
sub atan2 {
|
|
my ($z1, $z2, $inverted) = @_;
|
|
my ($re1, $im1, $re2, $im2);
|
|
if ($inverted) {
|
|
($re1, $im1) = ref $z2 ? @{$z2->cartesian} : ($z2, 0);
|
|
($re2, $im2) = @{$z1->cartesian};
|
|
} else {
|
|
($re1, $im1) = @{$z1->cartesian};
|
|
($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0);
|
|
}
|
|
if ($im2 == 0) {
|
|
return cplx(CORE::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
|
|
# ->display_format
|
|
#
|
|
# Set (fetch if no argument) display format for all complex numbers that
|
|
# 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.
|
|
#
|
|
# 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 = $format;
|
|
}
|
|
|
|
#
|
|
# (stringify)
|
|
#
|
|
# Show nicely formatted complex number under its cartesian or polar form,
|
|
# depending on the current display format:
|
|
#
|
|
# . If a specific display format has been recorded for this object, use it.
|
|
# . Otherwise, use the generic current default for all complex numbers,
|
|
# which is a package global variable.
|
|
#
|
|
sub stringify {
|
|
my ($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);
|
|
|
|
$x = int($x + ($x < 0 ? -1 : 1) * $eps)
|
|
if int(CORE::abs($x)) != int(CORE::abs($x) + $eps);
|
|
$y = int($y + ($y < 0 ? -1 : 1) * $eps)
|
|
if int(CORE::abs($y)) != int(CORE::abs($y) + $eps);
|
|
|
|
$re = "$x" if CORE::abs($x) >= $eps;
|
|
if ($y == 1) { $im = 'i' }
|
|
elsif ($y == -1) { $im = '-i' }
|
|
elsif (CORE::abs($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
|
|
#
|
|
# 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 <= $eps;
|
|
|
|
my $nt = $t / pit2;
|
|
$nt = ($nt - int($nt)) * pit2;
|
|
$nt += pit2 if $nt < 0; # Range [0, 2pi]
|
|
|
|
if (CORE::abs($nt) <= $eps) { $theta = 0 }
|
|
elsif (CORE::abs(pi-$nt) <= $eps) { $theta = 'pi' }
|
|
|
|
if (defined $theta) {
|
|
$r = int($r + ($r < 0 ? -1 : 1) * $eps)
|
|
if int(CORE::abs($r)) != int(CORE::abs($r) + $eps);
|
|
$theta = int($theta + ($theta < 0 ? -1 : 1) * $eps)
|
|
if ($theta ne 'pi' and
|
|
int(CORE::abs($theta)) != int(CORE::abs($theta) + $eps));
|
|
return "\[$r,$theta\]";
|
|
}
|
|
|
|
#
|
|
# Okay, number is not a real. Try to identify pi/n and friends...
|
|
#
|
|
|
|
$nt -= pit2 if $nt > pi;
|
|
|
|
if (CORE::abs($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 (CORE::abs($kpi/$n - $nt) <= $eps) {
|
|
$n = CORE::abs($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;
|
|
|
|
$r = int($r + ($r < 0 ? -1 : 1) * $eps)
|
|
if int(CORE::abs($r)) != int(CORE::abs($r) + $eps);
|
|
$theta = int($theta + ($theta < 0 ? -1 : 1) * $eps)
|
|
if ($theta !~ m(^-?\d*pi/\d+$) and
|
|
int(CORE::abs($theta)) != int(CORE::abs($theta) + $eps));
|
|
|
|
return "\[$r,$theta\]";
|
|
}
|
|
|
|
1;
|
|
__END__
|
|
|
|
=head1 NAME
|
|
|
|
Math::Complex - complex numbers and associated mathematical functions
|
|
|
|
=head1 SYNOPSIS
|
|
|
|
use Math::Complex;
|
|
|
|
$z = Math::Complex->make(5, 6);
|
|
$t = 4 - 3*i + $z;
|
|
$j = cplxe(1, 2*pi/3);
|
|
|
|
=head1 DESCRIPTION
|
|
|
|
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 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
|
|
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 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(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 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
|
|
I<as usual> when the imaginary part is zero (otherwise, it would not
|
|
be called an extension, would it?).
|
|
|
|
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.
|
|
|
|
z = a + bi
|
|
~z = a - bi
|
|
|
|
Simple... Now look:
|
|
|
|
z * ~z = (a + bi) * (a - bi) = a*a + b*b
|
|
|
|
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:
|
|
|
|
rho = abs(z) = sqrt(a*a + b*b)
|
|
|
|
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)
|
|
~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
|
|
numbers:
|
|
|
|
Re(z) = a
|
|
Im(z) = b
|
|
arg(z) = t
|
|
abs(z) = r
|
|
|
|
cbrt(z) = z ** (1/3)
|
|
log10(z) = log(z) / log(10)
|
|
logn(z, n) = log(z) / log(n)
|
|
|
|
tan(z) = sin(z) / cos(z)
|
|
|
|
csc(z) = 1 / sin(z)
|
|
sec(z) = 1 / cos(z)
|
|
cot(z) = 1 / tan(z)
|
|
|
|
asin(z) = -i * log(i*z + sqrt(1-z*z))
|
|
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) = 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))
|
|
tanh(z) = sinh(z) / cosh(z) = (exp(z) - exp(-z)) / (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))
|
|
|
|
acsch(z) = asinh(1 / z)
|
|
asech(z) = acosh(1 / z)
|
|
acoth(z) = atanh(1 / z) = 1/2 * log((1+z) / (z-1))
|
|
|
|
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.
|
|
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<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
|
|
|
|
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 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). (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 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
|
|
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";
|
|
|
|
$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
|
|
|
|
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 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 ...
|
|
|
|
or
|
|
|
|
atanh(-1): Logarithm of zero.
|
|
Died at...
|
|
|
|
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 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
|
|
|
|
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>> and
|
|
Jarkko Hietaniemi <F<jhi@iki.fi>>.
|
|
|
|
Extensive patches by Daniel S. Lewart <F<d-lewart@uiuc.edu>>.
|
|
|
|
=cut
|
|
|
|
1;
|
|
|
|
# eof
|