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Complex.pm 0000644 00000141621 15036325012 0006513 0 ustar 00 #
# Complex numbers and associated mathematical functions
# -- Raphael Manfredi Since Sep 1996
# -- Jarkko Hietaniemi Since Mar 1997
# -- Daniel S. Lewart Since Sep 1997
#
package Math::Complex;
{ use 5.006; }
use strict;
our $VERSION = 1.59_01;
use Config;
our ($Inf, $ExpInf);
our ($vax_float, $has_inf, $has_nan);
BEGIN {
$vax_float = (pack("d",1) =~ /^[\x80\x10]\x40/);
$has_inf = !$vax_float;
$has_nan = !$vax_float;
unless ($has_inf) {
# For example in vax, there is no Inf,
# and just mentioning the DBL_MAX (1.70141183460469229e+38)
# causes SIGFPE.
# These are pretty useless without a real infinity,
# but setting them makes for less warnings about their
# undefined values.
$Inf = "Inf";
$ExpInf = "Inf";
return;
}
my %DBL_MAX = # These are IEEE 754 maxima.
(
4 => '1.70141183460469229e+38',
8 => '1.7976931348623157e+308',
# AFAICT the 10, 12, and 16-byte long doubles
# all have the same maximum.
10 => '1.1897314953572317650857593266280070162E+4932',
12 => '1.1897314953572317650857593266280070162E+4932',
16 => '1.1897314953572317650857593266280070162E+4932',
);
my $nvsize = $Config{nvsize} ||
($Config{uselongdouble} && $Config{longdblsize}) ||
$Config{doublesize};
die "Math::Complex: Could not figure out nvsize\n"
unless defined $nvsize;
die "Math::Complex: Cannot not figure out max nv (nvsize = $nvsize)\n"
unless defined $DBL_MAX{$nvsize};
my $DBL_MAX = eval $DBL_MAX{$nvsize};
die "Math::Complex: Could not figure out max nv (nvsize = $nvsize)\n"
unless defined $DBL_MAX;
my $BIGGER_THAN_THIS = 1e30; # Must find something bigger than this.
if ($^O eq 'unicosmk') {
$Inf = $DBL_MAX;
} else {
local $SIG{FPE} = sub { };
local $!;
# We do want an arithmetic overflow, Inf INF inf Infinity.
for my $t (
'exp(99999)', # Enough even with 128-bit long doubles.
'inf',
'Inf',
'INF',
'infinity',
'Infinity',
'INFINITY',
'1e99999',
) {
local $^W = 0;
my $i = eval "$t+1.0";
if (defined $i && $i > $BIGGER_THAN_THIS) {
$Inf = $i;
last;
}
}
$Inf = $DBL_MAX unless defined $Inf; # Oh well, close enough.
die "Math::Complex: Could not get Infinity"
unless $Inf > $BIGGER_THAN_THIS;
$ExpInf = eval 'exp(99999)';
}
# print "# On this machine, Inf = '$Inf'\n";
}
use Scalar::Util qw(set_prototype);
use warnings;
no warnings 'syntax'; # To avoid the (_) warnings.
BEGIN {
# For certain functions that we override, in 5.10 or better
# we can set a smarter prototype that will handle the lexical $_
# (also a 5.10+ feature).
if ($] >= 5.010000) {
set_prototype \&abs, '_';
set_prototype \&cos, '_';
set_prototype \&exp, '_';
set_prototype \&log, '_';
set_prototype \&sin, '_';
set_prototype \&sqrt, '_';
}
}
my $i;
my %LOGN;
# Regular expression for floating point numbers.
# These days we could use Scalar::Util::lln(), I guess.
my $gre = qr'\s*([\+\-]?(?:(?:(?:\d+(?:_\d+)*(?:\.\d*(?:_\d+)*)?|\.\d+(?:_\d+)*)(?:[eE][\+\-]?\d+(?:_\d+)*)?))|inf)'i;
require Exporter;
our @ISA = qw(Exporter);
my @trig = qw(
pi
tan
csc cosec sec cot cotan
asin acos atan
acsc acosec asec acot acotan
sinh cosh tanh
csch cosech sech coth cotanh
asinh acosh atanh
acsch acosech asech acoth acotanh
);
our @EXPORT = (qw(
i Re Im rho theta arg
sqrt log ln
log10 logn cbrt root
cplx cplxe
atan2
),
@trig);
my @pi = qw(pi pi2 pi4 pip2 pip4 Inf);
our @EXPORT_OK = @pi;
our %EXPORT_TAGS = (
'trig' => [@trig],
'pi' => [@pi],
);
use overload
'=' => \&_copy,
'+=' => \&_plus,
'+' => \&_plus,
'-=' => \&_minus,
'-' => \&_minus,
'*=' => \&_multiply,
'*' => \&_multiply,
'/=' => \&_divide,
'/' => \&_divide,
'**=' => \&_power,
'**' => \&_power,
'==' => \&_numeq,
'<=>' => \&_spaceship,
'neg' => \&_negate,
'~' => \&_conjugate,
'abs' => \&abs,
'sqrt' => \&sqrt,
'exp' => \&exp,
'log' => \&log,
'sin' => \&sin,
'cos' => \&cos,
'atan2' => \&atan2,
'""' => \&_stringify;
#
# Package "privates"
#
my %DISPLAY_FORMAT = ('style' => 'cartesian',
'polar_pretty_print' => 1);
my $eps = 1e-14; # Epsilon
#
# 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)
#
# Die on bad *make() arguments.
sub _cannot_make {
die "@{[(caller(1))[3]]}: Cannot take $_[0] of '$_[1]'.\n";
}
sub _make {
my $arg = shift;
my ($p, $q);
if ($arg =~ /^$gre$/) {
($p, $q) = ($1, 0);
} elsif ($arg =~ /^(?:$gre)?$gre\s*i\s*$/) {
($p, $q) = ($1 || 0, $2);
} elsif ($arg =~ /^\s*\(\s*$gre\s*(?:,\s*$gre\s*)?\)\s*$/) {
($p, $q) = ($1, $2 || 0);
}
if (defined $p) {
$p =~ s/^\+//;
$p =~ s/^(-?)inf$/"${1}9**9**9"/e if $has_inf;
$q =~ s/^\+//;
$q =~ s/^(-?)inf$/"${1}9**9**9"/e if $has_inf;
}
return ($p, $q);
}
sub _emake {
my $arg = shift;
my ($p, $q);
if ($arg =~ /^\s*\[\s*$gre\s*(?:,\s*$gre\s*)?\]\s*$/) {
($p, $q) = ($1, $2 || 0);
} elsif ($arg =~ m!^\s*\[\s*$gre\s*(?:,\s*([-+]?\d*\s*)?pi(?:/\s*(\d+))?\s*)?\]\s*$!) {
($p, $q) = ($1, ($2 eq '-' ? -1 : ($2 || 1)) * pi() / ($3 || 1));
} elsif ($arg =~ /^\s*\[\s*$gre\s*\]\s*$/) {
($p, $q) = ($1, 0);
} elsif ($arg =~ /^\s*$gre\s*$/) {
($p, $q) = ($1, 0);
}
if (defined $p) {
$p =~ s/^\+//;
$q =~ s/^\+//;
$p =~ s/^(-?)inf$/"${1}9**9**9"/e if $has_inf;
$q =~ s/^(-?)inf$/"${1}9**9**9"/e if $has_inf;
}
return ($p, $q);
}
sub _copy {
my $self = shift;
my $clone = {%$self};
if ($self->{'cartesian'}) {
$clone->{'cartesian'} = [@{$self->{'cartesian'}}];
}
if ($self->{'polar'}) {
$clone->{'polar'} = [@{$self->{'polar'}}];
}
bless $clone,__PACKAGE__;
return $clone;
}
#
# ->make
#
# Create a new complex number (cartesian form)
#
sub make {
my $self = bless {}, shift;
my ($re, $im);
if (@_ == 0) {
($re, $im) = (0, 0);
} elsif (@_ == 1) {
return (ref $self)->emake($_[0])
if ($_[0] =~ /^\s*\[/);
($re, $im) = _make($_[0]);
} elsif (@_ == 2) {
($re, $im) = @_;
}
if (defined $re) {
_cannot_make("real part", $re) unless $re =~ /^$gre$/;
}
$im ||= 0;
_cannot_make("imaginary part", $im) unless $im =~ /^$gre$/;
$self->_set_cartesian([$re, $im ]);
$self->display_format('cartesian');
return $self;
}
#
# ->emake
#
# Create a new complex number (exponential form)
#
sub emake {
my $self = bless {}, shift;
my ($rho, $theta);
if (@_ == 0) {
($rho, $theta) = (0, 0);
} elsif (@_ == 1) {
return (ref $self)->make($_[0])
if ($_[0] =~ /^\s*\(/ || $_[0] =~ /i\s*$/);
($rho, $theta) = _emake($_[0]);
} elsif (@_ == 2) {
($rho, $theta) = @_;
}
if (defined $rho && defined $theta) {
if ($rho < 0) {
$rho = -$rho;
$theta = ($theta <= 0) ? $theta + pi() : $theta - pi();
}
}
if (defined $rho) {
_cannot_make("rho", $rho) unless $rho =~ /^$gre$/;
}
$theta ||= 0;
_cannot_make("theta", $theta) unless $theta =~ /^$gre$/;
$self->_set_polar([$rho, $theta]);
$self->display_format('polar');
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 {
return __PACKAGE__->make(@_);
}
#
# cplxe
#
# Creates a complex number from a (rho, theta) tuple.
# This avoids the burden of writing Math::Complex->emake(rho, theta).
#
sub cplxe {
return __PACKAGE__->emake(@_);
}
#
# pi
#
# The number defined as pi = 180 degrees
#
sub pi () { 4 * CORE::atan2(1, 1) }
#
# pi2
#
# The full circle
#
sub pi2 () { 2 * pi }
#
# pi4
#
# The full circle twice.
#
sub pi4 () { 4 * pi }
#
# pip2
#
# The quarter circle
#
sub pip2 () { pi / 2 }
#
# pip4
#
# The eighth circle.
#
sub pip4 () { pi / 4 }
#
# _uplog10
#
# Used in log10().
#
sub _uplog10 () { 1 / CORE::log(10) }
#
# i
#
# The number defined as i*i = -1;
#
sub i () {
return $i if ($i);
$i = bless {};
$i->{'cartesian'} = [0, 1];
$i->{'polar'} = [1, pip2];
$i->{c_dirty} = 0;
$i->{p_dirty} = 0;
return $i;
}
#
# _ip2
#
# Half of i.
#
sub _ip2 () { i / 2 }
#
# Attribute access/set routines
#
sub _cartesian {$_[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]->{c_dirty} = 0;
$_[0]->{'cartesian'} = $_[1] }
sub _set_polar { $_[0]->{c_dirty}++; $_[0]->{p_dirty} = 0;
$_[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 * CORE::cos($t), $r * CORE::sin($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 $self->{'polar'} = [CORE::sqrt($x*$x + $y*$y),
CORE::atan2($y, $x)];
}
#
# (_plus)
#
# Computes z1+z2.
#
sub _plus {
my ($z1, $z2, $regular) = @_;
my ($re1, $im1) = @{$z1->_cartesian};
$z2 = cplx($z2) unless ref $z2;
my ($re2, $im2) = ref $z2 ? @{$z2->_cartesian} : ($z2, 0);
unless (defined $regular) {
$z1->_set_cartesian([$re1 + $re2, $im1 + $im2]);
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};
$z2 = cplx($z2) unless ref $z2;
my ($re2, $im2) = @{$z2->_cartesian};
unless (defined $inverted) {
$z1->_set_cartesian([$re1 - $re2, $im1 - $im2]);
return $z1;
}
return $inverted ?
(ref $z1)->make($re2 - $re1, $im2 - $im1) :
(ref $z1)->make($re1 - $re2, $im1 - $im2);
}
#
# (_multiply)
#
# Computes z1*z2.
#
sub _multiply {
my ($z1, $z2, $regular) = @_;
if ($z1->{p_dirty} == 0 and ref $z2 and $z2->{p_dirty} == 0) {
# if both polar better use polar to avoid rounding errors
my ($r1, $t1) = @{$z1->_polar};
my ($r2, $t2) = @{$z2->_polar};
my $t = $t1 + $t2;
if ($t > pi()) { $t -= pi2 }
elsif ($t <= -pi()) { $t += pi2 }
unless (defined $regular) {
$z1->_set_polar([$r1 * $r2, $t]);
return $z1;
}
return (ref $z1)->emake($r1 * $r2, $t);
} else {
my ($x1, $y1) = @{$z1->_cartesian};
if (ref $z2) {
my ($x2, $y2) = @{$z2->_cartesian};
return (ref $z1)->make($x1*$x2-$y1*$y2, $x1*$y2+$y1*$x2);
} else {
return (ref $z1)->make($x1*$z2, $y1*$z2);
}
}
}
#
# _divbyzero
#
# Die on division by zero.
#
sub _divbyzero {
my $mess = "$_[0]: Division by zero.\n";
if (defined $_[1]) {
$mess .= "(Because in the definition of $_[0], the divisor ";
$mess .= "$_[1] " unless ("$_[1]" eq '0');
$mess .= "is 0)\n";
}
my @up = caller(1);
$mess .= "Died at $up[1] line $up[2].\n";
die $mess;
}
#
# (_divide)
#
# Computes z1/z2.
#
sub _divide {
my ($z1, $z2, $inverted) = @_;
if ($z1->{p_dirty} == 0 and ref $z2 and $z2->{p_dirty} == 0) {
# if both polar better use polar to avoid rounding errors
my ($r1, $t1) = @{$z1->_polar};
my ($r2, $t2) = @{$z2->_polar};
my $t;
if ($inverted) {
_divbyzero "$z2/0" if ($r1 == 0);
$t = $t2 - $t1;
if ($t > pi()) { $t -= pi2 }
elsif ($t <= -pi()) { $t += pi2 }
return (ref $z1)->emake($r2 / $r1, $t);
} else {
_divbyzero "$z1/0" if ($r2 == 0);
$t = $t1 - $t2;
if ($t > pi()) { $t -= pi2 }
elsif ($t <= -pi()) { $t += pi2 }
return (ref $z1)->emake($r1 / $r2, $t);
}
} else {
my ($d, $x2, $y2);
if ($inverted) {
($x2, $y2) = @{$z1->_cartesian};
$d = $x2*$x2 + $y2*$y2;
_divbyzero "$z2/0" if $d == 0;
return (ref $z1)->make(($x2*$z2)/$d, -($y2*$z2)/$d);
} else {
my ($x1, $y1) = @{$z1->_cartesian};
if (ref $z2) {
($x2, $y2) = @{$z2->_cartesian};
$d = $x2*$x2 + $y2*$y2;
_divbyzero "$z1/0" if $d == 0;
my $u = ($x1*$x2 + $y1*$y2)/$d;
my $v = ($y1*$x2 - $x1*$y2)/$d;
return (ref $z1)->make($u, $v);
} else {
_divbyzero "$z1/0" if $z2 == 0;
return (ref $z1)->make($x1/$z2, $y1/$z2);
}
}
}
}
#
# (_power)
#
# Computes z1**z2 = exp(z2 * log z1)).
#
sub _power {
my ($z1, $z2, $inverted) = @_;
if ($inverted) {
return 1 if $z1 == 0 || $z2 == 1;
return 0 if $z2 == 0 && Re($z1) > 0;
} else {
return 1 if $z2 == 0 || $z1 == 1;
return 0 if $z1 == 0 && Re($z2) > 0;
}
my $w = $inverted ? &exp($z1 * &log($z2))
: &exp($z2 * &log($z1));
# If both arguments cartesian, return cartesian, else polar.
return $z1->{c_dirty} == 0 &&
(not ref $z2 or $z2->{c_dirty} == 0) ?
cplx(@{$w->_cartesian}) : $w;
}
#
# (_spaceship)
#
# Computes z1 <=> z2.
# Sorts on the real part first, then on the imaginary part. Thus 2-4i < 3+8i.
#
sub _spaceship {
my ($z1, $z2, $inverted) = @_;
my ($re1, $im1) = ref $z1 ? @{$z1->_cartesian} : ($z1, 0);
my ($re2, $im2) = ref $z2 ? @{$z2->_cartesian} : ($z2, 0);
my $sgn = $inverted ? -1 : 1;
return $sgn * ($re1 <=> $re2) if $re1 != $re2;
return $sgn * ($im1 <=> $im2);
}
#
# (_numeq)
#
# Computes z1 == z2.
#
# (Required in addition to _spaceship() because of NaNs.)
sub _numeq {
my ($z1, $z2, $inverted) = @_;
my ($re1, $im1) = ref $z1 ? @{$z1->_cartesian} : ($z1, 0);
my ($re2, $im2) = ref $z2 ? @{$z2->_cartesian} : ($z2, 0);
return $re1 == $re2 && $im1 == $im2 ? 1 : 0;
}
#
# (_negate)
#
# Computes -z.
#
sub _negate {
my ($z) = @_;
if ($z->{c_dirty}) {
my ($r, $t) = @{$z->_polar};
$t = ($t <= 0) ? $t + pi : $t - pi;
return (ref $z)->emake($r, $t);
}
my ($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, -$im);
}
#
# (abs)
#
# Compute or set complex's norm (rho).
#
sub abs {
my ($z, $rho) = @_ ? @_ : $_;
unless (ref $z) {
if (@_ == 2) {
$_[0] = $_[1];
} else {
return CORE::abs($z);
}
}
if (defined $rho) {
$z->{'polar'} = [ $rho, ${$z->_polar}[1] ];
$z->{p_dirty} = 0;
$z->{c_dirty} = 1;
return $rho;
} else {
return ${$z->_polar}[0];
}
}
sub _theta {
my $theta = $_[0];
if ($$theta > pi()) { $$theta -= pi2 }
elsif ($$theta <= -pi()) { $$theta += pi2 }
}
#
# arg
#
# Compute or set complex's argument (theta).
#
sub arg {
my ($z, $theta) = @_;
return $z unless ref $z;
if (defined $theta) {
_theta(\$theta);
$z->{'polar'} = [ ${$z->_polar}[0], $theta ];
$z->{p_dirty} = 0;
$z->{c_dirty} = 1;
} else {
$theta = ${$z->_polar}[1];
_theta(\$theta);
}
return $theta;
}
#
# (sqrt)
#
# Compute sqrt(z).
#
# It is quite tempting to use wantarray here so that in list context
# sqrt() would return the two solutions. This, however, would
# break things like
#
# print "sqrt(z) = ", sqrt($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) = @_ ? $_[0] : $_;
my ($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 0 if $r == 0;
return (ref $z)->emake(CORE::exp(CORE::log($r)/3), $t/3);
}
#
# _rootbad
#
# Die on bad root.
#
sub _rootbad {
my $mess = "Root '$_[0]' illegal, root rank 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, $k) = @_;
_rootbad($n) if ($n < 1 or int($n) != $n);
my ($r, $t) = ref $z ?
@{$z->_polar} : (CORE::abs($z), $z >= 0 ? 0 : pi);
my $theta_inc = pi2 / $n;
my $rho = $r ** (1/$n);
my $cartesian = ref $z && $z->{c_dirty} == 0;
if (@_ == 2) {
my @root;
for (my $i = 0, my $theta = $t / $n;
$i < $n;
$i++, $theta += $theta_inc) {
my $w = cplxe($rho, $theta);
# Yes, $cartesian is loop invariant.
push @root, $cartesian ? cplx(@{$w->_cartesian}) : $w;
}
return @root;
} elsif (@_ == 3) {
my $w = cplxe($rho, $t / $n + $k * $theta_inc);
return $cartesian ? cplx(@{$w->_cartesian}) : $w;
}
}
#
# Re
#
# Return or set Re(z).
#
sub Re {
my ($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 0 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) = @_ ? @_ : $_;
return CORE::exp($z) unless ref $z;
my ($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()) { $t -= pi2 }
elsif ($t <= -pi()) { $t += pi2 }
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 $logn; # Cache log(n)
return &log($z) / $logn;
}
#
# (cos)
#
# Compute cos(z) = (exp(iz) + exp(-iz))/2.
#
sub cos {
my ($z) = @_ ? @_ : $_;
return CORE::cos($z) unless ref $z;
my ($x, $y) = @{$z->_cartesian};
my $ey = CORE::exp($y);
my $sx = CORE::sin($x);
my $cx = CORE::cos($x);
my $ey_1 = $ey ? 1 / $ey : Inf();
return (ref $z)->make($cx * ($ey + $ey_1)/2,
$sx * ($ey_1 - $ey)/2);
}
#
# (sin)
#
# Compute sin(z) = (exp(iz) - exp(-iz))/2.
#
sub sin {
my ($z) = @_ ? @_ : $_;
return CORE::sin($z) unless ref $z;
my ($x, $y) = @{$z->_cartesian};
my $ey = CORE::exp($y);
my $sx = CORE::sin($x);
my $cx = CORE::cos($x);
my $ey_1 = $ey ? 1 / $ey : Inf();
return (ref $z)->make($sx * ($ey + $ey_1)/2,
$cx * ($ey - $ey_1)/2);
}
#
# tan
#
# Compute tan(z) = sin(z) / cos(z).
#
sub tan {
my ($z) = @_;
my $cz = &cos($z);
_divbyzero "tan($z)", "cos($z)" if $cz == 0;
return &sin($z) / $cz;
}
#
# sec
#
# Computes the secant sec(z) = 1 / cos(z).
#
sub sec {
my ($z) = @_;
my $cz = &cos($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 $sz = &sin($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 $sz = &sin($z);
_divbyzero "cot($z)", "sin($z)" if ($sz == 0);
return &cos($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;
$z = cplx($z, 0) unless ref $z;
my ($x, $y) = @{$z->_cartesian};
return 0 if $x == 1 && $y == 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 (ref $z)->make($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;
$z = cplx($z, 0) unless ref $z;
my ($x, $y) = @{$z->_cartesian};
return 0 if $x == 0 && $y == 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 (ref $z)->make($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 $z;
my ($x, $y) = ref $z ? @{$z->_cartesian} : ($z, 0);
return 0 if $x == 0 && $y == 0;
_divbyzero "atan(i)" if ( $z == i);
_logofzero "atan(-i)" if (-$z == i); # -i is a bad file test...
my $log = &log((i + $z) / (i - $z));
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 $z == 0;
return ($z >= 0) ? CORE::atan2(1, $z) : CORE::atan2(-1, -$z)
unless ref $z;
_divbyzero "acot(i)" if ($z - i == 0);
_logofzero "acot(-i)" if ($z + i == 0);
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 $ex ? ($ex == $ExpInf ? Inf() : ($ex + 1/$ex)/2) : Inf();
}
my ($x, $y) = @{$z->_cartesian};
$ex = CORE::exp($x);
my $ex_1 = $ex ? 1 / $ex : Inf();
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) {
return 0 if $z == 0;
$ex = CORE::exp($z);
return $ex ? ($ex == $ExpInf ? Inf() : ($ex - 1/$ex)/2) : -Inf();
}
my ($x, $y) = @{$z->_cartesian};
my $cy = CORE::cos($y);
my $sy = CORE::sin($y);
$ex = CORE::exp($x);
my $ex_1 = $ex ? 1 / $ex : Inf();
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);
my $sz = sinh($z);
return 1 if $cz == $sz;
return -1 if $cz == -$sz;
return $sz / $cz;
}
#
# sech
#
# Computes the hyperbolic secant sech(z) = 1 / cosh(z).
#
sub sech {
my ($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;
my $cz = cosh($z);
return 1 if $cz == $sz;
return -1 if $cz == -$sz;
return $cz / $sz;
}
#
# cotanh
#
# Alias for coth().
#
sub cotanh { Math::Complex::coth(@_) }
#
# acosh
#
# Computes the area/inverse hyperbolic cosine acosh(z) = log(z + sqrt(z*z-1)).
#
sub acosh {
my ($z) = @_;
unless (ref $z) {
$z = cplx($z, 0);
}
my ($re, $im) = @{$z->_cartesian};
if ($im == 0) {
return CORE::log($re + CORE::sqrt($re*$re - 1))
if $re >= 1;
return cplx(0, CORE::atan2(CORE::sqrt(1 - $re*$re), $re))
if CORE::abs($re) < 1;
}
my $t = &sqrt($z * $z - 1) + $z;
# Try Taylor if looking bad (this usually means that
# $z was large negative, therefore the sqrt is really
# close to abs(z), summing that with z...)
$t = 1/(2 * $z) - 1/(8 * $z**3) + 1/(16 * $z**5) - 5/(128 * $z**7)
if $t == 0;
my $u = &log($t);
$u->Im(-$u->Im) if $re < 0 && $im == 0;
return $re < 0 ? -$u : $u;
}
#
# asinh
#
# Computes the area/inverse hyperbolic sine asinh(z) = log(z + sqrt(z*z+1))
#
sub asinh {
my ($z) = @_;
unless (ref $z) {
my $t = $z + CORE::sqrt($z*$z + 1);
return CORE::log($t) if $t;
}
my $t = &sqrt($z * $z + 1) + $z;
# Try Taylor if looking bad (this usually means that
# $z was large negative, therefore the sqrt is really
# close to abs(z), summing that with z...)
$t = 1/(2 * $z) - 1/(8 * $z**3) + 1/(16 * $z**5) - 5/(128 * $z**7)
if $t == 0;
return &log($t);
}
#
# atanh
#
# Computes the area/inverse 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 (1 - $z == 0);
_logofzero 'atanh(-1)' if (1 + $z == 0);
return 0.5 * &log((1 + $z) / (1 - $z));
}
#
# asech
#
# Computes the area/inverse hyperbolic secant asech(z) = acosh(1 / z).
#
sub asech {
my ($z) = @_;
_divbyzero 'asech(0)', "$z" if ($z == 0);
return acosh(1 / $z);
}
#
# acsch
#
# Computes the area/inverse hyperbolic cosecant acsch(z) = asinh(1 / z).
#
sub acsch {
my ($z) = @_;
_divbyzero 'acsch(0)', $z if ($z == 0);
return asinh(1 / $z);
}
#
# acosech
#
# Alias for acosh().
#
sub acosech { Math::Complex::acsch(@_) }
#
# acoth
#
# Computes the area/inverse hyperbolic cotangent acoth(z) = 1/2 log((1+z) / (z-1)).
#
sub acoth {
my ($z) = @_;
_divbyzero 'acoth(0)' if ($z == 0);
unless (ref $z) {
return CORE::log(($z + 1)/($z - 1))/2 if CORE::abs($z) > 1;
$z = cplx($z, 0);
}
_divbyzero 'acoth(1)', "$z - 1" if ($z - 1 == 0);
_logofzero 'acoth(-1)', "1 + $z" if (1 + $z == 0);
return &log((1 + $z) / ($z - 1)) / 2;
}
#
# acotanh
#
# Alias for acot().
#
sub acotanh { Math::Complex::acoth(@_) }
#
# (atan2)
#
# Compute atan(z1/z2), minding the right quadrant.
#
sub atan2 {
my ($z1, $z2, $inverted) = @_;
my ($re1, $im1, $re2, $im2);
if ($inverted) {
($re1, $im1) = ref $z2 ? @{$z2->_cartesian} : ($z2, 0);
($re2, $im2) = ref $z1 ? @{$z1->_cartesian} : ($z1, 0);
} else {
($re1, $im1) = ref $z1 ? @{$z1->_cartesian} : ($z1, 0);
($re2, $im2) = ref $z2 ? @{$z2->_cartesian} : ($z2, 0);
}
if ($im1 || $im2) {
# In MATLAB the imaginary parts are ignored.
# warn "atan2: Imaginary parts ignored";
# http://documents.wolfram.com/mathematica/functions/ArcTan
# NOTE: Mathematica ArcTan[x,y] while atan2(y,x)
my $s = $z1 * $z1 + $z2 * $z2;
_divbyzero("atan2") if $s == 0;
my $i = &i;
my $r = $z2 + $z1 * $i;
return -$i * &log($r / &sqrt( $s ));
}
return CORE::atan2($re1, $re2);
}
#
# display_format
# ->display_format
#
# Set (get if no argument) the display format for all complex numbers that
# don't happen to have overridden it via ->display_format
#
# When called as an object 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 %display_format = %DISPLAY_FORMAT;
if (ref $self) { # Called as an object method
if (exists $self->{display_format}) {
my %obj = %{$self->{display_format}};
@display_format{keys %obj} = values %obj;
}
}
if (@_ == 1) {
$display_format{style} = shift;
} else {
my %new = @_;
@display_format{keys %new} = values %new;
}
if (ref $self) { # Called as an object method
$self->{display_format} = { %display_format };
return
wantarray ?
%{$self->{display_format}} :
$self->{display_format}->{style};
}
# Called as a class method
%DISPLAY_FORMAT = %display_format;
return
wantarray ?
%DISPLAY_FORMAT :
$DISPLAY_FORMAT{style};
}
#
# (_stringify)
#
# Show nicely formatted complex number under its cartesian or polar form,
# depending on the current display format:
#
# . If a specific display format has been recorded for this object, use it.
# . Otherwise, use the generic current default for all complex numbers,
# which is a package global variable.
#
sub _stringify {
my ($z) = shift;
my $style = $z->display_format;
$style = $DISPLAY_FORMAT{style} unless defined $style;
return $z->_stringify_polar if $style =~ /^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);
my %format = $z->display_format;
my $format = $format{format};
if ($x) {
if ($x =~ /^NaN[QS]?$/i) {
$re = $x;
} else {
if ($x =~ /^-?\Q$Inf\E$/oi) {
$re = $x;
} else {
$re = defined $format ? sprintf($format, $x) : $x;
}
}
} else {
undef $re;
}
if ($y) {
if ($y =~ /^(NaN[QS]?)$/i) {
$im = $y;
} else {
if ($y =~ /^-?\Q$Inf\E$/oi) {
$im = $y;
} else {
$im =
defined $format ?
sprintf($format, $y) :
($y == 1 ? "" : ($y == -1 ? "-" : $y));
}
}
$im .= "i";
} else {
undef $im;
}
my $str = $re;
if (defined $im) {
if ($y < 0) {
$str .= $im;
} elsif ($y > 0 || $im =~ /^NaN[QS]?i$/i) {
$str .= "+" if defined $re;
$str .= $im;
}
} elsif (!defined $re) {
$str = "0";
}
return $str;
}
#
# ->_stringify_polar
#
# Stringify as a polar representation '[r,t]'.
#
sub _stringify_polar {
my $z = shift;
my ($r, $t) = @{$z->_polar};
my $theta;
my %format = $z->display_format;
my $format = $format{format};
if ($t =~ /^NaN[QS]?$/i || $t =~ /^-?\Q$Inf\E$/oi) {
$theta = $t;
} elsif ($t == pi) {
$theta = "pi";
} elsif ($r == 0 || $t == 0) {
$theta = defined $format ? sprintf($format, $t) : $t;
}
return "[$r,$theta]" if defined $theta;
#
# Try to identify pi/n and friends.
#
$t -= int(CORE::abs($t) / pi2) * pi2;
if ($format{polar_pretty_print} && $t) {
my ($a, $b);
for $a (2..9) {
$b = $t * $a / pi;
if ($b =~ /^-?\d+$/) {
$b = $b < 0 ? "-" : "" if CORE::abs($b) == 1;
$theta = "${b}pi/$a";
last;
}
}
}
if (defined $format) {
$r = sprintf($format, $r);
$theta = sprintf($format, $t) unless defined $theta;
} else {
$theta = $t unless defined $theta;
}
return "[$r,$theta]";
}
sub Inf {
return $Inf;
}
1;
__END__
=pod
=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> is
marked here as 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<y> or 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 a 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(y, x) = atan(y / x) # Minding the right quadrant, note the order.
The definition used for complex arguments of atan2() is
-i log((x + iy)/sqrt(x*x+y*y))
Note that atan2(0, 0) is not well-defined.
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 as 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)
You can return the I<k>th root directly by C<root(z, n, k)>,
indexing starting from I<zero> and ending at I<n - 1>.
The I<spaceship> numeric 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 the
C<make>, C<emake>, C<cplx>, and C<cplxe>: the appropriate component of
the argument will be used.
$z1 = cplx(-2, 1);
$z2 = cplx($z1, 4);
The C<new>, C<make>, C<emake>, C<cplx>, and C<cplxe> will also
understand a single (string) argument of the forms
2-3i
-3i
[2,3]
[2,-3pi/4]
[2]
in which case the appropriate cartesian and exponential components
will be parsed from the string and used to create new complex numbers.
The imaginary component and the theta, respectively, will default to zero.
The C<new>, C<make>, C<emake>, C<cplx>, and C<cplxe> will also
understand the case of no arguments: this means plain zero or (0, 0).
=head1 DISPLAYING
When printed, a complex number is usually shown under its cartesian
style I<a+bi>, but there are legitimate cases where the polar style
I<[r,t]> is more appropriate. The process of converting the complex
number into a string that can be displayed is known as I<stringification>.
By calling the class method C<Math::Complex::display_format> and
supplying either C<"polar"> or C<"cartesian"> as an argument, you
override the default display style, which is C<"cartesian">. Not
supplying any argument returns the current settings.
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 style for this number. Otherwise whatever you
specify will be the new display style 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 style attempts to emphasize arguments like I<k*pi/n>
(where I<n> is a positive integer and I<k> an integer within [-9, +9]),
this is called I<polar pretty-printing>.
For the reverse of stringifying, see the C<make> and C<emake>.
=head2 CHANGED IN PERL 5.6
The C<display_format> class method and the corresponding
C<display_format> object method can now be called using
a parameter hash instead of just a one parameter.
The old display format style, which can have values C<"cartesian"> or
C<"polar">, can be changed using the C<"style"> parameter.
$j->display_format(style => "polar");
The one parameter calling convention also still works.
$j->display_format("polar");
There are two new display parameters.
The first one is C<"format">, which is a sprintf()-style format string
to be used for both numeric parts of the complex number(s). The is
somewhat system-dependent but most often it corresponds to C<"%.15g">.
You can revert to the default by setting the C<format> to C<undef>.
# the $j from the above example
$j->display_format('format' => '%.5f');
print "j = $j\n"; # Prints "j = -0.50000+0.86603i"
$j->display_format('format' => undef);
print "j = $j\n"; # Prints "j = -0.5+0.86603i"
Notice that this affects also the return values of the
C<display_format> methods: in list context the whole parameter hash
will be returned, as opposed to only the style parameter value.
This is a potential incompatibility with earlier versions if you
have been calling the C<display_format> method in list context.
The second new display parameter is C<"polar_pretty_print">, which can
be set to true or false, the default being true. See the previous
section for what this means.
=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 CONSTANTS
=head2 PI
The constant C<pi> and some handy multiples of it (pi2, pi4,
and pip2 (pi/2) and pip4 (pi/4)) are also available if separately
exported:
use Math::Complex ':pi';
$third_of_circle = pi2 / 3;
=head2 Inf
The floating point infinity can be exported as a subroutine Inf():
use Math::Complex qw(Inf sinh);
my $AlsoInf = Inf() + 42;
my $AnotherInf = sinh(1e42);
print "$AlsoInf is $AnotherInf\n" if $AlsoInf == $AnotherInf;
Note that the stringified form of infinity varies between platforms:
it can be for example any of
inf
infinity
INF
1.#INF
or it can be something else.
Also note that in some platforms trying to use the infinity in
arithmetic operations may result in Perl crashing because using
an infinity causes SIGFPE or its moral equivalent to be sent.
The way to ignore this is
local $SIG{FPE} = sub { };
=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
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. atan2(0, 0) is undefined, and if the complex arguments
are used for atan2(), a division by zero will happen if z1**2+z2**2 == 0.
Note that because we are operating on approximations of real numbers,
these errors can happen when merely `too close' to the singularities
listed above.
=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>, C<atan2>).
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 SEE ALSO
L<Math::Trig>
=head1 AUTHORS
Daniel S. Lewart <F<lewart!at!uiuc.edu>>,
Jarkko Hietaniemi <F<jhi!at!iki.fi>>,
Raphael Manfredi <F<Raphael_Manfredi!at!pobox.com>>,
Zefram <zefram@fysh.org>
=head1 LICENSE
This library is free software; you can redistribute it and/or modify
it under the same terms as Perl itself.
=cut
1;
# eof
Trig.pm 0000644 00000051576 15036325012 0006022 0 ustar 00 #
# Trigonometric functions, mostly inherited from Math::Complex.
# -- Jarkko Hietaniemi, since April 1997
# -- Raphael Manfredi, September 1996 (indirectly: because of Math::Complex)
#
package Math::Trig;
{ use 5.006; }
use strict;
use Math::Complex 1.59;
use Math::Complex qw(:trig :pi);
require Exporter;
our @ISA = qw(Exporter);
our $VERSION = 1.23;
my @angcnv = qw(rad2deg rad2grad
deg2rad deg2grad
grad2rad grad2deg);
my @areal = qw(asin_real acos_real);
our @EXPORT = (@{$Math::Complex::EXPORT_TAGS{'trig'}},
@angcnv, @areal);
my @rdlcnv = qw(cartesian_to_cylindrical
cartesian_to_spherical
cylindrical_to_cartesian
cylindrical_to_spherical
spherical_to_cartesian
spherical_to_cylindrical);
my @greatcircle = qw(
great_circle_distance
great_circle_direction
great_circle_bearing
great_circle_waypoint
great_circle_midpoint
great_circle_destination
);
my @pi = qw(pi pi2 pi4 pip2 pip4);
our @EXPORT_OK = (@rdlcnv, @greatcircle, @pi, 'Inf');
# See e.g. the following pages:
# http://www.movable-type.co.uk/scripts/LatLong.html
# http://williams.best.vwh.net/avform.htm
our %EXPORT_TAGS = ('radial' => [ @rdlcnv ],
'great_circle' => [ @greatcircle ],
'pi' => [ @pi ]);
sub _DR () { pi2/360 }
sub _RD () { 360/pi2 }
sub _DG () { 400/360 }
sub _GD () { 360/400 }
sub _RG () { 400/pi2 }
sub _GR () { pi2/400 }
#
# Truncating remainder.
#
sub _remt ($$) {
# Oh yes, POSIX::fmod() would be faster. Possibly. If it is available.
$_[0] - $_[1] * int($_[0] / $_[1]);
}
#
# Angle conversions.
#
sub rad2rad($) { _remt($_[0], pi2) }
sub deg2deg($) { _remt($_[0], 360) }
sub grad2grad($) { _remt($_[0], 400) }
sub rad2deg ($;$) { my $d = _RD * $_[0]; $_[1] ? $d : deg2deg($d) }
sub deg2rad ($;$) { my $d = _DR * $_[0]; $_[1] ? $d : rad2rad($d) }
sub grad2deg ($;$) { my $d = _GD * $_[0]; $_[1] ? $d : deg2deg($d) }
sub deg2grad ($;$) { my $d = _DG * $_[0]; $_[1] ? $d : grad2grad($d) }
sub rad2grad ($;$) { my $d = _RG * $_[0]; $_[1] ? $d : grad2grad($d) }
sub grad2rad ($;$) { my $d = _GR * $_[0]; $_[1] ? $d : rad2rad($d) }
#
# acos and asin functions which always return a real number
#
sub acos_real {
return 0 if $_[0] >= 1;
return pi if $_[0] <= -1;
return acos($_[0]);
}
sub asin_real {
return &pip2 if $_[0] >= 1;
return -&pip2 if $_[0] <= -1;
return asin($_[0]);
}
sub cartesian_to_spherical {
my ( $x, $y, $z ) = @_;
my $rho = sqrt( $x * $x + $y * $y + $z * $z );
return ( $rho,
atan2( $y, $x ),
$rho ? acos_real( $z / $rho ) : 0 );
}
sub spherical_to_cartesian {
my ( $rho, $theta, $phi ) = @_;
return ( $rho * cos( $theta ) * sin( $phi ),
$rho * sin( $theta ) * sin( $phi ),
$rho * cos( $phi ) );
}
sub spherical_to_cylindrical {
my ( $x, $y, $z ) = spherical_to_cartesian( @_ );
return ( sqrt( $x * $x + $y * $y ), $_[1], $z );
}
sub cartesian_to_cylindrical {
my ( $x, $y, $z ) = @_;
return ( sqrt( $x * $x + $y * $y ), atan2( $y, $x ), $z );
}
sub cylindrical_to_cartesian {
my ( $rho, $theta, $z ) = @_;
return ( $rho * cos( $theta ), $rho * sin( $theta ), $z );
}
sub cylindrical_to_spherical {
return ( cartesian_to_spherical( cylindrical_to_cartesian( @_ ) ) );
}
sub great_circle_distance {
my ( $theta0, $phi0, $theta1, $phi1, $rho ) = @_;
$rho = 1 unless defined $rho; # Default to the unit sphere.
my $lat0 = pip2 - $phi0;
my $lat1 = pip2 - $phi1;
return $rho *
acos_real( cos( $lat0 ) * cos( $lat1 ) * cos( $theta0 - $theta1 ) +
sin( $lat0 ) * sin( $lat1 ) );
}
sub great_circle_direction {
my ( $theta0, $phi0, $theta1, $phi1 ) = @_;
my $lat0 = pip2 - $phi0;
my $lat1 = pip2 - $phi1;
return rad2rad(pi2 -
atan2(sin($theta0-$theta1) * cos($lat1),
cos($lat0) * sin($lat1) -
sin($lat0) * cos($lat1) * cos($theta0-$theta1)));
}
*great_circle_bearing = \&great_circle_direction;
sub great_circle_waypoint {
my ( $theta0, $phi0, $theta1, $phi1, $point ) = @_;
$point = 0.5 unless defined $point;
my $d = great_circle_distance( $theta0, $phi0, $theta1, $phi1 );
return undef if $d == pi;
my $sd = sin($d);
return ($theta0, $phi0) if $sd == 0;
my $A = sin((1 - $point) * $d) / $sd;
my $B = sin( $point * $d) / $sd;
my $lat0 = pip2 - $phi0;
my $lat1 = pip2 - $phi1;
my $x = $A * cos($lat0) * cos($theta0) + $B * cos($lat1) * cos($theta1);
my $y = $A * cos($lat0) * sin($theta0) + $B * cos($lat1) * sin($theta1);
my $z = $A * sin($lat0) + $B * sin($lat1);
my $theta = atan2($y, $x);
my $phi = acos_real($z);
return ($theta, $phi);
}
sub great_circle_midpoint {
great_circle_waypoint(@_[0..3], 0.5);
}
sub great_circle_destination {
my ( $theta0, $phi0, $dir0, $dst ) = @_;
my $lat0 = pip2 - $phi0;
my $phi1 = asin_real(sin($lat0)*cos($dst) +
cos($lat0)*sin($dst)*cos($dir0));
my $theta1 = $theta0 + atan2(sin($dir0)*sin($dst)*cos($lat0),
cos($dst)-sin($lat0)*sin($phi1));
my $dir1 = great_circle_bearing($theta1, $phi1, $theta0, $phi0) + pi;
$dir1 -= pi2 if $dir1 > pi2;
return ($theta1, $phi1, $dir1);
}
1;
__END__
=pod
=head1 NAME
Math::Trig - trigonometric functions
=head1 SYNOPSIS
use Math::Trig;
$x = tan(0.9);
$y = acos(3.7);
$z = asin(2.4);
$halfpi = pi/2;
$rad = deg2rad(120);
# Import constants pi2, pip2, pip4 (2*pi, pi/2, pi/4).
use Math::Trig ':pi';
# Import the conversions between cartesian/spherical/cylindrical.
use Math::Trig ':radial';
# Import the great circle formulas.
use Math::Trig ':great_circle';
=head1 DESCRIPTION
C<Math::Trig> defines many trigonometric functions not defined by the
core Perl which defines only the C<sin()> and C<cos()>. The constant
B<pi> is also defined as are a few convenience functions for angle
conversions, and I<great circle formulas> for spherical movement.
=head1 TRIGONOMETRIC FUNCTIONS
The tangent
=over 4
=item B<tan>
=back
The cofunctions of the sine, cosine, and tangent (cosec/csc and cotan/cot
are aliases)
B<csc>, B<cosec>, B<sec>, B<sec>, B<cot>, B<cotan>
The arcus (also known as the inverse) functions of the sine, cosine,
and tangent
B<asin>, B<acos>, B<atan>
The principal value of the arc tangent of y/x
B<atan2>(y, x)
The arcus cofunctions of the sine, cosine, and tangent (acosec/acsc
and acotan/acot are aliases). Note that atan2(0, 0) is not well-defined.
B<acsc>, B<acosec>, B<asec>, B<acot>, B<acotan>
The hyperbolic sine, cosine, and tangent
B<sinh>, B<cosh>, B<tanh>
The cofunctions of the hyperbolic sine, cosine, and tangent (cosech/csch
and cotanh/coth are aliases)
B<csch>, B<cosech>, B<sech>, B<coth>, B<cotanh>
The area (also known as the inverse) functions of the hyperbolic
sine, cosine, and tangent
B<asinh>, B<acosh>, B<atanh>
The area cofunctions of the hyperbolic sine, cosine, and tangent
(acsch/acosech and acoth/acotanh are aliases)
B<acsch>, B<acosech>, B<asech>, B<acoth>, B<acotanh>
The trigonometric constant B<pi> and some of handy multiples
of it are also defined.
B<pi, pi2, pi4, pip2, pip4>
=head2 ERRORS DUE TO DIVISION BY ZERO
The following functions
acoth
acsc
acsch
asec
asech
atanh
cot
coth
csc
csch
sec
sech
tan
tanh
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
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<tan>, C<sec>, C<tanh>, C<sech>, the argument cannot be I<pi/2 + k *
pi>, where I<k> is any integer.
Note that atan2(0, 0) is not well-defined.
=head2 SIMPLE (REAL) ARGUMENTS, COMPLEX RESULTS
Please note that some of the trigonometric functions can break out
from the B<real axis> into the B<complex plane>. For example
C<asin(2)> has no definition for plain real numbers but it has
definition for complex numbers.
In Perl terms this means that supplying the usual Perl numbers (also
known as scalars, please see L<perldata>) as input for the
trigonometric functions might produce as output results that no more
are simple real numbers: instead they are complex numbers.
The C<Math::Trig> handles this by using the C<Math::Complex> package
which knows how to handle complex numbers, please see L<Math::Complex>
for more information. In practice you need not to worry about getting
complex numbers as results because the C<Math::Complex> takes care of
details like for example how to display complex numbers. For example:
print asin(2), "\n";
should produce something like this (take or leave few last decimals):
1.5707963267949-1.31695789692482i
That is, a complex number with the real part of approximately C<1.571>
and the imaginary part of approximately C<-1.317>.
=head1 PLANE ANGLE CONVERSIONS
(Plane, 2-dimensional) angles may be converted with the following functions.
=over
=item deg2rad
$radians = deg2rad($degrees);
=item grad2rad
$radians = grad2rad($gradians);
=item rad2deg
$degrees = rad2deg($radians);
=item grad2deg
$degrees = grad2deg($gradians);
=item deg2grad
$gradians = deg2grad($degrees);
=item rad2grad
$gradians = rad2grad($radians);
=back
The full circle is 2 I<pi> radians or I<360> degrees or I<400> gradians.
The result is by default wrapped to be inside the [0, {2pi,360,400}[ circle.
If you don't want this, supply a true second argument:
$zillions_of_radians = deg2rad($zillions_of_degrees, 1);
$negative_degrees = rad2deg($negative_radians, 1);
You can also do the wrapping explicitly by rad2rad(), deg2deg(), and
grad2grad().
=over 4
=item rad2rad
$radians_wrapped_by_2pi = rad2rad($radians);
=item deg2deg
$degrees_wrapped_by_360 = deg2deg($degrees);
=item grad2grad
$gradians_wrapped_by_400 = grad2grad($gradians);
=back
=head1 RADIAL COORDINATE CONVERSIONS
B<Radial coordinate systems> are the B<spherical> and the B<cylindrical>
systems, explained shortly in more detail.
You can import radial coordinate conversion functions by using the
C<:radial> tag:
use Math::Trig ':radial';
($rho, $theta, $z) = cartesian_to_cylindrical($x, $y, $z);
($rho, $theta, $phi) = cartesian_to_spherical($x, $y, $z);
($x, $y, $z) = cylindrical_to_cartesian($rho, $theta, $z);
($rho_s, $theta, $phi) = cylindrical_to_spherical($rho_c, $theta, $z);
($x, $y, $z) = spherical_to_cartesian($rho, $theta, $phi);
($rho_c, $theta, $z) = spherical_to_cylindrical($rho_s, $theta, $phi);
B<All angles are in radians>.
=head2 COORDINATE SYSTEMS
B<Cartesian> coordinates are the usual rectangular I<(x, y, z)>-coordinates.
Spherical coordinates, I<(rho, theta, pi)>, are three-dimensional
coordinates which define a point in three-dimensional space. They are
based on a sphere surface. The radius of the sphere is B<rho>, also
known as the I<radial> coordinate. The angle in the I<xy>-plane
(around the I<z>-axis) is B<theta>, also known as the I<azimuthal>
coordinate. The angle from the I<z>-axis is B<phi>, also known as the
I<polar> coordinate. The North Pole is therefore I<0, 0, rho>, and
the Gulf of Guinea (think of the missing big chunk of Africa) I<0,
pi/2, rho>. In geographical terms I<phi> is latitude (northward
positive, southward negative) and I<theta> is longitude (eastward
positive, westward negative).
B<BEWARE>: some texts define I<theta> and I<phi> the other way round,
some texts define the I<phi> to start from the horizontal plane, some
texts use I<r> in place of I<rho>.
Cylindrical coordinates, I<(rho, theta, z)>, are three-dimensional
coordinates which define a point in three-dimensional space. They are
based on a cylinder surface. The radius of the cylinder is B<rho>,
also known as the I<radial> coordinate. The angle in the I<xy>-plane
(around the I<z>-axis) is B<theta>, also known as the I<azimuthal>
coordinate. The third coordinate is the I<z>, pointing up from the
B<theta>-plane.
=head2 3-D ANGLE CONVERSIONS
Conversions to and from spherical and cylindrical coordinates are
available. Please notice that the conversions are not necessarily
reversible because of the equalities like I<pi> angles being equal to
I<-pi> angles.
=over 4
=item cartesian_to_cylindrical
($rho, $theta, $z) = cartesian_to_cylindrical($x, $y, $z);
=item cartesian_to_spherical
($rho, $theta, $phi) = cartesian_to_spherical($x, $y, $z);
=item cylindrical_to_cartesian
($x, $y, $z) = cylindrical_to_cartesian($rho, $theta, $z);
=item cylindrical_to_spherical
($rho_s, $theta, $phi) = cylindrical_to_spherical($rho_c, $theta, $z);
Notice that when C<$z> is not 0 C<$rho_s> is not equal to C<$rho_c>.
=item spherical_to_cartesian
($x, $y, $z) = spherical_to_cartesian($rho, $theta, $phi);
=item spherical_to_cylindrical
($rho_c, $theta, $z) = spherical_to_cylindrical($rho_s, $theta, $phi);
Notice that when C<$z> is not 0 C<$rho_c> is not equal to C<$rho_s>.
=back
=head1 GREAT CIRCLE DISTANCES AND DIRECTIONS
A great circle is section of a circle that contains the circle
diameter: the shortest distance between two (non-antipodal) points on
the spherical surface goes along the great circle connecting those two
points.
=head2 great_circle_distance
You can compute spherical distances, called B<great circle distances>,
by importing the great_circle_distance() function:
use Math::Trig 'great_circle_distance';
$distance = great_circle_distance($theta0, $phi0, $theta1, $phi1, [, $rho]);
The I<great circle distance> is the shortest distance between two
points on a sphere. The distance is in C<$rho> units. The C<$rho> is
optional, it defaults to 1 (the unit sphere), therefore the distance
defaults to radians.
If you think geographically the I<theta> are longitudes: zero at the
Greenwhich meridian, eastward positive, westward negative -- and the
I<phi> are latitudes: zero at the North Pole, northward positive,
southward negative. B<NOTE>: this formula thinks in mathematics, not
geographically: the I<phi> zero is at the North Pole, not at the
Equator on the west coast of Africa (Bay of Guinea). You need to
subtract your geographical coordinates from I<pi/2> (also known as 90
degrees).
$distance = great_circle_distance($lon0, pi/2 - $lat0,
$lon1, pi/2 - $lat1, $rho);
=head2 great_circle_direction
The direction you must follow the great circle (also known as I<bearing>)
can be computed by the great_circle_direction() function:
use Math::Trig 'great_circle_direction';
$direction = great_circle_direction($theta0, $phi0, $theta1, $phi1);
=head2 great_circle_bearing
Alias 'great_circle_bearing' for 'great_circle_direction' is also available.
use Math::Trig 'great_circle_bearing';
$direction = great_circle_bearing($theta0, $phi0, $theta1, $phi1);
The result of great_circle_direction is in radians, zero indicating
straight north, pi or -pi straight south, pi/2 straight west, and
-pi/2 straight east.
=head2 great_circle_destination
You can inversely compute the destination if you know the
starting point, direction, and distance:
use Math::Trig 'great_circle_destination';
# $diro is the original direction,
# for example from great_circle_bearing().
# $distance is the angular distance in radians,
# for example from great_circle_distance().
# $thetad and $phid are the destination coordinates,
# $dird is the final direction at the destination.
($thetad, $phid, $dird) =
great_circle_destination($theta, $phi, $diro, $distance);
or the midpoint if you know the end points:
=head2 great_circle_midpoint
use Math::Trig 'great_circle_midpoint';
($thetam, $phim) =
great_circle_midpoint($theta0, $phi0, $theta1, $phi1);
The great_circle_midpoint() is just a special case of
=head2 great_circle_waypoint
use Math::Trig 'great_circle_waypoint';
($thetai, $phii) =
great_circle_waypoint($theta0, $phi0, $theta1, $phi1, $way);
Where the $way is a value from zero ($theta0, $phi0) to one ($theta1,
$phi1). Note that antipodal points (where their distance is I<pi>
radians) do not have waypoints between them (they would have an an
"equator" between them), and therefore C<undef> is returned for
antipodal points. If the points are the same and the distance
therefore zero and all waypoints therefore identical, the first point
(either point) is returned.
The thetas, phis, direction, and distance in the above are all in radians.
You can import all the great circle formulas by
use Math::Trig ':great_circle';
Notice that the resulting directions might be somewhat surprising if
you are looking at a flat worldmap: in such map projections the great
circles quite often do not look like the shortest routes -- but for
example the shortest possible routes from Europe or North America to
Asia do often cross the polar regions. (The common Mercator projection
does B<not> show great circles as straight lines: straight lines in the
Mercator projection are lines of constant bearing.)
=head1 EXAMPLES
To calculate the distance between London (51.3N 0.5W) and Tokyo
(35.7N 139.8E) in kilometers:
use Math::Trig qw(great_circle_distance deg2rad);
# Notice the 90 - latitude: phi zero is at the North Pole.
sub NESW { deg2rad($_[0]), deg2rad(90 - $_[1]) }
my @L = NESW( -0.5, 51.3);
my @T = NESW(139.8, 35.7);
my $km = great_circle_distance(@L, @T, 6378); # About 9600 km.
The direction you would have to go from London to Tokyo (in radians,
straight north being zero, straight east being pi/2).
use Math::Trig qw(great_circle_direction);
my $rad = great_circle_direction(@L, @T); # About 0.547 or 0.174 pi.
The midpoint between London and Tokyo being
use Math::Trig qw(great_circle_midpoint);
my @M = great_circle_midpoint(@L, @T);
or about 69 N 89 E, in the frozen wastes of Siberia.
B<NOTE>: you B<cannot> get from A to B like this:
Dist = great_circle_distance(A, B)
Dir = great_circle_direction(A, B)
C = great_circle_destination(A, Dist, Dir)
and expect C to be B, because the bearing constantly changes when
going from A to B (except in some special case like the meridians or
the circles of latitudes) and in great_circle_destination() one gives
a B<constant> bearing to follow.
=head2 CAVEAT FOR GREAT CIRCLE FORMULAS
The answers may be off by few percentages because of the irregular
(slightly aspherical) form of the Earth. The errors are at worst
about 0.55%, but generally below 0.3%.
=head2 Real-valued asin and acos
For small inputs asin() and acos() may return complex numbers even
when real numbers would be enough and correct, this happens because of
floating-point inaccuracies. You can see these inaccuracies for
example by trying theses:
print cos(1e-6)**2+sin(1e-6)**2 - 1,"\n";
printf "%.20f", cos(1e-6)**2+sin(1e-6)**2,"\n";
which will print something like this
-1.11022302462516e-16
0.99999999999999988898
even though the expected results are of course exactly zero and one.
The formulas used to compute asin() and acos() are quite sensitive to
this, and therefore they might accidentally slip into the complex
plane even when they should not. To counter this there are two
interfaces that are guaranteed to return a real-valued output.
=over 4
=item asin_real
use Math::Trig qw(asin_real);
$real_angle = asin_real($input_sin);
Return a real-valued arcus sine if the input is between [-1, 1],
B<inclusive> the endpoints. For inputs greater than one, pi/2
is returned. For inputs less than minus one, -pi/2 is returned.
=item acos_real
use Math::Trig qw(acos_real);
$real_angle = acos_real($input_cos);
Return a real-valued arcus cosine if the input is between [-1, 1],
B<inclusive> the endpoints. For inputs greater than one, zero
is returned. For inputs less than minus one, pi is returned.
=back
=head1 BUGS
Saying C<use Math::Trig;> exports many mathematical routines in the
caller environment and even overrides some (C<sin>, C<cos>). This is
construed as a feature by the Authors, actually... ;-)
The code is not optimized for speed, especially because we use
C<Math::Complex> and thus go quite near complex numbers while doing
the computations even when the arguments are not. This, however,
cannot be completely avoided if we want things like C<asin(2)> to give
an answer instead of giving a fatal runtime error.
Do not attempt navigation using these formulas.
L<Math::Complex>
=head1 AUTHORS
Jarkko Hietaniemi <F<jhi!at!iki.fi>>,
Raphael Manfredi <F<Raphael_Manfredi!at!pobox.com>>,
Zefram <zefram@fysh.org>
=head1 LICENSE
This library is free software; you can redistribute it and/or modify
it under the same terms as Perl itself.
=cut
# eof
BigInt/FastCalc.pm 0000644 00000010017 15053105225 0007733 0 ustar 00 package Math::BigInt::FastCalc;
use 5.006;
use strict;
use warnings;
use Math::BigInt::Calc 1.999801;
our @ISA = qw< Math::BigInt::Calc >;
our $VERSION = '0.5006';
##############################################################################
# global constants, flags and accessory
# announce that we are compatible with MBI v1.83 and up
sub api_version () { 2; }
# use Calc to override the methods that we do not provide in XS
require XSLoader;
XSLoader::load(__PACKAGE__, $VERSION, Math::BigInt::Calc->_base_len());
##############################################################################
##############################################################################
1;
__END__
=pod
=head1 NAME
Math::BigInt::FastCalc - Math::BigInt::Calc with some XS for more speed
=head1 SYNOPSIS
# to use it with Math::BigInt
use Math::BigInt lib => 'FastCalc';
# to use it with Math::BigFloat
use Math::BigFloat lib => 'FastCalc';
# to use it with Math::BigRat
use Math::BigRat lib => 'FastCalc';
=head1 DESCRIPTION
Math::BigInt::FastCalc inherits from Math::BigInt::Calc.
Provides support for big integer calculations. Not intended to be used by
other modules. Other modules which sport the same functions can also be used
to support Math::BigInt, like L<Math::BigInt::GMP> or L<Math::BigInt::Pari>.
In order to allow for multiple big integer libraries, Math::BigInt was
rewritten to use library modules for core math routines. Any module which
follows the same API as this can be used instead by using the following:
use Math::BigInt lib => 'libname';
'libname' is either the long name ('Math::BigInt::Pari'), or only the short
version like 'Pari'. To use this library:
use Math::BigInt lib => 'FastCalc';
=head1 STORAGE
Math::BigInt::FastCalc works exactly like Math::BigInt::Calc. Numbers are
stored in decimal form chopped into parts.
=head1 METHODS
The following functions are now implemented in FastCalc.xs:
_is_odd _is_even _is_one _is_zero
_is_two _is_ten
_zero _one _two _ten
_acmp _len
_inc _dec
__strip_zeros _copy
=head1 BUGS
Please report any bugs or feature requests to
C<bug-math-bigint-fastcalc at rt.cpan.org>, or through the web interface at
L<https://rt.cpan.org/Ticket/Create.html?Queue=Math-BigInt-FastCalc>
(requires login).
We will be notified, and then you'll automatically be notified of progress on
your bug as I make changes.
=head1 SUPPORT
You can find documentation for this module with the perldoc command.
perldoc Math::BigInt::FastCalc
You can also look for information at:
=over 4
=item * RT: CPAN's request tracker
L<https://rt.cpan.org/Public/Dist/Display.html?Name=Math-BigInt-FastCalc>
=item * AnnoCPAN: Annotated CPAN documentation
L<http://annocpan.org/dist/Math-BigInt-FastCalc>
=item * CPAN Ratings
L<http://cpanratings.perl.org/dist/Math-BigInt-FastCalc>
=item * Search CPAN
L<http://search.cpan.org/dist/Math-BigInt-FastCalc/>
=item * CPAN Testers Matrix
L<http://matrix.cpantesters.org/?dist=Math-BigInt-FastCalc>
=item * The Bignum mailing list
=over 4
=item * Post to mailing list
C<bignum at lists.scsys.co.uk>
=item * View mailing list
L<http://lists.scsys.co.uk/pipermail/bignum/>
=item * Subscribe/Unsubscribe
L<http://lists.scsys.co.uk/cgi-bin/mailman/listinfo/bignum>
=back
=back
=head1 LICENSE
This program is free software; you may redistribute it and/or modify it under
the same terms as Perl itself.
=head1 AUTHORS
Original math code by Mark Biggar, rewritten by Tels L<http://bloodgate.com/>
in late 2000.
Separated from BigInt and shaped API with the help of John Peacock.
Fixed, sped-up and enhanced by Tels http://bloodgate.com 2001-2003.
Further streamlining (api_version 1 etc.) by Tels 2004-2007.
Bug-fixing by Peter John Acklam E<lt>pjacklam@online.noE<gt> 2010-2016.
=head1 SEE ALSO
L<Math::BigInt::Lib> for a description of the API.
Alternative libraries L<Math::BigInt::Calc>, L<Math::BigInt::GMP>, and
L<Math::BigInt::Pari>.
Some of the modules that use these libraries L<Math::BigInt>,
L<Math::BigFloat>, and L<Math::BigRat>.
=cut