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-rw-r--r--ext/Math-Complex/lib/Math/Complex.pm2093
-rw-r--r--ext/Math-Complex/lib/Math/Trig.pm768
-rw-r--r--ext/Math-Complex/t/Complex.t1136
-rw-r--r--ext/Math-Complex/t/Trig.t393
-rw-r--r--ext/Math-Complex/t/underbar.t27
5 files changed, 0 insertions, 4417 deletions
diff --git a/ext/Math-Complex/lib/Math/Complex.pm b/ext/Math-Complex/lib/Math/Complex.pm
deleted file mode 100644
index 8475a2b5d0..0000000000
--- a/ext/Math-Complex/lib/Math/Complex.pm
+++ /dev/null
@@ -1,2093 +0,0 @@
-#
-# 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 strict;
-
-use vars qw($VERSION @ISA @EXPORT @EXPORT_OK %EXPORT_TAGS $Inf $ExpInf);
-
-$VERSION = 1.56;
-
-use Config;
-
-BEGIN {
- my %DBL_MAX =
- (
- 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} = { };
- 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;
- $ExpInf = exp(99999);
- }
- # print "# On this machine, Inf = '$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;
-
-@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
- );
-
-@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);
-
-@EXPORT_OK = @pi;
-
-%EXPORT_TAGS = (
- 'trig' => [@trig],
- 'pi' => [@pi],
-);
-
-use overload
- '+' => \&_plus,
- '-' => \&_minus,
- '*' => \&_multiply,
- '/' => \&_divide,
- '**' => \&_power,
- '==' => \&_numeq,
- '<=>' => \&_spaceship,
- 'neg' => \&_negate,
- '~' => \&_conjugate,
- 'abs' => \&abs,
- 'sqrt' => \&sqrt,
- 'exp' => \&exp,
- 'log' => \&log,
- 'sin' => \&sin,
- 'cos' => \&cos,
- 'tan' => \&tan,
- '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;
- $q =~ s/^\+//;
- $q =~ s/^(-?)inf$/"${1}9**9**9"/e;
- }
-
- return ($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;
- $q =~ s/^(-?)inf$/"${1}9**9**9"/e;
- }
-
- return ($p, $q);
-}
-
-#
-# ->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, $theta) 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>>
-
-=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
diff --git a/ext/Math-Complex/lib/Math/Trig.pm b/ext/Math-Complex/lib/Math/Trig.pm
deleted file mode 100644
index b7767bebcc..0000000000
--- a/ext/Math-Complex/lib/Math/Trig.pm
+++ /dev/null
@@ -1,768 +0,0 @@
-#
-# Trigonometric functions, mostly inherited from Math::Complex.
-# -- Jarkko Hietaniemi, since April 1997
-# -- Raphael Manfredi, September 1996 (indirectly: because of Math::Complex)
-#
-
-require Exporter;
-package Math::Trig;
-
-use 5.005;
-use strict;
-
-use Math::Complex 1.56;
-use Math::Complex qw(:trig :pi);
-
-use vars qw($VERSION $PACKAGE @ISA @EXPORT @EXPORT_OK %EXPORT_TAGS);
-
-@ISA = qw(Exporter);
-
-$VERSION = 1.20;
-
-my @angcnv = qw(rad2deg rad2grad
- deg2rad deg2grad
- grad2rad grad2deg);
-
-my @areal = qw(asin_real acos_real);
-
-@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);
-
-@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
-
-%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 $distance = great_circle_distance($theta0, $phi0, $theta1, $phi1);
-
- my $lat0 = pip2 - $phi0;
- my $lat1 = pip2 - $phi1;
-
- my $direction =
- acos_real((sin($lat1) - sin($lat0) * cos($distance)) /
- (cos($lat0) * sin($distance)));
-
- $direction = pi2 - $direction
- if sin($theta1 - $theta0) < 0;
-
- return rad2rad($direction);
-}
-
-*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>> and
-Raphael Manfredi <F<Raphael_Manfredi!at!pobox.com>>.
-
-=head1 LICENSE
-
-This library is free software; you can redistribute it and/or modify
-it under the same terms as Perl itself.
-
-=cut
-
-# eof
diff --git a/ext/Math-Complex/t/Complex.t b/ext/Math-Complex/t/Complex.t
deleted file mode 100644
index 687d6220df..0000000000
--- a/ext/Math-Complex/t/Complex.t
+++ /dev/null
@@ -1,1136 +0,0 @@
-#!./perl
-
-#
-# Regression tests for the Math::Complex pacakge
-# -- Raphael Manfredi since Sep 1996
-# -- Jarkko Hietaniemi since Mar 1997
-# -- Daniel S. Lewart since Sep 1997
-
-BEGIN {
- if ($ENV{PERL_CORE}) {
- chdir 't' if -d 't';
- #@INC = '../lib';
- }
-}
-
-use Math::Complex 1.54;
-
-use vars qw($VERSION);
-
-$VERSION = 1.92;
-
-my ($args, $op, $target, $test, $test_set, $try, $val, $zvalue, @set, @val);
-
-$test = 0;
-$| = 1;
-my @script = (
- 'my ($res, $s0,$s1,$s2,$s3,$s4,$s5,$s6,$s7,$s8,$s9,$s10,$z0,$z1,$z2);' .
- "\n\n"
-);
-my $eps = 1e-13;
-
-if ($^O eq 'unicos') { # For some reason root() produces very inaccurate
- $eps = 1e-10; # results in Cray UNICOS, and occasionally also
-} # cos(), sin(), cosh(), sinh(). The division
- # of doubles is the current suspect.
-
-while (<DATA>) {
- s/^\s+//;
- next if $_ eq '' || /^\#/;
- chomp;
- $test_set = 0; # Assume not a test over a set of values
- if (/^&(.+)/) {
- $op = $1;
- next;
- }
- elsif (/^\{(.+)\}/) {
- set($1, \@set, \@val);
- next;
- }
- elsif (s/^\|//) {
- $test_set = 1; # Requests we loop over the set...
- }
- my @args = split(/:/);
- if ($test_set == 1) {
- my $i;
- for ($i = 0; $i < @set; $i++) {
- # complex number
- $target = $set[$i];
- # textual value as found in set definition
- $zvalue = $val[$i];
- test($zvalue, $target, @args);
- }
- } else {
- test($op, undef, @args);
- }
-}
-
-#
-
-sub test_mutators {
- my $op;
-
- $test++;
-push(@script, <<'EOT');
-{
- my $z = cplx( 1, 1);
- $z->Re(2);
- $z->Im(3);
- print "# $test Re(z) = ",$z->Re(), " Im(z) = ", $z->Im(), " z = $z\n";
- print 'not ' unless Re($z) == 2 and Im($z) == 3;
-EOT
- push(@script, qq(print "ok $test\\n"}\n));
-
- $test++;
-push(@script, <<'EOT');
-{
- my $z = cplx( 1, 1);
- $z->abs(3 * sqrt(2));
- print "# $test Re(z) = ",$z->Re(), " Im(z) = ", $z->Im(), " z = $z\n";
- print 'not ' unless (abs($z) - 3 * sqrt(2)) < $eps and
- (arg($z) - pi / 4 ) < $eps and
- (Re($z) - 3 ) < $eps and
- (Im($z) - 3 ) < $eps;
-EOT
- push(@script, qq(print "ok $test\\n"}\n));
-
- $test++;
-push(@script, <<'EOT');
-{
- my $z = cplx( 1, 1);
- $z->arg(-3 / 4 * pi);
- print "# $test Re(z) = ",$z->Re(), " Im(z) = ", $z->Im(), " z = $z\n";
- print 'not ' unless (arg($z) + 3 / 4 * pi) < $eps and
- (abs($z) - sqrt(2) ) < $eps and
- (Re($z) + 1 ) < $eps and
- (Im($z) + 1 ) < $eps;
-EOT
- push(@script, qq(print "ok $test\\n"}\n));
-}
-
-test_mutators();
-
-my $constants = '
-my $i = cplx(0, 1);
-my $pi = cplx(pi, 0);
-my $pii = cplx(0, pi);
-my $pip2 = cplx(pi/2, 0);
-my $pip4 = cplx(pi/4, 0);
-my $zero = cplx(0, 0);
-my $inf = 9**9**9;
-';
-
-push(@script, $constants);
-
-
-# test the divbyzeros
-
-sub test_dbz {
- for my $op (@_) {
- $test++;
- push(@script, <<EOT);
- eval '$op';
- (\$bad) = (\$@ =~ /(.+)/);
- print "# $test op = $op divbyzero? \$bad...\n";
- print 'not ' unless (\$@ =~ /Division by zero/);
-EOT
- push(@script, qq(print "ok $test\\n";\n));
- }
-}
-
-# test the logofzeros
-
-sub test_loz {
- for my $op (@_) {
- $test++;
- push(@script, <<EOT);
- eval '$op';
- (\$bad) = (\$@ =~ /(.+)/);
- print "# $test op = $op logofzero? \$bad...\n";
- print 'not ' unless (\$@ =~ /Logarithm of zero/);
-EOT
- push(@script, qq(print "ok $test\\n";\n));
- }
-}
-
-test_dbz(
- 'i/0',
- 'acot(0)',
- 'acot(+$i)',
-# 'acoth(-1)', # Log of zero.
- 'acoth(0)',
- 'acoth(+1)',
- 'acsc(0)',
- 'acsch(0)',
- 'asec(0)',
- 'asech(0)',
- 'atan($i)',
-# 'atanh(-1)', # Log of zero.
- 'atanh(+1)',
- 'cot(0)',
- 'coth(0)',
- 'csc(0)',
- 'csch(0)',
- 'atan(cplx(0, 1), cplx(1, 0))',
- );
-
-test_loz(
- 'log($zero)',
- 'atan(-$i)',
- 'acot(-$i)',
- 'atanh(-1)',
- 'acoth(-1)',
- );
-
-# test the bad roots
-
-sub test_broot {
- for my $op (@_) {
- $test++;
- push(@script, <<EOT);
- eval 'root(2, $op)';
- (\$bad) = (\$@ =~ /(.+)/);
- print "# $test op = $op badroot? \$bad...\n";
- print 'not ' unless (\$@ =~ /root rank must be/);
-EOT
- push(@script, qq(print "ok $test\\n";\n));
- }
-}
-
-test_broot(qw(-3 -2.1 0 0.99));
-
-sub test_display_format {
- $test++;
- push @script, <<EOS;
- print "# package display_format cartesian?\n";
- print "not " unless Math::Complex->display_format eq 'cartesian';
- print "ok $test\n";
-EOS
-
- push @script, <<EOS;
- my \$j = (root(1,3))[1];
-
- \$j->display_format('polar');
-EOS
-
- $test++;
- push @script, <<EOS;
- print "# j display_format polar?\n";
- print "not " unless \$j->display_format eq 'polar';
- print "ok $test\n";
-EOS
-
- $test++;
- push @script, <<EOS;
- print "# j = \$j\n";
- print "not " unless "\$j" eq "[1,2pi/3]";
- print "ok $test\n";
-
- my %display_format;
-
- %display_format = \$j->display_format;
-EOS
-
- $test++;
- push @script, <<EOS;
- print "# display_format{style} polar?\n";
- print "not " unless \$display_format{style} eq 'polar';
- print "ok $test\n";
-EOS
-
- $test++;
- push @script, <<EOS;
- print "# keys %display_format == 2?\n";
- print "not " unless keys %display_format == 2;
- print "ok $test\n";
-
- \$j->display_format('style' => 'cartesian', 'format' => '%.5f');
-EOS
-
- $test++;
- push @script, <<EOS;
- print "# j = \$j\n";
- print "not " unless "\$j" eq "-0.50000+0.86603i";
- print "ok $test\n";
-
- %display_format = \$j->display_format;
-EOS
-
- $test++;
- push @script, <<EOS;
- print "# display_format{format} %.5f?\n";
- print "not " unless \$display_format{format} eq '%.5f';
- print "ok $test\n";
-EOS
-
- $test++;
- push @script, <<EOS;
- print "# keys %display_format == 3?\n";
- print "not " unless keys %display_format == 3;
- print "ok $test\n";
-
- \$j->display_format('format' => undef);
-EOS
-
- $test++;
- push @script, <<EOS;
- print "# j = \$j\n";
- print "not " unless "\$j" =~ /^-0(?:\\.5(?:0000\\d+)?|\\.49999\\d+)\\+0.86602540\\d+i\$/;
- print "ok $test\n";
-
- \$j->display_format('style' => 'polar', 'polar_pretty_print' => 0);
-EOS
-
- $test++;
- push @script, <<EOS;
- print "# j = \$j\n";
- print "not " unless "\$j" =~ /^\\[1,2\\.09439510\\d+\\]\$/;
- print "ok $test\n";
-
- \$j->display_format('style' => 'cartesian', 'format' => '(%.5g)');
-EOS
-
- $test++;
- push @script, <<EOS;
- print "# j = \$j\n";
- print "not " unless "\$j" eq "(-0.5)+(0.86603)i";
- print "ok $test\n";
-EOS
-
- $test++;
- push @script, <<EOS;
- print "# j display_format cartesian?\n";
- print "not " unless \$j->display_format eq 'cartesian';
- print "ok $test\n";
-EOS
-}
-
-test_display_format();
-
-sub test_remake {
- $test++;
- push @script, <<EOS;
- print "# remake 2+3i\n";
- \$z = cplx('2+3i');
- print "not " unless \$z == Math::Complex->make(2,3);
- print "ok $test\n";
-EOS
-
- $test++;
- push @script, <<EOS;
- print "# make 3i\n";
- \$z = Math::Complex->make('3i');
- print "not " unless \$z == cplx(0,3);
- print "ok $test\n";
-EOS
-
- $test++;
- push @script, <<EOS;
- print "# emake [2,3]\n";
- \$z = Math::Complex->emake('[2,3]');
- print "not " unless \$z == cplxe(2,3);
- print "ok $test\n";
-EOS
-
- $test++;
- push @script, <<EOS;
- print "# make (2,3)\n";
- \$z = Math::Complex->make('(2,3)');
- print "not " unless \$z == cplx(2,3);
- print "ok $test\n";
-EOS
-
- $test++;
- push @script, <<EOS;
- print "# emake [2,3pi/8]\n";
- \$z = Math::Complex->emake('[2,3pi/8]');
- print "not " unless \$z == cplxe(2,3*\$pi/8);
- print "ok $test\n";
-EOS
-
- $test++;
- push @script, <<EOS;
- print "# emake [2]\n";
- \$z = Math::Complex->emake('[2]');
- print "not " unless \$z == cplxe(2);
- print "ok $test\n";
-EOS
-}
-
-sub test_no_args {
- push @script, <<'EOS';
-{
- print "# cplx, cplxe, make, emake without arguments\n";
-EOS
-
- $test++;
- push @script, <<EOS;
- my \$z0 = cplx();
- print ((\$z0->Re() == 0) ? "ok $test\n" : "not ok $test\n");
-EOS
-
- $test++;
- push @script, <<EOS;
- print ((\$z0->Im() == 0) ? "ok $test\n" : "not ok $test\n");
-EOS
-
- $test++;
- push @script, <<EOS;
- my \$z1 = cplxe();
- print ((\$z1->rho() == 0) ? "ok $test\n" : "not ok $test\n");
-EOS
-
- $test++;
- push @script, <<EOS;
- print ((\$z1->theta() == 0) ? "ok $test\n" : "not ok $test\n");
-EOS
-
- $test++;
- push @script, <<EOS;
- my \$z2 = Math::Complex->make();
- print ((\$z2->Re() == 0) ? "ok $test\n" : "not ok $test\n");
-EOS
-
- $test++;
- push @script, <<EOS;
- print ((\$z2->Im() == 0) ? "ok $test\n" : "not ok $test\n");
-EOS
-
- $test++;
- push @script, <<EOS;
- my \$z3 = Math::Complex->emake();
- print ((\$z3->rho() == 0) ? "ok $test\n" : "not ok $test\n");
-EOS
-
- $test++;
- push @script, <<EOS;
- print ((\$z3->theta() == 0) ? "ok $test\n" : "not ok $test\n");
-}
-EOS
-}
-
-sub test_atan2 {
- push @script, <<'EOS';
-print "# atan2() with some real arguments\n";
-EOS
- my @real = (-1, 0, 1);
- for my $x (@real) {
- for my $y (@real) {
- next if $x == 0 && $y == 0;
- $test++;
- push @script, <<EOS;
-print ((Math::Complex::atan2($y, $x) == CORE::atan2($y, $x)) ? "ok $test\n" : "not ok $test\n");
-EOS
- }
- }
- push @script, <<'EOS';
- print "# atan2() with some complex arguments\n";
-EOS
- $test++;
- push @script, <<EOS;
- print (abs(atan2(0, cplx(0, 1))) < $eps ? "ok $test\n" : "not ok $test\n");
-EOS
- $test++;
- push @script, <<EOS;
- print (abs(atan2(cplx(0, 1), 0) - \$pip2) < $eps ? "ok $test\n" : "not ok $test\n");
-EOS
- $test++;
- push @script, <<EOS;
- print (abs(atan2(cplx(0, 1), cplx(0, 1)) - \$pip4) < $eps ? "ok $test\n" : "not ok $test\n");
-EOS
- $test++;
- push @script, <<EOS;
- print (abs(atan2(cplx(0, 1), cplx(1, 1)) - cplx(0.553574358897045, 0.402359478108525)) < $eps ? "ok $test\n" : "not ok $test\n");
-EOS
-}
-
-sub test_decplx {
-}
-
-test_remake();
-
-test_no_args();
-
-test_atan2();
-
-test_decplx();
-
-print "1..$test\n";
-#print @script, "\n";
-eval join '', @script;
-die $@ if $@;
-
-sub abop {
- my ($op) = @_;
-
- push(@script, qq(print "# $op=\n";));
-}
-
-sub test {
- my ($op, $z, @args) = @_;
- my ($baop) = 0;
- $test++;
- my $i;
- $baop = 1 if ($op =~ s/;=$//);
- for ($i = 0; $i < @args; $i++) {
- $val = value($args[$i]);
- push @script, "\$z$i = $val;\n";
- }
- if (defined $z) {
- $args = "'$op'"; # Really the value
- $try = "abs(\$z0 - \$z1) <= $eps ? \$z1 : \$z0";
- push @script, "\$res = $try; ";
- push @script, "check($test, $args[0], \$res, \$z$#args, $args);\n";
- } else {
- my ($try, $args);
- if (@args == 2) {
- $try = "$op \$z0";
- $args = "'$args[0]'";
- } else {
- $try = ($op =~ /^\w/) ? "$op(\$z0, \$z1)" : "\$z0 $op \$z1";
- $args = "'$args[0]', '$args[1]'";
- }
- push @script, "\$res = $try; ";
- push @script, "check($test, '$try', \$res, \$z$#args, $args);\n";
- if (@args > 2 and $baop) { # binary assignment ops
- $test++;
- # check the op= works
- push @script, <<EOB;
-{
- my \$za = cplx(ref \$z0 ? \@{\$z0->_cartesian} : (\$z0, 0));
-
- my (\$z1r, \$z1i) = ref \$z1 ? \@{\$z1->_cartesian} : (\$z1, 0);
-
- my \$zb = cplx(\$z1r, \$z1i);
-
- \$za $op= \$zb;
- my (\$zbr, \$zbi) = \@{\$zb->_cartesian};
-
- check($test, '\$z0 $op= \$z1', \$za, \$z$#args, $args);
-EOB
- $test++;
- # check that the rhs has not changed
- push @script, qq(print "not " unless (\$zbr == \$z1r and \$zbi == \$z1i););
- push @script, qq(print "ok $test\\n";\n);
- push @script, "}\n";
- }
- }
-}
-
-sub set {
- my ($set, $setref, $valref) = @_;
- @{$setref} = ();
- @{$valref} = ();
- my @set = split(/;\s*/, $set);
- my @res;
- my $i;
- for ($i = 0; $i < @set; $i++) {
- push(@{$valref}, $set[$i]);
- my $val = value($set[$i]);
- push @script, "\$s$i = $val;\n";
- push @{$setref}, "\$s$i";
- }
-}
-
-sub value {
- local ($_) = @_;
- if (/^\s*\((.*),(.*)\)/) {
- return "cplx($1,$2)";
- }
- elsif (/^\s*([\-\+]?(?:\d+(\.\d+)?|\.\d+)(?:[e[\-\+]\d+])?)/) {
- return "cplx($1,0)";
- }
- elsif (/^\s*\[(.*),(.*)\]/) {
- return "cplxe($1,$2)";
- }
- elsif (/^\s*'(.*)'/) {
- my $ex = $1;
- $ex =~ s/\bz\b/$target/g;
- $ex =~ s/\br\b/abs($target)/g;
- $ex =~ s/\bt\b/arg($target)/g;
- $ex =~ s/\ba\b/Re($target)/g;
- $ex =~ s/\bb\b/Im($target)/g;
- return $ex;
- }
- elsif (/^\s*"(.*)"/) {
- return "\"$1\"";
- }
- return $_;
-}
-
-sub check {
- my ($test, $try, $got, $expected, @z) = @_;
-
- print "# @_\n";
-
- if ("$got" eq "$expected"
- ||
- ($expected =~ /^-?\d/ && $got == $expected)
- ||
- (abs(Math::Complex->make($got) - Math::Complex->make($expected)) < $eps)
- ||
- (abs($got - $expected) < $eps)
- ) {
- print "ok $test\n";
- } else {
- print "not ok $test\n";
- my $args = (@z == 1) ? "z = $z[0]" : "z0 = $z[0], z1 = $z[1]";
- print "# '$try' expected: '$expected' got: '$got' for $args\n";
- }
-}
-
-sub addsq {
- my ($z1, $z2) = @_;
- return ($z1 + i*$z2) * ($z1 - i*$z2);
-}
-
-sub subsq {
- my ($z1, $z2) = @_;
- return ($z1 + $z2) * ($z1 - $z2);
-}
-
-__END__
-&+;=
-(3,4):(3,4):(6,8)
-(-3,4):(3,-4):(0,0)
-(3,4):-3:(0,4)
-1:(4,2):(5,2)
-[2,0]:[2,pi]:(0,0)
-
-&++
-(2,1):(3,1)
-
-&-;=
-(2,3):(-2,-3)
-[2,pi/2]:[2,-(pi)/2]
-2:[2,0]:(0,0)
-[3,0]:2:(1,0)
-3:(4,5):(-1,-5)
-(4,5):3:(1,5)
-(2,1):(3,5):(-1,-4)
-
-&--
-(1,2):(0,2)
-[2,pi]:[3,pi]
-
-&*;=
-(0,1):(0,1):(-1,0)
-(4,5):(1,0):(4,5)
-[2,2*pi/3]:(1,0):[2,2*pi/3]
-2:(0,1):(0,2)
-(0,1):3:(0,3)
-(0,1):(4,1):(-1,4)
-(2,1):(4,-1):(9,2)
-
-&/;=
-(3,4):(3,4):(1,0)
-(4,-5):1:(4,-5)
-1:(0,1):(0,-1)
-(0,6):(0,2):(3,0)
-(9,2):(4,-1):(2,1)
-[4,pi]:[2,pi/2]:[2,pi/2]
-[2,pi/2]:[4,pi]:[0.5,-(pi)/2]
-
-&**;=
-(2,0):(3,0):(8,0)
-(3,0):(2,0):(9,0)
-(2,3):(4,0):(-119,-120)
-(0,0):(1,0):(0,0)
-(0,0):(2,3):(0,0)
-(1,0):(0,0):(1,0)
-(1,0):(1,0):(1,0)
-(1,0):(2,3):(1,0)
-(2,3):(0,0):(1,0)
-(2,3):(1,0):(2,3)
-(0,0):(0,0):(1,0)
-
-&Re
-(3,4):3
-(-3,4):-3
-[1,pi/2]:0
-
-&Im
-(3,4):4
-(3,-4):-4
-[1,pi/2]:1
-
-&abs
-(3,4):5
-(-3,4):5
-
-&arg
-[2,0]:0
-[-2,0]:pi
-
-&~
-(4,5):(4,-5)
-(-3,4):(-3,-4)
-[2,pi/2]:[2,-(pi)/2]
-
-&<
-(3,4):(1,2):0
-(3,4):(3,2):0
-(3,4):(3,8):1
-(4,4):(5,129):1
-
-&==
-(3,4):(4,5):0
-(3,4):(3,5):0
-(3,4):(2,4):0
-(3,4):(3,4):1
-
-&sqrt
--9:(0,3)
-(-100,0):(0,10)
-(16,-30):(5,-3)
-
-&_stringify_cartesian
-(-100,0):"-100"
-(0,1):"i"
-(4,-3):"4-3i"
-(4,0):"4"
-(-4,0):"-4"
-(-2,4):"-2+4i"
-(-2,-1):"-2-i"
-
-&_stringify_polar
-[-1, 0]:"[1,pi]"
-[1, pi/3]:"[1,pi/3]"
-[6, -2*pi/3]:"[6,-2pi/3]"
-[0.5, -9*pi/11]:"[0.5,-9pi/11]"
-
-{ (4,3); [3,2]; (-3,4); (0,2); [2,1] }
-
-|'z + ~z':'2*Re(z)'
-|'z - ~z':'2*i*Im(z)'
-|'z * ~z':'abs(z) * abs(z)'
-
-{ (0.5, 0); (-0.5, 0); (2,3); [3,2]; (-3,2); (0,2); 3; 1.2; (-3, 0); (-2, -1); [2,1] }
-
-|'(root(z, 4))[1] ** 4':'z'
-|'(root(z, 5))[3] ** 5':'z'
-|'(root(z, 8))[7] ** 8':'z'
-|'(root(z, 8, 0)) ** 8':'z'
-|'(root(z, 8, 7)) ** 8':'z'
-|'abs(z)':'r'
-|'acot(z)':'acotan(z)'
-|'acsc(z)':'acosec(z)'
-|'acsc(z)':'asin(1 / z)'
-|'asec(z)':'acos(1 / z)'
-|'cbrt(z)':'cbrt(r) * exp(i * t/3)'
-|'cos(acos(z))':'z'
-|'addsq(cos(z), sin(z))':1
-|'cos(z)':'cosh(i*z)'
-|'subsq(cosh(z), sinh(z))':1
-|'cot(acot(z))':'z'
-|'cot(z)':'1 / tan(z)'
-|'cot(z)':'cotan(z)'
-|'csc(acsc(z))':'z'
-|'csc(z)':'1 / sin(z)'
-|'csc(z)':'cosec(z)'
-|'exp(log(z))':'z'
-|'exp(z)':'exp(a) * exp(i * b)'
-|'ln(z)':'log(z)'
-|'log(exp(z))':'z'
-|'log(z)':'log(r) + i*t'
-|'log10(z)':'log(z) / log(10)'
-|'logn(z, 2)':'log(z) / log(2)'
-|'logn(z, 3)':'log(z) / log(3)'
-|'sec(asec(z))':'z'
-|'sec(z)':'1 / cos(z)'
-|'sin(asin(z))':'z'
-|'sin(i * z)':'i * sinh(z)'
-|'sqrt(z) * sqrt(z)':'z'
-|'sqrt(z)':'sqrt(r) * exp(i * t/2)'
-|'tan(atan(z))':'z'
-|'z**z':'exp(z * log(z))'
-
-{ (1,1); [1,0.5]; (-2, -1); 2; -3; (-1,0.5); (0,0.5); 0.5; (2, 0); (-1, -2) }
-
-|'cosh(acosh(z))':'z'
-|'coth(acoth(z))':'z'
-|'coth(z)':'1 / tanh(z)'
-|'coth(z)':'cotanh(z)'
-|'csch(acsch(z))':'z'
-|'csch(z)':'1 / sinh(z)'
-|'csch(z)':'cosech(z)'
-|'sech(asech(z))':'z'
-|'sech(z)':'1 / cosh(z)'
-|'sinh(asinh(z))':'z'
-|'tanh(atanh(z))':'z'
-
-{ (0.2,-0.4); [1,0.5]; -1.2; (-1,0.5); 0.5; (1.1, 0) }
-
-|'acos(cos(z)) ** 2':'z * z'
-|'acosh(cosh(z)) ** 2':'z * z'
-|'acoth(z)':'acotanh(z)'
-|'acoth(z)':'atanh(1 / z)'
-|'acsch(z)':'acosech(z)'
-|'acsch(z)':'asinh(1 / z)'
-|'asech(z)':'acosh(1 / z)'
-|'asin(sin(z))':'z'
-|'asinh(sinh(z))':'z'
-|'atan(tan(z))':'z'
-|'atanh(tanh(z))':'z'
-
-&log
-(-2.0,0):( 0.69314718055995, 3.14159265358979)
-(-1.0,0):( 0 , 3.14159265358979)
-(-0.5,0):( -0.69314718055995, 3.14159265358979)
-( 0.5,0):( -0.69314718055995, 0 )
-( 1.0,0):( 0 , 0 )
-( 2.0,0):( 0.69314718055995, 0 )
-
-&log
-( 2, 3):( 1.28247467873077, 0.98279372324733)
-(-2, 3):( 1.28247467873077, 2.15879893034246)
-(-2,-3):( 1.28247467873077, -2.15879893034246)
-( 2,-3):( 1.28247467873077, -0.98279372324733)
-
-&sin
-(-2.0,0):( -0.90929742682568, 0 )
-(-1.0,0):( -0.84147098480790, 0 )
-(-0.5,0):( -0.47942553860420, 0 )
-( 0.0,0):( 0 , 0 )
-( 0.5,0):( 0.47942553860420, 0 )
-( 1.0,0):( 0.84147098480790, 0 )
-( 2.0,0):( 0.90929742682568, 0 )
-
-&sin
-( 2, 3):( 9.15449914691143, -4.16890695996656)
-(-2, 3):( -9.15449914691143, -4.16890695996656)
-(-2,-3):( -9.15449914691143, 4.16890695996656)
-( 2,-3):( 9.15449914691143, 4.16890695996656)
-
-&cos
-(-2.0,0):( -0.41614683654714, 0 )
-(-1.0,0):( 0.54030230586814, 0 )
-(-0.5,0):( 0.87758256189037, 0 )
-( 0.0,0):( 1 , 0 )
-( 0.5,0):( 0.87758256189037, 0 )
-( 1.0,0):( 0.54030230586814, 0 )
-( 2.0,0):( -0.41614683654714, 0 )
-
-&cos
-( 2, 3):( -4.18962569096881, -9.10922789375534)
-(-2, 3):( -4.18962569096881, 9.10922789375534)
-(-2,-3):( -4.18962569096881, -9.10922789375534)
-( 2,-3):( -4.18962569096881, 9.10922789375534)
-
-&tan
-(-2.0,0):( 2.18503986326152, 0 )
-(-1.0,0):( -1.55740772465490, 0 )
-(-0.5,0):( -0.54630248984379, 0 )
-( 0.0,0):( 0 , 0 )
-( 0.5,0):( 0.54630248984379, 0 )
-( 1.0,0):( 1.55740772465490, 0 )
-( 2.0,0):( -2.18503986326152, 0 )
-
-&tan
-( 2, 3):( -0.00376402564150, 1.00323862735361)
-(-2, 3):( 0.00376402564150, 1.00323862735361)
-(-2,-3):( 0.00376402564150, -1.00323862735361)
-( 2,-3):( -0.00376402564150, -1.00323862735361)
-
-&sec
-(-2.0,0):( -2.40299796172238, 0 )
-(-1.0,0):( 1.85081571768093, 0 )
-(-0.5,0):( 1.13949392732455, 0 )
-( 0.0,0):( 1 , 0 )
-( 0.5,0):( 1.13949392732455, 0 )
-( 1.0,0):( 1.85081571768093, 0 )
-( 2.0,0):( -2.40299796172238, 0 )
-
-&sec
-( 2, 3):( -0.04167496441114, 0.09061113719624)
-(-2, 3):( -0.04167496441114, -0.09061113719624)
-(-2,-3):( -0.04167496441114, 0.09061113719624)
-( 2,-3):( -0.04167496441114, -0.09061113719624)
-
-&csc
-(-2.0,0):( -1.09975017029462, 0 )
-(-1.0,0):( -1.18839510577812, 0 )
-(-0.5,0):( -2.08582964293349, 0 )
-( 0.5,0):( 2.08582964293349, 0 )
-( 1.0,0):( 1.18839510577812, 0 )
-( 2.0,0):( 1.09975017029462, 0 )
-
-&csc
-( 2, 3):( 0.09047320975321, 0.04120098628857)
-(-2, 3):( -0.09047320975321, 0.04120098628857)
-(-2,-3):( -0.09047320975321, -0.04120098628857)
-( 2,-3):( 0.09047320975321, -0.04120098628857)
-
-&cot
-(-2.0,0):( 0.45765755436029, 0 )
-(-1.0,0):( -0.64209261593433, 0 )
-(-0.5,0):( -1.83048772171245, 0 )
-( 0.5,0):( 1.83048772171245, 0 )
-( 1.0,0):( 0.64209261593433, 0 )
-( 2.0,0):( -0.45765755436029, 0 )
-
-&cot
-( 2, 3):( -0.00373971037634, -0.99675779656936)
-(-2, 3):( 0.00373971037634, -0.99675779656936)
-(-2,-3):( 0.00373971037634, 0.99675779656936)
-( 2,-3):( -0.00373971037634, 0.99675779656936)
-
-&asin
-(-2.0,0):( -1.57079632679490, 1.31695789692482)
-(-1.0,0):( -1.57079632679490, 0 )
-(-0.5,0):( -0.52359877559830, 0 )
-( 0.0,0):( 0 , 0 )
-( 0.5,0):( 0.52359877559830, 0 )
-( 1.0,0):( 1.57079632679490, 0 )
-( 2.0,0):( 1.57079632679490, -1.31695789692482)
-
-&asin
-( 2, 3):( 0.57065278432110, 1.98338702991654)
-(-2, 3):( -0.57065278432110, 1.98338702991654)
-(-2,-3):( -0.57065278432110, -1.98338702991654)
-( 2,-3):( 0.57065278432110, -1.98338702991654)
-
-&acos
-(-2.0,0):( 3.14159265358979, -1.31695789692482)
-(-1.0,0):( 3.14159265358979, 0 )
-(-0.5,0):( 2.09439510239320, 0 )
-( 0.0,0):( 1.57079632679490, 0 )
-( 0.5,0):( 1.04719755119660, 0 )
-( 1.0,0):( 0 , 0 )
-( 2.0,0):( 0 , 1.31695789692482)
-
-&acos
-( 2, 3):( 1.00014354247380, -1.98338702991654)
-(-2, 3):( 2.14144911111600, -1.98338702991654)
-(-2,-3):( 2.14144911111600, 1.98338702991654)
-( 2,-3):( 1.00014354247380, 1.98338702991654)
-
-&atan
-(-2.0,0):( -1.10714871779409, 0 )
-(-1.0,0):( -0.78539816339745, 0 )
-(-0.5,0):( -0.46364760900081, 0 )
-( 0.0,0):( 0 , 0 )
-( 0.5,0):( 0.46364760900081, 0 )
-( 1.0,0):( 0.78539816339745, 0 )
-( 2.0,0):( 1.10714871779409, 0 )
-
-&atan
-( 2, 3):( 1.40992104959658, 0.22907268296854)
-(-2, 3):( -1.40992104959658, 0.22907268296854)
-(-2,-3):( -1.40992104959658, -0.22907268296854)
-( 2,-3):( 1.40992104959658, -0.22907268296854)
-
-&asec
-(-2.0,0):( 2.09439510239320, 0 )
-(-1.0,0):( 3.14159265358979, 0 )
-(-0.5,0):( 3.14159265358979, -1.31695789692482)
-( 0.5,0):( 0 , 1.31695789692482)
-( 1.0,0):( 0 , 0 )
-( 2.0,0):( 1.04719755119660, 0 )
-
-&asec
-( 2, 3):( 1.42041072246703, 0.23133469857397)
-(-2, 3):( 1.72118193112276, 0.23133469857397)
-(-2,-3):( 1.72118193112276, -0.23133469857397)
-( 2,-3):( 1.42041072246703, -0.23133469857397)
-
-&acsc
-(-2.0,0):( -0.52359877559830, 0 )
-(-1.0,0):( -1.57079632679490, 0 )
-(-0.5,0):( -1.57079632679490, 1.31695789692482)
-( 0.5,0):( 1.57079632679490, -1.31695789692482)
-( 1.0,0):( 1.57079632679490, 0 )
-( 2.0,0):( 0.52359877559830, 0 )
-
-&acsc
-( 2, 3):( 0.15038560432786, -0.23133469857397)
-(-2, 3):( -0.15038560432786, -0.23133469857397)
-(-2,-3):( -0.15038560432786, 0.23133469857397)
-( 2,-3):( 0.15038560432786, 0.23133469857397)
-
-&acot
-(-2.0,0):( -0.46364760900081, 0 )
-(-1.0,0):( -0.78539816339745, 0 )
-(-0.5,0):( -1.10714871779409, 0 )
-( 0.5,0):( 1.10714871779409, 0 )
-( 1.0,0):( 0.78539816339745, 0 )
-( 2.0,0):( 0.46364760900081, 0 )
-
-&acot
-( 2, 3):( 0.16087527719832, -0.22907268296854)
-(-2, 3):( -0.16087527719832, -0.22907268296854)
-(-2,-3):( -0.16087527719832, 0.22907268296854)
-( 2,-3):( 0.16087527719832, 0.22907268296854)
-
-&sinh
-(-2.0,0):( -3.62686040784702, 0 )
-(-1.0,0):( -1.17520119364380, 0 )
-(-0.5,0):( -0.52109530549375, 0 )
-( 0.0,0):( 0 , 0 )
-( 0.5,0):( 0.52109530549375, 0 )
-( 1.0,0):( 1.17520119364380, 0 )
-( 2.0,0):( 3.62686040784702, 0 )
-
-&sinh
-( 2, 3):( -3.59056458998578, 0.53092108624852)
-(-2, 3):( 3.59056458998578, 0.53092108624852)
-(-2,-3):( 3.59056458998578, -0.53092108624852)
-( 2,-3):( -3.59056458998578, -0.53092108624852)
-
-&cosh
-(-2.0,0):( 3.76219569108363, 0 )
-(-1.0,0):( 1.54308063481524, 0 )
-(-0.5,0):( 1.12762596520638, 0 )
-( 0.0,0):( 1 , 0 )
-( 0.5,0):( 1.12762596520638, 0 )
-( 1.0,0):( 1.54308063481524, 0 )
-( 2.0,0):( 3.76219569108363, 0 )
-
-&cosh
-( 2, 3):( -3.72454550491532, 0.51182256998738)
-(-2, 3):( -3.72454550491532, -0.51182256998738)
-(-2,-3):( -3.72454550491532, 0.51182256998738)
-( 2,-3):( -3.72454550491532, -0.51182256998738)
-
-&tanh
-(-2.0,0):( -0.96402758007582, 0 )
-(-1.0,0):( -0.76159415595576, 0 )
-(-0.5,0):( -0.46211715726001, 0 )
-( 0.0,0):( 0 , 0 )
-( 0.5,0):( 0.46211715726001, 0 )
-( 1.0,0):( 0.76159415595576, 0 )
-( 2.0,0):( 0.96402758007582, 0 )
-
-&tanh
-( 2, 3):( 0.96538587902213, -0.00988437503832)
-(-2, 3):( -0.96538587902213, -0.00988437503832)
-(-2,-3):( -0.96538587902213, 0.00988437503832)
-( 2,-3):( 0.96538587902213, 0.00988437503832)
-
-&sech
-(-2.0,0):( 0.26580222883408, 0 )
-(-1.0,0):( 0.64805427366389, 0 )
-(-0.5,0):( 0.88681888397007, 0 )
-( 0.0,0):( 1 , 0 )
-( 0.5,0):( 0.88681888397007, 0 )
-( 1.0,0):( 0.64805427366389, 0 )
-( 2.0,0):( 0.26580222883408, 0 )
-
-&sech
-( 2, 3):( -0.26351297515839, -0.03621163655877)
-(-2, 3):( -0.26351297515839, 0.03621163655877)
-(-2,-3):( -0.26351297515839, -0.03621163655877)
-( 2,-3):( -0.26351297515839, 0.03621163655877)
-
-&csch
-(-2.0,0):( -0.27572056477178, 0 )
-(-1.0,0):( -0.85091812823932, 0 )
-(-0.5,0):( -1.91903475133494, 0 )
-( 0.5,0):( 1.91903475133494, 0 )
-( 1.0,0):( 0.85091812823932, 0 )
-( 2.0,0):( 0.27572056477178, 0 )
-
-&csch
-( 2, 3):( -0.27254866146294, -0.04030057885689)
-(-2, 3):( 0.27254866146294, -0.04030057885689)
-(-2,-3):( 0.27254866146294, 0.04030057885689)
-( 2,-3):( -0.27254866146294, 0.04030057885689)
-
-&coth
-(-2.0,0):( -1.03731472072755, 0 )
-(-1.0,0):( -1.31303528549933, 0 )
-(-0.5,0):( -2.16395341373865, 0 )
-( 0.5,0):( 2.16395341373865, 0 )
-( 1.0,0):( 1.31303528549933, 0 )
-( 2.0,0):( 1.03731472072755, 0 )
-
-&coth
-( 2, 3):( 1.03574663776500, 0.01060478347034)
-(-2, 3):( -1.03574663776500, 0.01060478347034)
-(-2,-3):( -1.03574663776500, -0.01060478347034)
-( 2,-3):( 1.03574663776500, -0.01060478347034)
-
-&asinh
-(-2.0,0):( -1.44363547517881, 0 )
-(-1.0,0):( -0.88137358701954, 0 )
-(-0.5,0):( -0.48121182505960, 0 )
-( 0.0,0):( 0 , 0 )
-( 0.5,0):( 0.48121182505960, 0 )
-( 1.0,0):( 0.88137358701954, 0 )
-( 2.0,0):( 1.44363547517881, 0 )
-
-&asinh
-( 2, 3):( 1.96863792579310, 0.96465850440760)
-(-2, 3):( -1.96863792579310, 0.96465850440761)
-(-2,-3):( -1.96863792579310, -0.96465850440761)
-( 2,-3):( 1.96863792579310, -0.96465850440760)
-
-&acosh
-(-2.0,0):( 1.31695789692482, 3.14159265358979)
-(-1.0,0):( 0, 3.14159265358979)
-(-0.5,0):( 0, 2.09439510239320)
-( 0.0,0):( 0, 1.57079632679490)
-( 0.5,0):( 0, 1.04719755119660)
-( 1.0,0):( 0 , 0 )
-( 2.0,0):( 1.31695789692482, 0 )
-
-&acosh
-( 2, 3):( 1.98338702991654, 1.00014354247380)
-(-2, 3):( 1.98338702991653, 2.14144911111600)
-(-2,-3):( 1.98338702991653, -2.14144911111600)
-( 2,-3):( 1.98338702991654, -1.00014354247380)
-
-&atanh
-(-2.0,0):( -0.54930614433405, 1.57079632679490)
-(-0.5,0):( -0.54930614433405, 0 )
-( 0.0,0):( 0 , 0 )
-( 0.5,0):( 0.54930614433405, 0 )
-( 2.0,0):( 0.54930614433405, 1.57079632679490)
-
-&atanh
-( 2, 3):( 0.14694666622553, 1.33897252229449)
-(-2, 3):( -0.14694666622553, 1.33897252229449)
-(-2,-3):( -0.14694666622553, -1.33897252229449)
-( 2,-3):( 0.14694666622553, -1.33897252229449)
-
-&asech
-(-2.0,0):( 0 , 2.09439510239320)
-(-1.0,0):( 0 , 3.14159265358979)
-(-0.5,0):( 1.31695789692482, 3.14159265358979)
-( 0.5,0):( 1.31695789692482, 0 )
-( 1.0,0):( 0 , 0 )
-( 2.0,0):( 0 , 1.04719755119660)
-
-&asech
-( 2, 3):( 0.23133469857397, -1.42041072246703)
-(-2, 3):( 0.23133469857397, -1.72118193112276)
-(-2,-3):( 0.23133469857397, 1.72118193112276)
-( 2,-3):( 0.23133469857397, 1.42041072246703)
-
-&acsch
-(-2.0,0):( -0.48121182505960, 0 )
-(-1.0,0):( -0.88137358701954, 0 )
-(-0.5,0):( -1.44363547517881, 0 )
-( 0.5,0):( 1.44363547517881, 0 )
-( 1.0,0):( 0.88137358701954, 0 )
-( 2.0,0):( 0.48121182505960, 0 )
-
-&acsch
-( 2, 3):( 0.15735549884499, -0.22996290237721)
-(-2, 3):( -0.15735549884499, -0.22996290237721)
-(-2,-3):( -0.15735549884499, 0.22996290237721)
-( 2,-3):( 0.15735549884499, 0.22996290237721)
-
-&acoth
-(-2.0,0):( -0.54930614433405, 0 )
-(-0.5,0):( -0.54930614433405, 1.57079632679490)
-( 0.5,0):( 0.54930614433405, 1.57079632679490)
-( 2.0,0):( 0.54930614433405, 0 )
-
-&acoth
-( 2, 3):( 0.14694666622553, -0.23182380450040)
-(-2, 3):( -0.14694666622553, -0.23182380450040)
-(-2,-3):( -0.14694666622553, 0.23182380450040)
-( 2,-3):( 0.14694666622553, 0.23182380450040)
-
-# eof
diff --git a/ext/Math-Complex/t/Trig.t b/ext/Math-Complex/t/Trig.t
deleted file mode 100644
index ae0b0fd1d6..0000000000
--- a/ext/Math-Complex/t/Trig.t
+++ /dev/null
@@ -1,393 +0,0 @@
-#!./perl
-
-#
-# Regression tests for the Math::Trig package
-#
-# The tests here are quite modest as the Math::Complex tests exercise
-# these interfaces quite vigorously.
-#
-# -- Jarkko Hietaniemi, April 1997
-
-BEGIN {
- if ($ENV{PERL_CORE}) {
- chdir 't' if -d 't';
- #@INC = '../lib';
- }
-}
-
-BEGIN {
- eval { require Test::More };
- if ($@) {
- # We are willing to lose testing in e.g. 5.00504.
- print "1..0 # No Test::More, skipping\n";
- exit(0);
- } else {
- import Test::More;
- }
-}
-
-plan(tests => 153);
-
-use Math::Trig 1.18;
-use Math::Trig 1.18 qw(:pi Inf);
-
-my $pip2 = pi / 2;
-
-use strict;
-
-use vars qw($x $y $z);
-
-my $eps = 1e-11;
-
-if ($^O eq 'unicos') { # See lib/Math/Complex.pm and t/lib/complex.t.
- $eps = 1e-10;
-}
-
-sub near ($$;$) {
- my $e = defined $_[2] ? $_[2] : $eps;
- my $d = $_[1] ? abs($_[0]/$_[1] - 1) : abs($_[0]);
- print "# near? $_[0] $_[1] : $d : $e\n";
- $_[1] ? ($d < $e) : abs($_[0]) < $e;
-}
-
-print "# Sanity checks\n";
-
-ok(near(sin(1), 0.841470984807897));
-ok(near(cos(1), 0.54030230586814));
-ok(near(tan(1), 1.5574077246549));
-
-ok(near(sec(1), 1.85081571768093));
-ok(near(csc(1), 1.18839510577812));
-ok(near(cot(1), 0.642092615934331));
-
-ok(near(asin(1), 1.5707963267949));
-ok(near(acos(1), 0));
-ok(near(atan(1), 0.785398163397448));
-
-ok(near(asec(1), 0));
-ok(near(acsc(1), 1.5707963267949));
-ok(near(acot(1), 0.785398163397448));
-
-ok(near(sinh(1), 1.1752011936438));
-ok(near(cosh(1), 1.54308063481524));
-ok(near(tanh(1), 0.761594155955765));
-
-ok(near(sech(1), 0.648054273663885));
-ok(near(csch(1), 0.850918128239322));
-ok(near(coth(1), 1.31303528549933));
-
-ok(near(asinh(1), 0.881373587019543));
-ok(near(acosh(1), 0));
-ok(near(atanh(0.9), 1.47221948958322)); # atanh(1.0) would be an error.
-
-ok(near(asech(0.9), 0.467145308103262));
-ok(near(acsch(2), 0.481211825059603));
-ok(near(acoth(2), 0.549306144334055));
-
-print "# Basics\n";
-
-$x = 0.9;
-ok(near(tan($x), sin($x) / cos($x)));
-
-ok(near(sinh(2), 3.62686040784702));
-
-ok(near(acsch(0.1), 2.99822295029797));
-
-$x = asin(2);
-is(ref $x, 'Math::Complex');
-
-# avoid using Math::Complex here
-$x =~ /^([^-]+)(-[^i]+)i$/;
-($y, $z) = ($1, $2);
-ok(near($y, 1.5707963267949));
-ok(near($z, -1.31695789692482));
-
-ok(near(deg2rad(90), pi/2));
-
-ok(near(rad2deg(pi), 180));
-
-use Math::Trig ':radial';
-
-{
- my ($r,$t,$z) = cartesian_to_cylindrical(1,1,1);
-
- ok(near($r, sqrt(2)));
- ok(near($t, deg2rad(45)));
- ok(near($z, 1));
-
- ($x,$y,$z) = cylindrical_to_cartesian($r, $t, $z);
-
- ok(near($x, 1));
- ok(near($y, 1));
- ok(near($z, 1));
-
- ($r,$t,$z) = cartesian_to_cylindrical(1,1,0);
-
- ok(near($r, sqrt(2)));
- ok(near($t, deg2rad(45)));
- ok(near($z, 0));
-
- ($x,$y,$z) = cylindrical_to_cartesian($r, $t, $z);
-
- ok(near($x, 1));
- ok(near($y, 1));
- ok(near($z, 0));
-}
-
-{
- my ($r,$t,$f) = cartesian_to_spherical(1,1,1);
-
- ok(near($r, sqrt(3)));
- ok(near($t, deg2rad(45)));
- ok(near($f, atan2(sqrt(2), 1)));
-
- ($x,$y,$z) = spherical_to_cartesian($r, $t, $f);
-
- ok(near($x, 1));
- ok(near($y, 1));
- ok(near($z, 1));
-
- ($r,$t,$f) = cartesian_to_spherical(1,1,0);
-
- ok(near($r, sqrt(2)));
- ok(near($t, deg2rad(45)));
- ok(near($f, deg2rad(90)));
-
- ($x,$y,$z) = spherical_to_cartesian($r, $t, $f);
-
- ok(near($x, 1));
- ok(near($y, 1));
- ok(near($z, 0));
-}
-
-{
- my ($r,$t,$z) = cylindrical_to_spherical(spherical_to_cylindrical(1,1,1));
-
- ok(near($r, 1));
- ok(near($t, 1));
- ok(near($z, 1));
-
- ($r,$t,$z) = spherical_to_cylindrical(cylindrical_to_spherical(1,1,1));
-
- ok(near($r, 1));
- ok(near($t, 1));
- ok(near($z, 1));
-}
-
-{
- use Math::Trig 'great_circle_distance';
-
- ok(near(great_circle_distance(0, 0, 0, pi/2), pi/2));
-
- ok(near(great_circle_distance(0, 0, pi, pi), pi));
-
- # London to Tokyo.
- my @L = (deg2rad(-0.5), deg2rad(90 - 51.3));
- my @T = (deg2rad(139.8), deg2rad(90 - 35.7));
-
- my $km = great_circle_distance(@L, @T, 6378);
-
- ok(near($km, 9605.26637021388));
-}
-
-{
- my $R2D = 57.295779513082320876798154814169;
-
- sub frac { $_[0] - int($_[0]) }
-
- my $lotta_radians = deg2rad(1E+20, 1);
- ok(near($lotta_radians, 1E+20/$R2D));
-
- my $negat_degrees = rad2deg(-1E20, 1);
- ok(near($negat_degrees, -1E+20*$R2D));
-
- my $posit_degrees = rad2deg(-10000, 1);
- ok(near($posit_degrees, -10000*$R2D));
-}
-
-{
- use Math::Trig 'great_circle_direction';
-
- ok(near(great_circle_direction(0, 0, 0, pi/2), pi));
-
-# Retired test: Relies on atan2(0, 0), which is not portable.
-# ok(near(great_circle_direction(0, 0, pi, pi), -pi()/2));
-
- my @London = (deg2rad( -0.167), deg2rad(90 - 51.3));
- my @Tokyo = (deg2rad( 139.5), deg2rad(90 - 35.7));
- my @Berlin = (deg2rad ( 13.417), deg2rad(90 - 52.533));
- my @Paris = (deg2rad ( 2.333), deg2rad(90 - 48.867));
-
- ok(near(rad2deg(great_circle_direction(@London, @Tokyo)),
- 31.791945393073));
-
- ok(near(rad2deg(great_circle_direction(@Tokyo, @London)),
- 336.069766430326));
-
- ok(near(rad2deg(great_circle_direction(@Berlin, @Paris)),
- 246.800348034667));
-
- ok(near(rad2deg(great_circle_direction(@Paris, @Berlin)),
- 58.2079877553156));
-
- use Math::Trig 'great_circle_bearing';
-
- ok(near(rad2deg(great_circle_bearing(@Paris, @Berlin)),
- 58.2079877553156));
-
- use Math::Trig 'great_circle_waypoint';
- use Math::Trig 'great_circle_midpoint';
-
- my ($lon, $lat);
-
- ($lon, $lat) = great_circle_waypoint(@London, @Tokyo, 0.0);
-
- ok(near($lon, $London[0]));
-
- ok(near($lat, $London[1]));
-
- ($lon, $lat) = great_circle_waypoint(@London, @Tokyo, 1.0);
-
- ok(near($lon, $Tokyo[0]));
-
- ok(near($lat, $Tokyo[1]));
-
- ($lon, $lat) = great_circle_waypoint(@London, @Tokyo, 0.5);
-
- ok(near($lon, 1.55609593577679)); # 89.16 E
-
- ok(near($lat, 0.36783532946162)); # 68.93 N
-
- ($lon, $lat) = great_circle_midpoint(@London, @Tokyo);
-
- ok(near($lon, 1.55609593577679)); # 89.16 E
-
- ok(near($lat, 0.367835329461615)); # 68.93 N
-
- ($lon, $lat) = great_circle_waypoint(@London, @Tokyo, 0.25);
-
- ok(near($lon, 0.516073562850837)); # 29.57 E
-
- ok(near($lat, 0.400231313403387)); # 67.07 N
-
- ($lon, $lat) = great_circle_waypoint(@London, @Tokyo, 0.75);
-
- ok(near($lon, 2.17494903805952)); # 124.62 E
-
- ok(near($lat, 0.617809294053591)); # 54.60 N
-
- use Math::Trig 'great_circle_destination';
-
- my $dir1 = great_circle_direction(@London, @Tokyo);
- my $dst1 = great_circle_distance(@London, @Tokyo);
-
- ($lon, $lat) = great_circle_destination(@London, $dir1, $dst1);
-
- ok(near($lon, $Tokyo[0]));
-
- ok(near($lat, $pip2 - $Tokyo[1]));
-
- my $dir2 = great_circle_direction(@Tokyo, @London);
- my $dst2 = great_circle_distance(@Tokyo, @London);
-
- ($lon, $lat) = great_circle_destination(@Tokyo, $dir2, $dst2);
-
- ok(near($lon, $London[0]));
-
- ok(near($lat, $pip2 - $London[1]));
-
- my $dir3 = (great_circle_destination(@London, $dir1, $dst1))[2];
-
- ok(near($dir3, 2.69379263839118)); # about 154.343 deg
-
- my $dir4 = (great_circle_destination(@Tokyo, $dir2, $dst2))[2];
-
- ok(near($dir4, 3.6993902625701)); # about 211.959 deg
-
- ok(near($dst1, $dst2));
-}
-
-print "# Infinity\n";
-
-my $BigDouble = 1e40;
-
-# E.g. netbsd-alpha core dumps on Inf arith without this.
-local $SIG{FPE} = sub { };
-
-ok(Inf() > $BigDouble); # This passes in netbsd-alpha.
-ok(Inf() + $BigDouble > $BigDouble); # This coredumps in netbsd-alpha.
-ok(Inf() + $BigDouble == Inf());
-ok(Inf() - $BigDouble > $BigDouble);
-ok(Inf() - $BigDouble == Inf());
-ok(Inf() * $BigDouble > $BigDouble);
-ok(Inf() * $BigDouble == Inf());
-ok(Inf() / $BigDouble > $BigDouble);
-ok(Inf() / $BigDouble == Inf());
-
-ok(-Inf() < -$BigDouble);
-ok(-Inf() + $BigDouble < $BigDouble);
-ok(-Inf() + $BigDouble == -Inf());
-ok(-Inf() - $BigDouble < -$BigDouble);
-ok(-Inf() - $BigDouble == -Inf());
-ok(-Inf() * $BigDouble < -$BigDouble);
-ok(-Inf() * $BigDouble == -Inf());
-ok(-Inf() / $BigDouble < -$BigDouble);
-ok(-Inf() / $BigDouble == -Inf());
-
-print "# sinh/sech/cosh/csch/tanh/coth unto infinity\n";
-
-ok(near(sinh(100), 1.3441e+43, 1e-3));
-ok(near(sech(100), 7.4402e-44, 1e-3));
-ok(near(cosh(100), 1.3441e+43, 1e-3));
-ok(near(csch(100), 7.4402e-44, 1e-3));
-ok(near(tanh(100), 1));
-ok(near(coth(100), 1));
-
-ok(near(sinh(-100), -1.3441e+43, 1e-3));
-ok(near(sech(-100), 7.4402e-44, 1e-3));
-ok(near(cosh(-100), 1.3441e+43, 1e-3));
-ok(near(csch(-100), -7.4402e-44, 1e-3));
-ok(near(tanh(-100), -1));
-ok(near(coth(-100), -1));
-
-cmp_ok(sinh(1e5), '==', Inf());
-cmp_ok(sech(1e5), '==', 0);
-cmp_ok(cosh(1e5), '==', Inf());
-cmp_ok(csch(1e5), '==', 0);
-cmp_ok(tanh(1e5), '==', 1);
-cmp_ok(coth(1e5), '==', 1);
-
-cmp_ok(sinh(-1e5), '==', -Inf());
-cmp_ok(sech(-1e5), '==', 0);
-cmp_ok(cosh(-1e5), '==', Inf());
-cmp_ok(csch(-1e5), '==', 0);
-cmp_ok(tanh(-1e5), '==', -1);
-cmp_ok(coth(-1e5), '==', -1);
-
-print "# great_circle_distance with small angles\n";
-
-for my $e (qw(1e-2 1e-3 1e-4 1e-5)) {
- # Can't assume == 0 because of floating point fuzz,
- # but let's hope for at least < $e.
- cmp_ok(great_circle_distance(0, $e, 0, $e), '<', $e);
-}
-
-print "# asin_real, acos_real\n";
-
-is(acos_real(-2.0), pi);
-is(acos_real(-1.0), pi);
-is(acos_real(-0.5), acos(-0.5));
-is(acos_real( 0.0), acos( 0.0));
-is(acos_real( 0.5), acos( 0.5));
-is(acos_real( 1.0), 0);
-is(acos_real( 2.0), 0);
-
-is(asin_real(-2.0), -&pip2);
-is(asin_real(-1.0), -&pip2);
-is(asin_real(-0.5), asin(-0.5));
-is(asin_real( 0.0), asin( 0.0));
-is(asin_real( 0.5), asin( 0.5));
-is(asin_real( 1.0), pip2);
-is(asin_real( 2.0), pip2);
-
-# eof
diff --git a/ext/Math-Complex/t/underbar.t b/ext/Math-Complex/t/underbar.t
deleted file mode 100644
index 643e86655c..0000000000
--- a/ext/Math-Complex/t/underbar.t
+++ /dev/null
@@ -1,27 +0,0 @@
-#
-# Tests that the standard Perl 5 functions that we override
-# that operate on the $_ will work correctly [perl #62412]
-#
-
-use Test::More;
-
-my @f = qw(abs cos exp log sin sqrt);
-
-plan tests => scalar @f;
-
-use strict;
-
-use Math::Complex;
-
-my %CORE;
-
-for my $f (@f) {
- local $_ = 0.5;
- $CORE{$f} = eval "CORE::$f";
-}
-
-for my $f (@f) {
- local $_ = 0.5;
- is(eval "Math::Complex::$f", $CORE{$f}, $f);
-}
-
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