require Exporter;
package Math::Complex;
+use 5.005_64;
use strict;
-use vars qw($VERSION @ISA @EXPORT %EXPORT_TAGS);
+our($VERSION, @ISA, @EXPORT, %EXPORT_TAGS);
my ( $i, $ip2, %logn );
-$VERSION = sprintf("%s", q$Id: Complex.pm,v 1.25 1998/02/05 16:07:37 jhi Exp $ =~ /(\d+\.\d+)/);
+$VERSION = sprintf("%s", q$Id: Complex.pm,v 1.26 1998/11/01 00:00:00 dsl Exp $ =~ /(\d+\.\d+)/);
@ISA = qw(Exporter);
#
# The number defined as pi = 180 degrees
#
-use constant pi => 4 * atan2(1, 1);
+sub pi () { 4 * CORE::atan2(1, 1) }
#
# pit2
#
# The full circle
#
-use constant pit2 => 2 * pi;
+sub pit2 () { 2 * pi }
#
# pip2
#
# The quarter circle
#
-use constant pip2 => pi / 2;
+sub pip2 () { pi / 2 }
#
# deg1
# One degree in radians, used in stringify_polar.
#
-use constant deg1 => pi / 180;
+sub deg1 () { pi / 180 }
#
# uplog10
#
# Used in log10().
#
-use constant uplog10 => 1 / log(10);
+sub uplog10 () { 1 / CORE::log(10) }
#
# i
my $self = shift;
my ($r, $t) = @{$self->{'polar'}};
$self->{c_dirty} = 0;
- return $self->{'cartesian'} = [$r * cos $t, $r * sin $t];
+ return $self->{'cartesian'} = [$r * CORE::cos($t), $r * CORE::sin($t)];
}
#
my ($x, $y) = @{$self->{'cartesian'}};
$self->{p_dirty} = 0;
return $self->{'polar'} = [0, 0] if $x == 0 && $y == 0;
- return $self->{'polar'} = [sqrt($x*$x + $y*$y), atan2($y, $x)];
+ return $self->{'polar'} = [CORE::sqrt($x*$x + $y*$y), CORE::atan2($y, $x)];
}
#
}
#
-# _zerotozero
-#
-# Die on zero raised to the zeroth.
-#
-sub _zerotozero {
- my $mess = "The zero raised to the zeroth power is not defined.\n";
-
- my @up = caller(1);
-
- $mess .= "Died at $up[1] line $up[2].\n";
-
- die $mess;
-}
-
-#
# (power)
#
# Computes z1**z2 = exp(z2 * log z1)).
#
sub power {
my ($z1, $z2, $inverted) = @_;
- my $z1z = $z1 == 0;
- my $z2z = $z2 == 0;
- _zerotozero if ($z1z and $z2z);
if ($inverted) {
- return 0 if ($z2z);
- return 1 if ($z1z or $z2 == 1);
+ return 1 if $z1 == 0 || $z2 == 1;
+ return 0 if $z2 == 0 && Re($z1) > 0;
} else {
- return 0 if ($z1z);
- return 1 if ($z2z or $z1 == 1);
+ 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);
+ my $w = $inverted ? CORE::exp($z1 * CORE::log($z2))
+ : CORE::exp($z2 * CORE::log($z1));
# If both arguments cartesian, return cartesian, else polar.
return $z1->{c_dirty} == 0 &&
(not ref $z2 or $z2->{c_dirty} == 0) ?
# (spaceship)
#
# Computes z1 <=> z2.
-# Sorts on the real part first, then on the imaginary part. Thus 2-4i > 3+8i.
+# Sorts on the real part first, then on the imaginary part. Thus 2-4i < 3+8i.
#
sub spaceship {
my ($z1, $z2, $inverted) = @_;
sub sqrt {
my ($z) = @_;
my ($re, $im) = ref $z ? @{$z->cartesian} : ($z, 0);
- return $re < 0 ? cplx(0, sqrt(-$re)) : sqrt($re) if $im == 0;
+ return $re < 0 ? cplx(0, CORE::sqrt(-$re)) : CORE::sqrt($re) if $im == 0;
my ($r, $t) = @{$z->polar};
- return (ref $z)->emake(sqrt($r), $t/2);
+ return (ref $z)->emake(CORE::sqrt($r), $t/2);
}
#
#
sub cbrt {
my ($z) = @_;
- return $z < 0 ? -exp(log(-$z)/3) : ($z > 0 ? exp(log($z)/3): 0)
+ return $z < 0 ? -CORE::exp(CORE::log(-$z)/3) : ($z > 0 ? CORE::exp(CORE::log($z)/3): 0)
unless ref $z;
my ($r, $t) = @{$z->polar};
- return (ref $z)->emake(exp(log($r)/3), $t/3);
+ return (ref $z)->emake(CORE::exp(CORE::log($r)/3), $t/3);
}
#
sub root {
my ($z, $n) = @_;
_rootbad($n) if ($n < 1 or int($n) != $n);
- my ($r, $t) = ref $z ? @{$z->polar} : (abs($z), $z >= 0 ? 0 : pi);
+ my ($r, $t) = ref $z ? @{$z->polar} : (CORE::abs($z), $z >= 0 ? 0 : pi);
my @root;
my $k;
my $theta_inc = pit2 / $n;
sub exp {
my ($z) = @_;
my ($x, $y) = @{$z->cartesian};
- return (ref $z)->emake(exp($x), $y);
+ return (ref $z)->emake(CORE::exp($x), $y);
}
#
my ($z) = @_;
unless (ref $z) {
_logofzero("log") if $z == 0;
- return $z > 0 ? log($z) : cplx(log(-$z), pi);
+ return $z > 0 ? CORE::log($z) : cplx(CORE::log(-$z), pi);
}
my ($r, $t) = @{$z->polar};
_logofzero("log") if $r == 0;
if ($t > pi()) { $t -= pit2 }
elsif ($t <= -pi()) { $t += pit2 }
- return (ref $z)->make(log($r), $t);
+ return (ref $z)->make(CORE::log($r), $t);
}
#
my ($z, $n) = @_;
$z = cplx($z, 0) unless ref $z;
my $logn = $logn{$n};
- $logn = $logn{$n} = log($n) unless defined $logn; # Cache log(n)
- return log($z) / $logn;
+ $logn = $logn{$n} = CORE::log($n) unless defined $logn; # Cache log(n)
+ return CORE::log($z) / $logn;
}
#
sub cos {
my ($z) = @_;
my ($x, $y) = @{$z->cartesian};
- my $ey = exp($y);
+ my $ey = CORE::exp($y);
my $ey_1 = 1 / $ey;
- return (ref $z)->make(cos($x) * ($ey + $ey_1)/2,
- sin($x) * ($ey_1 - $ey)/2);
+ return (ref $z)->make(CORE::cos($x) * ($ey + $ey_1)/2,
+ CORE::sin($x) * ($ey_1 - $ey)/2);
}
#
sub sin {
my ($z) = @_;
my ($x, $y) = @{$z->cartesian};
- my $ey = exp($y);
+ my $ey = CORE::exp($y);
my $ey_1 = 1 / $ey;
- return (ref $z)->make(sin($x) * ($ey + $ey_1)/2,
- cos($x) * ($ey - $ey_1)/2);
+ return (ref $z)->make(CORE::sin($x) * ($ey + $ey_1)/2,
+ CORE::cos($x) * ($ey - $ey_1)/2);
}
#
#
sub tan {
my ($z) = @_;
- my $cz = cos($z);
- _divbyzero "tan($z)", "cos($z)" if (abs($cz) < $eps);
- return sin($z) / $cz;
+ my $cz = CORE::cos($z);
+ _divbyzero "tan($z)", "cos($z)" if (CORE::abs($cz) < $eps);
+ return CORE::sin($z) / $cz;
}
#
#
sub sec {
my ($z) = @_;
- my $cz = cos($z);
+ my $cz = CORE::cos($z);
_divbyzero "sec($z)", "cos($z)" if ($cz == 0);
return 1 / $cz;
}
#
sub csc {
my ($z) = @_;
- my $sz = sin($z);
+ my $sz = CORE::sin($z);
_divbyzero "csc($z)", "sin($z)" if ($sz == 0);
return 1 / $sz;
}
#
sub cot {
my ($z) = @_;
- my $sz = sin($z);
+ my $sz = CORE::sin($z);
_divbyzero "cot($z)", "sin($z)" if ($sz == 0);
- return cos($z) / $sz;
+ return CORE::cos($z) / $sz;
}
#
#
sub acos {
my $z = $_[0];
- return atan2(sqrt(1-$z*$z), $z) if (! ref $z) && abs($z) <= 1;
+ return CORE::atan2(CORE::sqrt(1-$z*$z), $z) if (! ref $z) && CORE::abs($z) <= 1;
my ($x, $y) = ref $z ? @{$z->cartesian} : ($z, 0);
- my $t1 = sqrt(($x+1)*($x+1) + $y*$y);
- my $t2 = sqrt(($x-1)*($x-1) + $y*$y);
+ 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 $u = atan2(sqrt(1-$beta*$beta), $beta);
- my $v = log($alpha + sqrt($alpha*$alpha-1));
+ my $u = CORE::atan2(CORE::sqrt(1-$beta*$beta), $beta);
+ my $v = CORE::log($alpha + CORE::sqrt($alpha*$alpha-1));
$v = -$v if $y > 0 || ($y == 0 && $x < -1);
return $package->make($u, $v);
}
#
sub asin {
my $z = $_[0];
- return atan2($z, sqrt(1-$z*$z)) if (! ref $z) && abs($z) <= 1;
+ return CORE::atan2($z, CORE::sqrt(1-$z*$z)) if (! ref $z) && CORE::abs($z) <= 1;
my ($x, $y) = ref $z ? @{$z->cartesian} : ($z, 0);
- my $t1 = sqrt(($x+1)*($x+1) + $y*$y);
- my $t2 = sqrt(($x-1)*($x-1) + $y*$y);
+ 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 $u = atan2($beta, sqrt(1-$beta*$beta));
- my $v = -log($alpha + sqrt($alpha*$alpha-1));
+ my $u = CORE::atan2($beta, CORE::sqrt(1-$beta*$beta));
+ my $v = -CORE::log($alpha + CORE::sqrt($alpha*$alpha-1));
$v = -$v if $y > 0 || ($y == 0 && $x < -1);
return $package->make($u, $v);
}
#
sub atan {
my ($z) = @_;
- return atan2($z, 1) unless ref $z;
+ return CORE::atan2($z, 1) unless ref $z;
_divbyzero "atan(i)" if ( $z == i);
_divbyzero "atan(-i)" if (-$z == i);
- my $log = log((i + $z) / (i - $z));
+ my $log = CORE::log((i + $z) / (i - $z));
$ip2 = 0.5 * i unless defined $ip2;
return $ip2 * $log;
}
#
sub acot {
my ($z) = @_;
- _divbyzero "acot(0)" if (abs($z) < $eps);
- return ($z >= 0) ? atan2(1, $z) : atan2(-1, -$z) unless ref $z;
- _divbyzero "acot(i)" if (abs($z - i) < $eps);
- _logofzero "acot(-i)" if (abs($z + i) < $eps);
+ _divbyzero "acot(0)" if (CORE::abs($z) < $eps);
+ return ($z >= 0) ? CORE::atan2(1, $z) : CORE::atan2(-1, -$z) unless ref $z;
+ _divbyzero "acot(i)" if (CORE::abs($z - i) < $eps);
+ _logofzero "acot(-i)" if (CORE::abs($z + i) < $eps);
return atan(1 / $z);
}
my ($z) = @_;
my $ex;
unless (ref $z) {
- $ex = exp($z);
+ $ex = CORE::exp($z);
return ($ex + 1/$ex)/2;
}
my ($x, $y) = @{$z->cartesian};
- $ex = exp($x);
+ $ex = CORE::exp($x);
my $ex_1 = 1 / $ex;
- return (ref $z)->make(cos($y) * ($ex + $ex_1)/2,
- sin($y) * ($ex - $ex_1)/2);
+ return (ref $z)->make(CORE::cos($y) * ($ex + $ex_1)/2,
+ CORE::sin($y) * ($ex - $ex_1)/2);
}
#
my ($z) = @_;
my $ex;
unless (ref $z) {
- $ex = exp($z);
+ $ex = CORE::exp($z);
return ($ex - 1/$ex)/2;
}
my ($x, $y) = @{$z->cartesian};
- $ex = exp($x);
+ $ex = CORE::exp($x);
my $ex_1 = 1 / $ex;
- return (ref $z)->make(cos($y) * ($ex - $ex_1)/2,
- sin($y) * ($ex + $ex_1)/2);
+ return (ref $z)->make(CORE::cos($y) * ($ex - $ex_1)/2,
+ CORE::sin($y) * ($ex + $ex_1)/2);
}
#
sub acosh {
my ($z) = @_;
unless (ref $z) {
- return log($z + sqrt($z*$z-1)) if $z >= 1;
+ return CORE::log($z + CORE::sqrt($z*$z-1)) if $z >= 1;
$z = cplx($z, 0);
}
my ($re, $im) = @{$z->cartesian};
if ($im == 0) {
- return cplx(log($re + sqrt($re*$re - 1)), 0) if $re >= 1;
- return cplx(0, atan2(sqrt(1-$re*$re), $re)) if abs($re) <= 1;
+ return cplx(CORE::log($re + CORE::sqrt($re*$re - 1)), 0) if $re >= 1;
+ return cplx(0, CORE::atan2(CORE::sqrt(1-$re*$re), $re)) if CORE::abs($re) <= 1;
}
- return log($z + sqrt($z*$z - 1));
+ return CORE::log($z + CORE::sqrt($z*$z - 1));
}
#
#
sub asinh {
my ($z) = @_;
- return log($z + sqrt($z*$z + 1));
+ return CORE::log($z + CORE::sqrt($z*$z + 1));
}
#
sub atanh {
my ($z) = @_;
unless (ref $z) {
- return log((1 + $z)/(1 - $z))/2 if abs($z) < 1;
+ return CORE::log((1 + $z)/(1 - $z))/2 if CORE::abs($z) < 1;
$z = cplx($z, 0);
}
_divbyzero 'atanh(1)', "1 - $z" if ($z == 1);
_logofzero 'atanh(-1)' if ($z == -1);
- return 0.5 * log((1 + $z) / (1 - $z));
+ return 0.5 * CORE::log((1 + $z) / (1 - $z));
}
#
#
sub acoth {
my ($z) = @_;
- _divbyzero 'acoth(0)' if (abs($z) < $eps);
+ _divbyzero 'acoth(0)' if (CORE::abs($z) < $eps);
unless (ref $z) {
- return log(($z + 1)/($z - 1))/2 if abs($z) > 1;
+ return CORE::log(($z + 1)/($z - 1))/2 if CORE::abs($z) > 1;
$z = cplx($z, 0);
}
- _divbyzero 'acoth(1)', "$z - 1" if (abs($z - 1) < $eps);
- _logofzero 'acoth(-1)', "1 / $z" if (abs($z + 1) < $eps);
- return log((1 + $z) / ($z - 1)) / 2;
+ _divbyzero 'acoth(1)', "$z - 1" if (CORE::abs($z - 1) < $eps);
+ _logofzero 'acoth(-1)', "1 / $z" if (CORE::abs($z + 1) < $eps);
+ return CORE::log((1 + $z) / ($z - 1)) / 2;
}
#
($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0);
}
if ($im2 == 0) {
- return cplx(atan2($re1, $re2), 0) if $im1 == 0;
+ return cplx(CORE::atan2($re1, $re2), 0) if $im1 == 0;
return cplx(($im1<=>0) * pip2, 0) if $re2 == 0;
}
my $w = atan($z1/$z2);
my ($re, $im);
$x = int($x + ($x < 0 ? -1 : 1) * $eps)
- if int(abs($x)) != int(abs($x) + $eps);
+ if int(CORE::abs($x)) != int(CORE::abs($x) + $eps);
$y = int($y + ($y < 0 ? -1 : 1) * $eps)
- if int(abs($y)) != int(abs($y) + $eps);
+ if int(CORE::abs($y)) != int(CORE::abs($y) + $eps);
- $re = "$x" if abs($x) >= $eps;
+ $re = "$x" if CORE::abs($x) >= $eps;
if ($y == 1) { $im = 'i' }
elsif ($y == -1) { $im = '-i' }
- elsif (abs($y) >= $eps) { $im = $y . "i" }
+ elsif (CORE::abs($y) >= $eps) { $im = $y . "i" }
my $str = '';
$str = $re if defined $re;
my ($a, $b) = @_;
my $id = "$a $b";
-
+
unless (exists $gcd{$id}) {
$gcd{$id} = _gcd($a, $b);
$gcd{"$b $a"} = $gcd{$id};
$nt = ($nt - int($nt)) * pit2;
$nt += pit2 if $nt < 0; # Range [0, 2pi]
- if (abs($nt) <= $eps) { $theta = 0 }
- elsif (abs(pi-$nt) <= $eps) { $theta = 'pi' }
+ if (CORE::abs($nt) <= $eps) { $theta = 0 }
+ elsif (CORE::abs(pi-$nt) <= $eps) { $theta = 'pi' }
if (defined $theta) {
$r = int($r + ($r < 0 ? -1 : 1) * $eps)
- if int(abs($r)) != int(abs($r) + $eps);
+ if int(CORE::abs($r)) != int(CORE::abs($r) + $eps);
$theta = int($theta + ($theta < 0 ? -1 : 1) * $eps)
if ($theta ne 'pi' and
- int(abs($theta)) != int(abs($theta) + $eps));
+ int(CORE::abs($theta)) != int(CORE::abs($theta) + $eps));
return "\[$r,$theta\]";
}
$nt -= pit2 if $nt > pi;
- if (abs($nt) >= deg1) {
+ if (CORE::abs($nt) >= deg1) {
my ($n, $k, $kpi);
for ($k = 1, $kpi = pi; $k < 10; $k++, $kpi += pi) {
$n = int($kpi / $nt + ($nt > 0 ? 1 : -1) * 0.5);
- if (abs($kpi/$n - $nt) <= $eps) {
- $n = abs $n;
+ if (CORE::abs($kpi/$n - $nt) <= $eps) {
+ $n = CORE::abs($n);
my $gcd = gcd($k, $n);
if ($gcd > 1) {
$k /= $gcd;
$theta = $nt unless defined $theta;
$r = int($r + ($r < 0 ? -1 : 1) * $eps)
- if int(abs($r)) != int(abs($r) + $eps);
+ if int(CORE::abs($r)) != int(CORE::abs($r) + $eps);
$theta = int($theta + ($theta < 0 ? -1 : 1) * $eps)
if ($theta !~ m(^-?\d*pi/\d+$) and
- int(abs($theta)) != int(abs($theta) + $eps));
+ int(CORE::abs($theta)) != int(CORE::abs($theta) + $eps));
return "\[$r,$theta\]";
}
The division (/) and the following functions
log ln log10 logn
- tan sec csc cot
+ tan sec csc cot
atan asec acsc acot
tanh sech csch coth
atanh asech acsch acoth
=head1 AUTHORS
-Raphael Manfredi <F<Raphael_Manfredi@grenoble.hp.com>> and
+Raphael Manfredi <F<Raphael_Manfredi@pobox.com>> and
Jarkko Hietaniemi <F<jhi@iki.fi>>.
Extensive patches by Daniel S. Lewart <F<d-lewart@uiuc.edu>>.
projects / p5sagit/p5-mst-13.2.git / blobdiff