-# $RCSFile$
#
# Complex numbers and associated mathematical functions
-# -- Raphael Manfredi, September 1996
-# -- Jarkko Hietaniemi, March 1997
-
-require Exporter;
-package Math::Complex; @ISA = qw(Exporter);
+# -- Raphael Manfredi Since Sep 1996
+# -- Jarkko Hietaniemi Since Mar 1997
+# -- Daniel S. Lewart Since Sep 1997
+#
+
+package Math::Complex;
+
+our($VERSION, @ISA, @EXPORT, %EXPORT_TAGS, $Inf);
+
+$VERSION = 1.32;
+
+BEGIN {
+ unless ($^O eq 'unicosmk') {
+ my $e = $!;
+ # We do want an arithmetic overflow, Inf INF inf Infinity:.
+ undef $Inf unless eval <<'EOE' and $Inf =~ /^inf(?:inity)?$/i;
+ local $SIG{FPE} = sub {die};
+ my $t = CORE::exp 30;
+ $Inf = CORE::exp $t;
+EOE
+ if (!defined $Inf) { # Try a different method
+ undef $Inf unless eval <<'EOE' and $Inf =~ /^inf(?:inity)?$/i;
+ local $SIG{FPE} = sub {die};
+ my $t = 1;
+ $Inf = $t + "1e99999999999999999999999999999999";
+EOE
+ }
+ $! = $e; # Clear ERANGE.
+ }
+ $Inf = "Inf" if !defined $Inf || !($Inf > 0); # Desperation.
+}
use strict;
-use vars qw(@EXPORT $package $display
- $pi $i $ilog10 $logn %logn);
-
-@EXPORT = qw(
- pi i Re Im arg
- sqrt exp log ln
- log10 logn cbrt root
- tan
- cosec csc sec cotan cot
- asin acos atan
- acosec acsc asec acotan acot
- sinh cosh tanh
- cosech csch sech cotanh coth
- asinh acosh atanh
- acosech acsch asech acotanh acoth
- cplx cplxe
+my $i;
+my %LOGN;
+
+require Exporter;
+
+@ISA = qw(Exporter);
+
+my @trig = qw(
+ pi
+ tan
+ csc cosec sec cot cotan
+ asin acos atan
+ acsc acosec asec acot acotan
+ sinh cosh tanh
+ csch cosech sech coth cotanh
+ asinh acosh atanh
+ acsch acosech asech acoth acotanh
+ );
+
+@EXPORT = (qw(
+ i Re Im rho theta arg
+ sqrt log ln
+ log10 logn cbrt root
+ cplx cplxe
+ ),
+ @trig);
+
+%EXPORT_TAGS = (
+ 'trig' => [@trig],
);
use overload
'*' => \&multiply,
'/' => \÷,
'**' => \&power,
+ '==' => \&numeq,
'<=>' => \&spaceship,
'neg' => \&negate,
'~' => \&conjugate,
qw("" stringify);
#
-# Package globals
+# Package "privates"
#
-$package = 'Math::Complex'; # Package name
-$display = 'cartesian'; # Default display format
+my %DISPLAY_FORMAT = ('style' => 'cartesian',
+ 'polar_pretty_print' => 1);
+my $eps = 1e-14; # Epsilon
#
# Object attributes (internal):
# 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";
+}
+
#
# ->make
#
sub make {
my $self = bless {}, shift;
my ($re, $im) = @_;
- $self->{'cartesian'} = [$re, $im];
+ my $rre = ref $re;
+ if ( $rre ) {
+ if ( $rre eq ref $self ) {
+ $re = Re($re);
+ } else {
+ _cannot_make("real part", $rre);
+ }
+ }
+ my $rim = ref $im;
+ if ( $rim ) {
+ if ( $rim eq ref $self ) {
+ $im = Im($im);
+ } else {
+ _cannot_make("imaginary part", $rim);
+ }
+ }
+ $self->{'cartesian'} = [ $re, $im ];
$self->{c_dirty} = 0;
$self->{p_dirty} = 1;
+ $self->display_format('cartesian');
return $self;
}
sub emake {
my $self = bless {}, shift;
my ($rho, $theta) = @_;
- $theta += pi() if $rho < 0;
- $self->{'polar'} = [abs($rho), $theta];
+ my $rrh = ref $rho;
+ if ( $rrh ) {
+ if ( $rrh eq ref $self ) {
+ $rho = rho($rho);
+ } else {
+ _cannot_make("rho", $rrh);
+ }
+ }
+ my $rth = ref $theta;
+ if ( $rth ) {
+ if ( $rth eq ref $self ) {
+ $theta = theta($theta);
+ } else {
+ _cannot_make("theta", $rth);
+ }
+ }
+ if ($rho < 0) {
+ $rho = -$rho;
+ $theta = ($theta <= 0) ? $theta + pi() : $theta - pi();
+ }
+ $self->{'polar'} = [$rho, $theta];
$self->{p_dirty} = 0;
$self->{c_dirty} = 1;
+ $self->display_format('polar');
return $self;
}
#
sub cplx {
my ($re, $im) = @_;
- return $package->make($re, defined $im ? $im : 0);
+ return __PACKAGE__->make($re, defined $im ? $im : 0);
}
#
#
sub cplxe {
my ($rho, $theta) = @_;
- return $package->emake($rho, defined $theta ? $theta : 0);
+ return __PACKAGE__->emake($rho, defined $theta ? $theta : 0);
}
#
# pi
#
-# The number defined as 2 * pi = 360 degrees
+# The number defined as pi = 180 degrees
#
-sub pi () {
- $pi = 4 * atan2(1, 1) unless $pi;
- return $pi;
-}
+sub pi () { 4 * CORE::atan2(1, 1) }
+
+#
+# pit2
+#
+# The full circle
+#
+sub pit2 () { 2 * pi }
+
+#
+# pip2
+#
+# The quarter circle
+#
+sub pip2 () { pi / 2 }
+
+#
+# deg1
+#
+# One degree in radians, used in stringify_polar.
+#
+
+sub deg1 () { pi / 180 }
+
+#
+# uplog10
+#
+# Used in log10().
+#
+sub uplog10 () { 1 / CORE::log(10) }
#
# i
# The number defined as i*i = -1;
#
sub i () {
- $i = bless {} unless $i; # There can be only one i
+ return $i if ($i);
+ $i = bless {};
$i->{'cartesian'} = [0, 1];
- $i->{'polar'} = [1, pi/2];
+ $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
#
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, $x)];
}
#
#
sub plus {
my ($z1, $z2, $regular) = @_;
- $z2 = cplx($z2, 0) unless ref $z2;
my ($re1, $im1) = @{$z1->cartesian};
- my ($re2, $im2) = @{$z2->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 $z1;
#
sub minus {
my ($z1, $z2, $inverted) = @_;
- $z2 = cplx($z2, 0) unless ref $z2;
my ($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 $inverted ?
(ref $z1)->make($re2 - $re1, $im2 - $im1) :
(ref $z1)->make($re1 - $re2, $im1 - $im2);
+
}
#
# Computes z1*z2.
#
sub multiply {
- my ($z1, $z2, $regular) = @_;
- my ($r1, $t1) = @{$z1->polar};
- my ($r2, $t2) = ref $z2 ?
- @{$z2->polar} : (abs($z2), $z2 >= 0 ? 0 : pi);
- unless (defined $regular) {
- $z1->set_polar([$r1 * $r2, $t1 + $t2]);
+ my ($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()) { $t -= pit2 }
+ elsif ($t <= -pi()) { $t += pit2 }
+ 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);
+ }
}
- return (ref $z1)->emake($r1 * $r2, $t1 + $t2);
}
#
-# divbyzero
+# _divbyzero
#
# Die on division by zero.
#
-sub divbyzero {
- warn $package . '::' . "$_[0]: Division by zero.\n";
- warn "(Because in the definition of $_[0], $_[1] is 0)\n"
- if (defined $_[1]);
+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);
- my $dmess = "Died at $up[1] line $up[2].\n";
- die $dmess;
+
+ $mess .= "Died at $up[1] line $up[2].\n";
+
+ die $mess;
}
#
#
sub divide {
my ($z1, $z2, $inverted) = @_;
- my ($r1, $t1) = @{$z1->polar};
- my ($r2, $t2) = ref $z2 ?
- @{$z2->polar} : (abs($z2), $z2 >= 0 ? 0 : pi);
- unless (defined $inverted) {
- divbyzero "$z1/0" if ($r2 == 0);
- $z1->set_polar([$r1 / $r2, $t1 - $t2]);
- return $z1;
- }
- if ($inverted) {
- divbyzero "$z2/0" if ($r1 == 0);
- return (ref $z1)->emake($r2 / $r1, $t2 - $t1);
+ if ($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()) { $t -= pit2 }
+ elsif ($t <= -pi()) { $t += pit2 }
+ return (ref $z1)->emake($r2 / $r1, $t);
+ } else {
+ _divbyzero "$z1/0" if ($r2 == 0);
+ $t = $t1 - $t2;
+ if ($t > pi()) { $t -= pit2 }
+ elsif ($t <= -pi()) { $t += pit2 }
+ return (ref $z1)->emake($r1 / $r2, $t);
+ }
} else {
- divbyzero "$z1/0" if ($r2 == 0);
- return (ref $z1)->emake($r1 / $r2, $t1 - $t2);
+ my ($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);
+ }
+ }
}
}
#
sub power {
my ($z1, $z2, $inverted) = @_;
- return exp($z1 * log $z2) if defined $inverted && $inverted;
- return exp($z2 * log $z1);
+ if ($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}) : $w;
}
#
# (spaceship)
#
# Computes z1 <=> z2.
-# Sorts on the real part first, then on the imaginary part. Thus 2-4i > 3+8i.
+# Sorts on the real part first, then on the imaginary part. Thus 2-4i < 3+8i.
#
sub spaceship {
my ($z1, $z2, $inverted) = @_;
- $z2 = cplx($z2, 0) unless ref $z2;
- my ($re1, $im1) = @{$z1->cartesian};
- my ($re2, $im2) = @{$z2->cartesian};
+ 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 == $im2 ? 1 : 0;
+}
+
+#
# (negate)
#
# Computes -z.
my ($z) = @_;
if ($z->{c_dirty}) {
my ($r, $t) = @{$z->polar};
- return (ref $z)->emake($r, pi + $t);
+ $t = ($t <= 0) ? $t + pi : $t - pi;
+ return (ref $z)->emake($r, $t);
}
my ($re, $im) = @{$z->cartesian};
return (ref $z)->make(-$re, -$im);
#
# (abs)
#
-# Compute complex's norm (rho).
+# Compute or set complex's norm (rho).
#
sub abs {
- my ($z) = @_;
- return abs($z) unless ref $z;
- my ($r, $t) = @{$z->polar};
- return abs($r);
+ my ($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 -= pit2 }
+ elsif ($$theta <= -pi()) { $$theta += pit2 }
}
#
# arg
#
-# Compute complex's argument (theta).
+# Compute or set complex's argument (theta).
#
sub arg {
- my ($z) = @_;
- return ($z < 0 ? pi : 0) unless ref $z;
- my ($r, $t) = @{$z->polar};
- return $t;
+ my ($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 $theta;
}
#
#
# Compute sqrt(z).
#
+# It is quite tempting to use wantarray here so that in list context
+# sqrt() would return the two solutions. This, however, would
+# break things like
+#
+# print "sqrt(z) = ", sqrt($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) = @_;
- $z = cplx($z, 0) unless ref $z;
+ my ($re, $im) = ref $z ? @{$z->cartesian} : ($z, 0);
+ return $re < 0 ? cplx(0, CORE::sqrt(-$re)) : CORE::sqrt($re)
+ if $im == 0;
my ($r, $t) = @{$z->polar};
- return (ref $z)->emake(sqrt($r), $t/2);
+ return (ref $z)->emake(CORE::sqrt($r), $t/2);
}
#
#
# Compute cbrt(z) (cubic root).
#
+# Why are we not returning three values? The same answer as for sqrt().
+#
sub cbrt {
my ($z) = @_;
- return cplx($z, 0) ** (1/3) unless ref $z;
+ return $z < 0 ?
+ -CORE::exp(CORE::log(-$z)/3) :
+ ($z > 0 ? CORE::exp(CORE::log($z)/3): 0)
+ unless ref $z;
my ($r, $t) = @{$z->polar};
- return (ref $z)->emake($r**(1/3), $t/3);
+ 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 $mess;
}
#
#
sub root {
my ($z, $n) = @_;
- $n = int($n + 0.5);
- return undef unless $n > 0;
- my ($r, $t) = ref $z ? @{$z->polar} : (abs($z), $z >= 0 ? 0 : pi);
+ _rootbad($n) if ($n < 1 or int($n) != $n);
+ my ($r, $t) = ref $z ?
+ @{$z->polar} : (CORE::abs($z), $z >= 0 ? 0 : pi);
my @root;
my $k;
- my $theta_inc = 2 * pi / $n;
+ my $theta_inc = pit2 / $n;
my $rho = $r ** (1/$n);
my $theta;
- my $complex = ref($z) || $package;
+ my $cartesian = ref $z && $z->{c_dirty} == 0;
for ($k = 0, $theta = $t / $n; $k < $n; $k++, $theta += $theta_inc) {
- push(@root, $complex->emake($rho, $theta));
+ my $w = cplxe($rho, $theta);
+ # Yes, $cartesian is loop invariant.
+ push @root, $cartesian ? cplx(@{$w->cartesian}) : $w;
}
return @root;
}
#
# Re
#
-# Return Re(z).
+# Return or set Re(z).
#
sub Re {
- my ($z) = @_;
+ my ($z, $Re) = @_;
return $z unless ref $z;
- my ($re, $im) = @{$z->cartesian};
- return $re;
+ if (defined $Re) {
+ $z->{'cartesian'} = [ $Re, ${$z->cartesian}[1] ];
+ $z->{c_dirty} = 0;
+ $z->{p_dirty} = 1;
+ } else {
+ return ${$z->cartesian}[0];
+ }
}
#
# Im
#
-# Return Im(z).
+# Return or set Im(z).
#
sub Im {
- my ($z) = @_;
+ my ($z, $Im) = @_;
return 0 unless ref $z;
- my ($re, $im) = @{$z->cartesian};
- return $im;
+ 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(@_);
}
#
#
sub exp {
my ($z) = @_;
- $z = cplx($z, 0) unless ref $z;
my ($x, $y) = @{$z->cartesian};
- return (ref $z)->emake(exp($x), $y);
+ 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 $mess;
}
#
#
sub log {
my ($z) = @_;
- $z = cplx($z, 0) unless ref $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};
- my ($x, $y) = @{$z->cartesian};
- $t -= 2 * pi if ($t > pi() and $x < 0);
- $t += 2 * pi if ($t < -pi() and $x < 0);
- return (ref $z)->make(log($r), $t);
+ _logofzero("log") if $r == 0;
+ if ($t > pi()) { $t -= pit2 }
+ elsif ($t <= -pi()) { $t += pit2 }
+ return (ref $z)->make(CORE::log($r), $t);
}
#
#
# Compute log10(z).
#
+
sub log10 {
- my ($z) = @_;
- my $ilog10 = 1 / log(10) unless defined $ilog10;
- return log(cplx($z, 0)) * $ilog10 unless ref $z;
- my ($r, $t) = @{$z->polar};
- return (ref $z)->make(log($r) * $ilog10, $t * $ilog10);
+ return Math::Complex::log($_[0]) * uplog10;
}
#
sub logn {
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;
+ my $logn = $LOGN{$n};
+ $logn = $LOGN{$n} = CORE::log($n) unless defined $logn; # Cache log(n)
+ return &log($z) / $logn;
}
#
#
sub cos {
my ($z) = @_;
+ return CORE::cos($z) unless ref $z;
my ($x, $y) = @{$z->cartesian};
- my $ey = exp($y);
- my $ey_1 = 1 / $ey;
- return (ref $z)->make(cos($x) * ($ey + $ey_1)/2,
- sin($x) * ($ey_1 - $ey)/2);
+ 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 - $ey)/2);
}
#
#
sub sin {
my ($z) = @_;
+ return CORE::sin($z) unless ref $z;
my ($x, $y) = @{$z->cartesian};
- my $ey = exp($y);
- my $ey_1 = 1 / $ey;
- return (ref $z)->make(sin($x) * ($ey + $ey_1)/2,
- cos($x) * ($ey - $ey_1)/2);
+ 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 - $ey_1)/2);
}
#
#
sub tan {
my ($z) = @_;
- my $cz = cos($z);
- divbyzero "tan($z)", "cos($z)" if ($cz == 0);
- return sin($z) / $cz;
+ my $cz = &cos($z);
+ _divbyzero "tan($z)", "cos($z)" if $cz == 0;
+ return &sin($z) / $cz;
}
#
#
sub sec {
my ($z) = @_;
- my $cz = cos($z);
- divbyzero "sec($z)", "cos($z)" if ($cz == 0);
+ my $cz = &cos($z);
+ _divbyzero "sec($z)", "cos($z)" if ($cz == 0);
return 1 / $cz;
}
#
sub csc {
my ($z) = @_;
- my $sz = sin($z);
- divbyzero "csc($z)", "sin($z)" if ($sz == 0);
+ my $sz = &sin($z);
+ _divbyzero "csc($z)", "sin($z)" if ($sz == 0);
return 1 / $sz;
}
#
# cot
#
-# Computes cot(z) = 1 / tan(z).
+# Computes cot(z) = cos(z) / sin(z).
#
sub cot {
my ($z) = @_;
- my $sz = sin($z);
- divbyzero "cot($z)", "sin($z)" if ($sz == 0);
- return cos($z) / $sz;
+ my $sz = &sin($z);
+ _divbyzero "cot($z)", "sin($z)" if ($sz == 0);
+ return &cos($z) / $sz;
}
#
# Computes the arc cosine acos(z) = -i log(z + sqrt(z*z-1)).
#
sub acos {
- my ($z) = @_;
+ 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 $z;
- return ~i * log($z + (Re($z) * Im($z) > 0 ? 1 : -1) * sqrt($z*$z - 1));
+ 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, $v);
}
#
# Computes the arc sine asin(z) = -i log(iz + sqrt(1-z*z)).
#
sub asin {
- my ($z) = @_;
+ 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;
- return ~i * log(i * $z + sqrt(1 - $z*$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, $v);
}
#
#
sub atan {
my ($z) = @_;
- divbyzero "atan($z)", "i - $z" if ($z == i);
- return i/2*log((i + $z) / (i - $z));
+ return CORE::atan2($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 * $log;
}
#
#
sub asec {
my ($z) = @_;
+ _divbyzero "asec($z)", $z if ($z == 0);
return acos(1 / $z);
}
#
-# acosec
+# acsc
#
-# Computes the arc cosecant sec(z) = asin(1 / z).
+# Computes the arc cosecant acsc(z) = asin(1 / z).
#
-sub acosec {
+sub acsc {
my ($z) = @_;
+ _divbyzero "acsc($z)", $z if ($z == 0);
return asin(1 / $z);
}
#
-# acsc
+# acosec
#
-# Alias for acosec().
+# Alias for acsc().
#
-sub acsc { Math::Complex::acosec(@_) }
+sub acosec { Math::Complex::acsc(@_) }
#
# acot
#
-# Computes the arc cotangent acot(z) = -i/2 log((i+z) / (z-i))
+# Computes the arc cotangent acot(z) = atan(1 / z)
#
sub acot {
my ($z) = @_;
- divbyzero "acot($z)", "$z - i" if ($z == i);
- return i/-2 * log((i + $z) / ($z - i));
+ _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 / $z);
}
#
#
sub cosh {
my ($z) = @_;
- $z = cplx($z, 0) unless ref $z;
+ my $ex;
+ unless (ref $z) {
+ $ex = CORE::exp($z);
+ return $ex ? ($ex + 1/$ex)/2 : $Inf;
+ }
my ($x, $y) = @{$z->cartesian};
- my $ex = exp($x);
- my $ex_1 = 1 / $ex;
- return ($ex + $ex_1)/2 unless ref $z;
- return (ref $z)->make(cos($y) * ($ex + $ex_1)/2,
- sin($y) * ($ex - $ex_1)/2);
+ $ex = CORE::exp($x);
+ my $ex_1 = $ex ? 1 / $ex : $Inf;
+ return (ref $z)->make(CORE::cos($y) * ($ex + $ex_1)/2,
+ CORE::sin($y) * ($ex - $ex_1)/2);
}
#
#
sub sinh {
my ($z) = @_;
- $z = cplx($z, 0) unless ref $z;
+ my $ex;
+ unless (ref $z) {
+ return 0 if $z == 0;
+ $ex = CORE::exp($z);
+ return $ex ? ($ex - 1/$ex)/2 : "-$Inf";
+ }
my ($x, $y) = @{$z->cartesian};
- my $ex = exp($x);
- my $ex_1 = 1 / $ex;
- return ($ex - $ex_1)/2 unless ref $z;
- return (ref $z)->make(cos($y) * ($ex - $ex_1)/2,
- sin($y) * ($ex + $ex_1)/2);
+ 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 + $ex_1)/2);
}
#
sub tanh {
my ($z) = @_;
my $cz = cosh($z);
- divbyzero "tanh($z)", "cosh($z)" if ($cz == 0);
+ _divbyzero "tanh($z)", "cosh($z)" if ($cz == 0);
return sinh($z) / $cz;
}
sub sech {
my ($z) = @_;
my $cz = cosh($z);
- divbyzero "sech($z)", "cosh($z)" if ($cz == 0);
+ _divbyzero "sech($z)", "cosh($z)" if ($cz == 0);
return 1 / $cz;
}
sub csch {
my ($z) = @_;
my $sz = sinh($z);
- divbyzero "csch($z)", "sinh($z)" if ($sz == 0);
+ _divbyzero "csch($z)", "sinh($z)" if ($sz == 0);
return 1 / $sz;
}
sub coth {
my ($z) = @_;
my $sz = sinh($z);
- divbyzero "coth($z)", "sinh($z)" if ($sz == 0);
+ _divbyzero "coth($z)", "sinh($z)" if $sz == 0;
return cosh($z) / $sz;
}
#
sub acosh {
my ($z) = @_;
- $z = cplx($z, 0) unless ref $z; # asinh(-2)
- return log($z + sqrt($z*$z - 1));
+ 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 : $u;
}
#
# asinh
#
-# Computes the arc hyperbolic sine asinh(z) = log(z + sqrt(z*z-1))
+# Computes the arc hyperbolic sine asinh(z) = log(z + sqrt(z*z+1))
#
sub asinh {
my ($z) = @_;
- $z = cplx($z, 0) unless ref $z; # asinh(-2)
- return log($z + sqrt($z*$z + 1));
+ 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);
}
#
#
sub atanh {
my ($z) = @_;
- $z = cplx($z, 0) unless ref $z; # atanh(-2)
- divbyzero 'atanh(1)', "1 - $z" if ($z == 1);
- my $cz = (1 + $z) / (1 - $z);
- return log($cz) / 2;
+ 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 - $z));
}
#
#
sub asech {
my ($z) = @_;
- divbyzero 'asech(0)', $z if ($z == 0);
+ _divbyzero 'asech(0)', "$z" if ($z == 0);
return acosh(1 / $z);
}
#
sub acsch {
my ($z) = @_;
- divbyzero 'acsch(0)', $z if ($z == 0);
+ _divbyzero 'acsch(0)', $z if ($z == 0);
return asinh(1 / $z);
}
#
sub acoth {
my ($z) = @_;
- $z = cplx($z, 0) unless ref $z; # acoth(-2)
- divbyzero 'acoth(1)', "$z - 1" if ($z == 1);
- my $cz = (1 + $z) / ($z - 1);
- return log($cz) / 2;
+ _divbyzero 'acoth(0)' if ($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;
}
#
#
sub atan2 {
my ($z1, $z2, $inverted) = @_;
- my ($re1, $im1) = @{$z1->cartesian};
- my ($re2, $im2) = @{$z2->cartesian};
- my $tan;
- if (defined $inverted && $inverted) { # atan(z2/z1)
- return pi * ($re2 > 0 ? 1 : -1) if $re1 == 0 && $im1 == 0;
- $tan = $z2 / $z1;
+ my ($re1, $im1, $re2, $im2);
+ if ($inverted) {
+ ($re1, $im1) = ref $z2 ? @{$z2->cartesian} : ($z2, 0);
+ ($re2, $im2) = @{$z1->cartesian};
} else {
- return pi * ($re1 > 0 ? 1 : -1) if $re2 == 0 && $im2 == 0;
- $tan = $z1 / $z2;
+ ($re1, $im1) = @{$z1->cartesian};
+ ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0);
}
- return atan($tan);
+ if ($im2 == 0) {
+ return CORE::atan2($re1, $re2) if $im1 == 0;
+ return ($im1<=>0) * pip2 if $re2 == 0;
+ }
+ my $w = atan($z1/$z2);
+ my ($u, $v) = ref $w ? @{$w->cartesian} : ($w, 0);
+ $u += pi if $re2 < 0;
+ $u -= pit2 if $u > pi;
+ return cplx($u, $v);
}
#
# display_format
# ->display_format
#
-# Set (fetch if no argument) display format for all complex numbers that
-# don't happen to have overrriden it via ->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 a method, this actually sets the display format for
+# 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 $format = undef;
-
- if (ref $self) { # Called as a method
- $format = shift;
- } else { # Regular procedure call
- $format = $self;
- undef $self;
+ 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 (defined $self) {
- return defined $self->{display} ? $self->{display} : $display
- unless defined $format;
- return $self->{display} = $format;
+ if (ref $self) { # Called as an object method
+ $self->{display_format} = { %display_format };
+ return
+ wantarray ?
+ %{$self->{display_format}} :
+ $self->{display_format}->{style};
}
- return $display unless defined $format;
- return $display = $format;
+ # Called as a class method
+ %DISPLAY_FORMAT = %display_format;
+ return
+ wantarray ?
+ %DISPLAY_FORMAT :
+ $DISPLAY_FORMAT{style};
}
#
#
sub stringify {
my ($z) = shift;
- my $format;
- $format = $display;
- $format = $z->{display} if defined $z->{display};
+ my $style = $z->display_format;
+
+ $style = $DISPLAY_FORMAT{style} unless defined $style;
- return $z->stringify_polar if $format =~ /^p/i;
+ return $z->stringify_polar if $style =~ /^p/i;
return $z->stringify_cartesian;
}
my ($x, $y) = @{$z->cartesian};
my ($re, $im);
- $x = int($x + ($x < 0 ? -1 : 1) * 1e-14)
- if int(abs($x)) != int(abs($x) + 1e-14);
- $y = int($y + ($y < 0 ? -1 : 1) * 1e-14)
- if int(abs($y)) != int(abs($y) + 1e-14);
+ my %format = $z->display_format;
+ my $format = $format{format};
+
+ if ($x) {
+ if ($x =~ /^NaN[QS]?$/i) {
+ $re = $x;
+ } else {
+ if ($x =~ /^-?$Inf$/oi) {
+ $re = $x;
+ } else {
+ $re = defined $format ? sprintf($format, $x) : $x;
+ }
+ }
+ } else {
+ undef $re;
+ }
- $re = "$x" if abs($x) >= 1e-14;
- if ($y == 1) { $im = 'i' }
- elsif ($y == -1) { $im = '-i' }
- elsif (abs($y) >= 1e-14) { $im = $y . "i" }
+ if ($y) {
+ if ($y =~ /^(NaN[QS]?)$/i) {
+ $im = $y;
+ } else {
+ if ($y =~ /^-?$Inf$/oi) {
+ $im = $y;
+ } else {
+ $im =
+ defined $format ?
+ sprintf($format, $y) :
+ ($y == 1 ? "" : ($y == -1 ? "-" : $y));
+ }
+ }
+ $im .= "i";
+ } else {
+ undef $im;
+ }
- my $str = '';
- $str = $re if defined $re;
- $str .= "+$im" if defined $im;
- $str =~ s/\+-/-/;
- $str =~ s/^\+//;
- $str = '0' unless $str;
+ 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
#
my $z = shift;
my ($r, $t) = @{$z->polar};
my $theta;
- my $eps = 1e-14;
-
- return '[0,0]' if $r <= $eps;
-
- my $tpi = 2 * pi;
- my $nt = $t / $tpi;
- $nt = ($nt - int($nt)) * $tpi;
- $nt += $tpi if $nt < 0; # Range [0, 2pi]
- if (abs($nt) <= $eps) { $theta = 0 }
- elsif (abs(pi-$nt) <= $eps) { $theta = 'pi' }
+ my %format = $z->display_format;
+ my $format = $format{format};
- if (defined $theta) {
- $r = int($r + ($r < 0 ? -1 : 1) * $eps)
- if int(abs($r)) != int(abs($r) + $eps);
- $theta = int($theta + ($theta < 0 ? -1 : 1) * $eps)
- if ($theta ne 'pi' and
- int(abs($theta)) != int(abs($theta) + $eps));
- return "\[$r,$theta\]";
+ if ($t =~ /^NaN[QS]?$/i || $t =~ /^-?$Inf$/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;
+
#
- # Okay, number is not a real. Try to identify pi/n and friends...
+ # Try to identify pi/n and friends.
#
- $nt -= $tpi if $nt > pi;
- my ($n, $k, $kpi);
-
- for ($k = 1, $kpi = pi; $k < 10; $k++, $kpi += pi) {
- $n = int($kpi / $nt + ($nt > 0 ? 1 : -1) * 0.5);
- if (abs($kpi/$n - $nt) <= $eps) {
- $theta = ($nt < 0 ? '-':'').
- ($k == 1 ? 'pi':"${k}pi").'/'.abs($n);
- last;
+ $t -= int(CORE::abs($t) / pit2) * pit2;
+
+ 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;
}
+ }
}
- $theta = $nt unless defined $theta;
-
- $r = int($r + ($r < 0 ? -1 : 1) * $eps)
- if int(abs($r)) != int(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));
+ if (defined $format) {
+ $r = sprintf($format, $r);
+ $theta = sprintf($format, $theta) unless defined $theta;
+ } else {
+ $theta = $t unless defined $theta;
+ }
- return "\[$r,$theta\]";
+ return "[$r,$theta]";
}
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);
which is also expressed by this formula:
- z = rho * exp(i * theta) = rho * (cos theta + i * sin theta)
+ 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 polar notation (also known as the trigonometric
representation) is much more handy for performing multiplications and
divisions of complex numbers, whilst the cartesian notation is better
-suited for additions and substractions. Real numbers are on the I<x>
-axis, and therefore I<theta> is zero.
+suited for additions and subtractions. Real numbers are on the I<x>
+axis, and therefore I<theta> is zero or I<pi>.
All the common operations that can be performed on a real number have
been defined to work on complex numbers as well, and are merely
the number is within their definition set.
For instance, the C<sqrt> routine which computes the square root of
-its argument is only defined for positive real numbers and yields a
-positive real number (it is an application from B<R+> to B<R+>).
+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(z = [r,t]) = sqrt(r) * exp(i * t/2)
-Indeed, a negative real number can be noted C<[x,pi]>
-(the modulus I<x> is always positive, so C<[x,pi]> is really C<-x>, a
-negative number)
-and the above definition states that
+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
A I<new> operation possible on a complex number that is
the identity for real numbers is called the I<conjugate>, and is noted
-with an horizontal bar above the number, or C<~z> here.
+with a horizontal bar above the number, or C<~z> here.
z = a + bi
~z = a - bi
z1 * z2 = (r1 * r2) * exp(i * (t1 + t2))
z1 / z2 = (r1 / r2) * exp(i * (t1 - t2))
z1 ** z2 = exp(z2 * log z1)
- ~z1 = a - bi
- abs(z1) = r1 = sqrt(a*a + b*b)
- sqrt(z1) = sqrt(r1) * exp(i * t1/2)
- exp(z1) = exp(a) * exp(i * b)
- log(z1) = log(r1) + i*t1
- sin(z1) = 1/2i (exp(i * z1) - exp(-i * z1))
- cos(z1) = 1/2 (exp(i * z1) + exp(-i * z1))
- abs(z1) = r1
+ ~z = a - bi
+ abs(z) = r1 = sqrt(a*a + b*b)
+ sqrt(z) = sqrt(r1) * exp(i * t/2)
+ exp(z) = exp(a) * exp(i * b)
+ log(z) = log(r1) + i*t
+ sin(z) = 1/2i (exp(i * z1) - exp(-i * z))
+ cos(z) = 1/2 (exp(i * z1) + exp(-i * z))
atan2(z1, z2) = atan(z1/z2)
The following extra operations are supported on both real and complex
Re(z) = a
Im(z) = b
arg(z) = t
+ abs(z) = r
cbrt(z) = z ** (1/3)
log10(z) = log(z) / log(10)
tan(z) = sin(z) / cos(z)
- csc(z) = 1 / sin(z)
- sec(z) = 1 / 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 + sqrt(z*z-1))
+ 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) = -i/2 * log((i+z) / (z-i))
+ 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)
+ 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)
+ acsch(z) = asinh(1 / z)
+ asech(z) = acosh(1 / z)
acoth(z) = atanh(1 / z) = 1/2 * log((1+z) / (z-1))
-I<log>, I<csc>, I<cot>, I<acsc>, I<acot>, I<csch>, I<coth>,
-I<acosech>, I<acotanh>, have aliases I<ln>, I<cosec>, I<cotan>,
-I<acosec>, I<acotan>, I<cosech>, I<cotanh>, I<acosech>, I<acotanh>,
-respectively.
+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.
(root(z, n))[k] = r**(1/n) * exp(i * (t + 2*k*pi)/n)
-The I<spaceship> comparison operator 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.
+The I<spaceship> comparison operator, E<lt>=E<gt>, is also defined. In
+order to ensure its restriction to real numbers is conform to what you
+would expect, the comparison is run on the real part of the complex
+number first, and imaginary parts are compared only when the real
+parts match.
=head1 CREATION
$z = 3 + 4*i;
-if you like. To create a number using the trigonometric form, use either:
+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). (Mnmemonic: C<e> is used as a
-notation for complex numbers in the trigonometric form).
+(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 positive (it represents the distance to the origin in the complex
-plane).
+but that will be silently converted into C<[3,-3pi/4]>, since the
+modulus must be non-negative (it represents the distance to the origin
+in the complex plane).
+
+It is also possible to have a complex number as either argument of
+either the C<make> or C<emake>: the appropriate component of
+the argument will be used.
+
+ $z1 = cplx(-2, 1);
+ $z2 = cplx($z1, 4);
=head1 STRINGIFICATION
When printed, a complex number is usually shown under its cartesian
-form I<a+bi>, but there are legitimate cases where the polar format
+style I<a+bi>, but there are legitimate cases where the polar style
I<[r,t]> is more appropriate.
-By calling the routine C<Math::Complex::display_format> and supplying either
-C<"polar"> or C<"cartesian">, you override the default display format,
-which is C<"cartesian">. Not supplying any argument returns the current
-setting.
+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 format for this number. Otherwise whatever you
-specify will be the new display format for I<this> particular number.
+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 = (root(1, 3))[1];
+ print "j = $j\n"; # Prints "j = [1,2pi/3]"
$j->display_format('cartesian');
print "j = $j\n"; # Prints "j = -0.5+0.866025403784439i"
-The polar format attempts to emphasize arguments like I<k*pi/n>
-(where I<n> is a positive integer and I<k> an integer within [-9,+9]).
+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>.
+
+=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
$k = exp(i * 2*pi/3);
print "$j - $k = ", $j - $k, "\n";
-=head1 BUGS
+ $z->Re(3); # Re, Im, arg, abs,
+ $j->arg(2); # (the last two aka rho, theta)
+ # can be used also as mutators.
+
+=head1 ERRORS DUE TO DIVISION BY ZERO OR LOGARITHM OF ZERO
+
+The division (/) and the following functions
+
+ log ln log10 logn
+ tan sec csc cot
+ atan asec acsc acot
+ tanh sech csch coth
+ atanh asech acsch acoth
+
+cannot be computed for all arguments because that would mean dividing
+by zero or taking logarithm of zero. These situations cause fatal
+runtime errors looking like this
+
+ cot(0): Division by zero.
+ (Because in the definition of cot(0), the divisor sin(0) is 0)
+ Died at ...
+
+or
-Saying C<use Math::Complex;> exports many mathematical routines in the caller
-environment. This is construed as a feature by the Author, actually... ;-)
+ atanh(-1): Logarithm of zero.
+ Died at...
-The code is not optimized for speed, although we try to use the cartesian
-form for addition-like operators and the trigonometric form for all
-multiplication-like operators.
+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.
-The arg() routine does not ensure the angle is within the range [-pi,+pi]
-(a side effect caused by multiplication and division using the trigonometric
-representation).
+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>).
+This is construed as a feature by the Authors, actually... ;-)
All routines expect to be given real or complex numbers. Don't attempt to
use BigFloat, since Perl has currently no rule to disambiguate a '+'
operation (for instance) between two overloaded entities.
+In Cray UNICOS there is some strange numerical instability that results
+in root(), cos(), sin(), cosh(), sinh(), losing accuracy fast. Beware.
+The bug may be in UNICOS math libs, in UNICOS C compiler, in Math::Complex.
+Whatever it is, it does not manifest itself anywhere else where Perl runs.
+
=head1 AUTHORS
- Raphael Manfredi <F<Raphael_Manfredi@grenoble.hp.com>>
- Jarkko Hietaniemi <F<jhi@iki.fi>>
+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>>.
+
+=cut
+
+1;
+
+# eof
projects / p5sagit/p5-mst-13.2.git / blobdiff