# -- Daniel S. Lewart Since Sep 1997
#
-require Exporter;
package Math::Complex;
-use strict;
+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 vars qw($VERSION @ISA @EXPORT %EXPORT_TAGS);
+use strict;
-my ( $i, $ip2, %logn );
+my $i;
+my %LOGN;
-$VERSION = sprintf("%s", q$Id: Complex.pm,v 1.25 1998/02/05 16:07:37 jhi Exp $ =~ /(\d+\.\d+)/);
+require Exporter;
@ISA = qw(Exporter);
'*' => \&multiply,
'/' => \÷,
'**' => \&power,
+ '==' => \&numeq,
'<=>' => \&spaceship,
'neg' => \&negate,
'~' => \&conjugate,
# Package "privates"
#
-my $package = 'Math::Complex'; # Package name
-my $display = 'cartesian'; # Default display format
-my $eps = 1e-14; # Epsilon
+my %DISPLAY_FORMAT = ('style' => 'cartesian',
+ 'polar_pretty_print' => 1);
+my $eps = 1e-14; # Epsilon
#
# Object attributes (internal):
#
sub cplx {
my ($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);
}
#
#
# 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.
+#
+
+sub deg1 () { pi / 180 }
#
# uplog10
#
# Used in log10().
#
-use constant uplog10 => 1 / log(10);
+sub uplog10 () { 1 / CORE::log(10) }
#
# 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)];
}
#
if (defined $_[1]) {
$mess .= "(Because in the definition of $_[0], the divisor ";
- $mess .= "$_[1] " unless ($_[1] eq '0');
+ $mess .= "$_[1] " unless ("$_[1]" eq '0');
$mess .= "is 0)\n";
}
}
#
-# _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;
}
- return $inverted ? exp($z1 * log $z2) : exp($z2 * log $z1);
+ 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, $inverted) = @_;
}
#
+# (numeq)
+#
+# Computes z1 == z2.
+#
+# (Required in addition to spaceship() because of NaNs.)
+sub numeq {
+ my ($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 == $im2 ? 1 : 0;
+}
+
+#
# (negate)
#
# Computes -z.
#
sub abs {
my ($z, $rho) = @_;
- return $z unless ref $z;
+ unless (ref $z) {
+ if (@_ == 2) {
+ $_[0] = $_[1];
+ } else {
+ return CORE::abs($z);
+ }
+ }
if (defined $rho) {
$z->{'polar'} = [ $rho, ${$z->polar}[1] ];
$z->{p_dirty} = 0;
sub sqrt {
my ($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 0 if $r == 0;
+ return (ref $z)->emake(CORE::exp(CORE::log($r)/3), $t/3);
}
#
# Die on bad root.
#
sub _rootbad {
- my $mess = "Root $_[0] not defined, root must be positive integer.\n";
+ my $mess = "Root $_[0] illegal, root rank must be positive integer.\n";
my @up = caller(1);
sub root {
my ($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;
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}) : $w;
}
return @root;
}
#
sub Im {
my ($z, $Im) = @_;
- return $z unless ref $z;
+ return 0 unless ref $z;
if (defined $Im) {
$z->{'cartesian'} = [ ${$z->cartesian}[0], $Im ];
$z->{c_dirty} = 0;
sub exp {
my ($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), $t);
}
#
sub logn {
my ($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 (abs($cz) < $eps);
- return sin($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);
+ my $cz = &cos($z);
_divbyzero "sec($z)", "cos($z)" if ($cz == 0);
return 1 / $cz;
}
#
sub csc {
my ($z) = @_;
- my $sz = sin($z);
+ my $sz = &sin($z);
_divbyzero "csc($z)", "sin($z)" if ($sz == 0);
return 1 / $sz;
}
#
sub cot {
my ($z) = @_;
- my $sz = sin($z);
+ my $sz = &sin($z);
_divbyzero "cot($z)", "sin($z)" if ($sz == 0);
- return cos($z) / $sz;
+ return &cos($z) / $sz;
}
#
#
sub acos {
my $z = $_[0];
- return atan2(sqrt(1-$z*$z), $z) if (! ref $z) && abs($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);
+ 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 $u = atan2(sqrt(1-$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);
+ return (ref $z)->make($u, $v);
}
#
#
sub asin {
my $z = $_[0];
- return atan2($z, sqrt(1-$z*$z)) if (! ref $z) && abs($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);
+ 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 $u = atan2($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);
+ return (ref $z)->make($u, $v);
}
#
#
sub atan {
my ($z) = @_;
- return atan2($z, 1) unless ref $z;
+ return CORE::atan2($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 ( $z == i);
- _divbyzero "atan(-i)" if (-$z == i);
- my $log = log((i + $z) / (i - $z));
- $ip2 = 0.5 * i unless defined $ip2;
- return $ip2 * $log;
+ _logofzero "atan(-i)" if (-$z == i); # -i is a bad file test...
+ my $log = &log((i + $z) / (i - $z));
+ 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 $z == 0;
+ return ($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 / $z);
}
my ($z) = @_;
my $ex;
unless (ref $z) {
- $ex = exp($z);
- return ($ex + 1/$ex)/2;
+ $ex = CORE::exp($z);
+ return $ex ? ($ex + 1/$ex)/2 : $Inf;
}
my ($x, $y) = @{$z->cartesian};
- $ex = exp($x);
- my $ex_1 = 1 / $ex;
- 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 - $ex_1)/2);
}
#
my ($z) = @_;
my $ex;
unless (ref $z) {
- $ex = exp($z);
- return ($ex - 1/$ex)/2;
+ return 0 if $z == 0;
+ $ex = CORE::exp($z);
+ return $ex ? ($ex - 1/$ex)/2 : "-$Inf";
}
my ($x, $y) = @{$z->cartesian};
- $ex = exp($x);
- my $ex_1 = 1 / $ex;
- 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 + $ex_1)/2);
}
#
sub coth {
my ($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) = @_;
unless (ref $z) {
- return log($z + 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 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;
}
- return log($z + sqrt($z*$z - 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) = @_;
- 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) = @_;
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));
+ _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 / $z);
}
#
sub acoth {
my ($z) = @_;
- _divbyzero 'acoth(0)' if (abs($z) < $eps);
+ _divbyzero 'acoth(0)' if ($z == 0);
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)', "$z - 1" if ($z - 1 == 0);
+ _logofzero 'acoth(-1)', "1 + $z" if (1 + $z == 0);
+ return &log((1 + $z) / ($z - 1)) / 2;
}
#
($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0);
}
if ($im2 == 0) {
- return cplx(atan2($re1, $re2), 0) if $im1 == 0;
- return cplx(($im1<=>0) * pip2, 0) if $re2 == 0;
+ return CORE::atan2($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);
# display_format
# ->display_format
#
-# Set (fetch if no argument) display format for all complex numbers that
+# 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;
+ my $self = shift;
+ my %display_format = %DISPLAY_FORMAT;
- if (ref $self) { # Called as a method
- $format = shift;
- } else { # Regular procedure call
- $format = $self;
- undef $self;
+ 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) * $eps)
- if int(abs($x)) != int(abs($x) + $eps);
- $y = int($y + ($y < 0 ? -1 : 1) * $eps)
- if int(abs($y)) != int(abs($y) + $eps);
+ 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 $re;
+ }
+
+ if ($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;
+ }
- $re = "$x" if abs($x) >= $eps;
- if ($y == 1) { $im = 'i' }
- elsif ($y == -1) { $im = '-i' }
- elsif (abs($y) >= $eps) { $im = $y . "i" }
+ my $str = $re;
- my $str = '';
- $str = $re if defined $re;
- $str .= "+$im" if defined $im;
- $str =~ s/\+-/-/;
- $str =~ s/^\+//;
- $str = '0' unless $str;
+ 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 $str;
}
+
#
# ->stringify_polar
#
my ($r, $t) = @{$z->polar};
my $theta;
- return '[0,0]' if $r <= $eps;
-
- my $nt = $t / pit2;
- $nt = ($nt - int($nt)) * pit2;
- $nt += pit2 if $nt < 0; # Range [0, 2pi]
-
- if (abs($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 -= pit2 if $nt > pi;
- my ($n, $k, $kpi);
+ $t -= int(CORE::abs($t) / pit2) * pit2;
- 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($n);
- last;
+ if ($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
$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).
+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
=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
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
Note that because we are operating on approximations of real numbers,
these errors can happen when merely `too close' to the singularities
-listed above. For example C<tan(2*atan2(1,1)+1e-15)> will die of
-division by zero.
+listed above.
=head1 ERRORS DUE TO INDIGESTIBLE ARGUMENTS
use BigFloat, since Perl has currently no rule to disambiguate a '+'
operation (for instance) between two overloaded entities.
+In Cray UNICOS there is some strange numerical instability that results
+in root(), cos(), sin(), cosh(), sinh(), losing accuracy fast. Beware.
+The bug may be in UNICOS math libs, in UNICOS C compiler, in Math::Complex.
+Whatever it is, it does not manifest itself anywhere else where Perl runs.
+
=head1 AUTHORS
-Raphael Manfredi <F<Raphael_Manfredi@grenoble.hp.com>> and
+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>>.