# -- 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 1.26 1998/11/01 00:00:00 dsl 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 ? $theta : 0);
}
#
#
# The number defined as pi = 180 degrees
#
-use constant pi => 4 * CORE::atan2(1, 1);
+sub pi () { 4 * CORE::atan2(1, 1) }
#
# pit2
#
# The full circle
#
-use constant pit2 => 2 * pi;
+sub pit2 () { 2 * pi }
#
# pip2
#
# The quarter circle
#
-use constant pip2 => pi / 2;
+sub pip2 () { pi / 2 }
#
# deg1
# One degree in radians, used in stringify_polar.
#
-use constant deg1 => pi / 180;
+sub deg1 () { pi / 180 }
#
# uplog10
#
# Used in log10().
#
-use constant uplog10 => 1 / CORE::log(10);
+sub uplog10 () { 1 / CORE::log(10) }
#
# i
}
#
+# ip2
+#
+# Half of i.
+#
+sub ip2 () { i / 2 }
+
+#
# Attribute access/set routines
#
my ($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)];
+ 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";
}
return 1 if $z2 == 0 || $z1 == 1;
return 0 if $z1 == 0 && Re($z2) > 0;
}
- my $w = $inverted ? CORE::exp($z1 * CORE::log($z2))
- : CORE::exp($z2 * CORE::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 $z2->{c_dirty} == 0) ?
}
#
+# (numeq)
+#
+# Computes z1 == z2.
+#
+# (Required in addition to spaceship() because of NaNs.)
+sub numeq {
+ my ($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 $re < 0 ? cplx(0, CORE::sqrt(-$re)) : CORE::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 $z)->emake(CORE::sqrt($r), $t/2);
}
#
sub cbrt {
my ($z) = @_;
- return $z < 0 ? -CORE::exp(CORE::log(-$z)/3) : ($z > 0 ? CORE::exp(CORE::log($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 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} : (CORE::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 / $n;
#
sub Im {
my ($z, $Im) = @_;
- return $z unless ref $z;
+ return 0 unless ref $z;
if (defined $Im) {
$z->{'cartesian'} = [ ${$z->cartesian}[0], $Im ];
$z->{c_dirty} = 0;
sub logn {
my ($z, $n) = @_;
$z = cplx($z, 0) unless ref $z;
- my $logn = $logn{$n};
- $logn = $logn{$n} = CORE::log($n) unless defined $logn; # Cache log(n)
- return CORE::log($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 $ey = CORE::exp($y);
- my $ey_1 = 1 / $ey;
- return (ref $z)->make(CORE::cos($x) * ($ey + $ey_1)/2,
- CORE::sin($x) * ($ey_1 - $ey)/2);
+ 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 $ey = CORE::exp($y);
- my $ey_1 = 1 / $ey;
- return (ref $z)->make(CORE::sin($x) * ($ey + $ey_1)/2,
- CORE::cos($x) * ($ey - $ey_1)/2);
+ 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 $cz = CORE::cos($z);
- _divbyzero "tan($z)", "cos($z)" if (CORE::abs($cz) < $eps);
- return CORE::sin($z) / $cz;
+ my $cz = &cos($z);
+ _divbyzero "tan($z)", "cos($z)" if $cz == 0;
+ return &sin($z) / $cz;
}
#
#
sub sec {
my ($z) = @_;
- my $cz = CORE::cos($z);
+ my $cz = &cos($z);
_divbyzero "sec($z)", "cos($z)" if ($cz == 0);
return 1 / $cz;
}
#
sub csc {
my ($z) = @_;
- my $sz = CORE::sin($z);
+ my $sz = &sin($z);
_divbyzero "csc($z)", "sin($z)" if ($sz == 0);
return 1 / $sz;
}
#
sub cot {
my ($z) = @_;
- my $sz = CORE::sin($z);
+ my $sz = &sin($z);
_divbyzero "cot($z)", "sin($z)" if ($sz == 0);
- return CORE::cos($z) / $sz;
+ return &cos($z) / $sz;
}
#
#
sub acos {
my $z = $_[0];
- return CORE::atan2(CORE::sqrt(1-$z*$z), $z) if (! ref $z) && CORE::abs($z) <= 1;
- my ($x, $y) = ref $z ? @{$z->cartesian} : ($z, 0);
+ return CORE::atan2(CORE::sqrt(1-$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 $u = CORE::atan2(CORE::sqrt(1-$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 = $_[0];
- return CORE::atan2($z, CORE::sqrt(1-$z*$z)) if (! ref $z) && CORE::abs($z) <= 1;
- my ($x, $y) = ref $z ? @{$z->cartesian} : ($z, 0);
+ return CORE::atan2($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 $u = CORE::atan2($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 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 = CORE::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 (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);
+ _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 $ex;
unless (ref $z) {
$ex = CORE::exp($z);
- return ($ex + 1/$ex)/2;
+ return $ex ? ($ex + 1/$ex)/2 : $Inf;
}
my ($x, $y) = @{$z->cartesian};
$ex = CORE::exp($x);
- my $ex_1 = 1 / $ex;
+ 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) {
+ return 0 if $z == 0;
$ex = CORE::exp($z);
- return ($ex - 1/$ex)/2;
+ return $ex ? ($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 = 1 / $ex;
+ 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 $z) {
- return CORE::log($z + CORE::sqrt($z*$z-1)) if $z >= 1;
$z = cplx($z, 0);
}
my ($re, $im) = @{$z->cartesian};
if ($im == 0) {
- return cplx(CORE::log($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 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 CORE::log($z + CORE::sqrt($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 CORE::log($z + CORE::sqrt($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);
}
#
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 * CORE::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 (CORE::abs($z) < $eps);
+ _divbyzero 'acoth(0)' if ($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)', "$z - 1" if (CORE::abs($z - 1) < $eps);
- _logofzero 'acoth(-1)', "1 / $z" if (CORE::abs($z + 1) < $eps);
- return CORE::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(CORE::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;
- return $z->stringify_polar if $format =~ /^p/i;
+ $style = $DISPLAY_FORMAT{style} unless defined $style;
+
+ 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) * $eps)
- if int(CORE::abs($x)) != int(CORE::abs($x) + $eps);
- $y = int($y + ($y < 0 ? -1 : 1) * $eps)
- if int(CORE::abs($y)) != int(CORE::abs($y) + $eps);
-
- $re = "$x" if CORE::abs($x) >= $eps;
- if ($y == 1) { $im = 'i' }
- elsif ($y == -1) { $im = '-i' }
- elsif (CORE::abs($y) >= $eps) { $im = $y . "i" }
-
- my $str = '';
- $str = $re if defined $re;
- $str .= "+$im" if defined $im;
- $str =~ s/\+-/-/;
- $str =~ s/^\+//;
- $str =~ s/([-+])1i/$1i/; # Not redundant with the above 1/-1 tests.
- $str = '0' unless $str;
-
- return $str;
-}
-
+ my %format = $z->display_format;
+ my $format = $format{format};
-# Helper for stringify_polar, a Greatest Common Divisor with a memory.
+ if ($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 $re;
+ }
-sub _gcd {
- my ($a, $b) = @_;
+ if ($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;
+ }
- use integer;
+ my $str = $re;
- # Loops forever if given negative inputs.
+ 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) {
+ $str = "0";
+ }
- if ($b and $a > $b) { return gcd($a % $b, $b) }
- elsif ($a and $b > $a) { return gcd($b % $a, $a) }
- else { return $a ? $a : $b }
+ return $str;
}
-my %gcd;
-
-sub gcd {
- my ($a, $b) = @_;
-
- my $id = "$a $b";
-
- unless (exists $gcd{$id}) {
- $gcd{$id} = _gcd($a, $b);
- $gcd{"$b $a"} = $gcd{$id};
- }
-
- return $gcd{$id};
-}
#
# ->stringify_polar
my ($r, $t) = @{$z->polar};
my $theta;
- return '[0,0]' if $r <= $eps;
+ my %format = $z->display_format;
+ my $format = $format{format};
- my $nt = $t / pit2;
- $nt = ($nt - int($nt)) * pit2;
- $nt += pit2 if $nt < 0; # Range [0, 2pi]
-
- if (CORE::abs($nt) <= $eps) { $theta = 0 }
- elsif (CORE::abs(pi-$nt) <= $eps) { $theta = 'pi' }
-
- if (defined $theta) {
- $r = int($r + ($r < 0 ? -1 : 1) * $eps)
- if int(CORE::abs($r)) != int(CORE::abs($r) + $eps);
- $theta = int($theta + ($theta < 0 ? -1 : 1) * $eps)
- if ($theta ne 'pi' and
- int(CORE::abs($theta)) != int(CORE::abs($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 $nt > pi;
-
- if (CORE::abs($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 (CORE::abs($kpi/$n - $nt) <= $eps) {
- $n = CORE::abs($n);
- my $gcd = gcd($k, $n);
- if ($gcd > 1) {
- $k /= $gcd;
- $n /= $gcd;
- }
- next if $n > 360;
- $theta = ($nt < 0 ? '-':'').
- ($k == 1 ? 'pi':"${k}pi");
- $theta .= '/'.$n if $n > 1;
+ $t -= int(CORE::abs($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) * $eps)
- if int(CORE::abs($r)) != int(CORE::abs($r) + $eps);
- $theta = int($theta + ($theta < 0 ? -1 : 1) * $eps)
- if ($theta !~ m(^-?\d*pi/\d+$) and
- int(CORE::abs($theta)) != int(CORE::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
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
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 also mutators. The C<cbrt>
+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.
$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
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 the
+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
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
projects / p5sagit/p5-mst-13.2.git / blobdiff