#
# Complex numbers and associated mathematical functions
-# -- Raphael Manfredi September 1996
-# -- Jarkko Hietaniemi March-October 1997
-# -- Daniel S. Lewart September-October 1997
+# -- Raphael Manfredi Since Sep 1996
+# -- Jarkko Hietaniemi Since Mar 1997
+# -- Daniel S. Lewart Since Sep 1997
#
require Exporter;
package Math::Complex;
-$VERSION = 1.05;
+use strict;
-# $Id: Complex.pm,v 1.2 1997/10/15 10:08:39 jhi Exp $
+use vars qw($VERSION @ISA @EXPORT %EXPORT_TAGS);
-use strict;
+my ( $i, $ip2, %logn );
-use vars qw($VERSION @ISA
- @EXPORT %EXPORT_TAGS
- $package $display
- $i $ip2 $logn %logn);
+$VERSION = sprintf("%s", q$Id: Complex.pm,v 1.25 1998/02/05 16:07:37 jhi Exp $ =~ /(\d+\.\d+)/);
@ISA = qw(Exporter);
);
@EXPORT = (qw(
- i Re Im arg
+ i Re Im rho theta arg
sqrt log ln
log10 logn cbrt root
cplx cplxe
qw("" stringify);
#
-# Package globals
+# Package "privates"
#
-$package = 'Math::Complex'; # Package name
-$display = 'cartesian'; # Default display format
+my $package = 'Math::Complex'; # Package name
+my $display = 'cartesian'; # Default display format
+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) = @_;
+ 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 $self;
}
use constant pip2 => pi / 2;
#
+# deg1
+#
+# One degree in radians, used in stringify_polar.
+#
+
+use constant deg1 => pi / 180;
+
+#
# uplog10
#
# Used in log10().
return 0 if ($z1z);
return 1 if ($z2z or $z1 == 1);
}
- 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}) : $w;
}
#
#
# (abs)
#
-# Compute complex's norm (rho).
+# Compute or set complex's norm (rho).
#
sub abs {
- my ($z) = @_;
- my ($r, $t) = @{$z->polar};
- return $r;
+ my ($z, $rho) = @_;
+ return $z unless ref $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};
- if ($t > pi()) { $t -= pit2 }
- elsif ($t <= -pi()) { $t += pit2 }
- 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) = @_;
- return $z >= 0 ? sqrt($z) : cplx(0, sqrt(-$z)) unless ref $z;
- my ($re, $im) = @{$z->cartesian};
- return cplx($re < 0 ? (0, sqrt(-$re)) : (sqrt($re), 0)) if $im == 0;
+ my ($re, $im) = ref $z ? @{$z->cartesian} : ($z, 0);
+ return $re < 0 ? cplx(0, sqrt(-$re)) : sqrt($re) if $im == 0;
my ($r, $t) = @{$z->polar};
return (ref $z)->emake(sqrt($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 $z < 0 ? -exp(log(-$z)/3) : ($z > 0 ? exp(log($z)/3): 0)
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) = @_;
- return 0 unless ref $z;
- my ($re, $im) = @{$z->cartesian};
- return $im;
+ my ($z, $Im) = @_;
+ return $z unless ref $z;
+ if (defined $Im) {
+ $z->{'cartesian'} = [ ${$z->cartesian}[0], $Im ];
+ $z->{c_dirty} = 0;
+ $z->{p_dirty} = 1;
+ } else {
+ return ${$z->cartesian}[1];
+ }
+}
+
+#
+# rho
+#
+# Return or set rho(w).
+#
+sub rho {
+ Math::Complex::abs(@_);
+}
+
+#
+# theta
+#
+# Return or set theta(w).
+#
+sub theta {
+ Math::Complex::arg(@_);
}
#
sub tan {
my ($z) = @_;
my $cz = cos($z);
- _divbyzero "tan($z)", "cos($z)" if ($cz == 0);
+ _divbyzero "tan($z)", "cos($z)" if (abs($cz) < $eps);
return sin($z) / $cz;
}
#
sub acot {
my ($z) = @_;
+ _divbyzero "acot(0)" if (abs($z) < $eps);
return ($z >= 0) ? atan2(1, $z) : atan2(-1, -$z) unless ref $z;
- _divbyzero "acot(i)", if ( $z == i);
- _divbyzero "acot(-i)" if (-$z == i);
+ _divbyzero "acot(i)" if (abs($z - i) < $eps);
+ _logofzero "acot(-i)" if (abs($z + i) < $eps);
return atan(1 / $z);
}
#
sub acoth {
my ($z) = @_;
+ _divbyzero 'acoth(0)' if (abs($z) < $eps);
unless (ref $z) {
return log(($z + 1)/($z - 1))/2 if abs($z) > 1;
$z = cplx($z, 0);
}
- _divbyzero 'acoth(1)', "$z - 1" if ($z == 1);
- _logofzero 'acoth(-1)' if ($z == -1);
+ _divbyzero 'acoth(1)', "$z - 1" if (abs($z - 1) < $eps);
+ _logofzero 'acoth(-1)', "1 / $z" if (abs($z + 1) < $eps);
return log((1 + $z) / ($z - 1)) / 2;
}
my $z = shift;
my ($x, $y) = @{$z->cartesian};
my ($re, $im);
- my $eps = 1e-14;
$x = int($x + ($x < 0 ? -1 : 1) * $eps)
if int(abs($x)) != int(abs($x) + $eps);
$str .= "+$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 $str;
}
+
+# Helper for stringify_polar, a Greatest Common Divisor with a memory.
+
+sub _gcd {
+ my ($a, $b) = @_;
+
+ use integer;
+
+ # Loops forever if given negative inputs.
+
+ if ($b and $a > $b) { return gcd($a % $b, $b) }
+ elsif ($a and $b > $a) { return gcd($b % $a, $a) }
+ else { return $a ? $a : $b }
+}
+
+my %gcd;
+
+sub gcd {
+ my ($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 $z = shift;
my ($r, $t) = @{$z->polar};
my $theta;
- my $eps = 1e-14;
return '[0,0]' if $r <= $eps;
#
$nt -= pit2 if $nt > pi;
- my ($n, $k, $kpi);
- for ($k = 1, $kpi = pi; $k < 10; $k++, $kpi += pi) {
+ if (abs($nt) >= deg1) {
+ my ($n, $k, $kpi);
+
+ for ($k = 1, $kpi = pi; $k < 10; $k++, $kpi += pi) {
$n = int($kpi / $nt + ($nt > 0 ? 1 : -1) * 0.5);
if (abs($kpi/$n - $nt) <= $eps) {
- $theta = ($nt < 0 ? '-':'').
- ($k == 1 ? 'pi':"${k}pi").'/'.abs($n);
- last;
+ $n = 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;
+ last;
}
+ }
}
$theta = $nt unless defined $theta;
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
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))
+ ~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)
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 also 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.
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
$k = exp(i * 2*pi/3);
print "$j - $k = ", $j - $k, "\n";
-=head1 ERRORS DUE TO DIVISION BY ZERO
+ $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
- tan
- sec
- csc
- cot
- asec
- acsc
- atan
- acot
- tanh
- sech
- csch
- coth
- atanh
- asech
- acsch
- acoth
+ 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
Died at...
For the C<csc>, C<cot>, C<asec>, C<acsc>, C<acot>, C<csch>, C<coth>,
-C<asech>, C<acsch>, the argument cannot be C<0> (zero). For the
-C<atanh>, C<acoth>, the argument cannot be C<1> (one). For the
-C<atanh>, C<acoth>, the argument cannot be C<-1> (minus one). For the
-C<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>, C<sech>, the
-argument cannot be I<pi/2 + k * pi>, where I<k> is any integer.
+C<asech>, C<acsch>, the argument cannot be C<0> (zero). For the 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.
+
+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.
+
+=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
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
=cut
+1;
+
# eof
projects / p5sagit/p5-mst-13.2.git / blobdiff