#
# 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 $self;
}
#
# (abs)
#
-# Compute complex's norm (rho).
+# Compute or set complex's norm (rho).
#
sub abs {
- my ($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)
#
# 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);
my $z = shift;
my ($r, $t) = @{$z->polar};
my $theta;
- my $eps = 1e-14;
return '[0,0]' if $r <= $eps;
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
=cut
+1;
+
# eof
# $RCSfile: complex.t,v $
#
# Regression tests for the Math::Complex pacakge
-# -- Raphael Manfredi September 1996
-# -- Jarkko Hietaniemi March-October 1997
-# -- Daniel S. Lewart September-October 1997
-
-$VERSION = '1.05';
-
-# $Id: complex.t,v 1.1 1997/10/15 10:02:15 jhi Exp jhi $
+# -- Raphael Manfredi since Sep 1996
+# -- Jarkko Hietaniemi since Mar 1997
+# -- Daniel S. Lewart since Sep 1997
BEGIN {
chdir 't' if -d 't';
use Math::Complex;
+$VERSION = sprintf("%s", q$Id: complex.t,v 1.8 1998/02/05 16:03:39 jhi Exp $ =~ /(\d+\.d+)/);
+
my ($args, $op, $target, $test, $test_set, $try, $val, $zvalue, @set, @val);
$test = 0;
'my ($res, $s0,$s1,$s2,$s3,$s4,$s5,$s6,$s7,$s8,$s9,$s10, $z0,$z1,$z2);' .
"\n\n"
);
-my $eps = 1e-11;
+my $eps = 1e-13;
while (<DATA>) {
s/^\s+//;
}
}
+#
+
+sub test_mutators {
+ my $op;
+
+ $test++;
+push(@script, <<'EOT');
+{
+ my $z = cplx( 1, 1);
+ $z->Re(2);
+ $z->Im(3);
+ print 'not ' unless Re($z) == 2 and Im($z) == 3;
+EOT
+ push(@script, qq(print "ok $test\\n"}\n));
+
+ $test++;
+push(@script, <<'EOT');
+{
+ my $z = cplx( 1, 1);
+ $z->abs(3 * sqrt(2));
+ print 'not ' unless (abs($z) - 3 * sqrt(2)) < $eps and
+ (arg($z) - pi / 4 ) < $eps and
+ (Re($z) - 3 ) < $eps and
+ (Im($z) - 3 ) < $eps;
+EOT
+ push(@script, qq(print "ok $test\\n"}\n));
+
+ $test++;
+push(@script, <<'EOT');
+{
+ my $z = cplx( 1, 1);
+ $z->arg(-3 / 4 * pi);
+ print 'not ' unless (arg($z) + 3 / 4 * pi) < $eps and
+ (abs($z) - sqrt(2) ) < $eps and
+ (Re($z) + 1 ) < $eps and
+ (Im($z) + 1 ) < $eps;
+EOT
+ push(@script, qq(print "ok $test\\n"}\n));
+}
+
+test_mutators();
+
+my $constants = '
+my $i = cplx(0, 1);
+my $pi = cplx(pi, 0);
+my $pii = cplx(0, pi);
+my $pip2 = cplx(pi/2, 0);
+my $zero = cplx(0, 0);
+';
+
+push(@script, $constants);
+
+
# test the divbyzeros
sub test_dbz {
for my $op (@_) {
$test++;
-# push(@script, qq(print "# '$op'\n";));
- push(@script, qq(eval '$op';));
- push(@script, qq(print 'not ' unless (\$@ =~ /Division by zero/);));
- push(@script, qq( print "ok $test\\n";\n));
+ push(@script, <<EOT);
+eval '$op';
+print 'not ' unless (\$@ =~ /Division by zero/);
+EOT
+ push(@script, qq(print "ok $test\\n";\n));
}
}
for my $op (@_) {
$test++;
-# push(@script, qq(print "# '$op'\n";));
- push(@script, qq(eval '$op';));
- push(@script, qq(print 'not ' unless (\$@ =~ /Logarithm of zero/);));
- push(@script, qq( print "ok $test\\n";\n));
+ push(@script, <<EOT);
+eval '$op';
+print 'not ' unless (\$@ =~ /Logarithm of zero/);
+EOT
+ push(@script, qq(print "ok $test\\n";\n));
}
}
-my $minusi = cplx(0, -1);
-
test_dbz(
'i/0',
-# 'tan(pi/2)', # may succeed thanks to floating point inaccuracies
-# 'sec(pi/2)', # may succeed thanks to floating point inaccuracies
- 'csc(0)',
- 'cot(0)',
- 'atan(i)',
- 'atan($minusi)',
- 'asec(0)',
+ 'acot(0)',
+ 'acot(+$i)',
+# 'acoth(-1)', # Log of zero.
+ 'acoth(0)',
+ 'acoth(+1)',
'acsc(0)',
- 'acot(i)',
- 'acot($minusi)',
-# 'tanh(pi/2)', # may succeed thanks to floating point inaccuracies
-# 'sech(pi/2)', # may succeed thanks to floating point inaccuracies
- 'csch(0)',
- 'coth(0)',
- 'atanh(1)',
- 'asech(0)',
'acsch(0)',
- 'acoth(1)',
+ 'asec(0)',
+ 'asech(0)',
+ 'atan(-$i)',
+ 'atan($i)',
+# 'atanh(-1)', # Log of zero.
+ 'atanh(+1)',
+ 'cot(0)',
+ 'coth(0)',
+ 'csc(0)',
+ 'tan($pip2)',
+ 'csch(0)',
+ 'tan($pip2)',
);
-my $zero = cplx(0, 0);
-
test_loz(
'log($zero)',
+ 'acot(-$i)',
'atanh(-1)',
'acoth(-1)',
);
# test the 0**0
sub test_ztz {
- $test++;
+ $test++;
-# push(@script, qq(print "# 0**0\n";));
- push(@script, qq(eval 'cplx(0)**cplx(0)';));
- push(@script, qq(print 'not ' unless (\$@ =~ /zero raised to the/);));
- push(@script, qq( print "ok $test\\n";\n));
+ push(@script, <<'EOT');
+eval 'cplx(0)**cplx(0)';
+print 'not ' unless ($@ =~ /zero raised to the zeroth/);
+EOT
+ push(@script, qq(print "ok $test\\n";\n));
}
test_ztz;
for my $op (@_) {
$test++;
-# push(@script, qq(print "# root(2, $op)\n";));
- push(@script, qq(eval 'root(2, $op)';));
- push(@script, qq(print 'not ' unless (\$@ =~ /root must be/);));
- push(@script, qq( print "ok $test\\n";\n));
+ push(@script, <<EOT);
+eval 'root(2, $op)';
+print 'not ' unless (\$@ =~ /root must be/);
+EOT
+ push(@script, qq(print "ok $test\\n";\n));
}
}
$test++;
# check that the rhs has not changed
push @script, qq(print "not " unless (\$zbr == \$z1r and \$zbi == \$z1i););
- push @script, qq( print "ok $test\\n";\n);
+ push @script, qq(print "ok $test\\n";\n);
push @script, "}\n";
}
}
if (/^\s*\((.*),(.*)\)/) {
return "cplx($1,$2)";
}
+ elsif (/^\s*([\-\+]?(?:\d+(\.\d+)?|\.\d+)(?:[e[\-\+]\d+])?)/) {
+ return "cplx($1,0)";
+ }
elsif (/^\s*\[(.*),(.*)\]/) {
return "cplxe($1,$2)";
}
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