# $RCSFile$
#
# Complex numbers and associated mathematical functions
-# -- Raphael Manfredi, Sept 1996
+# -- Raphael Manfredi, September 1996
+# -- Jarkko Hietaniemi, March 1997
require Exporter;
package Math::Complex; @ISA = qw(Exporter);
+use strict;
+
+use vars qw(@EXPORT $package $display
+ $pi $i $ilog10 $logn %logn);
+
@EXPORT = qw(
pi i Re Im arg
+ sqrt exp log ln
log10 logn cbrt root
- tan cotan asin acos atan acotan
- sinh cosh tanh cotanh asinh acosh atanh acotanh
+ tan
+ cosec csc sec cotan cot
+ asin acos atan
+ acosec acsc asec acotan acot
+ sinh cosh tanh
+ cosech csch sech cotanh coth
+ asinh acosh atanh
+ acosech acsch asech acotanh acoth
cplx cplxe
);
use overload
- '+' => \&plus,
- '-' => \&minus,
- '*' => \&multiply,
- '/' => \÷,
+ '+' => \&plus,
+ '-' => \&minus,
+ '*' => \&multiply,
+ '/' => \÷,
'**' => \&power,
'<=>' => \&spaceship,
'neg' => \&negate,
- '~' => \&conjugate,
+ '~' => \&conjugate,
'abs' => \&abs,
'sqrt' => \&sqrt,
'exp' => \&exp,
'log' => \&log,
'sin' => \&sin,
'cos' => \&cos,
+ 'tan' => \&tan,
'atan2' => \&atan2,
qw("" stringify);
#
sub cplx {
my ($re, $im) = @_;
- return $package->make($re, $im);
+ return $package->make($re, defined $im ? $im : 0);
}
#
#
sub cplxe {
my ($rho, $theta) = @_;
- return $package->emake($rho, $theta);
+ return $package->emake($rho, defined $theta ? $theta : 0);
}
#
# Attribute access/set routines
#
-sub cartesian {$_[0]->{c_dirty} ? $_[0]->update_cartesian : $_[0]->{'cartesian'}}
-sub polar {$_[0]->{p_dirty} ? $_[0]->update_polar : $_[0]->{'polar'}}
+sub cartesian {$_[0]->{c_dirty} ?
+ $_[0]->update_cartesian : $_[0]->{'cartesian'}}
+sub polar {$_[0]->{p_dirty} ?
+ $_[0]->update_polar : $_[0]->{'polar'}}
sub set_cartesian { $_[0]->{p_dirty}++; $_[0]->{'cartesian'} = $_[1] }
sub set_polar { $_[0]->{c_dirty}++; $_[0]->{'polar'} = $_[1] }
#
sub plus {
my ($z1, $z2, $regular) = @_;
+ $z2 = cplx($z2, 0) unless ref $z2;
my ($re1, $im1) = @{$z1->cartesian};
- my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2);
+ my ($re2, $im2) = @{$z2->cartesian};
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};
- my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2);
+ my ($re2, $im2) = @{$z2->cartesian};
unless (defined $inverted) {
$z1->set_cartesian([$re1 - $re2, $im1 - $im2]);
return $z1;
sub multiply {
my ($z1, $z2, $regular) = @_;
my ($r1, $t1) = @{$z1->polar};
- my ($r2, $t2) = ref $z2 ? @{$z2->polar} : (abs($z2), $z2 >= 0 ? 0 : pi);
+ my ($r2, $t2) = ref $z2 ?
+ @{$z2->polar} : (abs($z2), $z2 >= 0 ? 0 : pi);
unless (defined $regular) {
$z1->set_polar([$r1 * $r2, $t1 + $t2]);
return $z1;
}
#
+# 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]);
+ my @up = caller(1);
+ my $dmess = "Died at $up[1] line $up[2].\n";
+ die $dmess;
+}
+
+#
# (divide)
#
# Computes z1/z2.
sub divide {
my ($z1, $z2, $inverted) = @_;
my ($r1, $t1) = @{$z1->polar};
- my ($r2, $t2) = ref $z2 ? @{$z2->polar} : (abs($z2), $z2 >= 0 ? 0 : pi);
+ my ($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;
}
- return $inverted ?
- (ref $z1)->emake($r2 / $r1, $t2 - $t1) :
- (ref $z1)->emake($r1 / $r2, $t1 - $t2);
+ if ($inverted) {
+ divbyzero "$z2/0" if ($r1 == 0);
+ return (ref $z1)->emake($r2 / $r1, $t2 - $t1);
+ } else {
+ divbyzero "$z1/0" if ($r2 == 0);
+ return (ref $z1)->emake($r1 / $r2, $t1 - $t2);
+ }
}
#
#
sub spaceship {
my ($z1, $z2, $inverted) = @_;
+ $z2 = cplx($z2, 0) unless ref $z2;
my ($re1, $im1) = @{$z1->cartesian};
- my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2);
+ my ($re2, $im2) = @{$z2->cartesian};
my $sgn = $inverted ? -1 : 1;
return $sgn * ($re1 <=> $re2) if $re1 != $re2;
return $sgn * ($im1 <=> $im2);
#
sub abs {
my ($z) = @_;
+ return abs($z) unless ref $z;
my ($r, $t) = @{$z->polar};
return abs($r);
}
#
sub arg {
my ($z) = @_;
- return 0 unless ref $z;
+ return ($z < 0 ? pi : 0) unless ref $z;
my ($r, $t) = @{$z->polar};
return $t;
}
#
# (sqrt)
#
-# Compute sqrt(z) (positive only).
+# Compute sqrt(z).
#
sub sqrt {
my ($z) = @_;
+ $z = cplx($z, 0) unless ref $z;
my ($r, $t) = @{$z->polar};
return (ref $z)->emake(sqrt($r), $t/2);
}
#
# cbrt
#
-# Compute cbrt(z) (cubic root, primary only).
+# Compute cbrt(z) (cubic root).
#
sub cbrt {
my ($z) = @_;
- return $z ** (1/3) unless ref $z;
+ return cplx($z, 0) ** (1/3) unless ref $z;
my ($r, $t) = @{$z->polar};
return (ref $z)->emake($r**(1/3), $t/3);
}
#
sub exp {
my ($z) = @_;
+ $z = cplx($z, 0) unless ref $z;
my ($x, $y) = @{$z->cartesian};
return (ref $z)->emake(exp($x), $y);
}
#
sub log {
my ($z) = @_;
+ $z = cplx($z, 0) unless ref $z;
my ($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), $t);
}
#
+# ln
+#
+# Alias for log().
+#
+sub ln { Math::Complex::log(@_) }
+
+#
# log10
#
# Compute log10(z).
#
sub log10 {
my ($z) = @_;
- $log10 = log(10) unless defined $log10;
- return log($z) / $log10 unless ref $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) / $log10, $t / $log10);
+ return (ref $z)->make(log($r) * $ilog10, $t * $ilog10);
}
#
#
sub logn {
my ($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) / log($n);
+ return log($z) / $logn;
}
#
my ($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);
+ return (ref $z)->make(cos($x) * ($ey + $ey_1)/2,
+ sin($x) * ($ey_1 - $ey)/2);
}
#
my ($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);
+ return (ref $z)->make(sin($x) * ($ey + $ey_1)/2,
+ cos($x) * ($ey - $ey_1)/2);
}
#
#
sub tan {
my ($z) = @_;
- return sin($z) / cos($z);
+ my $cz = cos($z);
+ divbyzero "tan($z)", "cos($z)" if ($cz == 0);
+ return sin($z) / $cz;
}
#
-# cotan
+# sec
+#
+# Computes the secant sec(z) = 1 / cos(z).
+#
+sub sec {
+ my ($z) = @_;
+ my $cz = cos($z);
+ divbyzero "sec($z)", "cos($z)" if ($cz == 0);
+ return 1 / $cz;
+}
+
+#
+# csc
+#
+# Computes the cosecant csc(z) = 1 / sin(z).
+#
+sub csc {
+ my ($z) = @_;
+ my $sz = sin($z);
+ divbyzero "csc($z)", "sin($z)" if ($sz == 0);
+ return 1 / $sz;
+}
+
#
-# Computes cotan(z) = 1 / tan(z).
+# cosec
#
-sub cotan {
+# Alias for csc().
+#
+sub cosec { Math::Complex::csc(@_) }
+
+#
+# cot
+#
+# Computes cot(z) = 1 / tan(z).
+#
+sub cot {
my ($z) = @_;
- return cos($z) / sin($z);
+ my $sz = sin($z);
+ divbyzero "cot($z)", "sin($z)" if ($sz == 0);
+ return cos($z) / $sz;
}
#
+# cotan
+#
+# Alias for cot().
+#
+sub cotan { Math::Complex::cot(@_) }
+
+#
# acos
#
# Computes the arc cosine acos(z) = -i log(z + sqrt(z*z-1)).
#
sub acos {
my ($z) = @_;
- my $cz = $z*$z - 1;
- $cz = cplx($cz, 0) if !ref $cz && $cz < 0; # Force complex if <0
- return ~i * log($z + sqrt $cz); # ~i is -i
+ $z = cplx($z, 0) unless ref $z;
+ return ~i * log($z + (Re($z) * Im($z) > 0 ? 1 : -1) * sqrt($z*$z - 1));
}
#
#
sub asin {
my ($z) = @_;
- my $cz = 1 - $z*$z;
- $cz = cplx($cz, 0) if !ref $cz && $cz < 0; # Force complex if <0
- return ~i * log(i * $z + sqrt $cz); # ~i is -i
+ $z = cplx($z, 0) unless ref $z;
+ return ~i * log(i * $z + sqrt(1 - $z*$z));
}
#
# atan
#
-# Computes the arc tagent atan(z) = i/2 log((i+z) / (i-z)).
+# Computes the arc tangent atan(z) = i/2 log((i+z) / (i-z)).
#
sub atan {
my ($z) = @_;
- return i/2 * log((i + $z) / (i - $z));
+ divbyzero "atan($z)", "i - $z" if ($z == i);
+ return i/2*log((i + $z) / (i - $z));
}
#
-# acotan
+# asec
+#
+# Computes the arc secant asec(z) = acos(1 / z).
+#
+sub asec {
+ my ($z) = @_;
+ return acos(1 / $z);
+}
+
+#
+# acosec
+#
+# Computes the arc cosecant sec(z) = asin(1 / z).
+#
+sub acosec {
+ my ($z) = @_;
+ return asin(1 / $z);
+}
+
+#
+# acsc
#
-# Computes the arc cotangent acotan(z) = -i/2 log((i+z) / (z-i))
+# Alias for acosec().
+#
+sub acsc { Math::Complex::acosec(@_) }
+
#
-sub acotan {
+# acot
+#
+# Computes the arc cotangent acot(z) = -i/2 log((i+z) / (z-i))
+#
+sub acot {
my ($z) = @_;
+ divbyzero "acot($z)", "$z - i" if ($z == i);
return i/-2 * log((i + $z) / ($z - i));
}
#
+# acotan
+#
+# Alias for acot().
+#
+sub acotan { Math::Complex::acot(@_) }
+
+#
# cosh
#
# Computes the hyperbolic cosine cosh(z) = (exp(z) + exp(-z))/2.
#
sub cosh {
my ($z) = @_;
- my ($x, $y) = ref $z ? @{$z->cartesian} : ($z);
+ $z = cplx($z, 0) unless ref $z;
+ my ($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);
+ return (ref $z)->make(cos($y) * ($ex + $ex_1)/2,
+ sin($y) * ($ex - $ex_1)/2);
}
#
#
sub sinh {
my ($z) = @_;
- my ($x, $y) = ref $z ? @{$z->cartesian} : ($z);
+ $z = cplx($z, 0) unless ref $z;
+ my ($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);
+ return (ref $z)->make(cos($y) * ($ex - $ex_1)/2,
+ sin($y) * ($ex + $ex_1)/2);
}
#
#
sub tanh {
my ($z) = @_;
- return sinh($z) / cosh($z);
+ my $cz = cosh($z);
+ divbyzero "tanh($z)", "cosh($z)" if ($cz == 0);
+ return sinh($z) / $cz;
}
#
-# cotanh
+# sech
+#
+# Computes the hyperbolic secant sech(z) = 1 / cosh(z).
+#
+sub sech {
+ my ($z) = @_;
+ my $cz = cosh($z);
+ divbyzero "sech($z)", "cosh($z)" if ($cz == 0);
+ return 1 / $cz;
+}
+
+#
+# csch
+#
+# Computes the hyperbolic cosecant csch(z) = 1 / sinh(z).
#
-# Comptutes the hyperbolic cotangent cotanh(z) = cosh(z) / sinh(z).
+sub csch {
+ my ($z) = @_;
+ my $sz = sinh($z);
+ divbyzero "csch($z)", "sinh($z)" if ($sz == 0);
+ return 1 / $sz;
+}
+
+#
+# cosech
+#
+# Alias for csch().
+#
+sub cosech { Math::Complex::csch(@_) }
+
#
-sub cotanh {
+# coth
+#
+# Computes the hyperbolic cotangent coth(z) = cosh(z) / sinh(z).
+#
+sub coth {
my ($z) = @_;
- return cosh($z) / sinh($z);
+ my $sz = sinh($z);
+ divbyzero "coth($z)", "sinh($z)" if ($sz == 0);
+ return cosh($z) / $sz;
}
#
+# cotanh
+#
+# Alias for coth().
+#
+sub cotanh { Math::Complex::coth(@_) }
+
+#
# acosh
#
# Computes the arc hyperbolic cosine acosh(z) = log(z + sqrt(z*z-1)).
#
sub acosh {
my ($z) = @_;
- my $cz = $z*$z - 1;
- $cz = cplx($cz, 0) if !ref $cz && $cz < 0; # Force complex if <0
- return log($z + sqrt $cz);
+ $z = cplx($z, 0) unless ref $z; # asinh(-2)
+ return log($z + sqrt($z*$z - 1));
}
#
#
sub asinh {
my ($z) = @_;
- my $cz = $z*$z + 1; # Already complex if <0
- return log($z + sqrt $cz);
+ $z = cplx($z, 0) unless ref $z; # asinh(-2)
+ return log($z + sqrt($z*$z + 1));
}
#
#
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);
- $cz = cplx($cz, 0) if !ref $cz && $cz < 0; # Force complex if <0
return log($cz) / 2;
}
#
-# acotanh
+# asech
+#
+# Computes the hyperbolic arc secant asech(z) = acosh(1 / z).
+#
+sub asech {
+ my ($z) = @_;
+ divbyzero 'asech(0)', $z if ($z == 0);
+ return acosh(1 / $z);
+}
+
+#
+# acsch
#
-# Computes the arc hyperbolic cotangent acotanh(z) = 1/2 log((1+z) / (z-1)).
+# Computes the hyperbolic arc cosecant acsch(z) = asinh(1 / z).
#
-sub acotanh {
+sub acsch {
my ($z) = @_;
+ divbyzero 'acsch(0)', $z if ($z == 0);
+ return asinh(1 / $z);
+}
+
+#
+# acosech
+#
+# Alias for acosh().
+#
+sub acosech { Math::Complex::acsch(@_) }
+
+#
+# acoth
+#
+# Computes the arc hyperbolic cotangent acoth(z) = 1/2 log((1+z) / (z-1)).
+#
+sub acoth {
+ my ($z) = @_;
+ $z = cplx($z, 0) unless ref $z; # acoth(-2)
+ divbyzero 'acoth(1)', "$z - 1" if ($z == 1);
my $cz = (1 + $z) / ($z - 1);
- $cz = cplx($cz, 0) if !ref $cz && $cz < 0; # Force complex if <0
return log($cz) / 2;
}
#
+# acotanh
+#
+# Alias for acot().
+#
+sub acotanh { Math::Complex::acoth(@_) }
+
+#
# (atan2)
#
# Compute atan(z1/z2).
sub atan2 {
my ($z1, $z2, $inverted) = @_;
my ($re1, $im1) = @{$z1->cartesian};
- my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2);
+ my ($re2, $im2) = @{$z2->cartesian};
my $tan;
if (defined $inverted && $inverted) { # atan(z2/z1)
return pi * ($re2 > 0 ? 1 : -1) if $re1 == 0 && $im1 == 0;
if (ref $self) { # Called as a method
$format = shift;
- } else { # Regular procedure call
+ } else { # Regular procedure call
$format = $self;
undef $self;
}
elsif ($y == -1) { $im = '-i' }
elsif (abs($y) >= 1e-14) { $im = $y . "i" }
- my $str;
+ my $str = '';
$str = $re if defined $re;
$str .= "+$im" if defined $im;
$str =~ s/\+-/-/;
my $z = shift;
my ($r, $t) = @{$z->polar};
my $theta;
+ my $eps = 1e-14;
- return '[0,0]' if $r <= 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) <= 1e-14) { $theta = 0 }
- elsif (abs(pi-$nt) <= 1e-14) { $theta = 'pi' }
+ if (abs($nt) <= $eps) { $theta = 0 }
+ elsif (abs(pi-$nt) <= $eps) { $theta = 'pi' }
if (defined $theta) {
- $r = int($r + ($r < 0 ? -1 : 1) * 1e-14)
- if int(abs($r)) != int(abs($r) + 1e-14);
- $theta = int($theta + ($theta < 0 ? -1 : 1) * 1e-14)
- if int(abs($theta)) != int(abs($theta) + 1e-14);
+ $r = int($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 "\[$r,$theta\]";
}
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) <= 1e-14) {
- $theta = ($nt < 0 ? '-':'').($k == 1 ? 'pi':"${k}pi").'/'.abs($n);
+ if (abs($kpi/$n - $nt) <= $eps) {
+ $theta = ($nt < 0 ? '-':'').
+ ($k == 1 ? 'pi':"${k}pi").'/'.abs($n);
last;
}
}
$theta = $nt unless defined $theta;
- $r = int($r + ($r < 0 ? -1 : 1) * 1e-14)
- if int(abs($r)) != int(abs($r) + 1e-14);
- $theta = int($theta + ($theta < 0 ? -1 : 1) * 1e-14)
- if int(abs($theta)) != int(abs($theta) + 1e-14);
+ $r = int($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));
return "\[$r,$theta\]";
}
logn(z, n) = log(z) / log(n)
tan(z) = sin(z) / cos(z)
- cotan(z) = 1 / tan(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))
atan(z) = i/2 * log((i+z) / (i-z))
- acotan(z) = -i/2 * log((i+z) / (z-i))
+
+ acsc(z) = asin(1 / z)
+ asec(z) = acos(1 / z)
+ acot(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)
- cotanh(z) = 1 / tanh(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)
+ 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))
- acotanh(z) = 1/2 * log((1+z) / (z-1))
-The I<root> function is available to compute all the I<n>th
+ 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.
+
+The I<root> function is available to compute all the I<n>
roots of some complex, where I<n> is a strictly positive integer.
There are exactly I<n> such roots, returned as a list. Getting the
number mathematicians call C<j> such that:
(root(z, n))[k] = r**(1/n) * exp(i * (t + 2*k*pi)/n)
-The I<spaceshift> operation 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 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 = 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).
-(Mnmemonic: C<e> is used as a notation for complex numbers in the trigonometric
-form).
+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).
It is possible to write:
use BigFloat, since Perl has currently no rule to disambiguate a '+'
operation (for instance) between two overloaded entities.
-=head1 AUTHOR
+=head1 AUTHORS
-Raphael Manfredi <F<Raphael_Manfredi@grenoble.hp.com>>
+ Raphael Manfredi <F<Raphael_Manfredi@grenoble.hp.com>>
+ Jarkko Hietaniemi <F<jhi@iki.fi>>
# $RCSfile$
#
# Regression tests for the new Math::Complex pacakge
-# -- Raphael Manfredi, Sept 1996
+# -- Raphael Manfredi, Septemeber 1996
+# -- Jarkko Hietaniemi Manfredi, March 1997
BEGIN {
chdir 't' if -d 't';
@INC = '../lib';
$test = 0;
$| = 1;
$script = '';
-$epsilon = 1e-10;
+my $eps = 1e-4; # for example root() is quite bad
while (<DATA>) {
- next if /^#/ || /^\s*$/;
- chop;
- $set_test = 0; # Assume not a test over a set of values
+ s/^\s+//;
+ next if $_ eq '' || /^\#/;
+ chomp;
+ $test_set = 0; # Assume not a test over a set of values
if (/^&(.*)/) {
$op = $1;
next;
next;
}
elsif (s/^\|//) {
- $set_test = 1; # Requests we loop over the set...
+ $test_set = 1; # Requests we loop over the set...
}
my @args = split(/:/);
- if ($set_test) {
+ if ($test_set == 1) {
my $i;
for ($i = 0; $i < @set; $i++) {
- $target = $set[$i]; # complex number
- $zvalue = $val[$i]; # textual value as found in set definition
+ # complex number
+ $target = $set[$i];
+ # textual value as found in set definition
+ $zvalue = $val[$i];
test($zvalue, $target, @args);
}
} else {
}
if (defined $z) {
$args = "'$op'"; # Really the value
- $try = "abs(\$z0 - \$z1) <= 1e-10 ? \$z1 : \$z0";
+ $try = "abs(\$z0 - \$z1) <= $eps ? \$z1 : \$z0";
$script .= "\$res = $try; ";
$script .= "check($test, $args[0], \$res, \$z$#args, $args);\n";
} else {
sub check {
my ($test, $try, $got, $expected, @z) = @_;
- if ("$got" eq "$expected" || ($expected =~ /^-?\d/ && $got == $expected)) {
+
+# print "# @_\n";
+
+ if ("$got" eq "$expected"
+ ||
+ ($expected =~ /^-?\d/ && $got == $expected)
+ ||
+ (abs($got - $expected) < $eps)
+ ) {
print "ok $test\n";
} else {
print "not ok $test\n";
[4,pi]:[2,pi/2]:[2,pi/2]
[2,pi/2]:[4,pi]:[0.5,-(pi)/2]
+&Re
+(3,4):3
+(-3,4):-3
+[1,pi/2]:0
+
+&Im
+(3,4):4
+(3,-4):-4
+[1,pi/2]:1
+
&abs
(3,4):5
(-3,4):5
+&arg
+[2,0]:0
+[-2,0]:pi
+
&~
(4,5):(4,-5)
(-3,4):(-3,-4)
(3,4):(3,4):1
&sqrt
+-9:(0,3)
(-100,0):(0,10)
(16,-30):(5,-3)
|'z - ~z':'2*i*Im(z)'
|'z * ~z':'abs(z) * abs(z)'
-{ (4,3); [3,2]; (-3,4); (0,2); 3; 1; (-5, 0); [2,1] }
+{ (2,3); [3,2]; (-3,2); (0,2); 3; 1.2; -3; (-3, 0); (-2, -1); [2,1] }
-|'exp(z)':'exp(a) * exp(i * b)'
+|'(root(z, 4))[1] ** 4':'z'
+|'(root(z, 5))[3] ** 5':'z'
+|'(root(z, 8))[7] ** 8':'z'
|'abs(z)':'r'
-|'sqrt(z) * sqrt(z)':'z'
-|'sqrt(z)':'sqrt(r) * exp(i * t/2)'
+|'acot(z)':'acotan(z)'
+|'acsc(z)':'acosec(z)'
+|'acsc(z)':'asin(1 / z)'
+|'asec(z)':'acos(1 / z)'
|'cbrt(z)':'cbrt(r) * exp(i * t/3)'
-|'log(z)':'log(r) + i*t'
-|'sin(asin(z))':'z'
|'cos(acos(z))':'z'
-|'tan(atan(z))':'z'
-|'cotan(acotan(z))':'z'
|'cos(z) ** 2 + sin(z) ** 2':1
-|'cosh(z) ** 2 - sinh(z) ** 2':1
|'cos(z)':'cosh(i*z)'
-|'cotan(z)':'1 / tan(z)'
-|'cotanh(z)':'1 / tanh(z)'
-|'i*sin(z)':'sinh(i*z)'
-|'z**z':'exp(z * log(z))'
-|'log(exp(z))':'z'
+|'cosh(z) ** 2 - sinh(z) ** 2':1
+|'cot(acot(z))':'z'
+|'cot(z)':'1 / tan(z)'
+|'cot(z)':'cotan(z)'
+|'csc(acsc(z))':'z'
+|'csc(z)':'1 / sin(z)'
+|'csc(z)':'cosec(z)'
|'exp(log(z))':'z'
+|'exp(z)':'exp(a) * exp(i * b)'
+|'ln(z)':'log(z)'
+|'log(exp(z))':'z'
+|'log(z)':'log(r) + i*t'
|'log10(z)':'log(z) / log(10)'
-|'logn(z, 3)':'log(z) / log(3)'
|'logn(z, 2)':'log(z) / log(2)'
-|'(root(z, 4))[1] ** 4':'z'
-|'(root(z, 8))[7] ** 8':'z'
+|'logn(z, 3)':'log(z) / log(3)'
+|'sec(asec(z))':'z'
+|'sec(z)':'1 / cos(z)'
+|'sin(asin(z))':'z'
+|'sin(i * z)':'i * sinh(z)'
+|'sqrt(z) * sqrt(z)':'z'
+|'sqrt(z)':'sqrt(r) * exp(i * t/2)'
+|'tan(atan(z))':'z'
+|'z**z':'exp(z * log(z))'
-{ (1,1); [1,0.5]; (-2, -1); 2; (-1,0.5); (0,0.5); 0.5; (2, 0) }
+{ (1,1); [1,0.5]; (-2, -1); 2; -3; (-1,0.5); (0,0.5); 0.5; (2, 0); (-1, -2) }
-|'sinh(asinh(z))':'z'
|'cosh(acosh(z))':'z'
+|'coth(acoth(z))':'z'
+|'coth(z)':'1 / tanh(z)'
+|'coth(z)':'cotanh(z)'
+|'csch(acsch(z))':'z'
+|'csch(z)':'1 / sinh(z)'
+|'csch(z)':'cosech(z)'
+|'sech(asech(z))':'z'
+|'sech(z)':'1 / cosh(z)'
+|'sinh(asinh(z))':'z'
|'tanh(atanh(z))':'z'
-|'cotanh(acotanh(z))':'z'
-{ (0.2,-0.4); [1,0.5]; -1.2; (-1,0.5); (0,-0.5); 0.5; (1.1, 0) }
+{ (0.2,-0.4); [1,0.5]; -1.2; (-1,0.5); 0.5; (1.1, 0) }
-|'asin(sin(z))':'z'
|'acos(cos(z)) ** 2':'z * z'
-|'atan(tan(z))':'z'
-|'asinh(sinh(z))':'z'
|'acosh(cosh(z)) ** 2':'z * z'
+|'acoth(z)':'acotanh(z)'
+|'acoth(z)':'atanh(1 / z)'
+|'acsch(z)':'acosech(z)'
+|'acsch(z)':'asinh(1 / z)'
+|'asech(z)':'acosh(1 / z)'
+|'asin(sin(z))':'z'
+|'asinh(sinh(z))':'z'
+|'atan(tan(z))':'z'
|'atanh(tanh(z))':'z'
+&sin
+( 2, 3):( 9.15449914691143, -4.16890695996656)
+(-2, 3):( -9.15449914691143, -4.16890695996656)
+(-2,-3):( -9.15449914691143, 4.16890695996656)
+( 2,-3):( 9.15449914691143, 4.16890695996656)
+
+&cos
+( 2, 3):( -4.18962569096881, -9.10922789375534)
+(-2, 3):( -4.18962569096881, 9.10922789375534)
+(-2,-3):( -4.18962569096881, -9.10922789375534)
+( 2,-3):( -4.18962569096881, 9.10922789375534)
+
+&tan
+( 2, 3):( -0.00376402564150, 1.00323862735361)
+(-2, 3):( 0.00376402564150, 1.00323862735361)
+(-2,-3):( 0.00376402564150, -1.00323862735361)
+( 2,-3):( -0.00376402564150, -1.00323862735361)
+
+&sec
+( 2, 3):( -0.04167496441114, 0.09061113719624)
+(-2, 3):( -0.04167496441114, -0.09061113719624)
+(-2,-3):( -0.04167496441114, 0.09061113719624)
+( 2,-3):( -0.04167496441114, -0.09061113719624)
+
+&csc
+( 2, 3):( 0.09047320975321, 0.04120098628857)
+(-2, 3):( -0.09047320975321, 0.04120098628857)
+(-2,-3):( -0.09047320975321, -0.04120098628857)
+( 2,-3):( 0.09047320975321, -0.04120098628857)
+
+&cot
+( 2, 3):( -0.00373971037634, -0.99675779656936)
+(-2, 3):( 0.00373971037634, -0.99675779656936)
+(-2,-3):( 0.00373971037634, 0.99675779656936)
+( 2,-3):( -0.00373971037634, 0.99675779656936)
+
+&asin
+( 2, 3):( 0.57065278432110, 1.98338702991654)
+(-2, 3):( -0.57065278432110, 1.98338702991654)
+(-2,-3):( -0.57065278432110, -1.98338702991654)
+( 2,-3):( 0.57065278432110, -1.98338702991654)
+
+&acos
+( 2, 3):( 1.00014354247380, -1.98338702991654)
+(-2, 3):( 2.14144911111600, -1.98338702991654)
+(-2,-3):( 2.14144911111600, 1.98338702991654)
+( 2,-3):( 1.00014354247380, 1.98338702991654)
+
+&atan
+( 2, 3):( 1.40992104959658, 0.22907268296854)
+(-2, 3):( -1.40992104959658, 0.22907268296854)
+(-2,-3):( -1.40992104959658, -0.22907268296854)
+( 2,-3):( 1.40992104959658, -0.22907268296854)
+
+&asec
+( 2, 3):( 1.42041072246703, 0.23133469857397)
+(-2, 3):( 1.72118193112276, 0.23133469857397)
+(-2,-3):( 1.72118193112276, -0.23133469857397)
+( 2,-3):( 1.42041072246703, -0.23133469857397)
+
+&acsc
+( 2, 3):( 0.15038560432786, -0.23133469857397)
+(-2, 3):( -0.15038560432786, -0.23133469857397)
+(-2,-3):( -0.15038560432786, 0.23133469857397)
+( 2,-3):( 0.15038560432786, 0.23133469857397)
+
+&acot
+( 2, 3):( 0.16087527719832, -0.22907268296854)
+(-2, 3):( -0.16087527719832, -0.22907268296854)
+(-2,-3):( -0.16087527719832, 0.22907268296854)
+( 2,-3):( 0.16087527719832, 0.22907268296854)
+
+&sinh
+( 2, 3):( -3.59056458998578, 0.53092108624852)
+(-2, 3):( 3.59056458998578, 0.53092108624852)
+(-2,-3):( 3.59056458998578, -0.53092108624852)
+( 2,-3):( -3.59056458998578, -0.53092108624852)
+
+&cosh
+( 2, 3):( -3.72454550491532, 0.51182256998738)
+(-2, 3):( -3.72454550491532, -0.51182256998738)
+(-2,-3):( -3.72454550491532, 0.51182256998738)
+( 2,-3):( -3.72454550491532, -0.51182256998738)
+
+&tanh
+( 2, 3):( 0.96538587902213, -0.00988437503832)
+(-2, 3):( -0.96538587902213, -0.00988437503832)
+(-2,-3):( -0.96538587902213, 0.00988437503832)
+( 2,-3):( 0.96538587902213, 0.00988437503832)
+
+&sech
+( 2, 3):( -0.26351297515839, -0.03621163655877)
+(-2, 3):( -0.26351297515839, 0.03621163655877)
+(-2,-3):( -0.26351297515839, -0.03621163655877)
+( 2,-3):( -0.26351297515839, 0.03621163655877)
+
+&csch
+( 2, 3):( -0.27254866146294, -0.04030057885689)
+(-2, 3):( 0.27254866146294, -0.04030057885689)
+(-2,-3):( 0.27254866146294, 0.04030057885689)
+( 2,-3):( -0.27254866146294, 0.04030057885689)
+
+&coth
+( 2, 3):( 1.03574663776500, 0.01060478347034)
+(-2, 3):( -1.03574663776500, 0.01060478347034)
+(-2,-3):( -1.03574663776500, -0.01060478347034)
+( 2,-3):( 1.03574663776500, -0.01060478347034)
+
+&asinh
+( 2, 3):( 1.96863792579310, 0.96465850440760)
+(-2, 3):( -1.96863792579310, 0.96465850440761)
+(-2,-3):( -1.96863792579310, -0.96465850440761)
+( 2,-3):( 1.96863792579310, -0.96465850440760)
+
+&acosh
+( 2, 3):( 1.98338702991654, 1.00014354247380)
+(-2, 3):( -1.98338702991653, -2.14144911111600)
+(-2,-3):( -1.98338702991653, 2.14144911111600)
+( 2,-3):( 1.98338702991654, -1.00014354247380)
+
+&atanh
+( 2, 3):( 0.14694666622553, 1.33897252229449)
+(-2, 3):( -0.14694666622553, 1.33897252229449)
+(-2,-3):( -0.14694666622553, -1.33897252229449)
+( 2,-3):( 0.14694666622553, -1.33897252229449)
+
+&asech
+( 2, 3):( 0.23133469857397, -1.42041072246703)
+(-2, 3):( -0.23133469857397, 1.72118193112276)
+(-2,-3):( -0.23133469857397, -1.72118193112276)
+( 2,-3):( 0.23133469857397, 1.42041072246703)
+
+&acsch
+( 2, 3):( 0.15735549884499, -0.22996290237721)
+(-2, 3):( -0.15735549884499, -0.22996290237721)
+(-2,-3):( -0.15735549884499, 0.22996290237721)
+( 2,-3):( 0.15735549884499, 0.22996290237721)
+
+&acoth
+( 2, 3):( 0.14694666622553, -0.23182380450040)
+(-2, 3):( -0.14694666622553, -0.23182380450040)
+(-2,-3):( -0.14694666622553, 0.23182380450040)
+( 2,-3):( 0.14694666622553, 0.23182380450040)
+
+# eof
projects / p5sagit/p5-mst-13.2.git / commitdiff