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complex.pd
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use strict;
use warnings;
use PDL::Types qw(ppdefs ppdefs_complex types);
my $R = [ppdefs()];
my $F = [map $_->ppsym, grep $_->real && !$_->integer, types()];
my $C = [ppdefs_complex()];
pp_core_importList('()');
pp_beginwrap; # required for overload to work
# pp_def functions go into the PDL::Complex namespace
# to avoid clashing with PDL::FFTW funcs of the same name that go
# into the PDL namespace
# it should be of no effect to the user of the module but you
# never know....
pp_bless('PDL::Complex');
pp_addpm {At => 'Top'}, <<'EOD';
use strict;
use warnings;
use Carp;
our $VERSION = '2.011';
=encoding utf8
=head1 NAME
PDL::Complex - handle complex numbers (DEPRECATED - use native complex)
=head1 SYNOPSIS
use PDL;
use PDL::Complex;
=head1 DESCRIPTION
This module is deprecated in favour of using "native complex" data types, e.g.:
use PDL;
my $complex_pdl = cdouble('[1+3i]');
print $complex_pdl * pdl('i'); # [-3+i]
This module features a growing number of functions manipulating complex
numbers. These are usually represented as a pair C<[ real imag ]> or
C<[ magnitude phase ]>. If not explicitly mentioned, the functions can work
inplace (not yet implemented!!!) and require rectangular form.
While there is a procedural interface available (C<< $x/$y*$c <=> Cmul
(Cdiv ($x, $y), $c) >>), you can also opt to cast your pdl's into the
C<PDL::Complex> datatype, which works just like your normal ndarrays, but
with all the normal perl operators overloaded.
The latter means that C<sin($x) + $y/$c> will be evaluated using the
normal rules of complex numbers, while other pdl functions (like C<max>)
just treat the ndarray as a real-valued ndarray with a lowest dimension of
size 2, so C<max> will return the maximum of all real and imaginary parts,
not the "highest" (for some definition)
=head2 Native complex support
2.027 added changes in complex number handling, with support for C99
complex floating-point types, and most functions and modules in the core
distribution support these as well.
PDL can now handle complex numbers natively as scalars. This has
the advantage that real and complex valued ndarrays have the same
dimensions. Consider this when writing code in the future.
See L<PDL::Ops/re>, L<PDL::Ops/im>, L<PDL::Ops/abs>, L<PDL::Ops/carg>,
L<PDL::Ops/conj> for more.
=head1 TIPS, TRICKS & CAVEATS
=over 4
=item *
C<i> is a function (not, as of 2.047, a constant) exported by this module,
which represents C<-1**0.5>, i.e. the imaginary unit. it can be used to
quickly and conveniently write complex constants like this: C<4+3*i>.
B<NB> This will override the PDL::Core function of the same name, which
returns a native complex value.
=item *
Use C<r2C(real-values)> to convert from real to complex, as in C<$r
= Cpow $cplx, r2C 2>. The overloaded operators automatically do that for
you, all the other functions, do not. So C<Croots 1, 5> will return all
the fifths roots of 1+1*i (due to broadcasting).
=item *
use C<cplx(real-valued-ndarray)> to cast from normal ndarrays into the
complex datatype. Use C<real(complex-valued-ndarray)> to cast back. This
requires a copy, though.
=back
=head1 EXAMPLE WALK-THROUGH
The complex constant five is equal to C<pdl(1,0)>:
pdl> p $x = r2C 5
5 +0i
Now calculate the three cubic roots of five:
pdl> p $r = Croots $x, 3
[1.70998 +0i -0.854988 +1.48088i -0.854988 -1.48088i]
Check that these really are the roots:
pdl> p $r ** 3
[5 +0i 5 -1.22465e-15i 5 -7.65714e-15i]
Duh! Could be better. Now try by multiplying C<$r> three times with itself:
pdl> p $r*$r*$r
[5 +0i 5 -4.72647e-15i 5 -7.53694e-15i]
Well... maybe C<Cpow> (which is used by the C<**> operator) isn't as
bad as I thought. Now multiply by C<i> and negate, then take the complex
conjugate, which is just a very expensive way of swapping real and
imaginary parts.
pdl> p Cconj(-($r*i))
[0 +1.70998i 1.48088 -0.854988i -1.48088 -0.854988i]
Now plot the magnitude of (part of) the complex sine. First generate the
coefficients:
pdl> $sin = i * zeroes(50)->xlinvals(2,4) + zeroes(50)->xlinvals(0,7)
Now plot the imaginary part, the real part and the magnitude of the sine
into the same diagram:
pdl> use PDL::Graphics::Gnuplot
pdl> gplot( with => 'lines',
PDL::cat(im ( sin $sin ),
re ( sin $sin ),
abs( sin $sin ) ))
An ASCII version of this plot looks like this:
30 ++-----+------+------+------+------+------+------+------+------+-----++
+ + + + + + + + + + +
| $$|
| $ |
25 ++ $$ ++
| *** |
| ** *** |
| $$* *|
20 ++ $** ++
| $$$* #|
| $$$ * # |
| $$ * # |
15 ++ $$$ * # ++
| $$$ ** # |
| $$$$ * # |
| $$$$ * # |
10 ++ $$$$$ * # ++
| $$$$$ * # |
| $$$$$$$ * # |
5 ++ $$$############ * # ++
|*****$$$### ### * # |
* #***** # * # |
| ### *** ### ** # |
0 ## *** # * # ++
| * # * # |
| *** # ** # |
| * # * # |
-5 ++ ** # * # ++
| *** ## ** # |
| * #* # |
| **** ***## # |
-10 ++ **** # # ++
| # # |
| ## ## |
+ + + + + + + ### + ### + + +
-15 ++-----+------+------+------+------+------+-----###-----+------+-----++
0 5 10 15 20 25 30 35 40 45 50
=head1 OPERATORS
The following operators are overloaded:
=over 4
=item +, += (addition)
=item -, -= (subtraction)
=item *, *= (multiplication; L</Cmul>)
=item /, /= (division; L</Cdiv>)
=item **, **= (exponentiation; L</Cpow>)
=item atan2 (4-quadrant arc tangent)
=item sin (L</Csin>)
=item cos (L</Ccos>)
=item exp (L</Cexp>)
=item abs (L</Cabs>)
=item log (L</Clog>)
=item sqrt (L</Csqrt>)
=item ++, -- (increment, decrement; they affect the real part of the complex number only)
=item "" (stringification)
=back
Comparing complex numbers other than for equality is a fatal error.
=cut
my $i;
BEGIN { $i = bless PDL->pdl(0,1) }
{
no warnings 'redefine';
sub i { $i->copy + (@_ ? $_[0] : 0) };
}
# sensible aliases from PDL::LinearAlgebra
*r2p = \&Cr2p;
*p2r = \&Cp2r;
*conj = \&Cconj;
*abs = \&Cabs;
*abs2 = \&Cabs2;
*arg = \&Carg;
*tan = \&Ctan;
*proj = \&Cproj;
*asin = \&Casin;
*acos = \&Cacos;
*atan = \&Catan;
*sinh = \&Csinh;
*cosh = \&Ccosh;
*tanh = \&Ctanh;
*asinh = \&Casinh;
*acosh = \&Cacosh;
*atanh = \&Catanh;
*tricpy = \&Ctricpy;
*mstack = \&Cmstack;
*augment = \&Caugment;
EOD
for (qw(Ctan Catan re im i cplx real)) {
pp_add_exported '', $_;
}
pp_addhdr <<'EOH';
#include <math.h>
#ifndef M_PI
# define M_PI 3.1415926535897932384626433832795029
#endif
#ifndef M_2PI
# define M_2PI (2. * M_PI)
#endif
#if __GLIBC__ > 1 && (defined __USE_MISC || defined __USE_XOPEN || defined __USE_ISOC9X)
# define CABS(r,i) hypot (r, i)
#else
static double
CABS (double r, double i)
{
double t;
if (r < 0) r = - r;
if (i < 0) i = - i;
if (i > r)
{
t = r; r = i; i = t;
}
if (r + i == r)
return r;
t = i / r;
return r * sqrt (1 + t*t);
}
#endif
#if __GLIBC__ >= 2 && __GLIBC_MINOR__ >= 1 && defined __USE_GNU
# define SINCOS(x,s,c) sincos ((x), &(s), &(c))
#else
# define SINCOS(x,s,c) \
(s) = sin (x); \
(c) = cos (x);
#endif
#define CSQRT(type,ar,ai,cr,ci) \
type mag = CABS ((ar), (ai)); \
type t; \
\
if (mag == 0) \
(cr) = (ci) = 0; \
else if ((ar) > 0) \
{ \
t = sqrt (0.5 * (mag + (ar))); \
(cr) = t; \
(ci) = 0.5 * (ai) / t; \
} \
else \
{ \
t = sqrt (0.5 * (mag - (ar))); \
\
if ((ai) < 0) \
t = -t; \
\
(cr) = 0.5 * (ai) / t; \
(ci) = t; \
}
#define CLOG(ar,ai,cr,ci) \
(cr) = log (CABS ((ar), (ai))); \
(ci) = atan2 ((ai), (ar));
EOH
pp_addpm <<'EOP';
=head2 from_native
=for ref
Class method to convert a native-complex ndarray to a PDL::Complex object.
=for usage
PDL::Complex->from_native($native_complex_ndarray)
=cut
sub from_native {
my ($class, $ndarray) = @_;
return $ndarray if UNIVERSAL::isa($ndarray,'PDL::Complex'); # NOOP if P:C
croak "not an ndarray" if !UNIVERSAL::isa($ndarray,'PDL');
croak "not a native complex ndarray" if $ndarray->type->real;
bless PDL::append($ndarray->re->dummy(0),$ndarray->im->dummy(0)), $class;
}
=head2 as_native
=for ref
Object method to convert a PDL::Complex object to a native-complex ndarray.
=for usage
$pdl_complex_obj->as_native
=cut
sub as_native {
PDL::Ops::czip(map $_[0]->slice("($_)"), 0..1);
}
=head2 cplx
=for ref
Cast a real-valued ndarray to the complex datatype.
The first dimension of the ndarray must be of size 2. After this the
usual (complex) arithmetic operators are applied to this pdl, rather
than the normal elementwise pdl operators. Dataflow to the complex
parent works. Use C<sever> on the result if you don't want this.
=for usage
cplx($real_valued_pdl)
=head2 complex
=for ref
Cast a real-valued ndarray to the complex datatype I<without> dataflow
and I<inplace>.
Achieved by merely reblessing an ndarray. The first dimension of the
ndarray must be of size 2.
=for usage
complex($real_valued_pdl)
=head2 real
=for ref
Cast a complex valued pdl back to the "normal" pdl datatype.
Afterwards the normal elementwise pdl operators are used in
operations. Dataflow to the real parent works. Use C<sever> on the
result if you don't want this.
=for usage
real($cplx_valued_pdl)
=cut
sub cplx($) {
return $_[0] if UNIVERSAL::isa($_[0],'PDL::Complex'); # NOOP if just ndarray
croak "first dimsize must be 2" unless $_[0]->dims > 0 && $_[0]->dim(0) == 2;
bless $_[0]->slice('');
}
sub complex($) {
return $_[0] if UNIVERSAL::isa($_[0],'PDL::Complex'); # NOOP if just ndarray
croak "first dimsize must be 2" unless $_[0]->dims > 0 && $_[0]->dim(0) == 2;
bless $_[0];
}
*PDL::cplx = \&cplx;
*PDL::complex = \&complex;
sub real($) {
return $_[0] unless UNIVERSAL::isa($_[0],'PDL::Complex'); # NOOP unless complex
bless $_[0]->slice(''), 'PDL';
}
=head2 t
=for usage
$pdl = $pdl->t(SCALAR(conj))
conj : Conjugate Transpose = 1 | Transpose = 0, default = 0;
=for ref
Convenient function for transposing real or complex 2D array(s).
For complex data, if conj is true returns conjugate transposed array(s).
Supports broadcasting. Not exported.
Originally by Grégory Vanuxem.
=cut
sub t {
my ($m, $conj) = @_;
my $ndims = $m->dims;
my $r = $ndims > 2 ? $m->xchg(1,2) :
$ndims > 1 ? $m->dummy(1) :
$m->dummy(1)->dummy(1);
$conj ? $r->conj : $r;
}
EOP
pp_def 'r2C',
Pars => 'r(); [o]c(m=2)',
Doc => 'convert real to complex, assuming an imaginary part of zero',
PMCode => << 'EOPM',
undef &PDL::r2C;
*PDL::r2C = \&PDL::Complex::r2C;
sub PDL::Complex::r2C {
return $_[0] if UNIVERSAL::isa($_[0],'PDL::Complex');
my $r = __PACKAGE__->initialize;
&PDL::Complex::_r2C_int($_[0], $r);
$r }
EOPM
Code => q!
$c(m=>0) = $r();
$c(m=>1) = 0;
!
;
pp_def 'i2C',
Pars => 'r(); [o]c(m=2)',
Doc => 'convert imaginary to complex, assuming a real part of zero',
PMCode => 'undef &PDL::i2C; *PDL::i2C = \&PDL::Complex::i2C; sub PDL::Complex::i2C { my $r = __PACKAGE__->initialize; &PDL::Complex::_i2C_int($_[0], $r); $r }',
Code => q!
$c(m=>0) = 0;
$c(m=>1) = $r();
!
;
pp_def 'Cr2p',
Pars => 'r(m=2); [o]p(m=2)',
Inplace => 1,
GenericTypes => $F,
Doc => 'convert complex numbers in rectangular form to polar (mod,arg) form. Works inplace',
Code => q!
$GENERIC() x = $r(m=>0);
$GENERIC() y = $r(m=>1);
$p(m=>0) = CABS (x, y);
$p(m=>1) = atan2 (y, x);
!
;
pp_def 'Cp2r',
Pars => 'r(m=2); [o]p(m=2)',
Inplace => 1,
GenericTypes => $F,
Doc => 'convert complex numbers in polar (mod,arg) form to rectangular form. Works inplace',
Code => q!
$GENERIC() m = $r(m=>0);
$GENERIC() a = $r(m=>1);
double s, c;
SINCOS (a, s, c);
$p(m=>0) = c * m;
$p(m=>1) = s * m;
!
;
pp_def 'Cadd', # this is here for a) completeness and b) not having to mess with PDL::Ops
Pars => 'a(m=2); b(m=2); [o]c(m=2)',
Doc => undef,
Code => q^
$GENERIC() ar = $a(m=>0), ai = $a(m=>1);
$GENERIC() br = $b(m=>0), bi = $b(m=>1);
$c(m=>0) = ar + br;
$c(m=>1) = ai + bi;
^
;
pp_def 'Csub', # this is here for a) completeness and b) not having to mess with PDL::Ops
Pars => 'a(m=2); b(m=2); [o]c(m=2)',
Doc => undef,
Code => q^
$GENERIC() ar = $a(m=>0), ai = $a(m=>1);
$GENERIC() br = $b(m=>0), bi = $b(m=>1);
$c(m=>0) = ar - br;
$c(m=>1) = ai - bi;
^
;
pp_def 'Cmul',
Pars => 'a(m=2); b(m=2); [o]c(m=2)',
Doc => 'complex multiplication',
Code => q^
$GENERIC() ar = $a(m=>0), ai = $a(m=>1);
$GENERIC() br = $b(m=>0), bi = $b(m=>1);
$c(m=>0) = ar*br - ai*bi;
$c(m=>1) = ar*bi + ai*br;
^
;
pp_def 'Cprodover',
Pars => 'a(m=2,n); [o]c(m=2)',
Doc => 'Project via product to N-1 dimension',
Code => q^
$GENERIC() br, bi, cr, ci,tmp;
cr = $a(m=>0,n=>0);
ci = $a(m=>1,n=>0);
loop(n=1) %{
br = $a(m=>0);
bi = $a(m=>1);
tmp = cr*bi + ci*br;
cr = cr*br - ci*bi;
ci = tmp;
%}
$c(m=>0) = cr;
$c(m=>1) = ci;
^
;
pp_def 'Cscale',
Pars => 'a(m=2); b(); [o]c(m=2)',
Doc => 'mixed complex/real multiplication',
Code => q^
$GENERIC() ar = $a(m=>0), ai = $a(m=>1);
$c(m=>0) = ar * $b();
$c(m=>1) = ai * $b();
^
;
pp_def 'Cdiv',
Pars => 'a(m=2); b(m=2); [o]c(m=2)',
GenericTypes => $F,
Doc => 'complex division',
Code => q^
$GENERIC() ar = $a(m=>0), ai = $a(m=>1);
$GENERIC() br = $b(m=>0), bi = $b(m=>1);
if (fabsl (br) > fabsl (bi))
{
$GENERIC() tt = bi / br;
$GENERIC() dn = br + tt * bi;
$c(m=>0) = (ar + tt * ai) / dn;
$c(m=>1) = (ai - tt * ar) / dn;
}
else
{
$GENERIC() tt = br / bi;
$GENERIC() dn = br * tt + bi;
$c(m=>0) = (ar * tt + ai) / dn;
$c(m=>1) = (ai * tt - ar) / dn;
}
^
;
pp_def 'Ceq',
Pars => 'a(m=2); b(m=2); [o]c()',
GenericTypes => $F,
Doc => "=for ref\n\nComplex equality operator.",
Code => q^
$c() = (($a(m=>0) == $b(m=>0)) && ($a(m=>1) == $b(m=>1)));
^,
PMCode => <<'EOF',
sub PDL::Complex::Ceq {
my @args = !$_[2] ? @_[1,0] : @_[0,1];
$args[1] = r2C($args[1]) if ref $args[1] ne __PACKAGE__;
PDL::Complex::_Ceq_int($args[0], $args[1], my $r = PDL->null);
$r;
}
EOF
;
pp_def 'Cconj',
Pars => 'a(m=2); [o]c(m=2)',
Inplace => 1,
Doc => 'complex conjugation. Works inplace',
Code => q^
$c(m=>0) = $a(m=>0);
$c(m=>1) = -$a(m=>1);
^
;
pp_def 'Cabs',
Pars => 'a(m=2); [o]c()',
GenericTypes => $F,
Doc => 'complex C<abs()> (also known as I<modulus>)',
PMCode => q^sub PDL::Complex::Cabs($) {
my $pdl= shift;
my $abs = PDL->null;
&PDL::Complex::_Cabs_int($pdl, $abs);
$abs;
}^,
Code => q^
$GENERIC() ar = $a(m=>0), ai = $a(m=>1);
$c() = CABS (ar, ai);
^
;
pp_def 'Cabs2',
Pars => 'a(m=2); [o]c()',
Doc => 'complex squared C<abs()> (also known I<squared modulus>)',
PMCode => q^sub PDL::Complex::Cabs2($) {
my $pdl= shift;
my $abs2 = PDL->null;
&PDL::Complex::_Cabs2_int($pdl, $abs2);
$abs2;
}^,
Code => q^
$GENERIC() ar = $a(m=>0), ai = $a(m=>1);
$c() = ar*ar + ai*ai;
^
;
pp_def 'Carg',
Pars => 'a(m=2); [o]c()',
GenericTypes => $F,
Doc => 'complex argument function ("angle")',
PMCode => q^sub PDL::Complex::Carg($) {
my $pdl= shift;
my $arg = PDL->null;
&PDL::Complex::_Carg_int($pdl, $arg);
$arg;
}^,
Code => q^
$c() = atan2 ($a(m=>1), $a(m=>0));
^
;
pp_def 'Csin',
Pars => 'a(m=2); [o]c(m=2)',
Inplace => 1,
GenericTypes => $F,
Doc => ' sin (a) = 1/(2*i) * (exp (a*i) - exp (-a*i)). Works inplace',
Code => q^
$GENERIC() ar = $a(m=>0), ai = $a(m=>1);
double s, c;
SINCOS (ar, s, c);
$c(m=>0) = s * cosh (ai);
$c(m=>1) = c * sinh (ai);
^
;
pp_def 'Ccos',
Pars => 'a(m=2); [o]c(m=2)',
Inplace => 1,
GenericTypes => $F,
Doc => ' cos (a) = 1/2 * (exp (a*i) + exp (-a*i)). Works inplace',
Code => q^
$GENERIC() ar = $a(m=>0), ai = $a(m=>1);
double s, c;
SINCOS (ar, s, c);
$c(m=>0) = c * cosh (ai);
$c(m=>1) = - s * sinh (ai);
^
;
pp_addpm <<'EOD';
=head2 Ctan
=for ref
Complex tangent
tan (a) = -i * (exp (a*i) - exp (-a*i)) / (exp (a*i) + exp (-a*i))
Does not work inplace.
=cut
sub Ctan($) { Csin($_[0]) / Ccos($_[0]) }
EOD
pp_def 'Cexp',
Pars => 'a(m=2); [o]c(m=2)',
Inplace => 1,
GenericTypes => $F,
Doc => ' exp (a) = exp (real (a)) * (cos (imag (a)) + i * sin (imag (a))). Works inplace',
Code => q^
$GENERIC() ar = $a(m=>0), ai = $a(m=>1);
$GENERIC() ex = exp (ar);
double s, c;
SINCOS (ai, s, c);
$c(m=>0) = ex * c;
$c(m=>1) = ex * s;
^
;
pp_def 'Clog',
Pars => 'a(m=2); [o]c(m=2)',
Inplace => 1,
GenericTypes => $F,
Doc => ' log (a) = log (cabs (a)) + i * carg (a). Works inplace',
Code => q^
$GENERIC() ar = $a(m=>0), ai = $a(m=>1);
CLOG (ar, ai, $c(m=>0), $c(m=>1));
^
;
pp_def 'Cpow',
Pars => 'a(m=2); b(m=2); [o]c(m=2)',
Inplace => ['a'],
GenericTypes => $F,
Doc => 'complex C<pow()> (C<**>-operator)',
Code => q^
$GENERIC() ar = $a(m=>0), ai = $a(m=>1);
$GENERIC() br = $b(m=>0), bi = $b(m=>1);
/* real ndarray (scalar or 1-ndarray) */
if($PDL(b)->dims[0]==0)
bi = 0;
double logr, logi, x, y;
double s, c;
if(ar == 0 && ai == 0){
if(br == 0 && bi == 0) {
$c(m=>0) = 1;
$c(m=>1) = 0;
}
else {
$c(m=>0) = 0;
$c(m=>1) = 0;
}
}
else {
CLOG (ar, ai, logr, logi);
x = exp (logr*br - logi*bi);
y = logr*bi + logi*br;
SINCOS (y, s, c);
$c(m=>0) = x * c;
if(ai == 0 && bi == 0) $c(m=>1) = 0;
else $c(m=>1) = x * s;
}
^
;
pp_def 'Csqrt',
Pars => 'a(m=2); [o]c(m=2)',
Inplace => 1,
GenericTypes => $F,
Doc => 'Works inplace',
Code => q^
$GENERIC() ar = $a(m=>0), ai = $a(m=>1);
CSQRT ($GENERIC(), ar, ai, $c(m=>0), $c(m=>1));
^
;
pp_def 'Casin',
Pars => 'a(m=2); [o]c(m=2)',
Inplace => 1,
GenericTypes => $F,
Doc => 'Works inplace',
Code => q^
$GENERIC() ar = $a(m=>0), ai = $a(m=>1);
$GENERIC() t1 = sqrt ((ar+1)*(ar+1) + ai*ai);
$GENERIC() t2 = sqrt ((ar-1)*(ar-1) + ai*ai);
$GENERIC() alpha = (t1+t2)*0.5;
$GENERIC() beta = (t1-t2)*0.5;
if (alpha < 1) alpha = 1;
if (beta > 1) beta = 1;
else if (beta < -1) beta = -1;
$c(m=>0) = atan2 (beta, sqrt (1-beta*beta));
$c(m=>1) = - log (alpha + sqrt (alpha*alpha-1));
if (ai > 0 || (ai == 0 && ar < -1))
$c(m=>1) = - $c(m=>1);
^
;
pp_def 'Cacos',
Pars => 'a(m=2); [o]c(m=2)',
Inplace => 1,
GenericTypes => $F,
Doc => 'Works inplace',
Code => q^
$GENERIC() ar = $a(m=>0), ai = $a(m=>1);
$GENERIC() t1 = sqrt ((ar+1)*(ar+1) + ai*ai);
$GENERIC() t2 = sqrt ((ar-1)*(ar-1) + ai*ai);
$GENERIC() alpha = (t1+t2)*0.5;
$GENERIC() beta = (t1-t2)*0.5;
if (alpha < 1) alpha = 1;
if (beta > 1) beta = 1;
else if (beta < -1) beta = -1;
$c(m=>0) = atan2 (sqrt (1-beta*beta), beta);
$c(m=>1) = log (alpha + sqrt (alpha*alpha-1));
if (ai > 0 || (ai == 0 && ar < -1))
$c(m=>1) = - $c(m=>1);
^
;
pp_addpm <<'EOD';
=head2 Catan
=for ref
Return the complex C<atan()>.
Does not work inplace.
=cut
sub Catan($) {
my $z = shift;
Cmul Clog(Cdiv (PDL::Complex::i()+$z, PDL::Complex::i()-$z)), PDL->pdl(0, 0.5);
}
EOD
pp_def 'Csinh',
Pars => 'a(m=2); [o]c(m=2)',
Inplace => 1,
GenericTypes => $F,
Doc => ' sinh (a) = (exp (a) - exp (-a)) / 2. Works inplace',
Code => q^
$GENERIC() ar = $a(m=>0), ai = $a(m=>1);
double s, c;
SINCOS (ai, s, c);
$c(m=>0) = sinh (ar) * c;
$c(m=>1) = cosh (ar) * s;
^
;
pp_def 'Ccosh',
Pars => 'a(m=2); [o]c(m=2)',
Inplace => 1,
GenericTypes => $F,
Doc => ' cosh (a) = (exp (a) + exp (-a)) / 2. Works inplace',
Code => q^
$GENERIC() ar = $a(m=>0), ai = $a(m=>1);
double s, c;
SINCOS (ai, s, c);
$c(m=>0) = cosh (ar) * c;
$c(m=>1) = sinh (ar) * s;
^
;
pp_def 'Ctanh',
Pars => 'a(m=2); [o]c(m=2)',
Inplace => 1,
GenericTypes => $F,
Doc => 'Works inplace',
Code => q^
$GENERIC() ar = $a(m=>0), ai = $a(m=>1);
double den;
double s, c;
SINCOS (2*ai, s, c);
den = cosh (2*ar) + c;
$c(m=>0) = sinh (2*ar) / den;
$c(m=>1) = s / den;
^
;
pp_def 'Casinh',
Pars => 'a(m=2); [o]c(m=2)',
Inplace => 1,
GenericTypes => $F,
Doc => 'Works inplace',
Code => q^
$GENERIC() ar = $a(m=>0), ai = $a(m=>1);
$GENERIC() yr = (ar-ai) * (ar+ai) + 1;
$GENERIC() yi = 2*ar*ai;
CSQRT ($GENERIC(), yr, yi, yr, yi)
yr += ar;
yi += ai;
CLOG (yr, yi, $c(m=>0), $c(m=>1));
^
;
pp_def 'Cacosh',
Pars => 'a(m=2); [o]c(m=2)',
Inplace => 1,
GenericTypes => $F,
Doc => 'Works inplace',
Code => q^
$GENERIC() ar = $a(m=>0), ai = $a(m=>1);
$GENERIC() yr = (ar-ai) * (ar+ai) - 1;
$GENERIC() yi = 2*ar*ai;
CSQRT ($GENERIC(), yr, yi, yr, yi)
yr += ar;
yi += ai;
CLOG (yr, yi, $c(m=>0), $c(m=>1));
^
;
pp_def 'Catanh',
Pars => 'a(m=2); [o]c(m=2)',
Inplace => 1,
GenericTypes => $F,
Doc => 'Works inplace',
Code => q^
$GENERIC() ar = $a(m=>0), ai = $a(m=>1);
double i2 = ai*ai;
double num = i2 + (1+ar) * (1+ar);
double den = i2 + (1-ar) * (1-ar);
$c(m=>0) = 0.25 * (log(num) - log(den));
$c(m=>1) = 0.5 * atan2 (2*ai, 1 - ar*ar - i2);
^
;
pp_def 'Cproj',
Pars => 'a(m=2); [o]c(m=2)',
Inplace => 1,
GenericTypes => $F,
Doc => 'compute the projection of a complex number to the riemann sphere. Works inplace',
Code => q^
$GENERIC() ar = $a(m=>0), ai = $a(m=>1);
double den = ar*ar + ai*ai + 1;
$c(m=>0) = 2*ar / den;
$c(m=>1) = 2*ai / den;
^
;
pp_def 'Croots',
Pars => 'a(m=2); [o]c(m=2,n)',
OtherPars => 'int n => n',
GenericTypes => $F,