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PDL-2.093
River stage three • 96 direct dependents • 105 total dependents
/ PDL::Complex

NAME

PDL::Complex - handle complex numbers (DEPRECATED - use native complex)

SYNOPSIS

use PDL;
use PDL::Complex;

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 [ real imag ] or [ 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 ($x/$y*$c <=> Cmul (Cdiv ($x, $y), $c)), you can also opt to cast your pdl's into the PDL::Complex datatype, which works just like your normal ndarrays, but with all the normal perl operators overloaded.

The latter means that sin($x) + $y/$c will be evaluated using the normal rules of complex numbers, while other pdl functions (like max) just treat the ndarray as a real-valued ndarray with a lowest dimension of size 2, so max will return the maximum of all real and imaginary parts, not the "highest" (for some definition)

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 "re" in PDL::Ops, "im" in PDL::Ops, "abs" in PDL::Ops, "carg" in PDL::Ops, "conj" in PDL::Ops for more.

TIPS, TRICKS & CAVEATS

  • i is a function (not, as of 2.047, a constant) exported by this module, which represents -1**0.5, i.e. the imaginary unit. it can be used to quickly and conveniently write complex constants like this: 4+3*i.

    NB This will override the PDL::Core function of the same name, which returns a native complex value.

  • Use r2C(real-values) to convert from real to complex, as in $r = Cpow $cplx, r2C 2. The overloaded operators automatically do that for you, all the other functions, do not. So Croots 1, 5 will return all the fifths roots of 1+1*i (due to broadcasting).

  • use cplx(real-valued-ndarray) to cast from normal ndarrays into the complex datatype. Use real(complex-valued-ndarray) to cast back. This requires a copy, though.

EXAMPLE WALK-THROUGH

The complex constant five is equal to 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 $r three times with itself:

pdl> p $r*$r*$r
[5 +0i  5 -4.72647e-15i  5 -7.53694e-15i]

Well... maybe Cpow (which is used by the ** operator) isn't as bad as I thought. Now multiply by 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

OPERATORS

The following operators are overloaded:

+, += (addition)
-, -= (subtraction)
*, *= (multiplication; "Cmul")
/, /= (division; "Cdiv")
**, **= (exponentiation; "Cpow")
atan2 (4-quadrant arc tangent)
sin ("Csin")
cos ("Ccos")
exp ("Cexp")
abs ("Cabs")
log ("Clog")
sqrt ("Csqrt")
++, -- (increment, decrement; they affect the real part of the complex number only)
"" (stringification)

Comparing complex numbers other than for equality is a fatal error.

FUNCTIONS

from_native

Class method to convert a native-complex ndarray to a PDL::Complex object.

PDL::Complex->from_native($native_complex_ndarray)

as_native

Object method to convert a PDL::Complex object to a native-complex ndarray.

$pdl_complex_obj->as_native

cplx

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 sever on the result if you don't want this.

cplx($real_valued_pdl)

complex

Cast a real-valued ndarray to the complex datatype without dataflow and inplace.

Achieved by merely reblessing an ndarray. The first dimension of the ndarray must be of size 2.

complex($real_valued_pdl)

real

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 sever on the result if you don't want this.

real($cplx_valued_pdl)

t

$pdl = $pdl->t(SCALAR(conj))
conj : Conjugate Transpose = 1 | Transpose = 0, default = 0;

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.

r2C

Signature: (r(); [o]c(m=2))

convert real to complex, assuming an imaginary part of zero

r2C does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

i2C

Signature: (r(); [o]c(m=2))

convert imaginary to complex, assuming a real part of zero

i2C does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

Cr2p

Signature: (r(m=2); float+ [o]p(m=2))

convert complex numbers in rectangular form to polar (mod,arg) form. Works inplace

Cr2p does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

Cp2r

Signature: (r(m=2); [o]p(m=2))

convert complex numbers in polar (mod,arg) form to rectangular form. Works inplace

Cp2r does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

Cmul

Signature: (a(m=2); b(m=2); [o]c(m=2))

complex multiplication

Cmul does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

Cprodover

Signature: (a(m=2,n); [o]c(m=2))

Project via product to N-1 dimension

Cprodover does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

Cscale

Signature: (a(m=2); b(); [o]c(m=2))

mixed complex/real multiplication

Cscale does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

Cdiv

Signature: (a(m=2); b(m=2); [o]c(m=2))

complex division

Cdiv does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

Ceq

Signature: (a(m=2); b(m=2); [o]c())

Complex equality operator.

Ceq does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

Cconj

Signature: (a(m=2); [o]c(m=2))

complex conjugation. Works inplace

Cconj does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

Cabs

Signature: (a(m=2); [o]c())

complex abs() (also known as modulus)

Cabs does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

Cabs2

Signature: (a(m=2); [o]c())

complex squared abs() (also known squared modulus)

Cabs2 does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

Carg

Signature: (a(m=2); [o]c())

complex argument function ("angle")

Carg does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

Csin

Signature: (a(m=2); [o]c(m=2))
sin (a) = 1/(2*i) * (exp (a*i) - exp (-a*i)). Works inplace

Csin does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

Ccos

Signature: (a(m=2); [o]c(m=2))
cos (a) = 1/2 * (exp (a*i) + exp (-a*i)). Works inplace

Ccos does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

Ctan

Complex tangent

tan (a) = -i * (exp (a*i) - exp (-a*i)) / (exp (a*i) + exp (-a*i))

Does not work inplace.

Cexp

Signature: (a(m=2); [o]c(m=2))
exp (a) = exp (real (a)) * (cos (imag (a)) + i * sin (imag (a))). Works inplace

Cexp does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

Clog

Signature: (a(m=2); [o]c(m=2))
log (a) = log (cabs (a)) + i * carg (a). Works inplace

Clog does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

Cpow

Signature: (a(m=2); b(m=2); [o]c(m=2))

complex pow() (**-operator)

Cpow does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

Csqrt

Signature: (a(m=2); [o]c(m=2))

Works inplace

Csqrt does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

Casin

Signature: (a(m=2); [o]c(m=2))

Works inplace

Casin does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

Cacos

Signature: (a(m=2); [o]c(m=2))

Works inplace

Cacos does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

Catan

Return the complex atan().

Does not work inplace.

Csinh

Signature: (a(m=2); [o]c(m=2))
sinh (a) = (exp (a) - exp (-a)) / 2. Works inplace

Csinh does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

Ccosh

Signature: (a(m=2); [o]c(m=2))
cosh (a) = (exp (a) + exp (-a)) / 2. Works inplace

Ccosh does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

Ctanh

Signature: (a(m=2); [o]c(m=2))

Works inplace

Ctanh does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

Casinh

Signature: (a(m=2); [o]c(m=2))

Works inplace

Casinh does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

Cacosh

Signature: (a(m=2); [o]c(m=2))

Works inplace

Cacosh does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

Catanh

Signature: (a(m=2); [o]c(m=2))

Works inplace

Catanh does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

Cproj

Signature: (a(m=2); [o]c(m=2))

compute the projection of a complex number to the riemann sphere. Works inplace

Cproj does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

Croots

Signature: (a(m=2); [o]c(m=2,n); int n => n)

Compute the n roots of a. n must be a positive integer. The result will always be a complex type!

Croots does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

re, im

Return the real or imaginary part of the complex number(s) given.

These are slicing operators, so data flow works. The real and imaginary parts are returned as ndarrays (ref eq PDL).

rCpolynomial

Signature: (coeffs(n); x(c=2,m); [o]out(c=2,m))

evaluate the polynomial with (real) coefficients coeffs at the (complex) position(s) x. coeffs[0] is the constant term.

rCpolynomial does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

Ctricpy

Signature: (A(c=2,m,n);[o] C(c=2,m,n); int uplo)

tricpy(PDL(A), int(uplo), PDL(C))

$c = $a->tricpy($uplo); # explicit uplo
$c = $a->tricpy;        # default upper

or

tricpy($a, $uplo, $c);  # modify c

Copy triangular part to another matrix. If uplo == 0 copy upper triangular part.

Originally by Grégory Vanuxem.

Ctricpy does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

Cmstack

Signature: (x(c=2,n,m);y(c,n,p);[o]out(c,n,q=CALC($SIZE(m)+$SIZE(p))))

Combine two 2D ndarrays into a single ndarray, along the second ("vertical") dim. This routine does backward and forward dataflow automatically.

Originally by Grégory Vanuxem.

Cmstack does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

Caugment

Signature: (x(c=2,n);y(c,p);[o]out(c,q=CALC($SIZE(n)+$SIZE(p))))

Combine two ndarrays into a single ndarray along the 0-th ("horizontal") dim. This routine does backward and forward dataflow automatically.

Originally by Grégory Vanuxem.

Caugment does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

AUTHOR

Copyright (C) 2000 Marc Lehmann <pcg@goof.com>. All rights reserved. There is no warranty. You are allowed to redistribute this software / documentation as described in the file COPYING in the PDL distribution.

SEE ALSO

perl(1), PDL.