The London Perl and Raku Workshop takes place on 26th Oct 2024. If your company depends on Perl, please consider sponsoring and/or attending.
 / 
PDL-2.008
River stage three • 96 direct dependents • 105 total dependents
/ PDL::Complex

NAME

PDL::Complex - handle complex numbers

SYNOPSIS

use PDL;
use PDL::Complex;

DESCRIPTION

This module features a growing number of functions manipulating complex numbers. These are usually represented as a pair [ real imag ] or [ angle phase ]. If not explicitly mentioned, the functions can work inplace (not yet implemented!!!) and require rectangular form.

While there is a procedural interface available ($a/$b*$c <=> Cmul (Cdiv $a, $b), $c)), you can also opt to cast your pdl's into the PDL::Complex datatype, which works just like your normal piddles, but with all the normal perl operators overloaded.

The latter means that sin($a) + $b/$c will be evaluated using the normal rules of complex numbers, while other pdl functions (like max) just treat the piddle as a real-valued piddle 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)

TIPS, TRICKS & CAVEATS

  • i is a constant exported by this module, which represents -1**0.5, i.e. the imaginary unit. it can be used to quickly and conviniently write complex constants like this: 4+3*i.

  • 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 threading).

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

  • This module has received some testing by Vanuxem Grégory (g.vanuxem at wanadoo dot fr). Please report any other errors you come across!

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 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, which is just a very expensive way of swapping real and imaginary parts.

pdl> p -($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

FUNCTIONS

cplx real-valued-pdl

Cast a real-valued piddle to the complex datatype. The first dimension of the piddle 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.

complex real-valued-pdl

Cast a real-valued piddle to the complex datatype without dataflow and inplace. Achieved by merely reblessing a piddle. The first dimension of the piddle must be of size 2.

real cplx-valued-pdl

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.

r2C

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

convert real to complex, assuming an imaginary part of zero

i2C

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

convert imaginary to complex, assuming a real part of zero

Cr2p

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

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

Cp2r

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

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

Cmul

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

complex multiplication

Cprodover

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

Project via product to N-1 dimension

Cscale

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

mixed complex/real multiplication

Cdiv

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

complex division

Ccmp

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

Complex comparison oeprator (spaceship). It orders by real first, then by imaginary. Hm, but it is mathematical nonsense! Complex numbers cannot be ordered.

Cconj

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

complex conjugation. Works inplace

Cabs

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

complex abs() (also known as modulus)

Cabs2

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

complex squared abs() (also known squared modulus)

Carg

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

complex argument function ("angle")

Csin

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

Ccos

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

Ctan a [not inplace]

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

Cexp

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

exp (a) = exp (real (a)) * (cos (imag (a)) + i * sin (imag (a))). Works inplace

Clog

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

log (a) = log (cabs (a)) + i * carg (a). Works inplace

Cpow

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

complex pow() (**-operator)

Csqrt

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

Works inplace

Casin

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

Works inplace

Cacos

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

Works inplace

Catan cplx [not inplace]

Return the complex atan().

Csinh

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

Ccosh

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

Ctanh

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

Works inplace

Casinh

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

Works inplace

Cacosh

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

Works inplace

Catanh

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

Works inplace

Cproj

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

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

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!

re cplx, im cplx

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 piddles (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.

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.