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PDL-2.020_02
River stage three • 93 direct dependents • 101 total dependents
52 ++
/ 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 [ 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 piddles, 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 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 conveniently 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

OPERATORS

The following operators are overloaded:

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

FUNCTIONS

cplx

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.

 cplx($real_valued_pdl)

complex

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.

 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)

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 piddles if the flag is set for any of the input piddles.

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 piddles if the flag is set for any of the input piddles.

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 piddles if the flag is set for any of the input piddles.

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 piddles if the flag is set for any of the input piddles.

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 piddles if the flag is set for any of the input piddles.

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 piddles if the flag is set for any of the input piddles.

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 piddles if the flag is set for any of the input piddles.

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 piddles if the flag is set for any of the input piddles.

Ccmp

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

Complex comparison operator (spaceship).

Ccmp orders by real first, then by imaginary. Hm, but it is mathematical nonsense! Complex numbers cannot be ordered.

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

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 piddles if the flag is set for any of the input piddles.

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 piddles if the flag is set for any of the input piddles.

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 piddles if the flag is set for any of the input piddles.

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 piddles if the flag is set for any of the input piddles.

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 piddles if the flag is set for any of the input piddles.

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 piddles if the flag is set for any of the input piddles.

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 piddles if the flag is set for any of the input piddles.

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 piddles if the flag is set for any of the input piddles.

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 piddles if the flag is set for any of the input piddles.

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 piddles if the flag is set for any of the input piddles.

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 piddles if the flag is set for any of the input piddles.

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 piddles if the flag is set for any of the input piddles.

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 piddles if the flag is set for any of the input piddles.

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 piddles if the flag is set for any of the input piddles.

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 piddles if the flag is set for any of the input piddles.

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 piddles if the flag is set for any of the input piddles.

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 piddles if the flag is set for any of the input piddles.

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 piddles if the flag is set for any of the input piddles.

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 piddles if the flag is set for any of the input piddles.

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 piddles if the flag is set for any of the input piddles.

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 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.

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

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.