Complex(3pm)

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

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 oeprator (spaceship). It 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 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

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 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

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 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.

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.