Complex(3)

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 intot he 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)":

   perldl> p $x = r2C 5
   [5 0]

Now calculate the three roots of of five:

   perldl> p $r = Croots $x, 3

   [
    [  1.7099759           0]
    [-0.85498797   1.4808826]
    [-0.85498797  -1.4808826]
   ]

Check that these really are the roots of unity:

   perldl> p $r ** 3

   [
    [             5              0]
    [             5 -3.4450524e-15]
    [             5 -9.8776239e-15]
   ]

Duh! Could be better. Now try by multiplying $r three times with itself:

   perldl> p $r*$r*$r

   [
    [             5              0]
    [             5 -2.8052647e-15]
    [             5 -7.5369398e-15]
   ]

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.

   perldl> p -($r*i)

   [
    [         -0   1.7099759]
    [  1.4808826 -0.85498797]
    [ -1.4808826 -0.85498797]
   ]

Now plot the magnitude of (part of) the complex sine. First generate the coefficients:

   perldl> $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:

   perldl> line im sin $sin; hold
   perldl> line re sin $sin
   perldl> line abs sin $sin

Sorry, but I didn't yet try to reproduce the diagram in this text. Just run the commands yourself, making sure that you have loaded "PDL::Complex" (and "PDL::Graphics::PGPLOT").

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

  Cp2r
  Signature: (r(m=2); [o]p(m=2))

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

  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

  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))

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

  cos (a) = 1/2 * (exp (a*i) + exp (-a*i))

  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)))

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

log (a) = log (cabs (a)) + i * carg (a)

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

complex "pow()" ("**"-operator)

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

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

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

  Catan cplx [not inplace]
Return the complex "atan()".

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

  sinh (a) = (exp (a) - exp (-a)) / 2

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

  cosh (a) = (exp (a) + exp (-a)) / 2

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

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

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

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

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

compute the projection of a complex number to the riemann sphere

  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.