Complex(3pm)

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 threading).

•   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.

•   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 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)

  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.

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.