Provided by: pdl_2.007-5_amd64 bug

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.