PDL::Complex - handle complex numbers

- NAME
- SYNOPSIS
- DESCRIPTION
- TIPS, TRICKS & CAVEATS
- EXAMPLE WALK-THROUGH
- FUNCTIONS

- cplx real-valued-pdl
- real cplx-valued-pdl
- r2C
- i2C
- Cr2p
- Cp2r
- Cmul
- Cscale
- Cdiv
- Ccmp
- Cconj
- Cabs
- Cabs2
- Carg
- Csin
- Ccos
- Ctan a [not inplace]
- Cexp
- Clog
- Cpow
- Csqrt
- Casin
- Cacos
- Catan cplx [not inplace]
- Csinh
- Ccosh
- Ctanh
- Casinh
- Cacosh
- Catanh
- Cproj
- Croots
- re cplx, im cplx
- rCpolynomial

- AUTHOR
- SEE ALSO

# 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.

- BEWARE: This module has not been extensively tested. Watch out and send me bugreports!

# 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.

## real cplx-valued-pdl

Cast a complex vlaued 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

## 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.

## 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), the PDL manpage.