PDL::Complex - handle complex numbers
use PDL; use PDL::Complex;
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.
[ real imag ]
[ angle phase ]
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.
$a/$b*$c <=> Cmul (Cdiv $a, $b), $c)
PDL::Complex
The latter means that sin($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)
sin($a) + $b/$c
max
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.
i
-1**0.5
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).
r2C(real-values)
$r = Cpow $cplx, r2C 2
Croots 1, 5
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.
cplx(real-valued-piddle)
real(complex-valued-piddle)
This module has received some testing by Vanuxem Grégory (g.vanuxem at wanadoo dot fr). Please report any other errors you come across!
The complex constant five is equal to pdl(1,0):
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:
$r
pdl> p $r*$r*$r [5 +0i 5 -4.72647e-15i 5 -7.53694e-15i]
Well... maybe Cpow (which is used by the ** operator) isn't as bad as I thought. Now multiply by i and negate, which is just a very expensive way of swapping real and imaginary parts.
Cpow
**
pdl> p -($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
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.
sever
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.
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.
Signature: (r(); [o]c(m=2))
convert real to complex, assuming an imaginary part of zero
convert imaginary to complex, assuming a real part of zero
Signature: (r(m=2); float+ [o]p(m=2))
convert complex numbers in rectangular form to polar (mod,arg) form. Works inplace
Signature: (r(m=2); [o]p(m=2))
convert complex numbers in polar (mod,arg) form to rectangular form. Works inplace
Signature: (a(m=2); b(m=2); [o]c(m=2))
complex multiplication
Signature: (a(m=2,n); [o]c(m=2))
Project via product to N-1 dimension
Signature: (a(m=2); b(); [o]c(m=2))
mixed complex/real multiplication
complex division
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.
Signature: (a(m=2); [o]c(m=2))
complex conjugation. Works inplace
Signature: (a(m=2); [o]c())
complex abs() (also known as modulus)
abs()
complex squared abs() (also known squared modulus)
complex argument function ("angle")
sin (a) = 1/(2*i) * (exp (a*i) - exp (-a*i)). Works inplace
cos (a) = 1/2 * (exp (a*i) + exp (-a*i)). Works inplace
tan (a) = -i * (exp (a*i) - exp (-a*i)) / (exp (a*i) + exp (-a*i))
exp (a) = exp (real (a)) * (cos (imag (a)) + i * sin (imag (a))). Works inplace
log (a) = log (cabs (a)) + i * carg (a). Works inplace
complex pow() (**-operator)
pow()
Works inplace
Return the complex atan().
atan()
sinh (a) = (exp (a) - exp (-a)) / 2. Works inplace
cosh (a) = (exp (a) + exp (-a)) / 2. Works inplace
compute the projection of a complex number to the riemann sphere. Works inplace
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!
n
a
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).
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.
coeffs
x
coeffs[0]
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.
perl(1), PDL.
To install PDL, copy and paste the appropriate command in to your terminal.
cpanm
cpanm PDL
CPAN shell
perl -MCPAN -e shell install PDL
For more information on module installation, please visit the detailed CPAN module installation guide.