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PDL::Complex - handle complex numbers |
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.
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)
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.
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).
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.
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).
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.
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
Signature: (r(); [o]c(m=2))
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
Signature: (r(m=2); [o]p(m=2))
convert complex numbers in polar (mod,arg) form to rectangular form
Signature: (a(m=2); b(m=2); [o]c(m=2))
complex multiplication
Signature: (a(m=2); b(); [o]c(m=2))
mixed complex/real multiplication
Signature: (a(m=2); b(m=2); [o]c(m=2))
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
Signature: (a(m=2); [o]c())
complex abs() (also known as modulus)
Signature: (a(m=2); [o]c())
complex squared abs() (also known squared modulus)
Signature: (a(m=2); [o]c())
complex argument function (``angle'')
Signature: (a(m=2); [o]c(m=2))
sin (a) = 1/(2*i) * (exp (a*i) - exp (-a*i))
Signature: (a(m=2); [o]c(m=2))
cos (a) = 1/2 * (exp (a*i) + exp (-a*i))
tan (a) = -i * (exp (a*i) - exp (-a*i)) / (exp (a*i) + exp (-a*i))
Signature: (a(m=2); [o]c(m=2))
exp (a) = exp (real (a)) * (cos (imag (a)) + i * sin (imag (a)))
Signature: (a(m=2); [o]c(m=2))
log (a) = log (cabs (a)) + i * carg (a)
Signature: (a(m=2); b(m=2); [o]c(m=2))
complex pow() (**-operator)
Signature: (a(m=2); [o]c(m=2))
Signature: (a(m=2); [o]c(m=2))
Signature: (a(m=2); [o]c(m=2))
Return the complex atan().
Signature: (a(m=2); [o]c(m=2))
sinh (a) = (exp (a) - exp (-a)) / 2
Signature: (a(m=2); [o]c(m=2))
cosh (a) = (exp (a) + exp (-a)) / 2
Signature: (a(m=2); [o]c(m=2))
Signature: (a(m=2); [o]c(m=2))
Signature: (a(m=2); [o]c(m=2))
Signature: (a(m=2); [o]c(m=2))
Signature: (a(m=2); [o]c(m=2))
compute the projection of a complex number to the riemann sphere
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!
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.
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.
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PDL::Complex - handle complex numbers |