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 conveniently 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!
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
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
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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))). 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.
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.
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.
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.
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.
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.
Return the complex "atan()".
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.