Complex(3pm)
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
- o "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". - o 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). - o 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. - o This module has received some testing by Vanuxem Gregory (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 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 of unity:
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> line im sin $sin; hold
pdl> line re sin $sin
pdl> 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. - 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
- 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
- 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
- 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))
- 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))
- 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)))
- 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)
- 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))
- 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))
- 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))
- 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
- 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
- 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))
- 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))
- 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))
- 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))
- 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
- 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.