Provided by: pdl_2.080-3_amd64 
NAME
PDL::Complex - handle complex numbers (DEPRECATED - use native complex)
SYNOPSIS
use PDL;
use PDL::Complex;
DESCRIPTION
This module is deprecated in favour of using "native complex" data types, e.g.:
use PDL;
my $complex_pdl = cdouble('[1+3i]');
print $complex_pdl * pdl('i'); # [-3+i]
This module features a growing number of functions manipulating complex numbers. These are
usually represented as a pair "[ real imag ]" or "[ magnitude phase ]". If not explicitly
mentioned, the functions can work inplace (not yet implemented!!!) and require rectangular
form.
While there is a procedural interface available ("$x/$y*$c <=> Cmul (Cdiv ($x, $y), $c)"),
you can also opt to cast your pdl's into the "PDL::Complex" datatype, which works just
like your normal ndarrays, but with all the normal perl operators overloaded.
The latter means that "sin($x) + $y/$c" will be evaluated using the normal rules of
complex numbers, while other pdl functions (like "max") just treat the ndarray as a real-
valued ndarray 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)
Native complex support
2.027 added changes in complex number handling, with support for C99 complex floating-
point types, and most functions and modules in the core distribution support these as
well.
PDL can now handle complex numbers natively as scalars. This has the advantage that real
and complex valued ndarrays have the same dimensions. Consider this when writing code in
the future.
See "re" in PDL::Ops, "im" in PDL::Ops, "abs" in PDL::Ops, "carg" in PDL::Ops, "conj" in
PDL::Ops for more.
TIPS, TRICKS & CAVEATS
• "i" is a function (not, as of 2.047, 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".
NB This will override the PDL::Core function of the same name, which returns a native
complex value.
• 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
broadcasting).
• use "cplx(real-valued-ndarray)" to cast from normal ndarrays into the complex
datatype. Use "real(complex-valued-ndarray)" to cast back. This requires a copy,
though.
EXAMPLE WALK-THROUGH
The complex constant five is equal to "pdl(1,0)":
pdl> p $x = r2C 5
5 +0i
Now calculate the three cubic roots 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*$r
[5 +0i 5 -4.72647e-15i 5 -7.53694e-15i]
Well... maybe "Cpow" (which is used by the "**" operator) isn't as bad as I thought. Now
multiply by "i" and negate, then take the complex conjugate, which is just a very
expensive way of swapping real and imaginary parts.
pdl> p Cconj(-($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
OPERATORS
The following operators are overloaded:
+, += (addition)
-, -= (subtraction)
*, *= (multiplication; "Cmul")
/, /= (division; "Cdiv")
**, **= (exponentiation; "Cpow")
atan2 (4-quadrant arc tangent)
sin ("Csin")
cos ("Ccos")
exp ("Cexp")
abs ("Cabs")
log ("Clog")
sqrt ("Csqrt")
++, -- (increment, decrement; they affect the real part of the complex number only)
"" (stringification)
Comparing complex numbers other than for equality is a fatal error.
FUNCTIONS
from_native
Class method to convert a native-complex ndarray to a PDL::Complex object.
PDL::Complex->from_native($native_complex_ndarray)
as_native
Object method to convert a PDL::Complex object to a native-complex ndarray.
$pdl_complex_obj->as_native
cplx
Cast a real-valued ndarray to the complex datatype.
The first dimension of the ndarray must be of size 2. After this the usual (complex)
arithmetic operators are applied to this pdl, rather than the normal elementwise pdl
operators. Dataflow to the complex parent works. Use "sever" on the result if you don't
want this.
cplx($real_valued_pdl)
complex
Cast a real-valued ndarray to the complex datatype without dataflow and inplace.
Achieved by merely reblessing an ndarray. The first dimension of the ndarray must be of
size 2.
complex($real_valued_pdl)
real
Cast a complex valued pdl back to the "normal" pdl datatype.
Afterwards the normal elementwise pdl operators are used in operations. Dataflow to the
real parent works. Use "sever" on the result if you don't want this.
real($cplx_valued_pdl)
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 ndarrays if
the flag is set for any of the input ndarrays.
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 ndarrays if
the flag is set for any of the input ndarrays.
Cr2p
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 ndarrays
if the flag is set for any of the input ndarrays.
Cp2r
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 ndarrays
if the flag is set for any of the input ndarrays.
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 ndarrays
if the flag is set for any of the input ndarrays.
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
ndarrays if the flag is set for any of the input ndarrays.
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 ndarrays
if the flag is set for any of the input ndarrays.
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 ndarrays
if the flag is set for any of the input ndarrays.
Ceq
Signature: (a(m=2); b(m=2); [o]c())
Complex equality operator.
Ceq does not process bad values. It will set the bad-value flag of all output ndarrays if
the flag is set for any of the input ndarrays.
Cconj
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 ndarrays
if the flag is set for any of the input ndarrays.
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 ndarrays
if the flag is set for any of the input ndarrays.
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 ndarrays
if the flag is set for any of the input ndarrays.
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 ndarrays
if the flag is set for any of the input ndarrays.
Csin
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 ndarrays
if the flag is set for any of the input ndarrays.
Ccos
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 ndarrays
if the flag is set for any of the input ndarrays.
Ctan
Complex tangent
tan (a) = -i * (exp (a*i) - exp (-a*i)) / (exp (a*i) + exp (-a*i))
Does not work inplace.
Cexp
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 ndarrays
if the flag is set for any of the input ndarrays.
Clog
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 ndarrays
if the flag is set for any of the input ndarrays.
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 ndarrays
if the flag is set for any of the input ndarrays.
Csqrt
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 ndarrays
if the flag is set for any of the input ndarrays.
Casin
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 ndarrays
if the flag is set for any of the input ndarrays.
Cacos
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 ndarrays
if the flag is set for any of the input ndarrays.
Catan
Return the complex "atan()".
Does not work inplace.
Csinh
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 ndarrays
if the flag is set for any of the input ndarrays.
Ccosh
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 ndarrays
if the flag is set for any of the input ndarrays.
Ctanh
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 ndarrays
if the flag is set for any of the input ndarrays.
Casinh
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 ndarrays
if the flag is set for any of the input ndarrays.
Cacosh
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 ndarrays
if the flag is set for any of the input ndarrays.
Catanh
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 ndarrays
if the flag is set for any of the input ndarrays.
Cproj
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 ndarrays
if the flag is set for any of the input ndarrays.
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 ndarrays
if the flag is set for any of the input ndarrays.
re, im
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 ndarrays (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
ndarrays if the flag is set for any of the input ndarrays.
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.