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
[ 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), $c)), you can also opt to cast your pdl's into the
PDL::Complex datatype, which works just like your normal piddles, but
with all the normal perl operators overloaded.
The latter means that sin($x) + $y/$c will be evaluated using the
normal rules of complex numbers, while other pdl functions (like max)
just treat the 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 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
The following operators are overloaded:
- +, += (addition)
-
- -, -= (subtraction)
-
-
- /, /= (division; /Cdiv)
-
-
- atan2 (4-quadrant arc tangent)
-
- <=> (nonsensical comparison operator; /Ccmp)
-
- sin (/Csin)
-
- cos (/Ccos)
-
- exp (/Cexp)
-
- abs (/Cabs)
-
- log (/Clog)
-
- sqrt (/Csqrt)
-
- <, <=, ==, !=, >=, > (just as nonsensical as /Ccmp)
-
- ++, -- (increment, decrement; they affect the real part of the complex number only)
-
- "" (stringification)
-
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.
cplx($real_valued_pdl)
Cast a real-valued piddle to the complex datatype without dataflow
and inplace.
Achieved by merely reblessing a piddle. The first dimension of the
piddle must be of size 2.
complex($real_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.
real($cplx_valued_pdl)
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 operator (spaceship).
Ccmp 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.
Complex tangent
tan (a) = -i * (exp (a*i) - exp (-a*i)) / (exp (a*i) + exp (-a*i))
Does not work inplace.
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().
Does not work inplace.
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.
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