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
- "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".
- 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 GrĂ©gory (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 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
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. 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.
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 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. 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.
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)). 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.
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 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))). 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.
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 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))
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.
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 piddles if the flag is set for any of the input piddles.
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 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. 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.
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 piddles if the flag is set for any of the input piddles.
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 piddles if the flag is set for any of the input piddles.
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 piddles if the flag is set for any of the input piddles.
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 piddles if the flag is set for any of the input piddles.
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 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. 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.
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.