Complex(x) User Contributed Perl Documentation Complex(x) NAME PDL::Complex - handle complex numbers SYNOPSIS use PDL; use PDL::Complex; DESCRIPTION This module features a growing number of functions manipu- lating complex numbers. These are usually represented as a pair "[ real imag ]" or "[ angle phase ]". If not explic- itly 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 rep- resents "-1**0.5", i.e. the imaginary unit. it can be used to quickly and conviniently write complex con- stants like this: "4+3*i". o Use "r2C(real-values)" to convert from real to com- plex, 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(com- plex-valued-piddle)" to cast back. This requires a copy, though. o BEWARE: This module has not been extensively tested. Watch out and send me bugreports! EXAMPLE WALK-THROUGH The complex constant five is equal to "pdl(1,0)": perldl> p $x = r2C 5 [5 0] Now calculate the three roots of of five: perldl> p $r = Croots $x, 3 [ [ 1.7099759 0] [-0.85498797 1.4808826] [-0.85498797 -1.4808826] ] Check that these really are the roots of unity: perldl> p $r ** 3 [ [ 5 0] [ 5 -3.4450524e-15] [ 5 -9.8776239e-15] ] Duh! Could be better. Now try by multiplying $r three times with itself: perldl> p $r*$r*$r [ [ 5 0] [ 5 -2.8052647e-15] [ 5 -7.5369398e-15] ] Well... maybe "Cpow" (which is used by the "**" operator) isn't as bad as I thought. Now multiply by "i" and negate, which is just a very expensive way of swapping real and imaginary parts. perldl> p -($r*i) [ [ -0 1.7099759] [ 1.4808826 -0.85498797] [ -1.4808826 -0.85498797] ] Now plot the magnitude of (part of) the complex sine. First generate the coefficients: perldl> $sin = i * zeroes(s)->xlinvals(2,4) + zeroes(s)->xlinvals(0,7) Now plot the imaginary part, the real part and the magni- tude of the sine into the same diagram: perldl> line im sin $sin; hold perldl> line re sin $sin perldl> 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::Graph- ics::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 opera- tors. Dataflow to the complex parent works. Use "sever" on the result if you don't want this. real cplx-valued-pdl Cast a complex vlaued 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 i2C Signature: (r(); [o]c(m=2)) convert imaginary to complex, assuming a real part of zero Cr2p Signature: (r(m=2); float+ [o]p(m=2)) convert complex numbers in rectangular form to polar (mod,arg) form Cp2r Signature: (r(m=2); [o]p(m=2)) convert complex numbers in polar (mod,arg) form to rectan- gular form Cmul Signature: (a(m=2); b(m=2); [o]c(m=2)) complex multiplication Cscale Signature: (a(m=2); b(); [o]c(m=2)) mixed complex/real multiplication Cdiv Signature: (a(m=2); b(m=2); [o]c(m=2)) complex division Ccmp Signature: (a(m=2); b(m=2); [o]c()) Complex comparison oeprator (spaceship). It orders by real first, then by imaginary. Cconj Signature: (a(m=2); [o]c(m=2)) complex conjugation Cabs Signature: (a(m=2); [o]c()) complex "abs()" (also known as modulus) Cabs2 Signature: (a(m=2); [o]c()) complex squared "abs()" (also known squared modulus) Carg Signature: (a(m=2); [o]c()) complex argument function ("angle") Csin Signature: (a(m=2); [o]c(m=2)) sin (a) = 1/(2*i) * (exp (a*i) - exp (-a*i)) Ccos Signature: (a(m=2); [o]c(m=2)) cos (a) = 1/2 * (exp (a*i) + exp (-a*i)) 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))) Clog Signature: (a(m=2); [o]c(m=2)) log (a) = log (cabs (a)) + i * carg (a) Cpow Signature: (a(m=2); b(m=2); [o]c(m=2)) complex "pow()" ("**"-operator) Csqrt Signature: (a(m=2); [o]c(m=2)) Casin Signature: (a(m=2); [o]c(m=2)) Cacos Signature: (a(m=2); [o]c(m=2)) Catan cplx [not inplace] Return the complex "atan()". Csinh Signature: (a(m=2); [o]c(m=2)) sinh (a) = (exp (a) - exp (-a)) / 2 Ccosh Signature: (a(m=2); [o]c(m=2)) cosh (a) = (exp (a) + exp (-a)) / 2 Ctanh Signature: (a(m=2); [o]c(m=2)) Casinh Signature: (a(m=2); [o]c(m=2)) Cacosh Signature: (a(m=2); [o]c(m=2)) Catanh Signature: (a(m=2); [o]c(m=2)) Cproj Signature: (a(m=2); [o]c(m=2)) compute the projection of a complex number to the riemann sphere Croots Signature: (a(m=2); [o]c(m=2,n); int n => n) Compute the "n" roots of "a". "n" must be a positive inte- ger. The result will always be a complex type! re cplx, im cplx Return the real or imaginary part of the complex number(r) given. These are slicing operators, so data flow works. The real and imaginary parts are returned as piddles (ref eq PDL). rCpolynomial Signature: (coeffs(s); x(c=2,m); [o]out(c=2,m)) evaluate the polynomial with (real) coefficients "coeffs" at the (complex) position(n) "x". "coeffs[0]" is the con- stant term. 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(l), PDL. perl v5.6.1 2002-04-08 Complex(x)