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Manpages ComplexSection: User Contributed Perl Documentation (3)Updated: 2002-04-08 Index Return to Main Contents NAMEPDL::Complex - handle complex numbersSYNOPSISuse PDL; use PDL::Complex; DESCRIPTIONThis 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
EXAMPLE WALK-THROUGHThe 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(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:
perldl> line im sin $sin; hold perldl> line re sin $sin perldl> line abs sin $sinSorry, 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"). FUNCTIONScplx real-valued-pdlCast 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.real cplx-valued-pdlCast 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.r2CSignature: (r(); [o]c(m=2))convert real to complex, assuming an imaginary part of zero i2CSignature: (r(); [o]c(m=2))convert imaginary to complex, assuming a real part of zero Cr2pSignature: (r(m=2); float+ [o]p(m=2))convert complex numbers in rectangular form to polar (mod,arg) form Cp2rSignature: (r(m=2); [o]p(m=2))convert complex numbers in polar (mod,arg) form to rectangular form CmulSignature: (a(m=2); b(m=2); [o]c(m=2))complex multiplication CscaleSignature: (a(m=2); b(); [o]c(m=2))mixed complex/real multiplication CdivSignature: (a(m=2); b(m=2); [o]c(m=2))complex division CcmpSignature: (a(m=2); b(m=2); [o]c())Complex comparison oeprator (spaceship). It orders by real first, then by imaginary. CconjSignature: (a(m=2); [o]c(m=2))complex conjugation CabsSignature: (a(m=2); [o]c())complex "abs()" (also known as modulus) Cabs2Signature: (a(m=2); [o]c())complex squared "abs()" (also known squared modulus) CargSignature: (a(m=2); [o]c())complex argument function (``angle'') CsinSignature: (a(m=2); [o]c(m=2)) sin (a) = 1/(2*i) * (exp (a*i) - exp (-a*i)) CcosSignature: (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)) CexpSignature: (a(m=2); [o]c(m=2))exp (a) = exp (real (a)) * (cos (imag (a)) + i * sin (imag (a))) ClogSignature: (a(m=2); [o]c(m=2))log (a) = log (cabs (a)) + i * carg (a) CpowSignature: (a(m=2); b(m=2); [o]c(m=2))complex "pow()" ("**"-operator) CsqrtSignature: (a(m=2); [o]c(m=2)) CasinSignature: (a(m=2); [o]c(m=2)) CacosSignature: (a(m=2); [o]c(m=2)) Catan cplx [not inplace]Return the complex "atan()".CsinhSignature: (a(m=2); [o]c(m=2)) sinh (a) = (exp (a) - exp (-a)) / 2 CcoshSignature: (a(m=2); [o]c(m=2)) cosh (a) = (exp (a) + exp (-a)) / 2 CtanhSignature: (a(m=2); [o]c(m=2)) CasinhSignature: (a(m=2); [o]c(m=2)) CacoshSignature: (a(m=2); [o]c(m=2)) CatanhSignature: (a(m=2); [o]c(m=2)) CprojSignature: (a(m=2); [o]c(m=2))compute the projection of a complex number to the riemann sphere CrootsSignature: (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! re cplx, im cplxReturn 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).rCpolynomialSignature: (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. AUTHORCopyright (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 ALSOperl(1), PDL.
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