Hello Santi Ponte, I am not any kind of expert on the QL iteration. All I can suggest is to check the references listed in the source file src/GaussQuadrature.jl at lines 65-71 and 303-306. Essentially, I just rewrote the original Fortran version gaussq.f first into Fortran 90 and then into Julia.
Regards, Bill. On Tuesday, March 25, 2014 4:45:50 AM UTC+11, Santi Ponte wrote: > > Hallo Bill; could you please explain or post some reference about that > specialised version you mention of the QL iteration to get directly the > first components of the normalised eigenvectors. As I am working with > complex, non-hermitian, symmetric eigenvalue problems it would be of great > help. Thanks! > > El miércoles, 16 de octubre de 2013 03:16:33 UTC+2, Bill McLean escribió: >> >> Steven, thanks for pointing out Base.gauss. My package relies, via eig, >> on the Lapack eigensystem routines for a symmetric tridiagonal matrix, but >> it would be possible to write such an eigensolver in Julia and so support >> higher precision. If I find the time I will try to do this. In fact, to >> generate the Gauss rules you need only the eigenvalues and the first >> component of each normalized eigenvector, and there is a specialised >> version of the QL iteration that does this without having to compute the >> other components of the eigenvectors. >> >> On Tuesday, October 15, 2013 6:38:54 AM UTC+11, Steven G. Johnson wrote: >>> >>> Note that this functionality (for constant weight functions) is already >>> in Base, e.g. >>> >>> x, w = Base.gauss(Float64, 17) >>> >>> gives a 17-point Gauss rule on [-1,1]. There is also Base.kronrod for >>> Gauss-Kronrod rules. (Currently, these are not documented; that >>> functionality use used internally by the quadgk function.) >>> >>> It is nice to have Gauss quadrature rules for different weight >>> functions, though. You might want to look at the Base implementation (in >>> base/quadgk.jl), however, and possibly exploit some of its subroutines, >>> since the Base implementation supports computation of points and weights in >>> arbitrary precision. >>> >>> On Saturday, October 12, 2013 9:05:53 PM UTC-4, Bill McLean wrote: >>>> >>>> I have written a Julia package to generate the points and weights of >>>> the classical Gauss quadrature rules. I did not succeed in following the >>>> instructions in the manual to add it to the list of available packages, >>>> but >>>> you can obtain the package from >>>> https://github.com/billmclean/GaussQuadrature.jl >>>> >>>>
