For some of our algorithms we present Fortran 77 LAPACK-style code and show the backward error of our updated factors is comparable to the error bounds of the QR factorization of Ã.
• F06x QF performs the downdating problem [ R1 αv T ] [ ˜R1 = Q 0 where R, ˜ R and ˜ Q are as above. The work is motivated by the problem of adaptive signal representation outside the orthogonal basis setting.
] , 536.4 Reichel and Gragg’s Algorithms Reichel and Gragg =-=-=- provide several Fortran 77 implementations of the algorithms discussed in  for updating the QR factorization, returning both ˜ Q and ˜R. The proposed techniques are shown to be relevant to the probl ..." A generalization of the Gram-Schmidt procedure is achieved by providing equations for updating and downdating oblique projectors.
The work is motivated by the problem of adaptive signal representation outside the orthogonal basis setting.
The proposed techniques are shown to be relevant to the problem of discriminating signals produced by different phenomena when the order of the signal model needs to be adjusted. when replacing vectors sequentially we need to allow for the recalculation of the corresponding projectors.
During the course of the incremental model construction, the algorithms are terminated using model selection principles such as the minimum descriptive length (MDL) and Akaike's information criterion (AIC).
Finally, experimental results on benchmark data are presented to demonstrate the competitiveness of the algorithms developed in this paper. Assume having a square matrix N 2 C with the characteristic polynomial p(z) = f(x; y) ig(x; y).
However, loss of orthogonality of the computed Krylov subspace basis can reduce the accuracy of the computed approximate eigenvalues. The Lanczos process is a well known technique for computing a few, say k, eigenvalues and associated eigenvectors of a large symmetric nn matrix.
The Lanczos process is a well known technique for computing a few, say k, eigenvalues and associated eigenvectors of a large symmetric nn matrix.
The method updates the initial Lanczos vector through an iterative scheme.