By Arieh Iserles

Acta Numerica surveys every year crucial advancements in numerical arithmetic and medical computing. the topics and authors of the important survey articles are selected by way of a distinct foreign editorial board to be able to record an important and well timed advancements in a fashion available to the broader group of execs with an curiosity in clinical computing. Acta Numerica volumes have proved to be a invaluable instrument not just for researchers and pros wishing to boost their figuring out of numerical recommendations and algorithms and keep on with new advancements, but additionally as a sophisticated educating reduction at faculties and universities. the various unique articles were used because the best source for graduate classes. this actual quantity was once initially released in 2004.

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Math. 14, 403-420. G. H. Golub and C. F. Van Loan (1996), Matrix Computations, 3rd edn, Johns Hopkins University Press, Baltimore. G. H. Golub and J. H. Wilkinson (1966), 'Note on the iterative refinement of least squares solution', Numer. Math. 9, 139-148. G. H. Golub, V. Klema and G. W. Stewart (1976), Rank degeneracy and least squares problems, Technical Report STAN-CS-76-559, August 1976, Computer Science Department, Stanford University, CA. G. H. Golub, F. T. Luk and M. Pagano (1979), A sparse least squares problem in photogrammetry, in Proceedings of the Computer Science and Statistics 12th Annual Symposium on the Interface (J.

5 we outlined the LSQR Krylov subspace algorithm for the least squares problem. A different way to compute the same sequence of approximations xk is to use the conjugate gradient method (CG), developed in the early 1950s. This has become a basic tool for solving large sparse symmetric positive definite linear systems. If applied to the normal equations it can also be used to solve linear least squares problems. The conjugate gradient algorithm for the normal equations generates approximations xk in the Krylov subspace xk€lCk(ATA,so), so = ATb.

17), which corresponds to using LSQR or the PLS method. There is some evidence that for ill-posed problems the Krylov subspaces often have better approximation properties than the singular subspaces used in TSVD; see Hanke (2001). Another widely used regularization method is Tikhonov regularization (Tikhonov 1963). In this method an approximate solution is obtained by solving a least squares problem with a quadratic constraint min ||Ax - b||2 subject to ||Ls|| 2 < 7, Le R p x n . 15) X Here 7 > 0 is the regularization parameter, which is used to find a balance between the size of the residual ||Aa; — 6JI2 and size of the solution as measured by the norm (or seminorm) ||La;||2.

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