By Arieh Iserles
Acta Numerica surveys each year an important advancements in numerical arithmetic and clinical computing. the themes and authors of the important survey articles are selected by means of a individual overseas editorial board with a view to file crucial and well timed advancements in a fashion obtainable to the broader group of execs with an curiosity in medical computing. Acta Numerica volumes have proved to be a helpful instrument not just for researchers and pros wishing to advance their realizing of numerical options and algorithms and stick to new advancements, but in addition as a sophisticated instructing reduction at faculties and universities. some of the unique articles were used because the top source for graduate classes. this actual quantity used to be initially released in 2004.
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Additional info for Acta Numerica 2004: Volume 13 (Acta Numerica)
Except for some details this is essentially the LSQR algorithm by Paige and Saunders (19826). 2. 27 on Tue Nov 09 11:59:42 GMT 2010. 5. Solving the sequence of bidiagonal problems The bidiagonal least squares problem can easily be solved by reducing Bk to upper bidiagonal form. The QR decomposition of Bk is computed by premultiplication with a sequence of Givens rotations. 23) The residual ATrk to the normal equations will be zero for the exact solution x and this quantity can therefore be used as a stopping criterion.
The matrix Q can be implicitly represented in terms of the Householder vectors of the factorization of the subblocks. This sequential Householder algorithm, which is also described in Lawson and Hanson (1995, Chapter 27), requires (m + 3n/2)w(w + 1) multiplications or about twice the work of the less stable Cholesky approach. A banded upper triangular matrix can be reduced to bidiagonal form using an algorithm similar to the one used by Schwarz (1968) for reducing a symmetric banded matrix to tridiagonal form.
2. Regularization of discrete ill-posed problems Inverse problems are problems where we want to determine the structure of a physical system from its measured behaviour. Such problems are often ill-posed in the sense that their solution does not depend continuously on the data. Inverse problems arise in many application such as astronomy, computerized tomography, geophysics, signal processing, etc. 14) that share a number of properties. The singular values of A decay gradually and cluster at zero, resulting in a huge condition number; cf.