 By Howard W. Eves

Booklet by means of Eves, Howard

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Additional info for Great Moments in Mathematics Before 1650

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In R. A reordering of the rows in A has no effect on the final R, but influences the sparsity of Q and the number of operations needed to perform the decomposition. Before computing R numerically, it is important to find a to find a column ordering that approximately minimizes the number of nonzero elements in R. ) The simplest ordering methods use a priori information, such as ordering the columns in order of increasing column count. Such orderings are usually inferior to ordering methods obtained from a symmetric ordering on the structure of the normal matrix, using minimum degree or nested dissection.

001 Cambridge Books Online © Cambridge University Press, 2010 34 A. BJORCK Sometimes only one-sided bounds apply. After a shift these can then be transformed into x > 0 and we have a least squares problems with nonnegativity constraints (NNLS): ||Aa;-6||2. , reconstruction problems in geodesy and tomography, contact problems for mechanical systems, control problems, etc. It can often be argued that a linear model is only realistic when the variables are constrained within meaningful intervals. To determine a unique solution for the BLS problem when rank(j4) < n, we may look for a solution to the problem min lla;^, xeM M = \x\ ^ min \\Ax-b\\2\.

Since many of these problems are neither sparse nor particularly large an implementation based on Householder bidiagonalization should be preferred. 8. 1. Constrained least squares problems In various applications the solution to a least squares problem is required to satisfy a subsystem of linear equations exactly. 27 on Tue Nov 09 11:59:42 GMT 2010. 001 Cambridge Books Online © Cambridge University Press, 2010 (8-1) THE CALCULATION OF LINEAR LEAST SQUARES PROBLEMS where A £ RmXn and B n>p.