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Modification methods for inverting matrices and solving systems of linear algebraic equations


Author: D. Goldfarb
Journal: Math. Comp. 26 (1972), 829-852
MSC: Primary 65F10; Secondary 65F30
DOI: https://doi.org/10.1090/S0025-5718-1972-0317527-4
MathSciNet review: 0317527
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Abstract: Modification methods for inverting matrices and solving systems of linear algebraic equations are developed from Broyden's rank-one modification formula. Several algorithms are presented that take as few, or nearly as few, arithmetic operations as Gaussian elimination and are well suited for the handling of data. The effect of rounding errors is discussed briefly.

Some of these algorithms are essentially equivalent to, or ``compact'' forms of, such known methods as Sherman and Morrison's modification method, Hestenes' biorthogonalization method, Gauss-Jordan elimination, Aitken's below-the-line elimination method, Purcell's vector method, and its equivalent, Pietrzykowski's projection method, and the bordering method. These methods are thus shown to be directly related to each other.

Iterative methods and methods for inverting symmetric matrices are also given, as are the results of some computational experiments.


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DOI: https://doi.org/10.1090/S0025-5718-1972-0317527-4
Article copyright: © Copyright 1972 American Mathematical Society

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