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UBC Theses and Dissertations

Iterative algorithms for the inversion of matrices on digital computers Harris, Arthur Dorian Shaw

Abstract

After a general discussion of matrix norms and digital operations, matrix inversion procedures based on power series expansions are examined. The general class of methods of which the Diagonal and Gauss-Seidel iterations are illustrative is studied in some detail with bounds for the error matrix being obtained assuming, both exact and digital operations. The concept of the condition of a matrix and its bearing on iterative inversion procedures is looked into. A similar derivation and examination is then made for Hotelling's algorithm. Hotelling's iteration is further examined with a view to modifying it. Higher-order formulae are obtained and criticized and a new variation of the algorithm called the Optimized Hotelling method is derived and commented on. Some schemes for constructing initial approximations in connection with Hotelling's iteration (and similar methods) are discussed and a new modification of a procedure proposed by Berger and Saibel is constructed. The final part of the thesis discusses a class of finite-step iterative inversions based on an identity of Householder's. Three members of the class, namely Jordan-type Completion, the Symmetric method and the Quasi-optimum method are defined and briefly discussed. The Quasi-optimum method is then examined in further detail and some of its properties derived for the special case with the unit matrix for an initial approximation.

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