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ABS projection algorithms: Mathematical techniques for linear and nonlinear equations. (English) Zbl 0691.65022
Chichester etc.: Ellis Horwood Limited; New York etc.: Halsted Press. 220 p. £39.95 (1989).
The ABS (Abaffy-Broyden-Spedicato) algorithm computes a solution \(x^+\) of the system \(Ax=b\) of m linear equations with n unknowns (m\(\leq n)\) as the \((m+1)\)-th iterate of a sequence of approximations \(x_ i\) to \(x^+\) with property that the approximation \(x_{i+1}\) obtained at the i-th iteration is a solution of the first i equations. The heuristics of the algorithm lies in the modification of the quasi-Newton method together with appropriate choice of its parameters. The step length \(\alpha_ i\) of the algorithm \(x_{i+1}=x_ i-\alpha_ ip_ i\) is chosen as \(\alpha_ i=(a^ T_ ix_ i-b_ i)/a^ T_ ip_ i,\) where \(a^ T_ i\in R^ n\) is the i-th row of the matrix A, \(p_ i=H^ T_ iz_ i,\) \(H_{i+1}=H_ i-H_ ia_ iw^ T_ iH_ i\) with \(w_ i\) arbitrary subject to \(w^ T_ iH_ ia_ i=1\) and \(z_ i\) taken arbitrary but nonorthogonal to \(H_ ia_ i.\)
In Chapter 3 the class of algorithms is reformulated in a slightly modified form, which is needed for dealing with the rank-deficient case, and a number of fundamental properties of the iterates is given. In Chapters 5, 6 modifications of Huang and Gauss-Cholesky algorithms are presented. Then the basic ABS algorithm is applied to the scaled system \(V^ TAx=V^ Tb\). Numerical experiments of several versions of the Huang algorithm on ill-conditioned linear systems and linear least squares are presented. The last Chapter formulates the scaled block ABS algorithm for nonlinear systems.
Reviewer: R.Lepp

65F20 Numerical solutions to overdetermined systems, pseudoinverses
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
65H10 Numerical computation of solutions to systems of equations
65F10 Iterative numerical methods for linear systems
90C30 Nonlinear programming
65K05 Numerical mathematical programming methods