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.

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

##### MSC:

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 |