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A Variable Metric Method for Approximating Generalized Inverses of Matrices
Author(s) -
Mohsen A.,
Stoer J.
Publication year - 2001
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/1521-4001(200107)81:7<435::aid-zamm435>3.0.co;2-5
Subject(s) - moore–penrose pseudoinverse , mathematics , positive definite matrix , rank (graph theory) , generalization , linear subspace , metric (unit) , matrix (chemical analysis) , sequence (biology) , combinatorics , variable (mathematics) , least squares function approximation , inverse , residual , algorithm , pure mathematics , mathematical analysis , eigenvalues and eigenvectors , statistics , physics , geometry , operations management , materials science , quantum mechanics , estimator , biology , economics , composite material , genetics
We present two variable metric update methods that have some attractive features: They deal with the problem of finding a least‐squares solution of a linear system Ax = b with an m × n ‐matrix of maximal rank, and can be viewed as a generalization of the DFP‐ and of the BFGS‐methods, respectively, to nonsymmetric matrices A : The methods generate a sequence of n × m ‐matrices H k so that the AH k are positive semidefinite and the H k approximate a right‐inverse of A if m ≤ n , and the Moore‐Penrose pseudoinverse of A , if m ≥ n . Thus for m = n the methods find approximations to A —1 that could be used as preconditioners in other methods for solving Ax = b when A is not positive definite. Both methods are related to cg ‐type algorithms minimizing the residual on Krylov‐subspaces.