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**An improved Newton iteration for the generalized inverse of a matrix, with applications.**
*(English)*
Zbl 0733.65023

Under the assumption that one has a parallel computer which is able to perform matrix-multiplications very efficiently, the good old Newton- Schulz iteration may be used to compute the inverse of an \(n\times n\) well-conditioned matrix A. The authors accelerate this method with a Chebyshev and an adaptive acceleration using cubic polynomials rather than the quadratics used in Newton’s method. Furthermore they ensure the stability of the method even when A is singular, they compute the matrix A(\(\epsilon\)) and its pseudoinverse \(A^+(\epsilon)\), the rank of A(\(\epsilon\)) and the orthogonal projection P(\(\epsilon\)) onto the range of A(\(\epsilon\)), where A(\(\epsilon\)) is obtained from A by setting to zero any of its singular values that are less than \(\epsilon\). The results have applications to signal processing and to computing the singular value decomposition (SVD) of a matrix. For some cases as ‘number of processors available \(<O(n^ 2)'\) more modestly parallel, but less costly methods for the SVD as the Kogbetliantz method or the cyclic Golub/Reinsch-method should be more efficient. The algorithms are given in some detail as well as experimental results.

Reviewer: N.Köckler (Paderborn)

### MSC:

65F20 | Numerical solutions to overdetermined systems, pseudoinverses |

65F10 | Iterative numerical methods for linear systems |

15A09 | Theory of matrix inversion and generalized inverses |