Computing generalized inverses using LU factorization of matrix product. (English) Zbl 1158.65029
The paper is a first attempt to compute \(\{2,3\},\) \(\{2,4\},\) \(\{1,2,3\},\) \(\{1,2,4\}\)-inverses and the Moore-Penrose inverse of a given rational matrix \(A.\)
The firsts section is an introduction in nature.
The second section concerns the characterization of classes \(A\{2,3\},\) \(A\{2,4\},\) \(A\{1,2,3\}\) and \(A\{1,2,4\}\) in terms of matrix products, involving other two rational matrices with appropriate dimensions and corresponding rank. Using these representations, one introduces a method for computing \(\{i,j,k\}\)-inverses of prescribed rank \(s\) of a given rational matrix \(A\). When \(A\) is a constant matrix, one gets an algorithm for computing the Moore-Penrose inverse. The algorithm is applicable to class of rational matrices if this one is implemented in symbolic programming languages like MATHEMATICA, MAPLE etc. The paper presents the implementation in MATHEMATICA. Due to problems with the simplification in rational expressions the above algorithm is not convenient for the implementation in high level programming languages, such as: C++, DELPHI, VISUALBASIC etc. Thus the presented implementation in language DELPHI is applicable only for constant matrices.
Examples illustrating symbolic implementation in MATHEMATICA are presented in the third section. One considers the practical case of implementation, which computes the Moore-Penrose inverse of a constant matrix, in comparison with several known methods such as: Grevile’s partitioning method, LeVerrier-Faddeev algorithm.
The fourth section contains the main conclusions.
The firsts section is an introduction in nature.
The second section concerns the characterization of classes \(A\{2,3\},\) \(A\{2,4\},\) \(A\{1,2,3\}\) and \(A\{1,2,4\}\) in terms of matrix products, involving other two rational matrices with appropriate dimensions and corresponding rank. Using these representations, one introduces a method for computing \(\{i,j,k\}\)-inverses of prescribed rank \(s\) of a given rational matrix \(A\). When \(A\) is a constant matrix, one gets an algorithm for computing the Moore-Penrose inverse. The algorithm is applicable to class of rational matrices if this one is implemented in symbolic programming languages like MATHEMATICA, MAPLE etc. The paper presents the implementation in MATHEMATICA. Due to problems with the simplification in rational expressions the above algorithm is not convenient for the implementation in high level programming languages, such as: C++, DELPHI, VISUALBASIC etc. Thus the presented implementation in language DELPHI is applicable only for constant matrices.
Examples illustrating symbolic implementation in MATHEMATICA are presented in the third section. One considers the practical case of implementation, which computes the Moore-Penrose inverse of a constant matrix, in comparison with several known methods such as: Grevile’s partitioning method, LeVerrier-Faddeev algorithm.
The fourth section contains the main conclusions.
Reviewer: R. Militaru (Craiova)
MSC:
| 65F20 | Numerical solutions to overdetermined systems, pseudoinverses |
| 15A09 | Theory of matrix inversion and generalized inverses |
| 68W30 | Symbolic computation and algebraic computation |
| 65F15 | Numerical computation of eigenvalues and eigenvectors of matrices |
Keywords:
Cholesky factorizations; generalized inverses; Moore-Penrose inverse; MATHEMATICA; DELPHI; numerical examples; comparison of methods; matrix products; rational matrices; algorithm; MAPLE; VISUALBASIC; Grevile’s partitioning method; LeVerrier-Faddeev algorithmReferences:
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