A finite algorithm for generalized inverses of polynomial and rational matrices. (English) Zbl 1028.65035
The author derives two dual algorithms for symbolic computation of outer inverses of a given one-variable rational matrix, including reflexive g-inverses, the Moore-Penrose inverse and the Drazin inverse. The algorithms are based on different modifications of the Leverrier-Faddeev algorithm. The implementation of the algorithm corresponding to rational matrices in the package MATHEMATICA is described.
Reviewer: Hang Tong Lau (St.Laurent/Quebec)
MSC:
| 65F20 | Numerical solutions to overdetermined systems, pseudoinverses |
| 15A09 | Theory of matrix inversion and generalized inverses |
| 15A54 | Matrices over function rings in one or more variables |
| 68W30 | Symbolic computation and algebraic computation |
Keywords:
polynomial matrix; Leverrier-Faddeev algorithm; generalized inverses; symbolic computation.; dual algorithms; outer inverses; rational matrix; reflexive \(g\)-inverses; Moore-Penrose inverse; Drazin inverse; package MATHEMATICASoftware:
MathematicaReferences:
| [1] | Decell, H. P., An application of the Cayley-Hamilton theorem to generalized matrix inversion, SIAM Rev., 7, 4, 526-528 (1965) · Zbl 0178.35504 |
| [2] | Grevile, T. N.E., The Souriau-Frame algorithm and the Drazin pseudoinverse, Linear Algebra Appl., 6, 205-208 (1973) · Zbl 0247.15004 |
| [3] | Hartwig, R. E., More on the Souriau-Frame algorithm and the Drazin inverse, SIAM J. Appl. Math., 31, 1, 42-46 (1976) · Zbl 0335.15008 |
| [4] | Ji, J., An alternative limit expression of Drazin inverse and its applications, Appl. Math. Comput., 61, 151-156 (1994) · Zbl 0805.65041 |
| [5] | Ji, J., A finite algorithm for the Drazin inverse of a polynomial matrix, Appl. Math. Comput., 130, 243-251 (2002) · Zbl 1026.65030 |
| [6] | Jones, J.; Karampetakis, N. P.; Pugh, A. C., The computation and application of the generalized inverse vai maple, J. Symbolic Comput., 25, 99-124 (1998) · Zbl 0894.68088 |
| [7] | Karampetakis, N. P., Computation of the generalized inverse of a polynomial matrix and applications, Linear Algebra Appl., 252, 35-60 (1997) · Zbl 0869.65028 |
| [8] | Karampetakis, N. P., Generalized inverses of two-variable polynomial matrices and applications, Circ. Syst. Signal Process., 16, 439-453 (1997) · Zbl 0882.93039 |
| [9] | Karampetakis, N. P.; Tzekis, P., On the computation of the generalized inverse of a polynomial matrix, Ima J. Math. Contr. Inform., 18, 83-97 (2001) · Zbl 0987.93023 |
| [10] | Rao, C. R., Generalized Inverses of Matrices and Applications (1971), Wiley: Wiley New York · Zbl 0236.15004 |
| [11] | P.S. Stanimirovic, N.P. Karampetakis, Symbolic implementation of Leverrier-Faddeev algorithm and applications, in: proceedings of the 8th IEEE Medit. Conference on Control and Automation, Patra, Greece, 2000; P.S. Stanimirovic, N.P. Karampetakis, Symbolic implementation of Leverrier-Faddeev algorithm and applications, in: proceedings of the 8th IEEE Medit. Conference on Control and Automation, Patra, Greece, 2000 |
| [12] | P.S. Stanimirovic, N.P. Karampetakis, On the computation of Drazin inverse of a rational matrix and applications, Technical Report, Department of Mathematics, Aristotle University of Thessaloniki, Thessaloniki 54006, Greece, 2000; P.S. Stanimirovic, N.P. Karampetakis, On the computation of Drazin inverse of a rational matrix and applications, Technical Report, Department of Mathematics, Aristotle University of Thessaloniki, Thessaloniki 54006, Greece, 2000 |
| [13] | P.S. Stanimirovic, M.B. Tasić, Drazin inverse of one-variable polynomial matrices, Filomat, Niš, in press; P.S. Stanimirovic, M.B. Tasić, Drazin inverse of one-variable polynomial matrices, Filomat, Niš, in press · Zbl 1224.65110 |
| [14] | Stanimirović, P. S., Limit representations of generalized inverses and related methods, Appl. Math. Comput., 103, 51-68 (1999) · Zbl 0927.65060 |
| [15] | Wang, G.; Lin, Y., A new extension of the Leverrier’s algorithm, Linear Algebra Appl., 180, 227-238 (1993) · Zbl 0782.65062 |
| [16] | Wolfram, S., Mathematica: a System for Doing Mathematics by Computer (1991), Addison-Wesley: Addison-Wesley Redwood City, CA · Zbl 0812.68063 |
| [17] | Wolfram, S., Mathematica Book, Version 3.0 (1996), Addison-Wesley: Addison-Wesley Redwood City, CA · Zbl 0878.65001 |
| [18] | Zielke, G., Report on test matrices for generalized inverses, Computing, 36, 105-162 (1986) · Zbl 0566.65026 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
