Milan B. Tasić

Symbolic computation of weighted Moore-Penrose inverse using partitioning method

Abstract We propose a method and algorithm for computing the weighted Moore–Penrose inverse of one-variable rational matrices. Continuing this idea, we develop an algorithm for computing the weighted Moore–Penrose inverse of one-variable polynomial matrix. These methods and algorithms are generalizations of the method for computing the weighted Moore–Penrose inverse for constant matrices, originated in Wang and Chen [G.R. Wang, Y.L. Chen, A recursive algorithm for computing the weighted Moore–Penrose inverse A MN † , J. Comput. Math. 4 (1986) 74–85], and the partitioning method for computing the Moore–Penrose inverse of rational and polynomial matrices introduced in Stanimirovic and Tasic [P.S. Stanimirovic, M.B. Tasic, Partitioning method for rational and polynomial matrices, Appl. Math. Comput. 155 (2004) 137–163]. Algorithms are implemented in the symbolic computational package MATHEMATICA .

Partitioning method for rational and polynomial matrices

We propose an extension of the Greviles partitioning method for computing the Moore-Penrose inverse, which is applicable to the set of rational matrices. Also, we develop an algorithm for computing the Moore-Penrose inverse of given one-variable polynomial matrix, which is based on the Greviles method. Major problems arising in the implementation of this method are repetitive recomputations of the same values and simplification of rational and polynomial expressions which contain unknown variable. These algorithms are implemented in the symbolic computational package MATHEMATICA.

Computing generalized inverses using LU factorization of matrix product

An algorithm for computing {2, 3}, {2, 4}, {1, 2, 3}, {1, 2, 4} -inverses and the Moore–Penrose inverse of a given rational matrix A is established. Classes A{2, 3} s and A{2, 4} s are characterized in terms of matrix products (R*A)† R* and T*(AT*)†, where R and T are rational matrices with appropriate dimensions and corresponding rank. The proposed algorithm is based on these general representations and the Cholesky factorization of symmetric positive matrices. The algorithm is implemented in programming languages MATHEMATICA and DELPHI, and illustrated via examples. Numerical results of the algorithm, corres-ponding to the Moore–Penrose inverse, are compared with corresponding results obtained by several known methods for computing the Moore–Penrose inverse.

Effective partitioning method for computing weighted Moore-Penrose inverse

We introduce a method and an algorithm for computing the weighted Moore-Penrose inverse of multiple-variable polynomial matrix and the related algorithm which is appropriated for sparse polynomial matrices. These methods and algorithms are generalizations of algorithms developed in [M.B. Tasic, P.S. Stanimirovic, M.D. Petkovic, Symbolic computation of weighted Moore-Penrose inverse using partitioning method, Appl. Math. Comput. 189 (2007) 615-640] to multiple-variable rational and polynomial matrices and improvements of these algorithms on sparse matrices. Also, these methods are generalizations of the partitioning method for computing the Moore-Penrose inverse of rational and polynomial matrices introduced in [P.S. Stanimirovic, M.B. Tasic, Partitioning method for rational and polynomial matrices, Appl. Math. Comput. 155 (2004) 137-163; M.D. Petkovic, P.S. Stanimirovic, Symbolic computation of the Moore-Penrose inverse using partitioning method, Internat. J. Comput. Math. 82 (2005) 355-367] to the case of weighted Moore-Penrose inverse. Algorithms are implemented in the symbolic computational package MATHEMATICA.

Symbolic and recursive computation of different types of generalized inverses

Abstract We propose a method and algorithm for recursive computation of different classes of generalized inverses of a given one-variable rational matrix and corresponding algorithm for polynomial matrix. These methods and algorithms are generalizations of the method for computing the generalized inverses for constant matrices, originated in [F.E. Udwadia, R.E. Kalaba, A unified approach for the recursive determination of generalized inverses, Comp. Math. Appl., 37 (1999), 125–130], and the partitioning method for computing the generalized inverses of rational and polynomial matrices introduced in [P.S. Stanimirovic, M.B. Tasic, Partitioning method for rational and polynomial matrices, Appl. Math. Comput., 155 (2004) 137–163; M.B. Tasic, P.S. Stanimirovic, M.D. Petkovic, Symbolic computation of weighted Moore–Penrose inverse using partitioning method, Appl. Math. Comput 189 (2007) 615–640]. Algorithms are implemented in the symbolic computational package MATHEMATICA.

Computing determinantal representation of generalized inverses

We investigate implementation of the determinantal representation of generalized inverses for complex and rational matrices in the symbolic package MATHEMATICA. We also introduce an implementation which is applicable to sparse matrices.

Computation of generalized inverses by using the LDL∗ decomposition

An efficient algorithm, based on the LDL∗ factorization, for computing {1,2,3} and {1,2,4} inverses and the Moore–Penrose inverse of a given rational matrix A, is developed. We consider matrix products A∗A and AA∗ and corresponding LDL∗ factorizations in order to compute the generalized inverse of A. By considering the matrix products (R∗A)†R∗ and T∗(AT∗)†, where R and T are arbitrary rational matrices with appropriate dimensions and ranks, we characterize classes A{1,2,3} and A{1,2,4}. Some evaluation times for our algorithm are compared with corresponding times for several known algorithms for computing the Moore–Penrose inverse.

About the generalized LM-inverse and the weighted Moore-Penrose inverse

The recursive method for computing the generalized LM-inverse of a constant rectangular matrix augmented by a column vector is proposed in Udwadia and Phohomsiri (2007) [16,17]. The corresponding algorithm for the sequential determination of the generalized LM-inverse is established in the present paper. We prove that the introduced algorithm for computing the generalized LM-inverse and the algorithm for the computation of the weighted Moore-Penrose inverse developed by Wang and Chen (1986) in [23] are equivalent algorithms. Both of the algorithms are implemented in the present paper using the package MATHEMATICA. Several rational test matrices and randomly generated constant matrices are tested and the CPU time is compared and discussed.

Differentiation of generalized inverses for rational and polynomial matrices

Our basic motivation is a direct method for computing the gradient of the pseudo-inverse of well-conditioned system with respect to a scalar, proposed in [13] by Layton. In the present paper we combine the Laytons method together with the representation of the Moore-Penrose inverse of one-variable polynomial matrix from [24] and developed an algorithm for computing the gradient of the Moore-Penrose inverse for one-variable polynomial matrix. Moreover, using the representation of various types of pseudo-inverses from [26], based on the Greviles partitioning method, we derive more general algorithms for computing {1}, {1,3} and {1,4} inverses of one-variable rational and polynomial matrices. Introduced algorithms are implemented in the programming language MATHEMATICA. Illustrative examples on analytical matrices are presented.

A problem in computation of pseudoinverses

In this paper we describe implementation of three methods for computing generalized inverses in the programming packages MATHEMATICA and DELPHI. Greviles method, the limit representation and the determinantal representation of generalized inverses are considered. A major problem arising in the implementation of these methods is very rapid increment of arithmetic operations. We describe a few different algorithms to avoid this common problem.