2016 EfficientGeneralizedInverseforS

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Subject Headings: Pseudo-Inverse Algorithm; Bose-Nguyen Generalized Inverse Algorithm.


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Solving large scale system of Simultaneous Linear Equations (SLE) has been (and continue to be) a major challenging problem for many real-world engineering and science applications. Solving SLE with singular coefficient matrices arises from various engineering and sciences applications [ 1]- [ 6 ]. In this paper, efficient numerical procedures for finding the generalized (or pseudo) inverse of a general (square / rectangle, symmetrical /unsymmetrical, non-singular / singular) matrix and solving systems of Simultaneous Linear Equations (SLE) are formulated and explained. The developed procedures and its associated computer software (under MATLAB [ 7 ] computer environment) have been based on "€a special Cholesky factorization schemes"€ (for a singular matrix). Test matrices from different fields of applications have been chosen, tested and compared with other existing algorithms. The results of the numerical tests have indicated that the developed procedures are far more efficient than the existing algorithms.

1. Introduction

In scientific computing, most computational time is spent on solving system of Simultaneous Linear Equations (SLE) which can be represented in matrix notations as

[math] Ax=b[/math] (1.1)

where [math]A \in R^{n \times n}[/math] is a singular/non-singular matrix, and [math]b[/math] is a given vector in [math]R^n[/math]. For practical engineering/ science applications, matrix [math]A[/math] can be either sparse (for most cases), or dense (for some cases). For a non-singular coefficient matrix [math]A[/math], direct methods (Cholesky factorization, LDLTalgorithm, LU decomposition, etc) or iterative methods (Conjugate Gradient algorithm, Bi-Conjugate Stabilization, GMRES, etc.) are used to solve Equation (1.1). If the coefficient matrix is singular or rectangular, the above mentioned direct and iterative methods cannot be used to solve Equation (1.1) and thus generalized inverse is needed to solve the unknown solution vector [math]x[/math] in Equation (1.1). The generalized (or pseudo) inverse of a matrix is an extension of the ordinary/regular square (non-singular) matrix inverse, which can be applied to any matrix (such as singular, rectangular etc.). The generalized inverse has numerous important engineering and sciences applications. Over the past decades, generalized inverses of matrices and its applications have been investigated by many researchers 1-6. Generalized inverse is also known as “Moore-Penrose inverse” or “g-inverse” or “pseudo-inverse” etc.

In this paper we introduce an efficient (in terms of computational time and computer memory requirement) generalized inverse formulation to solve SLE with full or deficient rank of the coefficient matrix. The coefficient matrix can be singular/non-singular, symmetric/unsymmetric, or square/rectangular. Due to popular MATLAB software, which is widely accepted by researchers and educators worldwide, the developed code from this work is written in MATLAB language.

The rest of this paper is organized as follows. In Section 2, we discuss background of generalized inverse. In Section 3, we give a description of the algorithm. This section also describes the efficient generalized inverse formulation (which uses modified Cholesky factorization). In Section 4, we present comparison of numerical performances of the proposed algorithm with other existing algorithms. Extensive set of coefficient matrices (including rectangular, square, symmetrical, non-symmetrical, singular, non-singular matrices) obtained from well-established/popular websites [8] [9] were tested and the numerical performance in terms of timings, error norm were compared with other algorithms. Finally, conclusions are drawn in Section 5.


  1. Nguyen, D.T. (2006) Finite Element Methods: Parallel-Sparse Statics and Eigen-Solutions. Springer Publisher.
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  3. Heath, M.T. (1997) Scientific Computing: An Introductory Survey. McGraw Hill Publisher.
  4. Hou, G. and Wang, Y. (2004) A Substructuring Technique for Design Modifications of Interface Conditions. Structural Dynamics & Materials Conference, Palm Springs, California, 19-22 April 2004. http://dx.doi.org/10.2514/6.2004-2010
  5. Farhat, C. and Roux, F.X. (1994) Implicit Parallel Processing in Structural Mechanics. Computational Mechanics Advances, 2, Elsevier Publisher.
  6. Pierre, C. (2005) Fast Computation of Moore-Penrose Inverse Matrices. Neural Information Processing—Letters and Reviews, 8.
  7. MATLAB, MATLAB—The Language of Technical Computing.
  8. Davis, T. University of South Florida Matrix Collection.
  9. SJSU, SJSU—Singular Matrix Database.;

 AuthorvolumeDate ValuetitletypejournaltitleUrldoinoteyear
2016 EfficientGeneralizedInverseforSS. Kadiam Bose
D. T. Nguyen
Efficient Generalized Inverse for Solving Simultaneous Linear Equations10.4236/jamp.2016.410032016
AuthorS. Kadiam Bose + and D. T. Nguyen +
doi10.4236/jamp.2016.41003 +
titleEfficient Generalized Inverse for Solving Simultaneous Linear Equations +
year2016 +