Examining Possible Lu Decompositions

Linear Algebra and its Applications

Not all matrices enjoy the existence of an LU factorization. For those that do not, a number of "repairs" are possible. For nonsingular matrices we offer here a permutation-free repair in which the matrix is factoredLŨ , withL andŨ collectively as near as possible to lower and upper triangular (in a natural sense defined herein). Such factorization is not generally unique in any sense. In the process, we investigate further the structure of matrices without LU factorization and permutations that produce an LU factorization.

Various algorithm such as Doolittle, Crouts and Cholesky s have been proposed to factor a square matrix into a product of L and U matrices, that is, to find L and U such that A = LU; where L and U are lower and upper triangular matrices respectively. These methods are derived by writing the general forms of L and U and the unknown elements of L and U are then formed by equati ng the corresponding entries in A and LU in a systematic way. This approach for computing L and U for larger values of n will involve many sum of products and will result in n 2 equations for a matrix of order n. In this paper, we propose a straightforward method based on multipliers derived from modification of Gaussion elimination algorithm.

ACM Communications in Computer Algebra, 2010

The definition of the LU factoring of a matrix usually requires that the matrix be invertible. Current software systems have extended the definition to non-square and rank-deficient matrices, but each has chosen a different extension. Two new extensions, both of which could serve as useful standards, are proposed here: the first combines LU factoring with full-rank factoring, and the second extension combines full-rank factoring with fraction-free methods. Amongst other applications, the extension to full-rank, fraction-free factoring is the basis for a fractionfree computation of the Moore-Penrose inverse.

Computers & Mathematics with Applications, 1985

A new parallel algorithm for the LU factorization of a given dense matrix A is described. The case of banded matrices is also considered. This algorithm can be combined with Sameh and Brent's [SIAM J. Numer. Anal. 14, 1101-I 113. (1977] to obtain the solution of a linear system of algebraic equations. The arithmetic complexity for the dense case is in' ($bn in the banded case), using 3(n -1) processors and no square roots.

Linear Algebra and its Applications, 1988

Results are given concerning the LU factorization of H-matrices, and Gaussian elimination with column-diagonaldominant pivoting is shown to be applicable to H-matrices. This algorithm, which uses a symmetric permutation to exchange the most diagonally dominant column of the unreduced submatrix into the pivotal position, is shown to be numerically stable by deriving an upper bound on the growth factor associated with the backward error analysis for Gaussian elimination. 1. INTRODUCTION An nXn realmatrix A=(a,.)isan M-matrix if aij<Oforall i# j and if ReXa for all x~o(A), & e spectrum of A. There are numerous *This author acknowledges support from the Graduate Research Council of Youngstown State University. 'This author acknowledges support from the Natural Sciences and Engineering Research Council under aant A-8214.

International Journal of Electrical and Computer Engineering (IJECE), 2021

Many systems like the control systems and in communication systems, there is usually a demand for matrix inversion solution. This solution requires many operations, which makes it not possible or very hard to meet the needs for real-time constraints. Methods were exists to solve this kind of problems, one of these methods by using the LU decomposition of matrix which is a good alternative to matrix inversion. The LU matrices are two matrices, the L matrix, which is a lower triangular matrix, and the U matrix, which is an upper triangular matrix. In this paper, a design of dual-core processor is used as the hardware of the work and certain software was written to enable the two cores of the dual-core processor to work simultaneously in computing the value of the L matrix and U matrix. The result of this work are compared with other works that using single-core processor, and the results found that the time required in the cores of the dual-core is more less than using single-core. The designed dual-core processor is invoked using the VHDL language. Keywords: Dual core Field programmable gate array LU decomposition MIPS processor Single core VHDL This is an open access article under the CC BY-SA license. 1. INTRODUCTION Many different systems require solving of matrix inversion, these systems like control or communication systems. The required time for solving the matrix inversion increases on the size of the matrix is become bigger. Hence, an alternative method were required in order to work in real-time, one of these methods is the LU decomposition [1]. In LU decomposition method the coefficient matrix [A] of the given system of equation [ ][ ] = [ ] is written as a product of a Lower triangular matrix (L) and an upper triangular matrix (U), such that [ ] = [ ][ ] where the elements of = (= 0 <) and the elements of = (= 0 >) that is, the matrices [L] and [U] look like [2, 3]. Following are set of equations for a 4x4 matrix.

2006

The objective of this work is to compare the developed LU factorization update with results from MINOS. This technique, use a matrix columns static reordering and they are rearranged in accordance with the increasing number of nonzero entries and triangularized, leading to sparse basis factorization without computational effort to reorder the columns. Only the columns factorizations actually modified by the change of basis are carried through due to matrix sparse structure. Computational results in Matlab for problems from the Netlib show that this is a very promising idea, since there is no need to refactorize the matrix in the tested problems.

BIT Numerical Mathematics, 1999

A new backward error analysis of LU factorization is presented. It allows to obtain a sharper upper bound for the forward error and a new definition of the growth factor that we compare with the well known Wilkinson growth factor for some classes of matrices. Numerical experiments show that the new growth factor is often of order approximately log 2 n whereas Wilkinson's growth factor is of order n or √ n.

This report is a result of a study about LU decomposition exploring partial pivoting with Matlab. In this work we'll gonna use two provided Matlab codes based on BlAS2 and BLAS3 and implement partial pivoting in both. The first one is called BLAS2LU.m wich applies a row permutation to matrix wich has m rows and n columns where m ≥ n. The second code provided is BLAS3LU.m wich applies a block LU factorization and calls BLAS2LU to perform multiple block factorization. Both codes initially without pivoting. The main goal of this work is modifying original codes and implement partial pivoting on BLAS2LU.m and BLAS3LU.m. Our partial pivoting implementation will call BLAS2LUPP and BLAS3LUPP respectively. On experimental component of this work we will test both codes with matrices generated randomly with different dimensions. For both solutions produced, we'll compute the numerical error using the permutation matrix and a speedup analysis to draw some conclusions. This is an academic work developed on University of Minho

Indagationes Mathematicae, 2012

LU-factorization has been an original motivation for the development of Semi-Separability (semiseparable systems of equations are sometimes called "quasi-separable") theory, to reduce the computational complexity of matrix inversion. In the case of infinitely indexed matrices, it got sidetracked in favor of numerically more stable methods based on orthogonal transformations and structural "canonical forms", in particular external (coprime) and outer-inner factorizations. This paper shows how these factorizations lead to what the author believes are new, closed and canonical expressions for the L and U factors, related existence theorems and a factorization algorithm for the case where the original system is invertible and the factors are required to have inverses of the same type themselves. The resulting algorithm is independent of the existence of the solution and has, in addition, the very nice property that it only uses orthogonal transformations. It succeeds in computing the subsequent partial Schur complements (the pivots) in a stable numerical way. c