BLU Factorization for Block Tridiagonal Matrices and Its Error Analysis

A block representation of the BLU factorization for block tridiagonal matrices is presented. Some properties on the factors obtained in the course of the factorization are studied. Simpler expressions for errors incurred at the process of the factorization for block tridiagonal matrices are considered.


Introduction
Tridiagonal matrices are connected with different areas of science and engineering, including telecommunication system analysis [1] and finite difference methods for solving partial differential equations [2][3][4].
The backward error analysis is one of the most powerful tools for studying the accuracy and stability of numerical algorithms.A backward analysis for the LU factorization and for the solution of the associated triangular linear systems is presented by Amodio and Mazzia [5].BLU factorization appears to have first been proposed for block tridiagonal matrices, which frequently arise in the discretization of partial differential equations.References relevant to this application include Isaacson and Keller [6], Bank and Rose [7], Mattheij [8], Concus, Golub and Meurant [9], Varah [10], Bank and Rose [11], and Yalamov and Plavlov [12].For a block dense matrix, Demmel and Higham [13] presented error analysis of BLU factorization, and Demmel, Higham and Shreiber [14] also extended earlier analysis.
This paper is organized as follows.We begin, in Section 2 by showing the representation of BLU factorization for block tridiagonal matrices.In Section 3 some properties on the factors associated with the factorization are presented.Finally, by the use of BLAS3 based on fast matrix multiplication techniques, an error analysis of the factorization is given in Section 4.
Throughout, we use the "standard model" of floatingpoint arithmetic in which the evaluation of an expression in floating-point arithmetic is denoted by (see Higham [15] for details).Here is the unit round-ing-off associated with the particular machine being used.Unless otherwise stated, in this section an unsubscripted norm denotes

Representation of BLU Factorization for Block Tridiagonal Matrices
Consider a nonsingular block tridiagonal matrix and with 1 and are arbitrary.We present the following factorization of A .The first step is represented as follows: where i I is the identity matrix of order , and The second step of the factorization is applied to in order to obtain a matrix with a sub-block , then Applying the method recursively, it follows that 1 .
and the factorization ends, we obtain where and From the proc- ess of the representation obtained, we get the results as follows: 1) Taking the second step for example, if is nonsingular then we can factor 1 and 1 in a similar manner, and this process can be continued recursively to obtain the complete block LU factorization; S 2) There exists obvious difference between partitioned LU factorization (see [15] for further details), GE and block LU factorization in this paper.

Some Properties on Factors of BLU Factorization
The usual property on Schur complements under BLU factorization for block diagonal dominance by rows is similar to that of point diagonal dominance, i.e., Schur complements under BLU factorization for block diagonal dominance by rows inherit the key property on original matrices.For the factors , and U , we have the following theorem.Proof.By the process of the factorization, it follows that Since i is block diagonally dominant, by the definition of block diagonal dominance, i preserves the same property as the matrix i .The proof for i and is as follows.The definition of block diagonal dominance, we have Thus the matrices 1 and 2 U are also block diagonally dominant.The result follows by induction, that is, i also preserves the same property as the matrix .For the matrix U, we have By the above proof, it follows that the matrix is also block diagonally dominant.□

U
The problem is whether the matrices i for all can inherit the same property as the matrix i .The result is negative.Take the following block tridiagonal matrix and , where 0.005

 
and i A , i and are  matrices.Since the following inequalities then the matrix A is block diagonally dominant by rows.Thus the matrix is also block diagonally dominant by rows.However, Therefore the matrices 1 and 2 are also block diagonally dominant by columns.Similarly, i for all block diagonally dominant by columns by induction.Then can also preserve the key property of .

Error Analysis
The use of BLAS3 based on fast matrix multiplication techniques affects the stability only insofar as it increases the constant terms in the normwise backward error bounds [13].We make assumption about the underlying level-3 BLAS (matrix-matrix operations).
p n  The computed solution K to the triangular systems JK Q  , with and , satisfies where denotes a constant depending on m and .In this section, we present the backward error analysis for the block LU factorization by applying BLAS3 based on fast matrix multiplication techniques.A in (1).Then Applying the standard analysis of errors, we can obtain the above result.Thus we omit it.□  .Since the factors i arising in the factorization in this paper are triangular matrices, from (2) we have where ˆˆî i i DU U   .Note that the multiplication do not produce error because of the structure of and .Then Compared to the proof of standard analysis of errors, there is a great different in form and the simpler proof of the latter embodies whose superiority.For the former, the error bound does not include ˆi U  , which makes the computation easier.
Applying the result of Theorem 4.1, we have the following theorem.
Proof.To save clutter we will omit " " from each bound.For the expression

Theorem 3 . 1 .
Let A in (1) be nonsingular and block diagonally dominant by rows (columns).Then the factors , andU also preserve the similar property.

L
the errors of j L and ˆj U can be repre-sented as J. M. Varah, "On the Solution of Block-Tridiagonal Systems Arising from Certain Finite-Difference Equations," Mathematics of Computation, Vol. 26, No. 120, 1972, pp.859-868.doi:10.1090/S0025-5718-1972-0323087-4Therefore the result holds.□ From Theorems 4.1 and 4.2, the bounds for , one of factors , 1 and of elements ij ˆi i L  ˆii U A , which make the computation of error bounds simpler.