Numerical algorithms for solving comrade linear systems based on tridiagonal solvers

Abstract

Recently, two efficient algorithms for solving comrade linear systems have been proposed by Karawia [A.A. Karawia, Two algorithms for solving comrade linear systems, Appl. Math. Comput. 189 (2007) 291–297]. The two algorithms are based on the LU decomposition of the comrade matrix. In this paper, two algorithms are presented for solving the comrade linear systems based on the use of conventional fast tridiagonal solvers and an efficient way of evaluating the determinant of the comrade matrix is discussed.

Introduction

We consider the solution of comrade linear systems of the form Ax = b, where x ≔ (x₁, x₂,…, x_n)^T, b ≔ (b₁, b₂,…, b_n)^T are vectors of length n and A is an n-by-n comrade matrix given by

$$A=\left(\begin{array}{ccccccc}\hfill -\frac{{\beta }_1}{{\alpha }_1}\hfill & \hfill \frac{1}{{\alpha }_1}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \dots \hfill & \hfill 0\hfill \\ \hfill \frac{{\gamma }_2}{{\alpha }_2}\hfill & \hfill -\frac{{\beta }_2}{{\alpha }_2}\hfill & \hfill \frac{1}{{\alpha }_2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \dots \hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{{\gamma }_3}{{\alpha }_3}\hfill & \hfill -\frac{{\beta }_3}{{\alpha }_3}\hfill & \hfill \frac{1}{{\alpha }_3}\hfill & \hfill 0\hfill & \hfill \dots \hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \ddots \hfill & \hfill \ddots \hfill & \hfill \ddots \hfill & \hfill \ddots \hfill & \hfill ⋮\hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill \ddots \hfill & \hfill \ddots \hfill & \hfill \ddots \hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \dots \hfill & \hfill 0\hfill & \hfill \frac{{\gamma }_{n-1}}{{\alpha }_{n-1}}\hfill & \hfill -\frac{{\beta }_{n-1}}{{\alpha }_{n-1}}\hfill & \hfill \frac{1}{{\alpha }_{n-1}}\hfill \\ \hfill -\frac{a_n}{{\alpha }_n}\hfill & \hfill -\frac{a_{n-1}}{{\alpha }_n}\hfill & \hfill \dots \hfill & \hfill -\frac{a_4}{{\alpha }_n}\hfill & \hfill -\frac{a_3}{{\alpha }_n}\hfill & \hfill \frac{{\gamma }_n-a_2}{{\alpha }_n}\hfill & \hfill \frac{-{\beta }_n-a_1}{{\alpha }_n}\hfill \end{array}\right)\text{.}$$

Multiplying Ax = b by the diagonal matrix D ≔ diag(α₁, …, α_n) from the left, i.e. DAx = Db, then we obtain the following linear systems:

$$\tilde{A}\mathit{x}=\tilde{\mathit{b}}\text{,}$$

where

$$\tilde{A}=\left(\begin{array}{ccccccc}\hfill -{\beta }_1\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \dots \hfill & \hfill 0\hfill \\ \hfill {\gamma }_2\hfill & \hfill -{\beta }_2\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \dots \hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\gamma }_3\hfill & \hfill -{\beta }_3\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill \dots \hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \ddots \hfill & \hfill \ddots \hfill & \hfill \ddots \hfill & \hfill \ddots \hfill & \hfill ⋮\hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill \ddots \hfill & \hfill \ddots \hfill & \hfill \ddots \hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \dots \hfill & \hfill 0\hfill & \hfill {\gamma }_{n-1}\hfill & \hfill -{\beta }_{n-1}\hfill & \hfill 1\hfill \\ \hfill -a_n\hfill & \hfill -a_{n-1}\hfill & \hfill \dots \hfill & \hfill -a_4\hfill & \hfill -a_3\hfill & \hfill {\gamma }_n-a_2\hfill & \hfill -{\beta }_n-a_1\hfill \end{array}\right)$$

and $\tilde{\mathit{b}}≔({\tilde{b}}_1\text{,}{\tilde{b}}_2\text{,}\dots \text{,}{\tilde{b}}_n{)}^{\mathrm{T}}=({\alpha }_1{b}_1\text{,}{\alpha }_2{b}_2\text{,}\dots \text{,}{\alpha }_n{b}_n{)}^{\mathrm{T}}$. Since the transformation does not lose generality for solving the original systems, i.e. it keeps the same solution as the original systems, we consider the solution of the above linear systems (1). The coefficient matrix of (1) is closely related to a tridiagonal matrix in that the matrix can be split as a tridiagonal matrix and a rank-one matrix of the form

$$\tilde{A}=T+{\mathit{e}}_n{\mathit{a}}^{\mathrm{T}}\text{,}$$

where e_n is the nth unit vector, i.e. the nth entry is one and others are all zeros, a ≔ (−a_n, −a_n−1,…, −a₁)^T, and

$$T≔\left(\begin{array}{ccccccc}\hfill -{\beta }_1\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \dots \hfill & \hfill 0\hfill \\ \hfill {\gamma }_2\hfill & \hfill -{\beta }_2\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \dots \hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\gamma }_3\hfill & \hfill -{\beta }_3\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill \dots \hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \ddots \hfill & \hfill \ddots \hfill & \hfill \ddots \hfill & \hfill \ddots \hfill & \hfill ⋮\hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill \ddots \hfill & \hfill \ddots \hfill & \hfill \ddots \hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \dots \hfill & \hfill 0\hfill & \hfill {\gamma }_{n-1}\hfill & \hfill -{\beta }_{n-1}\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \dots \hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\gamma }_n\hfill & \hfill -{\beta }_n\hfill \end{array}\right)\text{.}$$

Recently, Karawia has proposed numerical and symbolic algorithms for solving comrade liner systems [4]. Since the algorithms are based on the LU decomposition of $\tilde{A}$, it is natural and good approach. From the perturbation point of view, the matrix $\tilde{A}$ can be written as T + e_na^T. Hence, it may be also natural to use tridiagonal solvers for solving comrade linear systems. Efficient tridiagonal solvers have been given much attention and developed by many researchers such as TDMA or the Thomas algorithm see e.g. [5, p. 186], and the stable and reliable algorithms [1]. Therefore, it is worth considering numerical algorithms for comrade linear systems based on tridiagonal solvers.

The present paper is organized as follows: in the next section, we give two numerical algorithms for solving comrade linear systems. In Section 3, we present a numerical formula for evaluating the determinant of the comrade matrix. In Section 4, we show the result of a numerical experiment. Finally we make some concluding remarks in Section 5.