A New Symbolic Algorithm for Solving General Opposite-Bordered Tridiagonal Linear Systems

In the current article we propose a new efficient, reliable and breakdown-free algorithm for solving general opposite-bordered tridiagonal linear systems. An explicit formula for computing the determinant of an opposite-bordered tridiagonal matrix is investigated. Some illustrative examples are given.


Introduction
The n n × general tridiagonal matrix n T takes the form: The matrix in (1) frequently appears in many applications, for example, in parallel computing, telecommunication system analysis, solving differential equations using finite differences, heat conduction and fluid flow problems.The interested reader may refer to [1]- [12] and the references therein.
There are great interests in solving general opposite-bordered tridiagonal linear system, and hereafter it will be referred to as OBTLS, of the form: x f (5) in which the coefficient matrix A is given by: , , ,  This system frequently occurs in engineering computation and science, e.g. in the numerical solution of an ablation and heat transfer problem as referred in [24]- [28].The matrix A in (6) can be stored in 5 6 n − memory locations only.
In [28], the author presented a numeric algorithm for solving the linear system (5) with 0 2 0 n p q − = = .The algorithm is based on the elementary column operations (ECO's).It is noted that the numerical algorithm in [28] fails to solve some OBTLS of the form (5). Therefore, the main objective of the present paper is to construct a new symbolic and breakdown-free algorithm for solving the OBTLS in (5).
Throughout this paper, the word "simplify" means simplifying the algebraic expression under consideration to its simplest rational form.Also, λ is a formal parameter which can be treated as a symbolic name whose actual value is 0 as we will see later.
The organization of the paper is as follows.The main results are given in the next section.Some illustrative examples are given in Section 3. In Section 4, we present some concluding remarks.

Main Results
In this section, we are going to formulate a new algorithm for solving OBTLS of the form (5). We begin by considering the singly bordered tridiagonal linear systems of the form (7)-(8) below.

A Symbolic Algorithm for Solving Singly Bordered Tridiagonal Linear Systems
The k k × singly bordered tridiagonal linear system takes the form: where , , , , , , .
S is given by [1]: where and where the quantities , 1, 2, , , are given, respectively, by: , , for 1, 2, , 1 and It follows from ( 9)-( 11) that ( ) At this point, it should be mentioned that the above LU factorization is always possible even if the matrix k S is singular.The solution of the system in (7) reduces to solving the two standard linear systems: and .U = y z (17) We are now ready to formulate the following algorithm for solving the linear system (7).

A Symbolic Algorithm for Solving General OBTLS
In order to solve the general OBTLS (5) it is convenient to introduce the following notations: [ ] ( ) [ ] ( ) , , , , , , ˆ, , , , , the above notations, the linear system in (5) can be written in the partitioned form: The solution of the linear system (21), may be obtained by solving the two linear systems: 0 ˆ, It is not difficult to prove that: As can be seen from ( 24), we need to compute q By solving the following singly bordered systems with two right-hand sides we obtain the solution vectors v and w : Consequently, we have from ( 24) .
By substituting ( 26) and ( 27) into (22), it follows that Therefore, we get Hence, the solution vector of the OBTLS ( 5) is The proofs of the following result may be found in [29].Theorem 1. (Schur determinant identity) Let . ) By noticing (21), we see that the matrix A in (6) can be written in the partitioned form: Hence, by applying Theorem 1 on this matrix, we get the following result: Corollary 1.Let A be the n n × matrix given in (31), then the determinant of A is given by: ( ) ( ) The main result of the present paper may now be formulated as follows: This algorithm will be referred to as the OBS algorithm.The computational cost for OBS is 21 35 n − in terms of total number of flops, where each flop represents one of the four basic arithmetic floating point operations.
A MATLAB code based on the OBS algorithm is available upon request from the authors.

Conclusion
In this paper, we proposed a new efficient and reliable algorithm for solving general opposite-bordered gonal linear systems in linear time.An explicit formula for computing the determinant of an opposite-bordered tridiagonal matrix is obtained.Some illustrative examples are given.

1 .
by means of execution times and accuracy of the solutions in Example 2. Finally, we give Example 3 in order to demonstrate the validity of the OBS algorithm.All experiments were carried out using MATLAB 7.10.0.499 (R2010a) on a PC with Intel(R) Core(TM) i7-3770 CPU processor.Example Solve the opposite-bordered tridiagonal linear system:

Table 1 .
Mean value of the CPU times after 100 tests.

Table 2 .
Absolute errors and CPU times of Example 3 for the OBS algorithm.