On Solving Centrosymmetric Linear Systems

The current paper is mainly devoted for solving centrosymmetric linear systems of equations. Formulae for the determinants of tridiagonal centrosymmetric matrices are obtained explicitly. Two efficient computational algorithms are established for solving general centrosymmetric linear systems. Based on these algorithms, a MAPLE procedure is written. Some illustrative examples are given.


Throughout this paper, A and T
A denote the determinant and the transpose of the matrix A respectively.Also x     denotes the greatest integer less than or equal to .
Solving and analyzing linear systems of equations is a fundamental problem in science and engineering applications.The cost of solving any linear system using Gauss or Gauss-Gordan algorithms is The motivation of the current paper is to develop efficient algorithms for solving any centrosymmetric linear system having equations and unknowns provided that the coefficient matrix of the system is nonsingular.The cost of each algorithm depends on the solvers of two associated linear systems having smaller sizes than n.More precisely, if n = 2m, then each of the two associated linear systems consists of m equations.If n = 2m + 1, then we have one system having m equations and the other has (m + 1) equations.Consequently, if the two associated linear systems have special structures, then the cost of the centrosymmetric algorithm could be considerably reduced, in particular for large values of n.

. O n n n
The paper is organized as follows.In Section 2, some properties of the exchange and the rotate matrices are presented.Formulae for centrosymmetric tridiagonal determinants are obtained in Section 3. In Section 4, two computational algorithms for solving centrosymmetric linear systems are given.A MAPLE procedure is given in Section 5. Some illustrative examples are presented in Section 6.
Definition 1.1.The n n  matrix with 1's on the northeast-southeast diagonal and 0's elsewhere 0 0 is called the exchange matrix of order The subscript on n .n J is neglected whenever the size is obvious from the context.
Definition 1.3 [9].Let be an   , 1 Note that the bisymmetric matrix is both symmetric and centrosymmetric.It is also both symmetric and presymmetric.
The exchange matrix n J enjoys the following properties: where n I is the identity matrix of order .n

a version of the matrix
A that has been reflected in line at 0 degree to the horizontal measured counter-clockwise. The matrix product AJ is a version of the matrix A that has been reflected in line at 90 degrees to the horizontal measured counter-clockwise.

R JAJ A
 is a version of the matrix A that has been rotated counter-clockwise or clockwise by 180 degrees.Note that the exchange matrix J is equal to the identity matrix I but with the columns in reverse order.More precisely, , , , i j  It is also worth mentioned that the matrix J is sometimes called the counter-identity matrix or contra-identity matrix or per-identity matrix or the reflection matrix or the reversal matrix.
Exchange matrices are simple to construct in software platforms.For example, to construct 5 J in MAPLE, a single line of code can be used as follows: n := 5: J := array(1..n,1..n,sparse): for i to n do J[i, n + 1 -i] := 1 od: J n := op(J); or n := 5: J := matrix(n, n, 0): for i to n do J[i, n + 1i] := 1 od: J n := op(J); The rotate of , A R A of order satisfies: n , , , , In other words, the centrosymmetric matrix A is the same when read backwards as when read forwards.

Centrosymmetric Determinants
Centrosymmetric determinants take the form: , , , , , for each 1, 2, , . 2 In particular, centrosymmetric tridiagonal determinants are of special importance.For convenience of the reader, we present some definitions, notations and properties associated with tridiagonal matrices.
A tridiagonal matrix takes the form: These types of matrices frequently appear in many areas of science and engineering.For example in parallel computing, telecommunication system analysis and in solving differential equations using finite differences, see [22,23].A general n n  tridiagonal matrix of the form (4) can be stored in memory locations, rather than memory locations for a full matrix, by using three vectors This is always a good habit in computation in order to save memory space.
To study tridiagonal matrices it is very convenient to introduce a vector in the following way [24]: It is helpful to restate some important results concerning tridiagonal matrices of the form (4).For more details the interested reader may refer to [24][25][26][27][28].
Let and , , p q r s are positive integers such that and 1  .

Define the submatrix of ,
A denoted   .. , .. , A p q r s of order by .. , .. .p r p r p s p r p r p s q r q r q s a a a a a a A p q r s a a a In particular, let the leading principal minor of .A Theorem 3.1 [24].
Then the determinants in (8) satisfy a three-term recurrence where the initial values for k f are 0 1  a 0. f where are given by (6).Meanwhile, the , , , n c c c  le LU fact Doolitt orization [30] of T is given by: a singular ma Lemma 3.5 [24].

Algorithm 3.1 (DETGTRI [31]).
The determinant of the matrix in ( 4) can be computed using the following symbolic algorithm.
INPUT: Order of the matrix and the components, n

. T P 
The cost of the DETGTRI algorithm is   O n .The omputer Algebra algorithm is easy to implement in all C Sy , MATHEMATICA an nal matrix gi 0 .
stems (CAS) such as MACSYMA d MAPLE.Lemma 3.6.Consider the tridiago n ven by: U and n V be n n  matrices defined reectively as follows: 1 1 and , where k is a scalar quantity.Then by applying the DETGTRI algorithm, we see that: having used ( 12) and (13).Let   , 1 is an centrosymmetric matrix, then the three following facts are useful when we n n  deal with such matrices:

Fact (1): If then
In other words, the centrosymmetric matrix , , , , , A is the same when read backwards as when read forwards. ( facts, we may formulate the fo ng result wh will can be written rm: y: 2 0 ,  is orthogonal.Armed with the above llowi ose proof be omitted.
where , . w The determinant of the matrix in (20) is given by Concerning the inverse of centrosymmet the reader may refer to [8].
As an interesting special case of the Theorem 3.7, we give the following result.
Corollary 3.8.In Theorem 3.7, if R = T is centrotric tridiagon matrix, th ave ric matrices where k is the common value of the elements in positions (m centrosymmetric tridiagonal de , m+1) and (m+1, m) of the matrix R = T n when n = 2m.Proof.To compute the terminant of order .
n Two cases will be considered: In this case, the centrosymmetric tri diagonal matrix takes the form: Applying Theorem 3.7, we have By using Lemma 3.6, we obtain In this case, the centrosymmetric tridiagonal matrix takes the form: Applying Theorem 3.7, we obtain , By using the DETGTRI algorithm [31] together with ( 12) and ( 13), then we have   From ( 23) and ( 25), we see that in order to compute the determinant of a centrosymmetric tridiagonal matrix of order , then all we need is to compute if and if

Algorithms for Solving Centrosymmetric Linear Systems
Solving linear systems practically dominates scientific computing.In the present section, we focus on solving linear systems of centrosymmetric type.Two cases will be considered: For this case we are going to construct an algorithm for solving centrosymmetric linear systems of the form: Block multiplication is particularly useful when there are patterns in the matrices to be multi .Therefore it is convenient to rewrite (26) in the partitioned form w    and .
The system in (26) can also be written in matrix form as follows: (28) where here is the coefficient matrix of the system (26), and is the constant vector.Algorithm 4.1.An algorithm for solving centrosymmetric linear system of even order.
To solve the linear system of the form (26), we may proceed as follows: INPUT: The entries of the coefficient matrix and the constant vector in (28).

OUTPUT: Solution vector
Step 1: Construct the , , Step 3: Solve the two linear systems: ˆ, P  y b , , z z  and for and ectively.
Step 4: The solution vector is ven by The .1, will be refereed to as CENTRO- SYMM-I al al cost n n  additions/subtractions and no multiplica- The time complexity of Step 3 depends on the solvers of the two linear systems.For example, tridiagonal tions/subtractions and multiplicatio Step 3 is the step that leads to the reduction of the time complexity, because instead of solving a linear system of equations, we end up with two linear systems half the size of the original one.If th original system is solved with Gaussian elimination (GE) method, then the time complexity will be tions/divisions).linear systems can be solved in linear time (see [25]). .
used to solve th two systems in the third step, then the time complexity of our algorithm will be e     ethod more effici n the time com ity of our a will b see also [16]).

Case (ii):
3 wh significant reduction.ich is a If a m ent than the GE method is used, the plexlgorithm e less ( 21. m n  In this case the linear system to   be considered has the form: or equivalently, , metric linear system of odd ord The linear system (30) can also be written as: , R  x b (31) where is the coefficient matrix of the system (29), and is the constant vector.Algorithm 4.2.An algorithm for solving centrosym-er To solve the linear system of the form (29), we may proceed as follows: INPUT: The entries of the coefficient matrix R and the constant vector in (31).

OUTPUT: Solution vector
Step 1: Construct the matrices of orders , 1 Step 2: Compute , 2 , Step 4: The solution vector is given by , , , , The Algorithm will be refereed to as CENTRO- SYMM-II algorithm.
Concerning the computational cost of the CENTRO- SYMM-II rithm:  The time complexity of Step 3 depends on the solvers of the two linear systems. see also [16]).It may be convenient to fi owing result

Soluti
Here centrosymm, we ge on: ocedure 10 n  and 5. m  Using the pr t the following results: The so Example 6.2.Solve the centrosymmetric linear system Solution: Here 11 n  and Using the procedure centrosymm, we obtain the following results:

Solution:
Here 10 n  and 5. m  Using the procedure centrosymm, we obtain the following results: Error Singular Matrix !!!!This means that the given system has no solutions.

.
Step 1: Use (6) to compute the simplest fo components of the vector

:
The simplest rational form of the product