A Method for Solving Cyclic Block Penta-diagonal Systems of Linear Equations

Abstract

A method for solving cyclic block three-diagonal systems of equations is generalized for solving a block cyclic penta-diagonal system of equations. Introducing a special form of two new variables the original system is split into three block pentagonal systems, which can be solved by the known methods. As such method belongs to class of direct methods without pivoting. Implementation of the algorithm is discussed in some details and the numerical examples are present. The Maple and the Matlab programs are also present in appendices.

The use of the Sherman–Morrison–Woodbury formula to solve cyclic block tri-diagonal and cyclic block penta-diagonal linear systems of equations

“…This is again precisely the solution of the systems (9) in ref [2]. From (27) the final solution of ( 1) is ( )…”

“…is a matrix of the order 2 2 m m × which is assumed to be nonsingular. The solution x can therefore be written in the form (eq 8 in [2]) one can easily restore the system (17) in [2] for the determination of u and v. The equivalence between the method used in [2] and that using the generalized Woodbury formula is thus shown.…”

“…By ( 23) and (25) the matrix (21) has the form of matrix (5) in [2]. Now, by (16) the system (1) becomes ( )…”

“…The algorithms for solving the cyclic block tri-diagonal (CBTS) and cyclic block pentadiagonal systems (CBPS) of equations presented in ref [1], [2] are based on ad hoc introduction of the new unknown vector(s) which transform the original system into a non-cyclic system which can then be solved by standard methods. In this short paper the theoretical background of the algorithms based on the Woodbury formula ( [3], [4]) will be given.…”

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The article presents the theoretical background of the algorithms for solving cyclic block tridiagonal and cyclic block penta-diagonal systems of linear algebraic equations present in [1] and [2]. The theory is based on the Woodbury formula.

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