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2001•International Journal of Computer MathematicsRequires access

An iterative method for the positive real linear system

Changjun Li, Xiaoli Liang, David Evans

Open publisher page 8 citations

Abstract

In this paper a new iterative method is given for the linear system of equations Au = b, where A is large, sparse and nonsymmetrical and AT + A is symmetric and positive definite (SPD) or equivalently A is positive real. The new iterative method is based on a mixed-type splitting of the matrix A. The iterative method contains an auxiliary matrix D 1. It is shown that by proper chxoice of D 1 the new iterative method is convergent. It is also shown that by special choice of D 1, the new iterative method becomes the well-known (point) successive overrelaxiation (SOR) [1] method. Hence, it is shown that the (point) SOR method applied to the positive real system is convergent if the overrelaxiation parameter ω is in (0,ω U ). The upper bound ω U is also given in terms of the norm and smallest eigenvalue of related matrices (see Eq. (23)).

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What this paper is about

In this paper a new iterative method is given for the linear system of equations Au = b, where A is large, sparse and nonsymmetrical and AT + A is symmetric and positive definite (SPD) or equivalently A is positive real. The new iterative method is based on a mixed-type splitting of the matrix A. The iterative method contains an auxiliary matrix D 1. It is shown that by proper chxoice of D 1 the new iterative method is convergent. It is also shown that by special choice of D 1, the new iterative method becomes the well-known (point) successive overrelaxiation (SOR) [1] method. Hence, it is shown that the (point) SOR method applied to the positive real system is convergent if the overrelaxiation parameter ω is in (0,ω U ). The upper bound ω U is also given in terms of the norm and smallest eigenvalue of related matrices (see Eq. (23)).

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Available abstract

In this paper a new iterative method is given for the linear system of equations Au = b, where A is large, sparse and nonsymmetrical and AT + A is symmetric and positive definite (SPD) or equivalently A is positive real. The new iterative method is based on a mixed-type splitting of the matrix A. The iterative method contains an auxiliary matrix D 1. It is shown that by proper chxoice of D 1 the new iterative method is convergent. It is also shown that by special choice of D 1, the new iterative method becomes the well-known (point) successive overrelaxiation (SOR) [1] method. Hence, it is shown that the (point) SOR method applied to the positive real system is convergent if the overrelaxiation parameter ω is in (0,ω U ). The upper bound ω U is also given in terms of the norm and smallest eigenvalue of related matrices (see Eq. (23)).

Key concepts: Mathematics, Iterative method, Positive-definite matrix, Eigenvalues and eigenvectors, Matrix (chemical analysis), Linear system, Applied mathematics, Norm (philosophy)

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