An iterative method for the positive real linear system

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 A^T + 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)).

Keywords:

• Linear system of equations
• Iterative method
• SOR method
• Symmetric and definite positive (SPD) matrix
• Skew symmetric (SS) matrix
• Positive real matrix

AMS Classification:

• 65F10

C.R. Categories:

• G.13

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Notes

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