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By E. C. Titchmarsh

Eigenfunction Expansions linked to Second-Order Differential Equations half II (Two 2)

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Extra resources for Eigenfunction expansions associated with second-order differential equations. Part 2

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Further analysis of this procedure by Adams and Jordan [1984] showed that it is equivalent, in certain cases and up to transient effects, to carrying out SOR under a multicolor ordering. In particular, they show for a wide class of discretization stencils that the spectral radius of the SOR iteration matrix for certain multicolor orderings is the same as that of the iteration matrix for the natural rowwise ordering and hence, in these cases, the asymptotic rate of convergence for the multicolor orderings is the same as that for the natural ordering.

3; each sequence of numbers represents a grid point and indicates the times at which the corresponding unknown can be updated. Further analysis of this procedure by Adams and Jordan [1984] showed that it is equivalent, in certain cases and up to transient effects, to carrying out SOR under a multicolor ordering. In particular, they show for a wide class of discretization stencils that the spectral radius of the SOR iteration matrix for certain multicolor orderings is the same as that of the iteration matrix for the natural rowwise ordering and hence, in these cases, the asymptotic rate of convergence for the multicolor orderings is the same as that for the natural ordering.

1970] and was first analyzed as a parallel algorithm by Buzbee [1973]. 11) is discretized using the 5-point difference formula on an evenly spaced square grid one obtains the block tridiagonal system PARTIAL DIFFERENTIAL EQUATIONS ON VECTOR AND PARALLEL COMPUTERS 39 where A is an N X TV tridiagonal matrix whose jth row is (0, • • • ,0,1, - 4,1,0, • • • ,0), and Ut and fj are TV-vectors. Matrix decomposition is based on the fact that the eigensystem of A is known explicitly and thus the factorization VTAV= A is possible where A is a diagonal matrix whose entries are the eigenvalues of A.

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