Block iterative methods

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Block iterative methods

  1. 1. Block Iterative MethodsThe methods discussed so far are all subspace methods, that is, in every iteration they extend thedimension of the subspace generated. In fact, they generate an orthogonal basis for this subspace, byorthogonalizing the newly generated vector with respect to the previous basis vectors.However, in the case of nonsymmetric coefficient matrices the newly generated vector may be almostlinearly dependent on the existing basis. To prevent break-down or severe numerical error in suchinstances, methods have been proposed that perform a look-ahead step (see Freund, Gutknecht andNachtigal [99], Parlett, Taylor and Liu [168], and Freund and Nachtigal [100]).Several new, unorthogonalized, basis vectors are generated and are then orthogonalized with respect tothe subspace already generated. Instead of generating a basis, such a method generates a series of low-dimensional orthogonal subspaces.If conjugate gradient methods are considered to generate a factorization of a tridiagonal reduction ofthe original matrix, then look-ahead methods generate a block factorization of a block tridiagonalreduction of the matrix.Keeping the block size constant throughout the iteration leads to the Block Lanczos algorithm and theBlock Conjugate Gradient method (see OLeary [159]). In fact, one can show that the spectrum of thematrix is effectively reduced by the smallest eigenvalues, where is the block size.

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