Trigonometric Preconditioners for Block Toeplitz Systems

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DOI: 10.1007/978-3-0348-8871-4_18
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Abstract
This paper is concerned with the solution of a system of linear equations T M, N X=b, where T M, N denotes a positive definite doubly symmetric block-Toeplitz matrix with Toeplitz blocks arising from a generating function f of the Wiener class. We derive optimal and Strang-type trigonometric preconditioners P M, N of T M, N from the Fejér and Fourier sum of f, respectively. Using relations between trigonometric transforms and Toeplitz matrices, we prove that for all ε > 0 and sufficiently large M, N, at most O(M) + O(N) eigenvalues of lie outside the interval (1 — ε, l + ε) such that the preconditioned conjugate gradient method converges in at most O(M) + O(N) steps.
  • ... Then, the matrices A N (f) have real entries such that we restrict our attention to trigonometric preconditioners. Table 5 Both matrices are ill{conditioned and the CG{method without preconditioning, with Strang{type{preconditioning or with optimal trigonometric preconditioning converges very slow (see 22,25]). Our preconditioning determined by (4.7) leads to the number of iterations in Table 5.5. ...
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    This paper is concerned with the solution of systems of linear equations A Nx = b, where\(\{ A_N \} _{N \in \mathbb{N}}\) denotes a sequence of positive definite Hermitian ill-conditioned Toeplitz matrices arising from a (real-valued) nonnegative generating function f ∈ C 2π with zeros. We construct positive definite Hermitian preconditioners M N such that the eigenvalues of M N−1A N are clustered at 1 and the corresponding PCG-method requires only O(N log N) arithmetical operations to achieve a prescribed precision. We sketch how our preconditioning technique can be extended to symmetric Toeplitz systems, doubly symmetric block Toeplitz systems with Toeplitz blocks and non-Hermitian Toeplitz systems. Numerical tests confirm the theoretical expectations.
  • ... ρn is a block-Toeplitz-Toeplitz-block matrix. Note that we can avoid the complex arithmetic introduced by the FFT by using fast Toeplitz matrix times vector multiplications based on trigonometric transforms (see Algorithm 3 of [25]). ...
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  • ... As mentioned in the previous section, the optimal preconditioner in A F N coincides with our preconditioner (2.10) defined with respect to the Fejér kernel B 1,N and with w N = 0 in (2.7). It is easy to check (see [33]) that the optimal preconditioner in A O N , where O N ∈ {C IV N , S IV N } is equal to our preconditioner M N (f N , O N ) in (5.1) defined with respect to O N and with respect to the Fejér kernel. Unfortunately, the Fejér kernel preconditioners do not lead to a fast convergence of the PCG-method if the generating function f of A N has a zero of order 2s ≥ 2. In contrast to these results, the optimal preconditioners in A O N with O N defined by the DCT I -III or by the DST I -III do not coincide with the corresponding Fejér kernel preconditioner M N (f N , O N ) in (5.1). ...
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