An introduction to iterative Toeplitz solvers by Raymond Hon-Fu Chan, Xiao-Qing Jin

By Raymond Hon-Fu Chan, Xiao-Qing Jin

Toeplitz platforms come up in various purposes in arithmetic, medical computing, and engineering, together with numerical partial and usual differential equations, numerical recommendations of convolution-type indispensable equations, desk bound autoregressive time sequence in facts, minimum awareness difficulties on top of things conception, method id difficulties in sign processing, and snapshot recovery difficulties in photograph processing. This useful publication introduces present advancements in utilizing iterative equipment for fixing Toeplitz platforms in accordance with the preconditioned conjugate gradient process. The authors concentrate on the $64000 points of iterative Toeplitz solvers and provides particular recognition to the development of effective circulant preconditioners. functions of iterative Toeplitz solvers to useful difficulties are addressed, allowing readers to take advantage of the ebook s tools and algorithms to unravel their very own difficulties. An appendix containing the MATLAB® courses used to generate the numerical effects is integrated. scholars and researchers in computational arithmetic and medical computing will reap the benefits of this booklet.

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First, we claim that the preconditioners are positive definite. 1. The preconditioner Cn (Km,2r ∗ f ) is positive definite for f ∈ C+ 2π and for all positive integers m, n, and r. Proof. 1), Km,2r (x) is positive except at x = 2kπ/m, k = ±1, ±2, . . , ±(n − 1), and f ∈ C+ 2π is nonnegative and not identically zero. 11). 50 Chapter 4. Ill-conditioned Toeplitz systems For simplicity, we will use x to denote the function x defined on R in the following. 1 below. It is clear that x2p 2π ∈ C2π for any integer p.

197]. 5. 5. We remark that in general, Hα (x) is not a positive function in [−π, π]. 5, respectively. 5 as our generating functions. Eight different circulant preconditioners are tested. As before, the right-hand side b is the vector of all ones. 4 show the number of iterations required for convergence. 4, respectively. 3. 2. 5. 4. 5. Preconditioner used I Cn (Dm ∗ f ) Cn (Fn ∗ f ) Cn (Dn−1 ∗ f ) Modified Dirichlet de la Vall´ee Poussin von Hann Bernstein Hamming 32 18 11 12 12 12 11 11 12 11 64 29 14 13 14 14 14 12 14 13 128 44 16 14 16 16 15 13 14 14 n 256 66 16 15 17 16 16 15 16 16 512 67 15 14 15 15 15 15 15 15 1024 68 15 15 18 17 15 15 15 15 From the tables, we see that as n increases, the number of iterations increases for the original matrix Tn , while it stays almost the same for the preconditioned matrices.

2ν−2 |||e(0) ||| The theorem was proved by using Weyl’s theorem. R. Chan and Yeung later used Jackson’s theorems [35, pp. 5. 6 (R. Chan and Yeung [31]). Suppose f is a Lipschitz function of order ν for 0 < ν ≤ 1, or f has a continuous νth derivative for ν ≥ 1. Then there exists a constant c > 0 which depends only on f and ν such that for large n, k |||e(2k) ||| c log2 p ≤ . 2. 2 21 Optimal (circulant) preconditioner T. Chan in [33] proposed a specific circulant preconditioner called the optimal circulant preconditioner for solving Toeplitz systems.

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