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

By Raymond Hon-Fu Chan, Xiao-Qing Jin

Toeplitz structures come up in quite a few functions in arithmetic, medical computing, and engineering, together with numerical partial and traditional differential equations, numerical recommendations of convolution-type quintessential equations, desk bound autoregressive time sequence in data, minimum awareness difficulties on top of things conception, approach id difficulties in sign processing, and snapshot recovery difficulties in picture processing. This sensible e-book introduces present advancements in utilizing iterative tools for fixing Toeplitz structures in keeping with the preconditioned conjugate gradient technique. The authors specialize in the real facets of iterative Toeplitz solvers and provides certain cognizance to the development of effective circulant preconditioners. purposes of iterative Toeplitz solvers to useful difficulties are addressed, allowing readers to take advantage of the e-book s equipment and algorithms to unravel their very own difficulties. An appendix containing the MATLAB® courses used to generate the numerical effects is incorporated. scholars and researchers in computational arithmetic and clinical computing will make the most of this e-book.

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N − 1. Note that the choice of the grids requires some prior information about the zeros of f . 11) 32 Chapter 2. Circulant preconditioners where Fn is the Fourier matrix, Ωn = diag(1, ewn i , e2wn i , . . , e(n−1)wn i ), and Λn = diag(f (x0 ), f (x1 ), f (x2 ), . . , f (xn−1 )). 12) The preconditioner Pn has the following properties (see [68]): (i) Pn is Hermitian positive definite if f ≥ 0. (ii) Pn is an {enwn i }-circulant matrix [37]. Notice that {enwn i }-circulant matrices are Toeplitz matrices with the first entry of each column obtained by multiplying the last entry of the preceding column by enwn i .

88]. In the following, we will use the symbol K(x) to denote a generic kernel defined on [−π, π]. The notation Cn (K ∗ f ) denotes the circulant matrix with eigenvalues given by 2πj , 0 ≤ j ≤ n − 1. 11) λj (Cn (K ∗ f )) = (K ∗ f ) n Using this notation, we can rewrite Strang’s, T. Chan’s, and R. Chan’s circulant preconditioners as s(Tn (f )) = Cn (Dm ∗ f ), cF (Tn (f )) = Cn (Fn ∗ f ), r(Tn (f )) = Cn (Dn−1 ∗ f ), respectively. 1. Some kernels and their definitions. 4. Clustering properties 41 or otherwise by using the following construction process.

K=N +1 ∞ (N ) = Wn (N ) 1 · Wn 1. 1 2 ∞ Thus < . (N ) Hence the spectrum of Wn lies in (− , ). By Weyl’s theorem, we see that at most 2N eigenvalues of Bn = Tn − s(Tn ) have absolute values exceeding . 2, and using the fact that (s(Tn ))−1 Tn = In + (s(Tn ))−1 (Tn − s(Tn )), we have the following corollary. 3. Let f be a positive function in the Wiener class. Then for all > 0, there exist M and N > 0 such that for all n > N , at most M eigenvalues of (s(Tn ))−1 Tn − In have absolute values larger than .

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