By Lydia I. Kronsjo

ISBN-10: 0471912018

ISBN-13: 9780471912019

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**Sample text**

Let Wn(N ) ≡ Bn − Un(N ) . (N ) The leading (n − N )-by-(n − N ) block of Wn is the leading (n − N )-by-(n − N ) principal submatrix of Bn , and hence this block is a Toeplitz matrix. It is easy to (N ) see that the maximum absolute column sum of Wn is attained at the ﬁrst column (or the (n − N − 1)th column). Thus n−N −1 Wn(N ) 1 n−N −1 |bk | = = k=m+1 (N ) Since Wn k=m+1 (N ) is Hermitian, we have Wn Wn(N ) 2 n−N −1 |tk − tk−n | ≤ 2 ≤ Wn(N ) |tk | < . k=N +1 ∞ (N ) = Wn (N ) 1 · Wn 1. 1 2 ∞ Thus < .

9, we have the following corollary. 2. Let Tn be a Toeplitz matrix with a positive generating function f ∈ C2π . Then for all > 0, there exist M and N > 0 such that for all n > N , at most M eigenvalues of the matrix (cF (Tn ))−1 Tn − In have absolute values larger than . It follows that the convergence rate of the PCG method is superlinear. 4 for the convergence analysis of preconditioners derived by kernels. 15 for tF (Tn ) from the Wiener class to C2π ; see [6]. In the next section, we relate some of the circulant preconditioners discussed in Chapter 2 with well-known kernels in function theory.

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 . 11) is obtained, the products of Pn y and Pn−1 y for any vector y can be computed by FFTs in O(n log n) operations. 11), Pn can be constructed in O(n log n) operations.

### Algorithms: Their complexity and efficiency by Lydia I. Kronsjo

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