Circulant and skewcirculant matrices for solving Toeplitz matrix problems
Thomas Huckle Institut f¨ur Angewandte Mathematik und Statistik Universit¨at W¨urzburg Am Hubland D-8700 W¨urzburg, Federal Republic of Germany
0. Introduction
In recent papers G. Strang, R. Chan and T. Chan [15,4,5,3] studied the use of circulant matrices C for solving systems of linear equations Tnx = b with ⎛ ⎞ t0 t1 ...... tn−1 ⎜ ⎟ ⎜ . ⎟ ⎜ t1 t0 t1 . ⎟ ⎜ . . ⎟ ⎜ ...... ⎟ Tn := T (t0,t1, ..., tn−1):=⎜ . t1 . . . ⎟ ⎜ . .. .. ⎟ ⎝ . . . t1 ⎠
tn−1 ...... t1 t0 a symmetric positive definite Toeplitz matrix; thereby a symmetric circulant matrix C is de- fined by C = T (c0,c1, ..., ck−1,ck,ck−1, ..., c1) for even n and C = T (c0,c1, ..., ck,ck, ..., c1) for odd n with k = n/2 . Applying the preconditioned conjugate gradient method [10] for solving the system −1 Tnx = b one has to find a matrix C such that the eigenvalues of C Tn are clustered and Cy = d can be solved very fast. For circulant C the second condition is fulfilled by Fast Fourier Transform [7]. Thus G. Strang choose C := Cs := T (t0,t1,t2, ..., t2,t1) [15], and with R. Chan he proved in [4], that under certain assumptions Cs is again positive definite −1 and the spectrum of Cs Tn is clustered around 1. T. Chan introduced
(n − 1)t1 + tn−1 tn−1 +(n − 1)t1 C := T (t0, , ..., ). f n n
Cf minimizes C − Tn F over all symmetric circulant matrices, where F denotes the Frobenius norm. The spectrum of Cf is again clustered around 1 [see 3].
In this paper we introduce new circulant/skewcirculant approximations to Tn and study their properties. Thereby a symmetric matrix S is said to be skewcirculant,if S = T (s0,s1, ..., sk, −sk, ..., −s1)foroddn and S = T (s0,s1, ..., sk−1, 0, −sk−1, ..., −s1) for even n [7,8].
If Tn is indefinite any circulant approximation will be indefinite in general; but pre- conditioning is only possible for positive definite C. However from Cs and Cf one can
1 −1 always construct positive definite circulant matrices such that the spectrum of C Tn is approximately clustered around ±1.
Every n × n circulant matrix C has the same eigenvectors; thus approximating Tn by C can be considered as approximating the eigenvalues of Tn and ordering the eigenvectors of C such that they are near to those of Tn. Therefore these eigenvectors are efficient start vectors in the Lanczos algorithm for determining any, especially the maximal or minimal eigenvalue of Tn in O(nlog(n)) arithmetic operations. Finally in section 4 for some numerical examples we compare the clustering properties of the circulant/skewcirculant approximations and test the Lanczos method with the start vectors described above.
1. A new circulant/skewcirculant approximation to T
For symmetric Toeplitz matrices T,T1,T2 let us define a scalar product by (T1,T2)E := T T e1 T1T2e1 with e1 =(1, 0, ..., 0) , and the corresponding norm
n −1 2 2 2 T E := Te1 2 = ti . i=0
Similar to the best circulant Frobenius norm approximation to T one can search for a circulant matrix Ce with