ISSN:
1572-9125
Schlagwort(e):
65F10
;
65F15
;
Skew-Hermitian type Toeplitz matrix
;
circulant matrix
;
skew-circulant matrix
;
preconditioned conjugate gradient method
Quelle:
Springer Online Journal Archives 1860-2000
Thema:
Mathematik
Notizen:
Abstract We study the solutions of Toeplitz systemsA n x=b by the preconditioned conjugate gradient method. Then ×n matrixA n is of the forma 0 I+H n wherea 0 is a real number,I is the identity matrix andH n is a skew-Hermitian Toeplitz matrix. Such matrices often appear in solving discretized hyperbolic differential equations. The preconditioners we considered here are the circulant matrixC n and the skew-circulant matrixS n whereA n =1/2(C n +S n ). The convergence rate of the iterative method depends on the distribution of the singular values of the matricesC −1 n An andS −1 n A n . For Toeplitz matricesA n with entries which are Fourier coefficients of functions in the Wiener class, we show the invertibility ofC n andS n and prove that the singular values ofC −1 n A n andS −1 n A n are clustered around 1 for largen. Hence, if the conjugate gradient method is applied to solve the preconditioned systems, we expect fast convergence.
Materialart:
Digitale Medien
URL:
http://dx.doi.org/10.1007/BF01933178
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