No abstract

The shift-invert Arnoldi method is employed to generate an orthonormal basis from the Krylov subspace corresponding to a real Toeplitz matrix and an initial vector. The vectors and recurrence coefficients produced by this method are exploited to approximate the Toeplitz matrix exponential. Toeplitz matrix inversion formula and rapid Toeplitz matrix-vector multiplications are utilized to lower the computational costs. For convergence analysis, a sufficient condition is established to guarantee that the error bound is independent of the norm of the matrix. Numerical results are given to demonstrate the efficiency of the method.

SummaryA fast approximate inversion method is proposed for the block lower triangular Toeplitz with tri‐diagonal blocks (BL3TB) matrix. The BL3TB matrix is approximated by a block ϵ‐circulant matrix, which can be efficiently inverted using the fast Fourier transforms. The error estimation is given to show the high accuracy of the approximation. In applications, the proposed method is employed to solve the fractional sub‐diffusion equation whose discretized matrix by a finite difference method is a BL3TB matrix. Numerical experiments are carried out to demonstrate the efficiency of the proposed method. Copyright © 2015 John Wiley & Sons, Ltd.

The Lanczos method with shift-invert technique is exploited to approximate the symmetric positive semidefinite Toeplitz matrix exponential. The complexity is lowered by the Gohberg-Semencul formula and the fast Fourier transform. Application to the numerical solution of an integral equation is studied. Numerical experiments are carried out to demonstrate the effectiveness of the proposed method.

The numerical contour integral method with hyperbolic contour is exploited to solve space-fractional diffusion equations. By making use of the Toeplitz-like structure of spatial discretized matrices and the relevant properties, the regions that the spectra of resulting matrices lie in are derived. The resolvent norms of the resulting matrices are also shown to be bounded outside of the regions. Suitable parameters in the hyperbolic contour are selected based on these regions to solve the fractional diffusion equations. Numerical experiments are provided to demonstrate the efficiency of our contour integral methods.