Abstract
Based on the circulant-and-skew-circulant representation of Toeplitz matrix inversion and the divide-and-conquer technique, a fast numerical method is developed for solving N-by-N block lower triangular Toeplitz with M-by-M dense Toeplitz blocks system with \(\mathcal {O}(MN\log N(\log N+\log M))\) complexity and \(\mathcal {O}(NM)\) storage. Moreover, the method is employed for solving the linear system that arises from compact finite difference scheme for time-space fractional diffusion equations with significant speedup. Numerical examples are given to show the efficiency of the proposed method.
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This research was partially supported by the University of Macau [MYRG2016-00202-FST] and the Science and Technology Development Fund, Macao S.A.R (FDCT) [115/2013/A3].
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Huang, YC., Lei, SL. A fast numerical method for block lower triangular Toeplitz with dense Toeplitz blocks system with applications to time-space fractional diffusion equations. Numer Algor 76, 605–616 (2017). https://doi.org/10.1007/s11075-017-0272-6
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DOI: https://doi.org/10.1007/s11075-017-0272-6
Keywords
- Block lower triangular Toeplitz matrix with dense Toeplitz blocks
- Circulant-and-skew-circulant representation of Toeplitz matrix inversion
- Divide-and-conquer strategy
- Fast Fourier transform
- Time-space fractional partial differential equations