Can I convert a system of equations with a Toeplitz matrix to a system with a circulant matrix?

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Erik S.
Erik S. 2015-2-8
评论： Royi Avital 2020-1-11
Reading the following post it seems that it is possible to zero pad a Toeplitz matrix to a circulant.
http://scicomp.stackexchange.com/questions/7224/solving-linear-systems-by-fft?noredirect=1#comment27569_7224
How can I zero pad matrix A to a circulant matrix? When solving a system of equations how would I zero pad the right hand side?
Thank you! Erik

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Matt J
Matt J 2015-2-8
编辑：Matt J 2015-2-8
The idea of embedding a Toeplitz system of equations into a circulant system is to reduce the solution process to a series of FFT/IFFT operations. But I don't think that could really be possible as the StackExchange link claims. If it were that easy, there would be no need for the Levinson algorithm, which was expressly designed for solving Toeplitz systems.
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Chris Turnes
Chris Turnes 2015-7-21
编辑：Chris Turnes 2015-7-21
You actually can (sort of) do this, although it's not as simple as just inverting a circulant matrix. The idea is to add extra rows to make the system the first few columns of a circulant matrix, and then introduce a new unknown that you solve for (and an additional circulant submatrix into your system). If you're curious, the technique is described in the paper "Solving Toeplitz systems after extension and transformation."
It actually yields a more asymptotically-efficient algorithm for inversion (O(n log^2 n) instead of O(n^2)), but it has a higher overhead constant so it tends to really only be preferable for large systems.
EDIT: Just to clarify, that link will say the algorithm is O(n^2), which is true for the algorithm as described in that paper. But Van Barel gave an O(n log^2 n) method of solving the same problem with the same general approach in "A Stabilized Superfast Solver for Nonsymmetric Toeplitz Systems".

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