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

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

Answer 1

[Matt J]

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.

[Erik S.]

Ok, I see. Thanks Matt!

[Erik S.]

I think it is possible to convert the Toeplitz matrix to a ciculant. However, I don't know how to deal with the right hand side of a system Ax=b.

The following lines converts my Toeplitz to circulant

T = toeplitz([1,2,3,4,5]);

C = gallery('circul',c);

c = [ T(:,1); fliplr( T(1,:) )'];

[Matt J]

However, I don't know how to deal with the right hand side of a system Ax=b.

Because there is no way. That's why the Levinson algorithm is required.

You can do approximate things and get semi-accurate results, I'm sure. Extrapolate the right hand side smoothly down to zero somehow.

[Chris Turnes]

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".

[Royi Avital]

@Chris, do you know an extension of the Levinson Recursion to cases where the Matrix isn't rectangular?