Hi, everyone,
Do you know some good MATLAB subroutines (direct solvers) for solving nonsymmetric Toeplitz linear systems ? I note that most direct solvers are only suitable for symmetric Toeplitz matrix, but recently, during the implementation of my numerical simulations, I need to solve a series of Nonsymmetric Toeplitz linear systems, i.e., A*x_i = b_i. i = 1:s. the size of A is from 8 to 256. I feel that this size is not large, so I do not want to consider the iterative solvers, such as gmres with circulant preconditioners. I think that if the subroutines can be used like the following form, it will be great:
>> y = nonsymToeplitz(c,r,x) % c is the first column of Toeplitz matrix A, and r is the first row of A, x is the rhs, i.e., y = A^{-1} x;
If you know some information for this, please tell me.
@Yue Qiu, More precisely, It means that I want to find a direct solvers, (a MATLAB-like codes or a function), I can all it when I solve the series of Nonsymmetric Toeplitz linear systems. As you know, for Toeplitz problem, we almost do not need to store the whole matrix (i.e. store O(n^2) non-zeros entries) , we only need to store the elements of the first column and first row, (about O(n) non-zeros entries). In other words, I want to use a function (such as named "nonsymToeplitz.m"), which can be used as the following,
Input: elements of first column $c$ and first row $r$, s RHSs (b_1, b_2, ..., b_s)
Output: s solutions (x_i, i = 1:s)
for i = 1: s
x(:,i) = nonsymToeplitz(c, r, b(:,i));
end
Of course, we can also reformulate this problem as a matrix equation AX = B; B = [b_1, b_2,.., b_s]; A is the Toeplitz matrix. Then X is the solution.
I know Prof. Ming Gu recently worked out some solvers based on HSS/SSS matrix, but he do not give me his reply about this problem.
Netlib has a MATLAB toolbox (it is called "smt") for solving structured matrices, such as circulant and Toeplitz matrices. The toolbox is available at http://www.netlib.org/numeralgo/, file na30.zip.
LU decomposition gives a quick method for solving a Toeplitz system - You can apply stable Bareiss algorithm in Maple. Please see the link below.
http://www.maplesoft.com/support/help/Maple/view.aspx?path=LinearAlgebra/Generic/BareissAlgorithm
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