Regularizing preconditioners based on fit techniques in the image reconstruction problem

Regularizing preconditioners for the approximate solution by gradient-type methods of image restoration problems with two-level band Toeplitz structure, are examined. For problems having separable and positive definite matrices, the fit preconditioner, introduced in [6], has been shown to be effective in conjunction with CG. The cost of this preconditioner is of O(n^2) operations per iteration, where n^2 is the pixels number of the image, whereas the cost of the circulant preconditioners commonly used for this type of problems is of O(n^2 log n) operations per iteration. In this paper the extension of the fit preconditioner to more general cases is proposed: namely the nonseparable positive definite case and the symmetric indefinite case are treated. The major difficulty encountered in this extension concerns the factorization phase, where, unlike the separable case, a further approximation is required. Various approximate factorizations are proposed. The preconditioners thus obtained have still a cost of O(n^2) operations per iteration. A large numerical experimentation compares these preconditioners with the circulant Chan preconditioner, showing often better performances at a lower cost

Factorization of analytic functions by means of Koenig's theorem and Toeplitz computations

By providing a matrix version of Koenig's theorem we reduce the problem of evaluating the coefficients of a monic factor r(z) of degree h of a power series f(z) to that of approximating the first h entries in the first column of the inverse of an n x n Toeplitz matrix in block Hessenberg form for sufficiently large values of n. This matrix is reduced to a band matrix if f(z) is a polynomial. We prove that the factorization problem can be also reduced to solving a matrix equation phi (X) = 0 for an h x h matrix X, where phi is a matrix power series whose coefficients are Toeplitz matrices. The function phi is reduced to a matrix polynomial of degree 2 if f(z) is a polynomial of degree N and h greater than or equal to N/2. These reductions allow us to devise a suitable algorithm, based on cyclic reduction and on the concept of displacement rank, for generating a sequence of vectors v((2j)) that quadratically converges to the vector v having as components the coefficients of the factor r(z). In the case of a polynomial f(z) of degree N, the cost of computing the entries of v((2j)) given v((2j-1)) is O(N log N + theta (N)) arithmetic operations, where theta (N) = O(N log(2) N) is the cost of solving an N x N Toeplitz-like system. In the case of analytic functions the cost depends on the numerical degree of the power series involved in the computation. From the numerical experiments performed with several test polynomials and power series, the algorithm has shown good numerical properties and promises to be a good candidate for implementing polynomial root-finders based on recursive splitting strategies. Applications to solving spectral factorization problems and Markov chains are also shown

Factorization of Analytic Functions by means of Koenig's Theorem and Toeplitz Computations

By providing a matrix version of Koenig's theorem we reduce the problem of evaluating the coefficients of a monic factor $r(z)$ of degree $h$ of a power series $f(z)$ to that of approximating the first $h$ entries in the first row of the inverse of an infinite Toeplitz matrix in block Hessenberg form. This matrix is reduced to a band matrix if $f(z)$ is a polynomial. We devise a suitable algorithm, based on cyclic reduction and on the concept of displacement rank, for generating a sequence of vectors \B v^{(2^j)} that quadratically converges to the vector \B v having as components the coefficients of the factor $r(z)$. In the case of a polynomial $f(z)$ of degree $N$, the cost of computing the entries of \B v^{(2^j)} given \B v^{(2^{j-1})} is $O(N\log N+\theta(N))$, where $\theta(N)$ is the cost of solving an $N\times N$ Toeplitz-like system. In the case of analytic functions the cost depends on the numerical degree of the power series involved in the computation. The algorithm is strictly related to the Graeffe method for lifting the roots of a polynomial. From the numerical experiments performed with several test polynomials and power series, the algorithm has shown good numerical properties and promises to be a good candidate for implementing polynomial rootfinders based on recursive splittin