Based on the results obtained in [Hucht, J. Phys. A: Math. Theor. 50, 065201 (2017)], we show that partition function of anisotropic square lattice Ising model $L \times M$ rectangle, with open boundary conditions both directions, is given by determinant a $M/2 M/2$ Hankel matrix, equivalently can be written as Pfaffian skew-symmetric $M Toeplitz matrix. The $M-1$ independent matrix elements ma...

In this paper, we present several high performance variants of the classical Schur algorithm to factor various Toeplitz matrices. For positive definite block Toeplitz matrices, we show how hyperbolic Householder transformations may be blocked to yield a block Schur algorithm. This algorithm uses BLAS3 primitives and makes efficient use of a memory hierarchy. We present three algorithms for inde...

With the change of variables U = U*q, substitution into (21) results in UCU*q = Ud. Consequently, the solution vector U of (20) can be obtained from the solution vector q of (22) or q can be obtained from U. Note that if a system of real equations is Toeplitz-plus-Hankel (T + H) , where T i s symmetric Toeplitz and H i s skew-centrosym-metric Hankel, then the equations may be transformed into H...

A formula for the distance of a Toeplitz matrix to the subspace of {e}-circulant matrices is presented, and applications of {e}-circulant matrices to preconditioning of linear systems of equations with a Toeplitz matrix are discussed. Copyright c © 2006 John Wiley & Sons, Ltd. key words: Toeplitz matrix, circulant matrix, {e}-circulant, matrix nearness problem, distance to normality, preconditi...

A fast approximate inversion algorithm is proposed for two-level Toeplitz matrices (block Toeplitz matrices with Toeplitz blocks). It applies to matrices that can be sufficiently accurately approximated by matrices of low Kronecker rank and involves a new class of tensor-displacement-rank structured (TDS) matrices. The complexity depends on the prescribed accuracy and typically is o(n) for matr...

In this paper, we survey some of latest developments in using preconditioned conjugate gradient methods for solving mildly ill-conditioned Toeplitz systems where the condition numbers of the systems grow like O(n) for some > 0. This corresponds to Toeplitz matrices generated by functions having zeros of order. Because of the ill-conditioning, the number of iterations required for convergence in...

For a Toeplitz or Toeplitz-like matrix T, we define a preconditioning applied to the symmetrized matrix TnT, which decreases the condition number compared to the one of TnT and even the one of T. This enables us to accelerate the conjugate gradient algorithm for solving Toepiltz and Toeplitz-like linear systems, thus extending the previous results of [1], restricted to the Hermitian positive de...

Computations with Toeplitz and Toeplitz-like matrices are fundamental for many areas of algebraic and numerical computing. The list of computational problems reducible to Toeplitz and Toeplitz-like computations includes, in particular, the evaluation of the greatest common divisor (gcd), the least common multiple (lcm), and the resultant of two polynomials, computing Padé approximation and the ...

We extend the theory of Multigrid methods developed for PDE, Toeplitz and related matrices to the Block Toeplitz case. Prolongations and restrictions are defined depending on the zeroes of the generating function of the Block Toeplitz matrix. On numerical examples we compare different choices for prolongations and restrictions.

We compute the Dixmier trace of pseudo-Toeplitz operators on the Fock space. As an application we find a formula for the Dixmier trace of the product of commutators of Toeplitz operators on the Hardy and weighted Bergman spaces on the unit ball of C. This generalizes an earlier work of Helton-Howe for the usual trace of the anti-symmetrization of Toeplitz operators.