In the current study we investigate the two-stage iterative method for solving linear systems. Our new results shows which splitting generates convergence fast in iterative methods. Finally, we solve the Poisson-Block tridiagonal matrix from Poisson's equation which arises in mechanical engineering and theoretical physics. Numerical computations are presented based on a particular linear system...

Three algorithms providing rigourous bounds for the eigenvalues of a real matrix are presented. The first is an implementation of the bisection algorithm for a symmetric tridiagonal matrix using IEEE floating-point arithmetic. The two others use interval arithmetic with directed rounding and are deduced from the Jacobi method for a symmetric matrix and the Jacobi-like method of Eberlein for an ...

Let K denote a field and let V denote a vector space over K with finite positive dimension. We consider an ordered pair of linear transformations A : V → V and B : V → V which satisfy both (i), (ii) below. (i) There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing B is diagonal. (ii) There exists a basis for V with respe...

Abstract As an extension of Sylvester’s matrix, a tridiagonal matrix is investigated by determining both left and right eigenvectors. Orthogonality relations between eigenvectors are derived. Two determinants the matrices constructed evaluated in closed form.

In this paper we consider the application of polynomial root-finding methods to the solution of the tridiagonal matrix eigenproblem. All considered solvers are based on evaluating the Newton correction. We show that the use of scaled three-term recurrence relations complemented with error free transformations yields some compensated schemes which significantly improve the accuracy of computed r...

It is well known how any symmetric matrix can be reduced by an orthogonal similarity transformation into tridiagonal form. Once the tridiagonal matrix has been computed, several algorithms can be used to compute either the whole spectrum or part of it. In this paper, we propose an algorithm to reduce any symmetric matrix into a similar semiseparable one of semiseparability rank 1, by orthogonal...

Some properties of near-Toeplitz tridiagonal matrices with specific perturbations in the first and last main diagonal entries are considered. Applying the relation between the determinant and Chebyshev polynomial of the second kind, we first give the explicit expressions of determinant and characteristic polynomial, then eigenvalues are shown by finding the roots of the characteristic polynomia...

In this paper, we propose the use of complex-orthogonal transformations for nding the eigenvalues of a complex symmetric matrix. Using these special transformations can signiicantly reduce computational costs because the tridiagonal structure of a complex symmetric matrix is maintained.

Some properties of near-Toeplitz tridiagonal matrices with specific perturbations in the first and last main diagonal entries are considered. Applying the relation between the determinant and Chebyshev polynomial of the second kind, we first give the explicit expressions of determinant and characteristic polynomial, then eigenvalues are shown by finding the roots of the characteristic polynomia...

Let K denote a field, and let V denote a vector space over K with finite positive dimension. We consider a pair of linear transformations A : V → V and A : V → V which satisfy the following two conditions: (i) There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A is diagonal. (ii) There exists a basis for V with respe...