A real symmetric matrix of order n has a full set of orthogonal eigenvectors. The most used approach to compute the spectrum of such matrices reduces first the dense symmetric matrix into a symmetric structured one, i.e., tridiagonal matrices or semiseparable matrices. This step is accomplished in O(n 3) operations. Once the latter symmetric structured matrix is available, its spectrum is compu...

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 A : V → V which satisfy the following two properties: (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 wi...

A family of non-Hermitian but ${\cal PT}-$symmetric $2J$ by toy-model tridiagonal-matrix Hamiltonians $H^{(2J)}=H^{(2J)}(t)$ with $J=K+M=1,2,\ldots$ and $t<J^2$ is studied, for which a real $2K$ tridiagonal-submatrix component $C(t)$ the Hamiltonian assumed coupled to its other two complex Hermitian $M$ components $A(t)$ $B(t)$. By construction, (i) all submatrices get decoupled at $t=t_M=M\,(2...

This work essentially consists in inverting an exact, explicit, and original way the pentadiagonal Toeplitz matrix or tridiagonal block resulting from discretization of two-dimensional Laplace operator. method is algorithm facilitating resolution a large number problems governed by PDEs involving Laplacian two dimensions. It guarantees high precision efficiency solving various differential equa...

We propose efficient implementations of Seymour and Thomas algorithm which, given a planar graph and an integer β, decides whether the graph has the branchwidth at least β. The computational results of our implementations show that the branchwidth of a planar graph can be computed in a practical time and memory space for some instances of size about one hundred thousand edges. Previous studies ...

An algorithm for reducing a nonsymmetric matrix to tridiagonal form as a rst step toward nding its eigenvalues is described. The algorithm uses a variation of threshold pivoting, where at each step, the pivot is chosen to minimize the maximum entry in the transformation matrix that reduces the next column and row of the matrix. Situations are given where the tridiagonalization process breaks do...

Standard algorithms for computing the inverse of a tridiagonal matrix (or more generally, any Hines matrix) compute the entire inverse, which is not sparse. For some problems, only the elements of the inverse at locations corresponding to nonzero elements in the original matrix are required. We present an algorithm that efficiently computes only these elements in O(n) time and memory. This algo...

The application of the finite difference method to approximate the solution of an indefinite elliptic problem produces a linear system whose coefficient matrix is block tridiagonal and symmetric indefinite. Such a linear system can be solved efficiently by a conjugate residual method, particularly when combined with a good preconditioner. We show that specific incomplete block factorization exi...