Abstract: In this paper, we consider whose invereses are tridiagonal Z-matrices Based on a characterization of symmetric tridiagonal matirices by Gantmacher and Kein, we show that a matrix is the inverse of a tridiagonal Z-matrix if and only if up to a positive scaling of the rows,it is the Hadamard product of a so called weak type D matrix and a flipped weak type D matrix whose parameters satisfy certiain quadratic conditions we predict from these parameters to which class of Z-matices the inverse belings to In particular, we give a characterization of inverse tridiagonal M-matrices Moreover ,we charactetrize inverese of diagonal M-matrices that saftisfy certain row sum ceriteria. This leads to the cyclopses that are matrices constructed from type D and flipped type D matrices .we establish some properties of the cyclopses and provide explicit formulae for the entries of the inverse of a nonsingular cyclopses. we also shoe that the cyclopses are the only generalized ultrametric matrices whose inverses are tridiagonal

Abstract: We present one more algorithm to compute the condition number (for inversion) of an n X n tridiagonal matrix J in O(n) time. Previous O(n) algorithms for this task given by Higham [SIAM J. Sci. Statist. Comput., 7 (1986), pp. 150--165] are based on the tempting compact representation of the upper (lower) triangle of J-1 as the upper (lower) triangle of a rank-one matrix. However they suffer from severe overflow and underflow problems, especially on diagonally dominant matrices. Our new algorithm avoids these problems and is as efficient as the earlier algorithms.

Abstract: This paper shows how the symmetric eigenproblem, which is the computationally most demanding part of numerous scientific and industrial applications, can be solved much more efficiently than by using algorithms currently implemented in Lapack routines.

Abstract: In this paper we present three different pivoting strategies for solving general tridiagonal systems of linear equations. The first strategy resembles the classical method of Gaussian elimination with no pivoting and is stable provided a simple and easily checkable condition is met. In the second strategy, the growth of the elements is monitored so as to ensure backward stability in most cases. Finally, the third strategy also uses the right‐hand side vector to make pivoting decisions and is proved to be unconditionally backward stable.

Abstract: It is shown that maximal growth for Gaussian elimination with partial pivoting as measured in the 2-norm is achieved by orthogonal matrices. A precise bound on that growth is given.

Abstract: In this paper we present a new divide-and-conquer parallel algorithm to compute the eigenvalues of symmetric tridiagonal matrices. This algorithm combines the use of rank-one modifications in the division phase and the application of the Laguerre iteration in the updating phase. Our method is compared with one based on the same scheme but using rank-two modifications. A thorough experimental analysis in the Cray T3D parallel computer has been carried out. Special emphasis has been put on analysing the influence of the deflation phenomena on the computational cost of this kind of algorithm. Experimental results show that an adequate exploitation of the inherent parallelism in the divide-and-conquer scheme produces very efficient parallel algorithms. The obtained speedups clearly improve the best sequential algorithm, including the standard implementation of QR iteration in LAPACK.

Abstract: It is proved that any pseudo-Hermitian (or real pseudosymmetric) matrix can be brought to band form by a unitary (or orthogonal) similarity transformation. This band form is tridiagonal if the matrix is of the type (n - 1, 1) or (1, n - 1). A pseudounitary analogue of the QR algorithm for tridiagonal pseudo-Hermitian matrices is discussed.

Abstract: The authors use a recent c haracterization of the numerical range to obtain several conditions for a symmetric tridiagonal matrix with positive diagonal entries to be positive deenite.

Abstract: A new preconditioning strategy for symmetric positive definite banded circulant and Toeplitz systems is described. The optimal tridiagonal preconditioner for tridiagonal circulant systems is modified and applied to both circulant and Toeplitz banded systems. The strategy is extended to block tridiagonal systems. The parallelisation aspects of the PCG algorithm are discussed.

Abstract: Cyclic reduction is a powerful technique in numerical linear algebra for solving tridiagonal systems. It can be classified in the class of divide and conquer methods. Some of its surprising properties have been studied recently essentially in the case of diagonally dominant matrices. In this paper the case of Toeplitz matrices will be considered. A complete analysis of the properties of the cyclic reduction will be done by studing an associate system of nonlinear difference equations.

Abstract: Computing the eigenvalues and orthogonal eigenvectors of an $n\times n$ symmetric tridiagonal matrix is an important task that arises while solving any symmetric eigenproblem. All practical software requires $O(n\sp3)$ time to compute all the eigenvectors and ensure their orthogonality when eigenvalues are close. In the first part of this thesis we review earlier work and show how some existing implementations of inverse iteration can fail in surprising ways. The main contribution of this thesis is a new $O(n\sp2),$ easily parallelizable algorithm for solving the tridiagonal eigenproblem. Three main advances lead to our new algorithm. A tridiagonal matrix is traditionally represented by its diagonal and off-diagonal elements. Our most important advance is in recognizing that its bidiagonal factors are "better" for computational purposes. The use of bidiagonals enables us to invoke a relative criterion to judge when eigenvalues are "close". The second advance comes by using multiple bidiagonal factorizations in order to compute different eigenvectors independently of each other. Thirdly, we use carefully chosen dqds-like transformations as inner loops to compute eigenpairs that are highly accurate and "faithful" to the various bidiagonal representations. Orthogonality of the eigenvectors is a consequence of this accuracy. Only $O(n)$ work per eigenpair is needed by our new algorithm. Conventional wisdom is that there is usually a trade-off between speed and accuracy in numerical procedures, i.e., higher accuracy can be achieved only at the expense of greater computing time. An interesting aspect of our work is that increased accuracy in the eigenvalues and eigenvectors obviates the need for explicit orthogonalization and leads to greater speed. We present timing and accuracy results comparing a computer implementation of our new algorithm with four existing EISPACK and LAPACK software routines. Our test-bed contains a variety of tridiagonal matrices, some coming from quantum chemistry applications. The numerical results demonstrate the superiority of our new algorithm. For example, on a matrix of order 966 that occurs in the modeling of a biphenyl molecule our method is about 10 times faster than LAPACK's inverse iteration on a serial IBM RS/6000 processor and nearly 100 times faster on a 128 processor IBM SP2 parallel machine.

Abstract: We present an analytical solution to the generalized discrete Poisson equation, a matrix equation which has a tridiagonal matrix with fringes having an arbitrary value for the diagonal elements. The results are of relevance to a variety of physical problems, which require the numerical solution of the Poisson equation. As examples, the formula has been applied to the solution of the electrostatic problem of tunnelling junction arrays with two and three rows.