An algorithm for reducing a nonsymmetric matrix to tridiagonal form as a first step toward finding 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 down, and two recovery methods are presented for these situations. Although no existing tridiagonalization algorithm is guaranteed to succeed, this algorithm is found to be very robust and fast in practice. A gradual loss of similarity is also observed as the order of the matrix increases.

Reduction of a general matrix to tridiagonal form

The fastest parallel algorithm for a problem may be significantly less stable numerically than the fastest serial algorithm. We illustrate this phenomenon by a series of examples drawn from numerical linear algebra. We also show how some of these instabilities may be mitigated by better floating point arithmetic.

Trading Off Parallelism and Numerical Stability

A homotopy algorithm for the solution of general modified algebraic Riccati equations is presented. A continuous homotopy approach with discrete corrections at each homotopy step is used. The algorithm requires solution of several perturbed Lyapunov equations during each iteration; the rapid solution of sets of perturbed Lyapunov equations is discussed. >

Homotopy methods for the solution of general modified algebraic Riccati equations

In this paper two algorithms for the solution of large-order $(100 \leq n \leq 1000)$ Lyapunov equations $AX + XA^T + Q = 0$ are presented. First, a parallel version of the Hammarling algorithm for the solution of Lyapunov equations where the coefficient matrix A is large and dense is presented. Then a novel iterative parallel algorithm, called full-rank perturbed iteration (FRPI), is presented for the solution of Lyapunov equations where the matrix A is large and banded.

Parallel solution of large Lyapunov equations

The Lyapunov matrix equation AX+XA^@?=B is N-stable when all eigenvalues of the real nxn matrix A have positive real part. When the real nxn matrix B is spd the solution X is spd. It is of low rank when B=CC^@? where C is nxr with [email protected]?n. An efficient algorithm has been found for solving the low-rank equation. This algorithm is a result of over fifty years of research starting with seemingly unrelated development of alternating direction implicit (ADI) iterative solution of elliptical systems. The low rank algorithm may be applied to a full rank equation if one can approximate the right-hand side by a sum of low rank matrices. This may be attempted with the Lanczos algorithm.

Trail to a Lyapunov equation solver

This report presents several ideas for improving the stability of Gaussian reduction of an arbitrary real matrix to tridiagonal form. First, we analyze conditions under which reduction algorithms break down or become unstable. Second, we discuss how methods of threshold pivoting decrease the probability of these conditions occurring. Finally, we present new methods for recovering from breakdown when it does occur. The class of matrices that can be successfully reduced is significantly broadened by these new recovery algorithms. 16 refs., 4 figs.

Stabilized Gaussian reduction of an arbitrary matrix to tridiagonal form

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Stabilized Gaussian reduction of an arbitrary matrix to tridiagonal form