James R. Bunch

Rank-one modification of the symmetric eigenproblem

SummaryAn algorithm is presented for computing the eigensystem of the rank-one modification of a symmetric matrix with known eigensystem. The explicit computation of the updated eigenvectors and the treatment of multiple eigenvalues are discussed. The sensitivity of the computed eigenvectors to errors in the updated eigenvalues is shown by a perturbation analysis.

Direct Methods for Solving Symmetric Indefinite Systems of Linear Equations

Methods for solving symmetric indefinite systems are surveyed including a new one which is stable and almost as fast as the Cholesky method.

Some stable methods for calculating inertia and solving symmetric linear systems

Several decompositions ofsymmetric matrices for calculating inertia and solving systems of linear equations are discussed. New partial pivoting strategies for decomposing symmetric matrices are introduced and analyzed.

Updating the singular value decomposition

SummaryLetA be anm×n matrix with known singular value decomposition. The computation of the singular value decomposition of a matrixà is considered, whereà is obtained by appending a row or a column toA whenm≧n or by deleting a row or a column fromA whenm>n. An algorithm is also presented for solving the updated least squares problemà y−b≈, obtained from the least squares problemAx−b by appending an equation, deleting an equation, appending an unknown, or deleting an unknown.

The weak and strong stability of algorithms in numerical linear algebra

The stability of algorithms in numerical linear algebra is discussed. The concept of stability is extended to notions of weak stability and strong stability. Justifications are given for these extensions, and the implications of error analyses in terms of these definitions are discussed. The concept of weak stability helps to clarify some of the controversy which has arisen concerning the stability of algorithms for Toeplitz systems.

Decomposition of a symmetric matrix

SummaryAn algorithm is presented to compute a triangular factorization and the inertia of a symmetric matrix. The algorithm is stable even when the matrix is not positive definite and is as fast as Cholesky. Programs for solving associated systems of linear equations are included.

The strong stability of algorithms for solving symmetric linear systems

An algorithm for solving linear equations is stable on the class of nonsingular symmetric matrices or on the class of symmetric positive definite matrices if the computed solution solves a system that is near the original problem. Here it is shown that any stable algorithm is also strongly stable on the same matrix class if the computed solution solves a nearby problem that is also symmetric or symmetric positive definite.

Collinearity and Total Least Squares

The least squares (LS) and total least squares (TLS) methods are commonly used to solve the overdetermined system of equations

A computational method for the indefinite quadratic programming problem

Ax \approx b

A Note on the Stable Decomposition of Skew-Symmetric Matrices

. The main objective of this paper is to examine TLS when