Robert C. Ward

Numerical Computation of the Matrix Exponential with Accuracy Estimate

This paper presents and analyzes an algorithm for computing the exponential of an arbitrary

Computing the singular value decompostion of a product of two matrices

n \times n

Sparse Orthogonal Schemes for Structural Optimization Using the Force Method

matrix. Diagonal Pade table approximations are used in conjunction with several techniques for reducing the norm of the matrix. An important feature of the algorithm is that an estimate for the minimum number of digits accurate in the norm of the computed exponential matrix is returned to the user. In obtaining this estimate, several interesting results concerning rounding errors and Pade approximations are presented.

The Combination Shift

An algorithm is developed for computing the singular value decomposition of a product of two general matrices without explicitly forming the product. The algorithm is based on an earlier Jacobi-like method due to Kogbetliantz and uses plane rotations applied to the two matrices separately. A triangular variant of the basic algorithm is developed that reduces the amount of work required.

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Historically there are two principal methods of matrix structural analysis, the displacement (or stiffness) method and the force (or flexibility) method. In recent times the force method has been used relatively little because the displacement method has been deemed easier to implement on digital computers, especially for large sparse systems. The force method has theoretical advantages, however, for multiple redesign problems or nonlinear elastic analysis because it allows the solution of modified problems without restarting the computation from the beginning. In this paper we give an implementation of the force method which is numerically stable and preserves sparsity. Although it is motivated by earlier elimination schemes, in our approach each of the two main phases of the force method is carried out using orthogonal factorizaton techniques recently developed for linear least squares problems.

Algorithm

An extension of the

Balancing the Generalized Eigenvalue Problem

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An algorithm to compute a sparse basis of the null space

algorithm, called the combination shift

A L P S: Matrices with nonpositive off-diagonal entries

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An extension of the divide-and-conquer method for a class of symmetric block-tridiagonal eigenproblems

algorithm, is presented for solving the generalized matrix eigenvalue problem