Th.Lunde Johnsen

QR-factorization of partitioned matrices: Solution of large systems of linear equations with non-definite coefficient matrices

Abstract Methods of decomposing a partitioned rectangular matrix A into a product of an orthogonal matrix Q and an upper triangular matrix R are presented. The procedures can be applied to decompose a matrix stored in rectangular blocks on a random access second level storage.

On systems of linear equations of the form AtAx = b error analysis and certain consequences for structural applications

Abstract An error analysis of the solution of linear systems of the form A t Ax = b is presented. The analysis is based on the backward technique of J.H. Wilkinson applied to the QR method on A , which involves the use of elementary Hermitians. The accuracy of the method is compared to that of the classical Cholesky approach on the formed product K = A t A . Although the initial analysis shows no improved error bound for the QR approach compared to the Cholesky method, a more refined treatment reveals that the relative solution error using QR methods is usually considerably smaller than that using the Cholesky procedure, particularly for commonly occurring right-hand side vectors b . Practical error estimates are provided, both based on the spectral conditioning number measure as well as on the first correction in a special iterative improvement technique. As systems of equations similar to the above arise in the new natural factor technique of J.H. Argyris and O.E. Bronlund for matrix structural analysis, some discussion of structural applications in both statics and dynamics is presented. It is shown that the only case where the accuracy of the QR method is not considerably better than that for Cholesky reduction corresponds to an unusual physical situation. In addition, a natural factor formulation for the mixed method in finite element analysis is presented, extending the first writers previous work on this topic. Finally, several numerical examples are given, based on randomly generated data as well as on data from real structures, to demonstrate the validity and applicability of the method and the corresponding error analysis.

On numerical error in the finite element method

Abstract Various sources of errors, physical and numerical, in the finite element method are analysed. A new type of iterative improvement is introduced where the residual is calculated in single precision. The iteration scheme is analysed with respect to round-off errors and found to give significant improvement over existing direct approaches.

On the natural factor in nonlinear analysis

Abstract The natural factor approach as formulated for the linear displacement method is extended to nonlinear analysis. It is shown that the sparse population of the matrix factor may be efficiently utilized in the decomposition technique. Furthermore, it is demonstrated that in ill-conditioned cases the method performs numerically better than the standard Cholesky approach.

Some Hypermatrix Algorithms in Linear Algebra

The efficient solution of large linear matrix problems plays a central role in both linear and nonlinear structural analysis. Accordingly, a substantial effort has been allocated for the design of computer software in order to handle standard tasks like the solution of linear equations or eigenreduction.

Note on symmetric decomposition of some special symmetric matrices

Abstract The existence of the Cholesky decomposition of a class of symmetric non-definite matrices is proved, and an error bound for this decomposition is discussed. For a special class of such matrices two symmetric decompositions based on the QR factorization are suggested, and these are in general more accurate than the Cholesky decomposition and only slightly more time-consuming.

A new iterative solution for structures and continua with very stiff or rigid parts

Abstract Structures with widely different orders of stiffness or nearly incompressible continua are not uncommon in structural analysis and give considerable computational difficulties when using the matrix displacement method on a 32 or 36 bit word computer. This paper presents three approaches to overcome these problems. The first one is based on the idea of the natural method. The second one makes exclusively use of the widely used cartesian notation. Finally the third one considers structures with only a few rigid members. One central aim of this paper is to implement the developed theoretical tools in standard finite element packages. The paper has illustrative hand and computer solutions and the appendix presents the underlying rigorous mathematical proofs.

Galerkin-obrechkoff methods and hyperbolic initial boundary value problems with damping

The numerical solution of linear inhomogeneous but time-homogeneous hyperbolic initial value problems with damping is considered. A special class of high order Galerkin-Obrechkoff methods is investigated, and L2 error bounds are derived.

Hypermatrix generalization of the Jacobi- and Eberlein-method for computing eigenvalues and eigenvectors of hermitian or non-hermitian matrices

Abstract A generalization of the Jacobi and Eberlein methods for partitioned matrices is described. Whereas the generalization of the Jacobi method is fairly straightforward and obvious, the generalization of the Eberlein method involves the use of more advanced techniques. We thereby obtain an algorithm with a very modest core store requirement that uses the backing store in a highly efficient manner. An application to non-modal damping of vibration problems is discussed.

On accurate stress calculation in static and dynamic problems using the natural factor approach

Abstract The natural factor approach was introduced during the last few years as a tool for more accurate computations of displacements [1]. This paper presents an economic iterative procedure for the precise determination of stresses in numerically critical large scale problems. In addition, it furnishes a corresponding error analysis. Furthermore, we discuss the stress (and displacement) computations for unsupported structures under self-equilibrating loads as they may arise in dynamic analysis. Finally, some examples are given.