Showing papers on "Triangular matrix published in 1970"

Eigenvectors of real and complex matrices byLR andQR triangularizations

Abstract: In a recent paper [4] the triangularization of complex Hessenberg matrices using the LR algorithm was described. Denoting the Hessenberg matrix by H and the final triangular matrix by T we have $${P^{ - 1}}HP = T$$ (1) , where P is the product of all the transformation matrices used in the execution of the LR algorithm. In practice H will almost invariably have been derived from a general complex matrix A using the procedure comhes [3] and hence for some nonsingular S we have $${P^{ - 1}}{S^{ - 1}}ASP = T$$ (2) .

Optimization of Redundancy Selection in the Finite-Element Force Method

Abstract: Techniques for the automatic selection of redundancies in the matrix force method are well established. However, optimization of the selection is an area which still requires consider- able attention. The criterion for optimum selection requires the incorporation of the rela- tive element stiffnesses by a transformation of the variables and a suitable choice of pivots in the reduction of the transformed equilibrium equations. Two forms of weighting and a refined method of pivot selection have been developed in this paper. The incorporation of rigid elements is briefly discussed. The results of a numerical comparison between the vari- ous forms of weighting and pivot selection, including previously published methods, are pre- sented for a series of frame and beam problems. HE finite-elemen t force method of redundant analysis became fully automated with the introduction of tech- niques for the automatic selection of redundancies. These procedures, namely the Rank Technique and the Structure Cutter, are presented in detail in Refs. 1-3. Optimum selection of redundancies 4 depends on the rela- tive stiffnesses of the individual structural elements. Meth- ods of accounting for the effect of relative stiffnesses are given by Denke 3 and Contini and Haggenmacher.5 This paper presents the method of Ref. 5 as well as a new method based on sounder theoretical principles. All approaches use some method of transforming the inde- pendent generalized force variables and, therefore, the coefficient matrix in the equilibrium equations. Denke used the diagonal coefficients of the assembled element stiffness matrix and Contini and Haggenmacher the square roots of those coefficients as scalar factors applied to the independent generalized force variables. The use of the square roots was prompted by considerations of dimensional consistency. To remove the approximation involved in neglecting the off-diagonal terms of the stiffness matrix, Robinson expressed the matrix as the product of a lower triangular matrix, ob- tained by means of a Cholesky decomposition, and its trans- pose. This technique retains the dimensional consistency of the square root approach and results in transformed vari- ables which are linear combinations of the original variables within each structural element. All these methods break down if some of the structural elements are rigid, because the assembled element flexibility matrix is singular. It is necessary, therefore, to assign suit- able finite stiffnesses to the rigid elements in a rational manner. Having transformed the equilibrium equations, a reduction is made using an elimination procedure such as the Jordanian. In such a procedure, the selection of the pivot, from the cur- rent coefficient matrix, critically determines the statically determinate substructure and, consequently, the redundan- cies. It is essential to a meaningful pivot choice that all coefficients in each row of the coefficient matrix be dimen-

Estimates of the roundoff error in the solution of a system of conditional equations

Abstract: : The paper compares the roundoff error in the least-squares solution of a system of conditional equations Ax=f by two different methods, using backward error analysis. The first one entails solving the normal equations (A sup T)Ax = (A sup T)f and the second is one proposed by Faddeev, Faddeeva, and Kublanovskaya in 1966. This latter method involves multiplying the system by othogonal matrices to transform the matrix A into upper triangular form.