in this paper, the rayleigh's quotient and the inverse vector iteration method are presented. the latter approach helps to obtain the natural frequencies and mode shapes of a structure. inverse vector iteration method with shifting enables to determine the higher modes. some basic theorems of linear algebra are presented and extended to study the free vibration of structures. the variation...

This paper is devoted to the study of some preconditioners for the conjugate gradient algorithm used to solve large sparse linear and symmetric positive definite systems. The construction of a preconditioner based on the Gram–Schmidt orthogonalization process and the least squares method is presented. Some results on the condition number of the preconditioned system are provided. Finally, numer...

We discuss some features of the orthogonalization methods commonly applied to QSPR QSAR studies. We outline the well known multivariable linear regression analysis in vector form in order to compare mainly Randic and Gram-Schmidt orthogonalization procedures and also cast the basis for other approaches like Löwdin’s one. We expect that present review may become the starting point for future dev...

In this paper, we compare with the inverse iteration algorithms on PowerXCell 8i processor, which has been known as a heterogeneous environment. When some of all the eigenvalues are close together or there are clusters of eigenvalues, reorthogonalization must be adopted to all the eigenvectors associated with such eigenvalues. Reorthogonalization algorithms need a lot of computational cost. The...

Convergence acceleration by preconditioning is usually essential when solving the standard least squares problems by an iterative method. IMGS, is an incomplete modiied version of Gram-Schmidt orthogonalization to obtain an incomplete orthogonal factorization pre-conditioner M = R, where A = Q R + E is an approximation of a QR factorization, Q is an orthogonal matrix and R is upper triangular m...

The advent of supercomputers with hierarchical memory systems has imposed the use of block algorithms for the linear algebra algorithms. Although block algorithms may result in impressive improvements in performance, their numerical properties are quite different from their scalar counterpart and deserve an in-depth study. In this paper, the numerical stability ofblock Gram-Schmidt orthogonaliz...

It is well-known that orthogonalization of column vectors in a rectangular matrix B with respect to the bilinear form induced by a nonsingular symmetric indefinite matrix A can be seen as its factorization B = QR that is equivalent to the Cholesky-like factorization in the form BTAB = RTΩR, where Ω is some signature matrix. Under the assumption of nonzero principal minors of the matrix M = BTAB...