Showing papers on "Tridiagonal matrix algorithm published in 1974"
TL;DR: In this article, the alternative of simply altering the pivot element is examined and the alteration, which amounts to a rank one modification of the matrix, is undone at a later stage by means of the well-known formula for the inverse of a modified matrix.
Abstract: The rounding-error analysis of Gaussian elimination shows that the method is stable only when the elements of the matrix do not grow excessively in the course of the reduction. Usually such growth is prevented by interchanging rows and columns of the matrix so that the pivot element is acceptably large. In this paper the alternative of simply altering the pivot element is examined. The alteration, which amounts to a rank one modification of the matrix, is undone at a later stage by means of the well-known formula for the inverse of a modified matrix. The technique should prove useful in applications in which the pivoting strategy has been fixed, say to preserve sparseness in the reduction.
51 citations
TL;DR: In this paper, the maximum values in modulus of the pivots p 3, p 4 and p 5 in Gaussian elimination with complete pivoting are 2 1 4 and 4, respectively.
18 citations
TL;DR: An algorithm combining Gaussian elimination with the modified Gram-Schmidt (MGS) procedure is given for solving the linear least squares problem.
Abstract: An algorithm combining Gaussian elimination with the modified Gram-Schmidt (MGS) procedure is given for solving the linear least squares problem. The method is based on the operational efficiency of Gaussian elimination for LU decompositions and the numerical stability of MGS for unitary decompositions and is designed for slightly overdetermined linear systems.
17 citations
TL;DR: In this article, two new proofs are given of the following characterization theorem of M. Fiedler: if Cn, n⩾2, be the class of all symmetric, real matrices A of order n with the property that rank (A + D) ⩾ n - 1 for any diagonal real matrix D, then there exists a permutation matrix P such that PAPT is tridiagonal and irreducible.
13 citations
TL;DR: In this article, an inertia theorem for tridiagonal matrices is proved and used to deduce a criterion of Wall which relates root location of polynomials to continued fractions.
13 citations
TL;DR: A graph-theoretic algorithm which examines an arbitrary n × n matrix and determines whether or not it can be permuted into tridiagonal form is given and gives the explicit tridiagon form.
Abstract: Tridiagonalizing a matrix by similarity transformations is an important computational tool in numerical linear algebra. Consider the class of sparse matrices which can be tridiagonalized using only row and corresponding column permutations. The advantages of using such a transformation include the absence of round-off errors and improved computation time when compared with standard transformations. A graph-theoretic algorithm which examines an arbitrary n × n matrix and determines whether or not it can be permuted into tridiagonal form is given. The algorithm requires no arithmetic while the number of comparisons, the number of assignments, and the number of increments are linear in n. This compares very favorably with standard transformation methods. If the matrix is permutable into tridiagonal form, the algorithm gives the explicit tridiagonal form. Otherwise, early rejection will occur.
10 citations
3 citations
1 Jan 1974
2 citations
TL;DR: Givens bisection method for the calculation of the eigenvalues of a symmetric tridiagonal matrix is extended to directly decomposable matrices in this article, which is given an easy, algebraic correct algorithm, which is free of divisions.
Abstract: Givens well known bisection method for the calculation of the eigenvalues af a symmetric tridiagonal matrix is extended to directly decomposable matrices. It is given an easy, algebraic correct algorithm, which is free of divisions.