Showing papers on "Singular value decomposition published in 1971"
TL;DR: An algorithm for determining the triangular decomposition H R*DR of a Hankel matrix H using O(n') operations is derived and can be used to compute the three-term recurrence relation for orthogonal polynomials from a moment matrix.
Abstract: An algorithm for determining the triangular decomposition H R*DR of a Hankel matrix H using O(n') operations is derived. The derivation is based on the Lanczos algorithm and the relation between orthogonalization of vectors and the triangular decomposition of moment matrices. The algorithm can be used to compute the three-term recurrence relation for orthogonal polynomials from a moment matrix.
40 citations
TL;DR: A shifted QR Algebra for Hermitian Materials and its applications to Numerical Algebra and Mathematica.
19 citations
01 Jan 1971-Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences
1 citations
01 Apr 1971
TL;DR: The Penrose-Moore pseudo inverse as mentioned in this paper extends the notion of pseudo inverse for square nonsingular matrices to the class of all rectangular matrices and develops essential properties with applications to the theory of equations, constrained and unconstrained least squares, nonnegative definiteness, perturbation theory and the singular decomposition theorem.
Abstract: : The Penrose-Moore Pseudo Inverse extends the notion of 'inverse' for square nonsingular matrices to the class of all rectangular matrices. In the report the author develops the essential properties with applications to the theory of equations, constrained and unconstrained least squares, nonnegative definiteness, perturbation theory and the singular decomposition theorem. Various computational algorithms are developed and additional results are derived which apply to various statistical topics, such as the General Linear Hypothesis (BLUE'S, Orthogonal Designs, tests and confidence sets) Conditional Expectations for vector normal variables and Kalman Filtering. (Author)
1 citations
01 Jan 1971
TL;DR: This paper describes a scheme for iteratively improving the decomposition of a given matrix A into the product of two matrices B and C, which is most practically applicable in the LU and QR decompositions of a matrix.
Abstract: This paper describes a scheme for iteratively improving the decomposition of a given matrix A into the product of two matrices B and C Although B and C may be completely general matrices, the scheme is most practically applicable in the LU and QR decompositions of a matrix The proposed scheme is first shown to improve perturbed L and U factors of a given matrix A Next, it is shown that this process will allow for the solution of a badly conditioned system of linear equations when other methods, such as direct triangular decomposition, fail miserably Finally even when a method such as direct triangular decomposition works, but iterative improvement of the initial solution must be made, the proposed scheme may achieve comparable results and prove to be more economical under certain conditions