Delosme, J.-M

The Cholesky factorization, Schur complements, correlation coefficients, angles between vectors, and the QR factorization by J. -M Delosme ( Book )
1 edition published in 1988 in English and held by 4 libraries worldwide

The Cholesky Factorization, Schur Complements, Correlation Coefficients, Angles between Vectors, and the QR Factorization ( Book )
1 edition published in 1988 in English and held by 2 libraries worldwide
An m x m symmetric nonnegative definite matrix Sigma has Cholesky factorization Sigma = u-transpose u. By carrying out the factorization in a particular way for positive definite Sigma, the Schur complements of all the leading principal submatrices of Sigma are produced, as well as their Cholesky factors. It is shown how the same can be done for generalized Schur complements when Sigma is singular. When Sigma is the population covariance matrix of a multivariate random distribution, partial covariances and correlations can be defined in terms of the elements of such Schur complements. It follows that these can be produced efficiently and reliably from the Cholesky factorization. When n x m A is given and Sigma = A-transpose A, the Cholesky factor U may be found directly from the QR factorization A = Q1U, Q1-transpose Q1 = I, and this is preferable in many numerical computations. This QR factorization, or the modified Gram-Schmidt orthogonalization, produces projections of later columns of A onto spaces orthogonal to earlier columns. It is shown how the cosines of the angles between such projected vectors can be found using the elements of U. These cosines produced from A turn out to be the previously mentioned partial correlation coefficients produced from Sigma, when Sigma = A-transpose A. When A is obtained from observations of random variables, these are the sample correlation coefficients. It is shown how such correlation coefficients can be efficiently obtained when observations are added or deleted. This corresponds to altering all of A in a certain simple way, and adding or deleting rows.

ParLance: a para-functional programming environment for parallel and distributed computing by Yale University ( Book )
1 edition published in 1987 in English and held by 2 libraries worldwide

Efficient Systolic Arrays for the Solution of Toeplitz Systems: An Illustration of a Methodology for the Construction of Systolic Architectures in VLSI (Very Large Systems Integration) ( Book )
1 edition published in 1985 in English and held by 1 library worldwide
Advanced high resolution methods for real-time signal processing require fast multiprocessor architectures, realized as Very Large Scale Integrated systems, for their implementation. A class of parallel structures, 'systolic arrays', fulfills the demands of signal processing and at the same time conforms to the limitations of VLSI technology. The design of efficient systems of systolic arrays is an iterative process in which the information gathered from mapping algorithms onto hardware is influential in the development of the algorithms. Presently, the absence of mapping tools makes it extremely difficult to find good systolic implementations for many important problems. Often, one cannot do better than implementing structurally simple but numerically inferior algorithms, which is undesirable for most signal processing tasks. A methodology is proposed that automates the mapping of recurrence equations to processor arrays. Two aspects distinguish our methodology from extant work: (1) complex coupled systems of recurrence equations can be systematically treated and (2) the resulting systolic systems are optimal. The methodology takes as input recurrence equations describing the algorithm, along with certain desirable hardware features.

Efficient Parallel Solution of Linear Systems with Hyperbolic Rotations ( Book )
1 edition published in 1984 in English and held by 1 library worldwide
An algorithm based on hyperbolic rotations is presented for the solution of linear systems of equations, Ax = b, with symmetric positive definite coefficient matrix A. Forward elimination and backsubstitution are replaced by matrix vector multiplications, rendering the method most amenable to implementation on a variety of parallel and vector machines. The stability behaviour compares favourably with that of the best, known methods. The method can be simplified and formulated without square roots if A is also Toeplitz; a corresponding systolic architecture (in very large scale integrated circuits) for the resulting recurrence equations is more efficient than previously proposed pipelined Toeplitz system solvers. The hardware count becomes independent of the matrix size if its inverse is banded.

Transformation of broadcasts into propagations in systolic algorithms by Yale University ( Book )
1 edition published in 1989 in English and held by 1 library worldwide

Optimization of processor count for systolic arrays by Yale University ( Book )
1 edition published in 1989 in English and held by 1 library worldwide

Optimization of computation time for systolic arrays by Yale University ( Book )
1 edition published in 1989 in English and held by 1 library worldwide

Transformation of broadcasting into pipelining by Yale University ( Book )
1 edition published in 1987 in English and held by 1 library worldwide

Space and time bases for the sequentialization of systolic array designs by Yale University ( Book )
1 edition published in 1991 in English and held by 1 library worldwide