Matrix computations

Matrix computations can be very time consuming, particularly for matrices of large size. By performing the calculations in the most efficient way it is often possible to save a lot of time. Here, we provide some guidelines as to how certain matrix computations can most expediently be carried out. For definiteness, we consider the problem of computing the molar diffusion fluxes 7 from the equation 

Equation A.6.1 arises when the Maxwell-Stefan equations are solved for the case of steady-state, one-dimensional mass transfer, as discussed in Chapter 8. The matrices [A ] and [O] are as defined in Chapter 8, is the molar density of the mixture and a scalar, and (Ax) is a column matrix of mole fraction differences. All matrices in Eq. A.6.1 are of order n — 1 where n is the number of components in the mixture. For the purposes of this discussion we shall assume that the matrices [/ ] and [O] have already been calculated. The matrix function [0][exp[ I ] - [7]] denoted by [2], can be computed using Sylvester s expansion formula (see, however, below) so the immediate problem is the calculation of the column matrix (7) from 

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Gluob, G.H, and Van Loan, C.F., 1984. Matrix. Computations, Johns Hopkins University Press, Baltimore. 

Write the secular matrix, compute the eigenvalues, and draw the energy level diagram for fulvcnc. 

J. S. Quartemian, The Matrix Computer Netirorks and Conferencing Systems Digital Press, Bedford, Mass., 1989. 

Jennings, A., and J. J. McKeown. Matrix Computations for Engineers and Scientists, Wiley, New York (1992). 

The solution of the system may then be found by elimination or matrix methods if a solution exists (see Matrix Algebra and Matrix Computations ). 

See the section entitled Matrix Algebra and Matrix Computation. ... 

Not listed are two forthcoming books by J. H. Wilkinson that, however, are to be highly recommended as treatments of the techniques for matrix computations. 

Several software packages contain simple command lines for performing matrix computations directly and thus are conveniently capable of computing the correlation coefficient, for example as in r2 (Equation 59-8). 

The spin Hamiltonian also forms the theoretical basis for describing the temporal response of the spin system to a pulse sequence and/or mechanical manipulations of the sample via calculations of the evolution of the density matrix. Computer... 

The strategies developed by Crowe are described further in the next chapter, where other matrix computations for variable classification are analyzed. 

This approach can, in many instances, be extended even to cases where the basis is comprised of physicochemical, topological, or other such parameters. The similarity matrix is replaced in these cases by the correlation matrix computed with respect to the basis set of parameters (vide supra). 

G. H. Golub and C. F. Van Loan, Matrix Computations Johns Hopkins University Press, Baltimore, MD, 1983-... 

A theoretical approach is applied to elucidate the molecular conformations, associated flexibility, and dynamics of polylp-hydroxybenzoic acid) esters, pHB. Properties such as the radius of gyration and persistence length which are characteristic for the stiffness of a macromolecule are calculated on the basis of two different theoretical methods (a) Molecular dynamics and (b) the RIS model augmented by the more recent scheme for the matrix computations. The analysis of the results obtained by the latter method reflects a strong dependence on the choice of the structural parameters of the system. 

Some of the golden rules of modern matrix computations are ... 

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