Algorithm 343: eigenvalues and eigenvectors of a real general matrix

Abstract

Purpose. This subroutine finds all the eigenvalues and eigenvectors of a real general matrix. The eigenvalues are computed by the Qt~ double-step method and the eigenvectors by inverse iteration.Method. Firstly the following preliminary modifications are carried out to improve the accuracy of the computed results. (i) The matrix is scaled by a sequence of similarity transformations so that the absolute sums of corresponding rows and columns are roughly equal. (ii) The scaled matrix is normalized so that the value of the Euclidean norm is equal to one.The main part of the process commences with the reduction of the matrix to an upper-Hessenberg form by means of similarity transformations (Householder's method). Then the QR doublestep iterative process is performed on the Hessenberg matrix until all elements of the subdiagonal that converge to zero are in modulus less than 2 -t H HIIE , where t is the number of significant digits in the mantissa of a binary floating-point number. The eigenvalues are then extracted from this reduced form.Inverse iteration is performed on the upper-Hessenberg matrix until the absolute value of the largest component of the righthand side vector is greater than the bound 2t/(100 N), where N is the order of the matrix. Normally after this bound is achieved, one step more is performed to obtain the computed eigenvector, but at each step the residuals are computed, and if the residuals of one particular step are greater in absolute value than the residuals of the previous step, then the vector of the previous step is accepted as the computed eigenvector.

Analytical Design of Multispectral Sensors

“…The optimal sensor design problem may be solved on a digital computer using empirical data taken by field measurements Some approximations must be made in order to take into consideration some practical constraints First the response functions arc not available as continuous functions but aro obtained in the field by sampling the spectrum with an instrument that uses very narrow spectral windows Secondly, the parameters of the process arc not known a pnon, hence, it is necessary to estimate the mean and covanance functions using a representative sample from the ensemble Finally, because the data will be stored and processed digitally it is necessary to qu.mti7e the amplitude of the response at cacn of the spectral sample points Each of these constraints potentially can contribute to the representation error It has been shown that with reasonable care in selecting a sufficiently high spectral sampling rate, a large enough sample from the ensemble, and a large number of quantisation intervals that the contribution of these factors to the representation error is small [16] The integral equation (4) becomes the matrix equation 183 <s>r = (6) where <$> is the mamx of eigenvectors, I is the diagonal matrix of eigenvalues, K is the covanance matrix, and U'ls the diagonal matrix of weight coefficients…”

Analysis of three‐phase packed‐bed reactors

“…The roots are equivalent to the eigenvalues of the companion matrix of Equation (47) (Ralston and Wilf, 1960). Such eigenvalues were computed by the QR method described by Gard and Brebner (1968). The constants a1 .…”

Symbolic Solution of Finite-Difference Equations

“…The current capabilities of the program which performs this step are summarized in Table I. It should be remarked that, when dealing with homogeneous equations whose coefficients are numeric, it is possible to use strictly numeric routines such as [5], [7], [12] to replace the time-consuming symbolic computations of steps 2, 4, and 5.…”