No abstract

No 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.