Numerical Experiments using UBASIC

ABSTRACT – The multi-shift QR algorithm for approximating the eigenvalues of a full matrix is known to have convergence problems if the number of shifts used in one iteration is large. The mechanism by which the values of the shifts are being transmitted from one bulge matrix to another has been discovered. In the presence of round-off errors, however, the values of the shifts are blurred in certain bulge matrices causing the QR algorithm to miss the true eigenvalues of the matrix. In this paper, we give the maximum number of shifts that can be used in one iteration to keep the values of the shifts from blurring. We use the UBASIC language, and specify the minimum level of precision that maintains well-focused shifts. KEY WORDS – eigenvalues, QR algorithm, Schur upper triangular form, bulge-chase technique, Hessenberg form, blurring of shifts 1. The Problem and its Background Many mathematical softwares approximate the eigenvalues of a square matrix using the QR algorithm. The QR algorithm is an iterative algorithm that approximates the Schur upper triangular form of a matrix, and the eigenvalues of the matrix emerge along the main diagonal of the Schur form. Provide