A multishift QR iteration without computation of the shifts

Abstract

matrix computations, eigenvalues, QR algorithmEach iteration of the multishift QR algorithm of Bai and Demmel requires the computation of a "shift vector" defined by m shifts of the origin of the spectrum, which control the convergence of the process. A common choice of shifts consists of the eigenvalues of the trailing principal submatrix of order m, and current practice includes the computation of these eigenvalues in the determination of the shift vector. In this paper, we describe an algorithm based on the evaluation of the characteristic polynomial of a Hessenberg matrix, which directly produces the shift vector without computing eigenvalues. This algorithm is stable, more accurate, faster, and simpler than the current alternative. It also allows for a consistent shift strategy with dynamic adjustment of the number of shifts. ABSTRACTEach iteration of the multishift QR algorithm of Bai and Demmel requires the computation of a "shift vector" defined by m shifts of the origin of the spectrum, which control the convergence of the process. A common choice of shifts consists of the eigenvalues of the trailing principal submatrix of order m, and current practice includes the computation of these eigenvalues in the determination of the shift vector. In this paper, we describe an algorithm based on the evaluation of the characteristic polynomial of a Hessenberg matrix, which directly produces the shift vector without computing eigenvalues. This algorithm is stable, more accurate, faster, and simpler than the current alternative. It also allows for a consistent shift strategy with dynamic adjustment of the number of shifts.

Rational QR-iteration without inversion

“…In Equation (1), the matrix can be computed in two different ways, the first manner is referred to as the explicit QR-method, whereas the formula Q H AQ can lead to an implicit approach (see, e.g., [3,4]). The new algorithm presented in this manuscript admits both types of methods.…”

“…An implicit QR-method performed onto the structured matrix A, has only few building blocks (see also [3,4,22]):…”

Hyman’s method revisited

“…Here we point out again that Hyman's evaluation process is numerically stable (see Wilkinson [37,38], Parlett [27,28], Higham [12], Dubrulle, Golub [7]). …”

“…Dubrulle and Golub [7] suggested an algorithm based on Hyman's method, which directly produces the shift vector of the multishift QR iteration without computing the eigenvalues. They also confirm the stability and accuracy of Hyman's method in agreement with Wilkinson [37].…”

Computation of a Few Small Eigenvalues of a Large Matrix with Application to Liquid Crystal Modeling

“…The IRBL method can be regarded as a curtailed block QR algorithm for the symmetric eigenvalue problem; see, e.g., Bai and Demmel [4], and Dubrulle and Golub [10] for discussions of the latter. Similarly as in the block QR algorithm, the choice of shifts is important for the IRBL method.…”