Chasing Bulges or Rotations? A Metamorphosis of the QR-Algorithm

Abstract

The QR-algorithm is a renowned method for computing all eigenvalues of an arbitrary matrix. A preliminary unitary similarity transformation to Hessenberg form is indispensable for keeping the computational complexity of the subsequent QRsteps under control. When restraining computing time is the vital issue, we observe that the prominent role played by the Hessenberg matrix is sufficient but perhaps not necessary to fulfill this goal. In this paper, a whole new family of matrices, sharing the major qualities of Hessenberg matrices, will be put forward. This gives rise to the development of innovative implicit QR-type algorithms, pursuing rotations instead of bulges. The key idea is to benefit from the QR-factorization of the matrices involved. The prescribed order of rotations in the decomposition of the Q-factor uniquely characterizes several matrix types such as Hessenberg, inverse Hessenberg, and CMV matrices. Loosening the fixed ordering of these rotations provides us the class of matrices under consideration. Establishing a new implicit QRtype algorithm for these matrices requires a generalization of diverse well-established concepts. We consider the preliminary unitary similarity transformation, a proof of uniqueness of this reduction, an extension of the CMV -decomposition to a double Hessenberg factorization, and an explicit and implicit QRtype algorithm. A detailed complexity analysis illustrates the competitiveness of the novel method with the traditional Hessenberg approach. The numerical experiments show comparable accuracy for a wide variety of matrix types, but disclose an intriguing difference between the average number of iterations before deflation can be applied.

A Generalization of the Multishift QR Algorithm

“…In [13] it was shown that every complex n × n matrix is unitarily similar to a matrix in condensed form (2.1). The position vector p can be chosen arbitrarily.…”

“…Rational Krylov spaces associated with a condensed form. It was shown in [13] that the unitary similarity transformation that maps a matrix to a given condensed form (2.1) is essentially uniquely determined. The argument depends on a connection with a certain sequence of rational Krylov subspaces associated with the condensed form.…”

“…Generalizations of both the explicit and implicit QR algorithms were given in [13]. There it was shown that if the matrix A has condensed form with position vector p, then the matrix resulting from one iteration has a different position vectorp derived from p by moving each entry up by one.…”

“…The last two steps are shown in Figure 5.7 withp 19 = andp 20 = r. The position vector of the final configuration isp = [ , r, , r, , , , r, r, r, r, , , r, r, r, r, , , r]. In case m = 1 (single shift), the algorithm in this section reduces to that of [13]. The length of the limb becomes zero, and all operations involving the limb can be omitted.…”

“…Before we can apply Francis's algorithm to a given matrix, we must transform the matrix to a condensed form, usually upper Hessenberg form. In [13] a large family of condensed forms was introduced, of which upper Hessenberg form is just one special case. For each of those condensed forms, both an explicit and implicit QRlike algorithm were introduced.…”