Let H be an n × n unitary right Hessenberg matrix with positive subdiagonal elements. Using what we call the Schur parameterization of H, we show how one step of the shifted QR algorithm for H can be carried out in O(n) arithmetic operations. Coupled with the shift strategy of Eberlein and Huang [3], this will permit computation of the spectrum of H, to machine precision, in O(n2) operations. One potential application is the computation of Gauss-Szegö quadrature formulas [12], given the associated Schur parameters [7]. The weights can also be computed, by direct analogy with [6].

This paper shows that for unitary Hessenberg matrices the QR algorithm, with (an exceptional initial-value modification of) the Wilkinson shift, gives global convergence; moreover, the asymptotic rate of convergence is at least cubic, higher than that which can be shown to be quadratic only for Hermitian tridiagonal matrices, under no further assumption. A general mixed shift strategy with global convergence and cubic rates is also presented.

In applying the QR algorithm to compute the eigenvalues of a unitary Hessenberg matrix, a projected Wilkinson shift of unit modulus is proposed and proved to give global convergence with (at least) a quadratic asymptotic rate for the QR iteration. Experimental testing demonstrates that the unimodular shift produces more efficient numerical convergence.

The computation of zeros of polynomials is a classical computational problem. This paper presents two new zerofinders that are based on the observation that, after a suitable change of variable, any polynomial can be considered a member of a family of Szegő polynomials. Numerical experiments indicate that these methods generally give higher accuracy than computing the eigenvalues of the companion matrix associated with the polynomial.

Recently Laurie presented a new algorithm for the computation of (2n+1)-point Gauss-Kronrod quadrature rules with real nodes and positive weights. This algorithm first determines a symmetric tridiagonal matrix of order 2n + 1 from certain mixed moments, and then computes a partial spectral factorization. We describe a new algorithm that does not require the entries of the tridiagonal matrix to be determined, and thereby avoids computations that can be sensitive to perturbations. Our algorithm uses the consolidation phase of a divide-and-conquer algorithm for the symmetric tridiagonal eigenproblem. We also discuss how the algorithm can be applied to compute Kronrod extensions of Gauss-Radau and Gauss-Lobatto quadrature rules. Throughout the paper we emphasize how the structure of the algorithm makes efficient implementation on parallel computers possible. Numerical examples illustrate the performance of the algorithm.

As noted in the references the unitary Hessenberg QR algorithm is fundamental for statistical signal analysis, as is the closely related inverse algorithm (iuhqr). These algorithms are analogous with algorithms, tqr and itqr, for real symmetric tridiagonal matrices. No claim of numerical stability for uhqr was made when it was introduced in 1986. Indeed, an open problem was to make uhqr perform as well as tqr. Here we introduce a device which appears, on the basis of a large number of experiments, to solve the problem. However, given the history of tqr, we must say that this fact awaits proof.

Historically, the algorithm for completely solving the symmetric tridiagonal eigenvalue problem is the TQR algorithm. Rational variants of TQR were used when only the complete spectrum of eigenvalues was desired since they have the advantage of avoiding square roots. Several historical variants of TQR use fewer operations than the one used in LAPACK; however, these variants have known accuracy problems. In this paper a new variant of TQR is developed which is similar to parts of the variant proposed by Sack (1972). This variant has fewer operations than the one in LAPACK, which results in faster timings, and it avoids the accuracy problems of the other historical variants. In addition, it is shown how the eigenvectors can be recovered from the rational algorithms.

We consider the problem of reconstructing Jacobi matrices and real symmetric arrow matrices from two eigenpairs. Algorithms for solving these inverse problems are presented. We show that there are reasonable conditions under which this reconstruction is always possible. Moreover, it is seen that in certain cases reconstruction can proceed with little or no cancellation. The algorithm is particularly elegant for the tridiagonal matrix associated with a bidiagonal singular value decomposition.

\Ve consider a standard matrix flow on the set of unitary upper Hessenberg ma.trices ·with nonnegative subdia.gonal clements. The Schur parametrization of this set of ma.trices leads to ordinary differenlial equations for the >veights and the parameters lhal are analogous with the Toda. How as identilied with a How on Jacobi ma.trices. \Ve de rive explicit differential equations for the fluw on the Schur par;uneters of orthogonal Hessenberg nrntrices. \Ve also outline an efficient procedure for computing the solution of Jacobi fluws and Schur fluws.

We first review the basic relations between the regular formal orthogonal polynomials (FOPs) for a sequence of moments (Markov parameters), the nonsingular leading principal submatrices of the moment matrix M (which is an infinite Hankel matrix), the distinct entries on the main diagonal of the Padé table for the symbol of M (which is the generating function or z-transform of the moments), the corresponding continued fraction (which is a J-fraction or a P-fraction), and the Euclidean algorithm for power series in ζ-1, which in the generic case is seen to reduce to the Chebyshev algorithm. The underlying recurrences are a special case of the general recurrences that are the basis of the Cabay-Meleshko algorithm which, in contrast to the aforementioned tools, is (weakly) stable. While, in the Toeplitz solver terminology, the Cabay-Meleshko algorithm is of Levinson type, we also outline the corresponding O(N 2) Schur-type algorithm and a related O(N log2 N) algorithm. Finally, we sketch three look-ahead strategies of which two are applicable to the O(N log2 N) algorithm also.

Linear combinations of polynomials that are orthogonal with respect to an inner product defined on (part of) the real axis are commonly evaluated by the Clenshaw algorithm. We present an analogous algorithm for the evaluation of a linear combination Σnj=0αjφj of polynomials φj that are orthogonal with respect to an inner product defined on (part of) the unit circle. The φj are known as Szegő polynomials, and find applications, e.g., in signal processing. We also discuss how to express Σnj=0αjφj as a linear combination of monomials.

We show that the well-known Levinson algorithm for computing the inverse Cholesky factorization of positive definite Toeplitz matrices can be viewed as a special case of a more general process. The latter process provides a very efficient implementation of the Arnoldi process when the underlying operator is isometric. This is analogous with the case of Hermitian operators where the Hessenberg matrix becomes tridiagonal and results in the Hermitian Lanczos process. We investigate the structure of the Hessenberg matrices in the isometric case and show that simple modifications of them move all their eigenvalues to the unit circle. These eigenvalues are then interpreted as abscissas for analogs of Gaussian quadrature, now on the unit circle instead of the real line. The trapezoidal rule appears as the analog of the Gauss-Legendre formula.

Many algorithms for polynomial least-squares approximation of a real-valued function on a real interval determine polynomials that are orthogonal with respect to a suitable inner product defined on this interval. Analogously, it is convenient to compute Szego polynomials, i.e., polynomials that are orthogonal with respect to an inner product on the unit circle, when approximating a complex-valued function on the unit circle in the least-squares sense. It may also be appropriate to determine Szego polynomials in algorithms for least-squares approximation of real-valued periodic functions by trigonometric polynomials. This paper is concerned with Szego polynomials that are defined by a discrete inner product on the unit circle. We present a scheme for downdating the Szego polynomials and given least-squares approximant when a node is deleted from the inner product. Our scheme uses the QR algorithm for unitary upper Hessenberg matrices. We describe a data-fitting application that illustrates how our scheme can be combined with the fast-Fourier-transform algorithm when the given nodes are not equidistant. Application to sliding windows is discussed also.

Abstract : It is known that small relative perturbations in the entries of a bidiagonal matrix only cause small relative perturbations in its singular values, independent of the values of the matrix entries. In this paper we show that a matrix has this property if and only if its associated bipartite graph is acyclic. We also show how to compute the singular values of such a matrix to high relative accuracy. The same algorithm can compute eigenvalues of symmetric acyclic matrices with tiny component-wise relative backward error. This class includes tridragonal matfices, arrow matrices, and exponentially many others.

Schemes for the solution of linear initial or boundary value problems on a hypercube were developed by Katti and Neta [1] and tested and improved by Lustman, Neta and Katti [2]. Among other procedures for parallel computers, fully implicit Runge-Kutta methods were discussed by Jackson and Norsett [3] and Lie [4].Here, we develop a method based on extrapolation to the limit, which is useful even for nonlinear problems. Numerical experiments show excellent accuracy when low order schemes are combined with polynomial extrapolation.