The QR algorithm for unitary Hessenberg matrices

AbstractLet 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]

An Analogue for Szegő Polynomials of the Clenshaw Algorithm

NSF grant DMS 9002884National Research Council fellowshi

Inverse problems for orthogonal matrices, Toda flows, and signal processing

http://archive.org/details/inverseproblemsf00faybN

Downdating of Szego polynomials and data fitting applications

Many algorithms for polynomial least squares approximation of 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 computer 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 IIessenberg 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 alsohttp://archive.org/details/downdatingofszeg00gragN

Positive definite Toeplitz matrices, the Arnoldi process for isometric operators, and Gaussian quadrature on the unit circle

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.National Science FoundationNational Science Foundatio

Fortran subroutines for the evaluation of the confluent hypergeometric functions

In this report we list the Fortran subroutines for evaluating the confluent hypergeomet r i c functions M(a,b;x) and U(a,b;x) . These subroutines use the stable recurrence relations given e.g. in Wimp.Naval Postgraduate School, Monterey, CA.http://archive.org/details/fortransubroutin14gragO&MN, Direct FundingApproved for public release; distribution is unlimited

On singular values of Hankel operators of finite rank

Let H be a Hankel operator defined by its symbol rho = pi X Chi where is a monic polynomial of degree n and pi is a polynomial of degree less than n. Then H has rank n. We derive a generalized Takagi singular value problem defined by two n x n matrices, such that its n generalized Takagi singular values are the positive singular values of H. If rho is real, then the generalized Takagi singular value problem reduces to a generalized symmetric eigenvalue problem. The computations can be carried out so that the Lanczos method applied to the latter problem requires only 0(n log n) arithmetic operations for each iteration. If pi and chi are given in power form, then the elements of all n x n matrices required can be determined in 0(sq.n) arithmetic operationsNaval Postgraduate School and the National Science Foundation , Washington D.C.http://archive.org/details/onsingularvalues00gragO&MN, Direct fundin

FORTRAN subroutines for updating the QR decomposition

We present FORTRAN subroutines that update the QR decomposition in a numerically stable manner when A is modified by a matrix of rank one, or when a row or a column is inserted or deleted. These subroutines are modifications of the Algol procedures in Daniel et al. 5. We also present a subroutine that the elements in the lower right corner of R will generally be small if the columns of A are nearly linearly dependent. This subroutine is an implementation of the rank revealing QR decomposition scheme recently proposed by Chan (3). The subroutines have been written to perform well on a vector computer. Algorithms Additional Key Words and Phrases: QR decomposition, updating, subset selection. Computer programsPrepared for: Naval Postgraduate School and the National Science Foundation, Washington D.C.http://archive.org/details/fortransubroutin00gragO&MN, Direct FundingN

A divide and conquer method for unitary and orthogonal eigenproblems

Let H epsilon C be a unitary upper Hessenberg matrix whose eigenvalues, and possibly also eigenvectors, are to be determined. We describe how this eigenproblem can be solved by a divide and conquer method, in which the matrix H is split into two smaller unitary right Hessenberg matrices H1 and H2 by a rank-one modification of H. The eigenproblems for H1 and H2 can be solved independently, and the solutions of these smaller eigenproblems define a rational function, whose zeros on the unit circle are the eigenvalues of H. The eigenvectors of H can be determined from the eigenvalues of H and the eigenvectors of H1 and H2. The outlined splitting of unitary upper Hessenberg matrices into smaller such matrices is carried out recursively. This gives rise to a divide and conquer method that is suitable for implementation on a parallel computer. When H epsilon R sub nxn is orthogonal, the divide and conquer scheme simplifies and is described separately. Our interest in the orthogonal eigenproblem stems from applications in signal processing. Numerical examples for the orthogonal eigenproblem conclude the paperThis report was prepared in conjunction with research conducted for the National Science Foundation and for the Naval Postgraduate School Research Council and funded by the Naval Postgraduate School Research Council.http://archive.org/details/divideconquermet00gragO&MN, Direct FundingApproved for public release; distribution is unlimited