In this paper, methods which are guaranteed to converge to the minimum or maximum zeros are developed. The proposed methods are based on two approaches for generating fixed point functions of rational and radical forms. These include well known methods such as the Newton, and Halley methods as special cases, in addition to the rth root methods. Although these methods are only derived for polynomials, they are also applicable to some types of entire functions of finite number of zeros. Additionally, the proposed approach is useful to generate algorithms with any given rate of convergence.

Root iterations and the computation of minimum and maximum zeros of polynomials

High order root-finding algorithms are constructed based on some canonical conditions and a generalized Taylor series. The convergence order is automatically determined using these canonical conditions. The proposed approaches resulted in deriving methods of any desired order including the Newton, Halley, and Ostrowski iterations. It is also shown that, when zeros are simple, higher order methods may be obtained by applying lower order methods such as Newton Iteration to new functions which have same zeros as the original function. These functions are constructed so that the first few derivatives beyond the first vanish. Several examples are given for constructing methods of higher order for computing the zeros of the entire function sin(z).

On the Derivation of Higher Order Root-Finding Methods

The problem of computing the minimum eigenvector (the Pisarenko vector) of a covariance matrix is of considerable interest in signal processing and computational linear algebra. A unified algorithm for computing both the minimum and maximum eigenpairs by simply choosing the proper initial condition is proposed. In particular, the extremum eigenpairs are computed using higher order convergent methods which include the Newton method, the Halley method and root iterations. The advantage of these methods is that one can control where the methods converge by choosing a proper initial condition. Some of the methods are implemented using QR factorization to avoid matrix inversion. By appropriately choosing the initial condition, this approach can also be used to compute the largest eigenpair and thus can be applied for computing the minor and major subspaces of symmetric or Hermitian matrices. Procedures such as the double step Newton method for accelerating the developed methods are considered. Several randomly generated test problems are used to evaluate the performance of the methods.

An algorithm for computing the Pisarenko vector

In this paper we show that the Pisarenko vector for harmonic retrieval problems can be obtained without explicit eigendecomposition: The smallest eigenvalue and corresponding eigenvector of a covariance matrix are computed using higher order convergent methods which include the Newton method as special case. An implementation that relies on QR factorization and less on matrix inversion is presented. This approach can also be used to compute the largest eigenpair by appropriately choosing the initial condition. Additionally, an approach is proposed to accelerate the developed methods considerably by using the double step Newton method. Several randomly generated test problems are used to evaluate the performance and the computational cost of the methods.

New methods for computing the Pisarenko vector

There are numerous applications in sciences, engineering and mathematics that give rise to problems involving the computation of orthogonal projections onto selective invariant subspaces of matrices. Conventional algorithms for subspace estimation based upon eigenvalue decomposition (EVD) or singular value decomposition (SVD) are, however, both expensive to compute, and difficult to make recursive or implement in parallel. In contrast, algorithms based on the QR factorization have established pipelinable architectures. In this paper, we introduce novel matrix-inverse free algorithms for block matrix decomposition. They involve a combination of Newton's method and the QR factorization. Some of these methods can be shown to have cubic or quadratic convergence rates, while others can be of any desirable convergent rate.

Matrix inverse-free algorithms for the general eigenvalue problem

The derivation and implementation of many algorithms in signal/image processing and control involve some form of polynomial root-finding and/or matrix eigendecomposition. In this paper, higher order fixed point functions in rational and/or radical forms are developed. This set of iterations can be considered as extensions of known methods such as the Newton, Lagurre and Halley methods and can be applied to compute all zeros of a polynomial as well as all eigenvalues of a complex matrix. One of the main features of the proposed algorithms is that they could have any predetermined rate of convergence regardless of the multiplicity of the zeros or eigenvalues. Additionally, eigenvalues and eigenvectors are computed using fast matrix inverse free algorithms which are based on the QR factorization.

M.A. Hasan

Papers

Rational and radical fixed point functions for the eigenvalue problem and polynomials