Zhonggang Zeng

Computing the multiplicity structure in solving polynomial systems

This paper presents algorithms for computing the multiplicity structure of a zero to a polynomial system. The zero can be exact or approximate with the system being intrinsic or empirical. As an application, the dual space theory and methodology are utilized to analyze deflation methods in solving polynomial systems, to establish tighter deflation bound, and to derive special case algorithms.

Computing multiple roots of inexact polynomials

We present a combination of two algorithms that accurately calculate multiple roots of general polynomials. Algorithm I transforms the singular root-finding into a regular nonlinear least squares problem on a pejorative manifold, and it calculates multiple roots simultaneously from a given multiplicity structure and initial root approximations. To fulfill the input requirement of Algorithm I, we develop a numerical GCD-finder containing a successive singular value updating and an iterative GCD refinement as the main engine of Algorithm II that calculates the multiplicity structure and the initial root approximation. While limitations exist in certain situations, the combined method calculates multiple roots with high accuracy and consistency in practice without using multiprecision arithmetic, even if the coefficients are inexact. This is perhaps the first blackbox-type root-finder with such capabilities. To measure the sensitivity of the multiple roots, a structure-preserving condition number is proposed and error bounds are established. According to our computational experiments and error analysis, a polynomial being ill-conditioned in the conventional sense can be well conditioned with the multiplicity structure being preserved, and its multiple roots can be computed with high accuracy.

The approximate GCD of inexact polynomials

This paper presents an algorithm and its implementation for computing the approximate GCD (greatest common divisor) of multivariate polynomials whose coefficients may be inexact. The method and the companion software appear to be the first practical package with such capabilities. The most significant features of the algorithm are its robustness and accuracy as demonstrated in computational experiments. In addition, two variations of a squarefree factorization method for multivariate polynomials are proposed as an application of the GCD algorithm.

A Rank-Revealing Method with Updating, Downdating, and Applications

A new rank-revealing method is proposed. For a given matrix and a threshold for near-zero singular values, by employing a globally convergent iterative scheme as well as a deflation technique the method calculates approximate singular values below the threshold one by one and returns the approximate rank of the matrix along with an orthonormal basis for the approximate null space. When a row or column is inserted or deleted, algorithms for updating/downdating the approximate rank and null space are straightforward, stable, and efficient. Numerical results exhibiting the advantages of our code over existing packages based on two-sided orthogonal rank-revealing decompositions are presented. Also presented are applications of the new algorithm in numerical computation of the polynomial GCD as well as identification of nonisolated zeros of polynomial systems.

Multiple zeros of nonlinear systems

As an attempt to bridge between numerical analysis and algebraic geometry, this paper formulates the multiplicity for the general nonlinear system at an isolated zero, presents an algorithm for computing the multiplicity strucuture, proposes a depth-deflation method for accurate computation of multiple zeros, and introduces the basic algebraic theory of the multiplicity. Furthermore, this paper elaborates and proves some fundamental theorems of the multiplicity, including local finiteness, consistency, perturbation invarance, and depth-deflatability. The proposed algorithms can accurately compute the multiplicity and the multiple zeros using floating point arithmetic even if the nonlinear system is perturbed.

The quasi-Laguerre iteration

The quasi-Laguerre iteration has been successfully established, by the same authors, in the spirit of Laguerres iteration for solving the eigenvalues of symmetric tridiagonal matrices. The improvement in efficiency over Laguerres iteration is drastic. This paper supplements the theoretical background of this new iteration, including the proofs of the convergence properties.

A scalable eigenvalue solver for symmetric tridiagonal matrices

Abstract Both massively parallel computers and clusters of workstations are considered promising platforms for numerical scientific computing. This paper describes the first distributed-memory implementation of the split-merge algorithm, an eigenvalue solver for symmetric tridiagonal matrices that uses Laguerres iteration and exploits the separation property in order to create independent subtasks. Implementations of the split-merge algorithm on both an nCUBE-2 hypercube and a cluster of Sun Spare-10 workstations are described, with emphasis on load balancing, communication overhead, and interaction with other user processes. A performance study demonstrates the advantage of the new algorithm over a parallelization of the well-known bisection algorithm. A comparison of the performance of the nCUBE-2 and cluster implementations supports the claim that workstation clusters offer a cost-effective alternative to massively parallel computers for certain scientific applications.

ApaTools: a software toolbox for approximate polynomial algebra

Approximate polynomial algebra becomes an emerging area of study in recent years with a broad spectrum of applications. In this paper, we present a software toolbox ApaTools for approximate polynomial algebra. This package includes Maple and Matlab functions implementing advanced numerical algorithms for practical applications, as well as basic utility routines that can be used as building blocks for developing other numerical and symbolic methods in computational algebra.

A Rank-Revealing Method with Updating, Downdating, and Applications. Part II

As one of the basic problems in matrix computation, rank-revealing arises in a wide variety of applications in scientific computing. Although the singular value decomposition is the standard rank-revealing method, it is costly in both computing time and storage when the rank or the nullity is low, and it is inefficient in updating and downdating when rows and columns are inserted or deleted. Consequently, alternative methods are in demand in those situations. Following up on a recent rank-revealing algorithm by Li and Zeng for the low nullity case, we present a new rank-revealing algorithm for low rank matrices with efficient and straightforward updating/downdating capabilities. The method has been implemented in Matlab, and the numerical results show that the new algorithm appears to be efficient and robust.

The Homotopy Continuation Algorithm for the Real Nonsymmetric Eigenproblem: Further Development and Implementation

New development and implementation of a homotopy continuation algorithm for the nonsymmetric eigenvalue problem are presented, along with thorough investigation on the advantages and limitations of our code. Numerical results on substantial varieties of matrices arising in applications indicate excellent efficiency and stability of our algorithm and a drastic improvement over previous implementations.