Lihong Zhi

QR factoring to compute the GCD of univariate approximate polynomials

We present a stable and practical algorithm that uses QR factors of the Sylvester matrix to compute the greatest common divisor (GCD) of univariate approximate polynomials over /spl Ropf/[x] or /spl Copf/[x]. An approximate polynomial is a polynomial with coefficients that are not known with certainty. The algorithm of this paper improves over previously published algorithms by handling the case when common roots are near to or outside the unit circle, by splitting and reversal if necessary. The algorithm has been tested on thousands of examples, including pairs of polynomials of up to degree 1000, and is now distributed as the program QRGCD in the SNAP package of Maple 9.

Structured Low Rank Approximation of a Sylvester Matrix

The task of determining the approximate greatest common divisor (GCD) of univariate polynomials with inexact coefficients can be formulated as computing for a given Sylvester matrix a new Sylvester matrix of lower rank whose entries are near the corresponding entries of that input matrix. We solve the approximate GCD problem by a new method based on structured total least norm (STLN) algorithms, in our case for matrices with Sylvester structure. We present iterative algorithms that compute an approximate GCD and that can certify an approximate ∈-GCD when a tolerance ∈ is given on input. Each single iteration is carried out with a number of floating point operations that is of cubic order in the input degrees. We also demonstrate the practical performance of our algorithms on a diverse set of univariate pairs of polynomials.

Approximate greatest common divisors of several polynomials with linearly constrained coefficients and singular polynomials

We consider the problem of computing minimal real or complex deformations to the coefficients in a list of relatively prime real or complex multivariate polynomials such that the deformed polynomials have a greatest common divisor (GCD) of at least a given degree k. In addition, we restrict the deformed coefficients by a given set of linear constraints, thus introducing the linearly constrained approximate GCD problem. We present an algorithm based on a version of the structured total least norm (STLN) method and demonstrate on a diverse set of benchmark polynomials that the algorithm in practice computes globally minimal approximations. As an application of the linearly constrained approximate GCD problem we present an STLN-based method that computes a real or complex polynomial the nearest real or complex polynomial that has a root of multiplicity at least k. We demonstrate that the algorithm in practice computes on the benchmark polynomials given in the literature the known globally optimal nearest singular polynomials. Our algorithms can handle, via randomized preconditioning, the difficult case when the nearest solution to a list of real input polynomials actually has non-real complex coefficients.

Approximate factorization of multivariate polynomials via differential equations

The input to our algorithm is a multivariate polynomial, whose complex rational coefficients are considered imprecise with an unknown error that causes f to be irreducible over the complex numbers C. We seek to perturb the coefficients by a small quantitity such that the resulting polynomial factors over C. Ideally, one would like to minimize the perturbation in some selected distance measure, but no efficient algorithm for that is known. We give a numerical multivariate greatest common divisor algorithm and use it on a numerical variant of algorithms by W. M. Ruppert and S. Gao. Our numerical factorizer makes repeated use of singular value decompositions. We demonstrate on a significant body of experimental data that our algorithm is practical and can find factorizable polynomials within a distance that is about the same in relative magnitude as the input error, even when the relative error in the input is substantial (10-3).

Approximate factorization of multivariate polynomials using singular value decomposition

We describe the design, implementation and experimental evaluation of new algorithms for computing the approximate factorization of multivariate polynomials with complex coefficients that contain numerical noise. Our algorithms are based on a generalization of the differential forms introduced by W. Ruppert and S. Gao to many variables, and use singular value decomposition or structured total least squares approximation and Gauss-Newton optimization to numerically compute the approximate multivariate factors. We demonstrate on a large set of benchmark polynomials that our algorithms efficiently yield approximate factorizations within the coefficient noise even when the relative error in the input is substantial (10^-^3).

A complete symbolic-numeric linear method for camera pose determination

Camera pose estimation is the problem of determining the position and orientation of an internally calibrated camera from known 3D reference points and their images. We briefly survey several existing methods for pose estimation, then introduce our new complete linear method, which is based on a symbolic-numeric method from the geometric (Jet) theory of partial differential equations. The method is stable and robust. In particular, it can deal with the points near critical configurations. Numerical experiments are given to show the performance of the new method.

Pseudofactors of multivariate polynomials

We treat pseudo-factorization of a multivariate polynomial <italic>p</italic> over C as an overdetermined bilinear system for the coefficients of the factors. If the specified data (coefficients of <italic>p</italic>) are sufficiently close to the manifold of <italic>consistent</italic> data we project onto it by solving a well-chosen subsystem (otherwise we quit). On the manifold, we reduce the backward error by a <italic>linearized</italic> minimization procedure. The resulting algorithm appears to extend the range of treatable cases - in terms of number of variables and total degree - considerably.

On probabilistic analysis of randomization in hybrid symbolic-numeric algorithms

Algebraic randomization techniques can be applied to hybrid symbolic-numeric algorithms. Here we consider the problem of interpolating a sparse rational function from noisy values. We develop a new hybrid algorithm based on Zippels original sparse polynomial interpolation technique. We show experimentally that our algorithm can handle sparse polynomials with large degrees. We also give a (partial) mathematical justification why the Zippels algebraic randomization technique can be used with our approximate data: the randomly generated non-zero values are expected to be bounded away from zero. We show that the random Fourier-like matrices arising in our algorithm, have the desired rank property in the exact case, and appear usable numerically. Algebraic randomization techniques can be applied to hybrid symbolic-numeric algorithms. Here we consider the problem of interpolating a sparse rational function from noisy values. We develop a new hybrid algorithm based on Zippels original sparse polynomial interpolation technique. We show experimentally that our algorithm can handle sparse polynomials with large degrees. We also give a (partial) mathematical justification why the Zippels algebraic randomization technique can be used with our approximate data: the randomly generated non-zero values are expected to be bounded away from zero. We show that the random Fourier-like matrices arising in our algorithm, have the desired rank property in the exact case, and appear usable numerically.

Symbolic-numeric computation

The growing demand of speed, accuracy, and reliability in scientific and engineering computing has been accelerating the merging of symbolic and numeric computations, two types of computation coexisting in mathematics yet separated in traditional research of mathematical computation. This book presents 23 research articles on the integration and interaction of symbolic and numeric computation.

DISPLACEMENT STRUCTURE IN COMPUTING APPROXIMATE GCD OF UNIVARIATE POLYNOMIALS

We propose a fast algorithm for computing approximate GCD of univariate polynomials with coe‐cients that are given only to a flnite accuracy. The algorithm is based on a stabilized version of the generalized Schur algorithm for Sylvester matrix and its embedding. All computations can be done in O(n 2 ) operations, where n is the sum of the degrees of polynomials. The stability of the algorithm is also discussed.