Akira Terui

An iterative method for calculating approximate GCD of univariate polynomials

We present an iterative algorithm for calculating approximate greatest common divisor (GCD) of univariate polynomials with the real coefficients. For a given pair of polynomials and a degree, our algorithm finds a pair of polynomials which has a GCD of the given degree and whose coefficients are perturbed from those in the original inputs, making the perturbations as small as possible, along with the GCD. The problem of approximate GCD is transfered to a constrained minimization problem, then solved with a so-called modified Newton method, which is a generalization of the gradient-projection method, by searching the solution iteratively. We demonstrate that our algorithm calculates approximate GCD with perturbations as small as those calculated by a method based on the structured total least norm (STLN) method, while our method runs significantly faster than theirs by approximately up to 30 times, compared with their implementation. We also show that our algorithm properly handles some ill-conditioned problems with GCD containing small or large leading coefficient.

A formula for separating small roots of a polynomial

Let <i>P(x)</i> be a univariate polynomial over C, such that <i>P(x) = c<inf>n</inf>xⁿ + ... + c<inf>m+1</inf>x^m+1 + x^m + e<inf>m-1</inf>x^m-1 + ... + e<inf>0</inf>,</i> where max{<i>|c<inf>n</inf>|, ..., |c<inf>m+1</inf>|</i>} = 1 and <i>e</i> = max{<i>|e<inf>m-1</inf>|, |e<inf>m-2</inf>|^1/2, ..., |e<inf>0</inf>|^1/m</i>} << 1. <i>P(x)</i> has <i>m</i> small roots around the origin so long as <i>e</i> << 1. In 1999, we derived a formula that if <i>e</i> < 1/9 then <i>P(x)</i> has <i>m</i> roots inside a disc <i>D</i><inf>in</inf> of radius <i>R</i><inf>in</inf> and other <i>n - m</i> roots outside a disc <i>D</i><inf>out</inf> of radius <i>R</i><inf>out</inf>, located at the origin, where <i>R</i><inf>in(out)</inf> = [1 - (+) √1 - (16<i>e</i>)/(1 + 3<i>e</i>)²] × (1 + 3<i>e</i>)/4. Note that <i>R</i><inf>in</inf> = <i>R</i><inf>out</inf> if <i>e</i> = 1/9. Our formula is essentially the same as that derived independently by Yakoubsohn at almost the same time. In this short article, we introduce the formula and check its sharpness on many polynomials generated randomly.

GPGCD: An iterative method for calculating approximate GCD of univariate polynomials

We present an extension of our GPGCD method, an iterative method for calculating approximate greatest common divisor (GCD) of univariate polynomials, to polynomials with the complex coefficients. For a given pair of polynomials and a degree, our algorithm finds a pair of polynomials which has a GCD of the given degree and whose coefficients are perturbed from those in the original inputs, making the perturbations as small as possible, along with the GCD. In our GPGCD method, the problem of approximate GCD is transfered to a constrained minimization problem, then solved with a so-called modified Newton method, which is a generalization of the gradient-projection method, by searching the solution iteratively. While our original method is designed for polynomials with the real coefficients, we extend it to accept polynomials with the complex coefficients in this paper.

GPGCD, an iterative method for calculating approximate GCD, for multiple univariate polynomials

We present an extension of our GPGCD method, an iterative method for calculating approximate greatest common divisor (GCD) of univariate polynomials, to multiple polynomial inputs. For a given pair of polynomials and a degree, our algorithm finds a pair of polynomials which has a GCD of the given degree and whose coefficients are perturbed from those in the original inputs, making the perturbations as small as possible, along with the GCD. In our GPGCD method, the problem of approximate GCD is transferred to a constrained minimization problem, then solved with the so-called modified Newton method, which is a generalization of the gradient-projection method, by searching the solution iteratively. In this paper, we extend our method to accept more than two polynomials with the real coefficients as an input.

Computing clustered close-roots of univariate polynomials

Given a univariate polynomial having well-separated clusters of close roots, we give a method of computing close roots in a cluster simultaneously, without computing other roots. We first determine the position and the size of the cluster, as well as the number of close roots contained. Then, we move the origin to a near center of the cluster and perform the scale transformation so that the cluster is enlarged to be of size O(1). These operations transform the polynomial to a very characteristic one. We modify Durand-Kerners method so as to compute only the close roots in the cluster. The method is very efficient because we can discard most terms of the transformed polynomial. We also give a formula of quite tight error bound. We show high efficiency of our method by empirical experiments.

Recursive polynomial remainder sequence and its subresultants

Abstract We introduce concepts of “recursive polynomial remainder sequence (PRS)” and “recursive subresultant,” along with investigation of their properties. A recursive PRS is defined as, if there exists the GCD (greatest common divisor) of initial polynomials, a sequence of PRSs calculated “recursively” for the GCD and its derivative until a constant is derived, and recursive subresultants are defined by determinants representing the coefficients in recursive PRS as functions of coefficients of initial polynomials. We give three different constructions of subresultant matrices for recursive subresultants; while the first one is built-up just with previously defined matrices thus the size of the matrix increases fast as the recursion deepens, the last one reduces the size of the matrix drastically by the Gaussian elimination on the second one which has a “nested” expression, i.e. a Sylvester matrix whose elements are themselves determinants.

Durand-Kerner method for the real roots

Given a univariate real polynomial, this paper considers calculating all the real zero-points of the polynomial simultaneously. Among several numerical methods for calculating zero-points of a univariate polynomial, Durand-Kerner method is quite useful because it is most stable to converge to the zero-points. On the basis of Durand-Kerner method, we propose two methods for calculating the real zero-points of a univariate polynomial simultaneously. Convergence and error analysis of our methods are discussed. We compared our methods, the original Durand-Kerner method and Newton’s method and found that 1) our methods are more stable than Newton’s method but less than the original Durand-Kerner method, and 2) they are more efficient than the original Durand-Kerner method but less than Newton’s method. We conclude that our methods are useful when good initial values of the zero-points are known.

Developing Linear Algebra Packages on Risa/Asir for Eigenproblems

We are developing linear algebra packages on Risa/Asir, a computer algebra system. The aim is to provide programs for efficiently and exactly solving eigenproblems on the computer algebra system for large scale square matrices over integers or rational numbers. The software package consists of some programs. The followings are currently prepared for solving eigenproblems: computing eigenspaces, the spectral decomposition, Jordan chains and minimal annihilating polynomials.

An Automated Deduction and Its Implementation for Solving Problem of Sequence at University Entrance Examination

“Todai Robot Project” is a project of artificial intelligence launched by National Institute of Informatics for re-unifying the artificial intelligence field subdivided in 1980s and afterwards. We focus towards attaining a high score in National Center Test for University Admissions, and use Quantifier Elimination (QE) over the real closed fields as a main tool for solving problems in mathematics. However, it is not applicable for several kinds of problems such as one with sequence. In this article, we propose an algorithm for solving problems of sequence at the National Center Test for University Admissions.

An Extension and Efficient Calculation of the Horner’s Rule for Matrices

We propose an efficient method for calculating “matrix polynomials” by extending the Horner’s rule for univariate polynomials. We extend the Horner’s rule by partitioning it by a given degree to reduce the number of matrix-matrix multiplications. By this extension, we show that we can calculate matrix polynomials more efficiently than by using naive Horner’s rule. An implementation of our algorithm is available on the computer algebra system Risa/Asir, and our experiments have demonstrated that, at suitable degree of partitioning, our new algorithm needs significantly shorter computing time as well as much smaller amount of memory, compared to naive Horner’s rule. Furthermore, we show that our new algorithm is effective for matrix polynomials not only over multiple-precision integers, but also over fixed-precision (IEEE standard) floating-point numbers by experiments.