Matu-Tarow Noda

Approximate GCD and its application to ill-conditioned algebraic equations

Abstract We describe two algorithms of approximate GCD (greatest common divisor) for polynomials with coefficients of floating-point numbers. One is for univariate polynomials and the other for multivariate polynomials. The algorithms are careful extensions of the conventional Euclidean algorithm, and they are applied to solving ill-conditioned algebraic equations. Ill-conditioned univariate algebraic equations have multiple or close roots, and we can ill-conditioned system of algebraic equations which have an approprimate common divisor. Conventional numeric root-finding methods such as Newtons method give no satisfactory result for such ill-conditioned systems. By using an approximate GCD algorithm for multivariate polynomials, the system is transformed into a well-conditioned problems by a combination of algebraic and numeric methods; we call such algorithms hybrid algorithms. This paper shows the importance and possible fruitfulness of the hybrid algorithm.

APPROXIMATE GCD OF MULTIVARIATE POLYNOMIALS

Given two polynomials F and G in R[x1, . . . , xn], we are going to find the nontrivial approximate GCD C and polynomials F , G ∈ R[x1, . . . , xn] such that ||F − CF ′|| 1. Approximate GCD computation of univariate polynomials provides the basis for solving multivariate problem. But it is nontrivial to modify the techniques used in symbolic computation such as interpolation and Hensel lifting to compute the approximate GCD of multivariate polynomials. In section 2, we briefly review two methods 2,10 for computing GCD of polynomials with floating-point coefficients. In section 3, we focus on extending the Hensel lifting technique to polynomials with floating-point coefficients. The method involves the QR decomposition of a Sylvester matrix. We propose an efficient new algorithm which exploits the special structure of Sylvester matrix. A local optimization problem is also been proposed to improve the candidate approximate factors obtained from Hensel lifting. In order to compare the performance of the different methods, we implement all three methods in Maple. In section 4, we summarize the special problems we encounter when they are applied to polynomials with floating-point coefficients. A set of examples are given to show the efficiency and stability of the algorithms.

Hybrid method for computing the nearest singular polynomials

In this paper, we propose a combined symbolic-numeric algorithm for computing the nearest singular polynomial and its multiple zero. Explicit expressions of the minimal perturbation and the nearest singular polynomials are presented. A theoretical error bound and several numerical examples are given.

Hybrid Rational Function Approximation and Its Accuracy Analysis

We propose a rational function approximation method combining numeric and symbolic computations. Given functions or data are first interpolated by a rational function, i.e. the ratio of polynomials. Undesired poles appearing in the rational interpolant are removed by an approximate-GCD method. We call the rational approximation a Hybrid Rational Function Approximation and abbreviate it as HRFA. In this paper we give a short survey of the HRFA and then discuss its accuracy analysis by using the approximate-GCD proposed by Pan.

On the symbolic/numeric hybrid integration

A Hybrid Integration. Integrating a given function is one of the most important area in the mathematical computing. Both numerical and symbolic integration methods have been developed and widely used. Numerical methods, however, have some defects such as 1) formal integrals are not obtained, 2) wrong answers are given for pathological integrand and 3) error estimates depend on types of integrands. Symbolic methods have also difficulties on 1)restrictions on an integrand and 2)uses of wasteful big-number computation. To avoid difficulties, some attempts in which both methods are combined have been proposed. We call them hybrid methods. Here, we propose new hybrid integration method for a rational function (say q/r, q and r are polynomials) with floating point but real coefficients. An outline of Proposed Hybrid Algorithm. Our algorithm consists following four steps. 1)Approximate Horowitz step: The denominator of an integrand, r, is reduced to an approximate square-free form. In the step, the approximate-GCD algorithm proposed by Sasaki and Noda (Jour.Inf.Proc.l2(1989), 159-168) is used. The result is accurate up to the machine epsiron. 2)Numerical root-finding step: Zeros of r,rk, are computed by the numerical Durand-Kerner( DK) method which gives all zeros simultaneously. It is well known that zeros of a real coefficient polynomial are restricted to real or complex conjupairs. Then, let n the degree of r and m the number of real zeros, zeros are expressed as * rI, e-e, rm, rm+I=rm+a , *em, rn-I=rn * where rh* is a complex conjugate of rk. The error for LX-method is estimated by the Smith’s theorem. 3)Decomposition of q/r into partial fraction: The partial fraction decomposition of q/r is done by zeros,&, obtained in 2). Real zeros give terms whose denominators are linear order as ck / (xrk) k=l;*.,m . Complex conjugate zeros rk and rk -I are combined and give terms with quadratic denominater, ck/(x-rk>+ck*/(x-rk*) k=m+l,m+3...,n-1. coefficients, ck, are obtained symbolically by the use of residue theory as ck = q(rk) / r’(rk) k=l;..,n . 4)Obtaining formal integration: A linear and a quadratic terms obtained in 3) is transformed to a log part and a sum of a log and an arctan part of a formal integral, respectively. Here, we use known formulas of integrals. An example of the Hybrid Integration. We show how a formal integral is obtained by our algorithm. We consider an integral for s 4(x) r(x) dx = S 1.1 x= t x+ 0.001 x’+ 0.001 x’ 1.001 dx * Since r(x) and r’(x) has no approximate common factor, the integrand is decomposed as {4(x) rcxj dx= O+ S 1.1 xz + x t 0.001 x’ + 0.001 x’ 1.001 dx . Zeros of r(x) are one real( h= 1.0 ) and one complex conjugate pair(rz=-0.5005t0.8663139i , z-~= -0.5005-0.8663139,!) with numerical error at most lo1 4. Denominators of the partial fraction are, then, obtained as (x-r, )= x-1.0 and (x-r,)(x-r,)=~tl.OOlxtl.OO1 . The rational function is decomposed as 4(x) 0.69986675 0.4001332x + 0.6995666 r(x) = x1.0 + xz + 1.001 x t 1.001 , where coefficients of partial fractions are computed by the residue theory. Rules for formal integrals in 4) give the final integral as

Hybrid computation of Cauchy-type singular integral equations

Hybrid Rational Function Approximation (HRFA) [6] is one of important and interesting applications of approximate-GCD [1, 3, 5, 7, 8]. The HI:tFA has been already applied to da ta smoothing, integral of function and Cauchy principal value integral[4, 6]. Here we consider an application of the HI:tFA to Cauchy-type singular integral equations. Especially a method to approximate the solution of the dominant equation,

Algebraic methods for computing a generalized inverse

Generalized inverses have many applications in engineering problems, such as data analysis, electrical networks, character recognitions, and so on. The most frequently used generalized inverse is a Moore-Penrose types one. Let <i>A</i> be <i>m</i> x <i>n</i> matrix. The Moore-Penrose generalized inverse of <i>A,</i> i.e., <i>n</i> x <i>m</i> matrix <i>G,</i> satisfies the following definitions:<i>AGA = A</i><i>GAG = G</i>(<i>AG</i>)<i>^T = AG</i>(<i>GA</i>)<i>^T = GA</i>where <i>I</i> shows a unit matrix. The Moore-Penrose generalized inverse is uniquely determined for a given matrix <i>A.</i> We denote the Moore-Penrose generalized inverse of <i>A</i> as <i>G = A</i>⁺. Some properties of it are1. If <i>A</i> is regular, then <i>A</i>⁺ = <i>A</i>^-12. If <i>A</i> = 0, then <i>A</i>⁺ = 03. If <i>A</i> is an <i>m</i> x <i>n</i> matrix and its rank is <i>m,</i> then <i>A</i>⁺ = <i>A^T</i>(<i>AA^T</i>)^-1The problem is how to obtain <i>G</i> efficiently. Thus we wish to find a <i>fast</i> and <i>stable</i> method for the problem. The singular value decomposition (SVD) algorithm is well known numerical method which gives results quickly. However, on the accuracy of solutions, it is evident that an error-free method is preferable. Thus, here, we consider algebraic or symbolic methods for obtaining the Moore-Penrose generalized inverse.

A Simple Predictive Method for Discriminating Costly Classes Using Class Size Metric

Larger object classes often become more costly classes in the maintenance phase of object-oriented software. Consequently class would have to be constructed in a medium or small size. In order to discuss such desirable size, this paper proposes a simple method for predictively discriminating costly classes in version-upgrades, using a class size metric, Stmts. Concretely, a threshold value of class size (in Stmts) is provided through empirical studies using many Java classes. The threshold value succeeded as a predictive discriminator for about 73% of the sample Java classes.

Security Flaw in SAS-2 Protocol

SAS-2 is an alternative of a one-time password authentication protocol SAS, and is developed in order to reduce overhead due to the use of hash functions. The idea of both algorithms is sharing a similar secret number called the verifier that allows a client to be authenticated and that is changed for each new session. However, some of the combinations proposed in [1] to transmit the verifier may contain a security flaw, and the insecure combination results in vulnerability to impersonation attacks.

A hybrid integral for parametrized rational functions

We present a hybrid integral to obtain symbolic results of an indefinite integral where the integrand is an univariate rational function whose coeficients have a parameter. We consider calculating power series roots of the denominator polynomial by applying Hensel construction. Accurate numerical results for a definite integral are easily obtained by simple substitutions of upper and lower bounds of integral into obtained approximate symbolic results.Numerical experiments show that the hybrid integral works well around the expansion point of the power series roots.