Hiroshi Sekigawa

On the Location of Zeros of an Interval Polynomial

For an interval polynomial F, we provide a rigorous method for deciding whether there exists a polynomial in F that has a zero in a prescribed domain D. When D is real, we show that it is sufficient to examine a finite number of polynomials. When D is complex, we assume that the boundary C of D is a simple closed curve of finite length and C is represented by a piecewise rational function. The decision method uses the representation of C and the property that a polynomial in F is of degree one with respect to each coefficient regarded as a variable. Using the method, we can completely determine the set of real numbers that are zeros of a polynomial in F. For complex zeros, we can obtain a set X that contains the set Z(F), which consists of all the complex numbers that are zeros of a polynomial in F, and the difference between X and Z(F) can be as small as possible.

The Ratio Between the Toeplitz and the Unstructured Condition Number

Recently it was shown that the ratio between the normwise Toeplitz structured condition number of a linear system and the general unstructured condition number has a finite lower bound. However, the bound was not explicit, and nothing was known about the quality of the bound. In this note we derive an explicit lower bound only depending on the dimension n, and we show that this bound is almost sharp for all n.

Locating real multiple zeros of a real interval polynomial

For a real interval polynomial F, we provide a rigorous method for deciding whether there exists a polynomial in F that has a multiple zero in a prescribed interval in R. We show that it is sufficient to examine a finite number of edge polynomials in F. An edge polynomial is a real interval polynomial such that the number of coefficients that are intervals is one. The decision method uses the property that a univariate polynomial is of degree one with respect to each coefficient regarded as a variable. Using this method, we can completely determine the set of real numbers each of which is a multiple zero of some polynomial in F.

A new Gröbner basis conversion method based on stabilization techniques

We propose a new method for converting a Grobner basis w.r.t. one term order into a Grobner basis w.r.t. another term order by using the algorithm stabilization techniques proposed by Shirayanagi and Sweedler. First, we guess the support of the desired Grobner basis from a floating-point Grobner basis by exploiting the supportwise convergence property of the stabilized Buchbergers algorithm. Next, assuming this support to be correct, we use linear algebra, namely, the method of indeterminate coefficients to determine the exact values for the coefficients. Related work includes the FGLM algorithm and its modular version. Our method is new in the sense that it can be thought of as a floating-point approach to the linear algebra method. The results of Maple computing experiments indicate that our method can be very effective in the case of non-rational coefficients, especially the ones including transcendental constants.

The nearest polynomial with a zero in a given domain from a geometrical viewpoint

For a real univariate polynomial f and a closed complex domain D, whose boundary C is a simple curve parameterized by a univariate piecewise rational function, a rigorous method is given for finding a real univariate polynomial f such that f has a zero in D and |f - ~f|∞ is minimal. First, it is proved that the minimum distance between f and polynomials having a zero at α in C is a piecewise rational function of the real and imaginary parts of α. Thus, on C, the minimum distance is a piecewise rational function of a parameter obtained through the parameterization of C. Therefore, by using the property that ~f has a zero on C and computing the minimum distance on C, ~f can be constructed.

Quantum fourier transform over symmetric groups

This paper proposes an <i>O</i>(<i>n</i>⁴) quantum Fourier transform (QFT) algorithm over symmetric group <i>S_n</i>, the fastest QFT algorithm of its kind. We propose a fast Fourier transform algorithm over symmetric group <i>S_n</i>, which consists of <i>O</i>(<i>n</i>³) multiplications of unitary matrices, and then transform it into a quantum circuit form. The QFT algorithm can be applied to constructing the standard algorithm of the hidden subgroup problem.

Reducing exact computations to obtain exact results based on stabilization techniques

For a certain class of algebraic algorithms, we propose a new method that reduces the number of exact computational steps needed for obtaining exact results. This method is the floating-point interval method using zero rewriting and symbols. Zero rewriting, which is from stabilization techniques, rewrites an interval coefficient into the zero interval if the interval contains zero. Symbols are used to keep track of the execution path of the original algorithm with exact computations, so that the associated real coefficients can be computed by evaluating the symbols. The key point is that at each stage of zero rewriting, one checks to see if the zero rewriting is really correct by exploiting the associated symbol. This method mostly uses floating-point computations; the exact computations are only performed at the stage of zero rewriting and in the final evaluation to get the exact coefficients. Moreover, one does not need to check the correctness of the output.

The nearest polynomial to multiple given polynomials with a given zero

The following type of problems have been well-studied in the area of symbolic-numeric computation for about twenty years: Given a polynomial f ∈ C[x] and a point z ∈ C, find the nearest polynomial f̃ ∈ C[x] to f with f̃(z) = 0. A common framework for such problems is described in [7]. In previous works, for example [3, 4, 7, 6], problems for one given polynomial were considered. Here, we consider a problem for multiple given polynomials. Through observation or by using different numerical algorithms for a given input data, we may obtain multiple polynomials being equal in theory but being slightly different each other. Thus, it is worth considering the problem for multiple polynomials. In this abstract, after the preliminaries, we define the nearest polynomial to multiple given polynomials. In the definition, we use a pair of norms to measure the nearness between polynomials. We remark the difficulty of the problem of finding the nearest polynomial depends on the norm pair. Finally, we describe an algorithm for the problem when both of the norms are the ∞-norm.

On real factors of real interval polynomials

For a real multivariate interval polynomial <i>P</i> and a real multivariate polynomial <i>f</i>, we provide a rigorous method for deciding whether there exists a polynomial <i>p</i> in <i>P</i> such that <i>p</i> is divisible by <i>f</i>. When <i>P</i> is univariate, there is a well-known criterion for whether there exists a polynomial <i>p</i>(χ)in <i>P</i> such that <i>p</i>(α)=0 for a given real number α. Since <i>p</i>(α)=0 if and only if <i>p</i>(χ) is divisible by χ--α, our result is a generalization of the criterion to multivariate polynomials and higher degree factors.

On the Location of Pseudozeros of a Complex Interval Polynomial

Abstract.Given a univariate complex interval polynomial F, we provide a rigorous method for deciding whether there exists a pseudozero of F in a prescribed closed complex domain D. Here a pseudozero of F is defined to be a zero of some polynomial in F. We use circular intervals and assume that the boundary C of D is a simple curve and that C is the union of a finite number of arcs, each of which is represented by a rational function. When D is not bounded, we assume further that all the polynomials in F are of the same degree. Examples of such domains are the outside of an open disk and a half-plane with boundary. Our decision method uses the representation of C and the property that a polynomial in F is of degree 1 with respect to each coefficient regarded as a variable.