We are studying methods for estimating the relative positions of small swarm robots exploring asteroids by measuring the strength of radio waves they emit to each other. We are working on a method to efficiently solve the equations and evaluate the error by converting a system of nonlinear equations back to a system of linear equations through the substitution of variables.

Research term: 2010–present
Collaborators: Shinichi Tajima and Katsuyosi Ohara

For matrices whose components are given by integers or rational numbers, we are working on developing algorithms to efficiently compute general eigenspaces and Jordan chains using the minimum annihilating polynomials or pseudo minimum annihilating polynomials of the matrix based on the residue analysis on the resolvents.

Research term: 2005–present
Collaborator: Boming Chi

Approximate Greatest Common Divisor (GCD) is the oldest topic in symbolic-numeric computation still active in research. Among many approaches for calculating approximate GCDs, I'm developing an iterative algorithm that reduces the original problem into a constrained minimization problem.

We have successfully developed an algorithm that calculates approximate GCDs with perturbations as small as previously proposed algorithms, taking an optimization approach, and is extremely efficient than ever before (up to 30 times faster).

Research term: 2007–present
Collaborator: Jun Morita

We are working on characterizing the algebraic and combinatorial structure of words (strings) by means of formal power series, etc.

Participating term: 2016–2020
Participated in: The Reading Skills Project

In the project, we investigated and analyzed the correlations between the ability of junior high school and high school mathematics textbooks to read definition sentences and some other reading skills, as well as the differences in reading comprehension by reading text and relating it to a diagram.

Participating term: 2014–2016
Participated in: Todai Robot Project https://21robot.org/index-e.html

As a member of the science and mathematics team of the above-mentioned project, we participated in a research project on solving mathematical problems by automatic reasoning. We have developed an algorithm and implementation for solving problems of series in the National Center Test for University Admissions (NCT) using automatic reasoning and contributed to improving the performance in the mock examinations.

Research term: 2002–2008

For extracting “multiple zeros” of a given univariate polynomial, we calculate the GCD of the given polynomial and its derivative, then calculate the GCD of the just-acquired GCD and its derivative, and so on, which leads to a calculation called “squarefree decomposition.” We have named the polynomial remainder sequence (PRS) involved in the “recursive PRS” calculation and established the theory of subresultants for recursive PRS.

Research term: 2004–2007
Collaborator: Tateaki Sasaki

Computing multiple or closed zeros of univariate polynomials with an iterative method is difficult because of numerical errors. Focusing on a single cluster of multiple close zeros and assuming that the position and the multiplicity of the cluster have already been given by appropriate methods such as approximate squarefree decomposition, we have developed an iterative method calculating the zeros in the cluster simultaneously, accurately, and efficiently, with standard hardware arithmetic.

Research term: 1997–1999
Collaborator: Tateaki Sasaki

If the coefficients of a univariate polynomial have changed, the number of real zeros may also change. For given ranges of errors in the coefficients, we have developed a way of estimating the (range of) number of the real zeros.

Sturm sequence is an extension of the polynomial remainder sequence (PRS) and is used to calculate the number of real zeros. Some given polynomials may cause unstable behaviors in the Sturm sequence. Among such phenomena, we have focused our attention on the issue of a very small leading coefficient and established a condition such that we can calculate the number of real zeros appropriately while neglecting such a small leading coefficient in proceeding calculations.

Research term: 1995–1997
Collaborator: Tateaki Sasaki

Durand-Kerner's method is known as an iterative method to calculate all the zeros of univariate polynomials simultaneously. In this study, we have extended the D-K method as follows:

With the number of real zeros calculated in advance, calculate the real and the complex zeros while always distinguishing them,
By modifying the iterative formula, only the real zeros are calculated.