# Research projects

In this page, I briefly describe my present and past research projects (with research term and collaborators).

For outline of my overall research, please see Research page.

## Ongoing projects

### Iterative method for calculating approximate GCD of univariate polynomials

- Research term: 2005–present

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

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

## Completed projects

### "Recursive" PRS of univariate polynomials and its subresultants

- Research term: 2002–2008

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

### Iterative method for calculating single cluster of closed zeros of univariate polynomials

- Research term: 2004–2007
- Collaborator: Tateaki Sasaki

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

### Calculating the number of the real zeros of univariate polynomials with error terms

- Research term: 1997–1999
- Collaborator: Tateaki Sasaki

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

Sturm sequence is an extension of the polynomial remainder sequnece (PRS) and used to calculate the number of the 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 very small leading coefficient, and established a condition such that we can calculate the number of the real zeros appropriately with neglecting such small leading coefficient in proseeding calculations.

### Iterative method for calculating the real zeros of univariate polynomials

- Research term: 1995–1997
- Collaborator: Tateaki Sasaki

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

- With the number of the real zeros caluculated in advance, calculate the real and the copmlex zeros with always distinguishing them,
- With modifying the iterative formula, calculate only the real zeros.