Kiyoshi Shirayanagi

Remarks on Automatic Algorithm Stabilization

This note is about a methodology utilizing inexact computation in conjunction with exact computation where the exact input is known and exact output is desired. The inexact computation is used to help avert the growth of intermediate expressions. This growth frequently makes using exact computation infeasible. We mention several existing applications and also mention where the methodology is not useful. We propose new directions where one can make effective use of the stabilization methodology.

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.

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.

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.

A new idea on the interval-symbol method with correct zero rewriting for reducing exact computations

The ISCZ method (Interval-Symbol method with Correct Zero rewriting) was proposed in [2] based on Shirayanagi-Sweedler stabilization theory ([3]), to reduce the amount of exact computations as much as possible to obtain the exact results. The authors of [2] applied this method to Buchbergers algorithm which computes a Gröbner basis, but the effectiveness was not achieved except for a few examples. This is not only because the complex structure of Buchbergers algorithm causes symbols to significantly grow but also because we naively implemented the ISCZ method without any particular devices. In this poster, we propose a new idea for efficiency of the ISCZ method and show its effect by applying it to calculation of Frobenius canonical form of square matrices. Jordan canonical form is also well-known, but it requires an extension of the field containing the roots of its characteristic polynomial. On the other hand, Frobenius canonical form can be computed by using only basic arithmetic operations, but nevertheless has almost the same information as Jordan canonical form.

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.

Existence of the Exact CNOT on a Quantum Computer with the Exchange Interaction

We prove the existence of the exact CNOT gate on aquantum computer with the nearest-neighbor exchange interaction in the serial operation mode. Its existence has been an open problem, though a concrete sequence of exchange operations, which is approximately locally equivalent to the exact CNOT, has already been found. We found the exact values of time parameters (exchange rates between qubits) by using computer algebraic techniques such as Gröbner bases and resultants. These techniques have been widely used for finding rigorous solutions of simultaneous algebraic equations, and here are applied to finding quantum gates on the decoherence-free subsystem for the first time.

A Classification of Finite-dimensional Monomial Algebras

Monomial algebras are finitely presented algebras defined by monomials. This notion is considered the most fundamental of “standard” finitely presented algebras, in the sense that they have finite non-commutative Grobner bases as defining relations.

Knowledge Representation and Its Refinement in Go Programs

It is often said that, unlike chess, an exhaustive search of the game tree for deciding the next move for Go is impossible, because of the huge search space. Therefore, in Go search is limited to local fields only. For more global decisions, various strategic human knowledge should be applied. Almost all Go playing programs (for example, Zobrist 1970b; Ryder 1971; Sanechika et al. 1981) settle on an evaluation function to represent this knowledge. These programs decide on the next move by selecting candidates with maximum evaluation function values. However, Go programmers often judge that the program’s maximum-value moves are bad or unsuitable for a situation (a board state, or placement of stones on the board), so that the only way to get the proper result is to refine the evaluation function. But a temporary refinement for one situation may cause a contradiction for another.