Updated on 2022/11/26

OISHI, Shinichi

##### Scopus Paper Info
###### Paper Count: 0  Citation Count: 0  h-index: 8

Citation count denotes the number of citations in papers published for a particular year.

Affiliation
Faculty of Science and Engineering, School of Fundamental Science and Engineering
Job title
Professor
Homepage URL

### Concurrent Post

• Faculty of Science and Engineering   Graduate School of Fundamental Science and Engineering

### Research Institute

• 2020
-
2022

理工学術院総合研究所   兼任研究員

### Education

•
-
1981

Waseda University   Graduate School, Division of Science and Engineering

•
-
1981

Waseda University   Graduate School, Division of Science and Engineering

•
-
1976

Waseda University   Faculty of Science and Engineering

### Degree

• 早稲田大学   工学博士

• Dr. eng

• 早稲田大学   博士(工学)

### Research Experience

• 2014.09
-

Waseda University   Faculty of Science and Engineering   Senior Dean

• 2014.09
-

Waseda University   Faculty of Science and Engineering   Senior Dean

• 2010.09
-
2014.09

FacultyofScienceandEngineering

• 1989
-

Professor, Waseda University

• 1984
-
1985

Associate Professor, Waseda Unviersity

• 1984
-
1985

Present:Professor,DepartmentofAppliedMathematics,

• 1982
-
1984

Lecturer, Waseda University

• 1980
-
1982

Assistant Professor, Waseda University

• 1980
-
1982

Assistant Professor, Waseda University

•

•

JSST

•

Japan SIAM

•

IEICE

### Research Areas

• Basic mathematics

• Applied mathematics and statistics

• Theory of informatics

• Communication and network engineering

### Research Interests

• 数値数学、応用数学、計算理論、非線形理論・回路、情報理論、精度保証付き数値計算

### Papers

• Rigorous Numerical Enclosures for Positive Solutions of Lane-Emden's Equation with Sub-Square Exponents

Kazuaki Tanaka, Michael Plum, Kouta Sekine, Masahide Kashiwagi, Shin'ichi Oishi

NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION   43 ( 3 ) 322 - 349  2022.02

View Summary

The purpose of this paper is to obtain rigorous numerical enclosures for solutions of Lane-Emden's equation -Delta u = vertical bar u vertical bar(p-1)u with homogeneous Dirichlet boundary conditions. We prove the existence of a nondegenerate solution u nearby a numerically computed approximation (u) over cap together with an explicit error bound, i.e., a bound for the difference between u and (u) over cap: In particular, we focus on the sub-square case in which 1<p<2 so that the derivative p vertical bar u vertical bar(p-1) of the nonlinearity vertical bar u vertical bar(p-1)u is not Lipschitz continuous. In this case, it is problematic to apply the classical Newton-Kantorovich theorem for obtaining the existence proof, and moreover several difficulties arise in the procedures to obtain numerical integrations rigorously. We design a method for enclosing the required integrations explicitly, proving the existence of a desired solution based on a generalized Newton-Kantorovich theorem. A numerical example is presented where an explicit solution-enclosure is obtained for p = 3/2 on the unit square domain Omega = (0, 1)(2).

• Inverse norm estimation of perturbed Laplace operators and corresponding eigenvalue problems

Kouta Sekine, Kazuaki Tanaka, Shin'ichi Oishi

Computers & Mathematics with Applications   106   18 - 26  2022.01

• Makoto Mizuguchi, Mitsuhiro T. Nakao, Kouta Sekine, Shin’ichi Oishi

Journal of Scientific Computing   89 ( 2 )  2021.11

View Summary

<title>Abstract</title>In this paper, we propose <inline-formula><alternatives><tex-math>$$L^2(J;H^1_0(\Omega ))$$</tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:msup>
<mml:mi>L</mml:mi>
<mml:mn>2</mml:mn>
</mml:msup>
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:mi>J</mml:mi>
<mml:mo>;</mml:mo>
<mml:msubsup>
<mml:mi>H</mml:mi>
<mml:mn>0</mml:mn>
<mml:mn>1</mml:mn>
</mml:msubsup>
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:mi>Ω</mml:mi>
<mml:mo>)</mml:mo>
</mml:mrow>
<mml:mo>)</mml:mo>
</mml:mrow>
</mml:mrow>
</mml:math></alternatives></inline-formula> and <inline-formula><alternatives><tex-math>$$L^2(J;L^2(\Omega ))$$</tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:msup>
<mml:mi>L</mml:mi>
<mml:mn>2</mml:mn>
</mml:msup>
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:mi>J</mml:mi>
<mml:mo>;</mml:mo>
<mml:msup>
<mml:mi>L</mml:mi>
<mml:mn>2</mml:mn>
</mml:msup>
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:mi>Ω</mml:mi>
<mml:mo>)</mml:mo>
</mml:mrow>
<mml:mo>)</mml:mo>
</mml:mrow>
</mml:mrow>
</mml:math></alternatives></inline-formula> norm error estimates that provide the explicit values of the error constants for the semi-discrete Galerkin approximation of the linear heat equation. The derivation of these error estimates shows the convergence of the approximation to the weak solution of the linear heat equation. Furthermore, explicit values of the error constants for these estimates play an important role in the computer-assisted existential proofs of solutions to semi-linear parabolic partial differential equations. In particular, the constants provided in this paper are better than the existing constants and, in a sense, the best possible.

1
Citation
(Scopus)
• Numerical verification for asymmetric solutions of the Hénon equation on bounded domains

Taisei Asai, Kazuaki Tanaka, Shin’ichi Oishi

Journal of Computational and Applied Mathematics     113708 - 113708  2021.07

• Numerical verification methods for a system of elliptic PDEs, and their software library

Kouta Sekine, Mitsuhiro T. Nakao, Shin'ichi Oishi

IEICE NONLINEAR THEORY AND ITS APPLICATIONS   12 ( 1 ) 41 - 74  2021

View Summary

Since the numerical verification method for solving boundary value problems for elliptic partial differential equations (PDEs) was first developed in 1988, many methods have been devised. In this paper, existing verification methods are reformulated using a convergence theorem for simplified Newton-like methods in the direct product space V-h x V-perpendicular to of a computable finite-dimensional space V-h and its orthogonal complement space V-perpendicular to. Additionally, the Verified Computation for PDEs (VCP) library is provided, which is a software library written in the C++ programming language. The VCP library is introduced as a software library for numerical verification methods of solutions to PDEs. Finally, numerical examples are presented using the reformulated verification methods and VCP library.

• Kouta Sekine, Mitsuhiro T. Nakao, Shin’ichi Oishi

Numerische Mathematik   146 ( 4 ) 907 - 926  2020.12

View Summary

<title>Abstract</title>Infinite-dimensional Newton methods can be effectively used to derive numerical proofs of the existence of solutions to partial differential equations (PDEs). In computer-assisted proofs of PDEs, the original problem is transformed into the infinite-dimensional Newton-type fixed point equation <inline-formula><alternatives><tex-math>$$w = - {\mathcal {L } }^{-1} {\mathcal {F } }(\hat{u}) + {\mathcal {L } }^{-1} {\mathcal {G } }(w)$$</tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mi>w</mml:mi>
<mml:mo>=</mml:mo>
<mml:mo>-</mml:mo>
<mml:msup>
<mml:mrow>
<mml:mi>L</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mo>-</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msup>
<mml:mi>F</mml:mi>
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:mover>
<mml:mi>u</mml:mi>
<mml:mo>^</mml:mo>
</mml:mover>
<mml:mo>)</mml:mo>
</mml:mrow>
<mml:mo>+</mml:mo>
<mml:msup>
<mml:mrow>
<mml:mi>L</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mo>-</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msup>
<mml:mi>G</mml:mi>
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:mi>w</mml:mi>
<mml:mo>)</mml:mo>
</mml:mrow>
</mml:mrow>
</mml:math></alternatives></inline-formula>, where <inline-formula><alternatives><tex-math>$${\mathcal {L } }$$</tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>L</mml:mi>
</mml:math></alternatives></inline-formula> is a linearized operator, <inline-formula><alternatives><tex-math>$${\mathcal {F } }(\hat{u})$$</tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mi>F</mml:mi>
<mml:mo>(</mml:mo>
<mml:mover>
<mml:mi>u</mml:mi>
<mml:mo>^</mml:mo>
</mml:mover>
<mml:mo>)</mml:mo>
</mml:mrow>
</mml:math></alternatives></inline-formula> is a residual, and <inline-formula><alternatives><tex-math>$${\mathcal {G } }(w)$$</tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mi>G</mml:mi>
<mml:mo>(</mml:mo>
<mml:mi>w</mml:mi>
<mml:mo>)</mml:mo>
</mml:mrow>
</mml:math></alternatives></inline-formula> is a nonlinear term. Therefore, the estimations of <inline-formula><alternatives><tex-math>$$\Vert {\mathcal {L } }^{-1} {\mathcal {F } }(\hat{u}) \Vert$$</tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mrow>
<mml:mo>‖</mml:mo>
</mml:mrow>
<mml:msup>
<mml:mrow>
<mml:mi>L</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mo>-</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msup>
<mml:mi>F</mml:mi>
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:mover>
<mml:mi>u</mml:mi>
<mml:mo>^</mml:mo>
</mml:mover>
<mml:mo>)</mml:mo>
</mml:mrow>
<mml:mrow>
<mml:mo>‖</mml:mo>
</mml:mrow>
</mml:mrow>
</mml:math></alternatives></inline-formula> and <inline-formula><alternatives><tex-math>$$\Vert {\mathcal {L } }^{-1}{\mathcal {G } }(w) \Vert$$</tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mrow>
<mml:mo>‖</mml:mo>
</mml:mrow>
<mml:msup>
<mml:mrow>
<mml:mi>L</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mo>-</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msup>
<mml:mrow>
<mml:mi>G</mml:mi>
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:mi>w</mml:mi>
<mml:mo>)</mml:mo>
</mml:mrow>
<mml:mo>‖</mml:mo>
</mml:mrow>
</mml:mrow>
</mml:math></alternatives></inline-formula> play major roles in the verification procedures . In this paper, using a similar concept to block Gaussian elimination and its corresponding ‘Schur complement’ for matrix problems, we represent the inverse operator <inline-formula><alternatives><tex-math>$${\mathcal {L } }^{-1}$$</tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msup>
<mml:mrow>
<mml:mi>L</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mo>-</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msup>
</mml:math></alternatives></inline-formula> as an infinite-dimensional operator matrix that can be decomposed into two parts: finite-dimensional and infinite-dimensional. This operator matrix yields a new effective realization of the infinite-dimensional Newton method, which enables a more efficient verification procedure compared with existing Nakao’s methods for the solution of elliptic PDEs. We present some numerical examples that confirm the usefulness of the proposed method. Related results obtained from the representation of the operator matrix as <inline-formula><alternatives><tex-math>$${\mathcal {L } }^{-1}$$</tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msup>
<mml:mrow>
<mml:mi>L</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mo>-</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msup>
</mml:math></alternatives></inline-formula> are presented in the “Appendix”.

5
Citation
(Scopus)
• Yuta Matsushima, Kazuaki Tanaka, Shin’ichi Oishi

Journal of Advanced Simulation in Science and Engineering   7 ( 1 ) 136 - 150  2020  [Refereed]

• Estimation of Sobolev embedding constant on a domain dividable into bounded convex domains

Makoto Mizuguchi, Kazuaki Tanaka, Kouta Sekine, Shin'ichi Oishi

JOURNAL OF INEQUALITIES AND APPLICATIONS     1 - 18  2017.11  [Refereed]

View Summary

This paper is concerned with an explicit value of the embedding constant from W-1,W- q(Omega) to L-p(Omega) for a domain Omega subset of R-N (N is an element of N), where 1 &lt;= q &lt;= p &lt;=infinity. We previously proposed a formula for estimating the embedding constant on bounded and unbounded Lipschitz domains by estimating the norm of Stein's extension operator. Although this formula can be applied to a domain Omega that can be divided into a finite number of Lipschitz domains, there was room for improvement in terms of accuracy. In this paper, we report that the accuracy of the embedding constant is significantly improved by restricting Omega to a domain dividable into bounded convex domains.

18
Citation
(Scopus)
• A method for verifying the accuracy of numerical solutions of symmetric saddle point linear systems

Ryo Kobayashi, Takuma Kimura, Shin'ichi Oishi

NUMERICAL ALGORITHMS   76 ( 1 ) 33 - 51  2017.09  [Refereed]

View Summary

A fast numerical verification method is proposed for evaluating the accuracy of numerical solutions for symmetric saddle point linear systems whose diagonal blocks of the coefficient matrix are semidefinite matrices. The method is based on results of an algebraic analysis of a block diagonal preconditioning. Some numerical experiments are present to illustrate the usefulness of the method.

• Numerical verification for existence of a global-in-time solution to semilinear parabolic equations

Makoto Mizuguchi, Akitoshi Takayasu, Takayuki Kubo, Shin'ichi Oishi

Journal of Computational and Applied Mathematics   315   1 - 16  2017.05

10
Citation
(Scopus)
• Numerical validation of blow-up solutions of ordinary differential equations

Akitoshi Takayasu, Kaname Matsue, Takiko Sasaki, Kazuaki Tanaka, Makoto Mizuguchi, Shin'ichi Oishi

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS   314   10 - 29  2017.04  [Refereed]

View Summary

This paper focuses on blow-up solutions of ordinary differential equations (ODEs). We present a method for validating blow-up solutions and their blow-up times, which is based on compactifications and the Lyapunov function validation method. The necessary criteria for this construction can be verified using interval arithmetic techniques. Some numerical examples are presented to demonstrate the applicability of our method. (C) 2016 Elsevier B.V. All rights reserved.

17
Citation
(Scopus)
• Sharp numerical inclusion of the best constant for embedding H-0(1)(Omega) hooked right arrow L-p (Omega) on bounded convex domain

Kazuaki Tanaka, Kouta Sekine, Makoto Mizuguchi, Shin'ichi Oishi

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS   311   306 - 313  2017.02  [Refereed]

View Summary

In this paper, we propose a verified numerical method for obtaining a sharp inclusion of the best constant for the embedding H-0(1)(Omega) hooked right arrow L-p (Omega) on a bounded convex domain in R-2. We estimate the best constant by computing the corresponding extremal function using a verified numerical computation. Verified numerical inclusions of the best constant on a square domain are presented. (C) 2016 Elsevier B.V. All rights reserved.

8
Citation
(Scopus)
• Makoto Mizuguchi, Akitoshi Takayasu, Takayuki Kubo, Shin'ichi Oishi

SIAM Journal on Numerical Analysis   55 ( 2 ) 980 - 1001  2017.01

7
Citation
(Scopus)
• Error-free transformation of matrix multiplication with a posteriori validation

Katsuhisa Ozaki, Takeshi Ogita, Shin'ichi Oishi

NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS   23 ( 5 ) 931 - 946  2016.10  [Refereed]

View Summary

In this study, we examine the accurate matrix multiplication in floating-point arithmetic. We demonstrate the error-free transformations of matrix multiplication using high performance basic linear algebra subprograms. These transformations can be applied to accurately compute the product of two matrices using floating-point entries. A key technique for this calculation is error-free splitting in floating-point matrices. In this study, we improve upon our previous method by a posteriori validation using floating-point exception. In the method, we utilize the presence of overflow in a positive manner for detecting whether rounding error occurs. If overflow occurs, the result contains some exceptional values such as +/- and NaN, that is, the method fails by necessity. Otherwise, we can confirm that no rounding error occurs in the process. Therefore, reducing the possibility of overflow is important. The numerical results suggest that the proposed algorithm provides more accurate results compared with the original algorithm. Moreover, for the product of n x n matrices, when n5000, the new algorithm reduces the computing time for error-free transformation by an average of 20 % and up to 30 % compared with the original algorithm. Furthermore, the new algorithm can be used when matrix multiplication is performed using divide-and-conquer methods. Copyright (c) 2016 John Wiley & Sons, Ltd.

4
Citation
(Scopus)
• Simple floating-point filters for the two-dimensional orientation problem

Katsuhisa Ozaki, Florian Buenger, Takeshi Ogita, Shin'ichi Oishi, Siegfried M. Rump

BIT NUMERICAL MATHEMATICS   56 ( 2 ) 729 - 749  2016.06  [Refereed]

View Summary

This paper is concerned with floating-point filters for a two dimensional orientation problem which is a basic problem in the field of computational geometry. If this problem is only approximately solved by floating-point arithmetic, then an incorrect result may be obtained due to accumulation of rounding errors. A floating-point filter can quickly guarantee the correctness of the computed result if the problem is well-conditioned. In this paper, a simple semi-static floating-point filter which handles floating-point exceptions such as overflow and underflow by only one branch is developed. In addition, an improved fully-static filter is developed.

4
Citation
(Scopus)
• Makoto Mizuguchi, Akitoshi Takayasu, Takayuki Kubo, Shin'ichi Oishi

Nonlinear Theory and Its Applications, IEICE   7 ( 3 ) 386 - 394  2016

• Estimation of Sobolev-type embedding constant on domains with minimally smooth boundary using extension operator

Kazuaki Tanaka, Kouta Sekine, Makoto Mizuguchi, Shin'ichi Oishi

JOURNAL OF INEQUALITIES AND APPLICATIONS    2015.12  [Refereed]

View Summary

In this paper, we propose a method for estimating the Sobolev-type embedding constant from W-1,W-q(Omega) to L-p(Omega) on a domain Omega subset of R-n (n = 2,3, ... ) with minimally smooth boundary (also known as a Lipschitz domain), where p is an element of(n/(n - 1), infinity) and q = np/(n + p). We estimate the embedding constant by constructing an extension operator from W-1,W-q(Omega) to W-1,W-q(R-n) and computing its operator norm. We also present some examples of estimating the embedding constant for certain domains.

3
Citation
(Scopus)
• Improvement of error-free splitting for accurate matrix multiplication

Katsuhisa Ozaki, Takeshi Ogita, Shin'ichi Oishi

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS   288   127 - 140  2015.11  [Refereed]

View Summary

Recently, new algorithms for accurate matrix multiplication have been developed by the authors. A characteristic of the algorithms is a high dependency on level-3 BIAS routines, which are highly optimized for several architectures. An error-free splitting for floating-point matrices is a key technique in the algorithms. In this paper, an improvement of the error-free splitting is focused on. It is shown by numerical examples that the accuracy of computed results of matrix products can be improved by the modified error-free splitting, compared to that by the previous algorithms. (C) 2015 Elsevier B.V. All rights reserved.

• Mourning for Professor Kondo's Death(Obituary of Professor Jiro Kondo)

Oishi Shin'ichi

Bulletin of the Japan Society for Industrial and Applied Mathematics   25 ( 4 ) 180 - 180  2015

• Ozaki Katsuhisa, Ogita Takeshi, Bünger Florian, Oishi Shin'ichi

Nonlinear Theory and Its Applications, IEICE   6 ( 3 ) 364 - 376  2015

View Summary

This paper is concerned with real interval arithmetic. We focus on interval matrix multiplication. Well-known algorithms for this purpose require the evaluation of several point matrix products to compute one interval matrix product. In order to save computing time we propose a method that modifies such known algorithm by partially using low-precision floating-point arithmetic. The modified algorithms work without significant loss of tightness of the computed interval matrix product but are about 30% faster than their corresponding original versions. The negligible loss of accuracy is rigorously estimated.

• Newly Nominated JSIAM Fellows(<Special Article>JSIAM Fellows)

Oishi Shin'ichi

Bulletin of the Japan Society for Industrial and Applied Mathematics   25 ( 3 ) 98 - 98  2015

• Crisis of Mathematics Education in Science and Engineering Faculties

Oishi Shin'ichi

Bulletin of the Japan Society for Industrial and Applied Mathematics   25 ( 3 ) 97 - 97  2015

• Kazuaki Tanaka, Kouta Sekine, Makoto Mizuguchi, Shin'ichi Oishi

JSIAM Letters   7   73 - 76  2015

• Improved error bounds for linear systems with H-matrices

Atsushi Minamihata, Kouta Sekine, Takeshi Ogita, Siegfried M. Rump, Shin'ichi Oishi

IEICE NONLINEAR THEORY AND ITS APPLICATIONS   6 ( 3 ) 377 - 382  2015

View Summary

Improved componentwise error bounds for approximate solutions of linear systems are derived in the case where the coefficient of a given linear system is an H-matrix. One of the error bounds presented in this paper proves to be tighter than the existing error bound, which is effective especially for ill-conditioned cases. Numerical experiments are performed to illustrate the effect of the improvements.

• Convergence analysis of an algorithm for accurate inverse Cholesky factorization

Yuka Yanagisawa, Takeshi Ogita, Shin'ichi Oishi

JAPAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS   31 ( 3 ) 461 - 482  2014.11  [Refereed]

View Summary

This paper is concerned with factorization of symmetric and positive definite matrices which are extremely ill-conditioned. Following the results by Rump (1990), Oishi et al. (2007, 2009) and Ogita (2010), Ogita and Oishi (2012) derived an iterative algorithm for an accurate inverse matrix factorization based on Cholesky factorization for such ill-conditioned matrices. We analyze the behavior of the algorithm in detail and give reasons for convergency by the use of numerical error analysis. Main analysis is that each iteration reduces the condition number of a preconditioned matrix by a factor around the relative rounding error unit until convergence. This behavior is consistent with the numerical results.

3
Citation
(Scopus)
• Verified norm estimation for the inverse of linear elliptic operators using eigenvalue evaluation

Kazuaki Tanaka, Akitoshi Takayasu, Xuefeng Liu, Shin'ichi Oishi

JAPAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS   31 ( 3 ) 665 - 679  2014.11  [Refereed]

View Summary

This paper proposes a verified numerical method of proving the invertibility of linear elliptic operators. This method also provides a verified norm estimation for the inverse operators. This type of estimation is important for verified computations of solutions to elliptic boundary value problems. The proposed method uses a generalized eigenvalue problem to derive the norm estimation. This method has several advantages. Namely, it can be applied to two types of boundary conditions: the Dirichlet type and the Neumann type. It also provides a way of numerically evaluating lower and upper bounds of target eigenvalues. Numerical examples are presented to show that the proposed method provides effective estimations in most cases.

8
Citation
(Scopus)
• OISHI Shin'ichi

tits   19 ( 10 ) 10_58 - 10_60  2014

• OISHI Shin'ichi

IEICE Fundamentals Review   7 ( 4 ) 301 - 307  2014

View Summary

Mathematical consideration about foundation of circuit theory is explained.

• Characteristic Spaces Emerging from Primitive Chaos

Yoshihito Ogasawara, Shin'ichi Oishi

JOURNAL OF THE PHYSICAL SOCIETY OF JAPAN   83 ( 1 )  2014.01  [Refereed]

View Summary

This paper describes the emergence of two characteristic notions, nondegenerate Peano continuum and Cantor set, by the exploration of the essence of the existence of primitive chaos from a topological viewpoint. The primitive chaos is closely related to vital problems in physics itself and leads to chaotic features under natural conditions. The nondegenerate Peano continuum represents an ordinarily observed space, and the existence of a single nondegenerate Peano continuum guarantees the existence of infinite varieties of the primitive chaos leading to the chaos. This result provides an explanation of the reason why we are surrounded by diverse chaotic behaviors. Also, the Cantor set is a general or universal notion different from the special set, the Cantor middle-third set, and the existence of a single Cantor set guarantees infinite varieties of the primitive chaos leading to the chaos. This analogy implies the potential of the Cantor set for the method of new recognizing physical phenomena.

2
Citation
(Scopus)
• N. Yamanaka, S. Oishi

Nonlinear Theory and its Applications, IEICE.   51 ( 1 ) 15 - 34  2014

View Summary

An efficient format and fast algorithms of basic operations for 4-fold working precision are proposed. The proposed format is an unevaluated sum of four double precision numbers, capable of representing at least 203 bits of mantissa. Hence, it is slightly less accurate than quad-double format proposed by Hida et. al. [1], however presented algorithms based on the format are faster than those algorithms. By numerical experiments it is shown that the proposed algorithms are efficient.

• K. Sekine, A. Takayasu, S. Oishi

Nonlinear Theory and its Applications, IEICE.   5 ( 1 ) 64 - 79  2014

View Summary

This paper presents an algorithm of identifying parameters satisfying a sufficient condition of Plum's Newton-Kantorovich like theorem. Plum's theorem yields a numerical existence test of solutions for nonlinear partial differential equations. The sufficient condition of Plum's theorem is given by the nonemptiness of a region defined by one dimensional nonlinear inequalities. The aim of this paper is to develop a systematic method of constructing an inner inclusion of this region. If $\underline\rho \in {\mathbb R}^+$ is the minimum included in this region, $\underline\rho$ gives the minimum of the error bounds. Moreover, if $\overline\rho \in {\mathbb R}^+$ is the maximum included in this region, then $\overline\rho$ gives the maximum radius of a ball in which the exact solution is unique. In this paper, an algorithm is developed for finding $\rho_e$ and $\rho_u$ such that they belong to this region and become close approximations of $\underline\rho$ and $\overline\rho$, respectively. Finally, to illustrate features of Plum's theorem with our proposed algorithm, some numerical results compared with results by Plum's theorem with the Newton method are presented. In addition to this, Plum's theorem with our algorithm is also compared with Newton-Kantorovich's theorem. One of the most important facts found in this paper is that, for some examples, $\rho_e$ become smaller than error bounds obtained by Newton-Kantorovich's theorem. Moreover, also for these examples, $\rho_u$ become greater than regions indicating uniqueness of the exact solution derived by Newton-Kantorovich's theorem. This implies that Plum's theorem can be seen as a modification of Newton-Kantorovich's theorem.

• A. Takayasu, X. Liu, S. Oishi

Nonlinear Theory and its Applications, IEICE.   5 ( 1 ) 53 - 63  2014

View Summary

For Poisson's equation over a polygonal domain of general shape, the solution of which may have a singularity around re-entrant corners, we provide an explicit a priori error estimate for the approximate solution obtained by finite element methods of high degree. The method used herein is a direct extension of the one developed in preceding paper of the second and third listed authors, which provided a new approach to deal with the singularity by using linear finite elements. In the present paper, we also give a detailed discussion of the dependency of the convergence order on solution singularities, mesh sizes and degrees of the finite element method used.

• Fast verified solutions of sparse linear systems with H-matrices

A. Minamihata, K. Sekine, T. Ogita, S. Oishi

Reliable Computing    2014

• A modified algorithm for accurate inverse Cholesky factorization

Yuka Yanagisawa, Takeshi Ogita, Shin'ichi Oishi

Nonlinear Theory and Its Applications   5 ( 1 ) 34 - 46  2014

• Acknowledgement for Receiving JSIAM Fellow(<Special Article>JSIAM Fellows)

Oishi Shin'ichi

Bulletin of the Japan Society for Industrial and Applied Mathematics   23 ( 3 ) 98 - 99  2013

• A. Takayasu, X. Liu, S. Oishi

Nonlinear Theory and its Applications, IEICE.   4 ( 1 ) 34 - 61  2013

View Summary

In this paper, a numerical verification method is presented for second-order semilinear elliptic boundary value problems on arbitrary polygonal domains. Based on the Newton-Kantorovich theorem, our method can prove the existence and local uniqueness of the solution in the neighborhood of its approximation. In the treatment of polygonal domains with an arbitrary shape, which gives a singularity of the solution around the re-entrant corner, the computable error estimate of a projection into the finite-dimensional function space plays an essential role. In particular, the lack of smoothness of the solution makes classical error estimates fail on nonconvex domains. By using the Hyper-circle equation, an alternative error estimate of the projection has been proposed. Additionally, a new residual evaluation method based on the mixed finite element method works well. It yields more accurate evaluation than the existing method. The efficiency of our method is shown through illustrative numerical results on several polygonal domains.

• Y. Morikura, K. Ozaki, S. Oishi

Nonlinear Theory and its Applications, IEICE.   4 ( 1 ) 12 - 22  2013

View Summary

This paper is concerned with verification methods for numerical solutions of linear systems. Many methods for the verification require switches of rounding modes defined by the IEEE 754 standard. However, the switches cannot be supported in several computational environments. In such cases, Ogita-Rump-Oishi's method can work on such environments. Recently, Rump developed new error estimates of floating-point summation and dot product. The aim of this paper is to improve Ogita-Rump-Oishi's error estimates by using the error estimates by Rump. In addition, the computational cost of our method is comparable to that of Ogita-Rump-Oishi's method.

• K. Ozaki, T. Ogita, S. Oishi, S. M. Rump

Nonlinear Theory and its Applications, IEICE.   4 ( 1 ) 2 - 11  2013

View Summary

This paper is concerned with accurate numerical algorithms for matrix multiplication. Recently, an error-free transformation from a product of two floating-point matrices into an unevaluated sum of floating-point matrices has been developed by the authors. Combining this technique and accurate summation algorithms, new algorithms for accurate matrix multiplication could be investigated. In this paper, it is mentioned that the previous work is not the unique way to achieve an error-free transformation and the constraint of the error-free transformation is clarified. For the application, a new algorithm is developed reducing the number of matrix products compared to the previous algorithm.

• T. Nishi, S. Oishi, N. Takahashi

Nonlinear Theory and its Applications, IEICE.   4 ( 4 ) 430 - 450  2013

View Summary

The authors recently published a paper on some properties of the solution curves for the last n-1 equations of F(x)+Ax=b where x=[x1,x2,...,xn]T is a variable, F(x):Rn → Rn is a nonlinear function of which the first and second derivatives are strictly positive, A ∈ Rn × n is an Ω-matrix, and b ∈ Rn is a constant vector. In that paper, the authors showed that any solution curve possesses neither maximal points nor inflection points with respect to x1, by making use of a fundamental property of Ω-matrices, which is expressed in the form of inequality. However, the proof was a little unclear as shown in Introduction. The objective of this paper is to give an explicit proof for the property of Ω-matrices, which makes the author's previous result more rigorous.

• VERIFIED EIGENVALUE EVALUATION FOR THE LAPLACIAN OVER POLYGONAL DOMAINS OF ARBITRARY SHAPE

Xuefeng Liu, Shin'ichi Oishi

SIAM JOURNAL ON NUMERICAL ANALYSIS   51 ( 3 ) 1634 - 1654  2013  [Refereed]

View Summary

The finite element method (FEM) is applied to bound leading eigenvalues of the Laplace operator over polygonal domains. Compared with classical numerical methods, most of which can only give concrete eigenvalue bounds over special domains of symmetry, our proposed algorithm can provide concrete eigenvalue bounds for domains of arbitrary shape, even when the eigenfunction has a singularity. The problem of eigenvalue estimation is solved in two steps. First, we construct a computable a priori error estimation for the FEM solution of Poisson's problem, which holds even for nonconvex domains with reentrant corners. Second, new computable lower bounds are developed for the eigenvalues. Because the interval arithmetic is implemented throughout the computation, the desired eigenvalue bounds are expected to be mathematically correct. We illustrate several computation examples, such as the cases of an L-shaped domain and a crack domain, to demonstrate the efficiency and flexibility of the proposed method.

44
Citation
(Scopus)
• Guaranteed high-precision estimation for P-0 interpolation constants on triangular finite elements

Xuefeng Liu, Shin'ichi Oishi

JAPAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS   30 ( 3 ) 635 - 652  2013  [Refereed]

View Summary

We consider an explicit estimation for error constants from two basic constant interpolations on triangular finite elements. The problem of estimating the interpolation constants is related to the eigenvalue problems of the Laplacian with certain boundary conditions. By adopting the Lehmann-Goerisch theorem and finite element spaces with a variable mesh size and polynomial degree, we succeed in bounding the leading eigenvalues of the Laplacian and the error constants with high precision. An online demo for the constant estimation is also available at http://www.xfliu.org/onlinelab/.

8
Citation
(Scopus)
• Consideration of a Primitive Chaos

Yoshihito Ogasawara, Shin'ichi Oishi

JOURNAL OF THE PHYSICAL SOCIETY OF JAPAN   81 ( 10 )  2012.10  [Refereed]

View Summary

Since the chaos was discovered, it has been recognized that we are surrounded by diverse chaotic behaviors. The purpose of this study is to reconsider the implication of this fact through the notion of a primitive chaos. Under natural conditions, each primitive chaos leads to apparent chaotic features, such as the existence of a nonperiodic orbit, the existence of the periodic point whose prime period is n for any n is an element of N, the existence of a dense orbit, the density of periodic points, sensitive dependence on initial conditions, and topological transitivity, while infinite varieties of the primitive chaos are guaranteed by a nondegenerate Peano continuum.

3
Citation
(Scopus)
• A robust algorithm for geometric predicate by error-free determinant transformation

Katsuhisa Ozaki, Takeshi Ogita, Shin'ichi Oishi

INFORMATION AND COMPUTATION   216   3 - 13  2012.07  [Refereed]

View Summary

This paper concerns a robust algorithm for the 2D orientation problem which is one of the basic tasks in computational geometry. Recently, a fast and accurate floating-point summation algorithm is investigated by Rump, Ogita and Oishi in [S.M. Rump, T. Ogita, S. Oishi, Accurate floating-point summation. Part I: Faithful rounding, SIAM J. Sci. Comput. 31 (1) (2008) 189-2241, in which a new kind of an error-free transformation of floating-point numbers is used. Based on it, a new algorithm of error-free determinant transformation for the 2D orientation problem is proposed, which gives a correct result. Numerical results are presented for illustrating that the proposed algorithm has some advantage over preceding algorithms in terms of measured computing time. (C) 2012 Elsevier Inc. All rights reserved.

2
Citation
(Scopus)
• Oishi Shin'ichi, Ogita Takeshi, Ozaki Katsuhisa

Bulletin of the Japan Society for Industrial and Applied Mathematics   22 ( 3 ) 216 - 218  2012

• Error-free transformations of matrix multiplication by using fast routines of matrix multiplication and its applications

Katsuhisa Ozaki, Takeshi Ogita, Shin'ichi Oishi, Siegfried M. Rump

NUMERICAL ALGORITHMS   59 ( 1 ) 95 - 118  2012.01  [Refereed]

View Summary

This paper is concerned with accurate matrix multiplication in floating-point arithmetic. Recently, an accurate summation algorithm was developed by Rump et al. (SIAM J Sci Comput 31(1):189-224, 2008). The key technique of their method is a fast error-free splitting of floating-point numbers. Using this technique, we first develop an error-free transformation of a product of two floating-point matrices into a sum of floating-point matrices. Next, we partially apply this error-free transformation and develop an algorithm which aims to output an accurate approximation of the matrix product. In addition, an a priori error estimate is given. It is a characteristic of the proposed method that in terms of computation as well as in terms of memory consumption, the dominant part of our algorithm is constituted by ordinary floating-point matrix multiplications. The routine for matrix multiplication is highly optimized using BLAS, so that our algorithms show a good computational performance. Although our algorithms require a significant amount of working memory, they are significantly faster than 'gemmx' in XBLAS when all sizes of matrices are large enough to realize nearly peak performance of 'gemm'. Numerical examples illustrate the efficiency of the proposed method.

24
Citation
(Scopus)
• Fast algorithms for floating-point interval matrix multiplication

Katsuhisa Ozaki, Takeshi Ogita, Siegfried M. Rump, Shin'ichi Oishi

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS   236 ( 7 ) 1795 - 1814  2012.01  [Refereed]

View Summary

We discuss several methods for real interval matrix multiplication. First, earlier studies of fast algorithms for interval matrix multiplication are introduced: naive interval arithmetic, interval arithmetic by midpoint-radius form by Oishi-Rump and its fast variant by Ogita-Oishi. Next, three new and fast algorithms are developed. The proposed algorithms require one, two or three matrix products, respectively. The point is that our algorithms quickly predict which terms become dominant radii in interval computations. We propose a hybrid method to predict which algorithm is suitable for optimizing performance and width of the result. Numerical examples are presented to show the efficiency of the proposed algorithms. (C) 2011 Elsevier B.V. All rights reserved.

5
Citation
(Scopus)
• Matrix Multiplication with Guaranteed Accuracy by Level 3 BLAS

Katsuhisa Ozaki, Takeshi Ogita, Shin'ichi Oishi

INTERNATIONAL CONFERENCE OF COMPUTATIONAL METHODS IN SCIENCES AND ENGINEERING 2009 (ICCMSE 2009)   1504   1128 - 1133  2012  [Refereed]

View Summary

This paper is concerned with an accurate computing for matrix multiplication. We show that matrix multiplication can be transformed into a summation of floating-point matrices by mainly using level 3 operations in BLAS. We call it 'error-free transformation of matrix multiplication'. By combining this error-free transformation and accurate summation algorithms for floating-point numbers, we can obtain an accurate result of matrix multiplication. Numerical examples are presented to illustrate the efficiency of the proposed algorithm.

• T. Ogita, S. Oishi

Nonlinear Theory and Its Applications, IEICE   Vol. 3 ( No. 1 ) 103 - 111  2012.01

View Summary

In this paper, an algorithm for an accurate matrix factorization based on Cholesky factorization for extremely ill-conditioned matrices is proposed. The Cholesky factorization is widely used for solving a system of linear equations whose coefficient matrix is symmetric and positive definite. However, it sometimes breaks down by the presence of an imaginary root due to the accumulation of rounding errors, even if the matrix is actually positive definite. To overcome this, a completely stable algorithm named inverse Cholesky factorization is investigated, which never breaks down as long as the matrix is symmetric and positive definite. The proposed algorithm consists of standard numerical algorithms and an accurate algorithm for dot products. Moreover, it is shown that the algorithm can also verify the positive definiteness of a given real symmetric matrix. Numerical results are presented for illustrating the performance of the proposed algorithms.

• A Fast Verified Double Repeated Integration Algorithm using Double Exponential Formula

Naoya Yamanaka, aseda University, Shin'ichi Oishi, Waseda University, Takeshi Ogita, Tokyo Woman's, Christian University

Reliable Computing   Vol. 15 ( No.2 ) 156 - 167  2011.06

• Tetsuo Nishi, Waseda University, Siegflied M. Rump, Hamburg University of Technology (TUHH, Shin'ichi Oishi, Waseda University

NOLTA, IEICE   Vol. 2 ( No.2 ) 226 - 245  2011.04

View Summary

In this paper we study the generation of an ill-conditioned integer matrix A=[aij] with |aij|≤µ for some given constant µ. Let n be the order of A. We first give some upper bounds of the condition number of A in terms of n and µ. We next propose new methods to generate extremely ill-conditioned integer matrices. These methods are superior to the well-known method by Rump in some respects, namely, the former has a simple algorithm to generate a larger variety of ill-conditioned matrices. In particular we propose a method to generate ill-conditioned matrices with a choice of desirable singular value distributions as benchmark matrices.

• Tight and efficient enclosure of matrix multiplication by using optimized BLAS

Katsuhisa Ozaki, Takeshi Ogita, Shin'ichi Oishi

NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS   18 ( 2 ) 237 - 248  2011.03  [Refereed]

View Summary

This paper is concerned with the tight enclosure of matrix multiplication AB for two floating-point matrices A and B. The aim of this paper is to compute component-wise upper and lower bounds of the exact result C of the matrix multiplication AB by floating-point arithmetic. Namely, an interval matrix enclosing C is obtained. In this paper, new algorithms for enclosing C are proposed. The proposed algorithms are designed to mainly exploit the level 3 operations in BLAS. Although the proposed algorithms take around twice as much costs as a standard algorithm promoted by Oishi and Rump, the accuracy of the result by the proposed algorithms is better than that of the standard algorithm. At the end of this paper, we present numerical examples showing the efficiency of the proposed algorithms. Copyright (C) 2010 John Wiley & Sons, Ltd.

9
Citation
(Scopus)
• An algorithm for automatically selecting a suitable verification method for linear systems

Katsuhisa Ozaki, Takeshi Ogita, Shin&apos;ichi Oishi

NUMERICAL ALGORITHMS   56 ( 3 ) 363 - 382  2011.03  [Refereed]

View Summary

Several methods have been proposed to calculate a rigorous error bound of an approximate solution of a linear system by floating-point arithmetic. These methods are called &apos;verification methods&apos;. Applicable range of these methods are different. It depends mainly on the condition number and the dimension of the coefficient matrix whether such methods succeed to work or not. In general, however, the condition number is not known in advance. If the dimension or the condition number is large to some extent, then Oishi-Rump&apos;s method, which is known as the fastest verification method for this purpose, may fail. There are more robust verification methods whose computational cost is larger than the Oishi-Rump&apos;s one. It is not so efficient to apply such robust methods to well-conditioned problems. The aim of this paper is to choose a suitable verification method whose computational cost is minimum to succeed. First in this paper, four fast verification methods for linear systems are briefly reviewed. Next, a compromise method between Oishi-Rump&apos;s and Ogita-Oishi&apos;s one is developed. Then, an algorithm which automatically and efficiently chooses an appropriate verification method from five verification methods is proposed. The proposed algorithm does as much work as necessary to calculate error bounds of approximate solutions of linear systems. Finally, numerical results are presented.

2
Citation
(Scopus)
• Oishi Shin'ichi

Bulletin of the Japan Society for Industrial and Applied Mathematics   21 ( 1 ) 17 - 20  2011

• Ozaki Katsuhisa, Ogita Takeshi, Oishi Shin'ichi

Bulletin of the Japan Society for Industrial and Applied Mathematics   21 ( 3 ) 186 - 196  2011

View Summary

This paper is concerned with interval arithmetic, especially, an enclosure of a matrix product is focused on. Using level 3 operations of matrix computations, an algorithm outputting a tight enclosure for matrix multiplication is proposed. Most of the algorithms for this purpose require switches of rounding modes defined in the IEEE standard 754. However some programing enviroments have not supported them. Our proposed method demands only rounding-to-nearest mode, so that it is very portable.

• A Note on a Verified Automatic Integration Algorithm

N. Yamanaka, M. Kashiwagi, S. Oishi, T. Ogita

Reliable Computing   VOl. 15 ( No. 2 ) 156 - 167  2011

• Numerical Verification of Existence for Solutions of Dirichlet Boundary Value Problems of Semilinear Elliptic Equations

Shin’ichi Oishi, Akitoshi Takayasu

NOLTA, IEICE   Vol. 2 ( No.1 ) 74 - 89  2011.01

• Akitoshi Takayasu, Waseda University, Shin'ichi Oishi, Waseda University

Nonlinear Theory and Its Applications, IEICE   Vol. 2 ( No. 1 ) 74 - 89  2011.01

View Summary

Present authors have presented with Takayuki Kubo at University of Tsukuba a method of a computer assisted proof for the existence and uniqueness of solutions to two-point boundary value problems of nonlinear ordinary differential equations in the paper submitted for NOLTA, IEICE. This method uses piecewise linear finite element base functions and sometimes requires fine mesh. To overcome this difficulty, in this paper, an improved method is presented for the norm estimation of the residual to the operator equation. In this refined formulation, piecewise quadratic finite element base functions are used. A kind of the residual technique works sophisticatedly well. It is stated that the estimation of the residual can be expected smaller than that of the previous method. Finally, four examples are presented. Each result demonstrates that a remarkable improvement is achieved in accuracy of the guaranteed error estimation.

• Siegfried M. Rump, Shin’ichi Oishi

NOLTA, IEICE   Vol. 1 ( No. 1 ) 89 - 96  2010.10

View Summary

It is well known that it is an ill-posed problem to decide whether a function has a multiple root. For example, an arbitrarily small perturbation of a real polynomial may change a double real root into two distinct real or complex roots. In this paper we describe a computational method for the verified computation of a complex disc to contain exactly k roots of a univariate nonlinear function. The function may be given by some program. Computational results using INTLAB, the Matlab toolbox for reliable computing, demonstrate properties and limits of the method.

• S. M. Rump, T. Ogita, S. Oishi

NOLTA, IEICE   Vol. 1 ( No. 1 ) 2 - 24  2010.10

View Summary

Given a vector pi of floating-point numbers with exact sum s, we present a new algorithm with the following property: Either the result is a faithful rounding of s, or otherwise the result has a relative error not larger than epsKcond(∑pi) for K to be specified. The statements are also true in the presence of underflow, the computing time does not depend on the exponent range, and no extra memory is required. Our algorithm is fast in terms of measured computing time because it allows good instruction-level parallelism. A special version for K=2, i.e., quadruple precision is also presented. Computational results show that this algorithm is more accurate and faster than competitors such as XBLAS.

• Naoya Yamanaka, Tomoaki Okayama, Shin'ichi Oishi, Takeshi Ogita

NOLTA, IEICE   Vol. 1 ( No. 1 ) 119 - 132  2010.10

View Summary

A fast verified automatic integration algorithm is proposed for calculating univariate integrals over finite intervals. This algorithm is based on the double exponential formula proposed by Takahasi and Mori. The double exponential formula uses a certain trapezoidal rule. This trapezoidal rule is determined by fixing two parameters, the width h of a subdivision of a finite interval and the number n of subdivision points of this subdivision. A theorem is presented for calculating h and n as a function of a given tolerance of the verified numerical integration of a definite integral. An efficient a priori method is also proposed for evaluating function calculation errors including rounding errors of floating point calculations. Combining these, a fast algorithm is proposed for verified automatic integration. Numerical examples are presented for illustrating effectiveness of the proposed algorithm.

• Akitoshi Takayasu, Shin’ichi Oishi, Takayuki Kubo

NOLTA, IEICE   Vol. 1 ( No. 1 ) 105 - 118  2010.10

View Summary

In this paper, a numerical method is presented for verifying the existence and uniqueness of solutions to two-point boundary value problems of nonlinear ordinary differential equations. Taking into account every error of numerical computations such as the discretization error and the rounding error, this method also provides mathematically guaranteed error bounds between approximations obtained by numerical computations and the exact solution whose existence is proven by the numerical existence theorem, which is based on the Newton-Kantorovich theorem. Finally, illustrative numerical results are presented for showing the usefulness of the method.

• Fast Verification for All Eigenpairs in Symmetric Positive Definite Generalized Eigenvalue Problem

S. Miyajima, T. Ogita, S. M. Rump, S. Oishi

Reliable Computing,   VOl. 14   24 - 45  2010.06

• A Verified Automatic Contour Integration Algorithm

Naoya Yamanaka, Shin'ichi Oishi, Takeshi Ogita

REC 2010: PROCEEDINGS OF THE 4TH INTERNATIONAL WORKSHOP ON RELIABLE ENGINEERING COMPUTING: ROBUST DESIGN - COPING WITH HAZARDS, RISK AND UNCERTAINTY     149 - 158  2010  [Refereed]

View Summary

A verified automatic integration algorithm is proposed for calculating contour integration over complex field using numerical computations. The proposed algorithm is based on trapezoidal rule for angle. The error analysis of the method have been presented by several authors, however, these investigations are done basically for examining the rates of convergence, and several constants in these error formula were left unevaluated. In order to construct verified numerical integrator using the algorithm, the error formula is presented. To construct efficient verified numerical integrator, an efficient a priori method of evaluating function calculation errors is adopted. Combining these, a verified automatic integration algorithm is proposed. Numerical examples are presented for illustrating effectiveness of the proposed algorithm.

• Iterative Refinement for Ill-Conditioned Linear Systems

Shin&apos;ichi Oishi, Takeshi Ogita, Siegfried M. Rump

JAPAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS   26 ( 2-3 ) 465 - 476  2009.10  [Refereed]

View Summary

This paper treats a linear equation
Av = b,
where A is an element of F(nxn) and b is an element of F(n). Here, F is a set of floating point numbers. Let u be the unit round-off of the working precision and kappa(A) = parallel to A parallel to(infinity)parallel to A(-1)parallel to(infinity) be the condition number of the problem. In this paper, ill-conditioned problems with
1 &lt; u kappa(A) &lt; infinity
are considered and an iterative refinement algorithm for the problems is proposed. In this paper, the forward and backward stability will be shown for this iterative refinement algorithm.

• Adaptive and Efficient Algorithm for 2D Orientation Problem

Katsuhisa Ozaki, Takeshi Ogita, Siegfried M. Rump, Shin'ichi Oishi

JAPAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS   26 ( 2-3 ) 215 - 231  2009.10  [Refereed]

View Summary

This paper is concerned with a robust geometric predicate for the 2D orientation problem. Recently, a fast and accurate floating-point summation algorithm is investigated by Rump, Ogita and Oishi, which provably outputs a result faithfully rounded from the exact value of the summation of floating-point numbers. We optimize their algorithm for applying it to the 2D orientation problem which requires only a correct sign of a determinant of a 3 x 3 matrix. Numerical results illustrate that our algorithm works fairly faster than the state-of-the-art algorithm in various cases.

• Fast Verified Solutions of Linear Systems

Takeshi Ogita, Shin'ichi Oishi

JAPAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS   26 ( 2-3 ) 169 - 190  2009.10  [Refereed]

View Summary

This paper aims to survey fast methods of verifying the accuracy of a numerical solution of a linear system. For the last decade, a number of fast verification algorithms have been proposed to obtain an error bound of a numerical solution of a dense or sparse linear system. Such fast algorithms rely on the verified numerical computation using floating-point arithmetic defined by IEEE standard 754. Some fast verification methods for dense and sparse linear systems are reviewed together with corresponding numerical results to show the practical use and efficiency of the verified numerical computation as much as possible.

• International Activities of the Institute

Oishi Shin'ichi

Bulletin of the Japan Society for Industrial and Applied Mathematics   19 ( 1 ) 1 - 1  2009

• Tight Enclosures of Solutions of Linear Systems

T. Ogita, S. Oishi

International Series of Numerical Mathematics   157   167 - 178  2009

• Shin’ichi Oishi, Kunio Tanabe

JSIAM Letter   1   5 - 8  2009

View Summary

This paper concerns with the following linear programming problem: $\mbox{Maximize } c^tx, \mbox{ subject to } Ax \leqq b \mbox{ and } x\geqq 0,$ where $A \in \F^{m\times n}$, $b \in \F^m$ and $c, x \in \F^n$. Here, $\F$ is a set of floating point numbers. The aim of this paper is to propose a numerical method of including an optimum point of this linear programming problem provided that a good approximation of an optimum point is given. The proposed method is base on Kantorovich's theorem and the continuous Newton method. Kantorovich's theorem is used for proving the existence of a solution for complimentarity equation and the continuous Newton method is used to prove feasibility of that solution. Numerical examples show that a computational cost to include optimum point is about 4 times than that for getting an approximate optimum solution.

• Numerical Verification of Five Solutions in Two-transistor Circuits

Yusuke Nakaya, Tetsuo Nishi, Shin’ichi Oishi, Martin Claus

Japan J. Indust. Appl. Math.   Vol. 26 ( No. 2 ) 327 - 336  2009

• A parallel algorithm for accurate dot product

N. Yamanaka, T. Ogita, S. M. Rump, S. Oishi

PARALLEL COMPUTING   34 ( 6-8 ) 392 - 410  2008.07  [Refereed]

View Summary

Parallel algorithms for accurate summation and dot product are proposed, They are parallelized versions of fast and accurate algorithms of calculating sum and dot product using error-free transformations which are recently proposed by Ogita et al. [T. Ogita, S.M. Rump, S. Oishi, Accurate sum and dot product, SIAM J. Sci. Comput. 26 (6) (2005) 1955-1988]. They have shown their algorithms are fast in terms of measured computing time. However, due to the strong data dependence in the process of their algorithms, it is difficult to parallelize them. Similarly to their algorithms, the proposed parallel algorithms in this paper are designed to achieve the results as if computed in K-fold working precision with keeping the fastness of their algorithms. Numerical results are presented showing the performance of the proposed parallel algorithm of calculating dot product. (C) 2008 Elsevier B.V. All rights reserved.

12
Citation
(Scopus)
• OISHI Shin'ichi

IEICE Fundamentals Review   2 ( 2 ) 9 - 19  2008

• ACCURATE FLOATING-POINT SUMMATION PART II: SIGN, K-FOLD FAITHFUL AND ROUNDING TO NEAREST

Siegfried M. Rump, Takeshi Ogita, Shin'ichi Oishi

SIAM JOURNAL ON SCIENTIFIC COMPUTING   31 ( 2 ) 1269 - 1302  2008  [Refereed]

View Summary

In Part II of this paper we first refine the analysis of error-free vector transformations presented in Part I. Based on that we present an algorithm for calculating the rounded-to-nearest result of s := Sigma pi for a given vector of floating-point numbers pi, as well as algorithms for directed rounding. A special algorithm for computing the sign of s is given, also working for huge dimensions. Assume a floating-point working precision with relative rounding error unit eps. We define and investigate a K-fold faithful rounding of a real number r. Basically the result is stored in a vector Res(nu) of K nonoverlapping floating-point numbers such that Sigma Res(nu) approximates r with relative accuracy eps(K), and replacing Res(K) by its floating-point neighbors in Sigma Res(nu) forms a lower and upper bound for r. For a given vector of floating-point numbers with exact sum s, we present an algorithm for calculating a K-fold faithful rounding of s using solely the working precision. Furthermore, an algorithm for calculating a faithfully rounded result of the sum of a vector of huge dimension is presented. Our algorithms are fast in terms of measured computing time because they allow good instruction-level parallelism, they neither require special operations such as access to mantissa or exponent, they contain no branch in the inner loop, nor do they require some extra precision. The only operations used are standard floating-point addition, subtraction, and multiplication in one working precision, for example, double precision. Certain constants used in the algorithms are proved to be optimal.

54
Citation
(Scopus)
• Discretization principles for linear two-point boundary value problems, II

Tetsuro Yamamoto, Shin'ichi Oishi, Qing Fang

NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION   29 ( 1-2 ) 213 - 224  2008.01  [Refereed]

View Summary

Consider the boundary value problem Lu equivalent to -(pu ')' + qu ' + ru = f, a &lt;= x &lt;= b, u(a) = u(b) = 0. Let H(v)A(v)U = f and (A) over cap U-v = (f) over cap be its finite difference equations and piecewise linear finite element equations on partitions Delta(v) : a = x(0)(v) &lt; x(1)(v) &lt; ... &lt; x(nv+1)(v) = b, v = 1, 2.... with h(i)(v) = x(i)(v) - x(i-1)(v), h(v) = max(i)h(i)(v) -&gt; 0 as v -&gt; infinity, where H-v, are n(v) x n(v) diagonal matrices and A(v) as well as (A) over cap (v) are n(v) x n(v) tridiagonal. It is shown that the following three conditions are equivalent: (i) The boundary value problem has a unique solution u is an element of C-2[a, b]. (ii) For sufficiently large v &gt;= v(0), the inverse A(v)(-1) = (g(ij)(v)) exists and vertical bar g(ij)(v)vertical bar &lt;= M, for all(i,j) with a constant M &gt; 0 independent of h(v). (iii) For sufficiently large v &gt;= (v) over cap (0), (A(v)) over cap (-1) = ((g(ij)) over cap)(v) exists and vertical bar(g(ij)) over cap (v)vertical bar &lt;= (M) over cap, for all(i,j) with a constant (M) over cap &gt; 0 independent of h(v). It is also shown by a numerical example that the finite difference method with uniform nodes x(i+1) = x(i) + h, 0 &lt;= i &lt;= n, h = (b - a)/(n + 1) applied to the boundary value problem with no solution gives a ghost solution for every n.

2
Citation
(Scopus)
• ACCURATE FLOATING-POINT SUMMATION PART I: FAITHFUL ROUNDING

Siegfried M. Rump, Takeshi Ogita, Shin'ichi Oishi

SIAM JOURNAL ON SCIENTIFIC COMPUTING   31 ( 1 ) 189 - 224  2008  [Refereed]

View Summary

Given a vector of floating-point numbers with exact sum s, we present an algorithm for calculating a faithful rounding of s, i.e., the result is one of the immediate floating-point neighbors of s. If the sum s is a floating-point number, we prove that this is the result of our algorithm. The algorithm adapts to the condition number of the sum, i.e., it is fast for mildly conditioned sums with slowly increasing computing time proportional to the logarithm of the condition number. All statements are also true in the presence of underflow. The algorithm does not depend on the exponent range. Our algorithm is fast in terms of measured computing time because it allows good instruction-level parallelism, it neither requires special operations such as access to mantissa or exponent, it contains no branch in the inner loop, nor does it require some extra precision: The only operations used are standard floating-point addition, subtraction, and multiplication in one working precision, for example, double precision. Certain constants used in the algorithm are proved to be optimal.

124
Citation
(Scopus)
• ACCURATE FLOATING-POINT SUMMATION PART I: FAITHFUL ROUNDING

Siegfried M. Rump, Takeshi Ogita, Shin'ichi Oishi

SIAM JOURNAL ON SCIENTIFIC COMPUTING   31 ( 1 ) 189 - 224  2008  [Refereed]

View Summary

Given a vector of floating-point numbers with exact sum s, we present an algorithm for calculating a faithful rounding of s, i.e., the result is one of the immediate floating-point neighbors of s. If the sum s is a floating-point number, we prove that this is the result of our algorithm. The algorithm adapts to the condition number of the sum, i.e., it is fast for mildly conditioned sums with slowly increasing computing time proportional to the logarithm of the condition number. All statements are also true in the presence of underflow. The algorithm does not depend on the exponent range. Our algorithm is fast in terms of measured computing time because it allows good instruction-level parallelism, it neither requires special operations such as access to mantissa or exponent, it contains no branch in the inner loop, nor does it require some extra precision: The only operations used are standard floating-point addition, subtraction, and multiplication in one working precision, for example, double precision. Certain constants used in the algorithm are proved to be optimal.

124
Citation
(Scopus)
• Discretization principles for linear two-point boundary value problems, II

Tetsuro Yamamoto, Shin'ichi Oishi, Qing Fang

NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION   29 ( 1-2 ) 213 - 224  2008.01  [Refereed]

View Summary

Consider the boundary value problem Lu equivalent to -(pu ')' + qu ' + ru = f, a &lt;= x &lt;= b, u(a) = u(b) = 0. Let H(v)A(v)U = f and (A) over cap U-v = (f) over cap be its finite difference equations and piecewise linear finite element equations on partitions Delta(v) : a = x(0)(v) &lt; x(1)(v) &lt; ... &lt; x(nv+1)(v) = b, v = 1, 2.... with h(i)(v) = x(i)(v) - x(i-1)(v), h(v) = max(i)h(i)(v) -&gt; 0 as v -&gt; infinity, where H-v, are n(v) x n(v) diagonal matrices and A(v) as well as (A) over cap (v) are n(v) x n(v) tridiagonal. It is shown that the following three conditions are equivalent: (i) The boundary value problem has a unique solution u is an element of C-2[a, b]. (ii) For sufficiently large v &gt;= v(0), the inverse A(v)(-1) = (g(ij)(v)) exists and vertical bar g(ij)(v)vertical bar &lt;= M, for all(i,j) with a constant M &gt; 0 independent of h(v). (iii) For sufficiently large v &gt;= (v) over cap (0), (A(v)) over cap (-1) = ((g(ij)) over cap)(v) exists and vertical bar(g(ij)) over cap (v)vertical bar &lt;= (M) over cap, for all(i,j) with a constant (M) over cap &gt; 0 independent of h(v). It is also shown by a numerical example that the finite difference method with uniform nodes x(i+1) = x(i) + h, 0 &lt;= i &lt;= n, h = (b - a)/(n + 1) applied to the boundary value problem with no solution gives a ghost solution for every n.

2
Citation
(Scopus)
• Accurate Matrix Multiplication by using Level 3 BLAS Operation

K. Ozaki, T. Ogita, S. M. Rump, S. Oishi

2008 International Symposium on Nonlinear Theory and its Applications     508 - 511  2008

• Convergence of Rump's method for inverting arbitrarily ill-conditioned matrices

Shin'ichi Oishi, Kunio Tanabe, Takeshi Ogita, Siegfried M. Rump

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS   205 ( 1 ) 533 - 544  2007.08  [Refereed]

View Summary

In this paper, the problem of inverting regular matrices with arbitrarily large condition number is treated in double precision defined by IEEE 754 floating point standard. In about 1984, Rump derived a method for inverting arbitrarily ill-conditioned matrices. The method requires the possibility to calculate a dot product in higher precision. Rump's method is of theoretical interest. Rump made it clear that inverting an arbitrarily ill-conditioned matrix in single or double precision does not produce meaningless numbers, but contains a lot of information in it. Rump's method uses such inverses as preconditioners. Numerical experiments exhibit that Rump's method converges rapidly for various matrices with large condition numbers. Why Rump's method is so efficient for inverting arbitrarily ill-conditioned matrices is a little mysterious. Thus, to prove its convergence is an interesting problem in numerical error analysis. In this article, a convergence theorem is presented for a variant of Rump's method. (C) 2006 Elsevier B.V. All rights reserved.

19
Citation
(Scopus)
• A method of obtaining verified solutions for linear systems suited for Java

K. Ozaki, T. Ogita, S. Miyajima, S. Oishi, S. M. Rump

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS   199 ( 2 ) 337 - 344  2007.02  [Refereed]

View Summary

Recent development of Java's optimization techniques makes Java one of the most useful programming languages for numerical computations. This paper proposes a numerical method of obtaining verified approximate solutions of linear systems. Usual methods for verified computations use switches of rounding modes defined in IEEE standard 754. However, such switches of rounding modes have not been supported in Java. This method avoids using directed rounding, so that it is implementable on a wide range of programming languages including Java. Numerical experiments using Java illustrate that the method can give a very accurate error bound for an approximate solution of a linear system with almost same computational cost as that for calculating an approximate inverse by the Gaussian elimination. (c) 2005 Elsevier B.V. All rights reserved.

6
Citation
(Scopus)
• OISHI Shin'ichi

IEICE Fundamentals Review   1 ( 1 ) 2 - 3  2007

• Accurate Matrix Multiplication with Multiple Floating-point Numbers

K. Ozaki, T. Ogita, S. M. Rump, S. Oishi

Proceedings of 2007 International Symposium on Nonlinear Theory and its Applications     16 - 19  2007

• On three theorems of lees for numerical treatment of semilinear two-point boundary value problems

Tetsuro Yamamoto, Shin'ichi Oishi

JAPAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS   23 ( 3 ) 293 - 313  2006.10  [Refereed]

View Summary

This paper is concerned with semilinear tow-point boundary value problems of the form -(p(x)u')'+ f (x, u) = 0, a &lt;= x &lt;= b, alpha(0)u(a) - alpha(1)u'(a) = alpha, beta(0)u(b) + beta(1)u'(b) = beta, alpha(i) &gt;= 0, beta(i) &gt;= 0, i = 0, 1, alpha(0)+alpha(1) &gt; 0, beta(0)+beta(1) &gt; 0, alpha(0)+beta(0) &gt; 0. Under the assumption inf f(u) &gt; - lambda(1), where lambda(1) is the smallest eigenvalue of Lu = -(pu')' with the boundary conditions, unique existence theorems of solution for the continuous problem and a discretized. system with not necessarily uniform nodes are given as well as error estimates. The results generalize three theorems of Lees for u" = f (x, u), 0 &lt;= x &lt;= 1, u(0) = alpha, u(1) = beta.

• Numerical Verification for Each Eigenpair of Symmetric Matrix

Shinya Miyajima, Takeshi Ogita, Shin'ichi Oishi

Trans. JSIAM   16 ( 4 ) 535 - 552  2006

• Fast and robust algorithm for geometric predicates using floating-point arithmetic

K. Ozaki, T. Ogita, S.M. Rump, S. Oishi

Trans. JSIAM   16 ( 4 ) 553 - 562  2006

• 実対称行列の各固有値に対する精度保証付き数値計算法

宮島信也, 荻田武史, 大石進一

日本応用数理学会論文誌   15 ( 3 ) 253 - 268  2005.09

• 悪条件連立一次方程式の精度保証付き数値計算法

太田 貴久, 荻田 武史, S. M. Rump, 大石 進一

日本応用数理学会論文誌   15 ( 3 ) 269 - 286  2005.09

• Fast inclusion of interval matrix multiplication

Takeshi Ogita, Shin'ichi Oishi

Reliable Computing   11 ( 3 ) 191 - 205  2005.06

View Summary

This paper is concerned with interval matrix multiplication.New algorithms are proposed to calculate an inclusion of the product of interval matrices using rounding mode controlled computation. Thecomputational cost of the proposed algorithms is almost the same as that for calculating an inclusion of the product of point matrices.Numerical results are presented to illustrate that the new algorithms are much faster than the conventional algorithms and that the guaranteed accuracies obtained by the proposed algorithms are comparable to those of the conventional algorithms. © Springer 2005.

7
Citation
(Scopus)
• Numerical verification of solutions of Nekrasov's integral equation

Computing   26 ( 6 ) 1955 - 1988  2005.06

• 大規模連立一次方程式のための高速精度保証法

荻田 武史, 大石 進一

情報処理学会論文誌: 数理モデル化と応用   46:SIG10 (TOM12)   10 - 18  2005.06

• Numerical Verification for Each Eigenvalues of Symmetric Matrix

Shinya Miyajima, Takeshi Ogita, Shin'ichi Oishi

Trans. JSIAM   Vol. 15 ( 3 ) 253 - 268  2005.03

• Fast verification for respective eigenvalues of symmetric matrix

S Miyajima, T Ogita, S Oishi

COMPUTER ALGEBRA IN SCIENFIFIC COMPUTING, PROCEEDINGS   3718   306 - 317  2005  [Refereed]

View Summary

A fast verification algorithm of calculating guaranteed error bounds for all approximate eigenvalues of a real symmetric matrix is proposed. In the proposed algorithm, Rump's and Wilkinson's bounds are combined. By introducing Wilkinson's bound, it is possible to improve the error bound obtained by the verification algorithm based on Rump's bound with a small additional cost. Finally, this paper includes some numerical examples to show the efficiency of the proposed algorithm.

• Accurate sum and dot product

T Ogita, SM Rump, S Oishi

SIAM JOURNAL ON SCIENTIFIC COMPUTING   26 ( 6 ) 1955 - 1988  2005  [Refereed]

View Summary

Algorithms for summation and dot product of floating-point numbers are presented which are fast in terms of measured computing time. We show that the computed results are as accurate as if computed in twice or K-fold working precision, K &gt;= 3. For twice the working precision our algorithms for summation and dot product are some 40% faster than the corresponding XBLAS routines while sharing similar error estimates. Our algorithms are widely applicable because they require only addition, subtraction, and multiplication of floating-point numbers in the same working precision as the given data. Higher precision is unnecessary, algorithms are straight loops without branch, and no access to mantissa or exponent is necessary.

232
Citation
(Scopus)
• Numerical verification of solutions of periodic integral equations with a singular kernel

S Murashige, S Oishi

NUMERICAL ALGORITHMS   37 ( 1-4 ) 301 - 310  2004.12  [Refereed]

View Summary

This paper proposes the method of numerical verification of solutions of a periodic integral equation with a logarithmic singular kernel, which is typically found in some two-dimensional potential problems. The verification method utilizes a property of the singular integral for trigonometric polynomials, the periodic Sobolev space and Schauder's fixed point theorem.

• A Numrical Method of Proving the Existence of Solutions for Nonlinear ODEs Using Green Function Expression

11th GAMM-IMACS International Symposium on Scientific Coumputing, Computer Arithmetic, and Verified Numerics   Scan 2004  2004.10

• Verified Solutions of Linear Systems without Directed Rounding

11th GAMM-IMACS International Symposium on Scientific Coumputing, Computer Arithmetic, and Verified Numerics   Scan 2004  2004.10

• Nobuyo KASUGA,Katsuhito ITOH,Shin'ichi OISHI,Tomomasa NAGASHIMA:Study on Relationship between Technostress and Antisocial Behavior on Computers

IEICE Trans.   Vol.E87-D No.6 pp.1461-1465  2004.06

• On necessary and sufficient conditions for numerical verification of double turning points

K Tanaka, S Murashige, S Oishi

NUMERISCHE MATHEMATIK   97 ( 3 ) 537 - 554  2004.05  [Refereed]

View Summary

This paper describes numerical verification of a double turning point of a nonlinear system using an extended system. To verify the existence of a double turning point, we need to prove that one of the solutions of the extended system corresponds to the double turning point. For that, we propose an extended system with an additional condition. As an example, for a finite dimensional problem, we verify the existence and local uniqueness of a double turning point numerically using the extended system and a verification method based on the Banach fixed point theorem.

5
Citation
(Scopus)
• Activities of Special Interest Group on High Quality Computing

Oishi Shin'ichi

Bulletin of the Japan Society for Industrial and Applied Mathematics   14 ( 3 ) 288 - 289  2004

• Libraries, tools, and interactive systems for verified computations four case studies

RB Kearfott, M Neher, S Oishi, F Rico

NUMERICAL SOFTWARE WITH RESULT VERIFICATION   2991   36 - 63  2004  [Refereed]

View Summary

As interval analysis-based reliable computations find wider application, more software is becoming available. Simultaneously. the applications for which this software is designed are becoming more diverse. Because of this, the software itself takes diverse forms, ranging from libraries for application development to fully interactive systems. The target applications range from fairly general to specialized.
Here, we describe the design of four freely available software systems providing validated computations. Oishi provides Slab, a complete, high-performance system for validated linear algebra whose user interface mimics both Matlab's M-files and a large subset of Matlab's command-line functions. In contrast, CADNA (Fabien Rico) is a C++ library designed to give developers of embedded systems access to validated numeric computations. Addressing global constrained optimization and validated solution of nonlinear algebraic systems, Kearfott's GlobSol focuses on providing the most practical such system possible without specifying non-general problem structure, Kearfott's system has a Fortran-90 interface. Finally, Neher provides a mathematically sound stand-alone package ACETAF with an intuitive graphical user interface for computing complex Taylor coefficients and their bounds, radii of convergence, etc.
Overviews of each package's capabilities, use, and instructions for obtaining and installing appear.

• 実対称定値一般化固有値問題のすべての固有値の精度保証付き数値計算法

電子情報通信学会論文誌   Vol.J87-A, No.8  2004

• Highly Accurate Dot Product Calculation Algorithm and Applications

GAMM Seminar on Numerical Verification, Munchen    2003.11

• Apriori-error estimate and verification of numerical solutions of simultaneous linear equations

Dagstuhl Seminar 03421    2003.10

• Computation of sharp rigorous componentwise error bounds for the approximate solutions of systems of linear equations

Takeshi Ogita, Shin'ichi Oishi, Yasunori Ushiro

Reliable Computing   9 ( 3 ) 229 - 239  2003.06

View Summary

This paper is concerned with the problem of verifying the accuracy of approximate solutions of systems of linear equations. Recently, fast algorithms for calculating guaranteed error bounds of computed solutions of system's of linear equations have been proposed using the rounding mode controlled verification method and the residual iterative verification method. In this paper, a new verification method for systems of linear equations is proposed. Using this verification method, componentwise verified error bounds of approximate solutions of systems of linear equations can be calculated. Numerical results are presented to illustrate that it is possible to get very sharp error bounds of computed solutions of systems of linear equations whose coefficient matrices are symmetric and positive definite.

8
Citation
(Scopus)
• 待ち行列理論

コロナ社    2003.05

• Fast verification of solutions of matrix equations

S Oishi, SM Rump

NUMERISCHE MATHEMATIK   90 ( 4 ) 755 - 773  2002.02  [Refereed]

View Summary

In this paper. we are concerned with a matrix equation
Ax = b
where A is an a x n real matrix and x and b are n-vectors. Assume that an approximate solution (x) over tilde is given together with an approximate LU decomposition. We will present fast algorithms for proving nonsingularity of A and for calculating rigorous error bounds for parallel toA(-1)b-(x) over tilde parallel to(infinity), The emphasis is on rigour of the bounds. The purpose of this paper is to propose different algorithms, the fastest with 2/3n(3) flops computational cost for the verification step, the same as for the LU decomposition. The presented algorithms exclusively use library routines for LU decomposition and for all other matrix and vector operations.

37
Citation
(Scopus)
• Takeshi Ogita, Shin'ichi Oishi and Yasunori Ushiro: Fast inclusion and residual iteration for solutions of matrix equations

Computing, Supplement   16, pp.171-184  2002

• 中谷祐介,大石進一,柏木雅英,神澤雄智:変数分離型非線形方程式の解の非存在の厳密な数値的検証法と全解探索への応用

電子情報通信学会論文誌A   J84-A No.11 pp.1377-1384  2001.11

• Comuter Assisted Analysis of Nonlinear bynanucak systems(ハワイ)

SIAM    2001.08

• 線形数値計算の精度保証は数値解をもう1度計算する手間でできる

LA研究会    2001.03

• Fast verification methods in numerical linear algebra

2001.02

• Takeshi Ogita, Shin'ichi Oishi and Yasunori Ushiro: Fast verification of solutions for sparse monotone matrix equations

Computing, Supplement   15 pp.175-187  2001

• 数値線形代数の高速精度保証法

数値計算研究集会    2001.01

• 非線形問題を解く道具としての精度保証付数値計算

電子情報通信学会誌   1月号, 33  2001.01

• Shin'ichi Oishi: Fast enclosure of Matrix Eigenvalues and Singular Values via Rounding Mode Controlled Computation

Linear Algebra and its Applications   324; pp. 134-146  2001.01

• 線形問題については、精度保証付数値計算による、厳密な誤差評価が、近似計算の手間と同程度になるか、時には、精度保証の方が短くてすむ

GAMM    2000.12

• 高速精度保証

Workshop on Verified Numerical Computation    2000.11

• 固有値の高速精度保証と相対誤差精度保証

京都大学数理解析研究所 共同研究集会    2000.11

• 数値線形代数の高速精度保証

電気通信大学    2000.11

• 日本シュミレーション学会の国際ワークショップで精度保証付き数値計算をオーガナイズ

日本シュミレーション学会    2000.10

• 精度保証付き数値計算の入門

数理談話会    2000.10

• 実用に耐える精度保証付数値計算

日本応用数理学会    2000.10

• Accuracy of approximate solution of matrix equation can be verified with the cost of calculating it

Workshop on Numerical Analysis    2000.10

• 微積分とモデリングの数理

朝倉書店    2000.10

• 数値計算ツール

コロナ社    2000.10

• An Algorithm of Finding All Solutions with Guaranteed Accuracy for ODEs within Finite Steps

Proc. of 2000 International Symposium on Nonlinear Theory and its Applications (NOLTA2000)   pp. 621-624  2000.09

• A Numerical Method to Prove the Existence of Solutions for Nonlinear ODEs Using Affine Arithmetic &#65533;

Proc. of 2000 International Symposium on Nonlinear Theory and its Applications (NOLTA2000)   pp. 697-700  2000.09

• An Application of Oishi's Method to Verify Existence of Solution of Nonlinear Operator Equations with Non-Polynomial Nonlinear Term

Proc. of 2000 International Symposium on Nonlinear Theory and its Applications (NOLTA2000)   pp. 693-696  2000.09

• A Numerical Method of Proving the Existence of Solution for Nonlinear Equations with Guaranteed Accuracy

Proc. of 2000 International Symposium on Nonlinear Theory and its Applications (NOLTA2000)   pp. 689-691  2000.09

• 線形方程式はもう一度数値解を計算する手間で精度保証できる

日本シュミレーション学会誌   9月号19; 3  2000.09

• 実用段階に到達した精度保証付数値計算

第29回数値解析シンポジウム    2000.06

• パラメータ依存非線形方程式の解を含む区間の反復改良アルゴリズム

電子情報通信学会論文誌A   J83-A; 5, pp. 511-516  2000.05

• 数値線形代数の精度保証

計算工学   5; 4  2000.04

• 精度保証付数値計算の現状

電子情報通信学会 マイクロ波研究会招待講演    2000.03

• 精度保証付き数値計算

コロナ社    2000.01

• Y.Nakaya and S.Oishi, “Numerical Verification of Nonexistence of Solutions for Nonlinear Equations and its Application to All Solutions Algorithm"

Proc. of 1999 International Symposium on Nonlinear Theory and its Applications (NOLTA '99)   Vol.2, pp.835-838  1999.12

• K.Maruyama, T.Soma, S.Oishi and K.Horiuchi, “Numerical Comoputation with Guaranteed Accuracy of Periodic Solution of Ordinary Differential Equation Using Numerical Integration"

Proc. of 1999 International Symposium on Nonlinear Theory and its Applications (NOLTA '99)   Vol.2, pp.515-518  1999.12

• T.Miyata, T.Sato, Y.Kanzawa, M.Kashiwagi, S.Oishi, and K.Horiuchi, “A Numerical Method to Prove the Existence of Solutions for Ordinary Differential Equations Using Sobolev Norm"

Proc. of 1999 International Symposium on Nonlinear Theory and its Applications (NOLTA '99)   Vol.2, pp.511-514  1999.12

• T.Soma, S.Oishi, M.kashiwagi and K.Horiuchi, “An Algorithm of Finding All Solutions with Guaranteed Accuracy for Nonlinear Ordinary Differential Equations"

Proc. of 1999 International Symposium on Nonlinear Theory and its Applications (NOLTA '99)   Vol.2, pp.447-450  1999.12

• Y.Kanzawa and S.Oishi, “A Numerical Method to Prove the Existence of Solutions for Nonlinear OEDs Using Affine Arithmetic"

Proc. of 1999 International Symposium on Nonlinear Theory and its Applications (NOLTA '99)   Vol.2, pp.451-454  1999.12

• Fast Verified Numerical Computation

Dagstuhl Seminar on Self Validated Computation Computing    1999.12

• 高速精度保証付数値計算

日本応用物理数理学会 学会招待講演    1999.11

• 中谷祐介、大石進一、“非線形方程式の解の非存在の厳密な数値的検証法”

1999年電子情報通信学会ソサエティ大会講演論文集   A-2-7  1999.09

• 神沢雄智、大石進一、“階段関数近似を用いた非線形常微分方程式の解の数値的存在検証法”

1999年電子情報通信学会ソサエティ大会講演論文集   A-2-6  1999.09

• Imperfect singular solutions of nonlinear equations and a numerical method of proving their existence

Y Kanzawa, S Oishi

IEICE TRANSACTIONS ON FUNDAMENTALS OF ELECTRONICS COMMUNICATIONS AND COMPUTER SCIENCES   E82A ( 6 ) 1062 - 1069  1999.06  [Refereed]

View Summary

A new concept of "an imperfect singular solution" is defined as an approximate solution which becomes a singular solution by adding a suitable small perturbation to the original equations. A numerical method is presented for proving the existence of imperfect singular solutions of nonlinear equations with guaranteed accuracy. A few numerical examples are also presented for illustration.

• Calculating bifurcation points with guaranteed accuracy

J Kanzawa, S Oishi

IEICE TRANSACTIONS ON FUNDAMENTALS OF ELECTRONICS COMMUNICATIONS AND COMPUTER SCIENCES   E82A ( 6 ) 1055 - 1061  1999.06  [Refereed]

View Summary

This paper presents a method of calculating an interval including a bifurcation point. Turning points, simple bifurcation points, symmetry breaking bifurcation points and hysteresis points are calculated with guaranteed accuracy by the extended systems for them and by the Krawczyk-based interval validation method. Taking several examples, the results of validation are also presented.

• 神沢雄智、大石進一、“Affine Arithmeticを用いた非線形常微分方程式の解の数値的存在検証法”

電子情報通信学会技術研究報告   NLP99-6, pp.39-44  1999.05

• 精度保証付き数値計算のプログラミング

1999年電子情報通信学会総合大会講演論文集   TA-1-3,pp.555-556  1999.03

• 精度保証付き数値計算を高速に行うには

1999年電子情報通信学会総合大会講演論文集   TA-1-2,pp.553-554  1999.03

• 精度保証付き数値計算とは

1999年電子情報通信学会総合大会講演論文集   TA-1-1,pp.551-552  1999.03

• 非線形周期的常微分方程式の周期解の数値的存在検証法

1999年電子情報通信学会総合大会講演論文集   A-2-26  1999.03

• 非線形周期的常微分方程式の周期解の数値的存在検証法

電子情報通信学会技術研究報告   NLP98-112,pp.7-13  1999.03

• Sobolevノルムによる非線形常微分方程式の解の存在検証法

電子情報通信学会技術研究報告   NLP98-113,pp.15-21  1999.03

• 非線形周期的常微分方程式の周期解の数値的存在検証法

電子情報通信学会技術研究報告   NLP98-112,pp.7-13  1999.03

• 数値積分を用いた常微分方程式の周期解の精度保証付き数値計算

電子情報通信学会技術研究報告   NLP98-111,pp.  1999.03

• 数値計算

裳華房    1999.03

• 相馬隆郎,大石進一,堀内和夫: 精度保証付き数値計算法を用いた常微分方程式の任意精度反復改良法

電子情報通信学会論文誌A   Vol.J82-A No.1 pp.11-20  1999.01

• 精度保証数値計算

応用数理   8,3,pp.42-54  1998.12

• 非線形方程式の解の非存在検証法の効率化

1998年電子情報通信学会ソサエティ大会講演論文集   A-2-3  1998.09

• 周期的非線形微分方程式の周期解の存在検証法

1998年電子情報通信学会ソサエティ大会講演論文集   A-2-1  1998.09

• A Numerical Method of Proving the Existence of Periodic Solutions for Nonliear ODEs

Proc. of IMACS/GAMM International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics (SCAN-98)   pp.73-74  1998.09

• A Method to Prove the Existence of Periodic solutions for Nonlinear Ordinary Differential Equations

Proc. of 1998 International Symposium on Nonlinear Theory and its applications (NOLTA '98)   pp.983-986  1998.09

• An Efficient Method for Finding All Solutions of Nonlinear Equations with Guaranteed Accuracy

Proc. of 1998 International Symposium on NOnlinear Theory and its Applications (NOLTA '98)   pp.899/902  1998.09

• A Numerical Method to Prove the Existenc4e of Solutions for Ordinary Differential Equations

Proc. of 1998 International Symposium on Nonlinear Theory and its Applications (NOLTA '98)   pp.991-994  1998.09

• A Numerical Method of Proving Existence of Solutions for Nonlinear Ordinary Differential Equations Using Interval Newton Mappings

Proc. of 1998 International Symposium on Nonlinear Theory and its Applications (NOLTA '98)   pp.987-990  1998.09

• A method of proving the existence of simple turning points of two-point boundary value problems based on the numerical computation with guaranteed accuracy

T Soma, S Oishi, Y Kanzawa, K Horiuchi

IEICE TRANSACTIONS ON FUNDAMENTALS OF ELECTRONICS COMMUNICATIONS AND COMPUTER SCIENCES   E81A ( 9 ) 1892 - 1897  1998.09  [Refereed]

View Summary

This paper is concerned with the validation of simple turning points of two-point boundary value problems of nonlinear ordinary differential equations. Usually it is hard to validate approximate solutions of turning points numerically because of it's singularity. In this paper, it is pointed out that applying the infinite dimensional Krawcyzk-based interval validation method to enlarged system, the existence of simple turning points can be verified. Taking an example, the result of validation is also presented.

• 山口正樹、伊藤貴之、山田敦、大石進一、“メッシュデータの集合演算よる曲面形状の近似処理、”

1998年電子情報通信学会総合大会講演論文集   D-12-106  1998.03

• 青木康裕、大石進一、中谷祐介、“線形計画法を用いた非線形方程式の解の非存在検証法、”

1998年電子情報通信学会総合大会講演論文集   A-2-32  1998.03

• 大熊伸也、沼波秀晃、大石進一、“C++言語による精度保証付き数値計算ライブラリ、”

1998年電子情報通信学会総合大会講演論文集   A-2-30  1998.03

• 沼波秀晃、大石進一、“C++言語による精度保証ライブラリ、”

1998年電子情報通信学会総合大会講演論文集   A-2-29  1998.03

• 大上勝博、大石進一、“浮動小数点演算による中心と半径で表される区間の演算、”

1998年電子情報通信学会総合大会講演論文集   A-2-28  1998.03

• 神沢雄智、相馬隆郎、大石進一“非線形作用方程式の解の存在検証法、”

1998年電子情報通信学会総合大会講演論文集   A-2-26  1998.03

• 小田佳成、大石進一、神沢雄智、相馬隆郎、“区分関数を用いた非線形常微分方程式の境界値問題における精度保証、”

1998年電子情報通信学会総合大会講演論文集   A-2-25  1998.03

• 川野一成、神沢雄智、大石進一、“非線形常微分方程式の非孤立単純特異解の精度保証付き数値計算、”

1998年電子情報通信学会総合大会講演論文集   A-2-24  1998.03

• 同期技術と同期現象

日本物理学会誌   Vol.53,No.3, pp.200-204  1998.03

• 電子情報通信と数学

社団法人 電子情報通信学会    1998.02

• S. Oishi,“Numerical Verification Method of Existence of Connecting Orbits for Continuous Dynamical Systems,”

Jounal of Universal Computer Science   Vol.4, no.2, pp.193-201  1998.02

• Y. Nakaya and S. Oishi,“Finding All Solutions of Nonlinear Systems of Equations Using Linear Programming with Guaranteed Accuracy,”

Jounal of Universal Computer Science   Vol.4, no.2, pp.171-177  1998.02

• Y. Kanzawa, T. Souma and S. Oishi,“A Numerical Method to Prove the Existence of Solutions for Nonlinear Operator Equations,”

Proc. of 1997 International Symposium on Nonlinear Theory and its Applications (NOLTA '97)   pp.365-368  1997.11

• Y. Oda, S. Oishi, Y. Kanzawa and T. Souma,“Numerical validation for nonlinear boundary values problems using piecewise smooth function,”

Proc. of 1997 International Symposium on Nonlinear Theory and its Applications (NOLTA '97)   pp.361-364  1997.11

• H. Numanami and S. Oishi,“C++ Library for Numerical Caluculations with Guaranteed Accuracy,”

Proc. of 1997 International Symposium on Nonlinear Theory and its Applications (NOLTA '97)   pp.333-336  1997.11

• Y. Nakaya and S. Oishi,“A Numerical Method for Checking Nonexistence of Solution of Nonlinear Equations Using Optimization,”

Proc. of 1997 International Symposium on Nonlinear Theory and its Applications (NOLTA '97)   pp.313-316  1997.11

• 諸林操、牧野光則、大石進一、“レイトレーシング法を用いた虹の表現、”

1997年電子情報通信学会ソサエティ大会講演論文集   D-11-64  1997.09

• 武藤一、牧野光則、大石進一、“CGによる結晶の表現、”

1997年電子情報通信学会ソサエティ大会講演論文集   D-11-63  1997.09

• 中谷祐介、大石進一、“最適化手法による非線形方程式の解の非存在性の数値的検証法、”

1997年電子情報通信学会ソサエティ大会講演論文集   A-2-15  1997.09

• 青木康裕、大石進一、中谷祐介、“高分子溶液の多層平衡に関する非線形方程式の全解探索、”

1997年電子情報通信学会ソサエティ大会講演論文集   A-2-14  1997.09

• 神沢雄智、柏木雅英、大石進一“有理数演算を用いたパラメータ依存非線形方程式の解の区間反復改良、”

1997年電子情報通信学会ソサエティ大会講演論文集   A-2-13  1997.09

• 沼波秀晃、大石進一、“C++言語による精度保証ライブラリ”

1997年電子情報通信学会ソサエティ大会講演論文集   A-2-12  1997.09

• 大上勝博、大石進一、“円形複素領域を用いた非線形方程式の解の精度保証付き数値計算、”

1997年電子情報通信学会ソサエティ大会講演論文集   A-2-11  1997.09

• 寺岡秀礼、大石進一、神沢雄智、“非線形方程式の複素数解の精度保証付き数値計算、”

1997年電子情報通信学会ソサエティ大会講演論文集   A-2-10  1997.09

• 小田佳成、大石進一、神沢雄智、相馬隆郎、“非線形常微分方程式の境界値問題における精度保証の自動化、”

1997年電子情報通信学会ソサエティ大会講演論文集   A-2-9  1997.09

• 大熊伸也、大石進一、小田佳成、“ローレンツ方程式の初期値問題における近似解の精度保証、”

1997年電子情報通信学会ソサエティ大会講演論文集   A-2-8  1997.09

• 川野一成、神沢雄智、大石進一、“非線形常微分方程式の近似的特異解の精度保証付き数値計算、”

1997年電子情報通信学会ソサエティ大会講演論文集   A-2-7  1997.09

• S. Oishi,“Numerical Verification Method of Existence of Connecting Orbits for Continuous Dynamical Systems,”

Proc. of GAMM/IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics (SCAN '97)   pp.XIV-7-XIV-15  1997.09

• S. Oishi and Y. Nakaya,“Mathematical Programming Based Rigorous Numerical Nonexistence Test for Solutions of Nonlinear Equations,”

Proc. of GAMM/IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics (SCAN '97)   pp.VII-9-VII-12  1997.09

• Y. Nakaya and S. Oishi,“Finding All Solutions of Nonlinear Systems of Equations Using Linear Programming with Guaranteed Accuracy,”

Proc. of GAMM/IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics (SCAN '97)   pp.VII-5-VII-8  1997.09

• Y. Kanzawa and S. Oishi,“Approximate Singular Solutions of Nonlinear Equations and a Numerical Method of Proving their Existence,”

Proc. of GAMM/IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics (SCAN '97)   pp.VII-1-VII-4  1997.09

• 神沢雄智、柏木雅英、大石進一、中村晴幸、“有限ステップで停止する非線形方程式のすべての解を精度保証付きで求めるアルゴリズム”

電子情報通信学会論文誌(A)   Vol.J80-A, no.7, pp.1130-1137  1997.07

• 中谷祐介、大石進一、“化学平衡系の非線形方程式の精度保証付き数値計算、”

信学技報   NLP97-53, pp.103-109  1997.06

• 神沢雄智、柏木雅英、大石進一、“パラメータ依存非線形方程式のすべての解を精度保証付きで求めるアルゴリズム”

電子情報通信学会論文誌(A)   Vol.J80-A, no.6, pp.920-925  1997.06

• 神沢雄智、大石進一、“精度保証付き数値計算法を用いた非線形方程式の解曲線の存在検証法”

電子情報通信学会論文誌(A)   Vol.J80-A, no.6, pp.907-919  1997.06

• 非線形解析入門

コロナ社    1997.04

• Stability of synchronized states in one dimensional networks of second order PLLs

HA Tanaka, MD Vieira, AJ Lichtenberg, MA Lieberman, S Oishi

INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS   7 ( 3 ) 681 - 690  1997.03  [Refereed]

View Summary

Synchronous distributed timing clocks are the basic building blocks in digital communication systems. Conventional systems mainly employ a tree-like network of cascaded timing clocks for synchronous clocking. On the other hand, decentralized synchronous networks of timing clocks, which have been proposed from a very early stage of the digital communication, are gaining attention in the consumer communication networks and also recently in large, high-performance digital systems (such as multiprocessors) clocking. In this paper, we present a theoretical study of synchronous networks of timing clocks consisting of locally connected second order phase-locked loops (PLLs). We find a close connection between the stability properties of the first and second order networks. The particular examples of one way and two way nearest neighbor coupling, with a lag-lead filter and a triangular phase detector (PD) are analyzed in detail. Both the synchronized in-phase solution and the wave-like ''mode-lock'' solution are examined. A criterion is found for the stability of the one-way coupled network while the two-way coupled network is found to be always stable.

• First order phase transition resulting from finite inertia in coupled oscillator systems

HA Tanaka, AJ Lichtenberg, S Oishi

PHYSICAL REVIEW LETTERS   78 ( 11 ) 2104 - 2107  1997.03  [Refereed]

View Summary

We analyze the collective behavior of a set of coupled damped driven pendula with finite (large) inertia, and show that the synchronization of the oscillators exhibits a first order phase transition synchronization onset, substantially different from the second order transition obtained in the case of no inertia. There is hysteresis between two macroscopic states, a weakly and a strongly coherent synchronized state, depending on the coupling and the initial state of the oscillators. A self-consistent theory is shown to determine these cooperative phenomena and to predict the observed numerical data in specific examples.

• Self-synchronization of coupled oscillators with hysteretic responses

HA Tanaka, AJ Lichtenberg, S Oishi

PHYSICA D   100 ( 3-4 ) 279 - 300  1997.02  [Refereed]

View Summary

We analyze a large system of nonlinear phase oscillators with sinusoidal nonlinearity, uniformly distributed natural frequen cies and global all-to-all coupling, which is an extension of Kuramoto's model to second-order systems. For small coupling, the system evolves to an incoherent state with the phases of all the oscillators distributed uniformly. As the coupling is increased, the system exhibits a discontinuous transition to the coherently synchronized state at a pinning threshold of the coupling strength, or to a partially synchronized oscillation coherent state at a certain threshold below the pinning threshold. if the coupling is decreased from a strong coupling with all the oscillators synchronized coherently, this coherence can persist until the depinning threshold which is less than the pinning threshold, resulting in hysteretic synchrony depending on the initial configuration of the oscillators. We obtain analytically both the pinning and depinning threshold and also explain the discontinuous transition at the thresholds for the underdamped case in the large system size limit. Numerical exploration shows the oscillatory partially coherent state bifurcates at the depinning threshold and also suggests that this state persists independent of the system size. The system studied here provides a simple model for collective behaviour in damped driven high-dimensional Hamiltonian systems which can explain the synchronous firing of certain fireflies or neural oscillators with frequency adaptation and may also be applicable to interconnected power systems.

• 数理計算法に基づいた非線型方程式の解の非存在の数値的検証法

信学技法/電子情報通信学会   NLP96;111号  1996.12

• C+Tによる精度保証付き数値計算ライブラリ

信学技法/電子情報通信学会   NLP96;110号  1996.12

• 区分線型系微分方程式の周期解と分岐点の精度保証について

信学技法/電子情報通信学会   NLP96;109号  1996.12

• OISHI Shin'ichi

Journal of The Society of Instrument and Control Engineers   35 ( 10 ) 751 - 756  1996.10

• Numerical Validation for Nonlinear Boundary Value Problems Using Power Series Arithmetic

Theory and its Applications(NOLTA'96)/1996 International Symposium on Nonlinear    1996.10

• An Interative Refinment Method for Solutions of Nonlinear Ordinary Differential Equations with Arbitrarily Pricision

Theory and its Applications(NOLTA'96)/1996 International Symposium on Nonlinear    1996.10

• ある分岐点の精度保証つき数値計算法

信学技法/電子情報通信学会   CAS96;56号  1996.09

• ベギ級数演算を用いた非線形常微分方程式の境界値問題の近似解の精度保証について

信学技法/電子情報通信学会   CAS96巻55号  1996.09

• 精度保証付き数値計算を用いた常微分方程式の近似解の任意精度反復改良

信学技法/電子情報通信学会   CAS96巻54号  1996.09

• 有限ステップで停止することが証明されたパラメータ依存非線形方程式の全解探索アルゴリズム

信学技法/電子情報通信学会   CAS96巻41号  1996.09

• C+Tと浮動小数点数による精度保証付き数値計算ライブラリ

信学技法/電子情報通信学会   NLP96巻46号  1996.07

• 精度保証付数値計算法を用いた非線形方程式の解曲線の存在検証法

信学技法/電子情報通信学会   NLP96巻56号  1996.07

• 精度保証付数値計算を用いた常微分方程式の近似解の区間反復法

信学技法/電子情報通信学会   NLP96巻54号  1996.07

• 非線型方程式の近似的特異解とその数値的存在検証法

信学技法/電子情報通信学会   NLP96巻53号  1996.07

• 非線型方程式の複素数解の精度保証付数値計算

信学技法/電子情報通信学会   NLP96巻47号  1996.07

• Geometric structure of mutually coupled phase-locked loops

HA Tanaka, S Oishi, K Horiuchi

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I-FUNDAMENTAL THEORY AND APPLICATIONS   43 ( 6 ) 438 - 443  1996.06  [Refereed]

View Summary

Dynamical properties such as lock-in or out-of-lock condition of mutually coupled phase-locked loops (PLL's) are problems of practical interest, The present paper describes a study of such dynamical properties for mutually coupled PLL's incorporating lag filters and triangular phase detectors, The fourth-order ordinary differential equation (ODE) governing the mutually coupled PLL's is reduced to the equivalent third-order ODE due to the symmetry, where the system is analyzed in the context of nonlinear dynamical system theory, An understanding as to how and when lock-in can be obtained or out-of-lock behavior persists, is provided by the geometric structure of the invariant manifolds generated in the vector field from the third-order ODE. In addition, a connection to the recently developed theory on chaos and bifurcations from degenerated homoclinic points is also found to exist. The two-parameter diagrams of the one-homoclinic orbit are obtained by graphical solution of a set of nonlinear (finite dimensional) equations. Their graphical results useful in determining whether the system undergoes lock-in or continues out-of-lock behavior, are verified by numerical simulations.

• (解説)非線形現象の解析手法 {I} {II}-非線型現象の精度保証付き数値解析(1),(2)

電子情報通信学会誌/電子情報通信学会   79;2, 3  1996.02

• C++による精度保証付き数値計算システム

電子情報通信学会技術研究報告/電子情報通信学会   NLP95-55  1996.01

• An Approach to Trace Solution Curve of Nonlinear Equations

Proc. 1995 International Symposium on Nonlinear Theory and its Applications (NOLTA'95)/電子情報通信学会    1995.12

• Numerical Method of Calculating Hopf Bifurcation Point with Guaranteed Acuracy

Proc. 1995 International Symposium on Nonlinear Theory and its Applications (NOLTA'95)/電子情報通信学会    1995.12

• Numerical Verification of Existence of Connecting Orbits of Continu

Proc. 1995 International Symposium on Nonlinear Theory and its Applications (NOLTA'95)/電子情報通信学会    1995.12

• 粒子の成長を考慮した積乱雲の表現

第11回NICOGRAPH論文コンテスト論文集/コンピュータグラフィックス協会    1995.12

• An Interval Method of Proving Existence of Solutions for Nonlinear Boundary Value Problems

Numerical Analysis of Ordinary Differential Equations and its Applications (World Scientific, Proceedings)/World Scientific    1995.12

• 非線形作用素方程式のKrawczyk作用素と区間関数の積分理論による解の存在の数値的検証法

数理解析研究所講究録 928 短期共同研究数値計算における品質保証とその応用報告集/京都大学    1995.11

• 有限次元非線形方程式の全解探索アルゴリズム

数理解析研究所講究録 928 短期共同研究数値計算における品質保証とその応用報告集/京都大学    1995.11

• 連続力学系のコネクティングオービットの精度保証付き数値解法

数理解析研究所講究録 928 短期共同研究数値計算における品質保証とその応用報告集/京都大学    1995.11

• 非線形方程式の解曲線追跡のための一手法

電子情報通信学会技術研究報告/電子情報通信学会   NLP95-52  1995.10

• 連続力学系のコネクティングオービットの精度保証付き数値解法

電子情報通信学会技術研究報告/電子情報通信学会   NLP95-53  1995.10

• 区間を用いた解曲線追跡

1995年電子情報通信学会ソサイエティ大会講演文集/電子情報通信学会   A-41  1995.09

• Numerical Existence Theorems for Solutions of Nonlinear Boundary Value Problems of Ordinary Differential Equations

International Congress on Industrial and Applide Mathematics    1995.07

• Shin'ichi Oishi: Numerical verification of existence and inclusion of solutions for nonlinear operator equations

Journal of Computational and Applied Mathematics/North-Holland   60  1995.07

• 非線型関数方程式の解の数値的検証法

日本数学会大会    1995.04

• Hisa-Aki Tanaka, Toshiya Matsuda, Shin’ichi Oishi and Kazuo Horiuchi: Analytic Structure of Phase-Locked Loops in Complex Time

IEICE Trans. Fundamentals   Vol.E77-A, No.11, pp.1777-1782  1994.11

• Hisa-Aki Tanaka, Shin’ichi Oishi and Kazuo Horiuchi: Melnikov Analysis of a Second Order PLL in the Presence of a Weak CW Interference

IEICE Trans. Fundamentals   Vol.E77-A, No.11,pp.1887-1891  1994.11

• 区間解析と有理数演算による非線形方程式の近似解の精度保証

電子情報通信学会論文誌   Vol.J77-A, No.10, pp.1372-1382  1994.10

• 田中久陽、岡田淳、大石進一、堀内和夫: 多くのパラメーターを持つダイナミカルシステムの特異点解析-非対称結合神経回路網への応用-

電子情報通信学会論文誌   Vol.J77-A, No.7, pp.965-973  1994.07

• Shin'ichi Oishi: Two Topics in Nonlinear System Analysis through Fixed Point Theorems

IEICE Trans. Fundamentals   Vol.E77-A, No.7, pp.1144-1153  1994.07

• Fast and Accurate Numerical Verification Methods in Numerical Linear Algebra

11th GAMM-IMACS International Symposium on Scientific Coumputing, Computer Arithmetic, and Verified Numerics   Scan 2004

### Books and Other Publications

• 回路理論

大石進一

コロナ社  2013.05 ISBN: 9784339008494

• 待ち行列理論

大石進一

コロナ社  2003.05

• MATLABによる数値計算

大石進一

培風館  2001.07

• 数値計算ツール

コロナ社  2001

• 微積分とモデリングの数理

朝倉書店  2001

• 精度保証付き数値計算

コロナ社  2000

• 非線形解析入門

コロナ社  1998

• グラフィックス

日本評論社  1994

• 例にもとづく情報理論入門

講談社  1993

• フーリエ解析

岩波書店  1989

### Misc

• Tanaka Kazuaki, Sekine Kouta, Oishi Shin'ichi

数理解析研究所講究録   ( 2037 ) 125 - 140  2017.07

• Tanaka Kazuaki, Sekine Kouta, Oishi Shin'ichi

( 2037 ) 125 - 140  2017.07

View Summary

In this paper, we propose a numerical method for verifying the positivity of solutions to semilinear elliptic equations. We provide a sufficient condition for a solution to an elliptic equation to be positive in the domain of the equation, which can be checked numerically without requiring a complicated computation. We present some numerical examples.

•   ( 2037 ) 96 - 105  2017.07

• Yanagisawa Yuka, Ogita Takeshi, Oishi Shin'ichi

数理解析研究所講究録   ( 2005 ) 56 - 64  2016.11

• 小林領, 木村拓馬, 大石進一, 大石進一

日本応用数理学会年会講演予稿集(CD-ROM)   2015   ROMBUNNO.9GATSU9NICHI,13:30,G,  2015.09

• 小林領, 木村拓馬, 大石進一

日本応用数理学会年会講演予稿集(CD-ROM)   2014   ROMBUNNO.9GATSU3NICHI,11:00,G,  2014.08

• 29pAD-15 From Evolution of Interface to Evolution of Set

Ogasawara Yoshihito, Oishi Shin'ichi

Meeting abstracts of the Physical Society of Japan   69 ( 1 ) 344 - 344  2014.03

• OISHI Shin'ichi

IEICE technical report. Circuits and systems   113 ( 427 ) 45 - 47  2014.02

View Summary

In this note, it is discussed how to teach circuit theory logically understandable way putting Maxwell theory as axiom like Euclid's Elements.

• Validated Solutions for Symmetric Saddle Point Linear Systems

KOBAYASHI Ryo, KIMURA Takuma, KIMURA Takuma, OISHI Shin’ichi, OISHI Shin’ichi

International Conference on Simulation Technology (CD-ROM)   2014   90 - 91  2014

• On a characteristic property of the tent map (General and Geometric Topology today and their problems)

Ogasawara Yoshihito, Oishi Shin'ichi

RIMS Kokyuroku   1833   98 - 103  2013.05

• Mathematics of Simulation

Oishi Shin'ichi

Journal of the Japan Society for Simulation Technology   32 ( 1 ) 1 - 1  2013.03

• Oishi Shin'ichi

Journal of the Japan Society for Simulation Technology   31 ( 3 ) 136 - 138  2012.09

• 18pAC-8 On a chaotic property from the viewpoint of topology II

Ogasawara Yoshihito, Oishi Shin'ichi

Meeting abstracts of the Physical Society of Japan   67 ( 2 ) 232 - 232  2012.08

• 18pAC-7 On a chaotic property from the viewpoint of topology I

Ogasawara Yoshihito, Oishi Shin'ichi

Meeting abstracts of the Physical Society of Japan   67 ( 2 ) 232 - 232  2012.08

• 18pAC-9 On a chaotic property from the viewpoint of topology III

Ogasawara Yoshihito, Oishi Shin'ichi

Meeting abstracts of the Physical Society of Japan   67 ( 2 ) 232 - 232  2012.08

• AK-3-3 The Present State of the New Journal NOLTA, IEICE

Oishi Shin'ichi

Proceedings of the IEICE General Conference   2012   "SS - 11"  2012.03

• Sufficient Conditions for the Existence of a Primitive Chaotic Behavior (vol 79, 015002, 2010)

Yoshihito Ogasawara, Shin'ichi Oishi

JOURNAL OF THE PHYSICAL SOCIETY OF JAPAN   80 ( 6 )  2011.06

Other

• Oishi Shinich, Ogita Takeshi

Journal of the Japan Society for Simulation Technology   30 ( 1 ) 43 - 45  2011.04

• Liu Xuefeng, Oishi Shinichi

RIMS Kokyuroku   1733   31 - 39  2011.03

• OISHI Shinichi

49 ( 5 ) 273 - 278  2010.05

• Advances in Verified and Accurate Computations and Industrial Applications

OISHI Shinichi

49 ( 5 ) 271 - 272  2010.05

• OISHI Shin'ichi

The Journal of reliability engineering association of Japan   31 ( 4 ) 250 - 255  2009.06

• Yamanaka Naoya, Okayama Tomoaki, Oishi Shin'ichi, Ogita Takeshi

RIMS Kokyuroku   1638   146 - 158  2009.04

• 荻田 武史, Rump Siegfried M., 大石 進一

数理解析研究所講究録   1614   34 - 39  2008.10

• OZAKI Katsuhisa, OGITA Takeshi, OISHI Shin'ichi

RIMS Kokyuroku   1614   1 - 10  2008.10

• OISHI Shin'ichi

IEICE technical report   108 ( 103 ) 55 - 57  2008.06

View Summary

In this paper, we describe how to calculate the Bessel functions J_0(x) and J_1(x) with guaranteed error bound. We shall further present similar arguments for the Hankel functions H_0^<(1)>(x) and H_1^<(1)>(x).

• 荻田 武史, 尾崎 克久, 大石 進一

数理解析研究所講究録   1573   45 - 52  2007.11

• OISHI Shin'ichi, OGITA Takeshi

IPSJ Magazine   48 ( 10 ) 1103 - 1110  2007.10

• OISHI Shin'ichi

IEICE technical report   107 ( 184 ) 35 - 37  2007.07

View Summary

In this paper, we will present an accurate Cholesky decomposition algorithm. For the purpose, we develope multiple precision algorithms for the four basic arithmetic operations and the square root operation. Let A=[A_1, A_2, …, A_K] and B=[B_1, B_2, …, B_L] are K-tuple and L-tuple numbers. Here, A_i∈F for i=1, 2, …, K and B_i∈F for i=1, 2, …, L. For simplicity, we assume that K=max(K, L) Then, the following becomes multiple precision addition with K-fold accuracy :

• Fast Verification of Matrix Determinant

OGITA Takeshi, OZAKI Katsuhisa, OISHI Shin'ichi

26   225 - 228  2007.06

• Fast Verification for Solutions in Least Square Problem

MIYAJIMA Shinya, OGITA Takeshi, OISHI Shin'ichi

26   229 - 232  2007.06

• Fast and Adaptive Algorithm for 2D Orientation Problem

OZAKI Katsuhisa, OGITA Takeshi, OISHI Shin'ichi

26   221 - 224  2007.06

• OISHI Shin'ichi

IEICE technical report   107 ( 86 ) 59 - 61  2007.06

• A-2-27 Tracing Solution Curve of Nonlinear Equation using Affine Arithmetic

Kanazawa Yuchi, Oishi Shin'ichi

Proceedings of the IEICE General Conference   2007   74 - 74  2007.03

• Oishi Shin'ichi, Ogita Takeshi, Ohta Takahisa

Journal of the Japan Society for Simulation Technology   25 ( 3 ) 170 - 178  2006.09

View Summary

IEEE standard 754 is widely used as a standard of floating-point arithmetic. Most of CPUs in today's computers support IEEE standard 754. Using double precision arithmetic following IEEE standard 754, the authors have proposed fast methods of verifying the accuracy of a numerical solution of a linear system. In this paper, an accurate and fast verification method for a linear system is developed using residual iteration method. The residual iteration requires the availability of high precision computation. Up to now, extended precision, i.e. multiple precision and quadruple precision are used for the accurate computation of the residual. However, such higher precision arithmetic systems are not necessarily available on all computers. Therefore, the residual iteration using such systems does not have the portability. In this paper, an accurate, fast and portable method for a linear system using the fact that an algorithm of accurate dot product can portably be implemented and applied to the residual iteration. Finally, numerical results are presented showing the effectiveness of the proposed verification method.

• Ogita Takeshi, Oishi Shin'ichi

Journal of the Japan Society for Simulation Technology   25 ( 3 ) 179 - 184  2006.09

View Summary

In this paper, new verification methods for verifying the accuracy of a numerical solution of a linear system with a dense coefficient matrix are proposed. The proposed methods are based on a verification method which uses an approximate inverse of the coefficient matrix. It is possible to reduce the computational memory space for the verification drastically without slowing down its computational speed seriously. Numerical results are presented to illustrate that the proposed methods become more effective in larger problem size.

• Visualization of 2-dimensional objects in 4-dimensional space

OZAKI Katsuhisa, KORIYAMA Akira, OISHI Shin'ichi

25   73 - 76  2006.06

• Verified Solutions of Extremely Ill-conditioned Linear Systems

OHTA Takahisa, OGITA Takeshi, RUMP Siegfried M., OISHI Shin'ichi

24   225 - 228  2005.07

• Accurate, Verified and Portable Standard Functions and its Applications

OZAKI Katsuhisa, OGITA Takeshi, OISHI Shin'ichi

24   193 - 196  2005.07

• Verification for Each Eigenvalues of Symmetric Matrix

MIYAJIMA Shinya, OGITA Takeshi, OISHI Shin'ichi

24   229 - 232  2005.07

• Ozaki Katsuhisa, Ogita Takeshi, Miyajima Shinya, Oishi Shin'ichi

RIMS Kokyuroku   1441   75 - 88  2005.07

• 10. Productive ICT Academia Project

UEDA Kazunori, OISHI Shinichi, KATTO Jiro, NAKAJIMA Tatsuo, MURAOKA Yoichi, YAMANA Hayato

Journal of Information Processing Society of Japan   46 ( 4 ) 410 - 416  2005.04

• A-1-36 A Numerical Method of Proving Existence of Periodic Solution for Nonlinear ODE using Green's Function and Its comparison with Conventional Method

Kanzawa Yuchi, Oishi Shinichi

Proceedings of the IEICE General Conference   2005   36 - 36  2005.03

• Verification of Nonsingularity for Sparse Matrices using Direct Solution Methods

OGITA Takeshi, OISHI Shin'ichi

23   349 - 352  2004.06

• Numerical Verification Method for Simultaneous Linear Equations using Accurate Dot Product Calculation Algorithm

OHTA Takahisa, OISHI Shin'ichi, OGITA Takeshi, RUMP Siegfried M.

23   345 - 348  2004.06

• Moriyama Atsushi, Ogita Takeshi, Ushiro Yasunori, Oishi Shin'ichi

RIMS Kokyuroku   1362   47 - 55  2004.04

• Simulation is Everywhere

Oishi Shin'ichi

Journal of the Japan Society for Simulation Technology   22 ( 4 ) 225 - 225  2003.12

• 荻田 武史, 大石 進一, 後 保範

数理解析研究所講究録   1320   151 - 161  2003.05

• Fast Verification of Numerical Solutions of Matrix Equations by Iterative Solution Methods

OGITA Takeshi, OISHI Shin'ichi, USHIRO Yasunori

20   249 - 252  2001.06

• 荻田 武史, 後 保範, 大石 進一

数理解析研究所講究録   1198   161 - 169  2001.04

•   45 ( 1 ) 29 - 33  2001

• All Solution algorithm for Finite Dimensional Nonlinear Equations

KANAZAWA Yuchi, KASHIWAGI Masahide, OISHI Shin'ichi

10   47 - 50  2000.10

• OISHI Shin'ichi

Journal of the Japan Society for Simulation Technology   19 ( 1 ) 39 - 45  2000.03

View Summary

Programming techniques for numerical computation with guaranteed accuracy are surveyed. In particular, a programming library on Scilab is described in detail. Fast verification method is also described.

• Editorial Preface

OISHI Shin'ichi

The Journal of the Institute of Electronics, Information, and Communication Engineers   80 ( 11 ) 1103 - 1103  1997.11

• OISHI Shin'ichi

The Journal of the Institute of Electronics,Information and Communication Engineers   80 ( 11 ) 1139 - 1142  1997.11

• A Numerical Method for Calculating singular Solutions of Nonlinear Equations with Guaranteed Accuracy

KANZAWA Yuchi, Kawano Kazunari, OISHI Shin'ichi

Proceedings of the Society Conference of IEICE   1996   36 - 36  1996.09

• A Numerical method with Guaranteed Accuracy for Complex Number Solutions of Nonlinear Systems of Equations

TERAOKA Hideyuki, OISHI Shin'ichi, KANZAWA Yuchi

IEICE technical report. Nonlinear problems   96 ( 208 ) 9 - 15  1996.07

View Summary

Using the complex interval arithmetic, the Krawczyk method is extended to complex domain. Using this result, an algorithm is presented for finding all complex solutions of nonlinear equations. A few numerical examples are also reported to illustrate the validity of the method.

• OISHI Shin'ichi

The Journal of the Institute of Electronics,Information and Communication Engineers   79 ( 7 ) 693 - 695  1996.07

• Tracing Solution of ODE with one parameter by Krawczyk's Method

KANZAWA Yuchi, OISHI Shin'ichi

Proceedings of the IEICE General Conference   1996   98 - 98  1996.03

• Numerical Existense Theorem for Turning Points of Boundary-Value Problems

SOUMA Takao, KANZAWA Yuchi, OISHI Shin'ichi, HORIUCHI Kazuo

Proceedings of the IEICE General Conference   1996   99 - 99  1996.03

• Computer Assisted Proof in Chaos Theory

OISHI Shin'ichi

IEICE technical report. Information theory   95 ( 337 ) 25 - 30  1995.10

View Summary

Although the problem of proving the existence of connecting orbits for nonlinear ordinary differential equations is one of the most fun.

• A Numerical Method of Proving Existence of Connecting Orbits for Continuous Dynamical Systems

OISHI Shin'ichi

Proceedings of the IEICE General Conference   1995   409 - 410  1995.03

• SA-1-2 Numerical Verification of Existence of Solutions for Nonlinear Boundary Value Problems using Interval Analysis

Oishi Shin'ichi

1994   235 - 236  1994.09

• Numerical Verification of Existence of Periodic Solutions to the Duffing Equation

Oishi Shin'ichi

IEICE technical report. Nonlinear problems   93 ( 252 ) 9 - 16  1993.09

View Summary

It is shown that the existence of periodic solutions for the Duffing equation can be proved by the self-validating numerical simulation.A new type of the seaf-validating numerical simulation system is constructed,by which the existence of several kinds of periodic solutions for the Duffing equation are demonstrated.

• Numerical Validation Method for Nonlinear Equations Using Interval Method and Rational Arithnetic

Kashiwagi Masahide, Oishi Shin'ichi

IEICE technical report. Circuits and systems   93 ( 102 ) 83 - 90  1993.06

View Summary

In this report,we present a numerical validation method for finite dimensional nonlinear equations.In the first,we show an algorithm to obtain an interval that indudes exact solution using given approximate solution.In this algorithm we apply interval analysis and take representation error of equation into account.In the second,we show an interval iteration method,which can decrease the size of the interval to be arbitrarily small.In the iteration rational arithmetic is used and rounding of rational numbers is efficiently used.In the last,we report a trial implementation of these algorithms and some numerical examples.

• Numerical Verification of Existence of Periodic Solutions to the Duffing Equation

Oishi Shin'ichi

IEICE technical report. Circuits and systems   93 ( 102 ) 91 - 96  1993.06

View Summary

It is shown that the existence of periodic solutions for the Duffing equation can be proved by the self-validation numerical simulation.A new type of the self-validating numerical simulation system is constructed,by which the cxistence of several kinds of periodic solutions for the Duffing equation are demonstrated.

• Oishi Shin'ichi, Kashiwagi Masahide

RIMS Kokyuroku   831   115 - 128  1993.04

• Kashiwagi Masahide, Oishi Shin'ichi

RIMS Kokyuroku   787   72 - 94  1992.06

• HORIUCHI Kazuo, OISHI Shin'ichi

IPSJ Magazine   33 ( 4 ) 308 - 317  1992.04

• OISHI SHIN'ICHI

RIMS Kokyuroku   414   203 - 211  1981.01

### Awards

• 文化功労者

2020.11   文部科学省   精度保証付計算法と無誤差変換という画期的な数値計算法を編み出した．

• JSIAM Fellow

2013.06

• Best Paper Award, Nonlinear Theory and Its Applications , IEICE

2012.04

• 紫綬褒章

2012.04

• 日本応用数理学会業績賞

2012.03

• 科学技術分野における文部科学大臣表彰（研究部門)

2010.04

• 日本応用数理学会論文賞

2007.04

• 船井情報科学振興賞

2006.04

• 大川出版賞

2003.06

• 電子情報通信学会論文賞

1998

• 電子情報通信学会論文賞, 猪瀬賞

1995

• 電子情報通信学会論文賞

1992

• 電子通信学会学術奨励賞

1982.03

• 早稲田大学小野梓賞

1982.03

• 丹羽記念賞

1982.03

### Research Projects

• Project Year :

2008
-
2010

View Summary

Our research group is composed of scholars working in the areas of discrete mathematics, nonlinear differential equations, information theory and numerical computation. We have organized "Seminar on Digital Analysis" so that members can hold common understanding and insight on the fundamental theories and ideas of digital mathematics. As speakers of this seminar, we have invited 16 researchers who are highly active in the areas of discrete mathematics, mathematical modeling, information theory and numerical computation. We have succeeded in getting common understanding on digital analysis through exciting discussions in each lecture of the seminar.

• Project Year :

2005
-
2009

View Summary

Establishment of Verified Numerical Computation We have studied verified numerical computations for partial differential equations and systems of linear equations using digital computers. Calculating sum of a vector and dot product of two vectors with guaranteed high accuracy is ubiquitous in scientific computing. We have developed such algorithms for accurate sum and dot product, which are known to be the fastest so far. As applications, we have applied the fast and accurate algorithms to sparse matrix computations, computational geometry and so forth. Moreover, we have succeeded in proving the existence and uniqueness of a solution of a partial differential equation, and in calculating an error bound of its approximate solution.

• FDTDシミュレーションの精度

Project Year :

2004
-

• 通信における非線形性現象に関する研究

Project Year :

2000
-

• グラフィック・アルゴリズム及び精度保証付LSI設計支援システムの基礎的研究

Project Year :

1999
-

• 非線形通信回路技術の基礎研究

Project Year :

1998
-

• Study on Structures and Dynamics of Nonlinear Systems

View Summary

With the rapid advance of VLSI and the discovery of new nonlinear phenomena such as soliton and chaos, analysis of nonlinear dynamic systems becomes very important. The purpose of this project is to clarify the fundamental properties of both the nonlinear system structure and its dynamics, and to buid the foundation of their applications to the practical problems, by means of the new mathematical methods which the present researchers have developed.This project has been accomplished over three years, and its fruitful result is establishment and mutual recomposition of the following theories :1. Theory on evaluation of the overall dependence on internal structures and parameters of nonlinear systems, by making use of the original theory of nondeterministic operators.2. Theory on numerical analysis of nonlinear systems by the original methods based on homotopy or decomposition.3. New approach for the analysis of nonlinear phenomena such as soliton and chaos. Furthermore, using the above theories, we threw light on the followings :4. Observation of a non-periodic attractor from an extremely simple circuit, and the rigorous proof of the attractor to be chaotic.5. Modelling of parallel blower and multibody systems and analysis of their dynamics.6. Analysis of the models of optical transmission and biological systems

• 文書・図形・画像揚報に関するマルチメディア変換の研究

View Summary

昭和63年度は本研究のまとめの年として以下の事項について総括的な検討を加え、実用システムのプロトタイプを製作した。1.図画・文書の電子ファイルシステム対象図画として、地図・文書・一般図画をとり上げ、各々を図面解析し書式等を自動的に認識し生成するシステムのプロトタイプを作成した。これらはCCITT及びISOで検討されているドキュメントプーキテクチャの概念にのっとったものである。特に新聞紙面の自動解析技術においては、図面としてのレイアウトストラクチャからロジカルストラクチャを抽出する限界について深く検討した。その他、地図図面や他の図画についてのドキュメントアーキテクチャを検討し、これら文書・図面などはドキュメントアーキテクチャの思想の基に共通のデータ構造として表現できることがわかった。2.高詳細画像の電子ファイル化システム高詳細な画像データ及び動画などを、ディジタルストレージメディアあるいは高速のパケット網に伝送するデータ構造を提案した。この方式によれば論理的な共通方式のもとに画像情報、とりわけ動画情報を伝送することができる。またこの方式をドキュメントアーキテクチャの一構成要素としてとり込むことも可能であり、その為のコンテントアーキテクチャの整備を今後の課題として残している。3.文書図形・画像情報伝送におけるセキュリティ対策文書図形・画像情報特有の性質を利用した、ディジタル透かし方式を提案した。本方式は従来の情報暗号化方式とは性質を異なったもので、文書図形・画像情報をその構成要素と変換関数としてとらえ、この変換関数を他のものに変えて伝送する方式で、ドキュメント管理センタをおくことで、より安全性の高い情報伝送ができる

• 音声特徴抽出手法の高度化に関する研究

View Summary

本年度は、(1)相互情報量に基づく音韻性抽出、(2)非定常態を対象とした特徴抽出、(3)生成モデルに基づいた特徴抽出、(4)聴覚における時系列信号の特徴抽出、(5)音響的特徴の音韻環境依存性、(6)深い意味理解に必要な音響的特徴の抽出精度、の6点をテーマとして研究を行なった。(1)では、個々のフレームにおける音響的情報を多角的に把握し、それらを用いて音韻コンテキストを考慮した音韻性の特徴抽出を実現するために、特徴量相互の情報量を基準とした音韻性抽出手法について検討した。ここでは、複数の音響的特徴量を階層的に用いることにより、音響的類似性に基づいた分類、時系列的な出現パタンによる分類を行い、音韻性の記述は、まずフレーム単位に音韻候補及びその確からしさを示しさらに確実性のある音韻候補情報を用いて不確実な音韻候補の修正を行うことで実現した。(2)では、有声音を対象として、非定常波形の分析・特徴抽出を行なう構造モデルについて検討した。二連インパルス応答からなる非定常波形にPSE分析を適用したとき現われるFMスペクトルにスペクトル(ホルマント)の変化の方向性の情報が折り込まれていることが示された。(3)では、AbS型の調音パラメタ推定法を対象としてアルゴリズムの並列化を行なった。(4)では、人工内耳の基礎研究から、モルモットおよび重度難聴者の聴神経における時系列刺激の応答を調べ、人工喉頭の基礎研究から母音波形のゆらぎが音声の自然性にどのように寄与しているかを明確にした。(5)では、音響的特徴量が前後の音韻コンテキストから受ける影響について、数量化理論を非線形に拡張したモデルを用いて検討した。(6)では、地口の解釈を含む「深い意味理解」を実現するという高度な枠組みの中で必要とされる音響的特徴の抽出精度について検討した

• Studies on Modeling and Performance Analysis of Nonlinear Dynamic Systems

View Summary

This scientific research has been done during the period from May, 1988 to March 1981. The purpose of the research is to do systematic research on modeling and performance analysis of nonliear dynamic systems. The following are main results obtained by the research :1. Analysis of Fluctuation of Nonlinear Dynamic Systems and Studies on Infinite Dimensional Homotopy MethodA number of properties have been clarified on response of fluctuations to nonlinear systems by developing a theory of non-deterministic operators. Moreover, an infinite dimensional homotopy method has been established for numerically analyzing nonlinear systems including infinite dimensional systems.2. Local and Global Bifurcation Phenomena of Nonlinear CircuitsA number of properties have been clarified on the global bifurcation phenomena in nonlinear circuits analysis. Moreover, a number of interesting bifurcation phenomena have been observed by computer simulations and circuit experiments.3. Numerical Method of Nonlinear Dynamic SystemsAn efficient algorithm has been developed for caluculating solutions of nonlinear circuits, nonlinear programming, channels, and all solutions of a certain class of nonlinear equations.4. Modelig of Dynamical Systems and Its ApplicationBy using concept of the bond graph an automatic modeling method is presented for a class of dynamical systems. Based on this, an efficient modeling and a numerical integral method have been established for analyzing flexible multibody dynamics

• 自然言語と図形を用いた対話における意味理解と知識獲得に関する基礎研究

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1.自然語文や図形の意味を総合的に取り扱いうる認識表現の研究、及び自然語文・図形の解析と認識表現の生成を行うシステムの研究についてこれまでに構築したプロトタイプシステムの拡張のために、画像データ処理部についてアルゴリズムの再検討を行い、その結果に基いて図形処理ツールの拡張を行った。2.自然語文・図形の理解と問題解決プロセスに関する研究について本年度は特に、(1)数学文章題を解くシステム(2)高校化学の問題演習型知的CAIシステム(3)高校経済の教科書や新聞記事を読んで文章を理解するシステム 等を題材として、文章理解・対話制御・問題解決に関する諸問題の検討を行った。(1)については、数学の問題を解くために必要な知識を整理し、教師の助言を受け付ながら問題を解くシステムのプロトタイプを構築した。(2)については、特に教育への利用の便宜を考慮しつつ化学の教材知識について整理し、また問題解決プロセスについて検討した。更にこの求解結果を用いて対話指導を行う手法について考察し、以下の2つの能力を実現する対話制御手法を提案した。(1)外界からの情報を常時取り込み、得られた情報を直ちに行動に反映させる事が可能(2)状況に応じて多様な教授方法を実行可能これらの成果に基づいて問題演習型知的CAIシステムの問題解決部及び対話部を構築した。(3)については、人間が文章を読み取る際に生じるイメージをモデル化し、計算機が文章を読み取りながらこの様なイメージを構築することによって、概念間関係の推定や指示語の同定などを行って文章を理解する手法について検討した。また、この手法に基づいて新聞記事を読み取る文章理解システムのプロトタイプを構築した

• 自然言語と図形を用いた対話における意味理解と知識獲得に関する基礎的研究

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本年度は、昨年度に引続き高校化学を対象とする知的CAIシステムと学生の対話を題材として、人間-機械間のコミュニケ-ションに関する基礎的検討を行った。特に、学生に知識を効果的に伝授するための教師の発話内容と、そのような発話能力の実現方法について検討した。問題演習を通じて、学生により本質的な理解をさせるためには、求解に利用される知識がどの様な理由で成立するかを教える必要がある。ここではこの理由を知識の成立原理と呼ぶ、知識の成立原理を説明する能力を実現するため、まず高校化学の計算問題で利用される知識を分類した。さらに各タイプの知識毎に、その成立原理を説明するためにシステムが把握すべき事柄について整理した。その結果、利用される知識は数量間関係の知識、化学現象の知識、物質・物体に固有な構造・属性についての知識等に分類され、知識の成立原理として、問題で扱う系の各時点での状態(実体概念の属性値や実体概念間の位置関係、接続関係、包含関係等)、系に働く作用、系に生じる変化、及びそれらの相互関係等が重要な役割を果たす事が明らかになった。これらり事柄を教育システムが把握するためには、系の状態を、系に働く作用や、それによって生じる変化を考慮してシミュレ-トし、化学現象に関するモデルを構築する機構を持つ必要がある。そこで化学現象モデルについて検討を行い、具体的なモデル表現手法を提案した。提案されたモデルは、(1)演修問題を解くための知識と関係づけられており、その知識の成立原理を説明する際に利用できる、(2)系の状態・作用・変化の関係を陽に記述できる,等の特長を持つ。以上の成果をもとに,ワ-クステ-ション上に演習問題求解システムを構築し、昨年度の成果である対話システムと接続して、知識の成立原理を説明できる対話型教育システムを実現した

• STUDY ON MODELLING OF NONLINEAR SYSTEM AND SELF-VALIDATING MUMERICAL METHOD

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Recently, the study of nonlinear systems and nonlinear technologes has made a great advance. For example, the studies of optical fiber soliton communications, neural networks, fuzzy systems, analogue VLSI and so on have achieved much interests. Since such systems essentially use nonlinearity , new modelling techniques and reliable simulation techniques are required. Moreover, in the field of computer assisted design of nonlinear systems, it is very important to guarantee the accuracy of the result of calculation. For example, in the VLSI design, if we can validate the accuracy of modeling and numerical simulation, it enables to reduce the cost and period of design.In this study, taking the accuracy of modeling into consideration, and guarantteng the accuracy of numerical simulation of the modeling, we have develpoed a numerical validation method through the total simulation process of nonlinear system.In this year, we have improved the theory, algorithm and system developed until last year, and have made them more practical.1.Effectiveness of our fuzzy modeling theory is confirmed by numerical simulation.2.Automatic numerical validation method for general ordinary differential equations is developed. It provides a rigorous method for transient analysis.3.Extending the technique used in all solution method for nonlinear equations, we have developed a inclusion method for solution sets of set-valued functions. It gives a more rigorous system analysing method together with modeling method with guaranteed accuracy

• Co-operative Research of Analysis Method for Very High Speed Integrated Circuits Composed of Distributed and Lumped Elements

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We had 6 meetings on this research project during these two years. Main new results obtained by the members of this research within this period are given as follows.(1) We summarized the state of the art of this subject and classified systematically various analysis methods so far known.(2) Both the characteristics impedance function Zo (s) and a bilinear function of the phase characteristics rheta (s) of distributed elements have to be approximated by rational positive real functions. The approximation by means of these functions resulted in better property of convergence than the conventional Pade approximation method.(3) A new method of the transient analysis for multi-phase circuits in power system having multi-branches are presented and the validity by numerical simulation are shown.(4) For general numerical analysis, a numerical validation method using the interval analysis and rational arithmetic is presented and some further improvements of this method are given.(5) A Katzenelson-like algorithm was presented for solving piecewise-linear resistive network and its global and quadratic convergence were proved.(6) A transient analysis algorithm combining frequency-domain simulation for distributed constant circuit (transmission line) and time-domain simulation for lumped constant circuit (logic gate) was proposed. It utilized wavr relaxation method.(7) A new algorithm for general circuit analysis was implemented on parallel computer. The program enables us to treat fundamental block of integrated circuits simulation system for various circuits including transmission lines and nonlinear elements.(8) A new equivalent circuit for a dielectric filter was proposed, where the equivalent circuit consists of multiconductor transmission line and lumped constant elements and element values are determined from experimental results. The total characteristics of circuit are much coincident with physical results over wide frequency than usual model

• Research of Computer aided Nonlinear Analysis with Flexibility

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We had the purpose to establish fundamental theories and to constitute the elements of computer aided mathematical analyzing software. We developed our research as we planned as the following :(1) We proposed the fixed point theorem for fuzzy map which is obtained by modeling the system with uncertain property.(2) We make the theory to prove numerically the existence of solution for nonlinear operator equations.(3) Based on C++ and an object oriented language in which rational number arithmetic is implemented, we constituted 3 prototypes of object oriented software which can deal with various objects corresponding to interval analysis, automatic differentiation, function representation and so on.(4) By extending the methods to prove numerically the existence of bifurcation point, we developed the theory to cancel singular points. We also applied our theory to various types of bifurcation phenomena and indicated that we can prove the existence of actual bifurcation phenomena.(5) We proposed the theory to prove the existence of homoclinic orbits or heteroclinic orbits and prove their existence for actual examples.(6) We proposed an algorithm to prove the existence of all solutions in a bounded region for finite dimensional nonlinear equations and proved that this algorithm stops within finite steps under the certain conditions.(7) We proposed a method to prove the existence of all solutions with high speed in a bounded region for finite dimensional nonlinear equations with separability, whose example is VLSI circuit. (8) We could change the speed of calculation by the accuracy. Concretely, We proposed the method in which we can obtain the calculated results with super high speed when we demand its low accuracy and in which we can obtain the results with high speed even when we demand its high accuracy.(9) We realized the obtained techniques of numerical method with guaranteed accuracy on our prototypes of the software. We applied our software to various nonlinear functional equations and indicated its usefulness.(10) We combined the numerical method in the case that we demand its low accuracy and the numerical one in the case that we demand its high accuracy, by which we proposed the numerical method with high speed at our request of accuracy. We realized this method on our software and indicated its usefulness by the example of nonlinear circuit systems.(11) We integrated the above organized investigations and remade a prototype of the software for computer aided nonlinear analysis which can correspond to the changes of the problem or the accuracy. We also indicated its usefulness by applying it to nonlinear circuit problems

• Self-validating numerics with applications to computational science and technology

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In this research, we extended and improved the self-validating numerical methods which can be applied to wide mathematical and analytical problems as well as to particular problems such as equations in the mathematical fluid mechanics. The important research results done by investigators and co-investigators are as follows :1. (by Nakao, N. Yamamoto and Watanabe) Several refinements and extensions were established for the numerical verification methods of solutions for elliptic problems. Namely, the numerical computation with guaranteed error bounds for the eigenvalue problems of second order elliptic operator was established by using the techniques in the numerical verification method of solutions for nonlinear elliptic boundary value problems. We also formulated and obtained basic results for the self-validating method for solutions of elliptic variational inequalities,. Moreover, we presented a verified computation of solutions for the Navier-Stokes equation based on the a posteriori and constructive a priori error estimates for the finite element solutions of the Stokes problems. Additionally, we computed a turning point with rigorous error bound for the perturbed and parameterized Gelfand equation.2. (by Oishi) Some fast algorithms for the fundamental validated computations and the solutions of linear and nonlinear problems were presented.3. (by Kikuchi) Theoretical and numerical results were obtained for the error analysis of a special kind of finite element method for electro-magnetic problems.4. (by Sakai) Some applications of splines were presented for plane data approximation.5. (by Fujino) An efficient acceleration method was investigated for parallel machines.6. (by Mitsui) A self-validating method for ordinary differential equations with initial value problems was presented.7. (by T. Yamamoto) Some new error analysis was carried out for the Shortley-Weller type deference scheme for Dirichlet Problems.8. (by Tabata) Several error estimates were derived of the finite element method for the problem in fluid mechanics.9. (by Nishida) Some bifurcation phenomena in fluid dynamics were analyzed by the computer assisted proof.10. (by Murota) The reliability in the structural engineering was investigated by using the group theoretic bifurcation arguments

• Studies on fast numerical calculation with verification

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The present researcher has shown that the addition and the product of two matrices can be calculated with verification via tow times changes of rounding mode of floating point arithmetic. Namely he has proposed a vector interval arithmetic. Furthermore, utilizing perturbation theory, which gives a posteriori error estimate, it has shown that a rigorous error bound of an approximate solution of a system of linear equations can be calculated with the same cost as that of calculating such an approximate solution. It is around from 1,000 to 10,000 speed up compared with the previous method. This method can be extended to many problems of numerical linear algebra including matrix eigenvalue problems and singular value problems. As examples, from 1,000 to 30,000 dimensional full matrix systems have been solved via PC cluster system

• Synthetic approach for new developments of self-validating numerics

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In this research, we newly developed the self-validating numerical methods which can be applied to wide mathematical and analytical problems as well as extended or improved the existing techniques.And we actually applied these methods to particular problems such as equations in the mathematical fluid mechanics and oscillation problems. The important research results obtained by investigators and co-investigators are as follows :1. Nakao, N.Yamamoto, Watanabe established several refinements and extensions for the numerical verification methods of solutions for elliptic problems. Namely, they succeeded the numerical computation with guaranteed error bounds for the inverse eigenvalue problems of second order elliptic operator. They also obtained some results for enclosing the solutions for elliptic variational inequlities. Moreover, they computed an optimal constant with guaranteed accuracy appearing in the a priori error estimates for the finite element projection of the Poisson problem, which is an important contribution for the numerical verification for nonlinear elliptic problems.2. Nagatou and Minamoto obtained interesting computer assisted proofs for the Kolmogorov problem and for the perturbed Gelfand equation, respectively.3. Oishi established some fast algorithms for the fundamental validated computations for the solutions of linear equations.4. Nishida et al. computed with guaranteed error bounds for the non-trivial solution of heat convection problems, which is an important result for a computer assisted proof in the fluid mechanics.5. T. Yamamoto obtained some convergence results of the finite difference scheme for the singular solutions of two point boundary value problems

• Synthetic approach for the development of computer assisted analysis from the numerical verification methods

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In this research, we newly developed the numerical verification methods which can be applied to wide mathematical and analytical problems, as well as extended or improved the existing techniques.And we actually applied these methods to particular problems such as equations in the mathematical fluid mechanics and oscillation problems etc. The important research results obtained by investigators and co-investigators are as follows :1.Nakao, N.Yamamoto, Watanabe established several refinements and extensions for the constructive error estimates for the finite finite element projections of the Poisson and the bi-harmonic equations on various kinds of domains, particularly, on nonconvex polygonal domains. These results played important and essential roles for the numerical verification of solutions of nonlinear elliptic equations and the two dimensional stationary Navier-Stokes problems.2.Nagatou numerically proved the stability of the flow on the torus called Kolmogorov problem.3.Minamoto presented a formulation of the verification condition for the double turning point and applied it to the perturbed Gelfand equation.4.Oishi established some refinements on the fast algorithm for the solutions of linear equations.5.Nishida et al. presented the computed results with guaranteed error bounds for the symmetry breaking bifurcation point of the solution of two dimensional heat convection problems, as well as they formulated the numerical verification algorithm for the three dimensional problems with some prototypical verified examples.6.Chin obtained some numerical verification results on the existence of solutions and a posteriori error estimates for the linear complementarity problems

• Development of computer assisted analysis for complicated nonlinear phenomena

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We were working on the development and applications of the numerical verification methods for solutions of nonlinear partial differential equations, in particular, we succeeded in finding a new and very efficient verification principle for nonlinear evolutional problems. Also we extended and improved the existing verification methods for solutions of elliptic problems as well as we proved the effectiveness of the computer assisted proofs by applying our methods to resolve the actual nonlinear problems for which any theoretical approaches seem to be not useful to apply

### Specific Research

• 2014

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精度保証付き数値計算は研究代表者らが提案した，丸めモード制御方式による高速精度保証法や無誤差変換法の発展により，通常の近似計算の数倍の手間で，条件数に応じた高精度計算アルゴリズムを構築出来るようになった．これに伴い非常に広い範囲の理工学に現れる実数計算の問題を厳密に数値計算によって解けるようになり，様々な理工学の問題の解決を図ることができた．本年度は「ラプラス作用素の高精度固有値評価」，「非線形問題の線型化作用素に対する逆作用素ノルム評価」，「高精度逆コレスキー分解の収束解析」，「3次元多様体の双曲性に対する数値的分類定理」，「H行列を使用した線形方程式の新しい精度保証理論」に関して成果を得た．

• 2010

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精度保証付き数値計算学の展開：本研究では，非線形系に対する精度保証付き数値計算を展開するための基礎を構築し，その上で非線形偏微分方程式の計算機援用証明等の応用を展開した．1）大石・高安Newton-Kantorovichの定理を用いた計算機援用証明手法を確立するため，これまで不可能であった楕円型非線形偏微分方程式の解の存在と誤差の範囲内での一意性を証明した．計算で得られる近似解にある程度の滑らかさを仮定すると，従来よりもはるかに効果的な残差評価を適用でき，既存の過大評価を回避することができる．これによりEmden方程式の解などの非線形性が大きな解にも提案手法の適用範囲が拡大し，目標に向けて一歩前進した．2）劉・大石楕円型偏微分方程式を非凸な領域で考える場合，偏微分作用素は特異性により非常に扱いが難しい．従来法の多くは凸領域を仮定することが多いが，我々は混合型有限要素とHypercircleequationを用いて，任意多角形領域上でラプラス作用素の固有値評価を精度保証付きで求めるユニバーサルな手法を世界で初めて開発した．また提案手法を用いたWebアプリケーションを開発し，ユーザーがオンライン上でグラフィカルなシミュレーションを行えるようになっている．3）山中・大石精度保証付き数値積分では，全ての計算誤差を考慮し「ユーザーが要求する精度まで数学的に正しい結果を返すアルゴリズム」を提案した．この手法は計算に生じる公式誤差の上限を多重階微分値を利用したり，複素円盤領域上で事前に計算できる．一般的な近似解だけを求める数値積分アルゴリズムは許容誤差を満たすように再帰的にアルゴリズムが設計されていることが多いが，本手法を用いると許容誤差を満たす分点数が事前誤差評価によりあらかじめ計算できる．これを用いて，従来の近似計算アルゴリズムと同等程度（時に高速）な超高速かつ高信頼な精度保証付きアルゴリズムを開発した．

• 2002   柏木　雅英, 中谷　祐介, 宮田　高富

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精度保証付き数値計算を従来の近似計算と比べて２倍程度の手間で行うための理論体系の構築とソフトウエア開発を行った。主な成果をまとめると以下のようになる。(1) 理論的手法として、丸めの制御精度保証方式を考案した。これは、IEEE浮動小数点規格754に従うCPUにPortableに成立するアルゴリズム理論で、ベクトル区間演算を基礎としている。(2) この基礎理論にもとづき、数値線形代数の諸問題に対する、高速精度保証アルゴリズムを開発した。この中には、密係数行列をもつ連立一次方程式の精度保証理論、疎係数行列をもつ連立一次方程式の精度保証理論、固有値の高速精度保証法、残差反復解法の精度保証化理論などを含む。(3) 以上の成果に基づき、精度保証ライブラリを開発した。(4) これをSLABという精度保証付き数値計算モードをもつ数値計算ツールとしてソフトウエア化した。SLABは近似解の計算においてはMATLABと互換であるあるが、新たに精度保証モードをもち、この中で計算すると、解の存在と近似解の厳密な誤差を高速に実行する機能をもっている。SLABは現在GPLとして公開中である。

• 2000

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精度保証付き数値計算とは数値計算の結果得られた近似解の近くに真の解が存在することを保証し、その（局所的）一意性や真の解と数値解の誤差をシャープに評価することを目的として行われる数値計算のことである。したがって、数値計算におけるあらゆる誤差を数学的に正しく評価して、この目的を達成する必要がある。従来は、このようなことは理想ではあるが、理論的にも現実的にも難しく、精度保証付き数値計算は実質的に不可能であると考えられていた。本研究では、IEEE754の倍精度浮動小数点数規格にもとづき、浮動小数点数演算に於ける丸めのモードを適切に制御する手法を関数解析的な摂動理論（数値解析理論）を組み合わせることにより、精度保証付き数値計算が高速に実行できること明らかにする目的で実施された。以下、その成果の概要を述べる。１．連立一次方程式の数値解の高速精度保証　IEEE754の上への丸めと下への丸めのそれぞれのモードでベクトルの内積を２回実行することにより、ベクトルの内積の値を上下からシャープに評価できることを示した。これを用いて、連立一次方程式の数値解の精度保証を行うためのアルゴリズムを開発した。LU分解の事前誤差評価式を巧みに利用することで、数値解をガウスの消去法で求めるのと同じ手間でその精度保証ができることを示した。例えば1000x1000密行列を係数行列に持つ場合、Pentium III 800MHz CPUで最適化BLASとLAPACKにより、数値解は２秒で求まるが、その精度保証も２秒で実行可能であることを示している。２．行列の固有値の高速精度保証　Bauer-Fike型の固有値の摂動定理を利用して、行列の固有値を高速に精度保証する手法を１の技法を応用して確立した。この手法は、多重固有値をもつ一般複素行列に適用可能で、広い応用範囲をもつ。

• 1999   堀内　和夫, 川瀬　武彦, 吉村　浩明, 柏木　雅英, 神澤　雄智

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数値計算における丸め誤差および打切り誤差を勘案して、数学モデルとして与えられた方程式の数学的に厳密な意味での解の存在を数値計算により保証し、数値計算で得られた近似解(以下、数値解と略称する)と真の解との間の誤差のシャープな上限を数値計算することを精度保証付き数値計算という。精度保証付き数値計算は九州大学の須永教授によって1950年代の終わりに提案された区間解析がその基礎となっている日本発の技術である。区間解析では、実数は、その数を内部に含む、両端を浮動小数点数とする区間で近似される。そして、実数の四則演算は区間演算に置き換えられて実行される(区間演算単体では浮動小数点数の四則演算の２から４倍ほどの計算量)。須永の区間演算の提案は外国で認められ、アメリカ、ドイツを中心として精度保証付き数値計算の研究は進展してきた。精度保証付き数値計算の研究の発展は欧米で進められてきたともいえよう。これらの研究は、区間演算ごとに丸めの方向の切り替えをする前提であった。この方法では丸めの制御命令が四則演算ごとで加わることにより、高速化のために高度な調整を行っている従来の数値計算用のプログラム資産が活用できなくなるという欠点があった。本研究では、区間演算を行う際に必要となるCPUの丸め方向の変更命令を、できるだけ、プログラムの外へ出す方式を開発した。すなわち、通常の区間演算では演算ごとにCPUの丸めの方向が切り替えられていたが、本研究では、丸めの方向の切り替えを行列の積の演算の前後で行うことによって線形系の数値解の精度保証ができる方式を提案している。この方法では、連立一次方程式の数値解の精度保証などにおいては、区間演算ごとに丸めの方向が切り替えられていた従来方式に比べて、丸めの方向の切り替えの回数が数回というレベルに減るとともに、行列の積といったBLASの第３レベルの命令をそのまま使えるので、従来の計算機環境の中で、従来の最適化されたプログラムがそのまま使えるようになっている。これにより、精度保証付き数値計算が近似的な数値解を得るための従来の数値計算の計算時間に対して、実速度で７から８倍以内、早い場合には２倍程度で精度保証(近似解を求めることも含めての計算時間)ができることが示された。

### Syllabus

• School of Fundamental Science and Engineering

2022   full year

• School of Fundamental Science and Engineering

2022   full year

• School of Fundamental Science and Engineering

2022   fall semester

• School of Fundamental Science and Engineering

2022   fall semester

• School of Fundamental Science and Engineering

2022   fall semester

• School of Fundamental Science and Engineering

2022   spring semester

• School of Fundamental Science and Engineering

2022   spring semester

• School of Fundamental Science and Engineering

2022   fall semester

• School of Fundamental Science and Engineering

2022   fall semester

• School of Fundamental Science and Engineering

2022   fall semester

• School of Fundamental Science and Engineering

2022   fall semester

• School of Fundamental Science and Engineering

2022   spring semester

• School of Fundamental Science and Engineering

2022   spring semester

• School of Fundamental Science and Engineering

2022   fall semester

• School of Fundamental Science and Engineering

2022   fall semester

• School of Fundamental Science and Engineering

2022   spring semester

• School of Fundamental Science and Engineering

2022   spring semester

• School of Fundamental Science and Engineering

2022   spring semester

• School of Fundamental Science and Engineering

2022   spring semester

• School of Fundamental Science and Engineering

2022   spring semester

• School of Fundamental Science and Engineering

2022   spring semester

• School of Fundamental Science and Engineering

2022   spring semester

• School of Fundamental Science and Engineering

2022   fall semester

• School of Fundamental Science and Engineering

2022   fall semester

• School of Fundamental Science and Engineering

2022   fall semester

• School of Fundamental Science and Engineering

2022   spring semester

• School of Fundamental Science and Engineering

2022   spring semester

• School of Fundamental Science and Engineering

2022   spring semester

• School of Fundamental Science and Engineering

2022   fall semester

• School of Fundamental Science and Engineering

2022   fall semester

• School of Fundamental Science and Engineering

2022   spring semester

• School of Fundamental Science and Engineering

2022   spring semester

• Graduate School of Fundamental Science and Engineering

2022   fall semester

• Graduate School of Fundamental Science and Engineering

2022   spring semester

• Graduate School of Fundamental Science and Engineering

2022   full year

• Graduate School of Fundamental Science and Engineering

2022   fall semester

• Graduate School of Fundamental Science and Engineering

2022   spring semester

• Graduate School of Fundamental Science and Engineering

2022   fall semester

• Graduate School of Fundamental Science and Engineering

2022   spring semester

• Graduate School of Fundamental Science and Engineering

2022   spring semester

• Graduate School of Fundamental Science and Engineering

2022   full year

• Graduate School of Fundamental Science and Engineering

2022   full year

• Graduate School of Fundamental Science and Engineering

2022   fall semester

• Graduate School of Fundamental Science and Engineering

2022   spring semester

• Graduate School of Fundamental Science and Engineering

2022   fall semester

• Graduate School of Fundamental Science and Engineering

2022   spring semester

• Graduate School of Fundamental Science and Engineering

2022   spring semester

• Graduate School of Fundamental Science and Engineering

2022   full year

• Graduate School of Fundamental Science and Engineering

2022   spring semester

• Graduate School of Fundamental Science and Engineering

2022   summer quarter

• Graduate School of Fundamental Science and Engineering

2022   spring quarter

• Graduate School of Fundamental Science and Engineering

2022   full year

• Graduate School of Fundamental Science and Engineering

2022   full year

• Graduate School of Fundamental Science and Engineering

2022   fall semester

• Graduate School of Fundamental Science and Engineering

2022   spring semester

• Graduate School of Fundamental Science and Engineering

2022   fall semester

• Graduate School of Fundamental Science and Engineering

2022   spring semester

• Graduate School of Creative Science and Engineering

2022   fall semester

• Graduate School of Fundamental Science and Engineering

2022   full year

• Graduate School of Creative Science and Engineering

2022   spring semester

• Graduate School of Fundamental Science and Engineering

2022   full year

• Graduate School of Fundamental Science and Engineering

2022   spring semester

2022   spring semester

2022   fall semester

• Graduate School of Creative Science and Engineering

2022   spring semester

2022   spring semester

### Committee Memberships

•

IEICE  Trans. Fundamentals Editor

•

IEICE  Nonlinear Theory and its Applications Editor in Chief