Japan Journal of Industrial and Applied Mathematics Articles (Project Euclid)
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The latest articles from Japan Journal of Industrial and Applied Mathematics on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2010 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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Application of the Krawczyk--Moore--Jones Algorithm to Electric Circuit Analysis and Its Further Development
http://projecteuclid.org/euclid.jjiam/1265033776
<strong>Kohshi Okumura</strong><p><strong>Source: </strong>Japan J. Indust. Appl. Math., Volume 26, Number 2, 145--167.</p><p><strong>Abstract:</strong><br/>
This paper surveys applications of Krawczyk--Moore--Jones's algorithm and
presents its further developments.
The application is focused onto nonlinear electric circuit analysis.
The further developments are described on parallel KMJ algorithm,
Gray code KMJ algorithm and KMJ processor.
</p>projecteuclid.org/euclid.jjiam/1265033776_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTFast Verified Solutions of Linear Systems
http://projecteuclid.org/euclid.jjiam/1265033777
<strong>Takeshi Ogita</strong>, <strong>Shin'ichi Oishi</strong><p><strong>Source: </strong>Japan J. Indust. Appl. Math., Volume 26, Number 2, 169--190.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.jjiam/1265033777_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTAlgorithms for Accurate, Validated and Fast Polynomial Evaluation
http://projecteuclid.org/euclid.jjiam/1265033778
<strong>Stef Graillat</strong>, <strong>Philippe Langlois</strong>, <strong>Nicolas Louvet</strong><p><strong>Source: </strong>Japan J. Indust. Appl. Math., Volume 26, Number 2, 191--214.</p><p><strong>Abstract:</strong><br/>
We survey a class of algorithms to evaluate polynomials with floating
point coefficients and for computation performed with IEEE-754 floating
point arithmetic.
The principle is to apply, once or recursively, an error-free
transformation of the polynomial evaluation with the Horner algorithm
and to accurately sum the final decomposition.
These compensated algorithms are as accurate as the Horner
algorithm performed in $K$ times the working precision, for $K$ an
arbitrary positive integer.
We prove this accuracy property with an a priori error analysis.
We also provide validated dynamic bounds and apply these results to
compute a faithfully rounded evaluation.
These compensated algorithms are fast.
We illustrate their practical efficiency with numerical experiments on
significant environments. Comparing to existing alternatives these
$K$-times compensated algorithms are competitive for $K$ up to 4,
i.e., up to 212 mantissa bits.
</p>projecteuclid.org/euclid.jjiam/1265033778_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTAdaptive and Efficient Algorithm for 2D Orientation Problem
http://projecteuclid.org/euclid.jjiam/1265033779
<strong>Katsuhisa Ozaki</strong>, <strong>Takeshi Ogita</strong>, <strong>Siegfried M. Rump</strong>, <strong>Shin'ichi Oishi</strong><p><strong>Source: </strong>Japan J. Indust. Appl. Math., Volume 26, Number 2, 215--231.</p><p><strong>Abstract:</strong><br/>
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
$3\times 3$ matrix. Numerical results illustrate that our algorithm
works fairly faster than the state-of-the-art algorithm
in various cases.
</p>projecteuclid.org/euclid.jjiam/1265033779_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTA Numerical Verification Method for Two-Coupled Elliptic Partial Differential Equations
http://projecteuclid.org/euclid.jjiam/1265033780
<strong>Yoshitaka Watanabe</strong><p><strong>Source: </strong>Japan J. Indust. Appl. Math., Volume 26, Number 2, 232--247.</p><p><strong>Abstract:</strong><br/>
A numerical verification method of steady state solutions for a system of
reaction-diffusion equations is described.
Using a decoupling technique, the system is reduced to a single nonlinear
equation and a computer-assisted method for second-order elliptic boundary
value problems based on the infinite dimensional fixed-point theorem can be
applied.
Some numerical examples confirm the effectiveness of the method.
</p>projecteuclid.org/euclid.jjiam/1265033780_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTInversion of Extremely Ill-Conditioned Matrices in Floating-Point
http://projecteuclid.org/euclid.jjiam/1265033781
<strong>Siegfried M. Rump</strong><p><strong>Source: </strong>Japan J. Indust. Appl. Math., Volume 26, Number 2, 249--277.</p><p><strong>Abstract:</strong><br/>
Let an $n\times n$ matrix $A$ of floating-point numbers in some format
be given. Denote the relative rounding error unit of the given format
by $\mathtt{eps}$. Assume $A$ to be extremely ill-conditioned, that is
$\cond(A)\gg\mathtt{eps}^{-1}$. In about 1984 I developed an algorithm to
calculate an approximate inverse of $A$ solely using the given
floating-point format. The key is a multiplicative correction rather
than a Newton-type additive correction. I did not publish it because of
lack of analysis. Recently, in [9] a modification of the
algorithm was analyzed. The present paper has two purposes. The first
is to present reasoning how and why the original algorithm works. The
second is to discuss a quite unexpected feature of floating-point
computations, namely, that an approximate inverse of an extraordinary
ill-conditioned matrix still contains a lot of useful information. We
will demonstrate this by inverting a matrix with condition number
beyond $10^{300}$ solely using double precision. This is a workout of
the invited talk at the SCAN meeting 2006 in Duisburg.
</p>projecteuclid.org/euclid.jjiam/1265033781_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTDirect Methods for Linear Systems with Inexact Input Data
http://projecteuclid.org/euclid.jjiam/1265033782
<strong>Günter Mayer</strong><p><strong>Source: </strong>Japan J. Indust. Appl. Math., Volume 26, Number 2, 279--296.</p><p><strong>Abstract:</strong><br/>
We give a survey on direct methods for interval linear
systems. We also consider various kinds of solution sets and
show how the interval hull can be computed.
</p>projecteuclid.org/euclid.jjiam/1265033782_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTVerified Numerical Computation for Nonlinear Equations
http://projecteuclid.org/euclid.jjiam/1265033783
<strong>Goetz Alefeld</strong><p><strong>Source: </strong>Japan J. Indust. Appl. Math., Volume 26, Number 2, 297--315.</p><p><strong>Abstract:</strong><br/>
After the introduction basic properties of interval arithmetic are
discussed and different approaches are repeated by which one can
compute verified numerical approximations for a solution of
a nonlinear equation.
</p>projecteuclid.org/euclid.jjiam/1265033783_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTNumerical Verification Methods for Spherical $t$-Designs
http://projecteuclid.org/euclid.jjiam/1265033784
<strong>Xiaojun Chen</strong><p><strong>Source: </strong>Japan J. Indust. Appl. Math., Volume 26, Number 2, 317--325.</p><p><strong>Abstract:</strong><br/>
The construction of spherical $t$-designs with $(t+1)^2$ points
on the unit sphere $S^2$ in $\mathbb{R}^3$ can be reformulated as an
underdetermined system of nonlinear equations.
This system is highly nonlinear and involves the evaluation of
a degree $t$ polynomial in $(t+1)^4$ arguments.
This paper reviews numerical verification methods using the Brouwer
fixed point theorem and Krawczyk interval operator for solutions of
the underdetermined system of nonlinear equations.
Moreover, numerical verification methods for proving that a solution
of the system is a spherical $t$-design are discussed.
</p>projecteuclid.org/euclid.jjiam/1265033784_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTNumerical Existence Proof of Five Solutions for Certain Two-Transistor Circuit Equations
http://projecteuclid.org/euclid.jjiam/1265033785
<strong>Yusuke Nakaya</strong>, <strong>Tetsuo Nishi</strong>, <strong>Shin'ichi Oishi</strong>, <strong>Martin Claus</strong><p><strong>Source: </strong>Japan J. Indust. Appl. Math., Volume 26, Number 2, 327--336.</p><p><strong>Abstract:</strong><br/>
In this paper, we are concerned with the analysis of two-transistor
circuits. Applying technique for the numerical verification, we prove
rigorously the existence of five solutions in a two-transistor circuit.
The system of equations for a transistor circuit is obtained as
nonlinear equations, therefore Krawczyk's method is applied for proving
the existence of a solution.
</p>projecteuclid.org/euclid.jjiam/1265033785_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTOn Verified Numerical Computations in Convex Programming
http://projecteuclid.org/euclid.jjiam/1265033786
<strong>Christian Jansson</strong><p><strong>Source: </strong>Japan J. Indust. Appl. Math., Volume 26, Number 2, 337--363.</p><p><strong>Abstract:</strong><br/>
This survey contains recent developments for computing
verified results of convex constrained optimization problems, with
emphasis on applications. Especially, we consider the computation
of verified error bounds for non-smooth convex conic
optimization in the framework of functional analysis, for
linear programming, and
for semidefinite programming. A discussion of important problem
transformations to special types of convex problems and convex
relaxations is included. The latter are important for handling and
for reliability issues in global robust and combinatorial optimization.
Some remarks on numerical experiences, including also large-scale
and ill-posed problems, and software for verified computations
concludes this survey.
</p>projecteuclid.org/euclid.jjiam/1265033786_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTAn Application of Taylor Models to the Nakao Method on ODEs
http://projecteuclid.org/euclid.jjiam/1265033787
<strong>Nobito Yamamoto</strong>, <strong>Takashi Komori</strong><p><strong>Source: </strong>Japan J. Indust. Appl. Math., Volume 26, Number 2, 365--392.</p><p><strong>Abstract:</strong><br/>
The authors give short survey on validated computaion of initial value
problems for ODEs especially Taylor model methods. Then they propose
an application of Taylor models to the Nakao method which has been
developed for numerical verification methods on PDEs and apply it to
initial value problems for ODEs with some numerical experiments.
</p>projecteuclid.org/euclid.jjiam/1265033787_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTRecent Development in Rigorous Computational Methods in Dynamical Systems
http://projecteuclid.org/euclid.jjiam/1265033788
<strong>Zin Arai</strong>, <strong>Hiroshi Kokubu</strong>, <strong>Paweł Pilarczyk</strong><p><strong>Source: </strong>Japan J. Indust. Appl. Math., Volume 26, Number 2, 393--417.</p><p><strong>Abstract:</strong><br/>
We highlight selected results of recent development
in the area of rigorous computations which use interval arithmetic
to analyse dynamical systems. We describe general ideas and selected details
of different ways of approach and we provide specific sample applications
to illustrate the effectiveness of these methods.
The emphasis is put on a topological approach, which combined
with rigorous calculations provides a broad range of new methods
that yield mathematically reliable results.
</p>projecteuclid.org/euclid.jjiam/1265033788_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTComputer-Assisted Proofs for Semilinear Elliptic Boundary Value Problems
http://projecteuclid.org/euclid.jjiam/1265033789
<strong>Michael Plum</strong><p><strong>Source: </strong>Japan J. Indust. Appl. Math., Volume 26, Number 2, 419--442.</p><p><strong>Abstract:</strong><br/>
For second-order semilinear elliptic boundary value problems on
bounded or unbounded domains, a general computer-assisted method for
proving the existence of a solution in a ``close'' and explicit
neighborhood of an approximate solution, computed by numerical means,
is proposed. To achieve such an existence and enclosure result, we
apply Banach's fixed-point theorem to an equivalent problem for the
error, i.e., the difference between exact and approximate solution. The
verification of the conditions posed for the fixed-point argument
requires various analytical and numerical techniques, for example the
computation of eigenvalue bounds for the linearization at the
approximate solution. The method is used to prove existence and
multiplicity results for some specific examples.
</p>projecteuclid.org/euclid.jjiam/1265033789_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTNumerical Verification Method of Solutions for Elliptic Equations and Its Application to the Rayleigh--Bénard Problem
http://projecteuclid.org/euclid.jjiam/1265033790
<strong>Yoshitaka Watanabe</strong>, <strong>Mitsuhiro T. Nakao</strong><p><strong>Source: </strong>Japan J. Indust. Appl. Math., Volume 26, Number 2, 443--463.</p><p><strong>Abstract:</strong><br/>
We first summarize the general concept of our verification method
of solutions for elliptic equations.
Next, as an application of our method, a survey and future works on
the numerical verification method of solutions for
heat convection problems known as Rayleigh--Bénard problem
are described. We will give a method to verify the existence of
bifurcating solutions of the
two-dimensional problem and the bifurcation point itself.
Finally, an extension to the three-dimensional case and future works
will be described.
</p>projecteuclid.org/euclid.jjiam/1265033790_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTIterative Refinement for Ill-Conditioned Linear Systems
http://projecteuclid.org/euclid.jjiam/1265033791
<strong>Shin'ichi Oishi</strong>, <strong>Takeshi Ogita</strong>, <strong>Siegfried M. Rump</strong><p><strong>Source: </strong>Japan J. Indust. Appl. Math., Volume 26, Number 2, 465--476.</p><p><strong>Abstract:</strong><br/>
This paper treats a linear equation
\begin{equation*}
Av=b,
\end{equation*}
where $A \in \mathbb{F}^{n\times n}$ and $b \in \mathbb{F}^n$.
Here, $\mathbb{F}$ is a set of floating point numbers.
Let $\mathbf{u}$ be the unit round-off of the working precision and
$\kappa(A)=\|A\|_{\infty}\|A^{-1}\|_{\infty}$ be the condition number
of the problem. In this paper, ill-conditioned problems with
\begin{equation*}
1 < \mathbf{u}\kappa(A) < \infty
\end{equation*}
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.
</p>projecteuclid.org/euclid.jjiam/1265033791_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTNumerical Verification Method for Infinite Dimensional Eigenvalue Problems
http://projecteuclid.org/euclid.jjiam/1265033792
<strong>Kaori Nagatou</strong><p><strong>Source: </strong>Japan J. Indust. Appl. Math., Volume 26, Number 2, 477--491.</p><p><strong>Abstract:</strong><br/>
We consider an eigenvalue problem for differential operators, and
show how guaranteed bounds for eigenvalues
(together with eigenvectors) are obtained and how non-existence
of eigenvalues in a concrete region can be assured.
Some examples for several types of operators will be presented.
</p>projecteuclid.org/euclid.jjiam/1265033792_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTA Constructive A Priori Error Estimation for Finite Element Discretizations in a Non-Convex Domain Using Singular Functions
http://projecteuclid.org/euclid.jjiam/1265033793
<strong>Kenta Kobayashi</strong><p><strong>Source: </strong>Japan J. Indust. Appl. Math., Volume 26, Number 2, 493--516.</p><p><strong>Abstract:</strong><br/>
In solving elliptic problems by the finite element method
in a bounded domain which has a re-entrant corner,
the rate of convergence can be improved by adding a singular function
to the usual interpolating basis. When the domain is enclosed by line
segments which form a corner of $\pi/2$ or $3\pi/2$, we have obtained
an explicit a priori $H^{1}_{0}$ error estimation of $O(h)$
and an $L^{2}$ error estimation of $O(h^{2})$
for such a finite element solution of the Poisson equation.
Particularly, we emphasize that all constants in our error estimates
are numerically determined, which plays an essential role in the
numerical verification of solutions to non-linear elliptic problems.
</p>projecteuclid.org/euclid.jjiam/1265033793_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTComputer Algebra for Guaranteed Accuracy. How Does It Help?
http://projecteuclid.org/euclid.jjiam/1265033794
<strong>Masaaki Kanno</strong>, <strong>Hirokazu Anai</strong><p><strong>Source: </strong>Japan J. Indust. Appl. Math., Volume 26, Number 2, 517--530.</p><p><strong>Abstract:</strong><br/>
Today simulation technologies (based on numerical computation) are
definitely vital in many fields of science and engineering.
The accuracy of numerical simulations grows in importance as simulation
technologies develop and prevail, and many researches have been carried
out for establishing numerically stable algorithms.
In recent years, combining computer algebra and other guaranteed
accuracy approaches draws much attention as one of promising directions
for developing guaranteed accuracy algorithms for a wider class of
problems. This paper illustrates several typical usages of symbolic and
algebraic methods for guaranteed accuracy computation, highlighting
some of the recent applications in control problems.
</p>projecteuclid.org/euclid.jjiam/1265033794_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTPrefacehttp://projecteuclid.org/euclid.jjiam/1317758984<strong>Masaaki Sugihara</strong><p><strong>Source: </strong>Japan J. Indust. Appl. Math., Volume 26, Number 2-3, 123--123.</p>projecteuclid.org/euclid.jjiam/1317758984_Tue, 04 Oct 2011 16:09 EDTTue, 04 Oct 2011 16:09 EDTGuest editors' prefacehttp://projecteuclid.org/euclid.jjiam/1317758985<strong>Mitsuhiro T. Nakao</strong>, <strong>Shin'ichi Oishi</strong><p><strong>Source: </strong>Japan J. Indust. Appl. Math., Volume 26, Number 2-3, 123--124.</p>projecteuclid.org/euclid.jjiam/1317758985_Tue, 04 Oct 2011 16:09 EDTTue, 04 Oct 2011 16:09 EDTTheory of an interval algebra and its application to numerical analysis [Reprint of Res. Assoc. Appl. Geom. Mem. 2 (1958), 29–46]http://projecteuclid.org/euclid.jjiam/1317758986<strong>Teruo Sunaga</strong><p><strong>Source: </strong>Japan J. Indust. Appl. Math., Volume 26, Number 2-3, 125--143.</p><p><strong>Abstract:</strong><br/>
This is reprinted, with permission from the author and the publisher, from the original paper published in Research Association of Applied Geometry (RAAG) Memoirs, Vol. 2 (1958) pp. 29-46, published by Gakujutsu Bunken Fukyu-kai, Tokyo, Japan.
</p>projecteuclid.org/euclid.jjiam/1317758986_Tue, 04 Oct 2011 16:09 EDTTue, 04 Oct 2011 16:09 EDT