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Article
Open AccessA note on Oishi’s lower bound for the smallest singular value of linearized Galerkin equations
Recently Oishi published a paper allowing lower bounds for the minimum singular value of coefficient matrices of linearized Galerkin equations, which in turn arise in the computation of periodic solutions of n...
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Article
Open AccessLower bounds for the smallest singular values of generalized asymptotic diagonal dominant matrices
This article presents three classes of real square matrices. They are models of coefficient matrices of linearized Galerkin’s equations. These Galerkin’s equations are derived from first order nonlinear delay ...
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Article
Open AccessRigorous numerical inclusion of the blow-up time for the Fujita-type equation
Multiple studies have addressed the blow-up time of the Fujita-type equation. However, an explicit and sharp inclusion method that tackles this problem is still missing due to several challenging issues. In th...
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Article
Open AccessError Constants for the Semi-Discrete Galerkin Approximation of the Linear Heat Equation
In this paper, we propose \(L^2(J;H^1_0(\Omega ))\) L ...
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Article
Open AccessCorrection to: A note on Dekker’s FastTwoSum algorithm
The article “A note on Dekker’s FastTwoSum algorithm”, written by Marko Lange and Shin’ichi Oishi, was originally published Online First without Open Access.
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Article
Explicit a posteriori and a priori error estimation for the finite element solution of Stokes equations
For the Stokes equation over 2D and 3D domains, explicit a posteriori and a priori error estimation are novelly developed for the finite element solution. The difficulty in handling the divergence-free condition...
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Chapter and Conference Paper
Verified Numerical Computations and Related Applications
The author has been engaged in the study of numerical computations with result verification starting from 1990.
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Article
Open AccessA new formulation using the Schur complement for the numerical existence proof of solutions to elliptic problems: without direct estimation for an inverse of the linearized operator
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 ...
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Article
Open AccessA note on Dekker’s FastTwoSum algorithm
More than 45 years ago, Dekker proved that it is possible to evaluate the exact error of a floating-point sum with only two additional floating-point operations, provided certain conditions are met. Today the ...
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Article
Open AccessEstimation of Sobolev embedding constant on a domain dividable into bounded convex domains
This paper is concerned with an explicit value of the embedding constant from ...
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Article
A method for verifying the accuracy of numerical solutions of symmetric saddle point linear systems
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 ma...
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Article
Simple floating-point filters for the two-dimensional orientation problem
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 fl...
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Chapter and Conference Paper
Verified Error Bounds for the Real Gamma Function Using Double Exponential Formula over Semi-infinite Interval
An algorithm is presented for computing verified error bounds for the value of the real gamma function. It has been shown that the double exponential formula is one of the most efficient methods for calculatin...
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Chapter and Conference Paper
Verified Computations for Solutions to Semilinear Parabolic Equations Using the Evolution Operator
This article presents a theorem for guaranteeing existence of a solution for an initial-boundary value problem of semilinear parabolic equations. The sufficient condition of our main theorem is derived by a fi...
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Article
Open AccessEstimation of Sobolev-type embedding constant on domains with minimally smooth boundary using extension operator
In this paper, we propose a method for estimating the Sobolev-type embedding constant from ...
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Article
Verified norm estimation for the inverse of linear elliptic operators using eigenvalue evaluation
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 estimatio...
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Article
Convergence analysis of an algorithm for accurate inverse Cholesky factorization
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 ...
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Chapter and Conference Paper
High-Precision Eigenvalue Bound for the Laplacian with Singularities
For the purpose of bounding eigenvalues of the Laplacian over a bounded polygonal domain, we propose an algorithm to give high-precision bound even in the case that the eigenfunction has singularities around r...
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Article
Guaranteed high-precision estimation for P 0 interpolation constants on triangular finite elements
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 p...
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Article
Error-free transformations of matrix multiplication by using fast routines of matrix multiplication and its applications
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 te...
