The original form of the Sinc approximation is efficient for functions whose boundary values are zero, but not for other functions. The typical way to treat general boundary values is to introduce auxiliary basis functions, and in fact such an approach has been taken commonly in the literature. However, the approximation formula in each research is not exactly the same, and still other formulas can be derived as variants of existing formulas. The purpose of this paper is to sum up those existing formulas and new ones, and to give explicit proofs of those convergence theorems.

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Non-differentiability of the absolute value function at the origin can affect the accuracy of numerical computations for the $L^1$ norm. We present an example in which the accuracy does deteriorate, and we provide a convergence order for such situations. We propose a simple algorithm to improve the convergence order, confirming its effectiveness as in the example described above. Mesh-dependent integrands and applications to finite element method are also considered.

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MuPAT enables the users to easily treat quadruple and octuple precision arithmetics as well as double precision arithmetic on Scilab. Using external C routines, we have also developed a high speed implementation MuPAT_c for Windows, Mac OS, and Linux. MuPAT_c reduced the computation time especially for all octuple precision arithmetic and inner product of quadruple precision arithmetic. MuPAT_c can run 90-1200 times faster than MuPAT. We applied three different precisions to tridiagonalization by the Lanczos method and confirmed that a high precision arithmetic was essential for the Lanczos method to get accurate eigenvalues of real symmetric matrices.

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In this paper, we consider Euler-Maruyama approximations for 1-dimensional stochastic differential equations (SDEs) driven by rotation invariant (i.e. symmetric) $\alpha$ stable processes and discuss their rate of strong convergence by numerical simulations. We also study the relationship between the convergence rate and the index $\alpha$ of rotation invariant stable process and/or the exponent $\gamma$ of the Hölder continuity of the diffusion coefficient.

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This paper derives a new semi closed-form approximation formula for pricing an up-and-out barrier option under a certain type of stochastic volatility model including SABR model by applying a rigorous asymptotic expansion method developed by Kato, Takahashi and Yamada (2012). We also demonstrate the validity of our approximation method through numerical examples.

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Choice of suitable shifts strongly influences performance of numerical algorithms with shift for computing matrix eigenvalues or singular values. On the dqds (differential quotient difference with shifts) algorithm for singular values, a new shift strategy is proposed in this paper. The new shift strategy includes shifts obtained from an application of the Kato-Temple inequality on matrix eigenvalues. The dqds algorithm with the new shift strategy is shown to have a better performance in iteration number than that of the subroutine DLASQ in LAPACK.

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This paper is concerned with factorization of symmetric and positive definite matrices which are extremely ill-conditioned. Recently, Ogita and Oishi 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 explain its 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, which is consistent with the existing numerical results.

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We present an error estimation for the H1 gradient method, which provides numerical solutions to the shape-optimization problem of the domain in which a boundary value problem is defined. The main result is that if second-order elements are used for the solutions of the main and adjoint boundary value problems to evaluate the shape derivative, and the first-order elements are used for the solution of domain variation in the boundary value problem of the H1 gradient method, then we obtain first-order convergence of the solution of the domain variation with respect to the size of the finite elements.

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We obtained the best constants of Sobolev inequalities corresponding to complete low-cut filter. In the background, we have an $n$-dimensional boundary value problem and a one-dimensional periodic boundary value problem. The best constants of the corresponding Sobolev inequalities are equal to diagonal values of Green's functions for these boundary value problems.

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In this report we propose a simple new geometric integration approach for solving ordinary differential equations based on the concept of ``local'' invariants. The approach basically belongs to the class of invariants-preserving integrations, but it differs from any existing methods in that it can automatically detect enough number of invariants, and work even for non-conservative systems. Numerical examples show that the approach can in fact work.

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In this paper, we indicate that the Sakurai-Sugiura method with Rayleigh-Ritz projection technique, a numerical method for generalized eigenvalue problems, can be extended to nonlinear eigenvalue problems. The target equation is $T(\lambda)\bm{v}=0$, where $T$ is a matrix-valued function. The method can extract only the eigenvalues within a Jordan curve $\Gamma$ by converting the original problem to a problem with a smaller dimension. Theoretical validation of the method is discussed, and we describe its application using numerical examples.

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We improve our former implementation of key generation for a cryptosystem OTU2000 with resistance to quantum computers. First, we give a polynomial time algorithm to determine a prime number satisfying a secret key condition. Next, on another secret key condition to guarantee the uniqueness of decoding, we prove a weaker sufficient condition than that in original OTU2000 and give an algorithm by this new condition. These allow us to choose secret keys from more combinations than before. Experimental results including our improvements are also shown. This will lead us to use OTU2000 without quantum computers.

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In this letter, we propose a new method of multiple kernel learning (MKL) that utilizes an adaptively weighted regularization. The proposed method controls strength of penalty for each kernel depending on its importance so that important components are amplified and unimportant components are diminished. To show the effectiveness of the proposed method, a theoretical justification is provided based on the recently developed unifying framework for the learning rate of MKL. Numerical experiments are carried out to support the usefulness of the proposed method.

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We treat a continuous model of Thomson cable. The supremum of the absolute value of output voltage is estimated by the constant multiple of $L^2$ norm of input voltage. We obtain the best constants of the above estimations, which are expressed as rational functions of resistance, capacitance and conductance constants. In the background, we have an initial boundary value problem of heat equation.

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We consider computation of cumulative distribution functions from their corresponding characteristic functions. We may use some known formulas with singular integrals for the computation. It is, however, difficult to speed up the computation with such formulas, because the fast Fourier transform (FFT) cannot be applied directly to them. Based on existing works for pricing of financial derivatives, we propose a fast method for the computation with fractional FFT and obtain accurate results on the entire real line.

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The present paper describes a method by which to formulate a shape optimization problem of a linear elastic continuum for minimizing the maximum value of a strength measure, such as the von Mises stress. In order to avoid the irregularity of the shape derivative of the maximum value, the Kreisselmeier-Steinhauser function of the strength measure is used as the cost function. In the cost function, a parameter is used to control the regularity of the shape derivative. In the present paper, we propose a rule by which to appropriately determine the parameter. The effectiveness of the proposed rule is confirmed through a numerical example of a cantilever problem.

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We investigate the Block GMRES method for solving large and sparse linear systems with multiple right-hand sides. For solving linear systems with a single right-hand side, the Weighted GMRES method based on the weighted minimal residual condition has been proposed as an improvement of the GMRES method. In this paper, by applying the idea of the Weighted GMRES method to the Block GMRES method, we propose a Weighted Block GMRES method. The numerical experiments indicate that the Weighted Block GMRES($m$) method has higher performance for efficient convergence than the Block GMRES($m$) method.

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