In this paper, a fast and accurate algorithm for solving ill-conditioned linear systems is proposed. The proposed algorithm is based on a preconditioned technique using a result of an LU factorization, which requires less computational cost than a previous method using an approximate inverse. The algorithm can provide accurate numerical solutions for ill-conditioned problems beyond the limit of the working precision. Results of numerical experiments are presented for confirming the effectiveness of the proposed algorithm.

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The alternating least squares (ALS) method is frequently used for the computation of the canonical polyadic decomposition (CPD) of tensors. It generally gives accurate solutions, but demands much time. An alternative is the alternating slice-wise diagonalization (ASD) method, which provides an efficient way for third-order tensors, utilizing compression based on matrix singular value decomposition. In this paper, we propose a new simple algorithm, Reduced ALS, which employs the same compression procedure as ASD, but applies it more directly to ALS. Numerical experiments show that Reduced ALS runs as fast as ASD, avoiding instability ASD sometimes exhibits.

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We introduce an extended oqds algorithm for singular values of lower tridiagonal matrix which is a condensed form of inputted full matrix. Reduction to the lower tridiagonal matrix is able to be performed using cache-efficient block Householder method based on BLAS 2.5 routines. In this letter, we describe the implementation details of the latter algorithm such as the shift strategy and criteria for deflation and splitting. The effectiveness of our approach is demonstrated by numerical experiments.

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The joint singular value decomposition of multiple rectangular matrices is formulated as a Riemannian optimization problem on the product of two Stiefel manifolds. In this paper, the geometry of the objective function and the Riemannian manifold for this problem are studied to develop a Riemannian trust-region algorithm. The proposed algorithm globally and locally quadratically converges, and our numerical experiments demonstrate that it performs much better than the steepest descent method.

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We consider application of the discrete gradient method for the Webster equation, which models sound waves in tubes. Typically Hamilton equations are described by the use of gradients of the Hamiltonian and it is indispensable to introduce an inner product to define a gradient. We first apply the discrete gradient method to design an energy-preserving method by using a weighted inner product. Comparing with another scheme that is derived by a standard inner product, we show that the discrete gradient method has a geometric invariance, which implies that the method reflects the symplectic geometric aspect of mechanics.

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A family of curves with monotone curvature called the log-aesthetic curves (LAC) has been investigated in the field of industrial shape design. LAC has a radius of curvature in proportional to the power of a linear function of an arc-length parameter. In the present article we show that LAC can be naturally formulated in the similarity geometry and the Riccati equations satisfied by the similarity curvatures of LAC can be derived. Moreover, we clarify that certain generalizations of LAC (GLAC) can also be described in a uniform way.

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We introduce a model to evaluate credit value adjustment (CVA) for large derivative portfolios considering general wrong-way risk. First, we showed the empirical evidence that suggests the existence of wrong-way risks for interest rate swaps and foreign exchange forwards. Next, we formulate a model to calculate CVA considering the correlations between the probability of default of the counterparty, the interest rate curve, and the foreign exchange rate. Finally, we show numerical examples to estimate the effect of wrong-way risk, which can be related to the CVA stress testing.

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In this paper, we give detailed proofs of the transformation of the sum of the jump components that appears in Ito's formula for jump-diffusion processes into the stochastic integral with respect to a certain counting process. As applications of the transformed Ito's formula, the Black-Scholes equations in the compound Poisson process model and the jump-diffusion process model are discussed.

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The volume weighted average price (VWAP) execution strategy is well known and widely used in practice. In this study, we explicitly introduce a trading volume process into the Almgren-Chriss model, which is a standard model for optimal execution. We then show that the VWAP strategy is the optimal execution strategy for a risk-neutral trader. Moreover, we examine the case of a risk-averse trader and derive the first-order asymptotic expansion of the optimal strategy for a mean-variance optimization problem.

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In this article, we deal with the iterative methods for solving unsymmetric linear systems, especially BiCGSTAB. The introduced parameter in BiCGSTAB at each iteration is selected to minimize the 2-norm of the residual vector. Here, we suggest another way to select the parameter by the idea of weighting used in Weighted GMRES. By our procedure, more importance is assigned to the larger entry of the residual vector so that faster convergence can be expected. In numerical experiments, it is shown that our procedure is efficient compared with the original BiCGSTAB.

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The Dickson cryptosystem is a modification of the RSA and LUC based on the Dickson polynomial. In this paper, we consider Wiener's attack and Boneh-Durfee's algorithm on RSA to the Dickson cryptosystem. We then efficiently apply them when the secret exponent $d$ is sufficiently small compared to public modulus $n$. We show that if $d<(1/3\sqrt{2})n^{0.5}$, then Wiener's attack works. Furthermore, the bound on Boneh-Durfee's algorithm is extended up to $d<n^{0.585}$.

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We consider the discrete gradient method for dissipative linear-gradient systems, which strictly replicates the dissipation property, yielding a remarkable stability. However, it also replicates the nonlinearity of an original equation. To overcome this, we can employ multistep linearly implicit schemes as a relaxation; however, it can in turn destroy the originally aimed stability. Matsuo-Furihata (2014) introduced a dynamical systems viewpoint to understand the behavior for a toy scalar problem. In this letter, we show that their method can work also for the two-dimensional Duffing equation. There a new concept of semi-strong Lyapunov functionals is required.

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Iwata-Takamatsu (2013) showed that the maximum degree of minors in mixed polynomial matrices for a specified order can be computed by combinatorial relaxation type algorithm. In this letter, based on their algorithm, we propose an efficient combinatorial relaxation algorithm for computing the entire sequence of the maximum degree of minors. In our previous work, we dealt with a similar problem for rational function matrices, where the efficiency derived from the discrete concavity of valuated bimatroids. We follow the same line of discussion; but, technical details are different due to special characteristics of mixed matrices.

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To estimate the number of eigenvalues of a Hermitian matrix that are located in a given interval, existing methods include polynomial filtering and rational filtering. Both filtering approaches are based on stochastic approximations for matrix trace. In this paper, we analyze a rational filtering method that is based on polynomial filtering in which the solutions to the linear systems are approximated by a Krylov subspace method. Our analysis and numerical experiments indicate that the rational filtering method is effective when the eigenvalues of a given matrix are sparsely distributed in the target interval.

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In dislocation dynamics, dislocations can be regarded as open plane curves evolving according to the curvature flow with an external force. In the present paper, evolving curves connecting two circular obstacles are treated from both mathematical and numerical viewpoints: An exact solution curve is constructed with sliding endpoints along obstacles, and all important and typical phenomena including touching-splitting, non-touching and Orowan island can be treated numerically.

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We show conditional stability of the Lagrange-Galerkin scheme with numerical quadrature for the convection-diffusion equation. We consider the scheme under the assumption that quadrature points are inside the element and that the time increments are sufficiently small. Our analysis covers general triangular or tetrahedral meshes and arbitrary smooth velocities. We present some numerical examples that reflect the theoretical result.

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Recently, interior point methods are focused as an efficient strategy for large scale contact problems. In this paper, we present a method based on a predictor-corrector method for contact problems with multi-point constraints. Furthermore, we implement our algorithm into FrontISTR, which is an open-source and large scale finite element structural analysis software, and investigate the performance.

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De Klerk-Pasechnik (2002) showed a way to compute the stability number $\alpha(G)$ via copositive programming and proposed LP- and SDP-based approximation schemes for the copositive program. In this paper, we show that their LP-based approximation for the stable set problem is equivalent to a problem of minimizing a quadratic form over a rational grid on the simplex, which can be viewed as a discretized version of the Motzkin-Straus theorem. Furthermore, we provide an algorithm to recover a maximum stable set from an optimal solution of the LP-based approximation and propose a simple local search heuristics for the stable set problem.

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In this paper, we propose a numerical method for verifying the positiveness of solutions to semilinear elliptic boundary value problems. We provide a sufficient condition for a solution to an elliptic problem to be positive in the domain of the problem, which can be checked numerically without requiring a complicated computation. Although we focus on the homogeneous Dirichlet case in this paper (in fact, it is often possible that solutions are not positive near the boundary in this case), our method can be applied naturally to other boundary conditions. We present some numerical examples.

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We discuss the differences of local risk minimization (LRM) and delta hedging strategies, in exponential Lévy models, where delta hedging strategies in this paper are defined under the minimal martingale measures (MMM). First of all we give inequality estimations for the differences of LRM and delta hedging strategies, and then show numerical examples for the two typical exponential Lévy models, Merton models and variance Gamma (VG) models.

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