Abstract. The notion of efficient portfolios is re-stated in terms of data-measurable investment rates as well as a modified definition of the risk, in which the risk is evaluated by the unconditional mean of the squared deviation of the total return from its conditional mean. An optimization problem is mathematically solved to find the efficient portfolio that attains maximum expected return with the risk constrained to an arbitrarily given level. In addition, there is found a result similar to Tobin's one-fund theorem.

Abstract. We improve numerical verification results of semilinear elliptic problems through the verification of a certain linear problem. As a feature of the method described in this paper, the norm of the correction term of Newton’s method, which was conventionally evaluated by the product of an inverse-operator norm and the residual norm, is more precisely evaluated through numerical verification for the linear problem derived from an original problem. This not only improves error estimations of many approximate solutions but also enables us to verify low precision approximate solutions whose error bounds could not be evaluated.

Abstract. It would be very useful to have a flat-foldable box with panels as thick and rigid as bulky unbending cardboard, but there are no such industrial products just yet. Rigid folding, which is tackled in the Cauchy rigidity theorem and the bellow theorem, is a very interesting subject in the mathematical sciences. We work on boxes of thick and rigid materials with a flat-foldable structure. This article aims to solve this problem and check the validity of the solution by simulation.

Abstract. We give conditions such that the conjugate gradient (CG) method and the minimal residual (MINRES) method preconditioned by the stationary inner iterations with symmetric splitting matrices respectively determine a solution of a symmetric linear system including the singular case. These results are applied to the inner-iteration preconditioned CG and MINRES-type methods for solving least squares and minimum-norm solution problems in the rank-deficient case to complement the convergence theory of these methods in [Morikuni, Hayami, SIAM J. Matrix Appl. Anal., 34 (2013), 1–22].

Abstract. We propose a method to track the eigendecomposition of a time-dependent real symmetric matrix A(t) using Ogita-Aishima’s eigenvector refinement algorithm. In the tracking, the difference between the eigenpairs of A(t) and those of A(t + Δt) can become large depending on Δt. Also, convergence of the Ogita-Aishima algorithm deteriorates when multiple eigenvalues cross each other. In this paper, we improve Ogita-Aishima’s convergence theorem to enlarge the convergence radius and also propose a preprocessing method that improves the convergence property.

Abstract. We propose a numerical algorithm for the problem of computing the k-th singular triplet of a large sparse matrix. In the proposed algorithm, the problem is divided into subproblems, and then eigenvalue problems equivalent to the singular value problem are solved in each subproblem. We discuss the accuracy of the proposed algorithm and show that it can handle large matrices in numerical experiments.

Abstract. We consider polynomial preconditioners for block Krylov subspace methods with shift-invariance property for a solver to a shifted linear systems with multiple righthand sides and multiple shifts. In general, polynomial preconditioner for shifted Krylov subspace methods is not necessarily effective from the viewpoint of calculation time. In this paper, we show that applying polynomial preconditioner to block Krylov subspace methods with shift-invariance property is effective from the viewpoint of calculation time for solving shifted linear systems with multiple right-hand sides and multiple shifts.