We will discuss iterative methods for the solution of linear large sparse eigenproblems, such as the methods of Lanczos and Arnoldi. In particular, we will pay attention to the recently proposed class of Jacobi-Davidson methods, which represents versatile approaches for the solution of large sparse eigenproblems. In these methods correction equations have to be solved for a proper update of the approximations for an eigenpair. For the solution of these correction equations we may employ the arsenal of techniques available for iterative solution of linear sparse systems, including preconditioning. An overview of variants of the Jacobi-Davidson method for standard linear eigenproblems and generalized eigenproblems will be given. We will also describe how the approach can be used for the efficient computation of a number of eigenpairs.

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This paper surveys recent developments on numerical algorithms for high dimensional multiple integration. First, we present Wozniakowski's theorem published in 1991, which revealed a remarkable connection between the integration error and the discrepancy via the classical Wiener measure. Then, we introduce low-discrepancy sequences, by means of which one can compute the arithmetic mean of a number of sample values of the integrand as an approximation to the integration. As a concrete construction method of low-discrepancy sequences, we give the definition of generalized Niederreiter sequences and a brief introduction of Niederreiter-Xing sequences, which are constructed by using algebraic function fields. Finally, we describe Smolyak's algorithm, which is an algorithm computing the weighted mean of sample values of the integrand. Sample points that this algorithm uses are called hyperbolic cross points. An interesting result by Wasilkowski and Wozniakowski on this algorithm is presented.

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I overview the Fast Multipole Method (FMM) and the Barnes-Hut tree method. These algorithms evaluate mutual gravitational interaction between N particles in O(N) or O(N log N) times, respectively. I present basic algorithms as well as recent developments, such as Anderson's method of using Poisson's formula, the use of FFT, and other optimization techniques. I also summarize the current states of two algorithms. Though FMM with O(N) scaling is theoretically preferred over O(N log N) tree method, comparisons of existing implementations proved otherwise. This result is not surprizing, since the calculation cost of FMM scales as O(Np^2) where p is the order of expansion, while that of the tree method scales as O(N log Np).

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An approach is surveyed to show that verification of numerical computation can be done by a cost around 2〜4 times than that of usual computation. In this approach the floating point numbers arithmetic with rounding toward ±∞ is utilized. By considering inner products of vectors as units of computation, it is shown that drastic reduction of numbers of changing rounding direction in achieved. Moreover, by this approach programming of verification becomes extremely simple. Taking examples such as the matrix equation, polynomial evaluation, nonlinear equation, it is shown that verified solutions for these problems can be obtained by a cost about 2〜4 times than that of computation of approximate solutions.

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In past 10 years, a number of new Krylov subspace methods have been proposed, all of which are related to the Arnoldi process or the nonsymmetric Lanczos process, and can be implemented with low work and low storage per iteration. The primary aim of this paper is to highlight some of the most important developments of Krylov subspace methods which are almost mandatory for three-dimensional modeling. Here, Several practical Krylov subspace methods will be illustrated by using Ritz-Galerkin approach, Minimal Residual approach, Petrov-Galerkin approach respectively.

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