Yasuhiko Ikebe

The zeros of regular Coulomb wave functions and of their derivatives

A simple and efficient numerical method for computing the zeros of regular Coulomb wave functions and of their derivatives is presented. The method is based on the characterization of the zeros of the functions and of their derivatives in terms of eigenvalues of certain compact matrix operators. A similar approach has been reported for the computation of the zeros of Bessel functions and of their derivatives 19], [14).

The eigenvalue problem for infinite complex symmetric tridiagonal matrices with application

Abstract We consider an infinite complex symmetric (not necessarily Hermitian) tridiagonal matrix T whose diagonal elements diverge to ∞ in modulus and whose off-diagonal elements are bounded. We regard T as a linear operator mapping a maximal domain in the Hilbert space l2 into l2. Assuming the existence of T−1 we consider the problem of approximating a given simple eigenvalue λ of T by an eigenvalue λn of Tn, and nth order principal submatrix of T. Let x = [x(1),x(2),…]T be an eigenvector corresponding to λ. Assuming xTx ≠ 0 and f n+1 x (n+1) x (n) → 0 as n → ∞, we show that there exists a sequence {λn} of Tn such that λ − λ n = f n + 1 x (n+1) [1 + o(1)] (x T x) → 0 , where fn+1 represents the (n, n + 1) element of T. Application to the following problems is included: (a) solve Jv(z) = 0 for v, given z ≠ 0, and (b) compute the eigenvalues of the Mathieu equation. Fortunately, the existence of T−1 need not be verified for these examples since we may show that T + αI with α taken appropriately has an inverse.

Error analysis for the computation of zeros of regular Coulomb wave function and its first derivative

In 1975 one of the coauthors, Ikebe, showed that the problem of computing the zeros of the regular Coulomb wave functions and their derivatives may be reformulated as the eigenvalue problem for infinite matrices. Approximation by truncation is justified but no error estimates are given there. The class of eigenvalue problems studied there turns out to be subsumed in a more general problem studied by Ikebe et al. in 1993, where an extremely accurate asymptotic error estimate is shown. In this paper, we apply this error formula to the former case to obtain error formulas in a closed, explicit form.

FRACTAL IMAGE COMPRESSION USING LOCALLY REFINED PARTITIONS

In the present report, we show a practical fractal image compression method using locally refined partition which is generated automatically and controlled by the values of gradients in images, The method is similar to the ones by Barnsley and Hurd. In our method, using the quad-tree scheme, before the compression processes of the image, we locally refine the domain regions recursively until the size of the regions become smaller than the scale lengths of the gradients in the image or until the predefined minimum refinement size is reached. Based on our method, it is possible to assess the optimum-like set of domain regions for the desired file size.

“Interactive Multimedia Education at a Distance-Linear Algebra (IMED-LA)”: Its Present Status and Special Features of Its Content

The IMED-LA or Interactive Multimedia Education at a Distance — Linear Algebra is the ongoing international joint project by the collaboration of the Japan Team currently consisting of five members and the Center for Digital Innovation (CDI) of the University of California at Los Angeles (UCLA) headed by Director Maha Ashour-Abdalla. The project started in July, 1999. Its goal is the production of virtual university content in linear algebra, intended primarily for graduate students and working students who need a fast-paced study of the basic facts from linear algebra; hence the content can also effectively be used by the beginning college students under a proper guidance from the teacher. Our partially completed content is already available on the Internet at the following URL: http://www.cdi.ucla.edu/linearalgebra.