◆ Annual Review 2001

Mathematical Foundation of Computer Science Laboratory

Yasuhiko Ikebe

Nobuyoshi Asai
Assistant Professor
The MFSClab operates in collaboration with Assistant Professor Y. Kikuchi from Multimedia Systems Lab.
Jointly we supervised nearly 25 students, of which several were graduate students. Our educational contributions were made at a variety of fronts.
Our major research fronts may be described as follows:
1. Matrix-theoretic high-performance algorithm construction, with application to the numerical computation of critically important quantities associated with diAEcult special functions occurring from 3-D wave equations, such as regular Coulomb wave functions, Mathieu functions, spheroidal wave functions, Lame functions, ellipsoidal wave functions. The Bessel function of the first kind, an easier function serves as a model case, where two types of computational problems were solved in the complex domain as well as real domain: given a complex order Bessel function, compute a given number of zeros closest to the origin to a specified accuracy; and given a complex number find a specified number of orders of Bessel functions closest to the origin, to a specified accuracy. There are no comparable algorithms known elsewhere which give as accurate an error estimate as ours.
2. Virtual university content production: We have been collaborating with an foreign IT team (Center for Digital Innovation, the University of California at Los Angeles (UCLA)) to produce an interactive multimedia distance education content in linear algebra. Its first English version is available for public release at www.cdi.ucla.edu/ Linear algebra. The content is under rewrite at this writing. It has an outstanding feature in that it is organized round the Six Great Decompositions, so that a busystudent can learn the skeleton facts of the subject matter at the earliest possible timing, in fact, right after the student learned basic matrix operations. This completely new rearrangement of the facts is a result of Ikebe's research and represents a especially welcome fact to the busy working professionals who need to learn the basics of linear algebra FAST and at an industrial-strength level.
3. Construction of Hyper English word study dictionary: it is available at http://mnftv1.u-aizu.ac.jp/hidic/hdhome.htm. Again this represents a long-range project, aiming at compiling a database for the study of English words for Japanese students. The main feature is that the dictionary is organized to bring out the best fruits of comparative linguistics in the past century: the usefulness of Indo-European roots as presented in the wonderful book by Calvert Watkins, The American Heritage Dictionary of Indo-European Roots, Second Edition, Houghton Mi??in, 2000. Example: The organizing principle of the dictionary will tell you that the following words are best studied as a group: riddle, garble, crime, decree, discern, secret, hypocrisy (they are all derivatives from the same Indo-European root krei-, meaning to sieve, indicating that a group of seemingly unrelated words come from the same notion to sieve). Anotherexample: the usage of the so-called synonyms anger, rageandwrath canbe clearly explainedby tracingthemback to theirrespectiveIndo-Europeanroot: angerto angh-, meaningpainful, rage to rebh-, meaning violent, impetuous, and wrath to wreit-, meaning to turn, twist (anger is akin to angst, angina, rage is akin to rabid, rabies, and wrath is akin to wreath, writhe).
In summary, our Lab is as active as can be, to the pleasure of the involved researchers and of the students working under them.
Referred Journal Papers
[ikebe-001:2001]Y. Miyazaki, Y. Kikuchi, D.S. Cai, and Y. Ikebe. Error Analysis for the Computation of Zeros of Regular CoulombWave Function and Its First Derivative. Math. Comp., (70):1195{1204, 2001.
In 1975, one of the coauthors, Ikebe, showed that the problem of computing the zeros of the regular Coulomb wave functions and of their derivatives may be reformulatedasthe eigenvalueproblemfor in??nite matrices.Approximation by truncation is justi??ed but no error estimates are given there. The class of eigenvalue problems studied there turns out to be subsumed in a more general problemstudiedbythesameauthoret. al in 1993,whereanextremelyaccurate 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.
[nasai-001:2001] Y.Miyazaki, N. Asai, Y.Kikuchi, D.S. Cai,andY.Ikebe. Computation of Multiple Eigenvalues of In??nite Tri-Diagonal Matrices. Math. Comp., Accepted, 2001.
In this paper, it is ??rst given a necessary and suAEcient condition that in??- nite matrices of a certain type have double eigenvalues. The computation of such double eigenvalues is enabled by Newton method of two variables. The three-term recurrence relations obtained from its eigenvalue problem (EVP) subsume the well-known relations of (A) the zeros of J (z); (B) the zeros ?? 0 of zJ (z)+HJ (z); (C) the EVP of the Mathieu differential equation; and ?? ??(D) the EVP of the spheroidal wave equation. The results of experiments are shown for the three cases (A)-(C) for the computation of their `double pairs'.
Unrefereed Papers
[nasai-002:2001]S. Oode, I. Li, M.Abe, Y. Miyazaki, N. Asai, Y. Kikuchi, D. Cai, and Y. Ikebe. Pilot Experiment of Multimedia Distance Lecture. In Proceeding of 30th Numerical Analysis Symposium, pages 7-10, 2001.
[nasai-002:2001]Y. Ikebe, D. Cai, Y. Miyazaki, Y. Kikuchi, and N. Asai. Struc- turing Method of Content of Interactive Multimedia Education at a Dis- tance, IMED-LA. In Proceeding of 30th Numerical Analysis Symposium, pages 108-111, 2001.
[nasai-004:2001]Y. Ikebe, Y. Kikuchi, N. Asai, D. Cai, and Y. Miyazaki. Eigenvalue Problems of Spheroidal Wave Function, Lame Equation, and Ellipsoidal Wave Function, 2001-2002.