Nobuyoshi Asai

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.

The eigenvalue problem for infinite compact complex symmetric matrices with application to the numerical computation of complex zeros of J0(z) − iJ1(z) and of Bessel functions Jm(z) of any real order m

Abstract Consider computing simple eigenvalues of a given compact infinite matrix re- garded as operating in the complex Hilbert space l 2 by computing the eigenvalues of the truncated finite matrices and taking an obvious limit. In this paper we deal with a special case where the given matrix is compact, complex, and symmetric (but not necessarily Hermitian). Two examples of application are studied. The first is con- cerned with the equation J 0 ( z ) − iJ 1 ( z )=0 appearing in the analysis of the solitary-wave runup on a sloping beach, and the second with the zeros of the Bessel function J m ( z ) of any real order m . In each case, the problem is reformulated as an eigenvalue problem for a compact complex symmetric tridiagonal matrix operator in l 2 whose eigenvalues are all simple. A complete error analysis for the numerical solution by truncation is given, based on the general theorems proved in this paper, where the usefulness of the seldom used generalized Rayleigh quotient is demonstrated.

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.

On computing eigenvalues of Lamé equation and zeros of Whittaker function by matrix method

Consider computing zeros z of Whittaker function Mκ,µ (z) for given κ and µ, and eigenvalues of Lame equation for given ν and 0 < k′ < 1, by reformulating them as eigenvalue problems of infinite symmetric tridiagonal matrices.Miyazaki et al. have demonstrated a matrix method to compute zeros of Whittaker function of the first kind Mκ,µ (z), using the three-term recurrence relations, as a difference equation regarding µ, introduced by Boersma.This method, in which the coefficient matrix is considered as a compact operator from Hilbert space l2 to itself, also gives an asymptotically accurate error analysis.Erderyi et al. and Jansen have investigated matrix methods for computing eigenvalues of Lame equation. Eigenvalue problems of infinite symmetric tridiagonal matrices are considered where the matrix itself is NOT compact and its inverse is compact in Hilbert space l2, with their diagonal and super- and sub-diagonal elements diverging.In this paper, we attempted to rewrite the problem of computing zeros of...

Visual Zen art: aesthetic cognitive dissonance in Japanese dry stone garden measured in visual PageRank

One of the most famous Japanese landscape of dry stone Zen gardens in Ryoan temple established AC 1450 at Kyoto by the deputy Shogun Katsumoto Hosokawa, a UNESCO World Heritage Site, has the small 248 m2 empty rectangle where white sand was laid in between, in the abstract, placements of 15 rocks with mosses that seem to be scattered in seemingly haphazard. The landscape attracts many visitors, including Queen Elizabeth II, 1975, to hundreds of thousands for long years. The stone placements designed by the ancient anonymous landscape designer look random at a glance, however, the placement structure is hidden in white sand of the garden space in sophisticated manner, and that we are in perfect harmony with the temple buildings and landscapes. The stone structure looks puzzling at glance. However, the general stone placement structure can immediately be revealed by understanding the ancient garden design rule texts described in, for example, “Sakuteiki” (“Landscape Design” in Japanese, published around AC 1200). According to this instruction text, the stones are placed recursively and in fractal as an obtuse inequilateral triangle in different three level scales. The three stones are placed as vertices entitled “真 (very formal)” at the obtuse angle, “副(formal)” at the vertex closest to “very formal”, and “対(Casual)” at the remaining vertex. These three stones form a cluster and three clusters form a different obtuse inequilateral triangle, recursively, as shown in Image (lower). If we plot the size and the rank of the three level obtuse triangles in log scale, we obtain the perfect Zipf’s law[Gabaix 1999] or 1/f pink noise[Voss 1985]. This means that the stones are placed in a perfect fractal manner, and this fractal stone placement is completely understood by the garden viewers unconsciously in our eyetracking measurements shown in Image (upper). Eye tracking experiments are performed and the visual “PageRank”[Page et al. 1998] of eye movement are measured. We measured 10 testers’ eye movement while they were watching the landscape garden. All testers spend more than 90% of their time to watch the stone objects, and move their eyes from a stone vertex to another, following the triangles, recursively. Image (upper) shows a typical eye movement trajectory in 20 seconds, and the trajectory follows the three level of fractal triangle structure displayed in Image (lower). Five human figures 1 to 5 in different colors represent different viewing positions corresponding to different eye movement trajectories in their colors. We assume that the eye movement from one stone or one cluster to another is a forward “link”. Using the same technique as “PageRank”, the eye movement from one stone to another is taken as a directed graph. First, we generate an adjacency matrix of this graph structure with weighted sum of the number of eye visits. Next, we obtain the largest absolute eigenvalue of this matrix and its eigenvector. Third, the normalized component values of the eigenvector are the visual “PageRanks”, and now we rank the stones or the cluster in order. The visual “PageRank” in the Japanese dry stone Zen garden reveals the amazing hidden structure in this old landscape garden, which cannot be discovered by a simple hot spot diagram that only represents the length of fixation time. Figure 1 indicates that the “PageRanks” in two “真 (very formal)” stones (number 2-2 and 3-2) are significantly higher than the others, where the general fractal placement manner is somehow violated. At these stones circled in red in Image, the eye trajectories are strongly disturbed and some psychological tension caused by a disparity between what one expects to see and what one actually sees is observed. This is a well-known phenomenon in social psychology today called “cognitive dissonance” or “visual dissonance” although totally not known in 560 years ago when the temple was build. The cognitive dissonance happens when we perceive a discrepancy between our attitudes and our behaviors. Our eye sees the world of art with a thousand of expectations based on our personalities and our cognitive structure or knowledge system. Sometimes those expectations are fulfilled, sometimes not. In the case of unfulfilled expectations, the viewer is required to resolve his or her tension, or simply abandon the piece and consider another. An important part of human motivation is found in “dissonance reduction”, in that people do not normally choose to live in a state of psychological tension. In psychological terms, it is better to be avoided or resolved such an aversive state. The technique to produce unexpected visual forms is widely practiced by modern artists who seek to gain our attentions. However, the most striking feature here is that this cognitive dissonance was implemented in a very naive way and viewer observe this only unconsciously, not like many visual dissonance in modern arts. When people view this garden people feels deep serenity and perfect harmony with very small “disharmony” i. e. “dissonance” like in much beautiful music.

Computation of multiple eigenvalues of infinite tridiagonal matrices

In this paper, it is first given as a necessary and sufficient condition that infinite matrices of a certain type have double eigenvalues. The computation of such double eigenvalues is enabled by the 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 v (z); (B) the zeros of zJ f v(z) + HJ v (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.

“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.