Akiko Fukuda

The discrete hungry Lotka?Volterra system and a new algorithm for computing matrix eigenvalues

The discrete hungry Lotka?Volterra (dhLV) system is a generalization of the discrete Lotka?Volterra (dLV) system which stands for a prey?predator model in mathematical biology. In this paper, we show that (1) some invariants exist which are expressed by dhLV variables and are independent from the discrete time and (2) a dhLV variable converges to some positive constant or zero as the discrete time becomes sufficiently large. Some characteristic polynomial is then factorized with the help of the dhLV system. The asymptotic behaviour of the dhLV system enables us to design an algorithm for computing complex eigenvalues of a certain band matrix.

Development of test fixture for measurement of dielectric properties and its verification using animal tissues

The electromagnetic compatibility of implantable or wearable medical devices has often been evaluated using human phantoms to electrically mimic biological tissues. However, no currently existing test fixture can measure the electrical characteristics of gel-like materials. In this paper, we report the development of a new test fixture that consists of a coaxial tube whose outer conductor is divided along the axial direction into two sections, which facilitates filling and removal of gel-like materials in order to measure their electrical characteristics. Using this test fixture, we measured the electrical characteristics of a cows muscular tissues up to 1 h post-mortem; these measurements allowed us to obtain the relative permittivity and conductivity of the biological tissue, which should help to enable the design of new human phantoms.

Error analysis for matrix eigenvalue algorithm based on the discrete hungry Toda equation

Based on the integrable discrete hungry Toda (dhToda) equation, the authors designed an algorithm for computing eigenvalues of a class of totally nonnegative matrices (Ann Mat Pura Appl, doi:10.1007/s10231-011-0231-0). This is named the dhToda algorithm, and can be regarded as an extension of the well-known qd algorithm. The shifted dhToda algorithm has been also designed by introducing the origin shift in order to accelerate the convergence. In this paper, we first propose the differential form of the shifted dhToda algorithm, by referring to that of the qds (dqds) algorithm. The number of subtractions is then reduced and the effect of cancellation in floating point arithmetic is minimized. Next, from the viewpoint of mixed error analysis, we investigate numerical stability of the proposed algorithm in floating point arithmetic. Based on this result, we give a relative perturbation bound for eigenvalues computed by the new algorithm. Thus it is verified that the eigenvalues computed by the proposed algorithm have high relative accuracy. Numerical examples agree with our error analysis for the algorithm.

On a Variable Transformation Between Two Integrable Systems: The Discrete Hungry Toda Equation and the Discrete Hungry Lotka‐Volterra System

In the study of dynamical systems, it is often useful to find a variable transformation that maps the solution of one system to another system. Such transformation, known as the Backlund transformation, enables us to translate the knowledge on one system directly to the knowledge on the other system. In this paper, we present a Backlund transformation between two discrete integrable systems, namely, the discrete hungry Toda equation with origin shift and the discrete hungry Lotka‐Volterra system with finite step size.

An asymptotic analysis for an integrable variant of the Lotka–Volterra prey–predator model via a determinant expansion technique

The Hankel determinant appears in representations of solutions to several integrable systems. An asymptotic expansion of the Hankel determinant thus plays a key role in the investigation of asymptotic analysis of such integrable systems. This paper presents an asymptotic expansion formula of a certain Casorati determinant as an extension of the Hankel case. This Casorati determinant is then shown to be associated with the solution to the discrete hungry Lotka–Volterra (dhLV) system, which is an integrable variant of the famous prey–predator model in mathematical biology. Finally, the asymptotic behavior of the dhLV system is clarified using the expansion formula for the Casorati determinant.

Eigenvalue computation of totally nonnegative upper Hessenberg matrices based on a variant of the discrete hungry Toda equation

The integrable discrete hungry Toda (dhToda) equation has already been applied to design an eigenvalue algorithm which computes eigenvalues of totally nonnegative (TN) matrices with lower Hessenberg form. In this paper, an eigenvalue algorithm for computing eigenvalues of TN matrices with upper Hessenberg form is derived based on a variant of the dhToda equation.

Asymptotic analysis for an extended discrete Lotka-Volterra system related to matrix eigenvalues

The integrable discrete hungry Lotka–Volterra (dhLV) system is easily transformed to the qd-type dhLV system, which resembles the recursion formula of the qd algorithm for computing matrix eigenvalues. Some of the qd-type dhLV variables play a role for assisting the time evolution of the others. This property does not appear in the original dhLV system. In this article, we first show the existence of a centre manifold for the qd-type dhLV system. With the help of the centre manifold theory, we next investigate the local convergence of the qd-type dhLV system, and then clarify the monotonicity related to the qd-type dhLV variables at the final phase of the convergence.

Monotonic convergence to eigenvalues of totally nonnegative matrices in an integrable variant of the discrete lotka-volterra system

The Lotka-Volterra (LV) system describes a simple predator-prey model in mathematical biology. The hungry Lotka-Volterra (hLV) system assumed that each predator preys on two or more species is an extension; those involving summations and products of nonlinear terms are referred to as summation-type and product-type hLV systems, respectively. Time-discretizations of these systems are considered in the study of integrable systems, and have been shown to be applicable to computing eigenvalues of totally nonnegative (TN) matrices. Monotonic convergence to eigenvalues of TN matrices, however, has not yet been observed in the time-discretization of the product-type hLV system. In this paper, we show the existence of a center manifold associated with the time-discretization of the product-type hLV system, and then clarify how the solutions approach an equilibrium corresponding to the eigenvalues of TN matrices in the final phase of convergence.

Computation of Eigenvectors for a Specially Structured Banded Matrix

For a specially structured nonsymmetric banded matrix, which is related to a discrete integrable system, we propose a novel method to compute all the eigenvectors. We show that the eigenvector entries are arranged radiating out from the origin on the complex plane. This property enables us to efficiently compute all the eigenvectors. Although the intended matrix has complex eigenvalues, the proposed method can compute all the complex eigenvectors using only arithmetic of real numbers.