Emiko Ishiwata

• Tokyo University of Science

The discrete Toda (dToda) equation, which is a representative integrable system, is the recursion formula of the well-known quotient-difference algorithm for computing the eigenvalues of tridiagonal matrices. In other words, the dToda equation is related to the LR transformations of tridiagonal matrices. In this chapter, by extending the applicatio...

Global Krylov subspace methods are effective iterative solvers for large linear matrix equations. Several Lanczos-type product methods (LTPMs) for solving standard linear systems of equations have been extended to their global versions. However, the GPBiCGstab(L) method, which unifies two well-known LTPMs (i.e., BiCGstab(L) and GPBiCG methods), has...

The Toda equation is a famous integrable system studied in multiple fields, including mathematical physics and numerical computing. Forty years ago, Symes showed that the time-1 evolution in the Toda equation corresponds to the 1-step of the well-known QR algorithm whose target matrices are tridiagonal exponentials. The discrete Toda (dToda) equati...

The box and ball system (BBS) models the dynamics of balls moving among an array of boxes. The simplest BBS is derived from the ultradiscretization of the discrete Toda equation, which is one of the most famous discrete integrable systems. The discrete Toda equation can be extended to two types of discrete hungry Toda (dhToda) equations, one of whi...

To introduce time delay into integrable discrete systems, we present a time-delay version of the discrete Lotka-Volterra (dLV) system, which is a time-discretization of the famous predator-prey Lotka-Volterra system. Focusing on the LR transformations, which has been designed for solving symmetric eigenvalue problems, plays a key role in deriving t...

MuPAT, an interactive multiple precision arithmetic toolbox for use on MATLAB and Scilab, enables users to handle quadruple- and octuple-precision arithmetic operations. MuPAT uses the DD and QD algorithms, which require from 10 to 600 double-precision floating-point operations for each DD or QD operation, which entails corresponding execution time...

Short-recurrence Krylov subspace methods, such as conjugate gradient squared-type methods, often exhibit large oscillations in the residual norms, leading to a large residual gap and a loss of attainable accuracy for the approximate solutions. Residual smoothing is useful for obtaining smooth convergence for the residual norms, but it has been show...

In this paper, we clarify the periodic behaviour in the discrete hungry Toda (dhToda) equation, which is an extension of the famous integrable discrete Toda equation. The centre manifold theory, which is a classical analysis theory, plays a key role in analysing the dhToda equation. We first observe the asymptotic behaviour of the dhToda variable t...

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 i...

In general, when computing the eigenvalues of symmetric matrices, a matrix is tridiagonalized using some orthogonal transformation. The Householder transformation, which is a tridiagonalization method, is accurate and stable for dense matrices, but is not applicable to sparse matrices because of the required memory space. The Lanczos and Arnoldi me...

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-d...

In this paper, conserved quantities of the discrete hungry Lotka- Volterra (dhLV) system are derived. Our approach is based on the Lax representation of the dhLV system, which expresses the time evolution of the dhLV system as a similarity transformation on a certain square matrix. Thus, coefficients of the characteristic polynomial of this matrix...

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 Han...

Quasi-Minimal Residual (QMR)-type methods have been developed for improving the oscillations in the residual norms of the Bi-Conjugate Gradient (Bi-CG) method and the hybrid Bi-CG methods. The IDRstab method, which combines the Induced Dimension Reduction (IDR)(s) method with higher-order stabilizing polynomials, is often more effective than IDR(s)...

Double-double and Quad-double arithmetics are effective tools to reduce the round-off errors in floating-point arithmetic. However, the dense data structure for high-precision numbers in MuPAT/Scilab requires large amounts of memory and a great deal of the computation time. We implemented sparse data types ddsp and qdsp for double-double and quad-d...

The IDRStab method is often more effective than the IDR(s) method and the BiCGstab(@?) method for solving large nonsymmetric linear systems. IDRStab can have a large so-called residual gap: the convergence of recursively computed residual norms does not coincide with that of explicitly computed residual norms because of the influence of rounding er...

Recently, some of the authors designed an algorithm, named the dhLV algorithm, for computing complex eigenvalues of a certain class of band matrix. The recursion formula of the dhLV algorithm is based on the discrete hungry Lotka–Volterra (dhLV) system, which is an integrable system. One of the authors has proposed an algorithm, named the multiple...

The discrete hungry Toda (dhToda) equation is known as an integrable system which is derived from the study of the numbered box and ball system. In the authors’ paper (Fukuda et al., in Phys Lett A 375, 303–308, 2010), we associate the dhToda equation with a sequence of LR transformations for a totally nonnegative (TN) matrix, and then, in another...

MuPAT enables the users to easily treat quadruple and octuple precision arithmetics as well as double precision arithmetic on Scilab. Using external C routines, we have also developed a high speed implementation MuPAT_c for Windows, Mac OS, and Linux. MuPAT_c reduced the computation time especially for all octuple precision arithmetic and inner pro...

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 algorith...

To verify computation results of double precision arithmetic, a high precision arithmetic environment is needed. However, it is difficult to use high precision arithmetic in ordinary computing environments without any special hardware or libraries. Hence, we designed the quadruple precision arithmetic environment QuPAT on Scilab to satisfy the foll...

The IDRstab method is often more effective than both IDR(s) and BiCGstab(ℓ) for solving large nonsymmetric linear systems. However, the computational costs for vector updates are expensive on the original implementation of IDRstab. In this paper, we propose a variant of IDRstab to reduce the computational cost; vector updates are saved. Numerical e...

We derive a discretized SIR epidemic model with pulse vaccination and time delay from the original continuous model. The sufficient conditions for global attractivity of an infection-free periodic solution and permanence of our model are obtained. Improving discretization, our results are corresponding to those in the original continuous model.

The discrete hungry Toda (dhToda) equation and the discrete hungry Lotka–Volterra (dhLV) system are known as integrable discrete hungry systems. In this Letter, through finding the LR transformations associated with the dhToda equation and the dhLV system, we present a Bäcklund transformation between these integrable systems.

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 sy...

We derive a discretized SIRS epidemic model with time delay by applying a nonstandard finite difference scheme. Sufficient conditions for the global dynamics of the solution are obtained by improvements in discretization and applying proofs for continuous epidemic models. These conditions for our discretized model are the same as for the original c...

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 Bäcklund transformation, enables us to translate the knowledge on one system directly to the knowledge on the other system. In this paper, we present a Bäcklund transformat...

It has been clarified by numerical experiments that a variable preconditioned generalized conjugate residual GCR(m) method using the successive overrelaxation method is efficient for solving a sparse linear system. However, there are cases in which the residual norm of variable preconditioned GCR method stagnates. Then the inner iteration counts in...

The Generalized Conjugate Residual (GCR) method with a variable preconditioning is an efficient method for solving a large sparse linear system Ax=b. It has been clarified by some numerical experiments that the Successive Over Relaxation (SOR) method is more effective than Krylov subspace methods such as GCR and ILU(0) preconditioned GCR for perfor...

When floating point arithmetic is used in numerical computation, cancellation of significant digits, round-off errors and information loss cannot be avoided. In some cases it becomes necessary to use multiple precision arithmetic; however some operations of this arithmetic are difficult to implement within conventional computing environments. In th...

The discrete hungry Lotka-Volterra (dhLV) system is already shown to be applied to the matrix eigenvalue algorithm. In this paper, we discuss a form of the dhLV system named as the qd-type dhLV system and associate it with a matrix eigenvalue computation. Along a way similar to the dqd algorithm, we also design a new algorithm without cancellation...

To compute long term integrations for the pantograph differential equation with proportional delay qt, 0 < q ⩽ 1: y′(t) = ay(t) + by(qt) + f(t), y(0) = y 0, we offer two kinds of numerical methods using special mesh distributions, that is, a rational approximant with ‘quasi-uniform meshes’ (see E. Ishiwata and Y. Muroya [Appl. Math. Comput., 2007,...

In this paper we consider some classes of difference equations, including the well-known Clark model, and study the stability of their solutions. In order to do that we introduce a property, namely semicontractivity, and study relations between ‘semi-contractive’ functions and sufficient conditions for the solution of the difference equation to be...

In this paper, we first review our results in [Inverse Probl. 25, No. 1, Article ID 015007 (2009; Zbl 1161.35510); “On the qd-type discrete hungry Lotka-Volterra system and its application to the matrix eigenvalue algorithm”, JSIAM Lett. 1, 36–39 (2009)] that two discrete hungry integrable systems are related to matrix eigenvalue computation. Espec...

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 s...

We are concerned with the pantograph differential equation y′(t)=ay(t)+by(qt)+f(t),t>0,y(0)=y0y′(t)=ay(t)+by(qt)+f(t),t>0,y(0)=y0, with proportional delay qt,0<q<1qt,0<q<1. In the literatures, it is known that if we choose some proper m collocation points for m⩾2m⩾2, then collocation leads to a superconvergence result of order p∗=2m+1p∗=2m+1 at the...

Consider the following nonlinear difference equation with variable coefficients: where aj(n)⩾0, 0⩽j⩽m, and . We assume that there exists a strictly monotone increasing function f(x) on (−∞,+∞) such that f(0)=0, , x≠0, 0⩽j⩽m, and limx→−∞f(x) is finite if f(x)≠x. In this paper, we establish sufficient conditions for the zero solution of the above equ...

The Generalized Conjugate Residual (GCR) method with a variable preconditioning is an efficient method for solving a large sparse linear system Ax = b. It has been clarified by some numerical experiments that the Successive Over Relaxation (SOR) method is more effective than Krylov subspace methods such as GCR and ILU(0) preconditioned GCR for perf...

Consider the pantograph differential equation y′(t) = ay(t) + by(qt) + f(t), 0 ⩽ t ⩽ T, y(0) = y0, with proportional delay qt, 0 < q < 1. In this paper, we propose a piecewise (2m, m)-rational approximation with “quasi-uniform meshes” which corresponds to mth collocation method, and establish the global error analysis of O(h2m) on successive mesh p...

In this paper, we consider the following logistic equation with piecewise constant arguments: {dN(t)/dt=rN(t){1-∑j=0majN([t-j])}, t≥0, m≥1, N(0)=N0>0, N(-j)=N-j≥0, j=1,2,...,m, where r>0, a0,a1, ...,am≥0, ∑j=0m aj>0, and [x] means the maximal integer not greater than x. The sequence {Nn}n=0∞, where Nn=N(n), n=0,1,2,... satisfies the difference equa...

Consider the 'basic LUL factorization' of the matrices as the generalization of the LU factorization and the UL factorization, and using this LUL factorization of the matrices, we propose an "improved iterative method" such that the spectral radius of this iterative matrix is equal to zero, and this method converges at most n iterations. Our main c...

For the pantograph integro-differential equation (PIDE) with nonhomogeneous term: y'(t) = ay(t) + integral(0)(1) y(sigma(q)t) dmu(q) + integral(0)(1) y'(rho(q)t) dv(q) + f(t), t > 0, y(0) = y(0) with proportional delays sigma(q)t and rho(q)t, 0 < sigma(q), rho(q) less than or equal to 1, 0 < q less than or equal to 1, we consider the attainable ord...

To solve non-symmetric linear equations, we have proposed a generalized SOR method, named the improved SOR method with orderings, and for an $n\times n$ tridiagonal matrix, we have given $n$ selections of the multiple relaxation parameters which satisfy $\rho(\mathcal{L}_{\varPhi})=0$ and correspond to the reciprocal numbers of the pivots of Gaussi...

In this paper, we extend the recent results of H. Brunner in BIT (1997) for the DDE y′(t)= by(qt), y(0)=1 and the DVIE y(t)=1+∫0 t by(qs)ds with proportional delay qt, 0q≤1, to the neutral functional-differential equation (NFDE): and the delay Volterra integro-differential equation (DVIDE) : with proportional delays p i t and q i t, 0p i ,...

To analyze the attainable order of m-stage implicit (collocation-based) Runge-Kutta methods for the delay differential equation (DDE) y′(t) = by(qt), 0 < q ≤ 1 with y(0) = 1, and the delay Volterra integral equation (DVIE) y(t) = 1 + \(\tfrac{b}{q}\int {_0^{qt} }\) y(s) ds with proportional delay qt, 0 < q ≤ 1, our particular interest lies in the a...

Recently, a generalized SOR method with multiple relaxation parameters were considered for solving a linear system of equations and it was shown that if a pair of parameter values is computed from the pivots of the Gaussian elimination applied to the system, then the spectral radius of the iterative matrix is reduced to zero. A proper choice of ord...

A generalized SOR method with multiple relaxation parameters is considered for solving a linear system of equations. Optimal choices of the parameters are examined under the assumption that the coefficient matrix is tridiagonal and regular. It is shown that the spectral radius of the iterative matrix is reduced to zero for a pair of parameter value...

In this paper, on the basis of the results of Ishihara et al. (1997), we first discuss global convergence theorems for the improved SOR-Newton and block SOR-Newton methods with orderings applied to a system of mildly nonlinear equations, which includes as a special case the discretized version of the Dirichlet problem, for the equation ϵΔu + p(x)ux...

To solve non-symmetric linear equations, we have already proposed a generalized SOR method, named the “improved SOR method with orderings”, and if we use special relaxation parameters and proper orderings, then our method converges more rapidly and with fewer iterations than the usual SOR method. In this paper, we consider main convergence theorems...

In this paper, we give new necessary and sufficient conditions for an n × n matrix to be generalized diagonally dominant and propose those as practical criteria for the generalized diagonally dominant matrix. A numerical example shows their efficiency.

We consider the following non-autonomous and nonlinear difference equations with unbounded delays: {(xi + 1 = q xi - underover(∑, j = - ∞, i) ai, j fi - j (xj),, i = 0, 1, 2, ...,; xj = φ{symbol}j,, - ∞ < j ≤ 0,) where 0 < q < 1 and fj (x) (0 ≤ j < + ∞) are suitable functions. We establish sufficient conditions for the zero solution of the above eq...

Two kinds of discrete hungry Lotka-Volterra systems (dhLV) are known as discretiza-tions of the additive type hungry Lotka-Volterra system and the multiplicative one. By associating the dhLV of additive type (dhLVI) and the discrete hungry Toda equation (dhToda) with LR trans-formations, some of the authors give a Bäcklund transformation between th...