Quotient-difference Algorithm Research Articles

On the calculation of the poles of multivariate meromorphic functions using the symbolic-numeric two-point qd-algorithm

The so-called quotient-difference algorithm, or qd-algorithm, is used for determining the poles of a meromorphic function from its Taylor coefficients. A generalization of this algorithm to the univariate and multivariate two-point cases applied to a power series (positive or negative exponents) is presented. We describe also the symbolic-numeric two-point qd-algorithm to compute the poles of multivariate meromorphic functions in a given domain from its series expansion coefficients. This algorithm can be regarded as computing the parametrized eigenvalues for a tridiagonal matrix. Numerical examples are furnished to illustrate our results.

Multivariate homogeneous two-point Padé approximants and continued fractions

The multivariate homogeneous two-point Pade approximants have been defined and studied recently. In the current work, we consider higher-order approximants and derive error formulas of these approximants using orthogonality conditions. Diverse three-term recurrence relations satisfied by the monic orthogonal polynomials are presented. Various continued fractions provided by these relations and the quotient-difference algorithm applied to a power series (positive or negative exponents) are described in terms of their relationships with the multivariate homogeneous two-point Pade table. Numerical examples are furnished to illustrate our results.

Min-plus eigenvalue of tridiagonal matrices in terms of the ultradiscrete Toda equation

The discrete Toda molecule equation can be used to compute eigenvalues of tridiagonal matrices over conventional linear algebra, and is the recursion formula of the well-known quotient difference algorithm for tridiagonal eigenvalues. An ultradiscretization of the discrete Toda equation leads to the ultradiscrete Toda (udToda) equation, which describes motions of balls in the box and ball system. In this paper, we associate the udToda equation with eigenvalues of tridiagonal matrices over min-plus algebra, which is a semiring with two operation types: and . We also clarify an interpretation of the udToda variables in weighted and directed graphs consisting of vertices and edges.

Duality for discrete integrable systems II

We generalise the concept of duality to lattice equations. We derive a novel 3-dimensional lattice equation, which is dual to the lattice AKP equation. Reductions of this equation include Rutishauser’s quotient-difference (QD) algorithm, the higher analogue of the discrete time Toda (HADT) equation and its corresponding quotient–quotient-difference (QQD) system, the discrete hungry Lotka–Volterra system, discrete hungry QD, as well as the hungry forms of HADT and QQD. We provide three conservation laws, we conjecture the equation admits N-soliton solutions and that reductions have the Laurent property and vanishing algebraic entropy.

An application of the discrete-time Toda lattice to the progressive algorithm by Lanczos and related problems

The finite or the semi-infinite discrete-time Toda lattice has many applications to various areas in applied mathematics. The purpose of this paper is to review how the Toda lattice appears in the Lanczos algorithm through the quotient-difference algorithm and its progressive form (pqd). Then a multistep progressive algorithm (MPA) for solving linear systems is presented. The extended Lanczos parameters can be given not by computing inner products of the extended Lanczos vectors but by using the pqd algorithm with highly relative accuracy in a lower cost. The asymptotic behavior of the pqd algorithm brings us some applications of MPA related to eigenvectors.

Transient analysis on track circuits in crack state

The transient transformation of track circuit between the normal state and crack state is analyzed. Firstly, the model of crack state of track circuit in the complex frequency domain is established. According to the boundary conditions of crack state of track circuit, the expression of current traveling wave in the receiving end is obtained in complex frequency domain. Based on it, the numerical solution in the time domain is got by FFT&IFFT conjunction with quotient-difference (Q-D) algorithm. Then, the steady state of circuit in normal state is taken as state of zero moment of track circuit, the transient process from normal state transform into crack state is solved. Thirdly, the Electro-Magnetic Transient Program (EMTP) model of crack operation is established to prove that the transient analysis result is correct. It provides a power theoretical basis to distinguish the operation state of track circuit by transient analysis.

Accurate quotient-difference algorithm: Error analysis, improvements and applications

The compensated quotient-difference (Compqd) algorithm is proposed along with some applications. The main motivation is based on the fact that the standard quotient-difference (qd) algorithm can be numerically unstable. The Compqd algorithm is obtained by applying error-free transformations to improve the traditional qd algorithm. We study in detail the error analysis of the qd and Compqd algorithms and we introduce new condition numbers so that the relative forward rounding error bounds can be derived directly. Our numerical experiments illustrate that the Compqd algorithm is much more accurate than the qd algorithm, relegating the influence of the condition numbers up to second order in the rounding unit of the computer. Three applications of the new algorithm in the obtention of continued fractions and in pole and zero detection are shown.

Temporal behavior of an atom-cavity system in two distinct regimes

The time-dependent treatment of a three-level atom in the V configuration confined in an optical cavity as well as the dynamics of the cavity field in two different regimes, in the presence and absence of the nonlinear mirror, has been discussed. The time-dependent infinite coupled differential equations of motion are solved by the matrix continued fraction method. In deriving the numerical inverse Laplace transform, the quotient difference algorithm and the fast Fourier transform have been applied. The nonlinear effects of the system over the time evolution of the population inversion, the mean photon number and the second-order correlation function for some dimensionless parameters are delineated. Ultimately the reduction of the three-level atom to an effective two-level one under specific conditions and in a weak-driving limit has been performed.

About several classes of bi-orthogonal polynomials and discrete integrable systems

By introducing some special bi-orthogonal polynomials, we derive the so-called discrete hungry quotient-difference (dhQD) algorithm and a system related to the QD-type discrete hungry Lotka–Volterra (QD-type dhLV) system, together with their Lax pairs. These two known equations can be regarded as extensions of the QD algorithm. When this idea is applied to a higher analogue of the discrete-time Toda (HADT) equation and the quotient–quotient-difference (QQD) scheme proposed by Spicer, Nijhoff and van der Kamp, two extended systems are constructed. We call these systems the hungry forms of the higher analogue discrete-time Toda (hHADT) equation and the quotient-quotient-difference (hQQD) scheme, respectively. In addition, the corresponding Lax pairs are provided.

High-Accuracy Series Solution for Two-Dimensional Convection in a Horizontal Concentric Cylinder

High order series solutions of two-dimensional buoyancy-induced flow in the horizontal annulus between concentric cylinders are computed. A new efficient algorithm based on computation sequences and a detailed analysis of the shape of the solution is used to compute the solutions up to 30th order. The quotient-difference (QD) algorithm is applied to locate the singularities of the series solution, which gives accurate estimation of the radius of convergence. The condition number and the stability of the solution are also studied. All these resources obtained from the series solution make it a good reference to justify the results from more commonly used purely numerical methods, particularly for delicate solutions.

Detecting discontinuity points from spectral data with the quotient-difference (qd) algorithm

This paper introduces a new technique for the localization of discontinuity points from spectral data. Through this work, we will be able to detect discontinuity points of a 2π-periodic piecewise smooth function from its Fourier coefficients. This could be useful in detecting edges and reducing the effects of the Gibbs phenomenon which appears near discontinuities and affects signal restitution. Our approach consists in moving from a discontinuity point detection problem to a pole detection problem, then adapting the quotient-difference (qd) algorithm in order to detect those discontinuity points.

Higher analogues of the discrete-time Toda equation and the quotient-difference algorithm

The discrete-time Toda equation arises as a universal equation for the relevant Hankel determinants associated with one-variable orthogonal polynomials through the mechanism of adjacency, which amounts to the inclusion of shifted weight functions in the orthogonality condition. In this paper we extend this mechanism to a new class of two-variable orthogonal polynomials where the variables are related via an elliptic curve. This leads to a ‘higher order analogue of the discrete-time Toda’ (HADT) equation for the associated Hankel determinants, together with its Lax pair, which is derived from the relevant recurrence relations for the orthogonal polynomials. In a similar way as the quotient-difference (QD) algorithm is related to the discrete-time Toda equation, a novel quotient–quotient-difference (QQD) scheme is presented for the HADT equation. We show that for both the HADT equation and the QQD scheme, there exists well-posed s-periodic initial value problems, for almost all . From the Lax-pairs we furthermore derive invariants for corresponding reductions to dynamical mappings for some explicit examples.

Sparse multivariate polynomial interpolation via the quotient-difference algorithm

To reconstruct a black box multivariate sparse polynomial from its floating point evaluations, the existing algorithms need to know upper bounds for both the number of terms in the polynomial and the partial degree in each of the variables [2]. On the other hand, Rutishauser's quotient-difference algorithm [7, 3], or the qd-algorithm, is an iterative method that can be use to determine the poles of a meromorphic function from its Taylor coefficients. Combining the relation between the qd-algorithm, Prony's method and their connections to the Ben-Or/Tiwari algorithm [6], we present a new algorithm for reconstructing sparse multivariate polynomials in floating point arithmetic, in which neither the number of terms nor any partial degrees are required in the input [1]. For example, consider the bivariate black box polynomial function, f ( x 1 , x 2 ) = 1/3 x 6/1+4 x 4/2--2.1 x 4/1--4 x 2/2+ x 1 x 2 +4 x 2/1, termed the six-hump camel back function. Evaluate f (x1 , x2 ) at the non-equidistant s-th powers of (1/3, 1/5), in which 3 and 5 are relatively prime, and proceed with the qd-algorithm on the data f (1/3, 1/5), f (1/3 2 , 1/5 2 ), f (1/3 3 , 1/5 3 ), ....The magnitude of the values in the first five e -columns drops from 10 --2 to machine precision, while all values in the sixth e -column are of the order of machine precision. We conclude that the number of terms is 6 and obtain the multi-indices in the support directly from the q -values: 1/q 1 (S) → 9 = 3 2 . 1/q 2 (S) → 25 = 5 2 . 1/q 3 (S) → 625 = 5 4 . 1/q 2 (S) → 15 = 3 1 5 1 . 1/q 1 (S) → 81 = 3 4 . 1/q 1 (S) → 729 = 3 6 . After all six multi-indices in f are known, the coefficients in f can be recovered by a linear system formed by (at least six) evaluations of f . The convergence to zero of some e-columns in the qd-algorithm can be interpreted in an analogous way as the zero discrepancies in the early termination strategy [5, 4]. This is illustrated in Maple. Our sparse interpolation can be extended to polynomials represented in certain non-standard bases [4]. We show an extension to sparse interpolation of polynomials in the Pochhammer basis and present our current research on sparse polynomial interpolation in the Chebyshev basis, both in floating point arithmetic.

On Convergence of the DQDS Algorithm for Singular Value Computation

We prove global convergence, in exact arithmetic, for the differential quotient difference algorithm that is currently implemented in LAPACK for the computation of the singular values of a bidiagonal matrix. Our results cover any shift strategy that preserves positivity. We also show that the asymptotic rate for the Johnson shift is 3/2.

Accurate recovery of recursion coefficients from Gaussian quadrature formulas

We present an algorithm for the accurate recovery of recursion coefficients from quadrature formulas with positive weights, based on the differential form of the quotient-difference algorithm. The process is slightly faster than the Gragg–Harrod algorithm and is forward stable in the sense of componentwise relative error when the gaps between the nodes are available as data. This result shows constructively that the converse problem is well posed when the data are required to be accurate floating point numbers. This is not a contradiction of the examples given by Gragg and Harrod which are ill posed in a vector norm, because in that case very small numbers are not required to have any significant digits.

Mixture models in hazard rates estimation

We propose to use mixture of geometries to model the duration Ot a spell in discrete time. The discrete hazard rate can be determined by the moments of the mixing distribution through the qd(quotient difference) algorithm, By introducing a sequence of parameters called canonical moments, we can ensure the resulting estimate of the mixing distribution is supported on [0,1]. Using the bootstrap method, we can construct, instantaneous confidence intervals for the hazard rates. For the continuous time, a mixture of exponentials is proposed. By a simple transformation, the problem can be reduced to the case for mixture of geometries. The instantaneous confidence intervals for the hazard rates can be constructed.

A vector Chebyshev algorithm

We consider polynomials orthogonal relative to a sequence of vectors and derive their recurrence relations within the framework of Clifford algebras. We state sufficient conditions for the existence of a system of such polynomials. The coefficients in the above relations may be computed using a cross-rule which is linked to a vector version of the quotient-difference algorithm, both of which are proved here using designants. An alternative route is to employ a vector variant of the Chebyshev algorithm. This algorithm is established and an implementation presented which does not require general Clifford elements. Finally, we comment on the connection with vector Pade approximants.

Discrete dressing transformations and Painlevé equations

We propose a discrete analog of the dressing transformation. Our starting point is a variant of the quotient-difference algorithm which, in this case, corresponds to a linear problem with shifts in the eigenvalues. The proper periodicity conditions lead to one-dimensional systems which are discrete Painlevé equations. We obtain thus the alternate d-P II equation and a novel form for the discrete P IV equation.

Orthogonal polynomial approach to discrete Lax pairs for initial boundary-value problems of the QD algorithm

Using orthogonal polynomial theory, we construct the Lax pair for the quotient-difference algorithm in the natural Rutishauser variables. We start by considering the family of orthogonal polynomials corresponding to a given linear form. Shifts on the linear form give rise to adjacent families. A compatible set of linear problems is made up from two relations connecting adjacent and original polynomials. Lax pairs for several initial boundary-value problems are derived and we recover the discrete-time Toda chain equations of Hirota and of Suris. This approach allows us to derive a Backlund transform that relates these two different discrete-time Toda systems. We also show that they yield the same bilinear equation up to a gauge transformation. The singularity confinement property is discussed as well.

Singular rules for a multivariate quotient-difference algorithm

Using the multivariateqdg-algorithm developed in [5], it is possible to compute the partial numerators and denominators of a continued fraction representation associated with a descending staircase in a table of multivariate rational interpolants, more precisely, multivariate Newton-Pade approximants. The algorithm is only applicable if every three successive elements on the staircase are different. If a singularity occurs in the defining system of equations for the multivariate rational interpolant then singular rules must be developed. For the univariate Newton-Pade approximant this was done in [3] by Claessens and Wuytack. The idea to perturb the initial staircase and walk around the block structure in the table in order to avoid the singularity, is explored now in a multivariate setting. Another approach would be to use block bordering methods in combination with reverse bordering [4] in order to solve the rank deficient linear system of interpolation conditions (Newton-Pade approximation conditions) recursively. Since this last technique can also be used for scattered multivariate data exhibiting near-singularity, we describe the second approach in a separate paper [7]. Here we deal only with partially grid-structured data (satisfying the so-called rectangle rule or inclusion property).