Recursive Polynomial Research Articles

Cross section calculations for resonance-dominated photodissociation of HCN via the excited Ã1A″ electronic state

The multi-reference internally contracted configuration interaction (MRCI) method is used to generate the potential energy functions (PEF) of the ground X 1Σ+ and excited A 1A″ electronic states of HCN molecule, as well as the corresponding transition dipole moment function. The resonance dominated photodissociation cross section is calculated using a filter diagonalization method to determine the complex eigenvalues of the final quasi-bound states as well as the complex transition dipole matrix elements. An alternative iterative method, based on the recursive polynomial expansion of the absorbing-boundary conditions Green operator, was also tested in cross section calculations. All nonadiabatic effects are neglected and molecular rotations are ignored.

Equalization of recursive polynomial systems

This letter presents some theorems for the exact and pth-order equalization of nonlinear systems described by recursive polynomial input-output relationships. It is shown that the nonlinear equalizers derived on the basis of this theory have simple and computationally efficient structures. Furthermore, the pth-order equalizers can be shown to operate in a stable manner for a finite range of values of the input amplitude when the linear component of the nonlinear system being equalized has minimum phase properties.

Exact Power of Conditional Tests for Several Binomial Populations

An efficient recursive polynomial multiplication method is proposed for exact unconditional power calculation for unordered 2 x K contingency table with up to moderate sample size. Our method can be applied to the family of cell-additive statistics which includes the Freeman-Halton statistic, the Pearson X 2 statistic and the likelihood ratio statistic. We illustrate our proposed method by several numerical examples.

On the stability of discrete time recursive Volterra filters

Many real nonlinear systems are characterized by an infinite input signal memory. In such conditions, system modeling by means of recursive polynomial filters requires a much lower number of coefficients with respect to nonrecursive realizations. However, the main problem of recursive polynomial filters is their inherent instability. This letter describes simple sufficient stability conditions for a class of discrete-time nonlinear systems based on recursive Volterra filters of arbitrary orders.

Exact inference on stratified two-stage Markov chain models

The field of exact analysis of independent discrete outcomes data with covariates has come of age in the recent years. Development of efficient numerical algorithms has been the prime factor fueling the growth. Exact inference on correlated or time-varying discrete data, on the other hand, has garnered little attention. In this paper, we lay the foundation for exact analysis for one class of longitudinal data problems – Markov chain models with covariates. In particular, we focus on a two-state first-order stationary Markov chain model with a multi-level stratification factor and a binary covariate of interest. We show that exact distributions for such models derive from conditional convolutions of elemental Bose–Einstein distributions. We develop a recursive polynomial multiplication algorithm with checks for infeasibility to compute such distributions, and empirically demonstrate its efficiency. We utilize the algorithm to compare exact inference on Markov chain data with the traditional large sample method. Given the transition counts, the latter method is insensitive as to whether the data are generated from a few long chains or many short chains. The exact method takes this feature into account. Our findings point towards a potential problem with large sample inference, namely the use of observed instead of expected information in finite samples. This issue needs further study. It is also seen that with Markov chain data, inferences drawn from the exact method may differ from that from the asymptotic method at transition count values that may not be deemed small.

Fast Spectral Projection Algorithms for Density-Matrix Computations

We present a fast algorithm for the construction of a spectral projector. This algorithm allows us to compute the density matrix, as used in, e.g., the Kohn–Sham iteration, and so obtain the electron density. We compute the spectral projector by constructing the matrix sign function through a simple polynomial recursion. We present several matrix representations for fast computation within this recursion, using bases with controlled space–spatial-frequency localization. In particular we consider wavelet and local cosine bases. Since spectral projectors appear in many contexts, we expect many additional applications of our approach.

Entropy and uniform distribution of orbits in $$\mathbb{T}$$

We present a class of integer sequences {c n } with the property that for everyp-invariant and ergodic positive-entropy measure μ on L 2 $$\mathbb{T}$$ , {c n x (mod 1)} is uniformly distributed for μ-almost everyx. This extends a result of B. Host, who proved this for the sequence {q n }, forq relatively prime top. Our class of sequences includes, for instance, the sequencec n =Мf i (n)q i n , where the numbersq i are distinct and are relatively prime top andf i are any polynomials. More generally, recursion sequences for which the free coefficient of the recursion polynomial is relatively prime top are in this class as well, provided they satisfy a simple irreducibility condition. In the multi-dimensional case we derive sufficient conditions for a pair of endomorphisms $$\mathbb{T}^d $$ (withA diagonal) and anA-invariant and ergodic measure μ, such thatB-orbits of the form {B n ω} are uniformly distributed for μ-almost every $$\mathbb{T}^d $$ .

Small-sample study of the use of mid-p power divergence goodness-of-fit tests

Asymptotic goodness-of-fit tests based on large sample theory have been shown to be unreliable in small sample designs and sparse data structures. In these situations, it is widely believed that exact version tests are dependable. However, exact goodness-of-fit tests that are based on discrete probabilities are conservative and the calculation of the exact distribution may be computationally intractable. In this article, we present a method, which is based on recursive polynomial multiplication, for performing exact analysis of goodness-of-fit tests. This method has been shown to be more efficient and accurate than an existing fast Fourier transform method. To overcome the conservativeness problem, we suggest the use of mid–P version goodness–of–fit tests. The small sample properties of the asymptotic, the exact, and the mid–P version goodness–of–fit tests are examined via the recursive polynomial multiplication method. Four commonly used power divergence statistics and six null hypotheses ranging from the equiprobable to the extremely skew are considered. The actual significance level behaviors under various null hypotheses suggest that as an anti–conservative test, the mid–P version tests behave fairly well. Nevertheless, it is not the case for the asymptotic and the exact tests. Among the four statistics considered, the exact power comparisons under a specific alternative to various null hypotheses show that the power divergence statistic with λ= 2/3 and the Pearson’s chi–squared statistic (λ = 1) are generally the most preferable.

Analysis of Carleman Representation of Analytical Recursions

We study a general method to map a nonlinear analytical recursion onto a linear one. The solution of the recursion is represented as a product of matrices whose elements depend only on the form of the recursion and not on initial conditions. First we consider the method for polynomial recursions of arbitrary degree and then the method is generalized to analytical recursions. Some properties of these matrices, such as the existence of an inverse matrix and diagonalization, are also studied.

Discrete energy representation and generalized propagation of physical systems

This work discusses the discrete energy representation based on generalized propagation of a physical system. Here, the propagation is defined as a recursion scheme which generates a series of system states from a given initial state. Examples of such schemes include the time propagation and polynomial recursion. It is argued that each propagation determines a set of energy points, which form the discrete energy representation. A unitary transformation can be established between the discrete energy representation and the generalized time representation, much like the well-known transformation between the discrete variable representation and the finite basis representation. Such a collocation approach can be useful in calculating many properties that are local in the energy domain. Numerical examples are presented to demonstrate the utility in filter diagonalization.

Assessing fast Fourier transform algorithms

We gave a comprehensive description of the fast Fourier transform (FFT) algorithms for exact inference on discrete data in an earlier paper. With that as the base, we now present an assessment of the algorithms in terms of efficiency, accuracy and implementation. Our findings are: (i) little is known about the actual properties of the FFT methods for logistic models and multi-way contingency tables, and (ii) for problems where comparative studies have been done, the FFT methods are not as efficient or accurate as network and polynomial recursion methods. The shortfalls noted in our review help pinpoint areas for future research.

Polynomial Recursion Equations in Form Factors of ADE-Toda Field Theories

It is shown that the problem of calculating form factors in ADE affine Toda field theories can be reduced to the nonperturbative recursive calculation of polynomials symmetric in each sort of variables. We determine these recursion equations explicitly for the ADE series and characterize the polynomial solutions by an interplay between the weight space of the underlying Lie algebra and representations of the symmetric group.

GENERALISED TRANSFER FUNCTIONS OF NEURAL NETWORKS

When artificial neural networks are used to model non-linear dynamical systems, the system structure which can be extremely useful for analysis and design, is buried within the network architecture. In this paper, explicit expressions for the frequency response or generalised transfer functions of both feedforward and recurrent neural networks are derived in terms of the network weights. The derivation of the algorithm is established on the basis of the Taylor series expansion of the activation functions used in a particular neural network. This leads to a representation which is equivalent to the non-linear recursive polynomial model and enables the derivation of the transfer functions to be based on the harmonic expansion method. By mapping the neural network into the frequency domain information about the structure of the underlying non-linear system can be recovered. Numerical examples are included to demonstrate the application of the new algorithm. These examples show that the frequency response functions appear to be highly sensitive to the network topology and training, and that the time domain properties fail to reveal deficiencies in the trained network structure.

The Number of Permutations with Exactlyr132-Subsequences IsP-Recursive in the Size!

Proving a first nontrivial instance of a conjecture of Noonan and Zeilberger we show that the numberSr(n) of permutations of lengthncontaining exactlyrsubsequences of type 132 is aP-recursive function ofn. We show that this remains true even if we impose some restrictions on the permutations. We also show the stronger statement that the ordinary generating functionGr(x) ofSr(n) is algebraic, in fact, it is rational in the variablesxand1−4x. We use this information to show that the degree of the polynomial recursion satisfied bySr(n) isr.

The quantum resonance spectrum of the H3+ molecular ion for J=0. An accurate calculation using filter diagonalization

Three-dimensional variational calculations of bound and resonance spectra of the H 3 + molecular ion for the total angular momentum J = 0 are presented. All N E + N A1 + N A2 = 2 × 604 + 358 + 256 bound states of the MBB PES and some N E + N A1 + N A2 = 2 × 264 + 142 + 119 resonances with widths Γ < 100 cm −1 are computed. Such calculations are very challenging due to (i) a very deep potential-energy well which gives rise to a high density of bound and resonance states and (ii) the floppiness of the H 3 + molecular ion which leads to various numerical difficulties. The computation was done using the filter diagonalization method. Here a series of subspace diagonalizations is performed on a Hamiltonian with a complex absorbing potential, Ĥ − iŴ, represented in a relatively small adapted basis set. Each small basis set is obtained by acting with the ABC (absorbing boundary conditions) Green's function on an initial wavepacket, using the modified by ABC Chebyshev recursion polynomial expansion. The method is described in some detail. Additionally, we describe an efficient and simple procedure of using Jacobi coordinate DVR (discrete variable representation) for a floppy triatomic molecule resulting in a sparse Hamiltonian matrix with a small spectral range, which in particular, overcomes the slow convergence of iterative Krylov type methods. Comparison between the present version of the filter diagonalization and the Lanczos method is made. The level statistics of the H 3 + bound and resonance states are analysed showing the strongly chaotic nature of the spectra, starting already at very moderate energies, caused by the extreme floppiness of the system. The widths of computed resonances fluctuate by orders of magnitude. At energies just above the dissociation threshold the resonances are essentially very narrow and isolated, but start to overlap very quickly as energy increases. The number of resonance states per effective decay channel is sufficient to carry out a statistical analysis of decay rates by fitting their distribution to the χ 2 distribution with ν eff degrees of freedom, where ν eff is the effective number of decay channels. The result of this analysis is however negative, suggesting that the concept of average width on which it is partially based is not correct when resonances are overlapping, and that more reliable statistical theories of the effective Hamiltonian must be developed for the barrierless reactions.

Solving nonlinear recursions

A general method to map a polynomial recursion on a matrix linear one is suggested. The solution of the recursion is represented as a product of a matrix multiplied by the vector of initial values. This matrix is product of transfer matrices whose elements depend only on the polynomial and not on the initial conditions. The method is valid for systems of polynomial recursions and for polynomial recursions of arbitrary order. The only restriction on these recurrent relations is that the highest-order term can be written in explicit form as a function of the lower-order terms (existence of a normal form). A continuous analog of this method is described as well.

A stability condition for adaptive recursive second-order polynomial filters

Recursive polynomial filters require a much lower number of coefficients with respect to nonrecursive realizations. However, the main problem of recursive polynomial filters is their inherent instability. In this paper sufficient stability conditions for recursive quadratic polynomial filters are reported. Moreover, an application to nonlinear system identification is described.

A comparison of algorithms for exact analysis of unordered 2 × K contingency tables

We present a comparison of two efficient algorithms for exact analysis of an unordered 2 × K table. First, by considering conditional generating functions, we show that both the network algorithm of Mehta and Patel ( J. Amer. Statist. Assoc. 78 (1983)) and the fast Fourier transform (FFT) algorithm of Baglivo et al. ( J. Amer. Statist. Assoc. 82 (1992)) rest on the same foundation. This foundation is a recursive polynomial relation. We further show that the network algorithm is equivalent to a stage-wise implementation of this recursion while the FFT algorithm is based on performing the same recursion at complex roots of unity. Our empirical results for the Pearson X 2, likelihood ratio, and Freeman-Halton statistics show that the network algorithm, or equivalently, the recursive polynomial multiplication algorithm is superior to the FFT algorithm with respect to computing speed and accuracy.

Stability condition for certain recursive second-order polynomial filters

A stability condition for a time-invariant recursive second-order Volterra filter is presented. When the input is bounded by the derived sufficient stability condition, the output of the presented recursive nonlinear filter is guaranteed to be bounded.

Nonlinear system identification and adaptive control using polynomial networks

This work introduces a new nonlinear computational model called polynomial network for identification and adaptive control of nonlinear dynamical systems. The network approximates the Volterra systems or recursive polynomial systems. The approximation properties of this network are compared with the sigmoid networks. The results show that the polynomial network constructs a simpler and smaller model and requires less training data. Also, the model realized by the polynomial network is mathematically tractable. The feasibility of using this model for direct model reference adaptive control of a class of nonlinear systems is demonstrated.