Recursive Evaluation Research Articles

Recursive evaluation of 3 j and 6 j coefficients

Recursive evaluation of 3 j and 6 j coefficients

Further Results on Recursive Evaluation of Compound Distributions

A recent result by Panjer provides a recursive algorithm for the compound distribution of aggregate claims when the counting law belongs to a special recursive family. In the present paper we first give a characterization of this recursive family, then describe some generalizations of Panjer's result.

Recursive Evaluation of a Family of Compound Distributions

Compound distributions such as the compound Poisson and the compound negative binomial are used extensively in the theory of risk to model the distribution of the total claims incurred in a fixed period of time. The usual method of evaluating the distribution function requires the computation of many convolutions of the conditional distribution of the amount of a claim given that a claim has occurred. When the expected number of claims is large, the computation can become unwieldy even with modern large scale electronic computers.In this paper, a recursive definition of the distribution of total claims is developed for a family of claim number distributions and arbitrary claim amount distributions. When the claim amount is discrete, the recursive definition can be used to compute the distribution of total claims without the use of convolutions. This can reduce the number of required computations by several orders of magnitude for sufficiently large portfolios.Results for some specific distributions have been previously obtained using generating functions and Laplace transforms (see Panjer (1980) including discussion). The simple algebraic proof of this paper yields all the previous results as special cases.

A symbolic limit evaluation program in REDUCE

A method for the automatic evaluation of algebraic limits is described. It combines many of the techniques previously employed, including top-down recursive evaluation, power series expansion, and L'Hôpital's rule. It introduces the concept of a special algebraic form for limits. The method has been implemented in MODE-REDUCE.

Recursive evaluation of Fourier series

Recursive evaluation of Fourier series

On the calculation of the acyclic polynomial

Graphical methods are developed for recursive evaluation of the acyclic polynomial. Analytical formulas of the acyclic polynomials for several specific series of graphs are given. Mathematical properties of the derivatives of the acyclic polynomial are given.

Recursive evaluation of 3j and 6j coefficients

Recursive evaluation of 3j and 6j coefficients

Recursive identification of linear and non-linear systems

The paper develops recursive techniques for off-line identification of linear and nonlinear systems. It is shown that if the system is linear and time invariant, impulse response characterization of the system coupled with an orthogonal series approximation can be utilized for the purpose stated above. The techniques of adaptive Kalman filtering are shown to be applicable, which besides permitting recursive evaluation of the coefficients, lead to a number of important advantages. In the second part of the study, the proposed method is extended for recursive identification of a class of non-linear systems which can be represented as a cascade combination of a linear dynamical system and a non-linear zero memory system. The method of Volterra series representation of such systems is utilized. Results are illustrated through numerical examples in each case.

On radial weighting effects in gaussian expansions of self-consistent field atomic orbitals

Gaussian expansions of the SCF functions for the first row atoms, boron through fluorine, in ground and low-lying electronic states have been generated under a wide range of radial weighting conditions by a full least-squares procedure. Typical results are presented and the quality of the wavefunctions obtained are analyzed in terms of regional electron densities and a variety of expectation values including energies. A novel method for recursive evaluation of repeated integrals of the error function, F l (α,ζ), is adopted and analyzed. These integrals are central quantities in the least-squares procedure employed.

Recursive Evaluation of Padé Approximants for Matrix Sequences

An algorithm is described for calculating the existing Padé approximants to any sequence A <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> , A <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> , ⋅ ⋅ ⋅ of s × t-matrices. As an application the algorithm gives a new way for finding the minimal polynomial of any square matrix A and the inverse of the characteristic matrix xI − A.

SENSOR‐ARRAY PROCESSING WITH CHANNEL‐RECURSIVE BAYES TECHNIQUES

Arrays of seismometers, hydrophones, and electromagnetic receivers have several signal processing problems in common. This paper is concerned primarily with source location and secondarily with signal extraction. The basic problem can be described as follows: A transient signal from an event is detected in the outputs of the sensor array. We determine the location of the source from the temporal positions of the signal in the array outputs. Further, if the signal is unknown, we estimate it. The approach taken here differs from previous investigations in three ways: (i) a Bayes estimation approach is used, (ii) the estimates are evaluated recursively with respect to channels, and (iii) a time‐domain approach is used, as opposed to a frequency‐domain approach. The proposed estimation technique is optimum with respect to a large class of loss functions, since it is based on the expectation of the posterior distribution. Recursive evaluation of the posterior expectation has several advantages. At each step we have the optimum estimate of the unknown parameters and the corresponding covariance matrix. A channel selection rule and stopping rule are defined in terms of the covariance matrix. Further, having an optimum estimate at each step permits simplification of the processing; e.g., the search interval may be limited to the most probable region of the parameter space. Such techniques significantly decrease the processing time and increase the rate of convergence. Equations are developed for the known‐signal case with planar and spherical wavefronts, and results are presented from a computer simulation. Subsequently, equations for the unknown‐signal case are presented with simulation results.

The analytical and recursive evaluation of two-center integrals

The analytical and recursive evaluation of two-center integrals

The recursive and analytical evaluation of atomic correlation integrals

The recursive and analytical evaluation of atomic correlation integrals