Stable convergence acceleration using Laplace transforms

Summary We consider the general class of power series where the terms may be expressed as the Laplace transforms of known functions. The sum of the series can then be evaluated efficiently and accurately by means of quadrature schemes, recently published by Frank Stenger. The method works also far outside the region of convergence as will be illustrated by numerical examples.Dedicated to Professor Lothar Collatz on the occasion of his 75th birthday     

Rational chebyshev approximations of elementary functions

Explicit formulæ which should be useful in practical computations, are given for the functions e^–x, log x, (1 + x), arctg x/x, sin x/x, cos x and (2/x) arcsin (x/2) in the interval 0x1 except for log x, where the interval is 1/2x1 instead.     

SCF and limited CI calculations for assignment of the Auger spectrum and of the satellites in the soft X-ray spectrum of H2O

SCF and limited CI calculations have been performed for a number of excited states of doubly ionized water. The calculations allowed an assignment of both the Auger spectrum and most of the satellites in the soft X-ray spectrum. The assignment for the two highest peaks in the Auger spectrum was ambiguous. SCF calculations of the K emission spectrum of H₂O were also performed. It was found that limited CI calculations were of importance for some states with two open shells in one symmetry. Such effects are caused by large off-diagonal lagrangian multiplies connecting the open shells in the reference configuration.     

Valence electron shake-up in photoelectron spectra

Valence electron shake-up phenomena are important in ultraviolet photoelectron spectroscopy because extra bands are formed which may complicate the interpretation, and because the primary bands are displaced in energy which may invalidate comparisons with calculated ionization energies. It would be of value during experimental work to have access to simple methods for an approximate estimation of these effects. Such methods are presented here together with applications to the photoelectron spectra of ethylene and benzene.     

Hadronization of excited nucleons in nuclear collisions at ISR energies

Calorimeters downstream of the intersection of pα and αα beams in the ISR have been used to study the hadronization of excited nucleons. These data extend and support the conclusions from previous studies of the A-dependence of particle multiplicities in ultrarelativistic p-nucleus collisions, which provided evidence that the proton hadronizes outside the nuclear volume.     

Analysis of hashing with chaining in the prime area

The expected performance of hashing with chaining in the prime area is analyzed. The method analyzed is briefly characterized as hashing with chaining of overflow records in the prime storage area, using one or several noncoalescing chains per bucket, and with random search for empty space. The analysis is asymptotic, and it is assumed that deletions do not occur. Simple closed formulas cannot be obtained, except in the case of bucket size one, but numerical results are readily computed. The expected performance compares favorably with that of other methods for handling overflow records.     

A comparison of some algorithms for the nonlinear least squares problem

The problem of minimizing a sum of squares of nonlinear functions is studied. To solve this problem one usually takes advantage of the fact that the objective function is of this special form. Doing this gives the Gauss-Newton method or modifications thereof. To study how these specialized methods compare with general purpose nonlinear optimization routines, test problems were generated where parameters determining the local behaviour of the algorithms could be controlled. The order of 1000 test problems were generated for testing three algorithms: the Gauss-Newton method, the Levenberg-Marquardt method and a quasi-Newton method.     

On the prime zeta function

The basic properties of the prime zeta function are discussed in some detail. A certain Dirichlet series closely connected with the function is introduced and investigated. Its dependence on the structure of the natural numbers with respect to their factorization is particularly stressed.     

Convergence acceleration by means of numerical quadrature

By regarding a series as a Stieltjes integral to which classical numerical methods are applied, very accurate expressions for the sum are obtained. Convergence and stability are investigated in some cases of practical importance. When certain conditions are satisfied, realistic and strict error bounds for the sum can be found. Some generalizations are also indicated.