Jonas T. Holdeman

A method for the approximation of functions defined by formal series expansions in orthogonal polynomials

An algorithm is described for numerically evaluating functions defined by formal (and possibly divergent) series as well as convergent series of orthogonal functions which are, apart from a factor, orthogonal polynomials. When the orthogonal functions are polynomials, the approximations are rational functions. The algorithm is similar in some respects to the method of Pad6 approximants. A rational approximation involving Tchebychev polynomials due to H. Maehley and described by E. Kogbetliantz (1) is a special case of the algorithm. E 1. Introduction. The solutions to many physical problems are obtained as expansions in (infinite) series of orthogonal functions. When the series are con- vergent they can in principle be approximated to any accuracy by truncating the series at the proper point. When the series are weakly convergent or divergent, the procedures for their numerical evaluation become rather ad hoc and a more general approach would be useful. In the following sections we describe an approximation to functions defined as infinite series of orthogonal functions which are (apart from a factor) orthogonal polynomials. The approximation takes the form of the ratio of two functions, the denominator function being a polynomial. When the orthogonal functions are polynomials, the approximating function is rational. The derivation of the algorithm will be formal and no real proofs are given. Indication that the algorithm is at least sometimes valid is provided by the numer- ical examples in Section 6. While the derivations could probably be made rigorous in the case of absolutely convergent series, interesting cases occur with divergent series such as the examples of Section 6. Numerous other practical applications of the algorithm have been made in the past year at Los Alamos Scientific Laboratory. The success of these examples would justify an effort at finding a class of functions representable by divergent series for which the algorithm is applicable. 2. The Approximation. The approximations we shall discuss fall roughly into two problems. The first of these is the approximation problem, that is, given a function (or equivalently its expansion), find an easily calculated approximation to the function. The second problem is the summation of infinite series, that is, given an infinite series (which may not be convergent in the ordinary sense), assign a sum to the series which gives the same value as the ordinary summation method when the series is convergent. Clearly the two problems cannot be sharply separated. In this paper however we will lean more toward the second problem. For simplicity we assume the functions we will encounter are real on the real

Magnetic torque on a shielded superconducting gyroscope

The torque on a superconducting sphere rotating in an arbitrary magnetic field is calculated. The result is expressed in terms of the coefficients of the expansion of the magnetic field in spherical harmonic functions, but in general a boundary‐value problem must be solved to obtain these coefficients. The boundary‐value problem is solved and the torque calculated for configurations pertinent to the gyroscope relativity experiment, a proposed satellite experiment which is to test various theories of gravitation. Typical numerical results for these torques are given and compared with the predicted relativistic effects.

Support-electrode torque on a spherical superconducting gyroscope

In a proposed satellite experiment, a spherical superconducting gyroscope is to be levitated by servo-controlled voltages applied to superconducting electrodes. Gyroscope readout will utilize the magnetic field that is generated by a rotating superconductor. However exclusion of this field from the electrodes via the Meissner effect will produce a torque on the gyroscope. That torque is calculated in this paper.

SUMIT: a computer code to interpolate and sum single release atmospheric model results onto a master grid

This report describes a computer code for the Systematic Unification of Multiple Input Tables of data (SUMIT). This code is designed to be an integral part of the Computerized Radiological Risk Investigation System (CRRIS) for assessing the health impacts of airborne releases of radioactive pollutants. SUMIT reads radionuclide air concentrations and ground deposition rates for different release points and combines them over a specified master grid. The resulting SUMIT grid may be circular, rectangular, or consist of irregularly spaced points. SUMIT can apply a different scaling factor to all data from each source. This program is designed to sum data written by the CRRIS code ANEMOS. Of course, SUMIT could read any data organized in the same manner at ANEMOS output. Descriptions of the necessary user input and data files are provided along with a complete listing of the SUMIT code. 10 references, 4 figures, 2 tables.

PREPR2: a program to aid in the preparation of input data for the farout hydraulic transport code

A computer code PREPR2 was written to aid in the preparation of data sets for the FAROUT/FAR2D/FARTMP/FARCCH/FARCRD computer codes used for the assessment of enviornmental impacts in power plant sitings. These latter hydraulic transport codes use two types of discrete elements, a rectangular internal element and a more complicated boundary element which incorporates the irregular shape of the region being simulated. The data set is prepared from a map of the area to be simulated and a description of the discrete element structure to be used. The computer program, the requirements for its use, and its use at an actual plant site are described.