James N. Hanson

Generalized pearson distributions and nonlinear programing

A generalization of the Pearson curves is obtained as the solution to the differential equation which best fits a histogram in the mean square and satisfies certain statistical constraints, e.g., the mean and variance may be prescribed. Φ is the theoretical distribution defined on the intervel is a rational function with numerator and denomenator orders of mand n, respevtively. The values of the coefficients in are obtained from a Powell minimization of the mean-square value plus sum-of-squares constraints. Excellent fits are obtained effciently which, furthermore, are capable of providing an analytical representation of an infinite tail. An accurate initial approximation of for starting the Powell minimization is obtained from a symbol manipulation code in PL/1—Formac and based on the visual decomposition of a histogram into sums of normal curves.

Experiments with equation solutions by functional analysis algorithms and formula manipulation

The generalizations, due to Kantorovich et al. of the well-known numerical algorithms, successive approximations, steepest decent, and Newtons method; onto normed spaces, Hilbert spaces, and Banach spaces, respectively, have been tested on a variety of equations occuring in engineering and physics. Analytical (nonnumerical) solutions to a second-order partial differential equation, a nonlinear first-order ordinary differential equation, van der Pols equation, a nonlinear damping problem, and a nonlinear two-point boundary value problem were obtained by symbol manipulation as, for example, provided by FORMAC. These algorithms result in relatively simple forms, e.g., polynomials in sines and cosines, depending on the choice of the initial approximation, and yield high accuracy in a few iterations and in seconds-to-minutes of machine time. It is suggested, on the basis of these experiments, that functional analysis algorithms, as developed by Kantorovich, evaluated by automatic formula manipulation can yield analytical solutions of any desired accuracy to a variety of functional equations. In this way, analytical solutions are obtained providing qualitative information while subsequent numerical evaluation avoids much of the art and inaccuracy associated with numerical procedures.

Some data conversions for managing the internal and output form of formac constants

Three PL/I functions for managing Formac constants have been throughly tested and documented. These functions are: FLT&CHR for converting double precision floating point numbers to rational form as a PL/I character strings, TRUNCATE for truncating the numerator and denominator of a Formac rational constant to prescribed length, and EXT&FLT for constructing a PL/I character string containing an arbitrary length (precision) floating point equivalent of a Formac rational constant.

Planar motion of sliding cams by computer algebraic manipulation

Abstract The law of cosines, colinearity of tangents at the point of contact and Kennedys Theorem have been used to derive a system of three first order differential equations and the planar motion of sliding cams. The construction and solution of these equations and of auxilliary quantities, such as higher derivatives of angular motions and torques, have been accomplished by a PL/I-Formac code. This code includes a symbolic application of Newtons method for determining initial conditions. A complete example and computer output is presented. The emphasis of this paper is the use of computer aided algebraic and symbol manipulation requiring only the equations for the cam profiles and other analytic information as input, and the exact constitutive equations are generated and used as opposed to approximate methods such as replacement mechanisms.

Computer aided symbolic solution of multi-point boundary value problems occurring in physics and engineering

Abstract The author has elsewhere published papers demonstrating applicability of computer algebraic and symbol manipulation in obtaining solutions to ordinary and partial differential equations by Piccards method, steepest descent and various forms of the Newton-Kantorovich theorems and applying them to non-trivial problems in engineering, physics and, especially, celestial mechanics. In this paper, the Taylor series will be developed permitting expansion about any point and for any boundary conditions for any order derivative at arbitrary points, i.e., the general multi-point boundary valve problem (MPBVP) will be solved. The symbolic algorithm developed is written in PL/1-Formac and produces the Taylor series solution for any nonlinear differential equation in which the highest order derivative may be algebraically isolated. This program permits the continuation of this solution on intervals of the independent variable, in the manner of polynomial splines. This program permits symbolic solutions, e.g. in terms of a symbolic initial condition. However, such a solution requires enormous main storage. PL/1-Formac was used due to its general availability and its compatibility with relatively small main storage, even as small as 200 K bytes. Two parameters are available for attaining a given numerical accuracy; the order of the Taylor expansion and the number of continuation intervals into which the solution range is divided. Experiments show that high accuracy can be obtained by judiciously selecting these two parameters in order to counterbalance truncation error against numerical round-off error. Extensive additional documentation of this procedure has been performed on scores of problems occurring in the applied mathematics literature.

The computer generated symbolic approximations to systems of nonlinear ode's by matrix annihilation and the Newton-Kantorovich method

Abstract The symbolic solution, x = exp( At ) x (0), of the n -th order linear system, dx / dt = Ax , is approximated by using matrix annihilation of A , where the characteristic polynomial, but not its roots, is obtained by the methods of Leverrier or Krylov. This linear solution may be used to successively approximate the nonlinear case via Kantorovichs extension of Newtons method to Banach spaces. Symbolic solutions in the form of Taylor polynomials have been generated by the Formac extension to PL/I, to which very high order variable arithmetic has been added.

Fitting a curve to data when both variables are subject to error

A parameterized curve is optimized to best fit the multi-modal distribution formed by the sum of theoretical bivariant probability density functions representing individual data groups where each data group results from the repeated observations of the same measures of x and y.

COMPUTER SYMBOLIC SOLUTION OF NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS WITH ARBITRARY BOUNDARY CONDITIONS BY THE TAYLOR SERIES

Publisher Summary This chapter discusses the computer symbolic solution of nonlinear ordinary differential equations with arbitrary boundary conditions by Taylor series. Many theoretical tools are available for obtaining qualitative properties of the solutions of differential equations, such as solution existence over an interval of the independent variable, periodicity, quasi-periodicity, and boundedness. However, these same tools have seldom been used in actually exhibiting the analytic solution or an analytic approximation of the solution. The Taylor expansion power series can be generated by polynomial manipulation so that the size of expressions do not grow so rapidly. The computer can perform these manipulations readily. The analytical mathematical use of the digital computer is well established. There have been a number of studies dealing with the automatic computer generated analytic solution of ordinary differential equations.

Analysis of Irregular Reflectors

The flux distribution resulting from the light from an emitting surface being scattered by reflection from an irregular mirror surface is formulated. The particular case of the limb-darkened sun, a parabolic mirror, and a plane focal surface are analyzed using an example of geometrical scattering that occurs for thin metal optical surfaces possessing a show-through due to honeycomb backing. A matrix algorithm for computing the flux distribution is developed.

Analysis of the Thermoelastic Deformation of Optical Surfaces

The transient solution of the thermoelastic problem of an instantaneous point source of heat on the surface of a semi-infinite solid is derived by a somewhat heuristic procedure. This solution is adapted to the case of a very thin surface film such as an aluminum reflecting film. A number of approximations are incorporated so that the surface deflection becomes inversely proportional to the product of the radial distance from the point source and the third power of the time. The proportionality constant is a thermoelastic coefficient depending on the thermal, elastic, and emissive properties of the mirror. It is estimated that nearly instantaneous point sources of the order of milliwatts are to be expected and that they result in deflections of the order of seconds of arc for aluminum and quartz. On the basis of this analysis, metal is thermoelastically at least as good as quartz for a mirror material. Pyrex is significantly inferior to both. These results essentially agree with Couder’s.