Exact Polynomial Research Articles

The computer contouring of fabric diagrams

Abstract The problems associated with the computer contouring of fabric diagrams have been investigated with the aim of producing continuous, flowing, curved-line contours. This aim has been achieved by piecewise surface-fitting, using a 16-term exact fit 6th-order polynomial. Additional facilities provided by the new FORTRAN IV routines, written for a Honeywell 66/80 computer, include choice of projection (equal area or stereographic), choice of contour interval, and choice of contouring grid (square, stereographic equilinear, or stereographic equiangular).

Minimal error difference formulas

The usual formula for the r th difference of f(X), at intervals of h, may introduce an error of 2 rε, where ε is the |error| in f(X). When f(X) is either an exact polynomial of the n th degree, or very closely approximated by one within a finite interval, say [−1, 1], the r th difference, at X = X 0, is expressible as ∑ n+1 i=1 a i f(X i), where for certain points X i within [−1, 1], depending upon (X 0, h), ∑ n+1 i=1 |a i| may be very much less than 2 r. Nodes X i that minimize ∑ n+1 i=1|a i| are said to provide “minimal error difference formulas”. For very small h, close approximations to them are obtainable from similar derivative formulas. For other combinations (X 0, h), non-minimal formulas for equally spaced X i's, with a i's precomputed to higher accuracy than that in f(X), greatly reduce ∑ n+1 i=1|a i| from 2 r, ensure its approach to zero with h, and in many cases also yield more decimals and significant figures than the direct differencing of f(X). For r = 1, simple conditions for the non-existence of any expression ∑ n+1 i=1 a i f(X i), which improves ∑ n+1 i=1|a i| to be <2, are given for (X 0, h), expressed as h ≥ h 0 which depends upon X 0 and extrema of Chebyshev polynomials.

Algorithms for exact polynomial root calculation

This thesis discusses two sets of fully specified algorithms which, given a univariate polynomial with integer coefficients (with possible multiple roots) and a positive rational error bound, uses infinite-precision arithmetic and Sturm's Theorem to compute intervals containing the real roots of the polynomial and whose lengths are less than the given error bound, The algorithms also provide a simple means of determining the number of real roots in any interval.The first set of algorithms uses infinite-precision integer arithmetic to compute the sequence of polynomials required by Sturm's Theorem and also to do all the required polynomial evaluations. The second set uses congruence arithmetic to compute the images of the sequence of polynomials over the finite fields GF(p i ) for several primes p i and then uses the natural homomorphisms from the integers onto the finite fields GF(p i ) and the Chinese Remainder Theorem to do the evaluations. The primary advantage of the use of congruence arithmetic in the second set of algorithms is that the computing time to multiply any two elements of GF(p i ) is a constant whereas the time to multiply any two integers is proportional to the square of their lengths.The computing times of both sets of algorithms are analyzed in detail using the techniques of computing time analysis developed in the past few years by G. E. Collins and D. Knuth. It is shown that if P is a primitive, positive, univariate polynomial over the integers of positive degree, m, whose coefficients are bounded in magnitude by d, and which has L real roots, and ε is some positive rational number, 0 &lt; ε ≤ 1, then the computing time to approximate the roots of P to within is:[EQUATION]using either set of algorithms.The empirical results of applying the two sets of algorithms to random polynomials and the first ten Legendre and Chebyshev polynomials are reported. Neither set of algorithms proved to be faster for all the polynomials tested.

St. Venant flexure of pretwisted rectangular plates

An exact polynomial solution is obtained for the problem of St. Venant flexure of a pretwisted rectangular plate treated as a shallow hyperbolic paraboloidal shell. The results are generalizations of known simple solutions in plate theory and in the theory of generalized plane stress. The effect of the pretwist is discussed and the dependence of the dominant stress state on a pretwist parameter is delineated.

Accelerated Convergence, Divergence, Iteration, Extrapolation, and Curve Fitting

This paper discusses some applications of the epsilon algorithm (EA), a sequential procedure for calculating Padé approximants. The EA may be used to: (1) accelerate the convergence of slowly converging series and iterations; (2) obtain useful results from divergent series and iterations; (3) obtain the limits of iterated vector and matrix sequences; (4) aid in the solution of differential and integral equations; (5) carry out numerical integration in a new way; (6) extrapolate; (7) fit a curve to a polynomial or to a constant plus sum of exponentials. As an illustration of curve fitting and extrapolation, we present results obtained with exact polynomial data plus random noise combined additively or proportionately. For such nonstationary data, the results are comparable, and in some cases superior, to least squares in yielding good estimates of the exact polynomial coefficients. One important advantage of the EA is that it builds up polynomials whose lower-order coefficients are independent of higher-order ones. This property is valuable when the degree of the polynomial is unknown. Finally, a simple empirical equation is given relating the precision of least-squares-calculated polynomial coefficients to the degree of the fitted polynomial and the number of effective decimal digits carried in the calculation.