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Showing papers on "Quintic function published in 1989"


Journal ArticleDOI
TL;DR: In this article, the Hermite polynomials are used to preserve local positivity, monotonicity, and convexity of the data if we restrict their derivatives to satisfy constraints at the data points.
Abstract: The Hermite polynomials are simple, effective interpolants of discrete data. These interpolants can preserve local positivity, monotonicity, and convexity of the data if we restrict their derivatives to satisfy constraints at the data points. This paper de- scribes the conditions that must be satisfied for cubic and quintic Hermite interpolants to preserve these properties when they exist in the discrete data. We construct algorithms to ensure that these constraints are satisfied and give numerical examples to illustrate the effectiveness of the algorithms on locally smooth and rough data. 1. Introduction. Piecewise polynomial interpolants, especially those based on Hermite polynomials (polynomials determined by their values and values of one or more derivatives at both ends of an interval), have a number of desirable properties. They are easy to compute once the derivative values are chosen. If the derivative values are chosen locally (e.g., by finite difference methods), then the interpolant at a given point will depend only on the given data at nearby mesh points. If the derivatives are computed by spline methods, then the interpolant will have an extra degree of continuity at the mesh points. In either case, the interpolant is linear in the given function values and has excellent convergence properties as the mesh spacing decreases. These methods, however, do not necessarily preserve the shape of the given data. When the data arise from a physical experiment, it may be vital that the interpolant preserve nonnegativity (f(x) > 0), nonpositivity (f(x) 0 or f(x) 0), or concavity (f(x) < 0). In this and other cases, geometric considerations, such as preventing spurious behavior near rapid changes in the data, may be more important than the asymptotic accuracy of the interpolation method. One can construct a shape-preserving interpolant by constraining the derivatives for the Hermite polynomials to meet conditions which imply the desired properties ((4), (5), (8), (11)—(15), (20)), by adding new mesh points

147 citations


Journal ArticleDOI
TL;DR: In this paper, a new solution to the quintic polynomial was given, in which the transcendental inversion of the icosahedral map (due to Hermite and Kronecker) was replaced by a purely iterative algorithm.
Abstract: Equations that can be solved using iterated rational maps are characterized: an equation is ‘computable’ if and only if its Galois group is within A5 of solvable. We give explicitly a new solution to the quintic polynomial, in which the transcendental inversion of the icosahedral map (due to Hermite and Kronecker) is replaced by a purely iterative algorithm. The algorithm requires a rational map with icosahedral symmetries; we show all rational maps with given symmetries can be described using the classical theory of invariant polynomials.

112 citations


Journal ArticleDOI
L. Gagnon1
TL;DR: In this article, a large set of exact analytic traveling-wavesolutions for the equation iut + uxx = a1u|u|2 + a2u| u|4 that can be of interest in higher-order nonlinear-optical media is presented.
Abstract: We present a large set of exact analytic traveling-wavesolutions for the equation iut + uxx = a1u|u|2 + a2u|u|4 that can be of interest in higher-order nonlinear-optical media. This set includes all bright and dark solitary waves as well as periodic ones. Our procedure consists of direct integration; it is then purely algebraic and simple. Solutions are given in simple forms and tabulated.

62 citations


Journal ArticleDOI
TL;DR: In this article, the Savizky and Golay cubic/quadratic, quartic/quintic and Gaussian convolutional functions were investigated and it was shown that the most effective smoothing is achieved with a single pass with a number of points in the smooth equal to 1.7 times the FWHM (N channels) of the peak to be smoothed and that this improves the signal-to-noise ratio by 0.69 N0.5.

23 citations


Dissertation
01 Jan 1989
TL;DR: All the new methods set up here are capable of reproducing the solutions to the KdV equation efficiently and accurately, the best amongst these methods are collocation with quintic splines or Galerkin with quadratic splines.
Abstract: The main aim of this study is the construction of new, efficient, and accurate numerical algorithms based on the finite element method, for the solution of the Korteweg-de Vries equation. Firstly the theoretical background to the KdV equation is discussed, and existing numerical methods based mainly on finite differences are discussed. In the following chapters finite element methods based on Bubnov-Galerkin approach are set up. Initially we used cubic Hermite interpolation functions, and in later methods cubic spline and quadratic spline shape functions. The appropriate element matrices were determined algebraically using the computer algebra package REDUCE. Finally we set up a method based on collocation using quintic spline interpolation functions. The numerical algorithms have been validated by studying the motion, interaction and deve lopment of solitons. We have demonstrated that these algorithms can faithfully represent the amplitude of a single soliton over many time steps and predict the progress of the wave front with small error. In the interaction of two solitons the numerical algorithms faithfully reproduce the changes in amplitudes and phase shifts of the analytic solution. We compare, in detai 1 the L - and L -error norms of the 2 00 present algorithms with published results. The conservative properties of the algorithms are also examined in detail. The modified and generalised Korteweg-de Vries equation have also been solved using collocation method with quintic splines interpolation functions. Again, the solution method has been validated by studying the motion, interaction, and development of solitons. We have concluded that all the new methods set up here are capable of reproducing the solutions to the KdV equation efficiently and accurately, the best amongst these methods are collocation with quintic splines or Galerkin with quadratic splines. The collocation method is also very efficient and accurate for solving the modified KdV equation.

12 citations


Journal ArticleDOI
TL;DR: In this article, a method using quintic splines which provides 0(h 6) uniformly convergent approximations for the solution of fourth order two-point boundary value problems was presented.
Abstract: It has been known for a long time that quintic spline collocation for fourth order two-point boundary value problems: y (4)+p(x)y=q(x), a

10 citations


Journal ArticleDOI
TL;DR: In this article, the authors defined the minimal model of a normal quintic K-3 surface with only one triple point if and only if it contains a nonsingular curve of genus two and a nonsedular rational curve crossing each other transversally.
Abstract: If a normal quintic surface is birational to a K-3 surface then it must contain from one to three triple points as its only essential singularities. A K-3 surface is the minimal model of a normal quintic surface with only one triple point if and only if it contains a nonsingular curve of genus two and a nonsingular rational curve crossing each other transversally. The minimal models of normal quintic K-3 surfaces with several triple points can also be characterized by the existence of some special divisors.

4 citations


Journal ArticleDOI
TL;DR: A formulation for Bezier-type quintic curves is presented which yields a curve which is tangent to the middle side of the control polygon or allows the shape of a curve to be varied for a given controlpolygon.

2 citations


Book ChapterDOI
01 Jun 1989
TL;DR: In this article, a smooth bivariate piecewise polynomial interpolant for given function values at scattered points in IR 2 is determined in the spline space S m 2 m + 1 (δ).
Abstract: The algorithm presented here determines a smooth bivariate piecewise polynomial interpolant for given function values at scattered points in IR 2 . After choosing an appropriate triangulation δ, the interpolant will be determined in the spline space S m 2 m +1 (δ) so that a high degree of polynomial precision is achieved. For the case m = 2 (quintic splines of class C 2 ), several numerical tests are performed.

1 citations


Journal ArticleDOI
TL;DR: This paper studies quintic residuacity of primes p of the form for which the expression for 4 f modulo p becomes indeterminate, and replaces it by a much simpler expression.
Abstract: This paper studies quintic residuacity of primes p of the form for which the expression for 4 f modulo p given in the first volume of this journal becomes indeterminate, and replaces it by a much simpler expression.