Search results
Results from the WOW.Com Content Network
The original use of interpolation polynomials was to approximate values of important transcendental functions such as natural logarithm and trigonometric functions.Starting with a few accurately computed data points, the corresponding interpolation polynomial will approximate the function at an arbitrary nearby point.
With the n-th polynomial normalized to give P n (1) = 1, the i-th Gauss node, x i, is the i-th root of P n and the weights are given by the formula [3] = [′ ()]. Some low-order quadrature rules are tabulated below (over interval [−1, 1] , see the section below for other intervals).
The simplest interpolation method is to locate the nearest data value, and assign the same value. In simple problems, this method is unlikely to be used, as linear interpolation (see below) is almost as easy, but in higher-dimensional multivariate interpolation, this could be a favourable choice for its speed and simplicity.
For integrating f over [,] with Gauss–Legendre quadrature, the associated orthogonal polynomials are Legendre polynomials, denoted by P n (x). With the n-th polynomial normalized so that P n (1) = 1, the i-th Gauss node, x i, is the i-th root of P n and the weights are given by the formula [5]
Newton's form has the simplicity that the new points are always added at one end: Newton's forward formula can add new points to the right, and Newton's backward formula can add new points to the left. The accuracy of polynomial interpolation depends on how close the interpolated point is to the middle of the x values of the set of points used ...
The most common method for estimating the Gaussian parameters is to take the logarithm of the data and fit a parabola to the resulting data set. [7] [8] While this provides a simple curve fitting procedure, the resulting algorithm may be biased by excessively weighting small data values, which can produce large errors in the profile estimate.
In numerical analysis Chebyshev–Gauss quadrature is an extension of Gaussian quadrature method for approximating the value of integrals of the following kind:
In numerical analysis, Gauss–Hermite quadrature is a form of Gaussian quadrature for approximating the value of integrals of the following kind: ∫ − ∞ + ∞ e − x 2 f ( x ) d x . {\displaystyle \int _{-\infty }^{+\infty }e^{-x^{2}}f(x)\,dx.}