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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).
It is named after the mathematician Carl Friedrich Gauss. The graph of a Gaussian is a characteristic symmetric " bell curve " shape. The parameter a is the height of the curve's peak, b is the position of the center of the peak, and c (the standard deviation , sometimes called the Gaussian RMS width) controls the width of the "bell".
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In numerical analysis Gauss–Laguerre quadrature (named after Carl Friedrich Gauss and Edmond Laguerre) is an extension of the Gaussian quadrature method for approximating the value of integrals of the following kind: + (). In this case
Shoelace scheme for determining the area of a polygon with point coordinates (,),..., (,). The shoelace formula, also known as Gauss's area formula and the surveyor's formula, [1] is a mathematical algorithm to determine the area of a simple polygon whose vertices are described by their Cartesian coordinates in the plane. [2]
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The entry 4+2i = −i(1+i) 2 (2+i), for example, could also be written as 4+2i= (1+i) 2 (1−2i). The entries in the table resolve this ambiguity by the following convention: the factors are primes in the right complex half plane with absolute value of the real part larger than or equal to the absolute value of the imaginary part.
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]