In numerical analysis quadrature refers to the computation of definite integrals.

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1 Numerical Quadrature In numerical analysis quadrature refers to the computation of definite integrals. f(x) a x i x i+1 x i+2 b x A traditional way to perform numerical integration is to take a piece of graph paper and count the number of boxes or quadrilaterals lying below a curve of the integrand. For this reason numerical integration is also called numerical quadrature

2 Numerical Quadrature than simple box counting. The Riemann definition of an integral is the limit of the sum over boxes as the width h of the box approaches zero (Figure 6.1): b a f(x) dx = lim h 0 h (b a)/h i=1 f(x i ). (6.2) The numerical integral of a function f(x) is approximated as the equivalent of a finite sum over boxes of height f(x) and width w i : b a f(x) dx N f(x i )w i, (6.3) i=1 which is similar to the Riemann definition (6.2) except that there is no limit to an infinitesimal box size. Equation (6.3) is the standard form for all integration algorithms; the function f(x) is evaluated at N points in the interval [a, b], and the function values f i f(x i ) are summed with each term in the sum weighted by w i.

3 Euler, Midpoint and Trapezoidal

4 Simpson Rule If we approximate the function with a parabola we obtain a better approximation: and using, for the derivatives Pluggin in the integral Note that odd terms do not contribute to the integrals and the formula is actually correct for polynomials up to order 3.

5 Extended (or composite) Simpson rule If we apply the previous results to successive, nonoverlapping pairs of intervals, we get the extended Simpson rule: xn x 1 [ 1 f(x)dx = h 3 f f f f f N f N f N ] + O ( ) 1 N 4 The 2/3, 4/3 alternation continues throughout the interior of the evaluation. Many people believe that the wobbling alternation somehow contains deep information about the integral of their function that is not apparent to mortal eyes. In fact, the alternation is an artifact of using the building block.

6 On the choice of N In general, if the quadrature rule is given by an N-point formula the results is exact for polynomials of degree N-1. However, if N is odd, the formula is correct for polynomial up to degree N (the function can always be rewritten across a symmetric interval).

7 Integral as a weighted sum In general we can then write the integral in the form: If f(x) is a polynomial, then for each power p: The previous forms a linear system of (N+1) unknowns (the weights) to be determined for equally spaced abscissa. This is known as the method of undetermined coefficients

8 Example: For N = 2, we can find a quadrature of the form Which is exact for polynomials of degree 2. Since the quadrature has to be exact for all terms in the polynomials (f(x) = ax^2+bx+c), we obtain the following system

9 Gaussian Quadrature

10 Example 6.9 We are looking for a quadrature of the form Z 1 1 f(x)dx A 0 f(x 0 )+A 1 f(x 1 ). 2 A straightforward computation will amount to making this quadrature exact for the polynomials of degree 6 3. The linearity of the quadrature means that it is su cient to make the quadrature exact for 1, x, x 2, and x 3. Hence we write the system of equations Z 1 1 f(x)dx = Z 1 1 x i dx = A 0 x i 0 + A 1 x i 1, i =0, 1, 2, 3. From this we can write 8 A 0 + A 1 =2, >< A 0 x 0 + A 1 x 1 =0, A 0 x >: A 1 x 2 1 = 2, 3 A 0 x A 1 x 3 1 =0. Solving for A 1, A 2, x 0, and x 1 we get A 1 = A 2 =1, x 0 = x 1 = 1 p 3, so that the desired quadrature is Z 1 1 f(x)dx f 1 1 p3 + f p3. (6.31)

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13 Gauss-Legendre Quadrature Number of points, n Points, x i Weights, w i

14 Gauss-Legendre Quadrature Table 25.4 ABSCISSAS AND WEIGHT FACTORS FOR GAUSSIAN INTEGRATION Abscissas= iti (Zeros of Legendre Polynomials) i xi n= n= n= n= 'U) i 2= U I +xi b n= Weight Factors=wi I1 = 8 wi n= n = n= n

15 Change of Interval A simple linear change of variable can be used to provide integration between x=a and x=b: An integral over [a, b] must be changed into an integral over [ 1, 1] before applying the Gaussian quadrature rule. This change of interval can be done in the following way: Applying the Gaussian quadrature rule then results in the following approximation:

16 Other forms Gaussian integration can be formulated in a more general way by introducing positive weight function into the integral an exploiting different kind of orthogonal polynomials yielding Where x k are the zero of some orthogonal polynomial and w k are the weights. This can be useful when the integrand contains an (apparent) singularity or when the interval extends to

17 Interval ω(x) [ 1, 1] 1 ( 1, 1) ( 1, 1) [ 1, 1] Orthogonal polynomials Legendre polynomials Jacobi polynomials Chebyshev polynomials (first kind) Chebyshev polynomials (second kind) A & S (β = 0) [0, ) Laguerre polynomials [0, ) Generalized Laguerre polynomials (, ) Hermite polynomials For more information, see... See Gauss Legendre quadrature above Gauss Jacobi quadrature Chebyshev Gauss quadrature Chebyshev Gauss quadrature Gauss Laguerre quadrature Gauss Laguerre quadrature Gauss Hermite quadrature

18 Checking the accuracy