Lec11

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Numerical Integration (Quadrature) Another application for our interpolation tools!

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Lecture 11 2 Integration: Area under a curve Curve = data or function z Integrating data Finite number of data points—spacing specified Data may be noisy Must interpolate or regress a smooth between points z Integrating functions Non-integrable function (usually) As many points as you need Spacing arbitrary Today we will discuss methods that work on both of these problems () b a f xd x Area = Area Area x x ab y y
Lecture 11 3 Numerical Integration: The Big Picture z Virtually all numerical integration methods rely on the following procedure: Start from N+1 data points ( x i ,f i ), i = 0,…, N , or sample a specified function f ( x ) at N + 1x i values to generate the data set Fit the data set to a polynomial, either locally (piecewise) or globally Analytically integrate the polynomial to deduce an integration formula of the general form: z Numerical integration schemes are further categorized as either: Closed – the x i data points include the end points a and b of the interval Open – the x i data points are interior to the interval w i are the “weights” x i are the “abscissas”, “points”, or “nodes”

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Lecture 11 4 Further classification of numerical integration schemes z Newton-Cotes Formulas Use equally spaced abscissas Fit data to local order N polynomial approximants Examples: Trapezoidal rule, N=1 Simpson’s 1/3 rule, N=2 Errors are algebraic in the spacing h between points z Clenshaw-Curtis Quadrature Uses the Chebyshev abscissas Fit data to global order N polynomial approximants Errors can be spectral, ~exp(-N) ~ exp (-1/h), for smooth functions z Gaussian Quadrature Unequally spaced abscissas determined optimally Fit data to global order N polynomial approximants Errors can be spectral, and smaller than Clenshaw-Curtis
Lecture 11 5 Trapezoid rule: 1 interval, 2 points () b a f xd x Area = x f ( x ) ab x f ( x ) [] 1 ()( ) 2 f af bb a + Area = average height 1 2 f b + f b f a Approximate the function by a linear interpolant between the two end points, then integrate that degree-1 polynomial

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Lecture 11 6 Trapezoid rule: 2 intervals, 3 points () b a f xd x Area = x f ( x ) x 0 x 2 x f ( x ) ab [] 01 12 2 22 ()2 2 hh f xf x f x h fx ++ + =+ + Area = x 1 We should be able to improve accuracy by increasing the number of intervals: this leads to “composite” integration formulas
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