integrals

# integrals - Chapter 8 Numerical Integration b Problem...

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Chapter 8: Numerical Integration Problem compute : I(f) = f ( x ) dx a b Sometimes the analytic form of f(x) is known. Examples: Other times only a set of function values is given (or can be computed). Example: estimate the area of lake Michigan

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Midpoint, Trapezoid and Simpson’s rules Constant, linear and quadratic interpolants
Newton-Cotes quadrature rules use equally spaced nodes in interval [a, b] Midpoint rule Trapezoid rule Simpson’s rule a b

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Compute = 0.746824 M(f) = (1 0) exp( 1/4) = 0.778801 T(f) = (1/2)[exp(0) + exp( 1)] = 0.683940 S(f) = (1/6)[exp(0) + 4 exp( 1/4) + exp( 1)]=0.747180
Note: are all of the form Points x i are called the nodes w i are called the weights Integrand function is sampled at n points Rules are based on polynomial interpolation Computational work is measured by number of evaluations of integrand function required • Formulae of this type are called Newton-Cotes integration

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Quadrature rule is of degree 1 if it is exact for every polynomial of degree 1 (linear function) but not exact for some polynomial of degree 2 Quadrature rule is of degree 2 if it is exact for every polynomial of degree 2 (quadratic function) but not exact for some polynomial of degree 3
Midpoint rule is based on a polynomial of degree zero but it is a rule of degree 1 Trapezoidal rule is based on a polynomial of degree 1 and it is a rule of degree 1

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Quadrature rule is of degree d if it is exact for every polynomial of degree d , but not exact for some polynomial of degree d + 1 By construction, n-point interpolatory quadrature rule is of degree at least n 1 n-point Newton-Cotes rule is of degree n 1 if n is even, but of degree n if n is odd This phenomenon is due to cancellation of positive and negative errors
• Newton-Cotes quadrature rules are simple and often effective , but they have drawbacks • Using large number of equally spaced nodes may incur erratic behavior associated with high-degree polynomial interpolation (e.g., weights may be negative)

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integrals - Chapter 8 Numerical Integration b Problem...

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