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lecture12 - Numerical Methods in Engineering ENGR 391...

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Numerical Methods in Engineering ENGR 391 Faculty of Engineering and Computer Sciences Concordia University Numerical integration Chapter 7

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Numerical Integration Numerical Quadrature: •Basic method involved in approximating integrals • Sometimes even simple functions may not have exact formulas for their anti-derivatives. • Even when an exact formula for the anti-derivative does exist, it may be difficult to find. ?? 2 0 2 = dx e x b a dx x f ) (
Numerical Integration A numerical integration rule has the form: Recall that an integral is simply that area under the curve . In general, a numerical integration formula approximating a definite integrals can be obtained by: -a weighted sum of function values at points within the interval of integration (Riemann sum) [] = = n i i i x f C f Q 0 ) ( error truncation f E Coefficient C i depends on particular methods Quadrature formula [] [] f E f Q dx x f b a + = ) (

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Degree of precision Degree of precision The degree of precision of a numerical integration method (quadrature method) is the highest degree of the polynomial for which the error of the method is zero. [] [] 0 3 = = f E f Q dx x b a ex: Degree of precision = 3
Newton-Cotes Integration Formulas •T h e Newton-Cotes formulas are the most common numerical integration schemes. • They are based on the strategy of replacing a complicated function or tabulated data with an approximating function that is easy to integrate: n n n n n b a n b a x a x a x a a x f dx x f dx x f I + + + + = = 1 1 1 0 ) ( ) ( ) ( L Using interpolation polynomials

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• Based on the interpolation polynomial • Use function values at equally spaced points • Formula can be derived by approximating the function to be integrated by its Lagrange interpolating polynomial and then integrating the polynomial exactly degree one degree two Newton-Cotes Integration Formulas Conditions
The Trapezoidal Rule •T h e Trapezoidal rule is the first of the Newton-Cotes closed integration formulas, corresponding to the case where the polynomial is first order (linear) : • The area under this first order polynomial is an estimate of the integral of f(x) between the limits of a and b : = b a b a dx x f dx x f I ) ( ) ( 1 2 ) ( ) ( ) ( b f a f a b I + = Trapezoidal rule

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2 ) ( ) ( ) ( b f a f a b I + = The Trapezoidal rule gives exact result when applied to any function whose second derivative is identically equal to zero, that is, any polynomial of degree 1 or less
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lecture12 - Numerical Methods in Engineering ENGR 391...

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