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lecture 7 - Part Six Numerical Differentiation and...

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Part Six Numerical Differentiation and Integration
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Fig PT6.10
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Chapter 21 Newton-Cotes Integration Formulas
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Bungee Jumper Given the velocity of the bungee jumper, determine the vertical distance the jumper has fallen after time t Analytic integration (closed form solution) Numerical integration needed for more complex, nonlinear functions t m gc c m t z dt t m gc c gm dt t v t z t m gc c gm t v d d t 0 d d t 0 d d cosh ln ) ( tanh ) ( ) ( tanh ) (
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Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral
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Use of strips to approximate an integral
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Integration Weighted sum of functional values at specified points Newton-Cotes closed or open formulae -- evenly spaced points Approximate the function by Lagrange interpolation polynomial Integration of a simple interpolation polynomial Guassian Quadratures
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Basic Numerical Integration Weighted sum of function values ) ( ) ( ) ( ) ( ) ( n n 1 1 0 0 i n 0 i i b a x f c x f c x f c x f c dx x f x 0 x 1 x n x n-1 x f ( x )
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0 2 4 6 8 10 12 3 5 7 9 11 13 15 Numerical Integration Idea is to do integral in small parts, like the way you first learned integration - a summation Numerical methods just try to make it faster and more accurate
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Newton-Cotes formulas - based on idea dx x f dx x f I b a n b a ) ( ) ( Approximate f ( x ) by a polynomial n n 1 n 1 n 1 0 n x a x a x a a x f ) ( Numerical integration
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f n ( x ) can be linear f n ( x ) can be quadratic
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f n ( x ) can also be cubic or other higher-order polynomials
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Polynomial can be piecewise over the data
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Numerical Integration Newton-Cotes Closed Formulae -- Use both end points Trapezoidal Rule : Linear Simpson’s 1/3 -Rule : Quadratic Simpson’s 3/8 -Rule : Cubic Boole’s Rule : Fourth -order Higher-order methods Newton-Cotes Open Formulae -- Use only interior points midpoint rule Higher-order methods
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Closed and Open Formulae (a) End points are known (b) Extrapolation
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Trapezoidal Rule Straight-line approximation ) ( ) ( ) ( ) ( ) ( ) ( 1 0 1 1 0 0 i 1 0 i i b a x f x f 2 h x f c x f c x f c dx x f x 0 x 1 x f ( x ) L(x)
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Trapezoidal Rule Lagrange interpolation ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ; , , , ) ( ) ( ) ( b f a f 2 h 2 h b f 2 h a f d h b f d 1 h a f d L h dx x L dx x f b f a f 1 L 1 b x 0 a x a b h h dx d a b a x x b x a let x f x x x x x f x x x x x L 1 0 2 1 0 2 1 0 1 0 1 0 b a b a 1 0 1 0 1 0 0 1 0 1
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Example:Trapezoidal Rule Evaluate the integral Exact solution Trapezoidal Rule 926477 5216 1 x 2 e 4 1 e 4 1 e 2 x dx xe 1 0 x 2 4 0 x 2 x 2 4 0 x 2 . ) ( dx xe 4 0 x 2
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