lecture 7

lecture 7 - Part Six Numerical Differentiation and...

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Unformatted text preview: Part Six Numerical Differentiation and Integration Fig PT6.10 Chapter 21 Newton-Cotes Integration Formulas 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 d d t d d cosh ln ) ( tanh ) ( ) ( tanh ) ( Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral Use of strips to approximate an integral 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 Basic Numerical Integration Weighted sum of function values ) ( ) ( ) ( ) ( ) ( n n 1 1 i n i i b a x f c x f c x f c x f c dx x f x x 1 x n x n-1 x f ( x ) 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 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 n x a x a x a a x f ) ( Numerical integration f n ( x ) can be linear f n ( x ) can be quadratic f n ( x ) can also be cubic or other higher-order polynomials Polynomial can be piecewise over the data Numerical Integration Newton-Cotes Closed Formulae -- Use both end points Trapezoidal Rule : Linear Simpsons 1/3-Rule : Quadratic Simpsons 3/8-Rule : Cubic Booles Rule : Fourth-order Higher-order methods Newton-Cotes Open Formulae -- Use only interior points midpoint rule Higher-order methods Closed and Open Formulae (a) End points are known (b) Extrapolation Trapezoidal Rule Straight-line approximation ) ( ) ( ) ( ) ( ) ( ) ( 1 1 1 i 1 i i b a x f x f 2 h x f c x f c x f c dx x f x x 1 x f ( x ) L(x) 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 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 2 1 2 1 1 1 b a b a 1 1 1 1 1...
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lecture 7 - Part Six Numerical Differentiation and...

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