final - Final CVEN 302 December 3 2001 Lectures Goals...

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Final Final CVEN 302 December 3, 2001
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Lecture’s Goals Lecture’s Goals Interpolation Approximation Basic Numerical Integration Taylor, Euler, Runge Kutta Adam Bashforth Adam Moulton Systems of ODE
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Lecture 20 - Interpolation Methods CVEN 302 October 12, 2001
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Interpolation The Lagrange and Newton Interpolation are basically the same methods, but use different coefficients. The polynomials depend on the entire set of data points. Hermite Interpolation is a technique to calculate the values matches the function and first derivative.
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Interpolation The Rational function deals with fractional polynomials depend on the entire set of data points. Cubic Spline Interpolation is a piecewise technique to calculate the values matches the function and first derivative.
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Approximation The linear least squared method is straight forward to determine the coefficients of the line. 2 x xx x xy y xx 2 x xx y x xy N b N N a S S S S S S S S S S S - - = - - = b a + = x y
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Approximation The quadratic and higher order polynomial curve fits use a similar technique and involve solving a matrix of (n+1) x (n+1). n n 1 0 a a a x x y + + + = = = = = = = = = = N 1 i i n i N 1 i i i N 1 i i n 1 0 N 1 i 2n i N 1 i n i N 1 i i N 1 i n i N 1 i i a a a N Y x Y x Y x x x x x
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Approximation The higher order polynomials fit required that one selects the best fit for the data and a means of measuring the fit is the standard deviation of the results as a function of the degree of the polynomial. ( 29 = - - - = N 1 k 2 k k 1 1 y Y n N σ
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Basic Numerical Integration Basic Numerical Integration We want to find integration of functions of various forms of the equation known as the Newton Cotes integration formulas.
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Trapezoid Rule Trapezoid Rule Straight-line approximation [ ] ) x ( f ) x ( f 2 h ) x ( f c ) x ( f c ) x ( f c dx ) x ( f 1 0 1 1 0 0 i 1 0 i i b a + = + = = x 0 x 1 x f(x) L(x)
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Simpson’s 1/3-Rule Simpson’s 1/3-Rule Approximate the function by a parabola [ ] ) x ( f ) x ( f 4 ) x ( f 3 h ) x ( f c ) x ( f c ) x ( f c ) x ( f c dx ) x ( f 2 1 0 2 2 1 1 0 0 i 2 0 i i b a + + = + + = = x 0 x 1 x f(x) x 2 h h L(x)
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Simpson’s 3/8-Rule Simpson’s 3/8-Rule Approximate by a cubic polynomial [ ] ) x ( f ) x ( f 3 ) x ( f 3 ) x ( f 8 h 3 ) f(x c ) f(x c ) f(x c ) f(x c ) x ( f c dx ) x ( f 3 2 1 0 3 3 2 2 1 1 0 0 i 3 0 i i b a + + + = + + + = = x 0 x 1 x f(x) x 2 h h L(x) x 3 h
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Better Numerical Integration Better Numerical Integration Composite integration Composite Trapezoidal Rule Composite Simpson’s Rule Richardson Extrapolation Romberg integration
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Composite Trapezoid Rule Composite Trapezoid Rule [ ] [ ] [ ] [ ] ) x ( f ) x ( f 2 ) 2f(x ) f(x 2 ) f(x 2 h ) f(x ) f(x 2 h ) f(x ) f(x 2 h ) f(x ) f(x 2 h f(x)dx f(x)dx f(x)dx f(x)dx n 1 n i 1 0 n 1 n 2 1 1 0 x x x x x x b a n 1 n 2 1 1 0 + + + + + + = + + + + + + = + + + = - - - x 0 x 1 x f(x) x 2 h h x 3 h
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