# 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
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.
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
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
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)
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
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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