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Unformatted text preview: Review of Part III Methods and Formulas Polynomial Interpolation: An exact fit to the data. For n data points it is a n − 1 degree polynomial. Only good for very few, accurate data points. The coefficients are found by solving a linear system of equations. Spline Interpolation: Fit a simple function between each pair of points. Joining points by line segments is the most simple spline. Cubic is by far the most common and important. Cubic matches derivatives and second derivatives at data points. Simply supported and clamped ends are available. Good for more, but accurate points. The coefficients are found by solving a linear system of equations. Least Squares: Makes a “close fit” of a simple function to all the data. Minimizes the sum of the squares of the errors. Good for noisy data. The coefficients are found by solving a linear system of equations. Interpolation vs. Extrapolation: Polynomials, Splines and Least Squares are generally used for Interpolation , fitting between the data. Extrapolation , i.e. making fits beyond the data, is much more tricky. To make predictions beyond the data, you must have knowledge of the underlying process, i.e. what the function should be. Numerical Integration: Left Endpoint: L n = n summationdisplay i =1 f ( x i − 1 )Δ x i Right Endpoint: R n = n summationdisplay i =1 f ( x i )Δ x i . Trapezoid Rule: T n = n summationdisplay i =1 f ( x i − 1 ) + f ( x i ) 2 Δ x i . Midpoint Rule: M n = n summationdisplay i =1 f (¯ x i )Δ x i where ¯ x i = x i − 1 + x i 2 . 100 101 Numerical Integration Rules with Even Spacing: For even spacing: Δ x = b − a n where n is the number of subintervals, then: L n = Δ x n − 1 summationdisplay i =0 y i = b − a n n − 1 summationdisplay i...
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This note was uploaded on 01/15/2011 for the course MATH 345 taught by Professor Staff during the Spring '08 term at Ohio State.
- Spring '08