ENGM 880 Project.docx

Useful interpretation of the data can be better

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useful interpretation of the data can be better gleaned from the line fit, knowing or presuming the function should be linear. Method For a polynomial curve-fit, we require that the number of terms in the polynomial, n, are less than or equal to the number of data points we are given, N. If N=n+1, then the polynomial would pass through every line, and it could be described using an interpolating method such as Newton’s or Langrange’s method.[ CITATION Ger08 \l 1033 ] We assume the function will behave in the following form: y = a 0 + a 1 x + a 2 x 2 + + a n x n Equation 1 Where the a’s are constants which we will optimize. To optimize these constants, we must minimize the sum of squares equation. S = i = 1 N ( Y i a 0 a 1 x a 2 x 2 a n x n ) 2 Equation 2 This equation can be minimized by taking the partial derivative at each constant a. [ CITATION Ger08 \l 1033 ] This will give us n+1 equations for n+1 unknowns. The partial derivatives can be described as 2

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Least-Square Method ENGM 880 Jared Miller δS δ a k = 0 = i = 1 N 2 ( Y i a 0 a 1 x a 2 x 2 a n x n ) (− x i k ) Equation 3 or rewriting this in terms of the unknowns on the left and known variables on the right a 0 i = 1 N x i k + a 1 i = 1 N x i k + 1 + a 2 i = 1 N x i k + 2 + + a n i = 1 N x i k + n = i = 1 N x i k Y i Equation 4 for k=0,1,2…n. This could also be expressed in a matrix, as see in Equation 5. [ N x i x i 2 x i n x i x i 2 x i 3 x i n + 1 x i n x i n + 1 x i n + 2 x i 2 n ] [ a ] = [ Y i x i Y i x i n Y i ] Equation 5 It is thus possible to find the constants in [a], as this is a system of linear equations, which can be solved by using any number of methods, including MATLAB’s built-in function for solving matrices. Using these constants, one could construct the equation for the curve-fitted polynomial.
• Summer '09
• CraigJ.Eckhardt
• Quadratic equation, Elementary algebra, Quintic equation, Jared Miller

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