15 - 15 Curve Fitting with splines 15 Objectives Understand...

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15: Curve Fitting with splines 15: Curve Fitting with splines
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Nov 6, 2006 20:16 2 Objectives Understand that splines perform curve fitting while minimizing oscillations. Recognize that cubic polynomials are preferable to quadratic and to higher- order splines. Understand the conditions underlying a cubic spline fit. Understand the differences between natural, clumped, and not-a-knot end conditions. Fit a spline to data using MATLAB's built-in functions.
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Nov 6, 2006 20:16 3 Introduction For n data points, an ( n -1)th-order polynomial can be fit This may result in erroneous interpolation due to round-off error and oscillations. Instead, lower-order polynomials ( spline functions ) are used on subsets of data. Example: third-order curves curves connecting each pair of data points are called cubic splines.
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Nov 6, 2006 20:16 4 Why splines? Consider the step function on the right In Figures (a) through (c), n data points are obtained and an ( n -1)th-order polynomial is fit The greater the number of data points, the more oscillations we observe In Figure (d), linear splines are used to connect the data points; a closer fit is obtained Thus, splines usually provide a better approximation of the behaviour of functions with discontinuities
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Nov 6, 2006 20:16 5 Linear splines For n data points, there are n -1 intervals Each interval i has its own spline function, s i ( x ) For linear splines, s i ( x ) is a straight line with This is equivalent to using Newton's first-order polynomial to interpolate within each interval s i x  = a i b i x x i a i = f i b i = f i 1 f i x i 1 x i x 1 x 2 x i x i+1 x n-1 x n f i f i+1 s i (x)
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6 Linear splines – Example Consider the following data points Using linear splines, we have We can interpolate for the value of f at x=5 as follows i 1 3.0 2.5 2 4.5 1.0 3 7.0 2.5 4 9.0 0.5 x i f i s 1 x  = 2.5 1.0 2.5 4.5 3.0 x 3.0 s 2 x  = 1.0 2.5 1.0 7.0 4.5 x 4.5 s 3 x  = 2.5 0.5 2.5 9.0 7.0 x
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This note was uploaded on 11/13/2011 for the course CIVE 2*** taught by Professor - during the Spring '11 term at Carleton CA.

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15 - 15 Curve Fitting with splines 15 Objectives Understand...

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