NUMERICAL ANALYSIS CS 537
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Date Created: 10/23/15
C5537 Numerical Analysis Lecture 6 Approximation by Spline Functions Professor Jun Zhang Department of Computer Science University of Kentucky Lexington KY 402060046 October 30 2008 Lower Degree Splines Given a set of data it is possible to construct a polynomial to interpolate these data The disadvantage of higher order polynomials is its oscillation behavior A spline function is a function that consists of polynomial pieces joined together with certain smoothness conditions First degree spline function is a polygonal function with linear polynomials joined together to achieve continuity The points t0 t1 tn at which the function changes its character are called knots Note that knots do not have to be nodes High Order Polynomials 5 l a y y a Interpolation uses just one polynomial which may be oscillatory if the order ofthe polynomial is high Sp nes 5 J A spline function is a function that consists of polynomial pieces joined together with certain smoothness conditions Piecewise Linear Polynomial A piecewise linear function for the spline of degree 1 can be written as S0x x e t09t1 Sx S1c xet1t2 Sn 1x x e tn l tn where SiXalxbi Is a linear polynomial The knots t0 t1 tn and the coefficients at bi i 0 1 n 1 have to be known in order to evaluate Sx First to determine which interval x lies then evaluate the linear Tunction denned on that Interval ecew e L ear Polynom al 1 1 39inlervai x Hes then evaiuale i To eveidete 51x first to determine which the iineer function de ned on that inte Va ISpline of Degree 1 A function Sx is called a spline of degree 1 if 1 The domain ofS is an interval ab 2 S is continuous on ab 3 There is a partitioning of the interval a to lt t1 lt lt tn b such that S is a linear polynomial on each subinterval titl1 For outside part of the interval ab we define Sx S0x when x lt a and Sx Sn1x when x gt b It is important that the spline of degree 1 be continuous at the knots ie the left limit and the right limit are equal Inn fx hm fx m Determine if the following function is a first degree spline x 1SxS05 Sx 05x2x 05 05 S x S 2 x15 25x34 Each linear function is continuous on the subinterval it is defined We need to verify if they are continuous at the two interior knots x 05 and x 2 lim Sx lim x 05 x 05 x 05 lim Sx lim 05x 2x 05 025 x 05 x 05 The function is not a spline of degree 1 as lim Sx lim Sx x 05 x 05 ie Sx is discontinuous at the knotx 05 Construct Spline of Degree 1 Given a data set with t0 lt t1 lt lt tn 12 to t1 y yo yl tn yn A linear polynomial can be constructed using two pairs of neighboring data First compute the slope of the line as m yi1 yi ti1 ti The straight line equation is given by the pointslope formula as 500 yi mix ti It is easy to see that we have 2n degrees of freedom al and b and Zn conditions So the construction of first degree spline is guaranteed Modulus of Continuity The modulus of continuity of a functionfis defined as for a S u S v S b w h suplfu fvl lu v s h The quantity is the largest variation offover a small interval of size h It measures how muchfcan change in such an interval lffis continuous on ab then lim wfh 0 h 0 lffis differentiable on ab then fufvl lf39cuvl S Mlluvl S Mlh where M1 is the maximum value of f x on ab It follows that a f h s Mlh 10 Theorems pr is the first degree polynomial that interpolates a functionfat the end points of an interval ab then with h b a we have lfx pxl s wmh a s x s b Note that the linear function passes through ab can be written as x a b x Hence pltxbafbbafa fltxgt me limo fbl le x fal a b a Notethat 39 fxquot 39xfx x a b a Theresu foHomminnnedmtew 11
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