Transcription of Linear Interpolating Splines - USM
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Linear Interpolating Splines - USM
Jim LambersMAT 772 Fall Semester 2010-11 Lecture 17 NotesThese notes correspond to Sections , , and in the Interpolating SplinesWe have seen that high-degree polynomial interpolation can be problematic. However, if the fittingfunction is only required to have a few continuous derivatives, then one can construct apiecewisepolynomialto fit the data. We now precisely define what we mean by a piecewise (Piecewise polynomial) Let[ , ]be an interval that is divided into subintervals[ , +1],where = 0,.., 1, 0= and = . Apiecewise polynomialis a function ( )definedon[ , ]by ( ) = ( ), 1 , = 1,2,.., ,where, for = 1,2,.., , each function ( )is a polynomial defined on[ 1, ]. Thedegreeof ( )is the maximum degree of each polynomial ( ), for = 1,2,.., .It is essential to note that by this definition, a piecewise polynomial defined on [ , ] is equal tosome polynomial on each subinterval [ 1, ] of [ , ], for = 1,2.
Linear Interpolating Splines We have seen that high-degree polynomial interpolation can be problematic. However, if the tting function is only required to have a few continuous derivatives, then one can construct a piecewise polynomial to t the data. We now precisely de ne what we mean by a piecewise polynomial.
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