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Unformatted text preview: Chapter 7: Interpolation, Problem always has a solution y • Additional data might be prescribed, such as slope of interpolant at given points • Additional constraints might be imposed, such as smoothness or monotonicity of interpolant • f could be function of more than one variable, but we consider only 1dimensional case for Interpolation Applications • Plotting smooth curve through discrete data points • Reading between lines of table • Computing integrals and derivatives of functions • Replacing complicated function by simple one •Michael T. Heath Scientific Computing 4 / 54 y y y Too much energy Not smooth Interpolation vs approximation • By definition, interpolating function fits given data points exactly • Interpolation is NOT appropriate if data points subject to significant errors • It is usually preferable to smooth noisy data, for example by least squares approximation Families of functions used for interpolation include • Polynomials • Piecewise polynomials ( splines ) • Trigonometric functions • Exponential functions • Rational functions • We will focus on interpolation by polynomials and piecewise polynomials Functions for interpolation Polynomial Interpolation Given n interpolation points there is a polynomial is of degree (at most) n1 that interpolates the points n=2 points, use a line, Polynomial of degree 1 n=2 , use a quadratic Polynomial Interpolation Given n interpolation points, polynomial is of degree n1 We impose the conditions This determines the unknowns x i monomials Vandermonde system Theorem If the t i are distinct, Vandermonde matrix is nonsingular Solution exists for any data (x i ,y i ), and is unique In summary, there is a unique polynomial of degree n1 through n data points Lagrange interpolation Consider the quadratic function will correctly interpolate Similarly … The quadratic interpolant can be written as quadratic that vanishes at t 1 , t 3 quadratic that vanishes at t 1 , t 2 Lagrange basis functions: 5 data points, 4 th degree polynomial Horner’s method The polynomial Can be evaluated efficiently as Example:...
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This note was uploaded on 01/15/2012 for the course ECON 101 taught by Professor N/a during the Fall '11 term at Middlesex CC.
 Fall '11
 N/A
 Economics

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