interpolation - Chapter 7 Interpolation y Additional data...

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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 1-dimensional case
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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
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y y y Too much energy Not smooth
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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
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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
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Polynomial Interpolation Given n interpolation points there is a polynomial is of degree (at most) n-1 that interpolates the points n=2 points, use a line, Polynomial of degree 1 n=2 , use a quadratic
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Polynomial Interpolation Given n interpolation points, polynomial is of degree n-1 We impose the conditions This determines the unknowns x i monomials
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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 n -1 through n data points
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Lagrange interpolation Consider the quadratic function will correctly interpolate
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Similarly The quadratic interpolant can be written as quadratic that vanishes at t 1 , t 3 quadratic that vanishes at t 1 , t 2
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Lagrange basis functions: 5 data points, 4 th degree polynomial
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Horner’s method The polynomial Can be evaluated efficiently as
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