interpolation

interpolation - Chapter 7 Interpolation Problem always has...

Info iconThis preview shows pages 1–13. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 1-dimensional 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) n-1 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 n-1 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 n-1 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:...
View Full Document

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.

Page1 / 36

interpolation - Chapter 7 Interpolation Problem always has...

This preview shows document pages 1 - 13. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online