chap07_8up - Interpolation Polynomial Interpolation...

Info iconThis preview shows pages 1–2. 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
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Interpolation Polynomial Interpolation Piecewise Polynomial Interpolation Scientific Computing: An Introductory Survey Chapter 7 Interpolation Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign Copyright c 2002. Reproduction permitted for noncommercial, educational use only. Michael T. Heath Scientific Computing 1 / 56 Interpolation Polynomial Interpolation Piecewise Polynomial Interpolation Outline 1 Interpolation 2 Polynomial Interpolation 3 Piecewise Polynomial Interpolation Michael T. Heath Scientific Computing 2 / 56 Interpolation Polynomial Interpolation Piecewise Polynomial Interpolation Motivation Choosing Interpolant Existence and Uniqueness Interpolation Basic interpolation problem: for given data ( t 1 ,y 1 ) , ( t 2 ,y 2 ) , ... ( t m ,y m ) with t 1 < t 2 < < t m determine function f : R R such that f ( t i ) = y i , i = 1 ,...,m f is interpolating function , or interpolant , for given data Additional data might be prescribed, such as slope of interpolant at given points Additional constraints might be imposed, such as smoothness, monotonicity, or convexity of interpolant f could be function of more than one variable, but we will consider only one-dimensional case Michael T. Heath Scientific Computing 3 / 56 Interpolation Polynomial Interpolation Piecewise Polynomial Interpolation Motivation Choosing Interpolant Existence and Uniqueness Purposes for Interpolation Plotting smooth curve through discrete data points Reading between lines of table Differentiating or integrating tabular data Quick and easy evaluation of mathematical function Replacing complicated function by simple one Michael T. Heath Scientific Computing 4 / 56 Interpolation Polynomial Interpolation Piecewise Polynomial Interpolation Motivation Choosing Interpolant Existence and Uniqueness Interpolation vs Approximation By definition, interpolating function fits given data points exactly Interpolation is inappropriate if data points subject to significant errors It is usually preferable to smooth noisy data, for example by least squares approximation Approximation is also more appropriate for special function libraries Michael T. Heath Scientific Computing 5 / 56 Interpolation Polynomial Interpolation Piecewise Polynomial Interpolation Motivation Choosing Interpolant Existence and Uniqueness Issues in Interpolation Arbitrarily many functions interpolate given set of data points What form should interpolating function have? How should interpolant behave between data points? Should interpolant inherit properties of data, such as monotonicity, convexity, or periodicity? Are parameters that define interpolating function meaningful?...
View Full Document

Page1 / 7

chap07_8up - Interpolation Polynomial Interpolation...

This preview shows document pages 1 - 2. Sign up to view the full document.

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