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Introduction to Numerical Analysis I

Introduction to Numerical Analysis I. Interpolation. MATH/CMPSC 455. Chapter 3. Interpolation. A function is said to interpolate a set of data points if it passes through those points . Definition: The function interpolates the data sets if .

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Introduction to Numerical Analysis I

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  1. Introduction to Numerical Analysis I Interpolation MATH/CMPSC 455

  2. Chapter 3. Interpolation A function is said to interpolate a set of data points if it passes through those points

  3. Definition: The function interpolates the data sets if Note thatis required to be a function! Restriction on the data set:

  4. Interpolation Polynomial Mathematical Problem: (Interpolate points) Given n+1 points , we seek a polynomial of degree such that Main theorem of Polynomial interpolation: If are distinct, there is a unique polynomial of degree such that Mathematical Problem: (Interpolate a function) A function , assuming its values are known or computable at a set of n+1 points. we seek a polynomial of degree such that , How to find this polynomial?

  5. Lagrange Interpolation For a data set , the Lagrange form of the interpolation polynomial is

  6. Example: Example:

  7. How To? Method 1: Solving a linear system Determine coefficients Method 2: Lagrange Form of Interpolation Determine basis Method 3: Newton Form of Interpolation Use another basis which is easy to get, and has similar property as the basis for Lagrange form, and determine the coefficient easily.

  8. forms a basis of Newton form of interpolation polynomial: Determine the coefficients

  9. Newton’s Divided Differences Definition: Example:

  10. Newton Form of the Interpolation Polynomial Definition: Nested Form:

  11. Example:

  12. Example:

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