Introduction to Numerical Analysis I MATH/CMPSC 455 Interpolation.
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Transcript of Introduction to Numerical Analysis I MATH/CMPSC 455 Interpolation.
![Page 1: Introduction to Numerical Analysis I MATH/CMPSC 455 Interpolation.](https://reader034.fdocuments.net/reader034/viewer/2022051000/56649efb5503460f94c0d7d5/html5/thumbnails/1.jpg)
Introduction to Numerical Analysis I
MATH/CMPSC 455
Interpolation
![Page 2: Introduction to Numerical Analysis I MATH/CMPSC 455 Interpolation.](https://reader034.fdocuments.net/reader034/viewer/2022051000/56649efb5503460f94c0d7d5/html5/thumbnails/2.jpg)
CHAPTER 3. INTERPOLATION
A function is said to interpolate a set of data points if it passes through those points
![Page 3: Introduction to Numerical Analysis I MATH/CMPSC 455 Interpolation.](https://reader034.fdocuments.net/reader034/viewer/2022051000/56649efb5503460f94c0d7d5/html5/thumbnails/3.jpg)
Definition: The function interpolates the data sets if
Note that is required to be a function!
Restriction on the data set:
![Page 4: Introduction to Numerical Analysis I MATH/CMPSC 455 Interpolation.](https://reader034.fdocuments.net/reader034/viewer/2022051000/56649efb5503460f94c0d7d5/html5/thumbnails/4.jpg)
Main theorem of Polynomial interpolation:If are distinct, there is a unique polynomial of degree such that
How to find this polynomial?
INTERPOLATION POLYNOMIAL
Mathematical Problem: (Interpolate points)Given n+1 points , we seek a 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 ,
![Page 5: Introduction to Numerical Analysis I MATH/CMPSC 455 Interpolation.](https://reader034.fdocuments.net/reader034/viewer/2022051000/56649efb5503460f94c0d7d5/html5/thumbnails/5.jpg)
LAGRANGE INTERPOLATION
For a data set , the Lagrange form of the interpolation polynomial is
![Page 6: Introduction to Numerical Analysis I MATH/CMPSC 455 Interpolation.](https://reader034.fdocuments.net/reader034/viewer/2022051000/56649efb5503460f94c0d7d5/html5/thumbnails/6.jpg)
Example:
x 5 -7
y 1 -23
Example:
x
y
![Page 7: Introduction to Numerical Analysis I MATH/CMPSC 455 Interpolation.](https://reader034.fdocuments.net/reader034/viewer/2022051000/56649efb5503460f94c0d7d5/html5/thumbnails/7.jpg)
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.
![Page 8: Introduction to Numerical Analysis I MATH/CMPSC 455 Interpolation.](https://reader034.fdocuments.net/reader034/viewer/2022051000/56649efb5503460f94c0d7d5/html5/thumbnails/8.jpg)
forms a basis of
Newton form of interpolation polynomial:
Determine the coefficients
![Page 9: Introduction to Numerical Analysis I MATH/CMPSC 455 Interpolation.](https://reader034.fdocuments.net/reader034/viewer/2022051000/56649efb5503460f94c0d7d5/html5/thumbnails/9.jpg)
NEWTON’S DIVIDED DIFFERENCES
Definition:
Example:
![Page 10: Introduction to Numerical Analysis I MATH/CMPSC 455 Interpolation.](https://reader034.fdocuments.net/reader034/viewer/2022051000/56649efb5503460f94c0d7d5/html5/thumbnails/10.jpg)
NEWTON FORM OF THE INTERPOLATION POLYNOMIAL
Nested Form:
Definition:
![Page 11: Introduction to Numerical Analysis I MATH/CMPSC 455 Interpolation.](https://reader034.fdocuments.net/reader034/viewer/2022051000/56649efb5503460f94c0d7d5/html5/thumbnails/11.jpg)
Example:
![Page 12: Introduction to Numerical Analysis I MATH/CMPSC 455 Interpolation.](https://reader034.fdocuments.net/reader034/viewer/2022051000/56649efb5503460f94c0d7d5/html5/thumbnails/12.jpg)
Example:
x 0 2 3
f(x) 1 2 4