# interpolation1 - Interpolation Lagrange...

This preview shows pages 1–11. Sign up to view the full content.

Interpolation

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

View Full Document
Lagrange Interpolation . . . . . (x ,f ) (x 2 ,f 2 ) (x 3 ,f 3 ) (x 4 ,f 4 ) (x 5 ,f 5 ) f(x) (x 1 ,f 1 ) P(x)
Introduction The interpolation problem Given values of an unknown function f(x) at values x = x 0 , x 1 , …, x n , find approximate values of f(x) between these given values Polynomial interpolation – Find n th-order polynomial p n (x) that approximates the function f(x) and provides exact agreement at the n node points: ) ( ) ( ), ( ) ( ), ( ) ( 1 1 0 0 n n n n n x f x p x f x p x f x p = = = L The polynomial p n (x) is unique Interpolation: evaluate p n (x) for x 0 <= x <= x n Extrapolation: evaluate p n (x) for x 0 > x > x n

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

View Full Document
Purposes for Interpolation • Plotting smooth curve through discrete data points • Quick and easy evaluation of mathematical function • Replacing “difficult” function by “easy” one • “Reading between the lines” of table • Differentiating or integrating tabular data
Functions for Interpolation Some families of functions commonly used for interpolation include ° Polynomials ° Piecewise polynomials ° Trigonometric functions ° Exponentials ° Rational functions We focus on interpolation by polynomials and piecewise polynomials.

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

View Full Document
Polynomial Interpolation ) ( ... ) ( ) ( ) ( 2 2 1 1 x u a x u a x u a x P m m m + + + = ) 1 ( Y A V = V = u x u x u x u x u x u x u x u x u x n n n n n n 1 1 2 1 1 1 2 2 2 2 1 2 ( ) ( ) ... ( ) ( ) ( ) ... ( ) ... ... ... ... ( ) ( ) ... ( ) Vandermonde matrix n=m m i y x P i i m ,..., 2 , 1 ) ( , = = A Y = = a a a y y y n n 1 2 1 2 ... , ... Y V A 1 - = Extrapolation
Polynomial Interpolation Simplest and commonest type of interpolation uses polynomials. Unique polynomial of degree at most n – 1 passes through n data points ( t i , y i ), i = 1, . . . , n , where t i are distinct. There are many ways to represent or compute polynomial, but in theory all must give same result.

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

View Full Document
Lagrange
Lagrange Interpolation For given set of data points ( t i , y i ), i = 1, . . ., n , Lagrange basis functions are given by ) x )...(x x )(x x )...(x x )(x x (x ) x )...(x x )(x x )...(x x )(x x (x (x) l n j 1 j j 1 j j 2 j 1 j n 1 j 1 j 2 1 j - - - - - - - - - - = + - + - ) x (x n - which means that matrix of linear system Ax = y is identity. Thus, Lagrange polynomial interpolating data points ( t i , y i ) is given by p n –1 ( t ) = y 1 l 1 ( t ) + y 2 l 2 ( t ) + ... + y n l n ( t ). ) x (x k j k j k 1, k - = = = π

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

View Full Document
Examples: Lagrange Interpolation 1- Use Lagrange interpolation to find interpolating polynomial for three data points (–2, –27), (0, –1), 1, 0). Lagrange polynomial of degree two interpolating three points ( t 1 , y 1 ), ( t 2 , y 2 ), ( t 3 , y 3 ) is .
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern