# ch3 - Chapter 3 Interpolation and Polynomial Approximation...

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Chapter 3 Interpolation and Polynomial Approximation Per-Olof Persson [email protected] Department of Mathematics University of California, Berkeley Math 128A Numerical Analysis
Polynomial Interpolation Polynomials Polynomials P n ( x ) = a x x n + · · · a 1 x + a 0 are commonly used for interpolation or approximation of functions Benefits include efficient methods, simple differentiation, and simple integration Also, Weierstrass Approximation Theorem says that for each ε > 0 , there is a P ( x ) such that | f ( x ) - p ( x ) | < ε for all x in [ a, b ] for f C [ a, b ] . In other words, polynomials are good at approximating general functions.
The Lagrange Polynomial Theorem If x 0 , . . . , x n distinct and f given at these numbers, a unique polynomial P ( x ) of degree n exists with f ( x k ) = P ( x k ) , for each k = 0 , 1 , . . . , n The polynomial is P ( x ) = f ( x 0 ) L n, 0 ( x ) + . . . + f ( x n ) L n,n ( x ) = n X k =0 f ( x k ) L n,k ( x ) where L n,k ( x ) = ( x - x 0 )( x - x 1 ) · · · ( x - x k - 1 )( x - x k +1 ) · · · ( x - x n ) ( x k - x 0 )( x k - x 1 ) · · · ( x k - x k - 1 )( x k - x k +1 ) · · · ( x k - x n ) = Y i 6 = k ( x - x i ) ( x k - x i )
Lagrange Polynomial Error Term Theorem x 0 , . . . , x n distinct in [ a, b ] , f C n +1 [ a, b ] , then for x [ a, b ] there exists ξ ( x ) in ( a, b ) with f ( x ) = P ( x ) + f ( n +1) ( ξ ( x )) ( n + 1)! ( x - x 0 )( x - x 1 ) · · · ( x - x n ) where P ( x ) is the interpolating polynomial.
Recursive Generation of Lagrange Polynomials Definition Let f be defined at x 0 , . . . , x n , suppose m 1 , . . . , m k distinct integers with 0 m i n . The Lagrange polynomial that agrees with f ( x ) at x m 1 , x m 2 , . . . , x m k is denoted P m 1 ,m 2 ,...,m k ( x ) .