N π k 1 2 n that are located in 1 1 2 at z i cos iπ

This preview shows page 27 - 30 out of 36 pages.

n π , k = 1 , 2 , . . . , n that are located in ( - 1 , 1); (2) at z i = cos n , i = 0 , 1 , 2 , . . . , n the normalized Chebyshev polynomial ˜ T n ( x ) takes its extremal values ± 1 2 n - 1 alternatingly ( n + 1) times, i.e., ˜ T n ( z i ) = ( - 1) i 2 n - 1 , i = 0 , 1 , . . . , n. (3) The maximum value of ˜ T n ( x ) in [ - 1 , 1] is 1 2 n - 1 , i.e., max x [ - 1 , 1] | ˜ T n ( x ) | = 1 2 n - 1 . 27
Minimization of the approximation error of Lagrange polynomial interplant with respect to the max distance in the interval [ - 1 , 1] . If we use Lagrange interpolating polynomials as approximants, then a natural question is to try to choose the interpolat- ing nodes such that the approximation error is minimal. Suppose that we interpolate a function f ( x ) by its n -th Lagrange interpolating polynomial P n ( x ) based on n + 1 distinct interpolating nodes x 0 , x 1 , . . . , x n in [-1,1]. Then the approximation (interpolating) error is max x [ - 1 , 1] | f ( x ) - P n ( x ) | ≤ M n +1 ( n + 1)! max x [ - 1 , 1] | ( x - x 0 )( x - x 1 ) · · · ( x - x n ) | , where | f ( n +1 ( x ) | ≤ M n +1 , x [ - 1 , 1]. Now, our goal is to choose the interpolating nodes x * 0 , x * 1 , . . . , x * n such that min x 0 ,x 1 ,...,x n max x [ - 1 , 1] | ( x - x 0 )( x - x 1 ) · · · ( x - x n ) | = max x [ - 1 , 1] | ( x - x * 0 )( x - x * 1 ) · · · ( x - x * n ) | . In other words, we wish to find (to determine) interpolating nodes that minimize the approximation error and to use them in interpolating processes in order to obtain Lagrangian interpolating approximants having a minimal approximation error. Theorem 5.1. Given a function f ( x ) defined on the interval [ - 1 , 1] . Consider ˜ T n +1 ( x ) and denote its n + 1 zeros by x * 0 , x * 1 , . . . , x * n . Denote by P * n ( x ) the n -th Lagrange interpolating polynomial to f ( x ) by using as interpolating nodes the zeros of ˜ T n +1 ( x ) : x * 0 , x * 1 , . . . , x * n . Then, P * n ( x ) minimizes the approximation error in the interval [ - 1 , 1] and max x [ - 1 , 1] | f ( x ) - P * n ( x ) | ≤ M n +1 ( n + 1)! max x [ - 1 , 1] | ( x - x * 0 )( x - x * 1 ) · · · ( x - x * n ) | = M n +1 ( n + 1)! max x [ - 1 , 1] | ˜ T n +1 ( x ) | = M n +1 ( n + 1)! 1 2 n or in a concise form max x [ - 1 , 1 ] | f ( x ) - P * n ( x ) | ≤ M n + 1 ( n + 1 )! 1 2 n , where | f ( n + 1 ) ( x ) | ≤ M n + 1 for all x [ - 1 , 1 ] . Remark. The zeros x * 0 , x * 1 , . . . , x * n of T n +1 ( x ) are called Chebyshev interpolating nodes in the interval [ - 1 , 1] and the interpolation by using the Chebyshev in- terpolating nodes is called Chebyshev interpolation. 28
Proof of Theorem 5.1. Suppose to the contrary, that for some other n + 1 interpolating nodes x 0 , x 1 , . . . , x n the interpolating error is smaller, i.e., max x [ - 1 , 1] | ( x - x 0 )( x - x 1 ) · · · ( x - x n ) | < max x [ - 1 , 1] | ( x - x * 0 )( x - x * 1 ) · · · ( x - x * n ) | = 1 2 n , (1) where ( x - x * 0 )( x - x * 1 ) · · · ( x - x * n ) = ˜ T n +1 ( x ) = x n + q n ( x ) . Obviously Q n +1 ( x ) = ( x - x 0 )( x - x 1 ) · · · ( x - x n ) = x n +1 + r n ( x ) ˜ T n +1 ( x ) = ( x - x * 0 )( x - x * 1 ) · · · ( x - x * n ) = ˜ T n +1 ( x ) = x n +1 + q n ( x ) . Then ( x - x * 0 )( x - x * 1 ) · · · ( x - x * n ) - ( x - x 0 )( x - x 1 ) · · · ( x - x n ) = ˜ T n +1 ( x ) - Q n +1 ( x ) = q n ( x ) - r n ( x ) is a polynomial of degree n and at the n + 2 extremal points of ˜ T n +1 ( x ) it takes alternat- ingly positive and negative value. By the IVT ˜ T n +1 ( x ) - Q n +1 ( x ) has at least n + 1 zeros and being a polynomial of degree n from here ˜ T n +1 ( x ) - Q n +1 ( x ) 0 or equivalently ˜ T n +1 ( x ) Q n +1 ( x ) that is not possible because of (1).

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture