In fact it diverged quite dramatically toward the end points of the interval On

In fact it diverged quite dramatically toward the end

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. In fact it diverged quite dramatically toward the end points of the interval. On the other hand, we noticed fast and uniform convergence of p n to f when the nodes were given by x j = cos j n , j = 0 , . . . , n. (3.48)
44 CHAPTER 3. INTERPOLATION These points are called Chebyshev nodes or Gauss-Lobatto nodes and are the preferred choice in practice when high order, very accurate polynomial interpolation is needed. It is then natural to ask: Given any f 2 C [ a, b ], can we guarantee that if we choose the Chebyshev nodes k f - p n k 1 ! 0? The answer is no. Bernstein and Faber proved in 1914 that given any distribution of points, organized in a triangular array as x (0) 0 x (1) 0 , x (1) 1 x (2) 0 , x (2) 1 , x (2) 2 . . . (3.49) it is possible to construct a continuous function f for which its interpolating polynomial p n (corresponding to the nodes on the n -th row of (3.49)) will not converge uniformly to f as n ! 1 . However, if f is slightly smoother, for example f 2 C 1 [ a, b ], then for the Chebyshev array of nodes k f - p n k 1 ! 0. In one of the homework examples f ( x ) = e - x 2 and we noticed convergence of p n even with the equidistributed nodes. What is so special about this function? The function f ( z ) = e - z 2 , (3.50) z = x + iy is analytic in the entire complex plane. Using complex variables analysis it can be shown that if f is analytic in a su ffi ciently large region in the complex plane containing [ a, b ] then k f - p n k 1 ! 0. Just how large the region of analyticity needs to be? it depends on the asymptotic distribution of the nodes as n ! 1 . In the limit as n ! 1 , we can think of the nodes as a continuum with a density so that ( n + 1) Z x a ( t ) dt (3.51) is the total number of nodes in [ a, x ]. Take for example, [ - 1 , 1]. For equidistributed nodes ( x ) = 1 / 2 and for the Chebyshev nodes ( x ) = 1 / ( p 1 - x 2 ). It turns out that the relevant domain of analyticity is given in terms of the function φ ( z ) = - Z b a ( t ) ln | z - t | dt. (3.52)
3.8. PIECE-WISE LINEAR INTERPOLATION 45 Let Γ c be the level curve consisting of all the points z 2 C such that φ ( z ) = c for c constant. For very large and negative c , Γ c approximates a large circle. As c is increased, Γ c shrinks. We take the “smallest” level curve, Γ c 0 , which contains [ a, b ]. The relevant domain of analyticity is R c 0 = { z 2 C : φ ( z ) c 0 } . (3.53) Then, if f is analytic in R c 0 , k f - p n k 1 ! 0, not only in [ a, b ] but for at point in R c 0 . Moreover, | f ( x ) - p n ( x ) | Ce - N ( φ ( x ) - c 0 ) , (3.54) for some constant C . That is p n converges exponentially fast to f . For the Chebyshev nodes R c 0 approximates [ a, b ], so if f is analytic in any region containing [ a, b ], however thin this region might be, p n will converge uniformly to f . For equidistributed nodes, R c 0 looks like a football, with [ a, b ] as its longest axis. In the Runge example, the function is singular at z = ± i/ 5 which happens to be inside this football-like domain and this explain the observed lack of convergence for this particular function.

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