nodes are not equally spaced and tend to cluster toward the end points of the

# Nodes are not equally spaced and tend to cluster

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nodes are not equally spaced and tend to cluster toward the end points of the interval. 3.2 Connection to Best Uniform Approxima- tion Given a continuous function f in [ a, b ], its best uniform approximation p * n in P n is characterized by an error, e n = f - p * n , which equioscillates, as defined in (2.40), at least n + 2 times. Therefore e n has a minimum of n + 1 zeros and consequently, there exists x 0 , . . . , x n such that p * n ( x 0 ) = f ( x 0 ) , p * n ( x 1 ) = f ( x 1 ) , . . . p * n ( x n ) = f ( x n ) , (3.15) In other words, p * n is the polynomial of degree at most n that interpolates the function f at n + 1 zeros of e n . Of course, we do not construct p * n by finding these particular n +1 interpolation nodes. A more practical question is: given ( x 0 , f ( x 0 )) , ( x 1 , f ( x 1 )) , . . . , ( x n , f ( x n )), where x 0 , . . . , x n are distinct interpolation nodes in [ a, b ], how close is p n , the interpolating polynomial of degree at most n of f at the given nodes, to the best uniform approximation p * n of f in P n ? To obtain a bound for k p n - p * n k we note that p n - p * n is a polynomial of degree at most n which interpolates f - p * n . Therefore, we can use Lagrange formula to represent it p n ( x ) - p * n ( x ) = n X j =0 l j ( x )( f ( x j ) - p * n ( x j )) . (3.16) It then follows that k p n - p * n k Λ n k f - p * n k , (3.17)
42 CHAPTER 3. INTERPOLATION where Λ n = max a x b n X j =0 | l j ( x ) | (3.18) is called the Lebesgue Constant and depends only on the interpolation nodes, not on f . On the other hand, we have that k f - p n k = k f - p * n - p n + p * n k ≤ k f - p * n k + k p n - p * n k . (3.19) Using (3.17) we obtain k f - p n k (1 + Λ n ) k f - p * n k . (3.20) This inequality connects the interpolation error k f - p n k with the best approximation error k f - p * n k . What happens to these errors as we increase n ? To make it more concrete, suppose we have a triangular array of nodes as follows: x (0) 0 x (1) 0 x (1) 1 x (2) 0 x (2) 1 x (2) 2 . . . x ( n ) 0 x ( n ) 1 . . . x ( n ) n . . . (3.21) where a x ( n ) 0 < x ( n ) 1 < · · · < x ( n ) n b for n = 0 , 1 , . . . . Let p n be the interpolating polynomial of degree at most n of f at the nodes corresponding to the n + 1 row of (3.21). By the Weierstrass Approximation Theorem ( p * n is a better approxima- tion or at least as good as that provided by the Bernstein polynomial), k f - p * n k 0 as n → ∞ . (3.22) However, it can be proved that Λ n > 2 π 2 log n - 1 (3.23)
3.3. BARYCENTRIC FORMULA 43 and hence the Lebesgue constant is not bounded in n . Therefore, we cannot conclude from (3.20) and (3.22) that k f - p n k as n → ∞ , i.e. that the interpolating polynomial, as we add more and more nodes, converges uni- formly to f . That depends on the regularity of f and on the distribution of the nodes. In fact, if we are given the triangular array of interpolation nodes (3.21) in advance, it is possible to construct a continuous function f such that p n will not converge uniformly to f as n → ∞ . 3.3 Barycentric Formula The Lagrange form of the interpolating polynomial is not convenient for com- putations. If we want to increase the degree of the polynomial we cannot reuse the work done in getting and evaluating a lower degree one. How- ever, we can obtain a very efficient formula by rewriting the interpolating polynomial in the following way. Let