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A NEAR-MINIMAX APPROXIMATION METHOD Let f(x) be continuous on [a, b] = [-1, 1]. Consider
 

Summary: A NEAR-MINIMAX APPROXIMATION METHOD
Let f(x) be continuous on [a, b] = [-1, 1]. Consider
approximating f by an interpolatory polynomial of de-
gree at most n = 3. Let x0, x1, x2, x3 be interpo-
lation node points in [-1, 1]; let c3(x) be of degree
3 and interpolate f(x) at {x0, x1, x2, x3}. The in-
terpolation error is
f(x) - c3(x) =
(x)
4!
f(4)(x), -1 x 1 (1)
(x) = (x - x0)(x - x1)(x - x2)(x - x3) (2)
with x in [-1, 1]. We want to choose the nodes
{x0, x1, x2, x3} so as to minimize the maximum value
of |f(x) - c3(x)| on [-1, 1].
From (1), the only general quantity, independent of f,
is (x). Thus we choose {x0, x1, x2, x3} to minimize
max
-1x1
|(x)| (3)

  

Source: Atkinson, Kendall - Departments of Computer Science & Mathematics, University of Iowa

 

Collections: Computer Technologies and Information Sciences