For every absolutely continuous function on the sequence of interpolating polynomials constructed on Chebyshev nodes converges to " f " ( " x " ) uniformly.
12.
The disadvantage of Lagrange representation is that any additional point included will increase the order of the interpolating polynomial, leading to the need of recomputing all the fundamental polynomials.
13.
Does there exist a single table of nodes for which the sequence of interpolating polynomials converge to any continuous function " f " ( " x " )?
14.
Moreover, the interpolating polynomial is unique if and only if the number of adjustable coefficients is equal to the number of data points, i . e ., " N " = 2 " K " + 1.
15.
Lyness and Moler showed in 1966 that using undetermined coefficients for the polynomials in Neville's algorithm, one can compute the Maclaurin expansion of the final interpolating polynomial, which yields numerical approximations for the derivatives of the function at the origin.
16.
There is some subtlety in how one treats the a _ { N } coefficient in the integral, however to avoid double-counting with its alias it is included with weight 1 / 2 in the final approximate integral ( as can also be seen by examining the interpolating polynomial ):
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