Teaching MPPSC Assistant Professor Mock Test Series 2025 Mathematical Science Numerical Analysis Rate of Convergence
Let f be an infinitely differentiable real-valued function on a bounded interval I. Take n ≥ 1 interpolation points {x0, x1, ....., xn-1}. Take n additional interpolation points
xn+j = xj + ε, j = 0, 1, ....., n - 1
where ε > 0 is such that {x0, x1, ....., x2n-1} are all distinct.
Let p2n-1 be the Lagrange interpolation polynomial of degree 2n - 1 with the interpolation points {x0, x1, ....., x2n-1} for the function f.
Let q2n-1 be the Hermite interpolation polynomial of degree 2n - 1 with the interpolation points {x0, x1, ....., xn-1} for the function f. In the ε → 0 limit, the quantity
\(\sup _{x \in 1}\left|p_{2 n-1}(x)-q_{2 n-1}(x)\right|\)
1
does not necessarily converge
2
converges to \(\frac{1}{2 n}\)
3
converges to 0
4
converges to \(\frac{1}{2 n+1}\)