Use Theorem 3.9 or Algorithm 3.3 to construct an approximating polynomial for the following data.
ALGORITHM 3.3
Hermite Interpolation
To obtain the coefficients of the Hermite interpolating polynomial H(x) on the (n + 1) distinct numbers x0, …, xn for the function f:
INPUT numbers x0, x1, …, xn; values f (x0), ... , f (xn) and f′ (x0), ... , f′ (xn).
OUTPUT the numbers Q0, 0, Q1, 1, … , Q2n + 1, 2n + 1 where
Step 1 For i = 0, 1, … , n do Steps 2 and 3.
Step 2
Step 3 If i ≠ 0 then set
Step 4 For i = 2, 3, … , 2n + 1
for j = 2, 3, ... , i set
Step 5 OUTPUT (Q0, 0, Q1, 1, … , Q2n + 1, 2n + 1);
STOP.
Theorem 3.9 If f ∈ C1 [a, b] and x0, …, xn ∈ [a, b] are distinct, the unique polynomial of least degree agreeing with f and f′ at x0, …, xn is the Hermite polynomial of degree at most 2n + 1 given by
where, for Ln, j (x) denoting the jth Lagrange coefficient polynomial of degree n, we have
Moreover, if f ∈ C2n + 2 [a, b], then
for some (generally unknown) ξ(x) in the interval (a, b).
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