Lecture Notes Course à MA 190 Numerical Mathematics, First ...

6.4.2 Lagrange’s formulaTheorem 6.4.1 (Lagrange’s formula) Let the n nodes t i be distinct. ThenLagrange’s formula for the interpolating polynomial Q readsQ(t) =n∑ P (t)nf(t i )(t − t i )P ′ (t i ) , P (t) = ∏(t − t i ). (6.4)i=1i=1Proof. SetUsing L’Hôpital’s rule we findp i (t) = P (t)t − t i.lim p i (t) = P ′ (t i ).t→t iFor k ≠ i, we conclude, that p i (t k ) = 0. Since the points t i are distinct, wehave that P ′ (t i ) ≠ 0. Also, p i is a polynomial of exact degree n − 1. Thus (6.4)defines the interpolating polynomial as claimed.bulletRemark 6.4.2 Explicit form of Lagrange’s formula in a special case: We writeQ(t) =We give L i for n = 4n∑P (t)f(t i )L i (t), L i (t) =(t − t i )P ′ (t i )i=1L 1 (t) =L 2 (t) =L 3 (t) =L 4 (t) =(t − t 2 )(t − t 3 )(t − t 4 )(t 1 − t 2 )(t 1 − t 3 )(t 1 − t 4 )(t − t 1 )(t − t 3 )(t − t 4 )(t 2 − t 1 )(t 2 − t 3 )(t 2 − t 4 )(t − t 1 )(t − t 2 )(t − t 4 )(t 3 − t 1 )(t 3 − t 2 )(t 3 − t 4 )(t − t 1 )(t − t 2 )(t − t 3 )(t 4 − t 1 )(t 4 − t 2 )(t 4 − t 3 )6.4.3 Newton’s formula with divided differencesDivided differences with distinct arguments.:We define the divided differences recursively as follows. One argument:f[t i ] = f(t i ), i = 1, 2, . . . , n.Two arguments:f[t i , t j ] = f[t j] − f[t i ].(t j − t i )44