Problem 7.14. Let (æ1,...,am+1) be a sequence of pairwise distinct scalars in R and let (B1,...,Bm+1) be any sequence of scalars in R, not necessarily distinct. (1) Prove that there is a unique polynomial P of degree at most m such that P(a4) Bi, 1i
Problem 7.14. Let (æ1,...,am+1) be a sequence of pairwise distinct scalars in R and let (B1,...,Bm+1) be any sequence of scalars in R, not necessarily distinct. (1) Prove that there is a unique polynomial P of degree at most m such that P(a4) Bi, 1i
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Transcribed Image Text:Problem 7.14. Let (æ1,...,am+1) be a sequence of pairwise distinct scalars in R and let
(B1,...,Bm+1) be any sequence of scalars in R, not necessarily distinct.
(1) Prove that there is a unique polynomial P of degree at most m such that
P(a4) Bi, 1i <m+1.
Hint. Remember Vandermonde!
(2) Let Li(X) be the polynomial of degree m given by
(X - a1)(X- a1-1)(X -a441) (X -am+1)
(a-a1)(a- ai-1)(a-ai+1)
1 im
L&(X)
(a4-am+1)
The polynomials L'(X)
Lagrange polynomial interpolants. Prove that
are known as
Li(aj) 6 1 i,j <m+1
Prove that
Bm+1 Lm+1(X)
Р(X) — BiL1(X) +
is the unique polynomial of degree at most m such that
P(a4) Bi, 1< i <m+1
(3) Prove that L1(X),..., Lm+1(X) are linearly independent, and that they form a basis
of all polynomials of degree at most m
How is 1 (the constant polynomial 1) expressed
over the basis (L1(X),..., Lm+1(X) )?
Give the expression of every polynomial P(X) of degree at most m over the basis
(L1(X), ...,m+1(X)
(4) Prove that the dual basis (Li, ..., L1)of the basis (L1(X),..., Lm+1(X)) consists
of the linear forms L given by
L;(P) P(a)
for every polynomial P of degree at most m; this is simply evaluation at a
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