(a) Let xo, ₁,..., n be n+1 distinct points in [a, b], with co= a and cn = b, and f € C+¹[a, b]. Let P(x) = Po...(x) be the Lagrange polynomial interpolating the points to, 71, ..., that P(x) = f(x₁), for all i = 0, 1,..., n. In, such Express f(x) in terms of P(x) and a remainder term (the error formula). (b) Consider the case n = 2 and the data x To 0 ₁ 1 ₂ = 2 0 In 2 In 3 f(x) = ln(x+1) (i) Find polynomials L₁(x), i = 0, 1, 2, such that L,(x₁) = 1 and L₁(x) = 0 if i ‡ j. (ii) Deduce the Lagrange polynomial P(x)= Po,1,2(x) interpolating this data points. (iii) Write the error formula and find a bound for the error f(0.5) - P(0.5).
(a) Let xo, ₁,..., n be n+1 distinct points in [a, b], with co= a and cn = b, and f € C+¹[a, b]. Let P(x) = Po...(x) be the Lagrange polynomial interpolating the points to, 71, ..., that P(x) = f(x₁), for all i = 0, 1,..., n. In, such Express f(x) in terms of P(x) and a remainder term (the error formula). (b) Consider the case n = 2 and the data x To 0 ₁ 1 ₂ = 2 0 In 2 In 3 f(x) = ln(x+1) (i) Find polynomials L₁(x), i = 0, 1, 2, such that L,(x₁) = 1 and L₁(x) = 0 if i ‡ j. (ii) Deduce the Lagrange polynomial P(x)= Po,1,2(x) interpolating this data points. (iii) Write the error formula and find a bound for the error f(0.5) - P(0.5).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![(a) Let o, 1,..., n be n+1 distinct points in [a, b], with ro= a and cn = b, and f € C+¹[a, b].
Let P(x) = Po...(x) be the Lagrange polynomial interpolating the points ro, 1, ...,
that P(x) = f(x₁), for all i = 0, 1,..., n..
In, such
Express f(x) in terms of P(x) and a remainder term (the error formula).
(b) Consider the case n = 2 and the data
x
To 0 ₁ 1
0 In 2
₂ = 2
In 3
f(x) = ln(x+1)
(i) Find polynomials L, (x), i = 0, 1, 2, such that L₁(x) = 1 and L₁(x) = 0 if i ‡ j.
(ii) Deduce the Lagrange polynomial P(x)= Po,1,2(x) interpolating this data points.
(iii) Write the error formula and find a bound for the error f(0.5) - P(0.5).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fda0442f0-082c-40d3-95ab-8cd5dccad2c3%2F558a0785-ddc7-41a4-83cf-713e186db10c%2Fpm7tu5d_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(a) Let o, 1,..., n be n+1 distinct points in [a, b], with ro= a and cn = b, and f € C+¹[a, b].
Let P(x) = Po...(x) be the Lagrange polynomial interpolating the points ro, 1, ...,
that P(x) = f(x₁), for all i = 0, 1,..., n..
In, such
Express f(x) in terms of P(x) and a remainder term (the error formula).
(b) Consider the case n = 2 and the data
x
To 0 ₁ 1
0 In 2
₂ = 2
In 3
f(x) = ln(x+1)
(i) Find polynomials L, (x), i = 0, 1, 2, such that L₁(x) = 1 and L₁(x) = 0 if i ‡ j.
(ii) Deduce the Lagrange polynomial P(x)= Po,1,2(x) interpolating this data points.
(iii) Write the error formula and find a bound for the error f(0.5) - P(0.5).
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