Let P(x) be a polynomial that interpolates f(x) at x; for i = 0, 1,...,l−1,1+1, ---, n (i.e., excluding x₁) and Q(x) be another polynomial that interpolates f(x) at 2, for i = 0, 1,..., m - 1, m+ 1,,n (i.e., excluding m). Suppose l m (and of course 0≤l, m≤n). (a) Show that the linear interpolation of these two polynomials I-Im R(x) = P(x)- I-IL Im - Il τι – Tm interpolates f(x) at x₁ for i = 0, 1, 2, ..., n (i.e., R(x₁) = f (x₁) for i = 0, 1, 2, ---,n). 1 + Q(x) (b) Can you conclude that R(x) is the unique polynomial (of degree at most n) that interpolates f(x) at x; for i = 0, 1, 2,...,n? Explain in your own words.
Let P(x) be a polynomial that interpolates f(x) at x; for i = 0, 1,...,l−1,1+1, ---, n (i.e., excluding x₁) and Q(x) be another polynomial that interpolates f(x) at 2, for i = 0, 1,..., m - 1, m+ 1,,n (i.e., excluding m). Suppose l m (and of course 0≤l, m≤n). (a) Show that the linear interpolation of these two polynomials I-Im R(x) = P(x)- I-IL Im - Il τι – Tm interpolates f(x) at x₁ for i = 0, 1, 2, ..., n (i.e., R(x₁) = f (x₁) for i = 0, 1, 2, ---,n). 1 + Q(x) (b) Can you conclude that R(x) is the unique polynomial (of degree at most n) that interpolates f(x) at x; for i = 0, 1, 2,...,n? Explain in your own words.
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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Question
![3:08
Today
10:51 AM
[0₂3].
Let P(x) be a polynomial that interpolates f(x) at x; for i = 0, 1, ...‚l−1,l+1, ···‚ n
(i.e., excluding x₁) and Q(x) be another polynomial that interpolates f(x) at x₁ for
i = 0, 1,..., m − 1, m+ 1, - -
,...,n (i.e., excluding m). Suppose l m (and of course
0 ≤ 1, m≤n).
lating points for ating points f
(a) Show that the linear interpolation of these two polynomials
R(x) = P(x)-
HIIL
Im-Il
x-xm
τι – Tm
interpolates f(x) at x; for i = 0, 1, 2, …,n (i.e., R(xi) = f(xi) for i = 0, 1, 2, ---, n).
ad P(z) that int
matrix and then interpolating
f(1.4)- P
hat interpolates range interp
hat interpolates and error (
f(z). What do
1
LTE
+ Q(x);
(b) Can you conclude that R(x) is the unique polynomial (of degree at most n) that
interpolates f(x) at x, for i = 0, 1, 2,..., n? Explain in your own words.
i
Edit
KOBALT
目
Lagrange inte
0,1,2. Sketch
range interpol
. Sketch the for](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6b50cde9-6f11-4c6d-8f35-b591071c68b1%2F863d6675-1805-45b1-bf58-5ebe258e03a0%2Faxskpy9_processed.png&w=3840&q=75)
Transcribed Image Text:3:08
Today
10:51 AM
[0₂3].
Let P(x) be a polynomial that interpolates f(x) at x; for i = 0, 1, ...‚l−1,l+1, ···‚ n
(i.e., excluding x₁) and Q(x) be another polynomial that interpolates f(x) at x₁ for
i = 0, 1,..., m − 1, m+ 1, - -
,...,n (i.e., excluding m). Suppose l m (and of course
0 ≤ 1, m≤n).
lating points for ating points f
(a) Show that the linear interpolation of these two polynomials
R(x) = P(x)-
HIIL
Im-Il
x-xm
τι – Tm
interpolates f(x) at x; for i = 0, 1, 2, …,n (i.e., R(xi) = f(xi) for i = 0, 1, 2, ---, n).
ad P(z) that int
matrix and then interpolating
f(1.4)- P
hat interpolates range interp
hat interpolates and error (
f(z). What do
1
LTE
+ Q(x);
(b) Can you conclude that R(x) is the unique polynomial (of degree at most n) that
interpolates f(x) at x, for i = 0, 1, 2,..., n? Explain in your own words.
i
Edit
KOBALT
目
Lagrange inte
0,1,2. Sketch
range interpol
. Sketch the for
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