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
icon
Related questions
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
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
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 4 steps with 80 images

Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,