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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Hello, kindly assist me with the solution to Q1. I will appreciate it if you provide a very detailed solution, thanks
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
Expert Solution
Trending now
This is a popular solution!
Step by step
Solved in 10 steps with 5 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
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…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
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…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
