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 Q2. I will appreciate it if you provide a very detailed solution, thanks
![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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0282592e-adf8-40c1-9d31-7bca34586c75%2F47efe68a-5082-45ac-9149-e0135d707b5b%2Fil9v1gk.png&w=3840&q=75)
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
![](/static/compass_v2/shared-icons/check-mark.png)
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 4 steps with 4 images
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
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](https://www.bartleby.com/isbn_cover_images/9780470458365/9780470458365_smallCoverImage.gif)
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
![Numerical Methods for Engineers](https://www.bartleby.com/isbn_cover_images/9780073397924/9780073397924_smallCoverImage.gif)
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…](https://www.bartleby.com/isbn_cover_images/9781118141809/9781118141809_smallCoverImage.gif)
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
![Advanced Engineering Mathematics](https://www.bartleby.com/isbn_cover_images/9780470458365/9780470458365_smallCoverImage.gif)
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
![Numerical Methods for Engineers](https://www.bartleby.com/isbn_cover_images/9780073397924/9780073397924_smallCoverImage.gif)
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…](https://www.bartleby.com/isbn_cover_images/9781118141809/9781118141809_smallCoverImage.gif)
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
![Mathematics For Machine Technology](https://www.bartleby.com/isbn_cover_images/9781337798310/9781337798310_smallCoverImage.jpg)
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
![Basic Technical Mathematics](https://www.bartleby.com/isbn_cover_images/9780134437705/9780134437705_smallCoverImage.gif)
![Topology](https://www.bartleby.com/isbn_cover_images/9780134689517/9780134689517_smallCoverImage.gif)