(c) Verify that the Legendre polynomials {P2, P3, P4} are indeed mutually orthogonal.
(c) Verify that the Legendre polynomials {P2, P3, P4} are indeed mutually orthogonal.
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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Can you do Question 3(c)
![Question 3.
We would like to determine the expansion of
functions in terms of Legendre polynomials over a vector space £2[-1, 1] of complex-values
functions on the interval [-1, 1] together with the inner product (,) : L²[-1,1] → R defined
by
(9,1b) := [ 4(a)adx,
Consider the continuous analogue of the Vandermonde matrix
X = = [1|x|x²|x-¹]
with each column a continuous monomial function x L²[-1, 1] for j = 0, 1,..., n - 1 € No.
Suppose that it has QR factorisation X = QR where Q = [qo(x)|q1(x)| 9n-1(x)] and R =
[rir₂ rn].
(a) Determine an expression for the inner product (qi, q;) such that the matrix Q is unitary
(b) Use the Gram-Schmidt orthogonalisation framework to determine the first five polyno-
mials {90, 91, 92, 93, 94). How would you convert these into the Legendre polynomials
{Po, P1, P2, P3, P4}?
(c) Verify that the Legendre polynomials {P2, P3, P4} are indeed mutually orthogonal.
€²[1,1]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8bbfe58a-2e9f-4c9b-a6ba-b126483afbad%2Fef13947c-ee43-4b49-bf43-117173feeb3d%2Fy36lu4u_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Question 3.
We would like to determine the expansion of
functions in terms of Legendre polynomials over a vector space £2[-1, 1] of complex-values
functions on the interval [-1, 1] together with the inner product (,) : L²[-1,1] → R defined
by
(9,1b) := [ 4(a)adx,
Consider the continuous analogue of the Vandermonde matrix
X = = [1|x|x²|x-¹]
with each column a continuous monomial function x L²[-1, 1] for j = 0, 1,..., n - 1 € No.
Suppose that it has QR factorisation X = QR where Q = [qo(x)|q1(x)| 9n-1(x)] and R =
[rir₂ rn].
(a) Determine an expression for the inner product (qi, q;) such that the matrix Q is unitary
(b) Use the Gram-Schmidt orthogonalisation framework to determine the first five polyno-
mials {90, 91, 92, 93, 94). How would you convert these into the Legendre polynomials
{Po, P1, P2, P3, P4}?
(c) Verify that the Legendre polynomials {P2, P3, P4} are indeed mutually orthogonal.
€²[1,1]
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