Problem 1. Short problems: 1. True or False? Gaussian quadrature with 3 integration nodes is exact for polynomials of degree 7. 2. A Gaussian quadrature with how many points is required to integrate polynomials of degree 10 exactly? 3. What is the result obtained by using Simpson's rule to integrate f(x) = x² over [0, 1]? 4. True or False? For the nodes xo = = 0, x₁ = 1, and x2 = 2, the Lo(x) = 1- x²? 5. True or False? Let pn be the degree n Lagrange polynomial interpolating f. The interpolation error ||f - Pn||[a,b] always tends to zero as n → ∞. 6. How do you get the monomial basis coefficients of the cardinal basis coefficients? 7. Let (qo,..., 9n) = Pn and let ƒ € L²([a, b]). Let (f, g) denote the inner product on L²([a, b]). Write the linear system you should solve to find the coefficients of the polynomial in Pn approximating f in the qm basis with the minimum L2 norm.
Problem 1. Short problems: 1. True or False? Gaussian quadrature with 3 integration nodes is exact for polynomials of degree 7. 2. A Gaussian quadrature with how many points is required to integrate polynomials of degree 10 exactly? 3. What is the result obtained by using Simpson's rule to integrate f(x) = x² over [0, 1]? 4. True or False? For the nodes xo = = 0, x₁ = 1, and x2 = 2, the Lo(x) = 1- x²? 5. True or False? Let pn be the degree n Lagrange polynomial interpolating f. The interpolation error ||f - Pn||[a,b] always tends to zero as n → ∞. 6. How do you get the monomial basis coefficients of the cardinal basis coefficients? 7. Let (qo,..., 9n) = Pn and let ƒ € L²([a, b]). Let (f, g) denote the inner product on L²([a, b]). Write the linear system you should solve to find the coefficients of the polynomial in Pn approximating f in the qm basis with the minimum L2 norm.
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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![Problem 1. Short problems:
1. True or False? Gaussian quadrature with 3 integration nodes is exact for polynomials of degree 7.
2. A Gaussian quadrature with how many points is required to integrate polynomials of degree 10 exactly?
3. What is the result obtained by using Simpson's rule to integrate f(x) = x² over [0, 1]?
4. True or False? For the nodes xo
=
= 0, x₁ = 1, and ₂
2, the Lo(x) = 1 − x²?
=
5. True or False? Let pn be the degree n Lagrange polynomial interpolating f. The interpolation error
||f - Pn||[a,b] always tends to zero as n → ∞.
6. How do you get the monomial basis coefficients of the cardinal basis coefficients?
7. Let (90,..., an)
Pn and let ƒ € L²([a, b]). Let (f, g) denote the inner product on L²([a, b]). Write
the linear system you should solve to find the coefficients of the polynomial in Pª approximating ƒ in
the qm basis with the minimum L² norm.
=](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F725ee8ea-573b-4075-8ffe-8d3f36956d72%2F273b601b-a81a-4c86-b2bf-a0071a87bed6%2Fhwa515_processed.png&w=3840&q=75)
Transcribed Image Text:Problem 1. Short problems:
1. True or False? Gaussian quadrature with 3 integration nodes is exact for polynomials of degree 7.
2. A Gaussian quadrature with how many points is required to integrate polynomials of degree 10 exactly?
3. What is the result obtained by using Simpson's rule to integrate f(x) = x² over [0, 1]?
4. True or False? For the nodes xo
=
= 0, x₁ = 1, and ₂
2, the Lo(x) = 1 − x²?
=
5. True or False? Let pn be the degree n Lagrange polynomial interpolating f. The interpolation error
||f - Pn||[a,b] always tends to zero as n → ∞.
6. How do you get the monomial basis coefficients of the cardinal basis coefficients?
7. Let (90,..., an)
Pn and let ƒ € L²([a, b]). Let (f, g) denote the inner product on L²([a, b]). Write
the linear system you should solve to find the coefficients of the polynomial in Pª approximating ƒ in
the qm basis with the minimum L² norm.
=
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