Question 1: (a) Consider the n-point Newton-Cotes quadrature rule n [^ f(x)dx ~ Σ wif(xi). i=1 (1) Give the general form of the expected error term. By considering a constant function, show that n Σω; = h. i=1 wi (b) The quadrature rule ["^ f(x)dx = w₁f (x1) + w₁f (x2) + E (2) can be made exact for all cubic polynomials, by using a suitable choice of weights and points, respectively w; and xi, i = {1, 2}. • Write down four equations satisfied by w₁, W2, x1 and x2. • Combine two of these equations to show that x2 = -x₁ and hence that w₁ = W2. • Calculate the weights and then the value of the quadrature points. ⚫ Find the error term and write the quadrature rule (2). What class of quadrature rules does it belong to?
Question 1: (a) Consider the n-point Newton-Cotes quadrature rule n [^ f(x)dx ~ Σ wif(xi). i=1 (1) Give the general form of the expected error term. By considering a constant function, show that n Σω; = h. i=1 wi (b) The quadrature rule ["^ f(x)dx = w₁f (x1) + w₁f (x2) + E (2) can be made exact for all cubic polynomials, by using a suitable choice of weights and points, respectively w; and xi, i = {1, 2}. • Write down four equations satisfied by w₁, W2, x1 and x2. • Combine two of these equations to show that x2 = -x₁ and hence that w₁ = W2. • Calculate the weights and then the value of the quadrature points. ⚫ Find the error term and write the quadrature rule (2). What class of quadrature rules does it belong to?
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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Transcribed Image Text:Question 1:
(a) Consider the n-point Newton-Cotes quadrature rule
n
[^ f(x)dx ~ Σ wif(xi).
i=1
(1)
Give the general form of the expected error term. By considering a constant function,
show that
n
Σω; = h.
i=1
wi
(b) The quadrature rule
["^ f(x)dx = w₁f (x1) + w₁f (x2) + E
(2)
can be made exact for all cubic polynomials, by using a suitable choice of weights and
points, respectively w; and xi, i = {1, 2}.
• Write down four equations satisfied by w₁, W2, x1 and x2.
• Combine two of these equations to show that x2
=
-x₁ and hence that w₁ = W2.
• Calculate the weights and then the value of the quadrature points.
⚫ Find the error term and write the quadrature rule (2). What class of quadrature
rules does it belong to?
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