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The quadrature rule [“ f(x)dx = w₁ f(x1) + w2f(x2) + E (2) -h can be made exact for all cubic polynomials, by using a suitable choice of weights and points, respectively wi 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?

The quadrature rule [“ f(x)dx = w₁ f(x1) + w2f(x2) + E (2) -h can be made exact for all cubic polynomials, by using a suitable choice of weights and points, respectively wi 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
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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The quadrature rule [”, f(x)dx = w₁ f(x1) +w2f(x2) + E (2) -h can be made exact for all cubic polynomials, by using a suitable choice of weights and points, respectively w₁ 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?
Transcribed Image Text:The quadrature rule [”, f(x)dx = w₁ f(x1) +w2f(x2) + E (2) -h can be made exact for all cubic polynomials, by using a suitable choice of weights and points, respectively w₁ 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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