[1] [Gaussian quadrature over arbitrary intervals] In this exercise you will derive a change of variables so that the technique of Gaussian quadrature discussed in lecture can be applied to integrate a function f over an arbitrary interval ſå f(x)dx. (a) Define a linear function u(x) that maps [1,2] on to [−1,1]. Use the result to transform the integral into 2 log(x) dx L₁9(u) du What is g(u)? (b) The nodes and weights for the two-point Gaussian quadrature rule were derived using the ‘brute-force' method in HW6, exercise [5]; the approx- imation was given by [9(0)du = g() + 9 (3) g Use (1) and your answer from part (a) to approximate 2 f* log(a)da (1) and report the relative error. (c) Now derive a linear function u(x) that maps [a, b] on to [−1,1]. The map should have the property that u(a) = −1 and u(b) = 1. Use the result, as well as the change of variables (i.e. 'u-substitution') formula pb pu(b) dx [ f(x) dx = 0 F (x (1)) == du du u(a) g(u):= to express the integral ſå ƒ(x) dx over [a, b] as an integral over [−1,1]. What is g(u)? This is procedure one can use to enable the application of Gaussian quadrature to integrals over arbitrary intervals [a, b].
[1] [Gaussian quadrature over arbitrary intervals] In this exercise you will derive a change of variables so that the technique of Gaussian quadrature discussed in lecture can be applied to integrate a function f over an arbitrary interval ſå f(x)dx. (a) Define a linear function u(x) that maps [1,2] on to [−1,1]. Use the result to transform the integral into 2 log(x) dx L₁9(u) du What is g(u)? (b) The nodes and weights for the two-point Gaussian quadrature rule were derived using the ‘brute-force' method in HW6, exercise [5]; the approx- imation was given by [9(0)du = g() + 9 (3) g Use (1) and your answer from part (a) to approximate 2 f* log(a)da (1) and report the relative error. (c) Now derive a linear function u(x) that maps [a, b] on to [−1,1]. The map should have the property that u(a) = −1 and u(b) = 1. Use the result, as well as the change of variables (i.e. 'u-substitution') formula pb pu(b) dx [ f(x) dx = 0 F (x (1)) == du du u(a) g(u):= to express the integral ſå ƒ(x) dx over [a, b] as an integral over [−1,1]. What is g(u)? This is procedure one can use to enable the application of Gaussian quadrature to integrals over arbitrary intervals [a, b].
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Gaussian Quadrature
![### Gaussian Quadrature over Arbitrary Intervals
In this exercise, you will derive a change of variables to use the Gaussian quadrature technique to integrate a function \( f \) over an arbitrary interval \(\int_a^b f(x)dx\).
#### (a) Linear Function Definition
- **Objective**: Define a linear function \( u(x) \) that maps \([1, 2]\) onto \([-1, 1]\).
- **Application**: Use this mapping to transform the integral
\[
\int_1^2 \log(x)dx
\]
into
\[
\int_{-1}^1 g(u) \, du.
\]
- **Question**: What is \( g(u) \)?
#### (b) Two-Point Gaussian Quadrature
- **Method**: Nodes and weights for the two-point Gaussian quadrature rule were derived using the ‘brute-force’ method in HW6, exercise [5].
- **Approximation Formula**:
\[
\int_{-1}^1 g(u)du \approx g\left( -\frac{1}{\sqrt{3}} \right) + g\left( \frac{1}{\sqrt{3}} \right).
\]
- **Task**: Use the above approximation and your result from part (a) to approximate
\[
\int_1^2 \log(x)dx
\]
and report the relative error.
#### (c) Derivation of a Linear Function
- **Task**: Derive a linear function \( u(x) \) that maps \([a, b]\) onto \([-1, 1]\) with \( u(a) = -1 \) and \( u(b) = 1 \).
- **Integration Formula**:
\[
\int_a^b f(x)dx = \int_{u(a)}^{u(b)} \underbrace{f(x(u)) \frac{dx}{du}}_{g(u) :=} \, du
\]
- **Objective**: Express the integral \(\int_a^b f(x) \, dx\) over \([a, b]\) as an integral over \([-1, 1]\). Determine \( g(u) \).
This procedure enables the application of Gaussian](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7fd47556-f3ce-4f39-818d-be563d9523c8%2F876885ba-d449-4ef2-961d-141c912bd361%2Fhpkcdgm_processed.png&w=3840&q=75)
Transcribed Image Text:### Gaussian Quadrature over Arbitrary Intervals
In this exercise, you will derive a change of variables to use the Gaussian quadrature technique to integrate a function \( f \) over an arbitrary interval \(\int_a^b f(x)dx\).
#### (a) Linear Function Definition
- **Objective**: Define a linear function \( u(x) \) that maps \([1, 2]\) onto \([-1, 1]\).
- **Application**: Use this mapping to transform the integral
\[
\int_1^2 \log(x)dx
\]
into
\[
\int_{-1}^1 g(u) \, du.
\]
- **Question**: What is \( g(u) \)?
#### (b) Two-Point Gaussian Quadrature
- **Method**: Nodes and weights for the two-point Gaussian quadrature rule were derived using the ‘brute-force’ method in HW6, exercise [5].
- **Approximation Formula**:
\[
\int_{-1}^1 g(u)du \approx g\left( -\frac{1}{\sqrt{3}} \right) + g\left( \frac{1}{\sqrt{3}} \right).
\]
- **Task**: Use the above approximation and your result from part (a) to approximate
\[
\int_1^2 \log(x)dx
\]
and report the relative error.
#### (c) Derivation of a Linear Function
- **Task**: Derive a linear function \( u(x) \) that maps \([a, b]\) onto \([-1, 1]\) with \( u(a) = -1 \) and \( u(b) = 1 \).
- **Integration Formula**:
\[
\int_a^b f(x)dx = \int_{u(a)}^{u(b)} \underbrace{f(x(u)) \frac{dx}{du}}_{g(u) :=} \, du
\]
- **Objective**: Express the integral \(\int_a^b f(x) \, dx\) over \([a, b]\) as an integral over \([-1, 1]\). Determine \( g(u) \).
This procedure enables the application of Gaussian
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