[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].
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter4: Polynomial And Rational Functions
Section4.5: Rational Functions
Problem 54E
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Question
Gaussian Quadrature
![[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
L',
g(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(u) du = 9 (1) + 9 (13)
g(u)du
Use (1) and your answer from part (a) to approximate
2
a
log(x) dx
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
dx
[*1(x)dx= [(z(u)) == du
du
(1)
=
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].](/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:[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
L',
g(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(u) du = 9 (1) + 9 (13)
g(u)du
Use (1) and your answer from part (a) to approximate
2
a
log(x) dx
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
dx
[*1(x)dx= [(z(u)) == du
du
(1)
=
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].
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