GAUSS QUADRATURE RULEINTERACTIVE E-BOOK |
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Title |
An interactive e-book for illustrating Gauss Quadrature Rule for numerical integration. |
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Creator |
Autar K Kaw |
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Subject and Keywords |
Gauss Quadrature Rule, Numerical Integration, Mathcad, Maple, Mathematica, Matlab, Simulations. |
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Description |
An interactive E-book for illustrating Gauss Quadrature Rule. It includes links to simulations in Mathcad, Maple, Mathematica and Matlab for the algorithm, multiple choice quizzes, problem set, and a PowerPoint presentation. |
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Publisher |
Holistic Numerical Methods Institute, |
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Contributors |
Autar Kaw, Michael Keteltas |
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Format |
Text/HTML |
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Last Revised |
June 14, 2004 |
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Identifier |
http://numericalmethods.eng.usf.edu/ebooks/gaussquadrature_07int_ebook.pdf |
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Language |
English |
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Rights |
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Two-point Gaussian Quadrature Rule Higher point Gaussian Quadrature formulas Derivation of Gaussian Quadrature Rule Higher point Gaussian Quadrature formulas Argument and weighing factors for n-point Gauss Quadrature Rules Examples Example 1: Two-point Gauss Quadrature Example 2: Derive the 1-point Gauss Quadrature Example 3: Derivation of a formula Example 4: Two-point Gauss Quadrature application problem Example 5: Three-point Gauss Quadrature application problem Simulation Method ([MAPLE] [MATHCAD] [MATHEMATICA] [MATLAB]) Convergence ([MAPLE] [MATHCAD] [MATHEMATICA] [MATLAB]) Problem Sets |
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This is an example of how an instructor of a Numerical Methods course can develop an E-book on a single topic by using resources that are only available at the Holistic Numerical Methods Institute (HNMI) Website. Although one of the primary advantages of a web-based resource is that one can use documents outside of its own domain, we are particularly using this example as a way to illustrate the holistic nature of the resources available at HNMI. You are welcome to modify and edit this e-book to suit your purpose. Using the textbook document written in MS Word 2000 as a foundation, it was made interactive within MS Word 2000 by putting bookmarks and hyperlinks to background information, PowerPoint presentations, Mathcad simulations, historical anecdotes, multiple choice tests, problem sets, etc. Then the MS Word 2000 file was saved as a webpage and that is what you are about to read here. Back to TOC |
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Integration is the process of measuring the area under a function plotted on a graph. Why would we want to do so? Among the most common examples are finding the velocity of a body from acceleration functions, and displacement of a body from velocity data. Throughout the engineering fields, there are (what sometimes seems like) countless applications for integral calculus. Sometimes, the evaluation of expressions involving these integrals can become daunting, if not indeterminate. For this reason, a wide variety of numerical methods have been developed to simplify the integral. Here, we will discuss the Gauss Quadrature Rule of integral approximation.
Figure 1: Integration of a functionGAUSS QUADRATURE RULE: Background Previously, we developed Trapezoidal Rule and Simpson’s 1/3rd Rule by several methods. One of them was the method of undetermined coefficients. To derive Trapezoidal rule from the method of undetermined coefficients, we approximated
Let the right hand side be exact for integrals of a straight line, that is, for integrated form
So
But from Equation (1), we want
to give the same result as
Equation (2) for
Hence from Equations (2) and (4),
Since
Multiply Equation (5a) by a and subtracting from Equation (5b) gives
Substituting the above found
value of
Therefore
Derivation of two-point Gaussian Quadrature Rule The two-point
Gauss Quadrature Rule is an extension of the Trapezoidal Rule approximation
where the arguments of the function are not predetermined as
There are four unknowns
The formula would then give
Equating Equations (8) and (9) gives
Since in Equation (10), the
constants
Without proof (see Example 1 for proof of a related problem), we can find that the above four simultaneous nonlinear equations have only one acceptable solution
Hence
(13) We can derive the same formula
by assuming that the expression gives exact values for integrals of
individual integrals of These will give four equations as follows
These four simultaneous nonlinear Equations (14) can be solved with a single acceptable solution
Hence
Since two points are chosen, it is called the two-point Gauss Quadrature Rule. Higher point versions can also be developed.
Simulations Method ([MAPLE] [MATHCAD] [MATHEMATICA] [MATLAB]) Convergence ([MAPLE] [MATHCAD] [MATHEMATICA] [MATLAB])
Higher point Gaussian Quadrature Formulas For example
is called the three-point Gauss
Quadrature Rule. The coefficients
Arguments and weighing factors for n-point Gauss Quadrature Rules In handbooks (see Table 1), coefficients and arguments given for n-point Gauss Quadrature Rule are given for integrals of the form
Table
1: Weighting factors used in Gauss Quadrature formulas
So if the table is given for
The answer lies in that any
integral with limits of
If If such that
Solving these two simultaneous linear Equations (21) gives
Hence
Substituting our values of
For an integral
where
Solution Assuming the formula gives exact values for
integrals
Multiplying Equation (E1.3) by
The solution to the above equation is
I.
II.
III.
That leaves the solution of
Substituting (E1.7) in Equation (E1.3) gives
From Equations (E1.2) and (E1.8),
Equations (E1.4) and (E1.9) gives
Since Equation (E1.7) requires that the two results be of opposite sign, we get
Hence
For an integral, Solution The one-point Gaussian quadrature rule is
Assuming the formula gives
exact values for integrals
Since
Therefore, one-point Gauss Quadrature Rule can be expressed as
What would be the formula for
if you want the above formula
to give you exact values of Solution If the formula is exact for
linear combination of
Solving the two Equations (E3.1) simultaneously gives
So
Let us see if the formula works. Evaluate
The exact value of
Any surprises? Now evaluate
The exact value of
Because the formula will only
give exact values for linear combinations of Do you see now why we choose
gives us exact values? Use two-point Gauss Quadrature
Rule to approximate the distance covered by a rocket from
Also, find the absolute relative true error. Solution First, change the limits of
integration from
Next, get weighting factors and function argument values from Table 1 for the two point rule,
Now we can use the Gauss Quadrature formula
since
The absolute relative true
error,
Use three-point Gauss
Quadrature Rule to approximate the distance covered by a rocket from
Also, find the absolute relative true error. Solution: First, change the limits of
integration from
The weighting factors and function argument values are
and the formula is
since
The absolute relative true
error,
1..The upward velocity of a rocket is given by v(t)= 200 ln(t+1) ‑ 10 t, t>0, where t is given in seconds and v is given in meters per second. a) Use 2-point Gauss Quadrature rule to calculate the distance covered by the rocket form t=0 to t=5 s. b) What is the true value of the distance covered by the rocket form t=0 to t=5 s? c) What is the true error in part (a)? d) What is the relative true error in part (a)? e) What is the absolute relative true error in percentage for part (a). f) Use 3-point Gauss Quadrature rule to calculate the distance covered by the rocket form t=0 to t=5 s. Answer: a) 1035m b) 1025m c) –10m, d) -0.9756%, e) 0.9756% f) 1025 m
2..The velocity of a body is given by v(t)=(t +1), 0<t<2, and v(t)=33 2<t<5, where the velocity, v(t) is given in m/s and t is in seconds. a) Find the distance covered by the body between 1<t<1.9 seconds by using the two-point Gauss quadrature rule. Find the absolute relative true error. b) Find the distance covered by the body between 1<t<1.9 seconds by four-segment Trapezoidal rule. Find the absolute relative true error. c) Find the distance covered by the body between 1<t<3.9 seconds by using the two-point Gauss quadrature rule. Find the absolute relative true error. d) Find the distance covered by the body between 1<t<3.9 seconds by four-segment Trapezoidal rule. Find the absolute relative true error.
Answer: a) 2.205, 0% b) 2.205%, 0% c) 51.63m, 20.81% d)62.51m, 4.12%
3..A scientist develops an approximate formula for integration as
The values of c1 and c2 are found by assuming that the formula is exact for the functions of the form a0x+a1x2 polynomial. a) Find the values of c1 and c2 b) Verify the
formula works exactly for the integral of c) Verify the
formula does not work exactly for the integral of
Answer: a) c1= b) 447.7 c) 10 (exact) 12.976 (by formula)
*4..Find the
value of the integral Answer: 0.3473
*5..The one-point Gauss Quadrature rule is defined as
The values of c1 and x1 are found by assuming that the one-point formula is exact for any first order polynomial. Find c1 in the above one-point Gauss quadrature rule. Answer: b-a
*6..The one-point Gauss Quadrature rule is defined as
The values of c1 and x1 are found by assuming that the one-point formula is exact for any first order polynomial. Find x1 in the above one-point Gauss quadrature rule. Answer: (b+a)/2
*7..A scientist develops an approximate formula for integration as
The values of c1 and x1 are found by assuming that the formula is exact for the functions of the form a0x+a1x2 polynomial. Find c1 and x1. Answer: Multiple choice question examination
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(1)
(2)
(3)
.
(4)
and 
are
arbitrary constants for a general straight line
(5.a,b)
(6a) 
in Equation (5a)
gives
(6b)
(7)
and 
, but as unknowns 
and 
. So in the
two-point Gauss Quadrature Rule, the integral is approximated as
and 
. These are
found by assuming that the formula gives exact results for integrating a
general third order polynomial, 
.
Hence
(8)
(9)
(10)
and 
are
arbitrary, the coefficients of 
(11)
(12)
and 
. The reason
the formula can also be derived using this method is that the linear
combination of the above integrands is a general third order polynomial given
by
(14)
(16)
(17)
, and the function
arguments 
are
calculated by assuming the formula gives exact expressions for integrating a
fifth order polynomial 
.
General n-point rules would approximate the integral
(18)
(19)
and function
arguments 
integrals,
how does one solve 
?
can
be converted into an integral with limits 
. Let
(20)
then 
then 
(21)
(22)
into the
integral gives us
(23)
show that
the two-point Gauss Quadrature rule approximates to
(E1.1)
and 
.
Then 
(E1.2)
(E1.3)
(E1.4)
(E1.5)
and
subtracting from Equation (E1.5) gives
.
(E1.6)
or/and
or/and
or/and
.
is not
acceptable as Equations (E1.2-E1.5) reduce to 
and 
. But since
, then 
from 
, but 
. 
is not
acceptable as Equations (E1.2-E1.5) reduce to 
; 
.
is not acceptable
as Equations (E1.2-E1.5) reduce to 
. If 
, then 
gives 
and that
violates 
as the only
possible acceptable solution and in fact, it does not have violations (see it
for yourself)
(E1.8)
(E1.9)
(E1.10)
(E1.11)
derive the
one-point Gaussian Quadrature Rule.
(E2.1)
(E2.2)
the other
equation becomes
(E2.3)
(E2.4)
that
is a linear combination of 
.
(E3.1)
(E3.2)
(E3.3)
using the
above formula.
using the
above formula
is given by
as the
integrand for which the formula
to 
as given by
to 
.
, is (Exact value
= 11061.34m)
.
;c2= 
e-t dt
using 2-point Gaussian Quadrature rule.
, where a ≤
x1≤ b.