Simple detailed worked examples using Gaussian Quadrature method
Nasser Abbasi
We seek to find a numerical value for the definite integral of a real valued
function of a real variable over a specific range. In other words, to evaluate
Geometrically, this integral represents the area under
from
to
The following are few detailed step-by-step examples showing how to use Gaussian Quadrature (GQ) to solve this problem.
Few points to remember about GQ.
There are different versions of GQ depending on the basis polynomials it uses
which in turns determines the location of the integration points. We will only
use GQ based on Legendre polynomials. The integration points (called
)
are the roots of the Legendre polynomials.
GQ gives an exact answer when the function to be integrated is a polynomial of
order
where
is the number of integration points.
Since Legendre polynomials are defined over
![$[-1,1]$](graphics/numIntegration__9.png)
,
we need to map the range of the function to be integrated to be
![$[-1,1]$](graphics/numIntegration__10.png)
.
The actual integration is performed over
![$[-1,1]$](graphics/numIntegration__11.png)
but the values are mapped back to the original range using a transformation
rule. This diagram illustrates this.
See appendix to see how the transformation rule is derived.
To be consistent, we follow the book notations and call the user function to
be integrated as
and the mapped function as
.
Using GQ, we need to have access to a lookup table to obtain the values of the
weights (called Multipliers or
in the book) and the integration points
.
For each different value of
(the numbers of integration points, also called the order of GQ), there will
be a specific set of values
and
to use. Once we have
and
then the value of the integral is
This is an overview of the GQ algorithm. Next we will work out few examples to illustrate.
Evaluate
,
Use
We see that
,
Answer
step 1:
is given, go to step 2.
step 2: From lookup we see that
And
step 3: Evaluate the integral
LOOP over all the points from left to right.
But
Hence
Do the next point:
But
Hence
Do the next point
But
Hence
Do the next point
But
Hence
Do the next point
But
Hence
No more points. Add to obtain the final answer
To verify, use say Maple:
Evaluate
We see that
,
Answer:
step 1:
is not given. But since polynomial, we can determine minimum
Order of polynomial is
hence we need
step 2: From lookup we see that
And
step 3: Evaluate the integral
LOOP over all the points from left to right.
But
Hence
Do next point
But
Hence
Do next point
But
Hence
Hence the answer is
Verify
Evaluate
We see that
,
Answer:
step 1:
is not given. But since polynomial, we can determine minimum
Order of polynomial is
hence we need
step 2: From lookup we see that
And
step 3: Evaluate the integral
LOOP over all the points from left to right.
But
Hence
Do next point
But
Hence
Do next point
But
Hence
Hence the answer is
An easy way to find how the function changes when we change the range is to
align the ranges over each others and take the ratio between them as the scale
factor. This diagram shows this for a general case where we map
defined over
![$[a,b]$](graphics/numIntegration__88.png)
to a new range defined over
![$[c_{1},c_{2}]$](graphics/numIntegration__89.png)
We
see from the diagram that
But
The above is called the Jacobian of the transformation.
Now, From the diagram we see that
and
Hence (1) becomes
For the specific case when
and
the above expressions become
Which is the mapping used in the Gaussian Quadrature method.
It interesting to see the effect of this transformation on the shape of some
functions. Below I plotted some functions under this transformation. The left
plots are the original functions plotted over some range, in this case
![$[4,10]$](graphics/numIntegration__101.png)
and the left side plots show the new shape (the function
)
over the new range
![$[-1,1]$](graphics/numIntegration__103.png)
