EGM6321 - Principles of Engineering Analysis 1, Fall 2010
Mtg 45: Fri, 3 Dec10
Difference Eq.(1)p.44-5 :
|
(1)
|
Where:
and
Legendre differential equation Eq.(1)p.5-4 and Eq.(1)p.24-1
Gauss-Legendre (GL) quadrature
Important application, e.g. Finite Element
The word quadrature comes from quadrilateral (note use of quad in both words).
Greeks: measure areas, where Area equals sum of quadrilaterals
The word cubature comes from the word cube:
Volume is equal to the sum of cubes
|
(1)
|
|
(2)
|
with being roots of for Where the sub n is defined as degree "n"
THEOREM :
|
(3)
|
|
(4)
|
|
(5)
|
END THEOREM
Example: (2 integration points)
Eq.(4)p.36-2 :
Eq.(5)p.36-2 :
HW8.2 . More generally, verify the table for Gauss Legendre quadrature in wikipedia. Analyze expression for and , and ater verifying the expression for ,with . See Fall 2009 HW. pg.35-2 and HW7.9 P41-2 . Evaluate and and compare results. END HW8.2
NIST Handbook
Gaussian quadrature
Number of points, n |
Points, xi |
Weights, wi
|
1 |
0 |
2
|
2 |
|
1
|
3 |
0 |
8⁄9
|
|
5⁄9
|
4 |
|
|
|
|
5 |
0 |
128⁄225
|
|
|
|
|
HW8.4 Back to HW7.4 P38-5. Complete the first three non-zero coefficients using Gauss Legendre quadrature up to within 5% accuracy. END HW8.4
see Fall 2009 P34-2
Question: How does Gauss Legendre quadrature compare to other quadratures, e.g. trapezoidal rule?
Answer: Look at Eq.(5) P45-2.
Consider (set of polynomials of degree less than or equal )
i.e. can integrate exactly any polynomial of degree les than or equal to using only n integration points (almost half)
Example:
Trapezoidal Rule (more details in EGM6341)
Where n=number of trapezoidal panels
Trapezoidal rule can only integrate exactly a straight line. Also as ( ), even for a simple polynomial of degree 2 (parabola)
References