ME 413
Numerical Methods in Mechanical
Engineering
Instructor: Yaling Liu
ME 413
Course
Syllabus
Survey
Took
computer related course before?
Know
any computer language such as C,
Fortran, or Matlab?
Did
some programming before?
Have
some idea about n
Chebyshev
Economization
This provides a uniform approximation of a function of on the interval [ -1, +1]
Please note that if our function f(x) is defined over the interval [a, b] we can use
the change of variables:
x = a + (b a ) * ( + 1 ) / 2.0
Polynomia
Quadratic Eqs.
a x 2 b x c 0 is solved by the formula x1, 2
b b 2 4ac
2a
If a or c or both are small then inaccuracies may occur because of
loss of significance due to subtraction
To avoid these:
q - 12 b b 2 4ac
1) for b 0 use
then
1
x1
b
2a
1 b 2 b 2
Iterative methods
Gauss - Seidel
When the number of equations increases, direct methods
become impractical
Gauss - Seidel iteration converges to the solution when the set
of equations is diagonally dominant
example:
5 1 2 x1 5
1 6 2 x2 11
1 0 3 x 2
SOME REVIEWS OF LINEAR ALGEBRA
Multiplication between a matrix and a vector
a11
A
a21
a12
;
a22
x
b
y
SOME REVIEWS OF LINEAR ALGEBRA
To find y=Ab
a11
Ab
a21
a12 x
a22 y
a11 x a12 y
a21 x a22 y
Note: A ( ax + by ) = aA x + bA y,
where