Example of Ill-Conditioning in Numerical Problem

1
ME6758, Dr. Ferri
Example of Ill-Conditioning in Numerical Problem
Uniform force per unit length over last 90 of
length
Elongation of a tapered rod
F(x)
x
x
L
L
Governing equation
Approximate the solution by a series
2
Let
Get N equations in N unknowns
Now, solve system of equations for coefficients
a
As N goes to infinity, we expect that the
solution y(x) should approach the correct
answer
3
See that N5 is actually the best answer!
Exact answer should have y 0.95 and dy/dx 0
at xL. See that the solution is diverging as N
is increased. The phenomenon can be explained by
examining the condition number of the K-matrix.