Computational Techniques in Mechanical Engineering
ME 535

Spring 2015
Partial Differential Equations
Consider the non dimensional equation
2
=
+A
t
x2
1
(1)
Explicit Form
Expanding in a Taylor series
(x, t + t) = (x, t) +
1 2 2
t +
t + . . .
t
2! t2
from which we obtain
(x, t + t) (x, t)
1 2
=
+.
t
t
2! t2
(x, t + t) (x, t)
Computational Techniques in Mechanical Engineering
ME 535

Spring 2015
Chebyshev Polynomials
Suppose that we fit data using a polynomial over the range 1 x 1.
f(x) = pn (x) =
N
X
ai x i
(1)
i
Near x = 0, only the low order terms will be important, but near x = 1, the
high order terms will dominate. If f(x) has reasonable val
Computational Techniques in Mechanical Engineering
ME 535

Spring 2015
ODEEuler
A.F. Emery 1
Taylor Series and Eulers (One Step) Methods
To solve dy/dx = f(x, y) let us expand y in a Taylor series
h3
h2 00
y (x0 ) + y 000 (x0 ) + . . .
2!
3!
h2 0
h3 00
=y(x0 ) + hf(x0 , yo ) + f (x0 , y0 ) + f (x0 , y0 ) + . . .
2!
3!
y(x0
Computational Techniques in Mechanical Engineering
ME 535

Spring 2015
Solving Ax = b by optimizing a Quadratic
Function
It is possible to develop iterative methods based upon the ideas of optimization.
Consider the quadratic form
Q=
1 T
x Ax xT b
2
(1)
At the point, x + x, we have
Q(x + x) Q(x) =
1
xT Ax
2
(2)
and if A is p
Computational Techniques in Mechanical Engineering
ME 535

Spring 2015
Curve Fitting
Why do we want to curve fit? In general, we fit data points to produce a smooth
representation of the system whose response generated the data points. We do
this for a variety of reasons
1. to interpolate or extrapolate
2. to understand the
Computational Techniques in Mechanical Engineering
ME 535

Spring 2015
Condition Numbers, Norms, Eigenvalues and
Eigenvectors
1
Condition Number
The condition number is a measure of the effect on an approximate inverse of
a matrix A (or on an approximate solution of Ax = b), when the elements are
changed slightly. Several di
Computational Techniques in Mechanical Engineering
ME 535

Spring 2015
Linear Algebraic Equations
1
Fundamentals
Consider the set of linear algebraic equations
n
X
aij xi = bi represented by Ax = b
(1a)
j
with
[Ab ] [Ab]
and
r(A) rank of A
(1b)
Then Ax=b has a solution iff the rank of A and of Ab satisfy
r(A) = r(Ab )
(2)
I
Computational Techniques in Mechanical Engineering
ME 535

Spring 2015
Wavelet Examples
Consider the signal shown in Figure 1a that has 2048 sample points. A Haar
wavelet analysis was done with 11 levels. Figure 2b depicts the energy contained
in the Haar detail vectors. Note that the first several levels had very small
ener
Computational Techniques in Mechanical Engineering
ME 535

Spring 2015
1
Floating Point Numbers
Let F be the set of floating point numbers and let each number X in F be
represented by
X = (
d2
dn
d1
+ 2 + + n ) E
(1a)
where
0 di 1
LEU
(i = 1, , n)
(1b)
(1c)
If di 6= 0, then the floating point number system is said to be norm
Computational Techniques in Mechanical Engineering
ME 535

Spring 2015
1
1 FUNDAMENTALS
Iterative Methods for Ax=b
1
Fundamentals
consider the solution of the set of simultaneous equations Ax = b where A is a square matrix, n n and b
is a right hand vector. We write the iterative method as
xk+1 = Fk (A, b, xk , xk1, . . . ,