9/10/2015
EMAE 250: Week 4
This Weeks Topics:
Preliminary Math Review
Eigenvalues and eigenvectors
Matrix Norms
Linear Algebraic Equations Continued
Matrix Inversion using LU Decomposition
Matrix Norm
Condition Number and Errors in Matrix Inversion
L

11/5/2015
Ordinary Differential Equations (ODEs)
Runge-Kutta Methods
Text: Chapter 25
Motivation
Mass-spring system:
c
m
m
k
d 2x
dx
c kx 0
dt 2
dt
Inverted pendulum:
m
l
g
d 2 g
sin 0
dt 2 l
Motivation
http:/www.bmw.com/com/en/newvehicles/x3/x3/2
006/a

11/10/2015
Ordinary Differential Equations (ODEs)
Runge-Kutta Methods
Text: Chapter 25
Alternative Form of Heuns Method
f ( xi , yi ) f ( xi1 , yi01 )
h
2
0
yi 1 yi f ( xi , yi )h
1
yi 1 yi f ( xi , yi ) f ( xi h, yi f ( xi , yi ) h) h
2
yi 1 yi
Rewrite

10/5/2015
Curve Fitting: Interpolation
Chapter 18
Interpolation
Least-square regressions vs. Interpolation
Seek a function that provides a good approximation
to the unknown data value at intermediate points
Two interpolation methods:
Newton's Divided Diff

9/21/2015
EMAE 250: Week 5
Todays Topics:
1-D Optimization
Golden-Section Search
Quadratic Method
Newtons Method
Optimization:
1-D Unconstrained Optimization
Text: Chapter 13
What is Optimization?
Root location and optimization are both looking for a p

10/15/2015
Curve Fitting:
Fourier Approximation
Text CH. 19
Curve Fitting with Sinusoidal Functions
We often deal with systems that oscillate or vibrate.
Trigonometric functions play a fundamental role in modeling
these problems.
Period
Consider a functi

10/27/2015
Error for the Multiple-Application Trapezoidal Rule
n 1
I (b a )
Width
f ( x0 ) 2 f ( xi ) f ( xn )
i 1
2n
n 1
h
f ( x0 ) 2 f ( xi ) f ( xn )
2
i 1
Average Height
The local truncation error of a single application of the trapazoidal rule:
1

9/29/2015
Optimization:
N-D Unconstrained Optimization
Text: Chapter 14
Multi-Dimensional Unconstrained Optimization
Unconstrained optimization problems: finding the minimum
or maximum of a function f(x1,x2,xn) without any
constraints on the problem varia

11/19/2015
Boundary-Value and Eigenvalue Problems
Chapter 27
Bounded-Value Problems
Initial-value vs. bounded-value
Initial-value problem: all the conditions are specified at the same value
of the independent variable.
Bounded-value problem: the condition

8/25/2015
Course Information
EMAE 250 Computers in Mechanical Engineering
Schedule
Lectures: T & TH 1:15-2:05, Glennan 421
LAB:
T 4:30-6:30, Location: TBA
W 3:00-5:00, Location: TBA
Textbook
Numerical Methods for Engineering by S. C. Chapra and R. P. Ca

9/3/2015
EMAE 250: Week 3
This Weeks Topics:
Linear Algebraic Equations
Preliminary Math Review
Equations with Multi-Variables
Gauss Elimination
Gauss Jordan Method
LU Decomposition
Matrix Inversion
Review: Bisection Method
Step 1: Choose lower and upper