EMAE250: Homework 11
Due on April 23, 2:10pm
Use up to fourth decimal places for all numerical calculations. You may use a calculator or
MATLAB for numerical calculations, but must show the computational steps.
1. Given
dy
= 15(sin x + y) cos x
dx
(a) [1p

3/16/2015
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 functio

2/2/2015
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
Li

2/9/2015
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 variab

10.1 Gaussian Elimination with
Partial Pivoting
10.2 Iterative Methods for
Solving Linear Systems
10.3 Power Method for
Approximating Eigenvalues
10.4 Applications of Numerical
Methods
10
NUMERICAL
METHODS
C
Carl Gustav Jacob
Jacobi
1 8 0 4 1 8 5 1
arl Gu

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

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

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

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

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

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

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

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

10/5/2015
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

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

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

1/27/2015
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

4/9/2015
Alternative Form of Huens Method
f ( xi , yi ) f ( xi1 , yi01 )
h
2
yi01 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 the above equation:
yi1 yi hf ( x*, y*)
where
1
f ( xi , yi ) f ( xi h, yi f (

EMAE250: Homework 5
Due on February 19, 2:10pm
Use up to fourth decimal places for all numerical calculations. You may use a calculator or
MATLAB for numerical calculations, but must show the computational steps.
1. We will compute the condition number fo

EMAE250: Homework 1
Due on January 22, 2:10pm
Use up to fourth decimal places for all numerical calculations.
1. [2pts] Textbook pp. 108, Problem 4.1. The following infinite series can be used to
approximate ex :
x2 x3
xn
ex = 1 + x +
+
+ +
2
3!
n!
Prove

Table 1: Given four data points
i
0
1
2
3
x
-2
-1
0
1
f (x)
-15
0
3
0
EMAE250: Homework 7
Due on March 26, 2:10pm
Use up to fourth decimal places for all numerical calculations. You may use a calculator or
MATLAB for numerical calculations, but must show

EMAE250: Homework 2
Due on January 29, 2:10pm
Use up to fourth decimal places for all numerical calculations.
1. [6pts] Textbook pp. 173, Problem 6.2 - Modified. Determine the highest real root
of
f (x) = 2x3 11.7x2 + 17.7x 5
(a) Graphically.
(b) Bisectio

EMAE250: Homework 8
Due on April 2, 2:10pm
Use up to fourth decimal places for all numerical calculations. You may use a calculator or
MATLAB for numerical calculations, but must show the computational steps.
1. Review Textbook Box 21.3 on pp. 617.
2. Tex

EMAE250: Homework 9
Due on April 9, 2:10pm
Use up to fourth decimal places for all numerical calculations. You may use a calculator or
MATLAB for numerical calculations, but must show the computational steps.
1. Textbook pp. 653. Problem 22.3 Use Romberg

EMAE250: Homework 4
Due on February 12, 2:10pm
Use up to fourth decimal places for all numerical calculations.
1. Textbook pp. 297, Problem 10.2.
(a) Use naive Gauss elimination to decompose the following system (i.e., LU decomposition) according to the d

EMAE250: Homework 6
Due on March 3, 2:10pm
Use up to fourth decimal places for all numerical calculations. You may use a calculator or
MATLAB for numerical calculations, but must show the computational steps.
1. Consider the following function:
f (x, y) =

EMAE250: Homework 3
Due on February 5, 2:10pm
Use up to fourth decimal places for all numerical calculations.
1. Textbook pp. 276, Problem 9.3. Three matrices are defined as
"
#
"
#
1 6
1 3
2 2
C=
A = 3 10 B =
0.5 2
3 1
7 4
(a) [1pt] Perform all possible

3/31/2015
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