Computational Methods for
Engineering
Instructor: Dr. Nguyen Quang Hung
Week 13:
Matrix Inverse and Condition
Calculating Matrix Inverse with LU
Factorization
LU factorization:
qCalculating the first column of the matrix inverse:
Solution:
Calculating Mat
Computational Methods for
Engineering
Instructor: Dr. Nguyen Quang Hung
Week 13:
LU Factorization
Overview of LU Factorization
[A]cfw_x ! cfw_b = 0
vAfter the first step of Gauss elimination:
! [U]cfw_x " cfw_d = 0
vAssuming that there is a lower diagonal
Computational Methods for
Engineering
Instructor: Dr. Nguyen Quang Hung
Week 12:
Linear Algebraic Equations
and Matrices
Matrix Algebra
Matrix Notation:
A matrix consists of a rectangular array of elements
represented by a single symbol
vMatrices with row
Computational Methods for
Engineering
Instructor: Dr. Nguyen Quang Hung
Week 2:
Mathematical Modeling
Numerical Methods
Problem Solving
A Simple Mathematical Model
A mathematical model can be broadly defined as a
formulation or equation that expresses the
ComputationalMethods
forEngineering
Instructor: Dr. Nguyen
Week 13: Hung
Quang
Matrix Inverse and Condition
CalculatingMatrixInversewithLU
Factorization
LU factorization:
q
Calculating the first column of the matrix inverse:
Solution:
CalculatingMatrixInv
Computational Methods for
Engineering
Instructor: Dr. Nguyen Quang Hung
Week 12:
Gauss Elimination
Solving Small Numbers of Equations
The Graphical Method
!
For three simultaneous equations:
Each equation would be represented
by a plane in a three-dimens
Computational Methods for
Engineering
Instructor: Dr. Nguyen Quang Hung
Week 11:
Tutorials
Solution #6.1
clc; clear all; close all;
x=0.5; es=0.0001;
while (1)
xold=x;
x=sin(sqrt(x);
ea=abs(x-xold)/x)*100;
if ea<=es, break, end
end
disp(x);
Solution #6.2
Computational Methods for
Engineering
Instructor: Dr. Nguyen Quang Hung
Week 10:
Roots: Open Methods
Review of Last lecture
Bracketing Methods
Incremental Search
Bisection
False Position (Linear Interpolation)
Simple Fixed-Point Iteration
Fixed-point iter
Computational Methods for
Engineering
Instructor: Dr. Nguyen Quang Hung
Week 7:
Roundoff and Truncation errors
(continue)
Roundoff Errors
q Roundoff errors arise because digital computers
cannot represent some quantities exactly. They
are important to eng
Computational Methods for
Engineering
Instructor: Dr. Nguyen Quang Hung
Week 7:
Roundoff and Truncation errors
(continue)
Problem 4.4
For computers, the machine epsilon can also be thought of
as the smallest number that when added to one gives a
number gr
Computational Methods for
Engineering
Instructor: Dr. Nguyen Quang Hung
Week 8:
Root: Bracketing Methods
Graphical Method
2
x + 5x + 2 = 0
Incremental Search
f (xl ) f (xu ) < 0
f(x): real and
continuous in [xl , xu]
There is at least one real root betwe
Computational Methods for
Engineering
Instructor: Dr. Nguyen Quang Hung
Week 6:
Roundoff and Truncation errors
Accuracy and Precision
qAccuracy: refers to how closely a computed or
measured value agrees with the true value.
qPrecision: refers to how close
Computational Methods for
Engineering
Instructor: Dr. Nguyen Quang Hung
Week 6:
Roundoff and Truncation errors
(continue)
Roundoff Errors
q Roundoff errors arise because digital computers
cannot represent some quantities exactly. They
are important to eng
Computational Methods for
Engineering
Instructor: Dr. Nguyen Quang Hung
Week 2:
Matlab Fundamentals
MATLAB Fundamentals
Matlab Environment
Mathematical Operations
Use of Built-in Functions
Graphics
Case study
MATLAB Environment
MATLAB is a computer pr
Computational Methods for
Engineering
Instructor: Dr. Nguyen Quang Hung
Week 4:
Programming with MATLAB
Problem: Calculate the Integral
"
#
0
3
4
x
!
dx =
= 6.4939
x
e !1
15
Solution
clc; clear all;
sint=0;dx=0.01;
for x=0.00001:dx:100
sint=sint+x^3*dx/(e
Computational Methods for
Engineering
Instructor: Dr. Nguyen Quang Hung
Week 3:
Problem Solving
Problem Solving
1.8. The amount of a uniformly distributed radioactive
contaminant contained in a closed reactor is measured by its
concentration c (becquerel/
Computational Methods for
Engineering
Instructor: Dr. Nguyen Quang Hung
Week 5:
Programming with MATLAB
(continue)
Problem 3.2
An amount of money P is invested in an account where
interest is compounded at the end of the period. The future
worth F yielded
Computational Methods for
Engineering
Instructor: Dr. Nguyen Quang Hung
Week 1:
Course Overview
Introduction to the Course
What are computational (numerical)
methods ?
q Numerical methods are techniques by which
mathematical problems are formulated so tha
I HC TN TO
Nhng yu cu v chng trnh ging dy v bng cp
i hc Tn To l mt i hc Vit nam s dng chng trnh ging dy da trn m hnh gio dc ca
Hoa K.
Bng cp
i hc Tn To cp bng C nhn Khoa hc (Bachelor of Science) cho 7 chuyn ngnh: K ton, Ti
chnh, Qun l, Kinh doanh
Full name: Le Minh Tan Trung
Summarize Film: Slumdog Millionaire
Jamal Malik 18 years old take part in the game is who wants to be a millionaire.
He just only appears on the television which is program his girlfriend like very
much. But that is opportunit
Full name: Le Minh Tan Trung
1/ Page 242
He acquires a lot of prize.
Apparently, she is very beautiful.
It shows her capacity.
We can base on context to predict sentence mean.
I can take evidence to against him.
I am focusing for Toefl.
It is mascu
Full name: Le Minh Tan Trung
Homework:
Write correct sentences for page 242
Chapter 6 starred words page 131 #3, Page 134 #2; page 136 #1
Work on compare / contrast paragraph
1. Page 242 chapter 6.
This is our benefits.
They created a lot of grandiose inv
Full name: Le Minh Tan Trung.
Homework
Page 38-43 , # 1,2,3 Interraction2
Page 241 stared work
Page 30 - #7 complete the Supporting detail
Page 30
Main ideas
Curitiba created a Green Exchange to help the city
Details:
> Cleans the streets of trash
> Excha