MATH 3311 (L2): Introduction to Numerical Methods
Instructor:
Prof. Jianping Gan Office: 3478; Email: [email protected];
Tel: 7421; Office hour: Wed: 16:30-17:30
Class schedule:
16:30-17:20 Tue. and Thu.,
Venue: 2407
Teaching Assistants: Taylor Michael ([email protected]); Yejun Gu ([email protected])
TA sessions:
T2a, Mon. 17:50-18:20
rm5583 (Taylor Michael: mtaylor)
T2b, Thur., 15:90-15:50,
rm 2463(Yejun Gu: yguab);
T2c, Wed., 12:30-13:20,
rm 2463(Yejun Gu: yguab);
Textbook:
Numerical Analysis, 8th ed., by Burden, R.L. and Faires J. D., Thomson Brooks/Cole.
http://www.as.ysu.edu/~faires/Numerical-Analysis
Grading:
Mid-term :
30%, March 27, 2013
Final
:
70%, All materials taught in the whole semester will be tested, including
those already tested in the midterm exam. But focus will be on those topics not covered in
the midterm exam. (Exams are closed books and notes, and no formula sheets are provided. However, you are
allowed to bring one 3in x 5in note card (front and back) to write whatever you think helpful or necessary. Calculators
approved by Hong Kong Examinations and Assessment Authority (香港考試及評核局) are allowed. Use 5-digit
rounding arithmetic in all calculations.)
Assignments:
Assigned and graded, but not marked. Solutions will be provided.
MATH3311: Introduction to
Numerical Methods
• 1. Develop an understanding of the core ideas and
concepts of Numerical Methods.
• 2. Be able to recognize the power of abstraction and
generalization, and to carry out investigative
mathematical work with independent judgment.
• 3. Be able to apply rigorous, analytic, highly numerate
approach to analyze and solve problems using
Numerical Methods.
• 4. Be able to communicate problem solutions using
correct mathematical terminology and good English.
Topics:
Introduction to computational world (Chapter 1)
Root Finding (Chapter 2)
Bisection method (2.1)
Fixed-point iteration (2.2)
Newton's method, Secant method (2.3)
Interpolation (Chapter 3)
Interpolation and the Lagrange (interpolating) polynomial (3.1)
Divided differences and Newton's interpolatory divided-difference formula (3.2)
Least Squares Data Fitting (chapter 8)
Least squares data fitting--data fitting, modes, normal equation (8.1)
Numerical Differentiation and Integration (Chapter 4)
Numerical differentiation--forward, backward, and central finite differences (4.1)
Elements of numerical integration (4.3)
Composite rules (4.4)
Gaussian quadrature--Gaussian node points, weights (4.7)
Solution of Ordinary Differential Equations (Chapter 5)
Euler's method (5.2)
Solving Linear Systems (Chapters 6 & 7)
Gauss elimination--multipliers, Gauss elimination, back substitution, pivoting
(6.1 - 6.2)
LU factorization--LU, forward substitution (6.5)
Iterative methods--matrix splitting, Jacobi method, Gauss-Seidel method, SOR
method (7.3)
Error bounds and iterative refinement (7.4)
Chapter 1
A: Introduction to Computational
World
• Numerical Method?
Example 1:
λ
a0 = b0e +
c0
λ
λ
( e − 1)
λ can not be solved analytically or by algebraic
methods.
λ can be solved by numerical method.
Example 2:
df
f ( x0 + ∆x ) − f ( x0 )
= f ' ( x0 ) = lim
dx
∆x
∆x →0
f ( x0 + ∆x ) − f ( x0 )
f ' ( x0 ) ≈
∆x
f (x)
x = x0 − ∆x
x = x0
x = x0 + ∆x
∆x2
∆x3
f ( x0 ± ∆x) = f ( x0 ) ± f ' ( x0 )∆x + f ' ' ( x0 )
± f ' ' ' ( x0 )
± ...
2!
3!
f ( x0 + ∆x) − f ( x0 )
f ( x0 + ∆x) − f ( x0 )
f ' ( x0 ) =
+R≈
∆x
∆x
or
f ( x0 ) − f ( x0 − ∆x)
f ( x0 ) − f ( x0 − ∆x)
f ' ( x0 ) =
+R≈
∆x
∆x
Truncation error
Round-off Error
3• 3 =3
In a machine (computer), only a fixed and finite number
digits are used to represent
3 • 3 = 2.99999999...
The error that results from replacing a number with its
floating-point form is called round-off error.
For 64 bit-binary digits,
< 2 − 1022
Underflow:
Overflow:
> 2 1023 • ( 2 − 2
− 52
)
If p* is an approximation to p, then
p − p*
p − p*
p
Absolute error
Relative error
Algorithms
Example of Algorithms:
N
An algorithm to compute ∑ i =1 xi = x1 + x 2 + ...... + x N ,
Where N and the numbers x 1 , x 2 ,....., x N are given, is
Described by the following:
INPUT
N , x 1 , x 2 ,..., x N
OUTPUT
sum =
STEP1
STEP2
STEP3
∑
N
i =1
xi
set sum = 0
For i = 1 , 2 ,... N do
set sum = sum + x i (add the next term)
OUTPUT
STOP
( sum );
Stable and Unstable:
En (error)
Unstable exponential error
growth, En=CnE0
Stable exponential error
growth, En=CnE0
n (number of iteration)
Stable: An algorithm has feature that small changes in the initial data produce
correspondingly small changes in the final result
Example:
1 n
) + c2 3n
3
1
with p 0 = 1 , p 1 =
, we have c 1 = 1 , c 2 = 0
3
1
pn = ( )n
3
p n = c1 (
Using five-digit rounding arithmetic (note: if 5th+1 digit ≥5, add 1
to the 5th digit. Otherwise keep the first 5 digits.),
p0 = 1.0000, p1 = 0.3333, so, c1 = 1.0000, c2 = −0.125 × 10−5
1 n
pˆ n = 1.0000( ) − 0.125 × 10−5 (3)n
3
Round-off error:
pn − pˆ n = 0.125 × 10−5 (3)n
Growth exponentially (unstable)
Example:
pn = c1 + c2n;
1
2
with p0 = 1, p1 = , we have c1 = 1, c2 = − , so
3
3
2
pn = 1 − n
3
For five-digit rounding arithmetic,
p0 = 1.0000, p1 = 0.3333,c1 = 1.0000, c2 = −0.6667
pˆ n = 1.000 − 0.6667n
2
p − pˆ n = (0.6667 − )n
3
Growth linearly (stable)
Convergence:
Suppose
{β }
∞
n n =1
→ 0 , {α
}
∞
n n =1
→α
α n − α ≤ k β n for large n ,
k is a positive constant
∞
We say {α n }n =1 converges to α
O( βn )
In general,
βn =
with rate of convergence
1
, p >0
p
n
1
so α n = α + O ( p )
n
Example:
n +1
n+3
α n = 2 , αˆ n = 3
n
n
Both series approach zero as n reaches ∞. However,
4
α̂ n
2
n +1 n + n
1
≤
=
2
= 2βn
2
2
n
n
n
n + 3 n + 3n
1
=
4
= 4βn
αˆ n − 0 = 3 ≤
3
2
n
n
n
1
so, α n = 0 + O ( );
n
The convergence rate (to zero)
1
for
and are
and
,
αˆ n = 0 + O ( 2 )
respectively.
n
αn − 0 =
αn
B: MATLAB Introduction
The name MATLAB stands for matrix
laboratory.
Features:
•
•
•
•
•
•
•
Math and computation
Algorithm development
Data acquisition
Modeling, simulation, and prototyping
Data analysis, exploration, and visualization
Scientific and engineering graphics
Application development, including graphical
user interface building.
In short, MATLAB integrates data structure,
programming and graphical user interface
together.
On Windows platforms, start MATLAB by
double-clicking the MATLAB shortcut icon
on your Windows desktop.
MATLAB Desktop
• Use desktop tools to manage your work in MATLAB. You can also
use MATLAB functions to perform the equivalent of most of the
features found in the desktop tools.
Matlab major features and applications
• Desktop Tools and Development
Environment
• The MATLAB Mathematical Function
Library
• The MATLAB Language
• Graphics
• MATLAB External Interfaces
Command Window
• Use the Command Window to enter variables and to run functions
and M-file scripts.
Ex 1.
Ex 2.
Command History
• Statements you enter in the Command Window are logged in the
Command History. From the Command History, you can view and
search for previously run statements, as well as copy and execute
selected statements. You can also create an M-file from selected
statements.
Help Browser
• To open the Help browser, click the Help button
in the desktop
toolbar.
• The Help browser consists of two panes, the Help Navigator, which
you use to find information, and the display pane, where you view
the information.
Current Directory
Workspace Browser
• The MATLAB workspace consists of the set of variables (named
arrays) built up during a MATLAB session and stored in this
temporary memory. The features (e.g. dimension and data type) of
these variables can be viewed in the workspace and these stored
variables can be used in the subsequent calculation.
• To delete variables from the workspace, select the variables and
select Edit > Delete. Alternatively, use the clear function.
• The core of MATLAB is array and matrix to
conduct calculation and to store data.
Creating Matrices
Array Editor
• Double-click a variable in the Workspace browser, or use openvar
variablename, to see it in the Array Editor. Use the Array Editor to
view and edit a visual representation of variables in the workspace.
Adding and Subtracting Matrices
Working with Workspace
Use mathematical functions in Matlab
Polynomials
• Polynomials p(x) = x3 – 6x2 – 11x – 6 is represented as
• The roots function calculates the roots of a polynomial:
• The function poly returns the polynomial coefficients:
Interpolation
Interpolation
• Linear interpolation (default)
• Nearest neighbor interpolation
• Cubic spline interpolation
Running M-files in the Editor/Debugger