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Numerical Methods Intro to Numerical Methods & Intro to Matlab Programming By M Jamil Khan
Numerical Methods
Intro to Numerical Methods &
Intro to Matlab Programming
By M Jamil Khan
Numerical Methods
What are numerical
methods and why should
you study them?
Numerical Methods
“Numerical Methods are techniques by which
mathematical Problems are formulated so that they can be
solved with arithmetic and logical operations. Because
digital computers excel at performing such operations,
numerical methods are sometimes referred as computer
mathematics”.
Numerical Methods
In pre computer era , the time and effort of implementing
such calculations seriously limited their practical use.
However, with the advent of fast, inexpensive digital
computers, the role of numerical methods in engineering
and scientific problem solving has exploded.
Numerical Methods
Numerical methods figure so prominently in, much of our
work, I believe that numerical methods should be a part of
every engineer's and scientist's basic education. Just as we
all must have solid foundation in the other areas of
mathematics and science, we should also have a
fundamental understanding of numerical methods. In
particular, we should have a solid appreciation of both
their capabilities and their limitations.
Why?
Numerical Methods
Beyond contributing to your overall education. .there are
several additional reasons why you should study
numerical methods: 1
1Numerical Methods
Numerical methods greatly expand the types of , problems
you can address. They are capable of handling large
systems of equations nonlinearities, and complicated
geometries that are not uncommon in engineering and
science and that are often impossible to solve analytically
with standard calculus. As such they greatly enhance your
problem-solving skills.2
2Numerical Methods
Numerical methods allow you to use "canned“ software with insight. During your career, you will invariably have occasion to use commercially available pre-packaged computer programs that involve numerical methods. The intelligent use of these programs is greatly enhanced by an understanding of the basic theory underlying the methods. In the absence of such understanding you will be left to treat such packages as “black boxes“ with little critical insight into their inner workings or the validity of the results they produce.
3
3Numerical Methods
Many problems cannot be approached using canned
programs. If you are conversant with numerical methods,
and are adept at computer programming, you can design
your own programs to solve problems without having to
buy or commission expensive software.4
4Numerical Methods
Numerical methods are an efficient vehicle for learning to use computers. Because numerical methods are expressly designed for computer implementation, they are ideal for illustrating the computer's powers and limitations. When you successfully implement numerical methods on a computer, and then apply them to solve otherwise intractable problems, you will be provided with a dramatic demonstration of how computer can serve our professional development. At the same time, you will also learn to acknowledge and control the errors of approximation that are part and parcel of large-scale numerical calculations.
5
5Numerical Methods
Numerical methods provide a vehicle for you to reinforce
your understanding of mathematics. Because one function
of numerical methods is to reduce higher mathematics to
basic arithmetic operations. they get at the "nuts and bolts"
of some otherwise obscure topics. Enhanced
understanding and insight can result from this alternative
perspective.
Books
Reference Books
• A first Course in Numerical Analysis with C++ by Dr. Saeed
Akhtar Bhatti, 4th Edition.
• Applied Numerical Methods using Matlab by Won Young Yang.
• Applied Numerical Methods with Matlab for Engineers and
Scientists by Steven C. Chapra, 2nd Edition.
Grading Policy
Unannounced Quizzes 20%
Lab Assignments 20%
(Late and Missed Submissions are Not Applicable)
Mid Exam 20%
Final Exam 40%
+ Attendance
+ Discipline
Computer Usage
You should be able to write a computer
code in a programming language you
know such as MATLAB or C.
Some Useful Links
http://web.uettaxila.edu.pk/CMS/FA2011/teNMTbs/
http://www.library.nu
http://www.google.com
http://www.codeproject.com
http://numericalmethods.eng.usf.edu
http://math.gmu.edu/introtomatlab.htm
http://www.mathworks.com
Feedback Request
Please mail questions and constructive comments to
Your feedback will be most appreciated On style, contents, detail, examples, clarity, conceptual
problems, exercises, missing information, depth, etc.
Local course support website http://web.uettaxila.edu.pk/CMS/SP2011/teNMTbs/
Course Objective
This course will emphasize the development of numerical
algorithms to provide solutions to common problems
formulated in science and engineering. The primary
objective of the course is to develop the basic
understanding of the construction of numerical algorithms,
and perhaps more importantly, the applicability and limits
of their appropriate use.
Course Outline
Introduction to Numerical Methods
Introduction to Matlab Programming
Error Analysis
Finite Differences
Interpolation
Numerical Differentiation
Course Outline
Numerical Integration
Ordinary Differential Equations
Non Linear Equations
Linear System of Equations
Today Objectives
The primary objective of today’s discussion is to provide you with a concrete idea of what numerical methods are and how they relate to engineering and scientific problem solving. Specific objectives and topics covered are• Learning how mathematical models can be formulated
on the basis of scientific principles t o simulate the behavior of a simple physical system.
• Understanding how numerical methods afford a means to generate solutions in a manner that can be implemented on a digital computer.
Today Objectives
• Understanding the different types of conservation laws that lie beneath the models used in the various engineering disciplines and appreciating the difference between steady-state and dynamic solutions of these models.
• Learning about the different types of numerical methods we will cover in this course.
A Simple Mathematical Model
A mathematical model can be broadly defined as a
formulation or equation that expresses the essential
features of a physical system or process in
mathematical terms. Models can be represented by a functional relationship
between dependent variables, independent variables,
parameters, and forcing functions.
Model Function
Dependent variable - a characteristic that usually
reflects the behavior or state of the system Independent variables - dimensions, such as time and
space, along which the system’s behavior is being
determined.
Dependentvariable f independent
variables , parameters, forcingfunctions
Model Function
Parameters - constants reflective of the system’s
properties or composition Forcing functions - external influences acting upon the
system
Model Function ExampleAssuming a bungee jumper is in mid-flight, an
analytical model for the jumper’s velocity,
accounting for drag, is
Dependent variable - velocity v Independent variables - time t Parameters - mass m, drag coefficient cd Forcing function - gravitational acceleration g
v t gmcd
tanhgcd
mt
Model Results
Using a computer (or a calculator), the model can be used to generate a
graphical representation of the system. For example, the graph below
represents the velocity of a 68.1 kg jumper, assuming a drag
coefficient of 0.25 kg/m
Numerical Modeling
Some system models will be given as implicit functions or as differential
equations - these can be solved either using analytical methods or
numerical methods. Example - the bungee jumper velocity equation from before is the
analytical solution to the differential equation
where the change in velocity is determined by the gravitational forces
acting on the jumper versus the drag force.
dvdt
gcd
mv2
Numerical Methods
To solve the problem using a numerical method, note that the time rate
of change of velocity can be approximated as:
dvdt
vt
v ti1 v ti
ti1 ti
Numerical Results
The efficiency and accuracy of numerical methods will
depend upon how the method is applied. Applying the
previous method in 2 s intervals yields:
Analytical and Numerical Solutions:
Numerical methods are techniques by which
mathematical problems are formulated so that they can be
solved with arithmetic and logical operations.
Numerical Analytical
approximate exact
more intuitive less intuitive
easily coded not so easy
easy to get not so easy
Bases for Numerical Models
Basic Needs in the Numerical Methods: Practical:
Can be computed in a reasonable amount of time. Accurate:
Good approximate to the true value, Information about the approximation error (Bounds,
error order,… ).
Solution of Nonlinear Equations Some simple equations can be solved analytically:
Many other equations have no analytical solution:
31)1(2
)3)(1(444solution Analytic
0342
2
xandx
roots
xx
solution analytic No052 29
xexxx
Methods for Solving Nonlinear Equations
o Bisection Method
o Newton-Raphson Method
o Secant Method
Solution of Systems of Linear Equations
unknowns. 1000in equations 1000 have weif do What to123,2
523,3:asit solvecan We
523
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Cramer’s Rule is Not Practical
this.compute toyears 10 than more needscomputer super A
needed. are tionsmultiplica102.3 system, 30by 30 a solve To
tions.multiplica 1)N!1)(N(N need weunknowns, N with equations N solve To
problems. largefor practicalnot is Rule sCramer'But
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20
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Methods for Solving Systems of Linear Equationso Naive Gaussian Elimination
o Gaussian Elimination with Scaled Partial Pivoting
o Algorithm for Tri-diagonal Equations
Curve Fitting Given a set of data:
Select a curve that best fits the data. One choice is to find the curve so that the sum of the square of the error is minimized.
x 0 1 2 y 0.5 10.3 21.3
Interpolation Given a set of data:
Find a polynomial P(x) whose graph passes through all tabulated points.
xi 0 1 2 yi 0.5 10.3 15.3
tablein the is)( iii xifxPy
Methods for Curve Fitting o Least Squares
o Linear Regressiono Nonlinear Least Squares Problems
o Interpolationo Newton Polynomial Interpolationo Lagrange Interpolation
Integration Some functions can be integrated
analytically:
?
:solutions analytical no have functionsmany But
421
29
21
0
3
1
23
1
2
dxe
xxdx
ax
Methods for Numerical Integrationo Upper and Lower Sums
o Trapezoid Method
o Romberg Method
o Gauss Quadrature
Solution of Ordinary Differential Equations
only. cases special for available are solutions Analytical *
equations. thesatisfies that function a is
0)0(;1)0(0)(3)(3)(
:equation aldifferenti theosolution tA
x(t)xx
txtxtx
Solution of Partial Differential Equations
Partial Differential Equations are more difficult to solve than ordinary differential equations:
)sin()0,(,0),1(),0(
022
2
2
2
xxutututu
xu
Bases for Numerical Models
Conservation laws provide the foundation for many model
functions. Different fields of engineering and science
apply these laws to different paradigms within the field.
Among these laws are:
Conservation of mass
Conservation of momentum
Conservation of charge
Conservation of energy
