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CS213

Numerical Analysis and Computer Applications

Prof. Dr. Wafaa El-Haweet

Outlines Error analysis & numerical instabilities

Approximation of roots of equations

Simultaneous linear algebraic equations and matrix inversion

Numerical differentiation & integration

Interpolation and Extrapolation

Least square approximation

Eigenvalues and eigenvectors

Ordinary differential equations

References

Applied Numerical Methods with MATLAB for engineering and Scientists, Steven C. Chapra, McGraw Hill.

Numerical Methods for Engineering, Steven C. Chapra and Raymond P. Canale, McGraw Hill.

Numerical Methods using MATLAB, John H. Mathews and Kurtis D. Fink, Pearso Printice Halln.

Numerical Methods with Fortran IV case studies, Dorn & McCracken, Wiley

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Weighting of Assessments

Mid-Term Exam 12 % (15 pts)

Final-term Exam 72 % (90 pts)

Semester Work 16 % (20 pts)

Total 100% (125 pts)

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Numerical Methods ISSUES IN NUMERICAL ANALYSIS

WHAT IS NUMERICAL ANALYSIS? It is a way to do highly complicated mathematics problems on a computer.

It is also known as a technique widely used by scientists

and engineers to solve their problems.

TWO ISSUES OF NUMERICAL ANALYSIS:

How to compute? This corresponds to algorithmic aspects; How accurate is it? That corresponds to error analysis

aspects.

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ADVANTAGES OF NUMERICAL ANALYSIS:

It can obtain numerical answers of the problems that have no analytic solution.

It does NOT need special substitutions and integrations by

parts. It needs only the basic mathematical operations:

addition, subtraction, multiplication and division, plus making

some comparisons.

IMPORTANT NOTES:

Numerical analysis solution is always numerical.

Results from numerical analysis is an approximation

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NUMERICAL ERRORS

When we get into the real world from an ideal world and finite to infinite, errors arise.

SOURCES OF ERRORS:

Mathematical problems involving quantities of infinite precision.

Numerical methods bridge the precision gap by putting errors under firm control.

Computer can only handle quantities of finite precision.

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Summery Types of Errors

Usually we come across the following types of errors in numerical

analysis:

i.Inherent Errors.

These are the errors involved in the statement of a problem. When the problem is first presented to the numerical analysis it may contain

certain data or parameters.

ii. Analytic Error.

These are the errors introduced due to transforming a physical or

mathematical problem into a computational problem.

iii.Rounding and Chopping Errors

The most widely and important errors caused by applying

numerical methods is Error caused by chopping and rounding:

For example

1. 1/3=0.3333 where 1/3=0.333333. 2. e= 2.718 where e=2.7182818. 3. 1.0000 where 0.99995

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iv. Formulation Errors

When solving the problem using mathematical method, usually a simple model would be used to describe the source problem, there

for some of the factors will be put away which means simplifying the

problems which cause some error and this error is called Formulation Error. For example.

the second law Newton F= m.a where m is a mass of a particle, a is acceleration

In the fact

m0 is initial mass of particle

V is velocity

C is velocity of light

since V < C so V/C 0

and m=m0

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STABILITY AND CONVERGENCE STABILITY in numerical analysis refers to the trend of error change

iterative scheme. It is related to the concept of convergence.

It is stable if initial errors or small errors at any time remain small

when iteration progresses. It is unstable if initial errors or small

errors at any time get larger and larger, or eventually get

unbounded.

CONVERGENCE: There are two different meanings of convergence

in numerical analysis:

a. If the discretized interval is getting finer and finer after dicretizing the continuous problems, the solution is convergent to the true solution.

b. For an iterative scheme, convergence means the iteration will get

closer to the true solution when it progresses. 15

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Condition of a Problem

For a problem with input (data) x and output y, y=F(x). The problem is said to be well-conditioned if small changes in x, lead to small changes in y.

Otherwise, we say the problem is ill-conditioned.

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Stability of an Algorithm

Stability indicates the sensitivity of an algorithm for solving a problem.

An algorithm is said to be stable if small changes in the input x lead to small changes in the output y.

Otherwise, the algorithm is said to be unstable.

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Condition and Stability

Condition => data

Stability => algorithm

Ill-conditioned very hard to get a good result with even the best algorithm.

Stable given good data (not ill-conditioned), the algorithm will not yield drastically different results if round-off or small noise is added to the input data.

THE END