2 properties of numerical techniques

8/7/2019 2 Properties of Numerical techniques

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PROPERTIES OF NUMERICAL

TECHNIQUES

MR.KANNAN NATARAJAN, M.TECH.,

LECTURER, MIT, MANIPAL-576104.

E-MAIL: [email protected]

MOBILE: +91-9008418513.


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PROPERTIES OF NUMERICAL

METHODS

Convergence exact solution within the limits.

Accuracy within the tolerable error limits.

Efficiency amt of time required to solve.

Consistency measure of accuracy.

18 February 2011 2Mr.Kannan Natarajan, MIT,Manipal.


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ACCURACY AND PRECISION

Accuracyrefers to how closely a computed or measured

value agrees with the true value, whileprecision refers to

how closely individual computed or measured values

agree with each other.

a) inaccurate and imprecise

b) accurate and imprecise

c) inaccurate and precise

d) accurate and precise



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ERROR DEFINITIONS

Absolute error (|Et|): the absolute differencebetween the true value and theapproximation.

True relative error: the true error divided bythe true value.



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ROUNDOFF ERRORS

Roundoff errors arise because digital

computers cannot represent some quantities

exactly.

Two major facets of roundoff errors involvedin numerical calculations:

Digital computers have size and precision limits

on their ability to represent numbers. Certain numerical manipulations are highly

sensitive to roundoff errors.



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TRUNCATION ERRORS

Truncation errors are those that result from

using an approximation in place of an exact

mathematical procedure.



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OTHER ERRORS

Blunders - errors caused by malfunctions of

the computer or human imperfection.

Model errors - errors resulting from

incomplete mathematical models.

Data uncertainty - errors resulting from the

accuracy and/or precision of the data.



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A SIMPLE MATHEMATICAL

MODEL


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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 mathematicalterms.

Models can be represented by a functional

relationship between dependent variables,independent variables, parameters, and

forcing functions.



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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 systems behavior is being

determined

Parameters - constants reflective of the systemsproperties or composition

Forcing functions - external influences acting upon the

system

Dependentvariable

! f independentvariables

, parameters, forcingfunctions



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MODEL FUNCTION - EXAMPLE

Assuming a bungee jumper is in mid-flight, an analytical model for the jumpers

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 !gm

cdtanh

gcd

m t



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MODEL RESULTS

Using a computer (or a calculator), the model can be used

to generate a graphical representation of the system.



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NUMERICAL RESULTS

The efficiency and accuracy of numerical methods

will depend upon how the method is applied.



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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



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Still we were

discussing about

problem definition and

mathematical models

Problemdefinition

Mathematical model

Problem solving tools;statistics, numerical

methods, graphics, etc.,

Numeric resultsor Graphic results

Problem solving tools;computers

Implementation.


2 properties of numerical techniques

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