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.
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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.
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