Errors: approximation error and Round off error.

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Errors

• Prepared by B.Anil Kumar

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Source of errors.

• Errors caused by using different machines or
  instruments.
• Modelling Errors. I.e. The real life problems are
  sometimes not possible to represent actual
  problems in mathematics.
• Computer arithmetic. I.e. on digital computer
  data processed but due to limitation of
  representation of numbers, the data may under
  flow or over flow.
• Programming error All types of errors arises due
  to computer is known as programming error or
  bugging.
• Chopping error. Sometimes for calculations we are
  taking limited number of decimal digits by simply
  ignoring the rest of digits.

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Source of errors.(continue..)

• Round-off errors- These errors occur due to
  numbers having limited significant digits. I.e.
  rounding the number to some specified number of
  digits.
• Inherent error - The error which is inherent in
  a numerical method itself is called inherent
  error.
• Truncation Error The Truncation error arises due
  to the replacement of infinite process by a
  finite one.Example-

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True error calculations
True error can be calculated when true value is
known in advance
Absolute error ( True error). It is defined as
the difference between true value of a quantity
and its approximate value.
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True error calculations (continue)
Relative error (True fractional relative error) -
It is defined as
Relative percent error ( True percent relative
error) -
It is defined as
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Example
Example Find the absolute error, relative and
relative percent error for
Ans-
Et 2.7182818-2.71428570.003996
ER(0.003996/2.7182818)0.00147
etEpER1000.147.
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Example-
Example- suppose that the lengths of a bridge
and rivet are 99999 cm respectively. If the true
values are 10000 and 10 cm respectively, then
compute true error and true percent relative
error.
Solution-
Bridge-
XT10000
XA9999
et(1/10000)1000.01
Et10000-99991
Rivet-
XT10
XA9
et(1/10)10010
Et10-91
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Note-

• The absolute error of a result correct to n
  significant figures cannot be grater than half
  unit in the nth place.

Example- Round-off 81.535 up to 4 significant
digits and compute Et
XA81.54
XT81.535
Et81.535-81.54 0.005
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Approximation Error-

• When the true value is unknown and the
  approximated values are calculated in an
  iterative method. Then the approximation error is
  calculated as follow.

EAApprox. ErrorCurrent approximation- Previous
approximation
The process of finding approximate percent
relative error ea is repeated until it will meet
the following criteria.
eaes
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Approximation Error- (continue.)
Here es is given stopping criteria.
In case of number of significant digits specified
for approximation, then es can be formulated as
es(0.5102-n)
Here n is the no. of significant digits
specified
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Example-

• Let us consider in some calculation, for x we
  are getting following approximated values.

Then find the
In each approximated value.
Ans-
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Assignment
Find the approximate value of
at x0.5 for the function
up to three significant figures.
Also find approximate percent relative error in
each approximation.
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Thank You ALL

• Presented by
• Boina Anil Kumar
• Asst.Prof in Mathematics
• MITS, Rayagada.
• Odisha, India
• E-mail anil.anisrav_at_gmail.com
• Visit at http//www.mits.edu.in/academics.phpfac
  ulty_BSH-tab