Numerical Approximation
Errors, Types, Usage & Error Propagation Numerical
Approximation
Presentation EMT3201F.O.TLecturer: Ms. E. Trim Date: 24th
February, 2015 11Name Registration #Josiah Headley 13/0933/1640Mark
Ramdihal13/0933/2206Dolall Mangal13/0933/2031Sheik
Mobeen13/0933/2518Chabilall Mangal13/0933/2030Jaikeshan Takchandra
13/0933/1323Group Members Thursday, February 26, 201522
Introduction Concept of Mistakes Concepts of Errors Absolute Errors
Relative Errors Error Propagation Conclusion Related Problems
References
Outline of Presentation Thursday, February 26, 20153In any work
involving the measurement of physical quantities and any
calculations which are made using these measurements, it is
necessary to form at least one idea of the errors likely to be
involved. Henceforth the concept of numerical approximation
arises.Numerical Approximation is an inexact representation of a
numerical value that is still close enough to be useful.It allows
for the reduction in complexities in solving problems while still
yielding a fairly accurate solution, thereby introducing the
concept of mistakes and errors.
Introduction Thursday, February 26, 20154Thursday, February 26,
20155 In the language ofNumerical Analysis, a mistake (orblunder)
is not an error! A mistake is due tofallibility, (meaning that a
mistake is made by human and not by computers and calculators).
Mistakes may benegligible, with little or no effect on the
accuracy of the calculation.
Or they may be soseriousas to render the calculated results
quite wrong.
Concept of Mistakes Thursday, February 26, 20156Some common
mistakes include:Transposition of digits(for example, reading 8235
as 8325). Misreadingof repeated digits(for example, reading 74438
as 74338). Misreading of tables(for example, referring to a wrong
line or a wrong column). Incorrectly positioning a decimal
point;(for example, placing a decimal point at 19.438 as 194.38).
Overlooking signs(especially near sign changes).
Contd MistakesThursday, February 26, 20157Four ways by which
mistakes can be avoided are:Double check calculations.Care,
avoiding repetition.Ensure signs are clearly written.Knowledge of
the common sources of mistakes.How to avoid mistakes7Errors The
difference between the exact value and the approximate value
obtained is the error.
Hence, Error = Exact value - Approximate value
Concept of ErrorsThursday, February 26, 20158Errors in the
method of measurement used.Errors resulting from failure to
recognize, prevent or allow for changes in experimental variables
due to extraneous factors e.g. temperature, barometric pressure,
etc.Instrument errors.Errors in reading the instrument.Errors in
recording and tabulating data.Arithmetic errorsErrors arising from
use of approximate values for physical constants etc., or due to
shortening values to a given number of significant figures List
availability dates.An error may occur as a result of either of the
following: Thursday, February 26, 201599Round-off ErrorTruncation
ErrorNumber representation Error
Types of Errors Thursday, February 26, 201510Round-off errors
are introduced when a numerical value is converted to an
approximated value of a certain number of significant figures or
decimal places.
When rounding, add one to the last desired digit to the number
provided that the previous digit is 5 and above
If the previous digit is less than 5, no change will be made to
the desired significant digit.
Round-off Errors Thursday, February 26, 201511Truncation Error
refers to an error in a method, which occurs because some series
(finite or infinite) is truncated to a fewer number of terms. Such
errors are essentially algorithmic errors and we can predict the
extent of the error that will occur in the method.
Truncation Errors Thursday, February 26, 201512Number
representation errors are errors that occur when numbers cannot be
represented exactly by a finite number of digits.
For instance, the arithmetical operation of division often gives
a number which does not terminate; the decimal (base 10)
representation of 2/3 which is = 0.66666666.. Or
Numbers with infinite decimal places such as 2 which is =
1.414213562.. Or
Even a number such as 0.1 which terminates in decimal form but
would not terminate if expressed in binary form.
Number Representation Errors Thursday, February 26,
201513Absolute Errors Thursday, February 26, 201514For example, if
you measure the width of a book using a ruler with millimeter
marks, the best you can do is measure the width of the book to the
nearest millimeter. You measure the book and find it to be 75 mm.
You report the absolute error in the measurement as 75 mm 1 mm. The
absolute error is 1 mm.
Note that absolute error is reported in the same units as the
measurement.
Contd Absolute errors Thursday, February 26, 201515Relative
error expresses how large the absolute error is compared with the
total size of the object you are measuring.
It is therefore necessary to determine the absolute error in
order to calculate the relative error.
Relative errors are usually expressed as fraction or is
multiplied by 100 and expressed asa percent.
Relative Errors Thursday, February 26, 201516Formula for
relative error:Relative Error = Absolute Error / True Value
Example:
Contd relative errors A driver's speedometer says his car is
going 60 miles per hour (mph) when it's actually going 62 mph. The
absolute error of his speedometer is 62 mph - 60 mph = 2 mph. The
relative error of the measurement is 2 mph / 60 mph = 0.033 or
3.3%Thursday, February 26, 201517Thursday, February 26,
201518Reducing the effects of the errors mentioned can be
significantly influenced by working with more significant figures
(round off errors) or by retaining more terms (truncation
error).
Reducing the chances of mistakes will also impact the influences
of errors. How can we Reduce ErrorsOnce error estimates have been
assigned to each piece of data, we must then find out how these
errors contribute to the error in the result. The error in a
quantity may be thought of as a variation in the value of that
quantity.
Terminologies Determinate errors error which have an explicit
sign, indicating its true arithmetic value, e.g. negative (-) or
positive (+) displacement Indeterminate errors errors with an
unknown sign, e.g. average body temperature is 35C 0.5C (also can
be referred to an uncertainty)
Error Propagation Thursday, February 26, 201519Thursday,
February 26, 201520Rules for error propagation
Sum and difference rule Product rule Multiplication with a
constant Quotient rule Polynomial functionsGeneral formula
Examples
Contd error propagation Contd error propagation Thursday,
February 26, 201521Contd error propagation Thursday, February 26,
201522Contd error propagation Thursday, February 26,
201523Thursday, February 26, 201524Contd error propagation Error
Propagation Thursday, February 26, 201525Contd Error orders,
example Thursday, February 26, 201526Thursday, February 26,
201527In the field of engineering it is very important that errors
and mistakes are kept to a minimal since they can:Have an adverse
effect on the overall project in terms of miscalculations that can
lead to design flawsAn understanding of errors and mistakes and how
they can be minimized and corrected is important in achieving the
highest accuracies in engineering.
Conclusion A 1 meter long steel tube was measured by a
technician with a measuring tape. He recorded his reading as 99.7
cm. Calculate the absolute and relative errors.Absolute error =
0.3cm, Relative error = 0.3%
1.5 kg of material was weighed by a technician and found to be
1505 grams. Calculate the absolute and relative errors.Absolute
error = 5g, Relative error = 0.33%
Suppose the measured value of the temperature is Ta = 146.2, but
the true temperature is T=145.9. What are the absolute error and
the relative errors?Relative error = 2%, Absolute error = 0.3
Homework Problems Thursday, February 26, 201528Thursday,
February 26, 201529Suppose your first measurement of the
oscillation period of a pendulum ist1= 4.0 0.1 secondsand the
second ist2= 3.85 .05 seconds. To find if the two measurements are
consistent, calculate the positive differenceand the error on that
difference.Answer: Error = 0.11s
A pile driver of is used to drive a pile vertically into the
ground. The pile driver falls through a distance of 4 m 0.1 m
rebound and causes the pile move a distance of 0.5 m 0.1 m.
Determine the velocity of the driver immediately before impact
(Take g = 9.81 m/s2) Velocity = (2gs)1/2 = (2*9.81*4)1/2 =
8.85m/sError = constant* x = 9.81* 0.1 = 0.981 m/s
Homework Problems contdChemistry.com, relative and absolute
errors,
Chemistry.comhttp://chemistry.about.com/od/workedchemistryproblems/fl/Absolute-Error-and-Relative-Error-Calculation.htm
[2015-02-19]Harvard.edu, online lecture notes error propagation,
Harvard.edu
http://www.fas.harvard.edu/~scphys/nsta/error_propagation.pdf
[2015-02-20]Msu.edu, online lecture notes error propagation,
Msu.eduhttp://lectureonline.cl.msu.edu/~mmp/labs/error/e2.htm
[2015-02-20]
References Thursday, February 26, 201530
