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Sources of Error. Major: All Engineering Majors Authors: Autar Kaw, Luke Snyder http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM Undergraduates. Sources of Error http://numericalmethods.eng.usf.edu. Two sources of numerical error. Round off error

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sources of error

Sources of Error

Major: All Engineering Majors

Authors: Autar Kaw, Luke Snyder

http://numericalmethods.eng.usf.edu

Transforming Numerical Methods Education for STEM Undergraduates

two sources of numerical error
Two sources of numerical error
  • Round off error
  • Truncation error
round off error
Round off Error
  • Caused by representing a number approximately
problems created by round off error
Problems created by round off error
  • 28 Americans were killed on February 25, 1991 by an Iraqi Scud missile in Dhahran, Saudi Arabia.
  • The patriot defense system failed to track and intercept the Scud. Why?
problem with patriot missile
Problem with Patriot missile
  • Clock cycle of 1/10 seconds was represented in 24-bit fixed point register created an error of 9.5 x 10-8 seconds.
  • The battery was on for 100 consecutive hours, thus causing an inaccuracy of
problem cont
Problem (cont.)
  • The shift calculated in the ranging system of the missile was 687 meters.
  • The target was considered to be out of range at a distance greater than 137 meters.
effect of carrying significant digits in calculations http numericalmethods eng usf edu
Effect of Carrying Significant Digits in Calculationshttp://numericalmethods.eng.usf.edu
find the contraction in the diameter
Find the contraction in the diameter

Ta=80oF; Tc=-108oF; D=12.363”

α = a0+ a1T + a2T2

regressing data in excel general format
Regressing Data in Excel(general format)

α = -1E-05T2 + 0.0062T + 6.0234

observed and predicted values
Observed and Predicted Values

α = -1E-05T2 + 0.0062T + 6.0234

regressing data in excel scientific format
Regressing Data in Excel (scientific format)

α = -1.2360E-05T2 + 6.2714E-03T + 6.0234

observed and predicted values15
Observed and Predicted Values

α = -1.2360E-05T2 + 6.2714E-03T + 6.0234

observed and predicted values16
Observed and Predicted Values

α = -1.2360E-05T2 + 6.2714E-03T + 6.0234

α = -1E-05T2 + 0.0062T + 6.0234

truncation error
Truncation error
  • Error caused by truncating or approximating a mathematical procedure.
example of truncation error
Example of Truncation Error

Taking only a few terms of a Maclaurin series to

approximate

If only 3 terms are used,

another example of truncation error

secant line

P

tangent line

Q

Figure 1. Approximate derivative using finite Δx

Another Example of Truncation Error

Using a finite

to approximate

another example of truncation error22
Another Example of Truncation Error

Using finite rectangles to approximate an integral.

example 1 maclaurin series
Example 1 —Maclaurin series

Calculate the value of

with an absolute

relative approximate error of less than 1%.

6 terms are required. How many are required to get at least 1 significant digit correct in your answer?

example 2 differentiation
Example 2 —Differentiation

Find

for

using

and

The actual value is

Truncation error is then,

Can you find the truncation error with

example 3 integration
Example 3 — Integration

Use two rectangles of equal width to approximate the area under the curve for

over the interval

integration example cont
Integration example (cont.)

Choosing a width of 3, we have

Actual value is given by

Truncation error is then

Can you find the truncation error with 4 rectangles?

additional resources
Additional Resources

For all resources on this topic such as digital audiovisual lectures, primers, textbook chapters, multiple-choice tests, worksheets in MATLAB, MATHEMATICA, MathCad and MAPLE, blogs, related physical problems, please visit

http://numericalmethods.eng.usf.edu/topics/sources_of_error.html

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THE END

http://numericalmethods.eng.usf.edu