Home > Textbook notes on measuring errors

01.02. Chapter 01.02

Measuring Errors 01.02.

*After reading
this chapter, you should be able to:*

*find the true and relative true error,**find the approximate and relative approximate error,**relate the absolute relative approximate error to the number of significant digits at least correct in your answers, and**know the concept of significant digits.*

In any numerical analysis, errors will arise during the calculations. To be able to deal with the issue of errors, we need to

- identify where the error is coming from, followed by
- quantifying the error, and lastly
- minimize the error as per our needs.

In this chapter,
we will concentrate on item (B), that is, how to quantify errors.

**Q**: What
is true error?

**A**: True
error denoted by is the difference between the true value (also called
the exact value) and the approximate value.

True Error True value – Approximate value

The derivative
of a function at a particular value of can be approximately calculated
by

of For and , find

a) the approximate value of

b) the true value of

c) the true error for part (a)

a)

For and ,

b) The exact
value of can be calculated by using our knowledge of differential calculus.

So the true
value of is

c) True error is calculated as

= True value – Approximate value

The magnitude
of true error does not show how bad the error is. A true error
of may seem to be small, but if the function given in the Example 1
werethe true error in calculating with would be This value of
true error is smaller, even when the two problems are similar in that
they use the same value of the function argument, and the step size,
. This brings us to the definition of relative true error.

**Q**: What
is relative true error?

**A**:
Relative true error is denoted by and is defined as the ratio between
the true error and the true value.

Relative True Error

The derivative
of a function at a particular value of can be approximately calculated
by

For and , find the relative true error at .

From Example 1,

= True value – Approximate value

Relative true
error is calculated as

Relative true
errors are also presented as percentages. For this example,

Absolute relative
true errors may also need to be calculated. In such cases,

= 0.0758895

=

**Q**: What
is approximate error?

**A**: In
the previous section, we discussed how to calculate true errors.
Such errors are calculated only if true values are known. An example
where this would be useful is when one is checking if a program is in
working order and you know some examples where the true error is known.
But mostly we will not have the luxury of knowing true values as why
would you want to find the approximate values if you know the true values.
So when we are solving a problem numerically, we will only have access
to approximate values. We need to know how to quantify error for such
cases.

Approximate error is denoted by and is defined as the difference between the present approximation and previous approximation.

Approximate Error Present Approximation – Previous Approximation

The derivative
of a function at a particular value of can be approximately calculated
by

For and at , find the following

a) using

b) using

c) approximate error for the value of for part (b)

a) The approximate expression for the derivative of a function is

.

For and ,

b) Repeat the
procedure of part (a) with

For and ,

c) So the approximate error, is

Present Approximation – Previous Approximation

The magnitude
of approximate error does not show how bad the error is . An approximate
error of may seem to be small; but for , the approximate error in calculating
with would be . This value of approximate error is smaller, even when
the two problems are similar in that they use the same value of the
function argument, , and and . This brings us to the definition of
relative approximate error.

**Q**: What
is relative approximate error?

**A**: Relative
approximate error is denoted by and is defined as the ratio between
the approximate error and the present approximation.

Relative Approximate Error

The derivative
of a function at a particular value of can be approximately calculated
by

For , find the relative approximate error in calculating using values from and .

From Example 3, the approximate value of using and using .

Present Approximation – Previous Approximation

The relative
approximate error is calculated as

Relative approximate
errors are also presented as percentages. For this example,

=

Absolute relative
approximate errors may also need to be calculated. In this example

or 3.8942%

**Q**: While
solving a mathematical model using numerical methods, how can we use
relative approximate errors to minimize the error?

**A**: In
a numerical method that uses iterative methods, a user can calculate
relative approximate error at the end of each iteration. The
user may pre-specify a minimum acceptable tolerance called the pre-specified
tolerance, . If the absolute relative approximate error is less
than or equal to the pre-specified tolerance , that is, , then the acceptable
error has been reached and no more iterations would be required. Alternatively,
one may pre-specify how many significant digits they would like to be
correct in their answer. In that case, if one wants at least
significant digits to be correct in the answer, then you would need
to have the absolute relative approximate error, %.

If one chooses 6 terms of the Maclaurin series for to calculate , how many significant digits can you trust in the solution? Find your answer without knowing or using the exact answer.

Using 6 terms,
we get the current approximation as

Using 5 terms,
we get the previous approximation as

The percentage
absolute relative approximate error is

Since, at least
2 significant digits are correct in the answer of

**Q**: But
what do you mean by significant digits?

**A**: Significant
digits are important in showing the truth one has in a reported number.
For example, if someone asked me what the population of my county is,
I would respond, ��The population of the Hillsborough county area is
1 million��. But if someone was going to give me a $100 for every
citizen of the county, I would have to get an exact count. That
count would have been 1,079,587 in year 2003. So you can see that
in my statement that the population is 1 million, that there is only
one significant digit, that is, 1, and in the statement that the population
is 1,079,587, there are seven significant digits. So, how do we
differentiate the number of digits correct in 1,000,000 and 1,079,587?
Well for that, one may use scientific notation. For our data we show

to signify the correct number of significant digits.

Give some examples of showing the number of significant digits.

- 0.0459 has three significant digits
- 4.590 has four significant digits
- 4008 has four significant digits
- 4008.0 has five significant digits
- has four significant digits
- has five significant digits
- has six significant digits

INTRODUCTION, APPROXIMATION AND ERRORS | |

Topic | Measuring Errors |

Summary | Textbook notes on measuring errors |

Major | General Engineering |

Authors | Autar Kaw |

Date | November 18, 2009 |

Web Site | http://numericalmethods.eng.usf.edu |

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