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01.02. Chapter 01.02
Measuring Errors 01.02.
After reading this chapter, you should be able to:
In any numerical analysis, errors will arise during the calculations. To be able to deal with the issue of errors, we need to
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.
| 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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