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Chapter 01.
Measuring Errors
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 (A) identify where the error is coming from, followed by (B) quantifying the error, and lastly (C) 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 Et is the difference between the true value (also called the exact
value) and the approximate value. True Error ^ True value – Approximate value
Example 1
The derivative of a function f ( x ) at a particular value of x can be approximately calculated
by
h
f x h f x f x
of f (^2 )For f ( x ) 7 e^0.^5 x and h 0. 3 , find a) the approximate value of f ( 2 ) b) the true value of f ( 2 ) c) the true error for part (a) Solution
a) h
f x h f x f x
01.02.2 Chapter 01.
For x 2 and h 0. 3 ,
f f f
f ( 2. 3 ) f ( 2 )
7 e^0.^5 (^2.^3 ) 7 e^0.^5 (^2 )
b) The exact value of f (^2 )can be calculated by using our knowledge of differential calculus.
f ( x ) 7 e^0.^5^ x f ' ( x ) 7 0. 5 e^0.^5^ x 3. 5 e^0.^5^ x So the true value of f '( 2 )is
f ' ( 2 ) 3. 5 e^0.^5 (^2 ) 9. 5140 c) True error is calculated as E (^) t = True value – Approximate value 9. 5140 10. 265 0. 75061 The magnitude of true error does not show how bad the error is. A true error of Et 0. 722
may seem to be small, but if the function given in the Example 1
were f ( x ) 7 10 ^6 e^0.^5 x ,the true error in calculating f (^2 ) with h 0. 3 , would be
Et 0. 75061 10 ^6. 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, x 2 and the step size, h 0. 3. This brings us to the definition of relative true error.
Q : What is relative true error? A : Relative true error is denoted by t and is defined as the ratio between the true error and
the true value.
Relative True Error TrueValue
TrueError
Example 2
The derivative of a function f ( x ) at a particular value of x can be approximately calculated
by
h
f x h f x f x
For f ( x ) 7 e^0.^5 x and h 0. 3 , find the relative true error at x 2.
01.02.4 Chapter 01.
h
f x h f x f x
For x 2 and h 0. 3 ,
f f f
f ( 2. 3 ) f ( 2 )
7 e^0.^5 (^2.^3 ) 7 e^0.^5 (^2 )
b) Repeat the procedure of part (a) with h 0. 15 ,
h
f x h f x f x
For x 2 and h 0. 15 ,
f f f
f ( 2. 15 ) f ( 2 )
7 e^0.^5 (^2.^15 ) 7 e^0.^5 (^2 )
c) So the approximate error, E (^) a is
E (^) a Present Approximation – Previous Approximation 9. 8799 10. 265 0. 38474 The magnitude of approximate error does not show how bad the error is. An approximate
error of E (^) a 0. 38300 may seem to be small; but for f ( x ) 7 10 ^6 e^0.^5 x , the approximate
error in calculating f '( 2 ) with (^) h 0. 15 would be (^) Ea 0. 38474 10 ^6. 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, (^) x 2 , and (^) h 0. 15 and (^) h 0. 3. This brings us to the definition of relative approximate error.
Q : What is relative approximate error? A : Relative approximate error is denoted by a and is defined as the ratio between the
approximate error and the present approximation.
Relative Approximate Error PresentApproximation
ApproximateError
Measuring Errors 01.02.
Example 4
The derivative of a function f ( x ) at a particular value of x can be approximately calculated
by
h
f x h f x f x
For f ( x ) 7 e^0.^5 x , find the relative approximate error in calculating f (^2 )using values from
h 0. 3 and h 0. 15. Solution
From Example 3, the approximate value of f ( 2 ) 10. 263 using h 0. 3 and
f '^ ( 2 ) 9. 8800 using h 0. 15. E (^) a Present Approximation – Previous Approximation 9. 8799 10. 265 0. 38474 The relative approximate error is calculated as
a PresentApproximation
ApproximateError
Relative approximate errors are also presented as percentages. For this example, a 0. 038942 100 % = (^) 3. 8942 % Absolute relative approximate errors may also need to be calculated. In this example a | 0. 038942 | 0. 038942 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 a at the end of each iteration. The user may pre-specify a minimum
acceptable tolerance called the pre-specified tolerance, s. If the absolute relative
approximate error a is less than or equal to the pre-specified tolerance s , that is, | (^) a (^) | s ,
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 m significant digits to be correct in the answer, then you would need to have the absolute relative approximate error, m a
Measuring Errors 01.02.
e) 1. 079 10 3 has four significant digits f) 1. 0790 10 3 has five significant digits g) 1. 07900 10 3 has six significant digits
INTRODUCTION, APPROXIMATION AND ERRORS
