Advanced Numerical Analysis Lecture Notes
(
) Let y*=1,000,000 and y=999,996; then the error is Ey =|y*-y|=|1000000-999996|=4
Chapter 1:
(
and the relative error is
RE y
y* y y
4 1000000
0.000004
) Let z*=0.000012 and z=0.000009; then the error is Ez =|z*-z|=|0.000012-0.000009|=0.000003 and the relative error is
Error Analysis
1
REz
z* z z
0.000003 0.000012
4
0.25
In the practice of numerical analysis it is important to be aware that computed solutions are not exact mathematical solutions. The precision of a numerical solution can be diminished in several subtle ways. Understanding these difficulties can often guide the practitioner in the proper implementation and/or development of numerical algorithms. Definition 1.1 Suppose x is an approximation to x*. The absolute error is Ex = |x* -x|.
And the relative error is REx that x*≠0 .
2
In case ( ), there is not too much difference between Ex and REx, and either could be used to determine the accuracy of x. In case ( ), the value of y is of magnitude 106, the error Ey is large, and the relative error REx is small. In this case, y would probably be considered a good approximation to y*. In case ( ), z is of magnitude 10-6 and the error Ez is the smallest of all three cases, but the relative error REz is the largest. In terms of percentage, it amounts to 25%, and thus z is a bad approximation to z*.
5
x* x , provided x
The absolute error is simply the difference between the true value and the approximate value, whereas the relative error expresses the error as a percentage of the true value. Example 1.1 Find the error and relative error in the following three cases.
Observe that as |x*| moves away from 1 (greater than or less than) the relative error REx is a better indicator than Ex of the accuracy of the approximation. Relative error is preferred for floating-point representations since it deals directly with the mantissa. Definition 1.2 The number x is said to approximate x* to d significant
) Let y*=1,000,000 and y=999,996; then the error is Ey =|y*-y|=|1000000-999996|=4
Chapter 1:
(
and the relative error is
RE y
y* y y
4 1000000
0.000004
) Let z*=0.000012 and z=0.000009; then the error is Ez =|z*-z|=|0.000012-0.000009|=0.000003 and the relative error is
Error Analysis
1
REz
z* z z
0.000003 0.000012
4
0.25
In the practice of numerical analysis it is important to be aware that computed solutions are not exact mathematical solutions. The precision of a numerical solution can be diminished in several subtle ways. Understanding these difficulties can often guide the practitioner in the proper implementation and/or development of numerical algorithms. Definition 1.1 Suppose x is an approximation to x*. The absolute error is Ex = |x* -x|.
And the relative error is REx that x*≠0 .
2
In case ( ), there is not too much difference between Ex and REx, and either could be used to determine the accuracy of x. In case ( ), the value of y is of magnitude 106, the error Ey is large, and the relative error REx is small. In this case, y would probably be considered a good approximation to y*. In case ( ), z is of magnitude 10-6 and the error Ez is the smallest of all three cases, but the relative error REz is the largest. In terms of percentage, it amounts to 25%, and thus z is a bad approximation to z*.
5
x* x , provided x
The absolute error is simply the difference between the true value and the approximate value, whereas the relative error expresses the error as a percentage of the true value. Example 1.1 Find the error and relative error in the following three cases.
Observe that as |x*| moves away from 1 (greater than or less than) the relative error REx is a better indicator than Ex of the accuracy of the approximation. Relative error is preferred for floating-point representations since it deals directly with the mantissa. Definition 1.2 The number x is said to approximate x* to d significant