APROXIMATIONS AND ERRORS

Meanings and Example

Significant figures

It is also called significant digits and has been developed to formally designate the reliability of a value numerically. The number of significant figures is the number of digits plus one estimated digit that can be used with confidence. For example:

0.00001845
0.0001845
0.001845

The zeros are not always significant figures as they can be used alone to locate the decimal point. The numbers above have four significant figures. When zeros are included in large numbers, is not clear how many zeros are significant if any.
The concept of significant figures has two important implications for the study of numerical methods .

Accuracy and precision

T

o the number of significant figures representing a quantity 2) extending from the repeated readings of an instrument that measures some property physical .

For example:
When making some shots in a target shooting accuracy refers to the magnitude of the spread of the bullets. Accuracy refers to the approach of a number or measure the true value is supposed to represent. The inaccuracy (known as bias) is defined as a systematic departure from the truth.

Error Definitions

The numerical errors are generated with the use of approximations to represent the operations and quantities math . These errors include truncation, resulting from about a procedure exact mathematical and rounding errors that result from about exact numbers. For both types of errors, the relationship between the exact result and the approximate or true is given by
Estimated value = true value + error
Numerical error is equal to the difference between the true value and approximate value, ie
Eu = true value - approximate value
Example:
Suppose we have to measure the length of a bridge and a rivet, obtained 9999 and 9 cm, respectively. If the values are true 10 000 and 10 cm, calculate: a) wrong and B) the percentage relative error of each case:
Solution: a) the error in the measurement of the bridge is:
Eu = 10 000 - 9999 = 1 cm
And for the rivet is:
Eu = 10-9 = 1 cm
b) The percentage relative error for the bridge is:
Eu = 1 / 10 000 100% = 0.01%
And for the rivet is:
Eu = 1 / 10 100% = 10%
Therefore although both measures have an error of 1 cm, the relative error percentage is much larger rivet. We conclude that it has done a good job as far from the bridge, while the estimate for the fastener leaves much to be desired.
Fractional relative error = error.
True value
The relative error can also be multiplied by 100% to express it as:
Eu = true 100% error
True value
Where Eu denotes the percentage relative error
I) the approximation is larger than the true value (or the previous approach is greater than the current approach), the error is negative, if the approximation is less than the true value, the error is positive. Also in the equations , the denominator may be less than zero, which leads to a negative error.