Chapter 2

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Chapter 2: Numerical Approximation

By Erika Villarreal

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Error analysis

In this unit, we define the concept of error in terms of numerical methods. Because these strategies are to be used when approximation is not possible to apply analytical techniques or approximation in solving practical problems of implementation, it yields a result that differs from the true value being sought. This difference is called error. This concept is of vital importance in numerical methods, since it is used as reference for the selection and evaluation of numerical methods of approximation and stop them. Here are some key concepts to a full understanding of the concept of error.


Significant Figures

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.

It is the concept that has been developed to describe formally the reliability of a numerical value. Significant digits of a number, are those that can be employed reliably to describe a quantity. For example, suppose you have an instrument where your meter brand:

The instrument can handle two-digit accuracy. The third is estimated. So in general only have three significant digits for the instrument.It is important to note that the zeros are not always significant digits, these can be used to locate the decimal point, for example:

a) 0.00001845b) 0.0001845c) 0.001845

d) 0.0000180

have four significant digits, where the number 1 is the first significant digit (digit significant main or most significant digit),8 is the second significant digit, 4 is the third significant digit and 5 is the fourth.

has three significant digits, 1, 8 and 0


Significant Figures

The importance of the concept of significant figures in the study of numerical methods is mainly on two aspects

1. Criteria to determine the precision of a numerical method. Method is acceptable when it guarantees a certain number of significant figures in the result

2. Stop Criterion. As numerical methods are iterative techniques can be established that when it reaches a certain number of significant digits is sufficient condition to stop the method


Exactitude and precision

-Exactitude .- Whether the calculated value is close to the true value.

-Precision.- Whereas numerical methods are iterative techniques, expresses how close an approximation or an estimate value with respect to approximations or previous iterations of the same.

-Inexactitude .-. It's a systematic removal of real value to calculate.

.- Imprecision or uncertainty. It is the measure of distance between them at the various approximations to a true value.

By observing the above definitions, can be determined that the error associated with numerical methods to measure the degree of exactitude and precision of them.


Definition of error

Overall, the error of a numerical method is the difference between the true value being sought and the approximation obtained through a numerical technique

Rounding error

Truncation error


. Truncation error

When using the finite number of terms to calculate a value that requires an infinite number of terms. For example, an expression to accurately determine the value of the Euler number (base of natural logarithms) through a series of MacLaurin is:

However, an approximation of that value can be obtained through finite expression:

this finite expression is manageable computationally speaking, contrary to the formula set out in its infinite form..


. Rounding error

This is because a computer can only represent a finite number of terms. To express a quantity with an infinite decimal expansion, you have to do without most of them. For example, the number π = 3.14159265 ...., has a non-periodic infinite decimal expansion. Therefore, for purposes of calculation, only take some of their digits. This is accomplished through two strategies:

1. Rounding. It ignores a number of significant figures and make an adjustment on the final figure is not discarded: π ≈ 3.1416

2. Cutting or pruning: to forego a number of significant figures without making an adjustment to the latter figure does not discarded π ≈ 3.1415 In actuarial applications, science and engineering, we recommend rounding, since the cutting or pruning involves the loss of information.


Example

Consider the approximation of π ≈ 3.14159265. Perform cutting and rounding:

a) Two significant digits.b) three significant digits.c) Four significant digits.d) Five significant digits.e) Six significant digits.f) Seven significant digits.g) Eight significant digits.

Solution:

The respective cutting and rounding to the respective number of significant digits, is summarized in the following table:


Once established classification error (the two sources of error in numerical methods), we proceed to define the concepts of true absolute error, relative absolute error, approximate absolute error and approximate relative error estimate, all of them as a sum or a result of rounding and truncation errors. The following concepts can be used as error criteria for unemployment and measures of accuracy of numerical methods.


Remarks on tolerance t of a numerical method

If the following criterion is met, the result is correct to n significant digits:

ExampleConsider the MacLaurin series for the determination of :

Beginning with the first term and adding one term at a time, estimating the value . After adding each term, calculating the real and approximate relative error. The calculation ends until the absolute value of the approximate error is less than the t pre-determined criteria to ensure correct three significant digits


SolutionConsider the number 1.648721271 and the true value . If you want three correct significant digits, it must be n =3. Therefore:

is guaranteed at least three significant digits correct, must be met:

In this case,

Namely

The following table shows the development of exercise:


SolutionIn the sixth iteration satisfies the criteria for tolerance, ea <t, given that the sixth iteration, ea = 0.0158, which is less than the predetermined tolerance: t = 0.05%. This will have to estimate is 1.648697917, with at least three correct significant digits: 1.648697917.


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· Burden Richard L. & Faires J. Douglas, Análisis numérico. 2ª. ed., México, Grupo Editorial Iberoamérica, 1993.

· Chapra Steven C. & Canale Raymond P., Métodos numéricos para ingenieros. 4ª. ed., México, McGraw-Hill, 2003.

· Gerald Curtis F. & Wheatly Patrick O., Análisis numérico con aplicaciones. 6ª. ed., México, Prentice Hall, 2000.

· Maron Melvin J. & López Robert J., Análisis numérico, con enfoque práctico. México, Editorial CECSA, 1995.

· Mathews John H. & Fink Kurtis D., Métodos numéricos con MATLAB. 3ª. ed., España, Pearson-Prentice Hall, 2004.

. http://www.acatlan.unam.mx/acatlecas/mn/MN_01.htm

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