APPROACHES

The numerical methods are useful alternative procedures to solve math problems for which complicates the use of traditional analytical methods and, occasionally, are the only possible solution.


These are techniques by which a mathematical model is solved using only arithmetic operations, tedious arithmetic.


They are systematic techniques whose results are approximations of the true value that assumes the variable of interest, the consistent repetition of the technique, which is called iteration, is what you closer and closer to the desired value.


Numerical Approach


The numerical approximation is divided into: figures meanings, correctness, accuracy, convergence, stability.


- Numerical approximation X * means a figure that represents a number whose exact value is X. To the extent that the number X * is closer to the exact value X, is a better approximation of that number


- Examples:


3.1416 is a numerical approximation of p,
2.7183 is a numerical approximation of e,
1.4142 is a numerical approximation of O 2, and
0.333333 is a numerical approximation of one third.
Significant figures


The measurements are normally done through instruments, such as a speedometer to measure the speed of a car, or an odometer to measure the distance covered.
The number of significant figures is the number of digits t, which can be used with confidence to measure a variable, for example, three significant figures on the speedometer and 7 significant figures on the odometer.
The zeros are included in a number are not always significant figures, for example, the numbers 0.00001845, 0.001845, 184 500 1845 and apparently have four significant figures, but would have to know the context in which they are working on each case, to identify how many and zeros which should be considered as significant figures.
The management of significant figures can develop criteria to detect how accurate are the results obtained and assess levels of accuracy and precision with which they are expressed some numbers such as p, e ó Ö2.
Alternatively the number of significant figures, n is the number of digits in the mantissa, which indicates the number of numbers to consider, after the decimal point. In manual operations, the number of digits in the mantissa is still relevant, although it has been displaced gradually the number of significant figures, by design, manage calculators and computers.


Accuracy and precision.


Accuracy refers to the number of significant figures represents a quantity.
Accuracy refers to the approach of a number or measure the numerical value is supposed to represent.
Example: p is an irrational number, consisting of an infinite number of digits; 3.141592653589793 ... is a very good approximation of p, which may be considered which is its exact value. In considering the following approximations of p:
p = 3.15 is vague and inaccurate.


p = 3.14 is accurate but imprecise.


p = 3.151692 is precise but inaccurate.


p = 3.141593 is accurate and precise.


The numerical methods should provide sufficiently accurate and precise solutions. The term error is used to represent both the inaccuracy and to measure the uncertainty in the predictions.
Convergence and stability


Convergence of a numerical method is defined as ensuring that, when making a "good number" of iterations, the approximations obtained eventually move closer and closer to the true value sought.
To the extent that a numerical method requires fewer iterations than the other, to approach the desired value, is said to have a faster convergence.
Stability of a numerical method means the level of assurance of convergence and is that some numerical methods do not always converge and, by contrast, differ; that is, away from the more desired result.
To the extent that a numerical method, to a very wide range of possibilities of mathematical modeling, it is safer to converge than the other, is said to have greater stability.
It is common to find methods that converge quickly, but they are very unstable and, in contrast, very stable models, but slow convergence.