ROOTS OF EQUATIONS

The purpose of calculating the roots of an equation is to determine the values of x for which holds:

f (x) = 0 (28)

The determination of the roots of an equation is one of the oldest problems in mathematics and there have been many efforts in this regard. Its importance is that if we can determine the roots of an equation we can also determine the maximum and minimum, eigenvalues of matrices, solving systems of linear differential equations, etc ...

The determination of the solutions of equation (28) can be a very difficult problem. If f (x) is a polynomial function of grade 1 or 2, know simple expressions that allow us to determine its roots. For polynomials of degree 3 or 4 is necessary to use complex and laborious methods. However, if f (x) is of degree greater than four is either not polynomial, there is no formula known to help identify the zeros of the equation (except in very special cases).

There are a number of rules that can help determine the roots of an equation:

• Bolzano's theorem, which states that if a continuous function, f (x) takes on the ends of the interval [a, b] values of opposite sign, then the function accepts at least one root in that interval.

• In the case where f (x) is an algebraic function (polynomial) of degree n and real coefficients, we can say that will have n real roots or complex.

• The most important property to verify the rational roots of an algebraic equation states that if p / q is a rational root of the equation with integer coefficients:

then the denominator q divides the coefficient a and the numerator p divides the constant term a0.

Example: We intend to calculate the rational roots of the equation:

3x3 3x2 - x - 1 = 0

First, you make a change of variable x = y / 3:

and then multiply by 32:

3y2 y3-3y = -90

with candidates as a result of the polynomial are:

Substituting into the equation, we obtain that the only real root is y = -3, that is to say, (which is also the only rational root of the equation). Logically, this method is not as effective, so we can serve only as guidelines.

Most of the methods used to calculate the roots of an equation are iterative and are based on models of successive approximations. These methods work as follows: from a first approximation to the value of the root, we determine a better approximation by applying a particular rule of calculation and so on until it is determined the value of the root with the desired degree of approximation.

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.

MODELING:

The perception of the world can be described as a succession of phenomena. From the beginning of time man has sought to discover them, whether they understand them or no. It’s apparent that an interpretation of the world is necessary, which must be sufficiently abstract to avoid being affected by the dynamics of the world (the small changes) and should be robust enough to represent that the data and the world are related. A tool like this is called data model, which can represent more or less reasonable some reality. The data model allows abstractions of the world, allowing focus on the macros, without worrying about the specific, so our concern is focused on generating a representation scheme, and not the values of the data.

Models can be called somewhat verbal, physical, mathematical and resolution model, where the latter can be divided into two types, one as an analytical solution model and other as numerical solution model.

The analytical solution model is considered as direct and accurate for linear problems, simple geometry, low dimension, few variables and problems come to regard as ideal.

The numerical solution model is considered approximate, using iterations, nonlinear problems, complex geometries, various sizes, and depending on many variables, then it would be a solution to a real problem.

Another important model is the computational model, basically this is responsible for keeping a mathematical language model in a computer language for this to be processed on a computer.

In the oil industry the models we use as a tool in reservoir characterization, where this is divided into two classes: static and dynamic characterization.

The static characterization is responsible for the reservoir geology, it is considered that this does not change with time and manages the structural, stratigraphic, sedimentological, petrophysical and geo statistics models.

Structural model: is responsible for reservoir geometry, structure and x-ray of it.

Stratigraphic model: define the sequence of sedimentary rocks and their facies.

Sedimentological model: defines the type of environment in which were deposited or which formed it in sedimentary rocks.

Petrophysical model: analyzes the variables of petrophysical properties such as porosity, permeability, etc..

Geostatistical models: Find the petrophysical properties between wells.

The dynamic characterization is responsible for the properties of reservoir fluids, because they change over time.