calculation of roots of equations

Calculation of roots of equations


The purpose of calculating the roots of an equation to determine the values of x for which holds: f (x) = 0

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 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 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 coefficient

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:

y3 + 3y2-3y = -9

0with candidates as a result of the polynomial are:

Substituting into the equation, we obtain that the only real root is y = -3,

(Which is also the only rational root of the equation). Logically, this method is very powerful, 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.

GRAPHIC METHOD
This method is used primarily to locate an interval where the function
any root.
Example 1
Locate an interval where the function f (x) = e ^ (- x) - ln x
has a root.

Solution
To calculate the root of f (x) do f (x) = 0, where e ^ (- x) = ln x.
Therefore, the problem is finding the intersection point of the functions
g (x) = e ^ (- x) and h (x) = ln x.
We know these graphs:


From which, we conclude that an interval where the only root is [1,1.5].
In fact, we are not interested in being the finest in the search interval as then apply systematic methods to approximate better root. Let's say that the usefulness of the graphic method is in providing a range with which we start work

MORE INFORMATION

calculo de raices


metodo de newton rhapson



biseccion excel




punto fijo




falsa posicion

significant figures

SIGNIFICANT FIGURES

It is considered that significant numbers are those numbers that have real meaning or provide any information. The figures are not significant as a result of the calculations and have no meaning. The significant figures of a number are determined by its error. Significant figures are those that occupy a position equal to or above the order or position of the error.

When expressed a number should be avoided if the use of significant figures, becouse it can be a source of confusion. The numbers should be rounded so that contain only significant figures. It's called rounding to the process of eliminating non-significant digits of a number.The rule to be used in the rounding of numbers are:
• If the figure is ignored is less than 5, is eliminated without more.• If the figure is greater than 5 eliminated, it increases by one unit the last digit retained.• When the digit removed is 5, last figure is taken as the nearest even number, ie if the figure is even left on hold, and if it is odd, take the higher figure.








Accuracy and Precision.
Accuracy refers to how close it is measured or calculated value of the true value. Accuracy refers to how close an individual value is measured or calculated with respect to the others.The inaccuracy is defined as a systematic departure from the truth. Imprecision, on the other hand, refers to the magnitude of the spread of values.The numerical methods should be sufficiently accurate or no bias to meet the requirements of a particular engineering problem.


Error.



In general, for any error, the relationship between the exact number and obtained by approximation is defined as: Error = Actual value-estimated value Sometimes they know exactly the value of the error, denoted as Ev, or we estimate an approximate error. Now, to define the magnitude of error, or incidence on calculating the error detected, we can normalize its value: Ea = relative error (fraction) estimated error = true value I Since the value of Ea can be either positive or negative, in many cases we want to know more the magnitude of the error, in which case we will use the absolute value of this. An interesting case is an investigation conducted by Scarborough, which determined the number of significant figures contained in the error as:

If we replace it in Eq. We will get the number of significant figures is the approximate value obtained reliable. Thus, if our calculation has an error less than the criterion for two significant figures, we get numbers that correspond to a minimum:

Es = (0.5x 102-2)% = 0.5%

This will help us determine how many terms are needed on a rough estimate to make sure that the error is under the margin is specified in ROUNDING ERROR Many times, computers cut decimal numbers between e17 and 12th decimal thus introducing a rounding error For example, the value of "e" is known as 2.718281828 ... to infinity. If we cut the number 2.71828182 (8 significant digits after the decimal point) we are obtaining or failure E = -2.71828182 2.718281828 = 0.000000008 ...

However, as we do not consider the number that was cut was greater than 5, then we should have let the number as 2.71828183, in which case the error would only E = 2.118281828 = -0.000000002 -2.11828183 .. , Which in absolute terms is much smaller than the last. In general, the cutting error of the computers will be much lower than the error introduced by a user, usually cut to a smaller number of significant figures. Depending on the magnitude of the numbers with which you work, the rounding error can have a big impact very small in the final calculation.

For example, if we have a product of 502.23 m and a dollar price of U.S. $ 7.52, the total price of U.S. $ 3,776.7696 give us (in Chilean pesos to $ 1 = $ 500 gives us $ 1,888,384 , 8). Now, if we introduce a variation of 0.1% in meters of the product and calculate the total, we get 502.23 * 0.1% = 507, 54, in U.S. $ equivalent of U.S. $ 3,816.7008 (ie, $ 1,908,350.4 Chilean pesos, a difference of $ 19,965.6), which is no less important as a variation of 0.1% in the footage gives a product greater than 1.5% error in the final price Truncation error.

Truncation errors are related to the method of approach to be used because they generally face an infinite series of terms, will tend to cut the number of terms, introducing an error at that time, not to use the full set (which supposed to be exact). In one iteration, is understood as the error by not following iteration and continue moving towards the solution. In an interval that is subdivided for a series of calculations on it, is associated with crossing number result of dividing the interval "n" times.

TOTAL NUMBER ERROR

The total numerical error is defined as the sum of the rounding and truncation errors introduced in the calculation. But here comes a big problem. The more calculations would be carried out to obtain a result, the rounding error will be increased. But on the other hand, the truncation error can be minimized by including more terms in the equation, reduce or continue the iteration step (ie larger number of calculations and probably biggest mistake of rounding.) So what criteria do we use? ... The ideal would be to determine the point where errors begin to hide the advantage of considering a lower truncation error. But as I said, is ideal, in practice we must consider that computers today have a significant handling much larger numbers than before so the rounding error is minimized greatly, but should not be allowed to forget his contribution to the error total.

darcy's law

Darcy's law is a general expression to describe the fluid flow go through a porous medium. this equation is the mathematical model describing this phenomenon:


shows that the volumetric flow go through a porous medium, is a function of cross-sectional area of the elevation and the pressure of the fluid.

MATHEMATICAL MODELS

A mathematical model is a description of a system using mathematical language. The process of developing a mathematical model is termed mathematical modelling (also written modeling). Mathematical models are used not only in the natural sciences (such as physics, biology, earth science, meteorology) and engineering disciplines, but also in the social sciences (such as economics, psychology, sociology and political science); physicists, engineers, statisticians, operations research analysts and economists use mathematical models most extensively.

in this course you can help yourself with diferents books that are excellents in the area :