Mathematical modeling

Mathematical modeling

A mathematical model is a representation of a phenomenon in the real world, using mathematical concepts to describe it. This is built for the purpose of questioning in relation to one aspect of the real world.
The formulation of a mathematical model involves:

• Identification of factors (variables) that is specific to the system.
• Establish reasonable assumptions about the system.


In the formulation of hypotheses involving variables change, a reason which they are involved, thus variables that are altered due to other, that is to say, derivatives or differential equations.



CHARACTERISTICS OF A MODEL:
• Diagrams or sketches are for the most part, they represent reality.
• Its main objective is to explain and predict issues of a situation.


NUMERIACAL APPROXIMATION
An approximation is an inexact representation of something that is still close enough to be useful. Although approximation is most often applied to numbers, it is also frequently applied to such things as mathematical functions, shapes, and physical laws.

ERROR
In science and engineering in general an error is defined as a difference between the desired and actual performance or behavior of a system or object.

ERROR ROUNDING
The rounding error is the loss of mathematical precision that occurs when rounded to the decimal part of a number. It's the difference between the value of the original number and value of the number after you apply the rounding.



ERROS TRUNCATION
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 complete series.



SIGNIFICANT FIGURES
The significant figures of a number are those digits that carry meaning contributing to its precision.


TOTAL NUMBER ERROR
The total numerical error is defined as the sum of the rounding and truncation errors introduced in the calculation.

Bibliography
Javier Aracil Santonja 1986 editorial TECNOS S.A 1986 o’Donnel 27-Madrid -9.
http://www2.uca.es/matematicas/Docencia/2005-2006/FC/0206024/Apuntes/tema1_0506.pdf
http://html.rincondelvago.com/metodos-numericos_5.html
http://www.slideshare.net/nestorbalcazar/mtodos-numricos-03
http://idkernel.sytes.net/MetNum/pdfs/raices2.pdf
http://noosfera.indivia.net/metodos/secante.html
http://es.wikipedia.org/wiki/M%C3%A9todo_de_la_secante
http://www.itescam.edu.mx/principal/sylabus/fpdb/recursos/r45622.PDF

Graphical method

Closed methods

A simple method to obtain an approximation to the root of the equation F (x) = 0; is graph and see where it cuts the x-axis.

Example: Y = x^2 – 3;

using a software:
X1≈ 1.732
X2 ≈ -1.732

The bisection method

If f(x) is real and continuous in the range from xi to xf f(xf) have opposite signs, then there is at least one real root between intervals.
The bisection method is a type of incremental search in which the interval is always divided in half. If the value of the function changes sign on an interval, we evaluate the value of the function at the midpoint.

Root position is determined at the point half the subinterval in which a sign change occurs.
This process is repeated to achieve the best approximation.

The false position method

An alternative technique bisection method is a straight line joining f (xi) and f (xf).
The intersection of this line with the x-axis represents a better approximation to the root; the fact that the curve is replaced by a straight line gives a false position of the root.



From http://www.slideshare.net/nestorbalcazar/mtodos-numricos-03



From http://www.slideshare.net/nestorbalcazar/mtodos-numricos-03

Open methods

Open methods

Open methods are based on formulas that require a single initial value (fixed point or Newton Raphson) or starting with a couple of them (secante), not necessarily surround the root.

Fixed point

Open methods use a formula to predict the root. This formula can be developed as a simple fixed point iteration, to rearrange the equation f (x) = 0 so that x is the left side of the
equation.

Newton-Raphson’s Method

Perhaps, in the formulas to find roots, the Newton-Raphson formula is the most widely used. If the initial value of the root is then xi can extend a tangent from the point [x, f (x)] of the curve. The point where this tangent crosses the x-axis represents a better approximation of the root.


From http://www.slideshare.net/nestorbalcazar/mtodos-numricos-03


Example:

Secant’s method

This method is a modification of the Newton-Raphson’s method; since this is not necessary to calculate the derivative then you have the following expression.

f^' (x)=(f(x_(i-1) )-f(x_i))/(x_(i-1)-x_i ) (1)

Newton raphson

x_(i+1)= x_i-(f(x_i))/(f^' (x_i)) (2)

(1) in (2) Secante

x_(i+1)≈x_i-(f(x_i)(x_(i-1)-x_i ))/(f(x_(i-1))-f(x_i))

This expression is replaced in the Newton-Raphson’s equation and gives us the formula is called the secant method, and this is to approximate the slope of the straight line connecting the function evaluated at the point of study and at the point the previous iteration.

Example:

Muller’s method

This is a method for finding roots of polynomial equations of the general form:

Where n is the order of the polynomial and are constant coefficients. Continuing with the polynomials, they comply with the following rules:
•For equation of order n, there are n real or complex roots. It should be noted that these roots aren’t necessarily distinct.
•If n is odd, there is at least one real root
•If the roots are complex, there is a conjugate pair.

With the secant method by drawing a straight line crossing the three-point function,
In the muller’s method these points crossing the function but approximating to a parabola, then find the point which cuts the x-axis.

Bairstow’s method

Bairstow's method is an efficient algorithm for finding the roots of a polynomial of arbitrary degree. This method is more efficient as the others methods because it finds real and imaginary roots.
The method estimated the roots, using quadratic factors like:
x^(2 )- hx-d

Where the polynomial
p(x)=(x^2-hx-d) p_1 (x)

If x^(2 )- hx-d is not quadratic polynomial factor this would be expressed as follows:
p(x)=(x^2-hx-d) p_1 (x)+Ax+B

Where A and B are function of h and d’s, then looking for values of h and d where A and B are iqual to cero.
The use the newton-raphson`s method.