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jueves, 6 de mayo de 2010

ROOTS OF EQUATIONS

ROOTS OF EQUATIONS





Given a continuous function f (x), find the value x0 of x for which f (x0) = 0. We assume that both x and f (x) are real, although some of the algorithms that we see are valid for complex functions analytic complex variable. The values for which x0 satisfies f (x0) = 0 are called roots of the equation.



CLOSED METHODS

1. Graphic Methods


A simple method to obtain an approximation to the root of the equation f (x) = 0, is to plot the function and see where it crosses the x-axis.

Example: f(x)=x3+x2-3x+5


The graph shows the existence of some roots, including a double root at x = 4.2






There is a root in X= 4.23 and X=4.26

metodo grafico



2. The Bisection Method



In general, if f(x) is real and continuous in the interval from X1 to Xu and f(x1) and have f(Xu) opposite signs, that is

F(X1)F(Xu)<0>

then there is at least one real root between X1 and Xu.

METODOS CERRADOS


OPEN METHODS



3. Simple Fixed-point Iteration
Open methods employ a formula to predict the root. Such a formula can be develped for simple fixed-point iteration by rearranging the function f(x)=0, so that s is on the left-hand side of the equation:

x=g(x)






Figure 2.1: Graphical depiction of simple fixed-point method.



4. The Newton-Raphson Method


If the initial guess at the root is Xi, a tangent can be extended from the point (Xi, X(Xi)). The point where this tangent crosses the x axis usaually represents an improved estimate of the root.
The Newton-Raphson formula is

Xi+1=Xi –( f(Xi)/f’(Xi))



http://www.uv.es/diazj/cn_tema2.pdf



Open Methods



ROOTS OF POLYNOMINALS


Every polynomial P in x corresponds to a function, ƒ(x) = P (where the occurrences of x in P are interpreted as the argument of ƒ), called the polynomial function of P; the equation in x setting f(x) = 0 is the polynomial equation corresponding to P. The solutions of this equation are called the roots of the polynomial; they are the zeroes of the function ƒ (corresponding to the points where the graph of ƒ meets the x-axis). A number a is a root of P if and only if the polynomial x − a (of degree one in x) divides P. It may happen that x − a divides P more than once: some power (x − a)m divides P; the largest such power m is called the multiplicity of the root a in P. When P is the zero polynomial, the corresponding polynomial equation is trivial, and this case is usually excluded when considering roots: with the above definitions every number would be a root of the zero polynomial, with undefined (or infinite) multiplicity. With this exception made, the number of roots of P, even counted with their respective multiplicities, cannot exceed the degree of P.

A polynomial function in one real variable can be represented by a graph.

The graph of the zero polynomial
f(x) = 0
is the x-axis.
The graph of a degree 0 polynomial
f(x) = a0, where a0 ≠ 0,
is a horizontal line with y-intercept a0
The graph of a degree 1 polynomial (or linear function)
f(x) = a0 + a1x , where a1 ≠ 0,
is an oblique line with y-intercept a0 and slope a1.
The graph of a degree 2 polynomial
f(x) = a0 + a1x + a2x2, where a2 ≠ 0
is a parabola.



The graph of a degree 3 polynomial
f(x) = a0 + a1x + a2x2, + a3x3, where a3 ≠ 0
is a cubic curve.



The graph of any polynomial with degree 2 or greater
f(x) = a0 + a1x + a2x2 + ... + anxn , where an ≠ 0 and n ≥ 2
is a continuous non-linear curve.
The graph of a non-constant (univariate) polynomial always tends to infinity when the variable increases indefinitely (in absolute value).

Polynomial graphs are analyzed in calculus using intercepts, slopes, concavity, and end behavior.

The illustrations below show graphs of polynomials.





Roots of Polynomials


miércoles, 5 de mayo de 2010

NUMERICAL APPROXIMATION

Numerical Approximation

Approximation usually occurs when an exact form or an exact numerical number is unknown or difficult to obtain. However some known form may exist and may be able to represent the real form so that no significant deviation can be found. It also is used when a number is not rational, such as the number π, which often is shortened to 3.14, or √7 as ≈ 2.65. Numerical approximations sometimes result from using a small number of significant digits.

Significant Digits
Significant digits give an indication of the accuracy of a number. A digit which is 0 is significant if it is not a place holder.

Accuracy and Precision
Accuracy
refers to the number of significant digits in a number.
Precision refers to the decimal position of the last significant digit.

Example:
Comparing the two numbers 0.041 and 7.673, we see that 7.673 is more accurate because it has four significant digits, where 0.041 only has two.
The numbers have the same precision, as the last significant digit is in the thousandths position for both.

Mathematical error

In applied mathematics, the difference between a true value and an estimate, or approximation, of that value. In statistics, a common example is the difference between the mean of an entire population and the mean of a sample drawn from that population. In numerical analysis, round-off error is exemplified by the difference between the true value of the irrational number π and the value of rational expressions such as 22/7, 355/113, 3.14, or 3.14159. Truncation error results from ignoring all but a finite number of terms of an infinite series.

http://www.intmath.com/Numbers/5_Approximate-numbers.php
http://en.wikipedia.org/wiki/Approximation

MODELING

MODELING

Mathematical model

A mathematical model is an abstract model that uses mathematical language to describe the behaviour of a system. It is defined as ‘'a representation of the essential aspects of an existing system, which presents knowledge of that system in usable form’’.
Mathematical models are used particularly in the natural sciences and engineering disciplines, it can take many forms, including but not limited to dynamical systems, statistical models, differential equations, or game theoretic models.
These models can overlap, with a given model involving a variety of abstract structures. There are six basic groups of variables: decision variables, input variables, state variables, exogenous variables, random variables, and output variables. Since there can be many variables of each type, the variables are generally represented by vectors.

A mathematical model usually describes a system by a set of variables and a set of equations that establish relationships between the variables. The values of the variables can be practically anything; real or integer numbers, boolean values or strings.

Classifying mathematical models
1. Linear vs. nonlinear
2. Deterministic vs. probabilistic (stochastic)
3. Static vs. Dynamic

Example of mathematical models

Model of a particle in a potential-field. In this model we consider a particle as being a point of mass m which describes a trajectory in space which is modeled by a function x : R → R3 giving its coordinates in space as a function of time. The potential field is given by a function V : R3 → R and the trajectory is a solution of the differential equation

m .(d^2/dt^2).X(t)=-grad(V)(X(t))

Darcy's law


In fluid dynamics and hydrology, Darcy's law is a phenomenologically derived constitutive equation that describes the flow of a fluid through a porous medium.
Darcy's law is a simple proportional relationship between the instantaneous discharge rate through a porous medium, the viscosity of the fluid and the pressure drop over a given distance.

Q=-kA.(Pb-Pa)/mL

The total discharge, Q (units of volume per time, e.g., ft³/s or m³/s) is equal to the product of the permeability (κ units of area, e.g. m²) of the medium, the cross-sectional area (A) to flow, and the pressure drop (Pb − Pa), all divided by the dynamic viscosity m (in SI units e.g. kg/(m•s) or Pa•s), and the length L the pressure drop is taking place over. The negative sign is needed because fluids flow from high pressure to low pressure.


http://en.wikipedia.org/wiki/Modelos matematicos
http://www.unesco.org.uy/phi/libros/obrashidraul/Cap7a.html



Serie de Taylor

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