NUMERICAL METHODS: mayo 2010

ROOTS OF EQUATIONS

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

f (x) = 0 (1)

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 ...

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:
As 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.

• GRAPHICS METHODS: is to plot the function and see where it crosses the x-axis This item, which represents the value of x for which f (x) = 0, provides an initial approximation of the root.

• OPEN METHODS: based on formulas that require only one value of x for a couple of them but do not necessarily contain the root. As such sometimes diverge or move away from the root as it grows the number of iterations. However, when the open methods in general do converge faster than methods that use ranges.

Calculation of multiple roots.
A multiple root corresponds to a point where a function is tangential to the x axis For example, two resulting from repeated roots

f (x) = (x-3) (x-1) (x-1) or (2)

Multiplying terms f (x) = x3-5x2 +7 x-3 (3)

The equation has a double root because a value of x cancels two terms of equation (2).

REFERENCES: NUMERICAL METHODS FOR ENGINEERS WITH PERSONAL COMPUTER APPLICATIONS. STEVEN C. Chapra / Raymond P. CANALE

NUMERICAL APPROACH

SIGNIFICANT FIGURES
When using a number in a calculation, there must be assurance that can be used with confidence. The significant figures (or significant digits) represent the use of a level of uncertainty under certain approximations. The concept of significant figures has been developed to formally designate the reliability of a numerical value. The number of significant figures is the number of digits plus one estimated digit that can be used with confidence.

The concept of significant figures has two important implications in the study of numerical methods.

1. The numerical methods obtain approximate results. Therefore it should develop criteria to specify how accurate are the results, with significant figures, we can say that the approach is acceptable if it is correct to four significant figures.

2. Certain quantities such as ℮ and Π represent specific numbers, you can not exactly express a number as they tend to infinity. The concept of significant figures is very important in the definition of accuracy and precision.

references picture: 21style.nireblog.com/pag_4/

PRECISION
Refers to the number of significant figures representing a number or extension in the repeated readings of an instrument that measures some physical property.

ACCURACY
It refers to the approach of a number or measure the true value is supposed to represent.

DEFINITIONS OF ERROR
The numerical errors are generated with the use of approximations for representing mathematical operations. These include truncation errors resulting from about exact mathematical procedure and the rounding errors that result from about exact numbers
.
For both types of errors the relationship between the exact result and the approximate or true is given by:

Approximate value = true value + error

ROUNDING ERROR
Occur because the computers just to hold a finite number of significant figures in a calculation. Computers perform this function in different ways. For example if you only saved seven significant figures computers can store and use π as π = 3.141592, omitting the remaining terms and generating a rounding error.

Truncation error
Are those that result when using an approximation rather than an exact mathematical procedure. For example, the approximate speed of a parachutist falling through the differential equation of the form:
dv / (dt) ≈ Δv / Dt = (v (t (i +1))-v (ti)) / (t (i +1) - t (i))

It introduces a truncation error in the numerical solution of differential equation because only approximates the true value of the derivative.

REFERENCES: NUMERICAL METHODS FOR ENGINEERS WITH PERSONAL COMPUTER APPLICATIONS. STEVEN C. Chapra / Raymond P. CANALE

MATHEMATICAL MODEL

MATHEMATICAL MODEL
Can be defined generally as a formulation or equation expressing the fundamental characteristics of a system or physical process in mathematical terms. The models are classified from simple algebraic relationships to large, complicated systems of differential equations.

Mathematical modeling, besides being an activity that fits into applications of mathematics and provides a link between these and other scientific and technical areas within the scope of the investigation, has also emerged as a powerful educational tool.

Imagen tomada: dcb.fi-c.unam.mx/users/gustavorb/MN.html

The implementation of a mathematical model is the way it uses science and technology to approach reality and solve problems. As a result, mathematical objects are necessary to understand the technological and scientific concepts through constructions and results. Mathematical models provide a framework in which are interrelated concepts of different sciences and in this sense, modeling appears as an important tool to teach math and science. Moreover, the use of models can help reinforce basic science concepts, enriching the learning. (1)

START MAKING YOUR PARTY:
Identification of variables causing the change in the system
Establishment of reasonable assumptions about a model system for mental, verbal and graphic

CHARACTERISTICS OF A MODEL
1. Describes a natural process or system
2. Represents a simplification and an idealization of reality
3. Leads to reproducible results

SIMULATED ENVIRONMENTS
What is really a numerical method of simulation?
Is trying to represent a physical phenomenon based on a small portion of the total being analyzed.
Its purpose is to simulate actual conditions scale, today facilmento is achieved with computer graphics.
The following video shows how automated are the oil rigs. Computer Graphing

DARCY'S LAW
Darcy's Law describes, based on laboratory experiments the characteristics of the flow of fluid through a porous medium.

The simplest formulation of the law (for linear systems) can be considered almost "intuitive": The flow of a fluid flowing through a porous medium depends linearly:

The geometrical properties of the system: Area (A) and length (L).
The fluid characteristics: Mainly its viscosity (μ).
The flow conditions: Differential Pressure (DP) between the edges of the system.

This makes it almost "obvious" that, equality of the other system variables, the flow rate (Q) flowing through a porous medium grows directly with the applied pressure difference and the available flow area and decreases when increases the length and fluid viscosity.

Analytically this dependence is expressed by the following formula: