The following files explain all the direct methods for solving this kind of systems. I recomend you to check all the files out, because one of them doesn`t have what the other one does. I mean, although some of the methods are repeated, you`ll find extra and different information on each presentation.
Here you`ll find all the methods with some theory and demostrations. cholesky Bibliography: http://www.lcc.uma.es/~villa/tn/tema04.pdf
Over here you have the opportunity to solve some purposed exercises and also you`ll see step by step how you can find the solution of different systems. sistema ecuaciones mdf Bibliography: http://www.eueti.uvigo.es/files/curso_cero/material/sistemas_de_ecuaciones.pdf
One method is missing, it`s called THOMAS and it`s used to solve tridiagonal systems (tridiagonal matrix). This last presentation has detailed explaination about Thomas method, solving an example using it.
Years ago, you learned to use the quadratic formula
To solve
The values calculated with (1) are called the “roots” of (2). They represent the values of x that make (2) equal to zero. Thus, we can define the root of an equation as the value of x that makes f(x)=0. For this reason, roots are sometimes called the zeros of the equation.
Although the quadratic formula is handy for solving (2), there are many other functions for which the root cannot be determined so easily. For these cases, the numerical methods provide efficient means to obtain the answer.
The importance of error was introduced in our discussion of the falling parachutist in Chapter 1. Recall that we determined the velocity of a falling parachutist by both analytical and numerical methods. Although the numerical technique yielded estimates that were close to exact analytical solution, there was a discrepancy, or error, because the numerical method involved an approximation. Actually, we were fortunate in that case because the availability of an analytical solution allowed us to compute the error exactly. For many applied engineering problems, we cannot obtain analytical solutions. Therefore, we cannot compute exactly the errors associated with our numerical methods. In these cases, we must settle for approximations or estimates of the errors.
Although perfection is a laudable goal, it is rarely, if ever, attained. For example, despite the fact that the model developed from Newton`s second law is an excellent approximation, it would never in practice exactly predict the parachutist`s fall. A variety of factors such us winds and slight variations in air resistance would result in deviations from the prediction. If these deviations are systematically high or low, then we might need to develop a new model. However, if they are randomly distributed and tightly grouped around the prediction, then the deviations might be considered negligible and the model deemed adequate. Numerical approximations also introduce similar discrepancies into the analysis. Again, the question is: How much error is present in our calculations and is it tolerable?
SIGNIFICANT FIGURE
The concept of a significant figure, or digit, has been developed to formally designate the reliability of a numerical value. The significant digits of a number are those that can be used with confidence. They correspond to the number of certain digits plus one estimated digit. It is conventional to set the estimated digit at one-half of the smallest scale division on the measurement device.
Although it is usually a straightforward procedure to ascertain the significant figures of a number, some cases can lead to confusion. For example zeros are not always significant figures because they may be necessary just to locate a decimal point. The numbers 0.00001845, 0.0001845, and 0.001845 all have four significant figures. Similarly, when trailing zeros are used in large numbers, it is not clear how many, if any, of the zeros are significant. For example, at face value the number 45,300 may have three, four or five significant digits, depending on whether the zeros are known with confidence. Such uncertainty can be resolved by using scientific notation, where 4.53 x10^4, 4.530 x10^4, 4.5300 x10^4 designate that the number is known to three, four and five significant figures, respectively.
ACCURACY AND PRECISION
The errors associated with both calculations and measurements can be characterized with regard to their accuracy and precision. Accuracy refers to how closely a computed or measured value agrees with the true value. Precision refers to how closely individual computed or measured values agree with each other. These concepts can be illustrated graphically with the following pictures:
Numerical methods should be sufficiently accurate or unbiased to meet the requirements of a particular engineering problem. They also should be precise enough for adequate engineering design. In the next terms that I am going to introduce in this blog, you should understand the term error as both inaccuracy and the imprecision of predictions.
ERROR DEFINITIONS
Both documents bellow have some important information about the main kind of errors in numerical methods. Just take some time to watch them.
The condition of a mathematical problem relates to its sensitivity to changes in its input values. We say that a computation is numerically unstable if the uncertainty of the input values is grossly magnified by the numerical method.
These ideas can be studied using a first order Taylor series
This relationship can be employed to estimate the relative error of f(x) as in
The relative error of x is given by
A condition number can be defined as the ratio of these relative errors
The condition number provides a measure of the extent to which an uncertainty in x is magnified by f(x). A value of 1 tells us that the function`s relative error is identical to the relative error in x. A value greater than 1 tells us that the relative error is amplified, whereas a value less than 1 tells us that it is attenuated. Functions with very large values are said to be ill-conditioned. Any combination of factors in the last equation, that increases the numerical value of the condition number will tend to magnify uncertainties in the computation of f(x).
Total Numerical Error
The total numerical error is the summation of the truncation and round-off errors. In general, the only way to minimize round-off errors is to increase the number of significant figures of the computer. Further, we have noted that round-off error will increase due to subtractive cancellation or due to an increase in the number of computations in an analysis. In contrast, the truncation error can be reduced by decreasing the step size. Because a decrease in step size can lead to subtractive cancellation or to increase in computations, the truncation errors are decreased as the round-off errors are increased. Therefore, we are faced by the following dilemma: The strategy for decreasing one component of the total error leads to an increase of the other component. In a computation, we could conceivably decrease the step size to minimize truncation errors only to discover that in doing so, the round-off error begins to dominate the solution and the total error grows! Thus, our remedy becomes our problem (the figure bellow shows this behavior). One challenge that we face is to determine an appropriate step size for a particular computation. We would like to choose a large step size in order to decrease the amount of calculations and round-off error begins to negate the benefits of step-size reduction.
In actual cases, however, such situations are relatively un common because most computers carry enough significant figures that round-off errors do not predominate. Nevertheless, they sometimes do occur and suggest a sort of “numerical uncertainty principle” that places an absolute limit on the accuracy that may be obtained using certain computerized numerical methods.
BIBLIOGRAPHY Numerical Methods for Engineering. Steven C. Chapra-Raymond P. Canale. McGrawHill fifth edition.
El concepto de modelos matemáticos se aplica para ayudadr a definir lo que se entiende por métodos numéricos y para ilustrar cómo pueden facilitar la solución de problemas en ingenieria. Asi pues, un modelo matemático está definido como una formualción o ecuación que expresa las características fundamentales de un sistema o proceso físico en términos matemáticos. los modelos se clasifican desde simples relaciones algebraicas hasta grandes y complicados sistemas de ecuaciones diferenciales. Si recordamos a Newton para este ejemplo, la expresion matemática, o modelo, de su segunda ley es la bien conocida ecuación F=ma (1), donde F es la fuerza neta que actúa sobre el cuerpo, m es la masa de objeto y a es su aceleración. dicha expresión tiene características comunes de los modelos matemáticos del mundo físico:
1. Describe un proceso natural en términos matemáticos.
2. Representa una idealización de la realidad. Es por esto que la segunda ley no incluye los efectos de la relatividad, que no son relevantes cuando se aplican a objetos y fuerzas que interactúan sobre o alrededor de la Tierra a escalas visibles a los seres humanos.
3. Conduce a resultados predecibles, por lo que puede emplearse para propósitos de predicción. Si se conoce la fuerza aplicada sobre un objeto y se tiene su masa, entonces se puede usar la expresion F=ma para predecir su aceleración. Como tiene una forma algebraica sencilla, puede despejarse directamente:
a=F/m (2)
De este modo, la aceleración puede calcularse fácilmente. Sin embargo, los modelos de otros fenómenos físicos pueden ser mucho más complejos y no pueden resolverse pues requieren de técnicas matemáticas más complejas para su solución. Un ejemplo de esto puede ser la segunda ley de Newton para determinar la velocidad final de un cuerpo en caída libre cerca de la superficie terrestre. El cuerpo en descenso será un paracaidista como se muestra en la siguiente figura
Para este caso puede crearse un modelo al expresar la aceleración como la razón de cambio de la velocidad con respecto al tiempo (dv/dt) y sustituir en la ecuación (1) de la siguiente manera:
m(dv/dt)=F (3) Donde v es la velocidad y así, la masa multiplicada por la razón de cambio de la velocidad es igual a la suma de las fuerzas que actúan sobre el cuerpo. Si la fuerza total es positiva, el objeto acelera. Si es negativa, el objeto permanecerá a un nivel constante.
Para un cuerpo que cae dentro del perímetro de la Tierra la fuerza total está compuesta por 2 fuerzas contrarias: la atracción hacia abajo debida a la gravedad Fd y la fuerza hacia arriba debida a la resistencia del aire Fu.
F=Fd+Fu (4)
Si a la fuerza hacia abajo se le asigna un signo positivo, se puede usar la segunda ley para formular la fuerza debida a la gravedad como FD=mg , donde g es la constante gravitacional o la aceleración debida a la gravedad que es aproximadamente 980 cm/s2. Se puede suponer que la resistencia del aire es linealmente proporcional a la velocidad Fu= -cv (6), donde c es una constante de proporcionalidad llamada coeficiente de arrastre. El parámetro c toma en cuenta las propiedades del objeto descendente, tales como su forma o la aspereza de su superficie, que afectan la resistencia del aire. Para este caso “c” estaría en función del tipo del traje del paracaidista por ejemplo.
La fuerza total es la diferencia entre las fuerzas hacia abajo y hacia arriba. Por tanto, las ecuaciones (3) y (6) pueden combinarse obteniendo:
m (dv/dt)= mg- cv (7)
o, dividiendo cada lado entre m,
dv/dt = g – (c/m)v (8)
Esta última ecuación es un modelo que relaciona la aceleración de un cuerpo que cae, con las fuerzas que actúan sobre él. Así mismo es una ecuación diferencial porque está escrita en términos de la razón de cambio diferencial (dv/dt) de la variable que nos interesa predecir. En contraste con la solución dada por la segunda ley de Newton (2), la solución exacta de la ecuación (8) para la velocidad del paracaidista que cae, no puede obtenerse mediante simples manipulaciones algebraicas; en vez de eso, deberán aplicarse técnicas del cálculo para obtener la solución exacta. Por ejemplo, si el paracaidista inicialmente está en reposo (v=0 en t=0), se puede usar el cálculo para resolver la ecuación (8) así
BIBLIOGRAPHY Métodos Numéricos para Ingenieros-Con Aplicaciones en Computadoras Personales. Steven C. Chapra, Ph.D-Raymond P. Canale, Ph.D. McGRAW-HILL. 1998.
I'm a student of petroleum engineering. Actually I study at Santander Industrial University and this is a kind of project to learn about the main topics of an important subject known as numerical methods for engineering. I hope you to enjoy my blog and I want you to take advantage of all the information and tools of this place.
