ITERATIVE METHODS FOR SOLUTION OF SYSTEMS OF LINEAR EQUATIONS

ITERATIVE METHODS FOR SOLUTION OF SYSTEMS OF LINEAR EQUATIONS

In general, these methods are based on fixed point method and the process is iterated so substitute in a formula.

Iterative methods compared with direct, we do not guarantee a better approach, however, are more efficient when working with large matrices.

In computational mathematics, is an iterative method to solve a problem (as an equation or a system of equations) by successive approximations to the solution, starting from an initial estimate. This approach contrasts with the direct methods, which attempt to solve the problem once (like solving a system of equations Ax = b by finding the inverse of the matrix A). Iterative methods are useful for solving problems involving a large number of variables (sometimes in the millions), where direct methods would be prohibitively expensive even with better computer power available.

• JACOBI METHOD

The basis of the method is to construct a convergent sequence defined iteratively. The limit of this sequence is precisely the solution of the system. For practical purposes if the algorithm stops after a finite number of steps leads to an approximation of the value of x in the solution of the system.

• GAUSS SEIDEL METHOD

The methods of Gauss and Cholesky methods are part of direct or finite. After a number of operations nito, in the absence of errors rounding, we get x solution of the system Ax = b. The Gauss-Seidel method is part of the so-called indirect methods or iterative. They start with x0 = (x01, X02,:::; x0 n), an approximation initial solution. Since x0 is building a new approach of the solution, x1 = (x11, x12;:::; x1n).

From built x1 x2 (here the superscript indicates the iteration and does not indicate a power). So on construye fxkg a sequence of vectors, with the aim, not always guaranteed to quelimk! 1xk = x:Generally, indirect thods are a good option when the matrix is very large and dispersed or sparse (sparse), ie when the number of onzero elements is small compared to n2, total number of elements A.

In these cases you must use an appropriate data structure lets you store only the nonzero elements. In each iteration of the Gauss-Seidel method, there are n subiteraciones. In ca first subiteracion be amended only x1. The other coordinates x2, x3, ..., xn are not modified can.

fuente

• Wladimiro Diaz Villanueva 1998-05-11
• Método de Jacobi de www.math-linux.com
• Método de Jacobi en el mundo de la matemáticas

DIRECT METHOD FOR SOLVING SYSTEMS OF LINEAR EQUATIONS

DIRECT METHOD FOR SOLVING SYSTEMS OF LINEAR EQUATIONS

In this lesson we study the solution of a Cramer system Ax = B, which means that A is ° c

invertible regular or verify ° c or det (A) 6 = 0, using a direct method. Since this is any
allowing, in the absence of errors, through a number of steps nito obtain the exact solution.
In property, this does not happen in general because of the inevitable rounding errors.

• Gauss- Jordan Elimination
• 
  In mathematics, Gaussian elimination, Gaussian elimination or Gauss-Jordan elimination, so named because Carl Friedrich Gauss and Wilhelm Jordan, are linear algebra algorithms to determine the solutions of a system of linear equations, matrices and inverse finding. Asystem of equations is solved by the Gauss when their solutions are obtained by reducing an equivalent system given in which each equation has one fewer variables than the last. When applying this process, the resulting matrix is known as "stagger."

• ALGORITHM GAUSS JORDAN
• 
  1. Go to the far left column is not zero
  2. If the first line has a zero in this column, swap it with another that does not have
  3. Then, get below zero this item forward, adding appropriate multiples of row than h e row below it
  4. Cover the top row and repeat the above process with the remaining submatrix. Repeat with the rest of the lines (at this point the array is in the form of step)
  5. Starting with the last line is not zero, move up: for each row get a 1 up front and introduce zero multiples of this sum for the rows corresponding
  
  An interesting variant of Gaussian elimination is what we call Gauss-Jordan, (due to Gauss and Wilhelm Jordan mentioned), this is to be a front for getting the steps 1 to 4 (called direct path) and the time these completed and will result in the reduced echelon form matrix.

• LU DESCOMPOSITION

ts name is derived from the English words "Lower" and "Upper", which in Spanish translates as "Bottom" and "Superior." Studying the process followed in the LU decomposition is possible to understand why this name, considering how original matrix is decomposed into two triangular

• LU decomposition involves only operations on the coefficient matrix [A], providing an efficient means to calculate the inverse matrix or solving systems of linear algebra.
  
  First you must obtain the matrix [L] and the matrix [U]..
  
  [L] is a diagonal matrix with numbers less than 1 on .the diagonal. [U] is an upper diagonal matrix on the diagonal which does not necessarily have to be number one.
  
  The first step is to break down or transform [A] [L] and [U], ie to obtain the lower triangular matrix [L] and the upper triangular matrix [U]

• INVERSE MATRIX

Is the matrix we get from chang ing rows by columns. The transpose of that represented by AT.

In mathematics, especially in linear algebra, a square matrix of order n is said to be invertible, nonsingular, nondegenerate or regular if there is another square matrix of order n, called the inverse matrix of A and represented as A-1matrices.


fuente

• Shen Kangshen et al. (ed.) (1999). Nine Chapters of the Mathematical Art, Companion and Commentary, Oxford University Press. cited byOtto Bretscher (2005).

• Linear Algebra and Its Applications, Thomson Brooks/Cole, pp. 46

MATHEMATICAL APPROXIMATION

MATHEMATICAL APPROXIMATION

Numerical approximation is defined as a figure reoresenta to a number whose exact value is the twelfth to the extent that the figure was closer to the exact value, it will be a better approximation of that number.

SIGNIFICANT FIGURES

The significant figures (or significant digits) represent the use of a level of uncertainty under certain approximations The use of these considers the last digit of approach is uncertain, for example, to determine the volume of a liquid using a graduated cylinder with a precision of 1 ml, implies an uncertainty range of 0.5 ml. It may be said that the volume of 6ml of 5.5 ml will be really to 6.5 ml. The previous volume is represented as (6.0 ± 0.5) ml. For specific values closer would have to use other instruments of greater precision, for example, a specimen finest divisions and thus obtain (6.0 ± 0.1) ml or something more satisfying as the required accuracy.

ACCURACY AND PRECISION

Accuracy refers to the dispersion of the set of values from repeated measurements of a magnitude. The lower the spread the greater the accuracy. A common measure of variability is the standard deviation of measurements and precision can be estimated as a function of it. Accuracy refers to how close the actual value is the measured value. In statistical terms, accuracy is related to the bias of an estimate. The smaller the bias is a more accurate estimate. When we express the accuracy of a result is expressed by the absolute error is the difference between the experimental value and the true value.

NUMERICAL STABILITY

In the mathematical subfield of numerical analysis, numerical stability is a property of numerical algorithms. Describe how errors in the input data are propagated through the algorithm. In a stable method, errors due to approximations are mitigated as appropriate computing. In an unstable method, any error in the processing is magnified as the calculation applicable. Methods unstable quickly generate waste and are useless for numerical processing.

CONVERGENCE

In mathematical analysis, the concept of convergence refers to the property they own some numerical sequences tend to a limit. This concept is very general and depending on the nature of the set where the sequence is defined, it can take several forms.

fuente

• George E. Forsythe, Michael A. Malcolm, and Cleve B. Moler. Computer Methods for Mathematical Computations. Englewood Cliffs, NJ: Prentice-Hall, 1977. (See Chapter 5.)

• William H. Press, Brian P. Flannery, Saul A. Teukolsky, William T. Vetterling. NumericalRecipes in C. Cambridge, UK: Cambridge University Press, 1988. (See Chapter 4.)

MATHEMATICAL MODEL

MATHEMATICAL MODEL

A product model is an abstraction of a real system, eliminating the complexities and making relevant assumptions, applies a mathematical technique and obtained a symbolic representation of it.

A mathematical model comprises at least three basic sets of elements:

• Decision variables and parameters

The decision variables are unknowns to be determined from the model solution. The parameters represent the values known to the system or that can be controlled.

• Restrictions

Constraints are relations between decision variables and magnitudes that give meaning to the solution of the problem and delimit values feasible. For example if one of the decision variables representing the number employees of a workshop, it is clear that the value of that variable can not be negative.

• Objective Function

The objective function is a mathematical relationship between variables decision parameters and a magnitude representing the target or product system. For example if the objective is to minimize system costs operation, the objective function should express the relationship between cost and decision variables. The optimal solution is obtained when the value of cost is minimal for a set of feasible values of the variables. Ie there to determine the variables x1, x2, ..., xn that optimize the value of Z = f (x1, x2, ..., xn) subject to constraints of the form g (x1, x2, ..., xn) b. Where x1, x2, ..., Xn are the decision variables Z is the objective function, f is a function mathematics.

HOW TO DEVELOP A MATEMATICAL MODEL

1. Find a real world problem.

2. Formulate a mathematical model of the problem, identifying variables (dependent and independent) and establishing hypotheses simple enough to be treated mathematically.

3. Apply mathematical knowledge that has to reach mathematical conclusions.

4. Compare the data obtained as predictions with real data. If the data are different, the process is restarted.

CLASSIFICATION OF MODELS

• Heuristic Models: (Greek euriskein 'find, invent'). Are those that are based on the explanations of natural causes or mechanisms that give rise to the phenomenon studied.

• Empirical models: (Greek empeirikos on the 'experience'). They are using direct observations or the results of experiments studied phenomenon.

Mathematical models are also different names in various applications. The following are some types to which you can adapt a mathematical model of interest. According to its scope models:

• Conceptual models :Are those that reproduce by mathematical formulas and algorithms more or less complex physical processes that occur in nature.

• Mathematical model of optimization :Mathematical optimization models are widely used in various branches of engineering to solve problems that are by nature indeterminate, ie have more than one possible solution.

CATEGORIES FOR ITS APPLICATION

For use commonly used in the following three areas, however there are many others such as finance, science and so on.

• Simulation: In situations accurately measurable or random, for example linear programming aspects precisely when, and probabilistic or heuristic when it is random.

• Optimization :To determine the exact point to resolve any administrative problems, production, or other status. When the optimization is complete or nonlinear, combination, refers to mathematical models little predictable, but they can fit into any existing alternative and approximate quantification.

• Control: To find out precisely how is something in an organization, research, area of operation, etc..

fuente:

• http://www.investigacion-operaciones.com/Formulacion%20Problemas.htm

• Ríos, Sixto (1995). Modelización. Alianza Universidad.

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 (28)

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 (28) 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).

In general, the methods for finding the real roots of algebraic equations and transcendental methods are divided into intervals and open methods.

• INTERVAL METHODS: exploit the fact that typically a function changes sign in the vicinity of a root. They get this name because it needs two initial values to be "encapsulated" to the root. Through such methods will gradually reduce the size of the interval so that the repeated application of the methods always produce increasingly close approximations to the actual value of the root, so methods are said to be convergen.

In Figure 2.1 is seen as the function changes + f (x) a - f (x), as it passes through the root c. This is because f (c) = 0 and necessarily pass function of positive to negative quadrant x. In some cases, to be seen later this does not happen, for now it will be assumed as shown. The methods they use open sign changes in order to place the root (point c), but it must then establish a range (such as [a, b]).

Similarly happens when the function passes through the point e, the change occurs-f (x) + f (x), to find the root of the method requires an interval as [d, f].


The main methods are Interval:

a. Graphical Method

b. Bisection Method

c. Linear Interpolation Method

d. methods of false position

• OPEN METHODS: in contrast, are based on formulas that require a single initial value x (initial approach to the root). Sometimes these methods away from the real value of the root grows the number of iterations.
  

Open the main methods are:

a. Newton Raphson method.

b. Secant method

c. Multiple roots
.
fuente:
· Burden Richard L. & Faires J. Douglas, Análisis numérico. 2ª. ed., México, Grupo Editorial Iberoamérica, 1993.
· Chapra Steven C. & Canale Raymond P., Métodos numéricos para ingenieros. 4ª. ed., México, McGraw-Hill, 2003