GAUSS SEIDEL
The Gauss- Seidel method is the most commonly used iterative method.Assume that we are given a set of n equations:
[A]{X}={B}
Suppose that for concisensess we limit ourselves to a 3x3 set of equations. If the diagonal elements are all nonzero, the first equation can be solved for x1, the second for x2, and the third for x3. Now, we can start the solution prcess by choosing guesses for the x's. A simple way to obtain initial guesses is to assume that they are all zero. these zeros can be susbtituid into first equation, which can be used to calculate a new value for x1. Then, we substitute this new value for x1 along with the previous guess of zero for x3 into next equation to compute a new value for x2. The Process is repeated to calculate a new estimate for x3. Then we return to the firs equation an repeat the entire procedure until our solution converges closely enough to the true values.
As each new x value is computed for the Gauss- Seidel method it is immediately used in the next equation to determine another x value. Thus, if the solution is converging, the best available estimates will be employed. An alternative approach, called JACOBI ITERATION ( Excel) utilizes a somewhat different tactic. Rather than using the latest available x's this technique estimates a set of new x's on the basis of a set of old x's. Thus, as new values are generated, they are not inmediately used but rather are retained for the next iteration.
Improvement of convergence using relaxation
Relaxation represents a slight modification of the Gauss- Seidel method and is designed to enhance convergence. After each new value of x is computed, that value is modified by a weighted average of the results of the previous and the present iterations:
Xi(new)= w*xi(new) + (1-w) xi(old)
where w is a weighting factor that is assigned a value between 0 and 2.
If w=1, (1-w) is equal to 0 and the result is unmodified. However, if w is set at a value between 0 and 1, the result is a weighted average of the present and the previous results. This type of modification is called underrelaxation. It is typically employed to make a nonconvergent system or to hasten convergence by dampening out oscilations.
For values of w from 1 to 2, extra weight is placed on the present value. In this instance, there is an implicit assumption that the new value is moving int he correct direction toward the true solution but at too slow a rate. Thus, the added weight of w is intented to improve the estimate by pushing it closer to the truth.Hence, this type of modification, which is called overrelaxation, is designed to accelarate the convergence of an already convergent system. The approach is also called successive or simultaneous overrelaxation, or SOR.
Graphical depiction
See more
Exercise Gauss Seidel - SOR Excel
Direct Methods for Solving Linear Equations Systems
Matrix Notation
A matrix is a rectangular array of elements represented by a single simbols. A set of horizontal elements called line ( or row ) and vertical , column.
The First subscript i always designates the number of the row in which is the element. The Second subscript j designates the column. ViewPDF
Matrix Operating Rules
Addition of two matrices, say, [A] and [B], is accomplished by adding corresponding terms in each matrix. The elements of the resulting matrix [C] are :
A matrix is a rectangular array of elements represented by a single simbols. A set of horizontal elements called line ( or row ) and vertical , column.
The First subscript i always designates the number of the row in which is the element. The Second subscript j designates the column. ViewPDF
Matrix Operating Rules
Addition of two matrices, say, [A] and [B], is accomplished by adding corresponding terms in each matrix. The elements of the resulting matrix [C] are :
cij= aij + bij
Similarly, the Substraction of two matrices, say, [E] minus [F] is obtain by subtracting corresponding terms:
dij= eij-fij
The addition and Subtraction can be performed only between matrices having the same dimensions.
Both Addition and subtraction are Conmutative:
[A] + [B] = [B] + [A]
Addition and Subtraction are also associative, that is,
([A] + [B]) + [C] = [A] + ([B] +[C])
NAIVE GAUSS ELIMINATION
Elimination of unknows
The Procedure:
1. The Ecuations were manipulated to eliminate one of the unknows from the ecuations.The result of this elimination step was that we had one equation with one unknown.
2. Consequently, this equation could be solved directly and the result back- substituted into one of the original equations to solve for the reaming unknown.
This basic approuch can be extended to large sets of equation by developing a systematic scheme or algorithm to eliminate unknows and to back- sustitute. Gauss Elimination is the most basic of these schemes ( The first phase is designed to reduce the set of equations to an upper triangular system).
PITFALLS OF ELIMINATION METHODS
Some pitfalls:
* Division by Zero. The primary reason that the foregoing technique is called " naive" is that
during both the elimination and the back-substitution phases, it is possible that a division by zero can occur.
* ill-Conditioned Systems. Well conditional systems are those where a small change in one or more of the coefficients results in a similar small change in the solution. Ill- Conditional systems are those where small changes in coefficients result in large changes in the solution.
Roots of Equations
Introduction
The purpose of calculating the roots of an equation to determine the values of x for which holds:
f(x) =0
The determination of the roots of an equation is one of the oldest problem and there have been many efforts in this regard. His importance is that if we can determine the roots of an equation. we can also determine the maximum and minimum, own values of a counterfoils, solve system of liner and differential equations.
Some methods used to obtain roots of an equation are:
Bracketing Methods
These techniques are called Bracketing methods because two initial guesses for the root are required.As the name implies , these guesses must bracket or be on either side of the root. The particular methods described herein employ different strategies to systematically reduce the width of the bracket and, hence , home in on correct answer.
-Graphical Method
-The Bisection Method
Open Methods
The open methods described are based on formulas that require only a single starting value of x or two starting values that do not necessarily bracket the root. As such, they sometimes diverge or move away from the root.
-Newton Raphson Method
-The Secant Method
Mathematical Approximation
When uising a number in a calculation ther must be assurance that can be used with confidence.The Concept of Significant figures has two important implications in the study of numerical methods.
- The Numerical Methods obtain approximate results. Therefore criterion should be developed to specify how accurate are the results obtained.
- Although some numbers represent specific number can be expressed exactly.
The significant figures of a number are those that can be used with confidence. They correspond to the numbers of certain digits plus one estimated digit.
Example
pi= 3,14159265358979......
Because the computer retains only a finite numbers of signicant figures, such number can ever be represent exactly. The omission of the remaining figures is called round-off error.
Accuracy and Precision ( Watch Video)
The error in calculations and measures can be characterized with regard to their accuracy and precision. Accuracy refers to how closely a computed or measured value agrees with the tre value. Precision refers to how closely individual computed or measured values agree with each other.
Inaccuracy is defined as systematic deviation from the truth. The Imprecision refers to the dispersion.
Error Definitions and The Taylor Series ( Document ) ( Excel )
Numerical Methods
Numerical Methods are techniques by which mathematical problems can be formulated in such a way tha can be solved using arithmetic.The numerical analysis to devise methods to " aproximate" efficiently solutions to problems espressed mathematically. The Main objective of numerical analysis is to find solutions " approximate" to complex problems using only the simplest operations.
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Mathematicals Model
Mathematical Approximation
Roots of Equations
MATHEMATICALS MODELS
In the first place we can define a Mathematical Model as a formulation or equation that expresses the main features of a Physical system or a Process in Mathematical terms.
Mathematical models are used particularly in natural sciences and engineering disciplines ( Physics, Biology, Meteorology, and Electrical Engineering) but also in social sciences ( Economics, Sociology and Political Science), physicists, engineers, computer scientists,and economists use mathematicals models mos extensively.
Mathematical models can take many forms,including but not limited dynamical systems, statistical models, differential equations, or game - theoretical models. These and other types of models can overlap with a given model involving a variety of abstract structures.
In General the model can be represented with the next functional relationship :
dependent variable = f (independent variable, parameters, forcing functions)
The dependent Variable is a feature that usually represent the behaviour or system's state, the Independents variables are dimensions as the time and space through which is determinate the system's behaviour. The Parameters are reflective of properties or system's composition. Finally the forcing functions are external influences acting upon the system.
we can obtain analytical or numerical solutions of a problem.The Firsts usally are accurate and the latter are aproximate.
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