Universidad Autónoma de Querétaro
Facultad de Ingeniería

“Iterative Methods”
“Gauss and Gauss-Seidel”

Profesor| | Nieves Fonseca Ricardo|
Mentado Camacho Félix
Navarro Escamilla Erandy
Péloquin Blancas María José
Rubio Miranda Ana Luisa

Abstract

Many real life problems give us several simultaneous linear equations to solve. And we have to find a common solution for each of them. There are several techniques to use. Instead of using methods that provide a solution to a set of linear equations after a finite number of steps, we can use a series of algorithms with fewer steps, but its accuracy depends on the number of times it is applied (also known as iterative methods). For large systems they may be a lot faster than direct methods. We will expand on two important methods to find numerical solutions to linear systems of equations. There will be an introduction to each method, besides detailed explanations on each of them. Normally each process is long, so they are ideal for programming.

Keywords

Iterative, algorithm, linear equation, convergence.

Objective

Understand the concepts of iterative methods, and convergence, besides the difference and usefulness between direct and iterative methods. To give a clear and understandable idea of Gauss

and Gauss-Seidel methods to solve systems of linear equations, and show how to apply them.

Investigation

Iterative method

An iterative method is one that computes approximations in a progressive way of the solution of a mathematical problem. The same improvement process is repeated on the approximate solution until a solution closer to the real value is obtained. Differently than a direct method, which entails finishing the process to get an answer, on an iterative one the process can be stopped when an iteration is finished to get an approximation to the solution. Advantages and disadvantages

The bad thing about iterative methods is that they...

...NumericalMethods Questions
1 f(x) = x3 – 2x – 5
a) Show that there is a root β of f(x) = 0 in the interval [2,3]. The root β is to be estimated using the iterative formula
,2 5 2 0 2 1 1 = =++ x x x n n
b) Calculate the values of x1, x2, x3, and x4, giving your answers to 4 sig fig.
c) Prove that, to 5 significant figures, β is 2.0946
2 Use the iterative formula
n
n n cox x x − =+ 1 1 With x1 = 0.5 to find the...

...Department of Mathematics
Subject Name: NumericalMethods
Subject Code: MA1251
Unit I
1) Write the Descartes rule of signs
Sol:
1) An equation f ( x) = 0 cannot have more number of positive roots than there are
changes of sign in the terms of the polynomial f ( x) .
2)An equation f ( x) = 0 cannot have more number of positive roots than there are
changes of sign in the terms of the polynomial f ( x) .
2) What is the order of convergence of Newton...

...Carl Gauss was a man who is known for making a great deal breakthroughs in the wide variety of his work in both mathematics and physics. He is responsible for immeasurable contributions to the fields of number theory, analysis, differential geometry, geodesy, magnetism, astronomy, and optics, as well as many more. The concepts that he himself created have had an immense influence in many areas of the mathematic and scientific world.
Carl Gauss was born Johann...

...Gauss-Jordan Matrix Elimination
-This method can be used to solve systems of linear equations involving two or more variables. However, the system must be changed to an augmented matrix. -This method can also be used to find the inverse of a 2x2 matrix or larger matrices, 3x3, 4x4 etc. Note: The matrix must be a square matrix in order to find its inverse. An Augmented Matrix is used to solve a system of linear equations. a1 x + b1 y + c1 z = d1 a 2 x...

...suited
for hardware implementation due to its simplicity. Moreover,
the coefﬁcient matrices of LSEs are not required to have any
special properties like being symmetric positive deﬁnite2 as
this is the case for example for the conjugate gradient methods [12]. However, compared to software, a direct implementation of Gaussian elimination in hardware does not yield the
desired advantage in efﬁciency. By slightly modifying the
logic of the algorithm and parallelizing element...

...Gauss Markov Theorem
In the mode [pic]is such that the following two conditions on the random vector [pic]are met:
1. [pic]
2. [pic]
the best (minimum variance) linear (linear functions of the [pic]) unbiased estimator of [pic]is given by least squares estimator; that is, [pic]is the best linear unbiased estimator (BLUE) of [pic].
Proof:
Let [pic]be any [pic]constant matrix and let [pic]; [pic] is a general linear function of [pic], which we shall take as an estimator of...

...Greetings, my fellows! I am Gauss, Johann Carl Friedrich Gauss. I am a German mathematician and I contributed significantly to maths and physics. I was born on the 30th April 1777 in Braunschweig, Germany. Unfortunately, my mother was not well educated and could not read or write and could not record my date of birth. The only she remembered was that I was born on a Wednesday, eight days before the Feast of Ascension, which occurs 40 days after Easter. I was...

...Carl Friedrich Gauss
Carl Friedrich Gauss was a German mathematician and scientist who
dominated the mathematical community during and after his lifetime. His
outstanding work includes the discovery of the method of least squares, the
discovery of non-Euclidean geometry, and important contributions to the theory
of numbers.
Born in Brunswick, Germany, on April 30, 1777, Johann Friedrich Carl
Gauss showed early and unmistakable signs...