ITERATIVE METHODS

PRESENTED BY: KATHERINE SILVA

NUMERICAL METHODS IN PETROLEUM ENGINEERING

1. JACOBI METHOD

The Jacobi method consists of a sequence of orthogonal transformations. Each transformation we will call a Jacobi rotation, which seeks to eliminate an element of the array. Thus we successively rotating the array until the error is small enough to be considered diagonal.

we choose a matrix Q is diagonal and whose diagonal elements are the same as those of the matrix A.

general equation can be written as

Qx(k) = (Q-A)x(k-1) + b

The sequence is constructed by decomposing the system matrix as follows:

whereD, a diagonal matrix.L, is a lower triangular matrix.U, is an upper triangular matrix

Starting from Ax=b, we can rewrite this equation as:

then

If aii ≠ 0 for each i. For the iterative rule, the definition of the Jacobi method can be expressed as:

where k is the iteration counter, finally we have:

EXAMPLE• Solve the following system:

• Solving

• Whit E= 0.001%

If x(0) [0,0.0]T

1. iteration

2. iteration

3. iteration

4. iteration

1. GAUSS SEIDEL METHOD

The Gauss-Seidel iteration is defined by taking Q as the lower triangular part of A including the diagonal elements:

define the matrix R = A-Q

and the equation can be written as:

Any component, i, the vector Qx (k) is given by the equation:

If we consider the peculiar form of the matrices Q and R, is that all the summands for which j> i on the left are zero, while the right side are zero for all summands that. We can write then:

clearing xi (k), we obtain:

• EXAMPLE:

10 2 -1 0 x1 26 1 20 -2 3 x2 = -15-2 1 30 0 x3 53 1 2 3 20 x4 47

starting from x0 = (1, 2, 3, 4).

)0466.2,9725.1,175.1,5.2(

0466.220

)9725.1*3)175.1(*25.2*1(47

)4,975.1,175.1,5.2(

9725.130

)4*0)175.1(*15.2*2(53

)4,3,175.1,5.2(

20

175.1)4*33*)2(5.2*1(15

)4,3,2,5.2(

5.210

)4*03*)1(2*2(26

4

4

4

3

3

3

2

2

2

1

1

1

x

x

x

x

x

x

x

x

Once an iteration has been done on complete is usedthe last x obtained as approximations on initial and start again, isx1 calculated so that it meets on the first equation, then calculatedx2 ...

Theoretically, the method of Gauss-Seidel can be a process in nito. Inpractice the process ends when the xk to xk + No changes aresmall ± os. This means that the current x is almost the solution

1. GAUSS SEIDEL METHOD WITH RELAXATION

Relaxation is a modification of Gauss Seidel method, and is designed to improve convergence. After each new estimated value of x, this value is modified by a weighted average of the results of previous and current iterations.If is equal to zero and the result is unchanged.

MODIFICATION OF SUBRELAJACION

values between 0 and 1The result is a weighted average of current and previous resultsmakes a non-convergent, converges

MODIFICATION OF OVERRELAXATION  

values between 1 and 2There is an implicit assumption that the new value moves in the right direction towards the real solution, but with a speed too slow, therefore we expect the additional weighting [improve the estimation to take it closer to the trueAccelerates the convergence of a system that already

•http://es.wikipedia.org/wiki/M%C3%A9todo_de_Jacobi

•http://html.rincondelvago.com/metodos-numericos_4.html

•http://www.uv.es/diaz/mn/node36.html•http://

matematicas.unal.edu.co/hmora/mnq.pdf•CHAPRA, Steven C. “Métodos Numéricos

para Ingenieros”. Edit. McGraw Hil.

BIBLIOGRAPHY