ch 7 gausseliminations

7/25/2019 Ch 7 GaussEliminations

1/8

Gauss-Seidel

Iterative or approximate methods provide analternative to the elimination methods. The Gauss-Seidel method is the most commonly used iterative

method.

The system [A]{X}={B} is reshaped by solving thefirst equation for x1, the second equation for x2, andthe third for x3, and n

th equation for xn. For

conciseness, we will limit ourselves to a 3x3 set ofequations.

PPS-UB-PAT-TM-2010 Chapter 11 1


2/8

33

2321313

1

22

32312122

11

31321211

a

xaxab

x

a

xaxabx

a

xaxabx


Now we can start the solution process by choosing

guesses for the xs. A simple way to obtain initialguesses is to assume that they are zero. These zeros

can be substituted into x1equation to calculate a new

x1=b1/a11.


3/8

New x1 is substituted to calculatex2 and x3. The

procedure is repeated until the convergence criterion

is satisfied:

sj

i

j

i

j

iia

x

xx %100

1

,


For all i, where j and j-1 are the present and previous

iterations.


4/8

Fig. 11.4



5/8

Convergence Criterion for Gauss-Seidel

Method

The Gauss-Seidel method has two fundamental

problems as any iterative method:

It is sometimes nonconvergent, and

If it converges, converges very slowly. Recalling that sufficient conditions for convergence

of two linear equations, u(x,y) and v(x,y) are

1

1

y

v

x

v

y

u

x

u



6/8

Similarly, in case of two simultaneous equations,the Gauss-Seidel algorithm can be expressed as

0

0

),(

),(

222

21

1

11

12

21

1

22

21

22

221

2

11

12

11

121

x

v

a

a

x

v

a

a

x

u

x

u

xaa

abxxv

xa

a

a

bxxu



7/8

Substitution into convergence criterion of two linear

equations yield:

1122

21

11

12

a

a

a

a

n

ij

j

jiaaii

aa

aa

1

,

2122

1211

:equationsnFor


In other words, the absolute values of the slopes

must be less than unity for convergence:


8/8

Figure 11.5


ch 7 gausseliminations

Documents