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ITERATIVE METHODS FOR SOLVING LINEAR EQUATIONS

ITERATIVE METHODS FOR SOLVING linear equations
There are other methods that can be used to solve a set of linear equations that are based
on iteration. In these cases, an initial estimate of the parameters is estimated and then the
equations are solved, yielding an updated version of the parameters. These new values
are then inserted back into the equations and the process continues until the desired
solution is reached . The two iterative methods discussed here are the Jacobi method and
the Gauss-Seidel method.

Jacobi Ite ration Method

Given a set of linear equations,


the problem is one of solving for x1, x2, …, xn. The right hand side of these equations, si,
re presents the solution . We begin by rearranging these equations in the form of solving
for the unknown parameters one equation at a time. Thus,

The superscript (0) indicates the initial estimate of the parameters. For the first pass,
these parameters are given the value zero . The equations are then solved which results in
an updated value of the parameters. These current estimates are then inserted back into
the equations and a newer set of parameters is arrived at by solving these equations. The
process continues until the solution converges.

As an example, take the fol lowing linear equations:


Rearrange these equations

Use as the initial estimates: Insert these estimates into these
equations yielding new estimates of the parameters.

Insert these updated estimates back into original equation again, yielding

This process is continued until the desired results are obtained. The following table
shows the solutions arrived at after each iteration. These results are from the attached
Fortran program. The output shows x(i) that are the parameters xi. Also output is the
change in the parameters , dx(i), between each iteration

How good are these results? Lets take our equations and put them into an augmented
matrix and solve using Gauss- Jordan elimination .

As one can see, the values using the Jacobi iterative method are very close. Following is
a Fortran program that can be used to use the Jacobi iteration to solve a set of equations.
The limitation now is that it is restricted to only a 3 x 3 matrix, due to formatting
procedures currently used in the program.

c Program Jacobi
c Program to solve a linear equation using the Jacobi Iteration
c method
c
IMPLICIT none
REAL*8 coef(3,4), d, dx(3), x(3,4), xn(3), xnp(3)
INTEGER i, iter, iterate,j, no, nv
DATA iterate /0/
c
c The data are entered into the program using a data file called
c jacobi.dat. It has the following row values
c number of equations
c number of variables
c x(1) x(2) x(3) solution for the first equation
c x(1) x(2) x(3) solution for the second equation
c x(1) x(2) x(3) solution for the third equation
c
OPEN (4, file = 'jacobi.dat')
OPEN (6, file = 'results')
c
c no is the number of equations and nv is the number of variables
c
read(4,*) no
do 5 i=1,no
xn(i) = 0.d0
5 continue
read(4,*) nv
write(6,901)
c
c The coefficients for the variables are read in the matrix x with
c the solution to the equations being the last column
c
do 10 i=1,no
read(4,*)(x(i,j),j=1,no+1)
c
c d is the coefficient for the variable that is being solved for
c it forms the denominator to compute the real number for the
c remaining coefficients
c
d = x(i,i)
do 7 j=1,no+1
coef(i,j) = x(i,j)/d
7 end do
c
c Because the Jacobi method solves for the unknown variable with
c respect to the current estimates of the other variables, the
c coefficient for the variable is made to be zero for subsequent
c use in the loop to compute the adjusted estimates
c
coef(i,i) = 0.d0
write(6,900)(x(i,j),j=1,nv+1)
10 end do
write(6,902)
do 13 i=1,no
write(6,900)(coef(i,j),j=1,nv+1)
13 end do
write(6,903)
15 iter = 0
c
c iterate is just a counter to keep track of the number of iterations
c
iterate = iterate+1
c
c Solve for the estimate of the unknown parameters
c
do 20 i=1,no
xnp(i) = coef(i,nv+1)
do 18 j=1,nv
xnp(i) = xnp(i) - coef(i,j)*xn(j)
18 end do
20 end do
c
c dx is a vector showing the change in the estimate of the variable
c with respect to the estimated value used in the previous iteration
c
do 50 i=1,no
dx(i) = xnp(i) - xn(i)
c
c Test to see if the change is greater than the threshold
c If it is, then the variable iter is made equal to 1
c At the beginning of each loop, this value is made equal to 0
c If iter is 1 then this means to iterate again
c
if (dabs(dx(i)).gt.0.01d0) iter = 1
c
c Update the estimated parameter value
c
xn(i) = xnp(i)
50 continue
write(6,904)iterate,(xn(i),i=1,nv),(dx(i),i=1,nv)
if (iter.gt.0) go to 15
900 FORMAT(5x,4(f10.4,5x))
901 FORMAT(15x,'SOLVING LINEAR EQUATION USING THE JACOBI ITERATION MET
1HOD',//,'The coefficients to the equations with the solution at th
1e end are:',/)
902 FORMAT(//,'Rearranging the equations to solve for the unknown vari
1ables yields',/,5x,'the following coefficients: ',/)
903 FORMAT(//,'The estimated results after each iteration are shown as
1:'//,'Iteration',2x,'x(1)',9x,'x(2)',9x,'x(3)',7x,'dx(1)',7x,'dx(2
2)',7x,'dx(3)')
904 FORMAT(i3,4x,6(f10.5,2x))
stop
end
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