The Jacobi method is a iterative method of solving the square system of linear equations.

1. Solving System of Equations
using
Jacobi Iterative
Method
Luckshay Batra
luckybatra17@gmail.com

2. About the Method
The Jacobi method is a iterative method of solving the square system of linear equations. This methods
makes two assumptions (i) the system given by
has a unique solution and (ii) the coefficient matrix A has no zeros on its main diagonal, namely, a11, a22, a33
are non-zeros.
Basic Idea of the Method : In Jacobi we solve these equations for x1, x2, x3. We solve Ist equation for x1, IInd
equation for x2, ans so on to obtain the rewritten equations as :
Then make an initial guess of the solution . Substitute these values into
the right hand side the of the rewritten equations to obtain the first approximation
This accomplishes one iteration.
In the same way, the second approximation is computed by substituting the
first approximation’s x-values into the right hand side of the rewritten equations.
By repeated iterations, we form a sequence of approximations
For each generate the components generate the components of from by
for i=1,2,3,...
1

3. Jacobi Method in Matrix Form : Consider to solve nXn size system of linear equations Ax=b with
We split Ainto
Dx=(L+U)x+bAx=b is transformed into (D-L-U)x=b,
Assume D-1 exists and
x(k+1)=D-1(L+U)x(k)+D-1b
2
or x(k+1)=Hx(k)+c
Then, x=D-1(L+U)x+D-1b
And the matrix form of Jacobi iterative method is
where H=D-1(L+U) and c=D-1b. k=1,2,3,...
Limitation of the Method
Inflexible : For many matrices they don't converge. In general these methods are applicable only for
special matrices something like “strong diagonal" or matrices with non-zero diagonal entries. Even if
they converge, they find only one solution. Usually they perform badly for degenerate matrices (we just
learned them for regular square matrices, but again, there are generalizations).
Large Set-Up Time : The Jacobi method cannot immediately begin producing results. Before it can begin
its iteration, a matrix −D−1(L+U) must be computed. For large input matrices, this may not be a trivial
operation, as it takes O(n2) time to perform this matrix multiplication. The result is a significant lag before
any results can be output.

4. 3
Algorithm
Step 1 : Input the coefficient matrix “A”, vector “b”, tolerance level “tol”, initial approximation “x”,
number of iterations “n”.
Step 2 :
Step 3 :
Decompose the coefficient matrix into Diagonal Matrix “D”, Lower Triangular Matrix “L” and
Upper Triangular Matrix “U”.
Calculate the values of H= −D−1(L+U) and c= D−1b.
Step 4 : For k = 1, k ≤ n, do steps 5,6,7.
Step 5 : x(k+1)=H*x(k)+cCalculate and B=max|x(k+1) - x(k)|
Step 6 :
Step 7 :
If B<tol, then
Output : “Method Successful.”
Output : “ Condition max|x(k+1) - x(k)| < tol was met after k iterations.”
Output : Solution of system of equations after k iterations.
End.
Else, calculate the next iteration k=k+1.
Step 8 : If B>tol or k>n
Output : “Maximum number of iterations reached without satisfying tolerance condition.”
Output : Value of x for k iterations performed.

5. 4
Matlab Code
Function Code :
function jacobi(A, b,tol,x,n)
D=diag(diag(A));
L=tril(-A,-1);
U=triu(-A,1);
H=inv(D)*(L+U);
c=inv(D)*b;
k=1;
while k<=n
x(:,k+1)=H*x(:,k)+c;
B=max(abs(x(:,k+1)-x(:,k)));
if B<tol
disp('Jacobi Method was successful')
disp('Condition max|x^(k+1)-x^(k)|<tol was met after k iterations'); disp(k);
disp('x = ');disp(x(:,k+1));
break
end
k = k+1;
end
if B>tol||k>n
disp('Maximum number of iterations reached without satisfying tolerance condition:')
disp(x');
end
Program Code :
disp('Jacobi Method for Solving System of Linear Equations')
A=input('Enter the coefficient matrix A : ');
b=input('Enter the right hand side vector b : ');
tol=input('Enter tolerance level : ');
x=input('Enter initial approximation : ');
n=input('No. of iterations : ');
jacobi(A,b,tol,x,n)

6. Examples
1. Consider the system of equations
0.2x1 + 0.1x2 +x3 +x4 = 1
0.1x1 + 4x2 -x3 +x4- x5= 2 x1
- x2 + 60x3-2x5= 3
x1+ x2 + 8x4 +4x5=4
-x2 -2 x3 +4x4+700x5=5
with initial guess x(0)=[0000 0] T . Tolerance =0.00005
Solution in Matlab
2 4 700]
>> jacobimethodfinalinputsolve
Jacobi Method for Solving System of Linear Equations
Enter the coefficient matrix A : [0.2 0.1 1 1 0 ; 0.1 4 -1 1 -1 ; 1 -1 60 0 -2 ; 1 1 0 8 4 ; 0 -1 -
Enter the right hand side vector b : [ 1 ; 2 ; 3 ; 4 ; 5 ]
Enter tolerance level : 0.00005
Enter initial approximation : [ 0 ; 0 ; 0 ; 0 ; 0]
No. of iterations : 91
Jacobi Method was successful
Condition max|x^(k+1)-x^(k)|<tol was met after k iterations
91
x =
7.8597
0.4229
-0.0736
-0.5406
0.0106
2. Consider the system of equations
2x1 - x2 = 7
-x1 + 2x2 - x3 = 1
-x2 + 2x3 =3
with initial guess x(0)=[0 0 0] T . Tolerance = 1x10-4
Solution in Matlab
>> jacobimethodfinalinputsolve
Jacobi Method for Solving System of Linear Equations
Enter the coefficient matrix A : [2 -1 0 ; -1 2 -1 ; 0 -1 2]
Enter the right hand side vector b : [7 ; 1 ; 3]
Enter tolerance level : 0.0001
Enter initial approximation : [0 ; 0 ; 0]
No. of iterations : 100
Jacobi Method was successful
Condition max|x^(k+1)-x^(k)|<tol was met after k iterations
31
x =
6.4999
5.9998
4.4999
5

7. 3. Consider the system of equations
2.55509x1 + 0.85822x2 + 0.92679x3 + 0.23732x4 + 0.17628x5 = 0.66360
0.38558x1 + 2.55646x2 + 0.40726x3 + 0.88843x4 + 0.89019x5 = 0.04431
0.44825x1 + 0.37596x2 + 2.21124x3 + 0.61538x4 + 0.61754x5 = 0.00939
0.53346x1 + 0.29330x2 + 0.23489x3 + 2.51195x4 + 0.57504x5 = 0.48425
0.33918x1 + 0.58728x2 + 0.93330x3 + 0.45368x4 + 2.98108x5 = 0.66024
with initial guess x(0)=[0.21771 0.08108 0.65971 0.68427 0.72107] T . Tolerance = 4x10-3
Solution in Matlab
>> jacobimethodfinalinputsolve
Jacobi Method for Solving System of Linear Equations
Enter the coefficient matrix A : [ 2.55509 0.85822 0.92679 0.23732 0.17628 ;
0.38558 2.55646 0.40726 0.88843 0.89019 ; 0.44825 0.37596 2.21124 0.61538 0.61754 ;
0.53346 0.29330 0.23489 2.51195 0.57504 ; 0.33918 0.58728 0.93330 0.45368 2.98108 ]
Enter the right hand side vector b : [0.66360 ; 0.04431; 0.00939; 0.48425; 0.66024]
Enter tolerance level : 0.00001
Enter initial approximation : [0.21771; 0.08108; 0.65971; 0.68427; 0.72107]
No. of iterations : 80
Jacobi Method was successful
Condition max|x^(k+1)-x^(k)|<tol was met after k iterations
67
x =
0.3252
-0.1265
-0.1331
0.0968
0.2363
6