Spectral Radius Analysis

Given,

`A = [[5,1,2],[-3,9,4],[1,2,-7]], x=[(x_1),(x_2),(x_3)]` and `B = [(10),(-14),(33)]`

by Jacobi method: `x = -D^(−1) .(L + U).x_(old)+D^(−1).B`

by Gauss Siedal method: `x = -(D + L)^(−1).U.x_(old)+(D + L)^(−1).B`

by SOR method: `x = -(D + omega.L)^(−1).(U+(1-omega.L)).x_(old)+(D +omega. L)^(−1).B`

PLOTS Obtained:

RESULTS Obtained:

 Jacobi Eigen Valuees are:

 0.00975629542759200 + 0.101574120390764i
 0.00975629542759184 - 0.101574120390764i
-0.0195125908551837 + 0.00000000000000i

 Spectral Radius by Jacobi Method:

 1.02041595605066	0.408166382420264	0.255103989012665	0.185530173827393	 
 0.145773708007237	0.120048936005960	0.102041595605066

 Gauss-Siedal Eigen Values are:

 0
-0.0164951313255698
 0.00615862338906188

 Spectral Radius by Gauss-Siedal Method:

 1.38147205381097	0.210848106058203	0.0859319326010081	0.0477751236256374	 
 0.0309353883808418	0.0219255520717438	0.0164951313255698

 SOR Eigen Values are:

-0.0596986667793144
 0.0272079209467433
 0.0200137934516188

 Spectral Radius by SOR Method:

 1.89861498760412	0.369337723456480	0.186441695984047	0.122645312829869	 
 0.0909346516323290	0.0721164345895917	0.0596986667793144

Printed Comments on Command Window:

Error!! Solution is unable to converge...
For Jacobi Method with Diagonal Magnify 0.5 and Spectral Radius 1.02042

Error!! Solution is unable to converge...
For Gauss Siedal Method with Diagonal Magnify 0.5 and Spectral Radius 1.38147

Error!! Solution is unable to converge...
For SOR Method with Diagonal Magnify 0.5 and Spectral Radius 1.89861

Conclusion:

As Diagonal Magnification increases, Spectral Radius decreases.

Again with increasing Spectral Radius, Number of Iterations increases.

And For Spectral Radius >1, Solution is not able to converge.