Gauss seidel method of convergence

In summary, the Gauss-Seidel method is an iterative numerical method used to solve linear equations and is commonly used in scientific and engineering applications. It works by repeatedly solving equations using updated values from the previous iteration until a solution within a desired accuracy is reached. The method has advantages such as faster convergence and higher efficiency for larger systems, but may not converge for all systems and requires careful selection of parameters for improved convergence.
  • #1
lavster
217
0
hi,

im trying to calculate the spectral radius (gauss seidel method) to test whether or not its converges. i no that if the modulus is less than one it converges. however, my answer gives a mixture of solutions which are above and below one - 0.45 and 2.15 and 0. so is this divergent or convergent? does all three need to be below 1 in order for it to converge?

thanks
 
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  • #2
I can't make heads or tails of your question, but the spectral radius is one number only.
 
  • #3
yeah, i was confused. but i get it now :)
 

What is the Gauss-Seidel method of convergence?

The Gauss-Seidel method is an iterative numerical method used to solve a system of linear equations. It is an improvement over the Jacobi method and is commonly used in scientific and engineering applications.

How does the Gauss-Seidel method work?

The method works by repeatedly solving equations from the system, using updated values from the previous iteration. This process continues until the calculated values converge to a solution within a desired accuracy.

What are the advantages of using the Gauss-Seidel method?

The Gauss-Seidel method typically converges faster than the Jacobi method, especially for larger systems of equations. It also requires less memory and is more efficient for sparse matrices.

What are the limitations of the Gauss-Seidel method?

The method may not converge for all systems of equations, particularly if the matrix is not diagonally dominant. It also may not converge if the system has multiple solutions or no solution at all.

How can the convergence of the Gauss-Seidel method be improved?

The convergence of the method can be improved by using a different initial guess, choosing a different iteration formula, or using a relaxation parameter. It is also important to check for convergence and adjust the accuracy tolerance as needed.

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