What if Gauss Seidel is not diagonally dominant?
What can be done? If the coefficient matrix is not originally diagonally dominant, the rows can be rearranged to make it diagonally dominant. This particular problem is an example of a system of linear equations that cannot be solved using the Gauss-Seidel method.
What is diagonally dominant condition?
In mathematics, a square matrix is said to be diagonally dominant if, for every row of the matrix, the magnitude of the diagonal entry in a row is larger than or equal to the sum of the magnitudes of all the other (non-diagonal) entries in that row.
Which of the following condition holds for diagonally dominant?
If the diagonal element of every row is greater or equal to the sum of the non-diagonal elements of the same row, then the matrix is a diagonally dominant matrix.
Is Gauss-Seidel faster than Jacobi?
The Gauss-Seidel method is like the Jacobi method, except that it uses updated values as soon as they are available. In general, if the Jacobi method converges, the Gauss-Seidel method will converge faster than the Jacobi method, though still relatively slowly.
Can a non square matrix be diagonally dominant?
A strictly diagonally dominant matrix is non-singular, i.e. has an inverse. This is known as the Levy-Desplanques theorem; a proof of the theorem can be found here. A symmetric diagonally dominant real matrix with non-negative diagonal entries is positive semidefinite (PSD).
What makes a matrix diagonally dominant?
In words, a diagonally dominant matrix is a square matrix such that in each row, the absolute value of the term on the diagonal is greater than or equal to the sum of absolute values of the rest of the terms in that row. A strictly diagonally dominant matrix is non-singular, i.e. has an inverse.
Why is Sor better than Gauss-Seidel?
Successive Overrelaxation (SOR) can be derived from the Gauss-Seidel method by introducing an extrapolation parameter . For the optimal choice of , SOR may converge faster than Gauss-Seidel by an order of magnitude.
Is (strictly) diagonally dominant by rows?
It is strictly diagonally dominant by rows if strict inequality holds in (2) for all . is (strictly) diagonally dominant by columns if is (strictly) diagonally dominant by rows. Diagonal dominance on its own is not enough to ensure nonsingularity, as the matrix (1) shows. Strict diagonal dominance does imply nonsingularity, however. Theorem 1.
When is a matrix generalized diagonally dominant by rows?
A matrix is generalized diagonally dominant by rows if is diagonally dominant by rows for some diagonal matrix with for all , that is, if It is easy to see that if is irreducible and there is strictly inequality in (6) for some then is nonsingular by Theorem 2.
Is the inverse of a strictly row diagonally dominant matrix?
It is interesting to note that the inverse of a strictly row diagonally dominant matrix enjoys a form of diagonal dominance, namely that the largest element in each column is on the diagonal.
Is diagonal dominance enough to ensure nonsingularity?
Diagonal dominance plus two further conditions is enough to ensure nonsingularity. We need the notion of irreducibility. A matrix is irreducible if there does not exist a permutation matrix such that.