Description
The Jacobi method is based on solving for every variable locally with respect to the other variables; one iteration corresponds to solving for every variable once. It is easy to understand and implement, but convergence is slow.
Jacobi Method.
The Jacobi method is a method of solving a matrix equation on a matrix that has no zeros along its main diagonal. Each diagonal element is solved for, and an approximate value plugged in. The process is then iterated until it converges. The Jacobi method is easily derived by examining each of the n equations in the linear system of equations Ax = b in isolation
The Gauss-Seidel method is similar to the Jacobi method except that it uses updated values as soon as they are available. It generally converges faster than the Jacobi method, although still relatively slowly.Gauss-Seidel Method Iterative Method for Linear System.
The method is an improved version of the Jacobi method. It is defined on matrices with non-zero diagonals, but convergence is only guaranteed if the matrix is either diagonally dominant, or symmetric and (semi) positive definite.
The successive over-relaxation method can be derived from the Gauss-Seidel method by introducing an extrapolation parameter omega. This method can converge faster than Gauss-Seidel by an order of magnitude. California State University, East Bay Numerical Analysis Iterative Techniques for Solving Linear Systems.♣ Successive Over-relaxation Method (SOR). This method of solving a linear system of equations Ax = b derived by extrapolating the Gauss-Seidel method. This extrapolation takes the form of a weighted average between the previous iterate and the computed Gauss-Seidel iterate successively for each component.
If ω = 1, the SOR method simplifies to the Gauss-Seidel method. A theorem due to Kahan (1958) shows that SOR fails to converge if ω is outside of the interval (0, 2). In general, it is not possible to compute in advance the value of ω that will maximize the rate of convergence of SOR.
Finally, the symmetric successive over-relaxation method is useful as a pre-conditioner for non-stationary methods. However, it has no advantage over the successive over-relaxation method as a stand-alone iterative method. Neumann Lemma. If A is an n × n matrix with ρ(A).
Symmetric Successive Over-relaxation Method (SSOR).
If the matrix A is symmetric, then
the Symmetric Successive Over-relaxation method, combines two SOR sweeps together in such a way
that the resulting iteration matrix is similar to a symmetric matrix. Specifically, the first SOR sweep is
carried out as in SOR, but in the second sweep the unknowns are updated in the reverse order. That is, SSOR is a forward SOR sweep followed by a backward SOR sweep. The similarity of the SSOR iteration
matrix to a symmetric matrix permits the application of SSOR as a pre-conditioner for other iterative
schemes for symmetric matrices. Indeed, this is the primary motivation for SSOR since its convergence
rate , with an optimal value, is usually slower than the convergence rate of SOR with an optimal value.
Explanation & Answer
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The Jacobi strategy depends on explaining each variable locally regarding alternate factors;
one iteration compares to solving for each variable once. This technique is straightforward to
actualizes. However, convergence is moderate.
Jacobi Method
The Jacobi technique is a strategy for solving a matrix equation on a matrix that has no zeros
along its fundamental diagonal. Every diagonal component is illuminated for, and an
approximate value is fixed. The procedure is then iterated until the point when it meets. The
Jacobi technique is effectively determined by inspecting each of the n equations in the linear
system of equations Ax = b in isolation.
The Gauss-Seidel strategy is like the Jacobi technique with the exception that it utilizes
updated values when they are accessible. It also converges faster but relatively slower as
compared to Jacobi strategy.
Gauss-Seidel Method Iterative Method for Linear Systems
This strategy is an enhanced version of the Jacobi technique. It is described on matrices with
non-zero diagonals. However convergence is only ensured if the matrix is either a diagonally
prevailing, or symmetric and (semi) positive ...
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