If the solution is converging and updated information is available for some of.The Gauss-Seidel and Jacobi algorithms Introduction The Gauss-Seidel and Jacobi algorithms are iterative algorithms for solving linear equations A x b. One of the equations is then used to obtain the revised value of a particular variable by substituting inStephen Andrilli, David Hecker, in Elementary Linear Algebra (Fourth Edition), 2010 Comparing Iterative and Row Reduction MethodsIntuitively, the Gauss-Seidel method seems more natural than the Jacobi method. To start with, a solution vector is assumed, based on guidance from practical experience in a physical situation. Gauss- Seidel method The Gauss Seidel Method (GS) is an iterative algorithm for solving a set of non-linear algebraic equations.
If so, each step of the iterative process is relatively easy. When a linear system has a sparse matrix, each equation in the system may involve very few variables. Such matrices are said to be sparse. The only roundoff error that we need to consider with an iterative method is the error involved in the most recent step.Also, in many applications, the coefficient matrix for a given system contains a large number of zeroes. Seidel method which is also known as the Liebmann method or the method of.When are iterative methods useful? A major advantage of iterative methods is that roundoff errors are not given a chance to “accumulate,” as they are in Gaussian elimination and the Gauss-Jordan Method, because each iteration essentially creates a new approximation to the solution.
■In partial pivoting, as work begins on a new pivot column, the entries in this column below the pivot row are examined, and we switch rows, if necessary, to place the entry having the highest absolute value into the pivot position. New VocabularyPartial pivoting is used to avoid roundoff errors that could be caused by dividing every entry of a row by a pivot value that is relatively small compared to the rest of its remaining row entries. But even if the coefficient matrix is not sparse, iterative methods often give more accurate answers when large matrices are involved because fewer arithmetic operations are performed overall.On the other hand, when iterative methods take an extremely large number of steps to stabilize or do not stabilize at all, it is much better to use the Gauss-Jordan Method or Gaussian elimination.
■The Gauss-Seidel Method differs from the Jacobi Method in that immediately after a new x i value is obtained from the ith equation, it is used in place of the old value in successive substitutions. ■In each iteration of the Jacobi Method, the most recently obtained values for x 1, x 2,…, x n are substituted into every equation in the system simultaneously to obtain the next set of values for x 1, x 2,…, x n. ■Before applying the Jacobi Method or the Gauss-Seidel Method, the equations are rearranged so that in the ith equation the coefficient of x i is nonzero, and so that x i is expressed in terms of the other variables. The methods are successful if the values for x 1, x 2,…, x n eventually stabilize, thereby producing the actual solution.
In the Gauss–Seidel method, instead of always using previous iteration values for all terms of the right-hand side of Eq. Another advantage of iterative methods is that roundoff errors are not compounded.The Gauss–Seidel method is also a point-wise iteration method and bears a strong resemblance to the Jacobi method, but with one notable exception. ■Iterative methods are often effective on sparse matrices.
(3.17b), only z assumes a previous iteration value. However, when y is determined using Eq. (3.17a), both y and z assume previous iteration values. Thus, for the 3×3 example system considered earlier when x is determined using Eq.
Essentially, this implies that only two out of the four terms on the right-hand side of the update formula are treated explicitly, as shown in Fig. 3.3(a), then, by the time it is node O’s turn to get updated, nodes W and S have already been updated, and these updated values must be used. In the context of solution of a 2D PDE on a structured mesh, if the node by node update pattern (or sweeping pattern) is from left to right and bottom to top, as shown in Fig.
For boundary conditions other than the Dirichlet type, appropriate values of the link coefficients must be derived from the nodal equation at that boundary, and an update formula must be used.Step 3: Compute the residual vector using ϕ ( n+1), and then compute R2 ( n+1).Step 4: Monitor convergence, i.e., check if R2 ( n+1) < ɛ tol? If YES, then go to Step 5. For the interior nodes, this yields ϕ i , j ( n + 1 ) = S i , j − a E ϕ i + 1 , j ( n ) − a W ϕ i − 1 , j ( n + 1 ) − a N ϕ i , j + 1 ( n ) − a S ϕ i , j − 1 ( n + 1 ) a O, where the link coefficients are given by Eq. If any of the boundaries have Dirichlet boundary conditions, the guessed values for the boundary nodes corresponding to that boundary must be equal to the prescribed boundary values.Step 2: Set ϕ ( n+1) = ϕ ( n) and apply the Gauss–Seidel update formula, Eq. We denote these values as ϕ (0).
Gauss Seidel Method Serial Method For
Version of 4.210, with efficiencyPetkovic and Stefanovic (1990) show that a forward-backward variation on the G.-S. Methods, the most efficient of which is the G.-S. 1626Petkovic, Stefanovic and Marjanovic (1992) and (1993) give several G.-S. However, their major shortcoming is that both schemes scale poorly, and the number of iterations go up by a factor of four when the number of nodes is increased by a factor of four.Again, it is seen that there is not much advantage in the serial method for large n.Hansen et al (1977) give a serial version of the modified Laguerre method 4.171 of order > 4.Petkovic and Stefanovic (1986B) give a Gauss-Seidel version of their “Square root iteration with Halley correction”, given by 4.185 and 4.188, having convergence order G and hence efficiencyPetkovic, Milovanovic and Stefanovic (1986) give a version of the above for multiple roots, again of efficiency. Although the convergence is slow, the cost per iteration of both methods is also very low, making them attractive choices.
Since the fine-grid solution is what we are ultimately interested in, it is sufficient to initialize (guess) the solution at the fine-grid nodes. The next step in the algorithm is to initialize the dependent variable on the fine grid. The order is at least 6, and so the efficiency is(4.55b) 2 ( Δ x C ) 2 + 2 ( Δ y C ) 2 ϕ I , J C − 1 ( Δ x C ) 2 ϕ I + 1 , J C − 1 ( Δ x C ) 2 ϕ I − 1 , J C − 1 ( Δ y C ) 2 ϕ I , J + 1 C − 1 ( Δ y C ) 2 ϕ I , J − 1 C = − S I , J.
In other words, the overall iteration count is not dictated by the solver but rather by the multigrid treatment of the errors. Rather, the multigrid framework is. This is because in the context of the GMG algorithm, the solver is not solely responsible for reducing the errors. While any solver, discussed in Chapter 3, may be used for this purpose, it is important to choose one that is easy to implement and whose computational workload per iteration is small. If Dirichlet boundary conditions are used, the initial guess at the boundary nodes should be set equal to the prescribed boundary value.Next, the algebraic equations on the fine grid are solved using a solver of choice, but only to partial convergence.
Based on the criterion of low computational workload per iteration, it is customary to use classical iterative solvers for the smoothing operation rather than fully implicit solvers, such as the Krylov subspace solvers.
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