GaussSeidel method
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:<math> T_i = \frac{1}{2}(T_{i1}+T_{i+1}).</math>  :<math> T_i = \frac{1}{2}(T_{i1}+T_{i+1}).</math>  
  Then, stepping through the solution vector from <math>i=2</math> to <math>i=n1</math>, we can compute the next iterate from the two surrounding values. Note that (in this scheme), <math>T_{i+1}</math> is from the previous iteration, while <math>T_{i1}</math> is from the current iteration.  +  Then, stepping through the solution vector from <math>i=2</math> to <math>i=n1</math>, we can compute the next iterate from the two surrounding values. Note that (in this scheme), <math>T_{i+1}</math> is from the previous iteration, while <math>T_{i1}</math> is from the current iteration: 
+  
+  :<math> T_i^{k+1} = \frac{1}{2}(T_{i1}^{k+1}+T_{i+1}^k).</math>  
+  
+  The following table gives the results of 10 iterations with 5 nodes (3 interior and 2 boundary) as well as <math>L_2</math> norm error.  
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  In this situation, the direction that we "sweep" is important  if we step though the solution vector in the opposite direction, the solution moves away from the chosen initial condition (zero everywhere in the interior) more quickly.  +  In this situation, the direction that we "sweep" is important  if we step though the solution vector in the opposite direction, the solution moves away from the chosen initial condition (zero everywhere in the interior) more quickly. The iteration is defined by 
+  
+  :<math> T_i^{k+1} = \frac{1}{2}(T_{i1}^{k}+T_{i+1}^{k+1}),</math>  
+  
+  and this gives us (slightly) faster convergence, as shown in the table below.  
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Revision as of 07:36, 8 April 2006
Introduction
We seek the solution to set of linear equations:
In matrix terms, the the GaussSeidel iteration can be expressed as
where , , and represent the diagonal, lower triangular, and upper triangular parts of the coefficient matrix and is the iteration count. This matrix expression is mainly of academic interest, and is not used to program the method. Rather, an elementbased approach is used:
Note that the computation of uses only those elements of that have already been computed and only those elements of that have yet to be advanced to iteration . This means that no additional storage is required, and the computation can be done in place ( replaces ). While this might seem like a rather minor concern, for large systems it is unlikely that every iteration can be stored. Thus, unlike the Jacobi method, we do not have to do any vector copying should we wish to use only one storage vector. The iteration is generally continued until the changes made by an iteration are below some tolerance.
Algorithm
 Chose an initial guess
 for k := 1 step 1 until convergence do
 for i := 1 step until n do

 for j := 1 step until i1 do
 end (jloop)
 for j := i+1 step until n do
 end (jloop)

 end (iloop)
 check if convergence is reached
 for i := 1 step until n do
 end (kloop)
Example Calculation
In some cases, we need not even explicitly represent the matrix we are solving. Consider the simple heat equation problem
subject to the boundary conditions and . The exact solution to this problem is . The standard secondorder finite difference discretization is
where is the (discrete) solution available at uniformly spaced nodes. In matrix terms, this can be written as
However, for any given for , we can write
Then, stepping through the solution vector from to , we can compute the next iterate from the two surrounding values. Note that (in this scheme), is from the previous iteration, while is from the current iteration:
The following table gives the results of 10 iterations with 5 nodes (3 interior and 2 boundary) as well as norm error.
Iteration  error  

0  0.0000E+00  0.0000E+00  0.0000E+00  0.0000E+00  1.0000E+00  1.0000E+00 
1  0.0000E+00  0.0000E+00  0.0000E+00  5.0000E01  1.0000E+00  6.1237E01 
2  0.0000E+00  0.0000E+00  2.5000E01  6.2500E01  1.0000E+00  3.7500E01 
3  0.0000E+00  1.2500E01  3.7500E01  6.8750E01  1.0000E+00  1.8750E01 
4  0.0000E+00  1.8750E01  4.3750E01  7.1875E01  1.0000E+00  9.3750E02 
5  0.0000E+00  2.1875E01  4.6875E01  7.3438E01  1.0000E+00  4.6875E02 
6  0.0000E+00  2.3438E01  4.8438E01  7.4219E01  1.0000E+00  2.3438E02 
7  0.0000E+00  2.4219E01  4.9219E01  7.4609E01  1.0000E+00  1.1719E02 
8  0.0000E+00  2.4609E01  4.9609E01  7.4805E01  1.0000E+00  5.8594E03 
9  0.0000E+00  2.4805E01  4.9805E01  7.4902E01  1.0000E+00  2.9297E03 
10  0.0000E+00  2.4902E01  4.9902E01  7.4951E01  1.0000E+00  1.4648E03 
In this situation, the direction that we "sweep" is important  if we step though the solution vector in the opposite direction, the solution moves away from the chosen initial condition (zero everywhere in the interior) more quickly. The iteration is defined by
and this gives us (slightly) faster convergence, as shown in the table below.
Iteration  error  

0  0.0000E+00  0.0000E+00  0.0000E+00  0.0000E+00  1.0000E+00  1.0000E+00 
1  0.0000E+00  1.2500E01  2.5000E01  5.0000E01  1.0000E+00  3.7500E01 
2  0.0000E+00  1.8750E01  3.7500E01  6.2500E01  1.0000E+00  1.8750E01 
3  0.0000E+00  2.1875E01  4.3750E01  6.8750E01  1.0000E+00  9.3750E02 
4  0.0000E+00  2.3438E01  4.6875E01  7.1875E01  1.0000E+00  4.6875E02 
5  0.0000E+00  2.4219E01  4.8438E01  7.3438E01  1.0000E+00  2.3438E02 
6  0.0000E+00  2.4609E01  4.9219E01  7.4219E01  1.0000E+00  1.1719E02 
7  0.0000E+00  2.4805E01  4.9609E01  7.4609E01  1.0000E+00  5.8594E03 
8  0.0000E+00  2.4902E01  4.9805E01  7.4805E01  1.0000E+00  2.9297E03 
9  0.0000E+00  2.4951E01  4.9902E01  7.4902E01  1.0000E+00  1.4648E03 
10  0.0000E+00  2.4976E01  4.9951E01  7.4951E01  1.0000E+00  7.3242E04 
For this toy example, there is not large penalty for choosing the wrong sweep direction. For some of the more complicated variants of GaussSeidel, there is a substantial penalty  the sweep direction determines (in a vague sense) the direction in which information travels.