Jacobi method
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  +  The Jacobi method is an algorithm in linear algebra for determining the solutions of a system of linear equations with largest absolute values in each row and column dominated by the diagonal element. Each diagonal element is solved for, and an approximate value plugged in. The process is then iterated until it converges. This algorithm is a strippeddown version of the Jacobi transformation method of matrix diagonalization. The method is named after German mathematician [http://en.wikipedia.org/wiki/Carl_Gustav_Jakob_Jacobi Carl Gustav Jakob Jacobi].  
We seek the solution to set of linear equations: <br>  We seek the solution to set of linear equations: <br>  
Line 16:  Line 16:  
</math>  </math>  
  Note that the computation of <math>\phi^{(k+1)}_i</math> requires each element in <math>\phi^{(k)}</math> except itself. Then, unlike in the [[GaussSeidel method]], we can't overwrite <math>\phi^{(k)}_i</math> with <math>\phi^{(k+1)}_i</math>, as that value will be needed by the rest of the computation. This  +  Note that the computation of <math>\phi^{(k+1)}_i</math> requires each element in <math>\phi^{(k)}</math> except itself. Then, unlike in the [[GaussSeidel method]], we can't overwrite <math>\phi^{(k)}_i</math> with <math>\phi^{(k+1)}_i</math>, as that value will be needed by the rest of the computation. This difference between the Jacobi and GaussSeidel methods complicates matters somewhat. Generally, two vectors of size <math>n</math> will be needed, and a vectortovector copy will be required. If the form of <math>A</math> is known (e.g. tridiagonal), then the additional storage should be avoidable with careful coding. 
== Algorithm ==  == Algorithm ==  
  +  Choose an initial guess <math>\phi^{0}</math> to the solution <br>  
  : for k := 1 step 1  +  : for k := 1 step 1 until convergence do <br> 
:: for i := 1 step until n do <br>  :: for i := 1 step until n do <br>  
::: <math> \sigma = 0 </math> <br>  ::: <math> \sigma = 0 </math> <br>  
Line 34:  Line 34:  
==Convergence==  ==Convergence==  
  +  The method will always converge if the matrix A is strictly or irreducibly diagonally dominant. Strict row diagonal dominance means that for each row, the absolute value of the diagonal term is greater than the sum of absolute values of other terms:  
+  
:<math>\left  a_{ii} \right  > \sum_{i \ne j} {\left  a_{ij} \right } </math>  :<math>\left  a_{ii} \right  > \sum_{i \ne j} {\left  a_{ij} \right } </math>  
  
  +  The Jacobi method sometimes converges even if this condition is not satisfied. It is necessary, however, that the diagonal terms in the matrix are greater (in magnitude) than the other terms.  
+  
+  == Example Calculation ==  
+  As with [[GaussSeidel methodGaussSeidel]], Jacobi iteration lends itself to situations in which we need not explicitly represent the matrix. Consider the simple heat equation problem  
+  
+  :<math>\nabla^2 T(x) = 0,\ x\in [0,1]</math>  
+  
+  subject to the boundary conditions <math>T(0)=0</math> and <math>T(1)=1</math>. The exact solution to this problem is <math>T(x)=x</math>. The standard secondorder finite difference discretization is  
+  
+  :<math> T_{i1}2T_i+T_{i+1} = 0,</math>  
+  
+  where <math>T_i</math> is the (discrete) solution available at uniformly spaced nodes (see [[GaussSeidel method#Example Calculationthe GaussSeidel example]] for the matrix form). For any given <math>T_i</math> for <math>1 < i < n</math>, we can write  
+  
+  :<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. For a proper Jacobi iteration, we'll need to use values from the previous iteration on the righthand side:  
+  
+  :<math> T_i^{k+1} = \frac{1}{2}(T_{i1}^{k}+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.  
+  { align=center border=1  
+  + Jacobi Solution  
+  ! Iteration !! <math>T_1</math> !! <math>T_2</math> !! <math>T_3</math> !! <math>T_4</math> !! <math>T_5</math> !! <math>L_2</math> 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  
+   5.0000E01  
+   1.0000E+00  
+   4.3301E01  
+    
+   3  
+   0.0000E+00  
+   1.2500E01  
+   2.5000E01  
+   6.2500E01  
+   1.0000E+00  
+   3.0619E01  
+    
+   4  
+   0.0000E+00  
+   1.2500E01  
+   3.7500E01  
+   6.2500E01  
+   1.0000E+00  
+   2.1651E01  
+    
+   5  
+   0.0000E+00  
+   1.8750E01  
+   3.7500E01  
+   6.8750E01  
+   1.0000E+00  
+   1.5309E01  
+    
+   6  
+   0.0000E+00  
+   1.8750E01  
+   4.3750E01  
+   6.8750E01  
+   1.0000E+00  
+   1.0825E01  
+    
+   7  
+   0.0000E+00  
+   2.1875E01  
+   4.3750E01  
+   7.1875E01  
+   1.0000E+00  
+   7.6547E02  
+    
+   8  
+   0.0000E+00  
+   2.1875E01  
+   4.6875E01  
+   7.1875E01  
+   1.0000E+00  
+   5.4127E02  
+    
+   9  
+   0.0000E+00  
+   2.3438E01  
+   4.6875E01  
+   7.3438E01  
+   1.0000E+00  
+   3.8273E02  
+    
+   10  
+   0.0000E+00  
+   2.3438E01  
+   4.8438E01  
+   7.3438E01  
+   1.0000E+00  
+   2.7063E02  
+  }  
+  
  ==External  +  ==External links== 
  *[http://en.wikipedia.org/wiki/Jacobi_method Wikipedia  +  *[http://www.mathlinux.com/spip.php?article49 Jacobi method from www.mathlinux.com] 
+  *[http://mathworld.wolfram.com/JacobiMethod.html Jacobi Method at Math World]  
+  *[http://en.wikipedia.org/wiki/Jacobi_method Jacobi method at Wikipedia] 
Latest revision as of 09:15, 3 January 2012
The Jacobi method is an algorithm in linear algebra for determining the solutions of a system of linear equations with largest absolute values in each row and column dominated by the diagonal element. Each diagonal element is solved for, and an approximate value plugged in. The process is then iterated until it converges. This algorithm is a strippeddown version of the Jacobi transformation method of matrix diagonalization. The method is named after German mathematician Carl Gustav Jakob Jacobi.
We seek the solution to set of linear equations:
In matrix terms, the definition of the Jacobi method 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 requires each element in except itself. Then, unlike in the GaussSeidel method, we can't overwrite with , as that value will be needed by the rest of the computation. This difference between the Jacobi and GaussSeidel methods complicates matters somewhat. Generally, two vectors of size will be needed, and a vectortovector copy will be required. If the form of is known (e.g. tridiagonal), then the additional storage should be avoidable with careful coding.
Contents 
Algorithm
Choose an initial guess to the solution
 for k := 1 step 1 until convergence do
 for i := 1 step until n do

 for j := 1 step until n do
 if j != i then
 end if
 if j != i then
 end (jloop)

 end (iloop)
 check if convergence is reached
 for i := 1 step until n do
 end (kloop)
Convergence
The method will always converge if the matrix A is strictly or irreducibly diagonally dominant. Strict row diagonal dominance means that for each row, the absolute value of the diagonal term is greater than the sum of absolute values of other terms:
The Jacobi method sometimes converges even if this condition is not satisfied. It is necessary, however, that the diagonal terms in the matrix are greater (in magnitude) than the other terms.
Example Calculation
As with GaussSeidel, Jacobi iteration lends itself to situations in which we need not explicitly represent the matrix. 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 (see the GaussSeidel example for the matrix form). 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. For a proper Jacobi iteration, we'll need to use values from the previous iteration on the righthand side:
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  5.0000E01  1.0000E+00  4.3301E01 
3  0.0000E+00  1.2500E01  2.5000E01  6.2500E01  1.0000E+00  3.0619E01 
4  0.0000E+00  1.2500E01  3.7500E01  6.2500E01  1.0000E+00  2.1651E01 
5  0.0000E+00  1.8750E01  3.7500E01  6.8750E01  1.0000E+00  1.5309E01 
6  0.0000E+00  1.8750E01  4.3750E01  6.8750E01  1.0000E+00  1.0825E01 
7  0.0000E+00  2.1875E01  4.3750E01  7.1875E01  1.0000E+00  7.6547E02 
8  0.0000E+00  2.1875E01  4.6875E01  7.1875E01  1.0000E+00  5.4127E02 
9  0.0000E+00  2.3438E01  4.6875E01  7.3438E01  1.0000E+00  3.8273E02 
10  0.0000E+00  2.3438E01  4.8438E01  7.3438E01  1.0000E+00  2.7063E02 