Jacobi method

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Revision as of 01:22, 19 December 2005

Introduction

We seek the solution to set of linear equations:

$A \phi = b$

In matrix terms, the definition of the Jacobi method can be expressed as :

$\phi^{(k+1)} = D^{ - 1} \left[\left( {L + U} \right)\phi^{(k)} + b\right]$

where $D$ , $L$ , and $U$ represent the diagonal, lower triangular, and upper triangular parts of the coefficient matrix $A$ and $k$ is the iteration count. This matrix expression is mainly of academic interest, and is not used to program the method. Rather, an element-based approach is used:

$\phi^{(k+1)}_i = \frac{1}{a_{ii}} \left(b_i -\sum_{j\ne i}a_{ij}\phi^{(k)}_j\right),\, i=1,2,\ldots,n.$

Note that the computation of $\phi^{(k+1)}_i$ requires each element in $\phi^{(k)}$ except itself. Then, unlike in the Gauss-Seidel method, we can't overwrite $\phi^{(k)}_i$ with $\phi^{(k+1)}_i$ , as that value will be needed by the rest of the computation. This is the most meaningful difference between the Jacobi and Gauss-Seidel methods. The minimum ammount of storage is two vectors of size $n$ , and explicit copying will need to take place.

Algorithm

Chose an initial guess $\phi^{0}$ to the solution

for k := 1 step 1 untill convergence do

for i := 1 step until n do

$\sigma = 0$

for j := 1 step until n do

if j != i then

$\sigma = \sigma + a_{ij} \phi_j^{(k-1)}$

end if

end (j-loop)

$\phi_i^{(k)} = {{\left( {b_i - \sigma } \right)} \over {a_{ii} }}$

end (i-loop)

check if convergence is reached

end (k-loop)

@@ Line 1: / Line 1: @@
+== Introduction ==
 We seek the solution to set of linear equations: <br>
-:<math> A \cdot \Phi = B </math> <br>
+:<math> A \phi = b </math> <br>
 In matrix terms, the definition of the Jacobi method can be expressed as : <br>
-<math>
+:<math>
-\phi^{(k)}  = D^{ - 1} \left( {L + U} \right)\phi^{(k - 1)}  + D^{ - 1} B
+\phi^{(k+1)}  = D^{ - 1} \left[\left( {L + U} \right)\phi^{(k)}  +  b\right]
 </math><br>
-Where '''D''','''L''' and '''U''' represent the diagonal, lower triangular and upper triangular matrices of coefficient matrix '''A''' and k is iteration counter.<br>
-=== Algorithm ===
+where <math>D</math>, <math>L</math>, and <math>U</math> represent the diagonal, lower triangular, and upper triangular parts of the coefficient matrix <math>A</math> and <math>k</math> is the iteration count.  This matrix expression is mainly of academic interest, and is not used to program the method.  Rather, an element-based approach is used:
-----
-:    Chose an intital guess <math>\Phi^{0}</math> to the solution <br>
+:<math>
+\phi^{(k+1)}_i  = \frac{1}{a_{ii}} \left(b_i -\sum_{j\ne i}a_{ij}\phi^{(k)}_j\right),\, i=1,2,\ldots,n.
+</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 [[Gauss-Seidel 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 is the most meaningful difference between the Jacobi and Gauss-Seidel methods.  The minimum ammount of storage is two vectors of size <math>n</math>, and explicit copying will need to take place.
+== Algorithm ==
+Chose an initial guess <math>\phi^{0}</math> to the solution <br>
 :    for k := 1 step 1 untill convergence do <br>
 ::   for i := 1 step until n do <br>
@@ Line 24: / Line 32: @@
 ::   check if convergence is reached
 :    end (k-loop)
-----
-'''Note''': The major difference between the Gauss-Seidel method and Jacobi method lies in the fact that for Jacobi method the values of solution of previous iteration (here k) are used, where as in Gauss-Seidel method the latest available values of solution vector <math>\Phi</math> are used. <br>
-----
-<i> Return to [[Numerical methods | Numerical Methods]] </i>

Jacobi method

From CFD-Wiki

Revision as of 01:22, 19 December 2005

Introduction

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