2.6 Iterative methods for linga Systems
2.6 Iterative Methods for Linear Systems
1.6.1 Jacobi Iteration
1.6.1 Jacobi Iteration
Example 3. 26. Consider the system of equations 4-y+ 4c-8+2=21 2x+y+52=15
Table 3. 2 Convergence Jacobi iteration for the Linear System(1) k k k 1.0 2.0 2.0 1 1.75 3.375 3.0 1.84375 3.875 3.025 1.9625 3.925 2.9625 41.990625003.97656250300000000 51.994140633995312503.00093750 151999999933.999999852.99999993 192.00000004.0000000030000000
Example 3. 27. Let the linear system(1) be rearranged as follows 2c+y+52=-15 4x-8y+2=-21 4c-y+2=7
Table 3.3 Divergent Jacobi iteration for the Linear System(4) k k k k0123 1.0 2.0 2.0 1.5 3.375 5.0 6.6875 2.5 16.375 34.6875 8.015625 17.25 446.617188 17.8125 123.73438 5307.92968836.15039121128125 6502.62793124.9296881202.56836
3.6.2 Gauss-Seidel iteration
3.6.2 Gauss-Seidel Iteration
Example 3. 28. Consider the system of equations given in(1)and the Gauss-Seidel iterative process suggested by(2) k+1 4 1+4k+1+ 9k+1 15+2xk+1-9k k+1
Table 3.2 Convergence Gauss-Seidel iteration for the Linear System(1) k a k y 2k 2.0 11.75 3.75 2.95 2195 3.96875 298625 319956253996093752993125 81.999993199999829999969 9199999983999993.000 102.000000.00003000000
Definition 3.6. A matrix A of dimension N X N is said to be strictly diagonally dominant provided that j=1 j* 1 for k=1.2