2.6 Iterative methods for linear 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 4x-y+2=-7 4x-8y+2=21 2x+y+52=15
Table 3.2 Convergence Jacobi iteration for the Linear System(1) k k 2k 1.0 2.0 2.0 1.75 3.375 3.0 21.84375 3.875 3.025 3 1.9625 3.925 2.9625 41.990625003.97656250300000000 51.994140633.995312503.00093750 151.99999933.99999852.9999993 192000000001.00000000000000
Example 3. 27. Let the linear system(1) be rearranged as follows 2x+y+52=-15 4x-8y+2=-21 4-y+2=7
Table 3.3 Divergent Jacobi iteration for the Linear System(4) k k yk k 1.0 2.0 2.0 1 1.5 3.375 5.0 26.6875 2.5 16.375 334.6875 8.015625 17.25 446.617188 17.8125 123.73438 5307.92968836.150391211.28125 6502.627931249296881202.56836
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) 7-k-2 k+1 21+4k+1+ k+1 15+ 2k+1-9k+1 k+1
Table 3. 2 Convergence Gauss-Seidel iteration for the Linear System( k a k y 2 k 2.0 2.0 1175 3.75 295 2195 3.96875 298625 31.9956253.9960937529903125 81.99931999989299996 91999999900 102.000000000030000
Definition 3.6. A matrix A of dimension N X N is said to be strictly diagonally dominant provided that ak>∑ for k=1.2..N j=1=k
