西安建筑科技大学：《高等数学计算方法》课程教学资源（PPT课件讲稿）Chapter 2.3 2.6 Iterative Methods for Linear Systems

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