《数值分析》课程PPT教学课件（英文版）Chapter 02 Numerical Integration 2.6 Iterative Methods for Linear Systems

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