By R. K. Jain, S.R.K. Iyengar
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9999. 0. 51 SOLUTION OF EQUATIONS AND EIGEN VALUE PROBLEMS REVIEW QUESTIONS 1. Define an iterative procedure for solving a system of algebraic equations Ax = b. What do we mean by convergence of an iterative procedure? Solution A general linear iterative method for the solution of the system of equations Ax = b can be written in matrix form as x(k+1) = Hx(k) + c, k = 0, 1, 2, ... where x(k+1) and x(k) are the approximations for x at the (k + 1)th and kth iterations respectively. H is called the iteration matrix depending on A and c, which is a column vector depends on A and b.
R2 – 2R1, R3 – R1 : 0 0 1 3 5 0 0 1 LM MN 1 R2 ↔ R3. Then, R2/2 : 0 0 LM MN 1 R1 – R2, R3 + R2 : 0 0 0 1 0 1 1 −1 − 1/4 7/4 − 1/4 3/2 7/4 −2 3/4 − 1/4 − 5/4 1/2 − 1/4 −1 0 0 1 1 −1 2 OP PQ 0 0 0 1/2 . 1 0 OP PQ − 1/2 1/2 . 1/2 3/2 −2 7/2 1/2 −1 − 1/2 0 1 0 OP PQ 0 0 . 1 38 NUMERICAL METHODS R3 /(– 1/4) : LM1 MN00 0 1 0 − 1/4 7/4 1 3/4 − 1/4 5 LM MN 1 R1 + (1/4)R3, R2 – (7/4)R3 : 0 0 0 0 −4 0 1 0 0 0 1 OP PQ − 1/2 1/2 . −2 2 −9 5 −1 7 −4 Therefore, the inverse of the given matrix is given by LM 2 MN− 95 −1 7 −4 OP PQ −1 4 .
However, such manual verification or exchange of equations may not be possible for large systems that we obtain in application problems. The necessary and sufficient condition for convergence is that the spectral radius of the iteration matrix H is less than one unit, that is, ρ(H) < 1, where ρ(H) is the largest eigen value in magnitude of H. Testing of this condition is beyond the scope of the syllabus. Remark 21 How do we find the initial approximations to start the iteration? If the system is diagonally dominant, then the iteration converges for any initial solution vector.