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Tham khảo tài liệu 'burden - numerical analysis 5e (pws, 1993) episode 2 part 8', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 418 CHAPTER 7 Iterative Techniques in Matrix Algebra Theorem 7.24 Ostrowski Reich. If A is a positive definite matrix and 0 Ct 2 then the SOR method converges for any choice of initial approximate vector x 0 . E H B Theorem 7.25 If A is positive definite and tridiagonal then p T8 p 7 2 1 and the optimal choice of 0 for the SOR method is 2 Ct ----- A 1 Vi - p rg With this choice of Ú p TJ Ct 1. 1 -2 Vi - p 7 F E s B EXAMPLE 4 In Example 3 the matrix A was given by 4 3 .0 A - 3 4 -1 0 -1 4 Tf D-1 This matrix is positive definite and tridiagonal so Theorem 7.25 applies. Since L ZZ 1 4 0 0 0 -3 0 0 -0.75 0 0 1 4 0 -3 0 1 -0.75 0 0.25 _0 0 1 4_ 0 1 0_ 0 0.25 0 we have T - AZ -A -0.75 0 0 0.25 so Thus -0.75 -A 0.25 A A2 - 0.625 . 2 det 7 - AZ p Ty - Vo.625 2 2 3T1H 111 111 III. I I -III .11 __ I I T- 1 OzL 1 Vl - p Tg 1 Vl - p 7 P 1 Vl - 0.625 This explains the rapid convergence obtained in Example 3 by using Cl 1.25. s We close this section with Algorithm 7.3 for the SOR method. SOR ALGORITHM To solve Ax - b given the parameter Ct and an initial approximation XCO INPUT the number of equations and unknowns zz entries Oịị 1 z j n of the matrix A the entries bị 1 s i n of b the entties XOi 1 Ì n of xo X C4 the parameter Ct tolerance TOL maximum number of iterations N. 7.3 Iterative Techniques for Solving Linear Systems 419 OUTPUT the approximate solution Xj . x or a message that the number of iterations was exceeded. Step 1 Set k 1. Step 2 While k -3 ĂỌ do Steps 3-6. Step 3 For i 1 . . . n i 1 n S a Xj - y ciyXOj 3- bJ j i j i i setx - 1 ứ XOj 3----------------------------------. an Step 4 If IIX XO TOL then OUTPUT Xj . . x Procedure completed successfully. STOP. Step 5 Set k k 1. Step 6 For i 1 . . n setXOị xz. Step 7 OUTPUT Maximum number of iterations exceeded Procedure completed unsuccessfully. STOP. EXERCISESET73 1. Find the first two iterations of the Jacobi method for the following linear systems using X 0 a. 3xj x2 x3 1 3xj 6x2 2x3 0 3xj 3x2 7x3 4. b. lOX
