10. HAFTA BLM323 NUMERIC ANALYSIS. Okt. Yasin ORTAKCI.

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1 1. HAFTA NUMERIC ANALYSIS Okt. Yasin ORTAKCI Karabük Üniversitesi Uzaktan Eğitim Uygulama ve Araştırma Merkezi

2 2 2- ITERATIVE METHODS Jacobi ve Gauss Seidel are frequently used methods of solving lineer equation system. Both methods find the solution with sequantial calculation.. As you remember, a lineer equation system notation is: a 11 x 1 + a 12 x a 1n x n b 1 a 21 x 1 + a 22 x a 2n x n b 2... a n1 x 1 + a n2 x a nn x n b n To find each x value following formula can be used: x 1 b 1 (a 12 x 2 + a 13 x a 1n x n ) a 11 x 2 b 2 (a 21 x 1 + a 23 x a 2n x n ) a x n b n (a n1 x 1 + a n2 x a n(n 1) x n 1 ) a nn a. Jacobi Iteration Method The Jacobi iterative method is obtained by solving the ith equation in Ax b for x i to obtain (provided a ii )

3 3 x i k 1 a ii [ n b i (a ij x i k 1 ) j1 j i ] for i 1, 2,..., n k: iteration number and k 1 We first solve equation Ei for xi, for each i 1, 2,...,n to obtain x 1 1 b 1 (a 12 x 2 + a 13 x a 1n x n ) a 11 x 2 1 b 2 (a 21 x 1 + a 23 x a 2n x n ) a 22 x n 1 b n (a n1 x 1 + a n2 x a n(n 1) x n 1 ) a nn From the initial approximation x 1, x 2,., x n New x 1 1, x 2 1,, x n 1 values are calculated by using initial approximation values. x 1 1 : The value of x 1 after first iteration Additional iterates, x 1 k, x 2 k,, x n k are generated in a similar manner. x 1 k : The value of x 1 after kth iteration At the end of each iteration error check is made. For every x i ; ε i x i k+1 x i k < ε ref must be satisfied i 1,2,., n and k: iteration number EXAMPLE:Solve the following lineer equation system by taking ε ref.1 8x 1 + 2x 2 + 3x 3 3 x 1 9x 2 + 2x 3 1 2x 1 + 3x 2 + 6x 3 31

4 4 SOLUTION: Taking aproximate values of x as; x 1, x 2, x 3 x 1 3 (2x 2 + 3x 3 ) 8 x 2 1 (x 1 + 2x 3 ) 9 x 3 31 (2x 1 + 3x 2 ) 6 3 (2 + 3 ) ( + 2 ) (2 + 3 )

5 5 Algoritması INPUT the number of equations and unknowns n; the entries ai j, 1 i, j n of the matrix A; the entries bi, 1 i n of b; the entries XOi, 1 i < n of XO x ; tolerance TOL; maximum number of iterations N max. OUTPUT the approximate solution x1,..., xn or a message that the number of iterations was exceeded. Step 1 Set k 1. Step 2 While (k N max ) do Steps 3 6. Step 3 For i 1,..., n n set xi 1 b a i (a ij XO j ) ii j1 [ j i ] Step 4 If all x i XO i < TOL then OUTPUT (x1,..., xn); (The procedure was successful. ) STOP. Step 5 Set k k + 1. Step 6 For i 1,..., n set XOi xi. Step 7 OUTPUT ( Maximum number of iterations exceeded ); (The procedure was unsuccessful. ) STOP.

6 6 b.gauss-sidel-method A possible improvement in Jacobi Iteration can be seen by reconsidering Jacobi Iteration formula. All the x k values are used to compute new x k+1. But, for i > 1, x k+1 1, x k+1 k+1 2,,, x n values have already been computed and are expected to be better approximations to the actual solutions x 1, x 2,, x i 1 than are x k 1, x k 2,,, x k n. It seems reasonable, then, to compute x i k+1 using these most recently calculated values. From the initial approximation x 1, x 2,., x n New x 1 1, x 2 1,, x n 1 values are calculated by using initial approximation values. x 1 1 b 1 (a 12 x 2 +a 13 x a 1n x n ) a 11 x 2 1 b 2 (a 21 x a 23 x a 2n x n ) a x n 1 b n (a n1 x a n2 x a n(n 1) x n 1 1 ) a nn At the end of each iteration error check is made. For every x i ; ε i x i k+1 x i k < ε ref must be satisfied i 1,2,., n and k: iteration number

7 7 EXAMPLE: Solve the following lineer equation system with Gauss Siedel mehtod by taking ε ref.1. Solution: 5x 1 + 2x 2 + x 3 12 x 1 + 2x 2 + x 3 8 2x 1 + x 2 + 4x 3 16 Taking aproximate values of x as; x 1, x 2, x 3 Iteration 1 x 1 12 (2 x 2 + x 3 ) 5 12 (2 + ) x 2 8 (1 x x 3 ) 2 8 (2.4 + ) x 3 16 (2 x x 2 1 ) 4 16 ( ) Iteration 2 x 1 12 ( ) 5.86 x 2 8 ( ) x 3 16 ( ) 4 Following table shows the whole iteration results: 2.94

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9 9 INPUT the number of equations and unknowns n; the entries ai j, 1 i, j n of the matrix A; the entries bi, 1 i n of b; the entries XOi, 1 i < n of XO x ; tolerance TOL; maximum number of iterations N max. OUTPUT the approximate solution x1,..., xn or a message that the number of iterations was exceeded. Step 1 Set iteration_no 1. Step 2 While (iteration_no N max ) do Steps 3 6. Step 3 For i 1,..., n i 1 set xi 1 [b a i [ (a ij x j ) + (a ij XO j )]] ii j1 n ji+1 Step 4 If all x i XO i < TOL then OUTPUT (x1,..., xn); (The procedure was successful. ) STOP. Step 5 Set iteration_no iterasyon_no + 1 Step 6 For i 1,..., n set XOi xi. Step 7 OUTPUT ( Maximum number of iterations exceeded ); (The procedure was unsuccessful. ) STOP. Kaynakça Ders Notları",Yrd. Doç. Dr. Mustafa Sönmez Doç. Dr. İbrahim UZUN, (24), "Numarik Analiz Beta Yayıncılık.