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5 using the GMRES method. 88) ⎣ 2 6 −12 2 ⎦ x = ⎣ 3 ⎦ 4 3 1 0 −8 with x0 = [0 0 0 0]T . Let ε = 10−3 . 5993) is greater than ε, j = j + 1 and repeat. 7708) is greater than ε, j = j + 1 and repeat. 3473) is greater than ε, j = j + 1 and repeat. 0000 T Since error (= |s(5)| = 0), the iteration has converged. 91) ⎡ Solving for x from yields ⎡ which is the same as the previous example. 7 Problems 1. Show that the number of multiplications and divisions required in the LU factorization of an n × n square matrix is n(n2 − 1)/3.
38) 3. If j = n, then stop. 4. Assuming that qjj = 0, set the j th row of Q (j th row of U ) as (j) qjk = ajk qjj for k = j + 1, . . 39) 5. Update A(j+1) from A(j) as (j+1) aik (j) = aik − qij qjk for i = j + 1, . . , n, and k = j + 1, . . 40) 6. Set j = j + 1. Go to step 2. 22 Computational Methods for Electric Power Systems This factorization algorithm gives rise to the same number of multiplications and divisions as Crout’s algorithm for LU factorization. Crout’s algorithm uses each entry of the A matrix only once, whereas Gauss’ algorithm updates the A matrix each time.
70000]T and explain what happens. 9. Solve the system of equations in problem 2 using the conjugate gradient method. 10. Solve the system of equations in problem 2 using the GMRES method. 11. Consider an n × n tridiagonal matrix of the form ⎤ ⎡ a −1 ⎥ ⎢ −1 a −1 ⎥ ⎢ ⎥ ⎢ −1 a −1 ⎥ ⎢ Ta = ⎢ ⎥ −1 a −1 ⎥ ⎢ ⎣ −1 a −1 ⎦ −1 a 42 Computational Methods for Electric Power Systems where a is a real number. (a) Verify that the eigenvalues of Ta are given by λj = a − 2 cos (jθ) j = 1, . . , n where θ= π n+1 (b) Let a = 2.