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Extra info for Computational Methods for Electric Power Systems, Second Edition
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. i. Will the Jacobi iteration converge for this matrix? ii. Will the Gauss-Seidel iteration converge for this matrix?
Sn . . 0 ⎥ ⎥ .. . ⎥ . ⎥ ⎥ . . −sn . . cs . . 0 ⎥ ⎥ .. . ⎥ . ⎦ . 0 ... 0 ... 0 ... 1 1 ⎢ .. ⎢. ⎢ ⎢0 ⎢ ⎢ .. ⎢. ⎢ ⎢0 ⎢ ⎢. ⎣ .. 87) with properly chosen cn = cos(φ) and sn = sin(φ) for some rotation angle φ can be used to zero the element Aki . One of the diﬃculties with the GMRES methods is that as k increases, the number of vectors requiring storage increases as k and the number of multiplications as 12 k 2 n (for a n × n matrix). e. it can be restarted every m steps, where m is some ﬁxed integer parameter.
N where θ= π n+1 (b) Let a = 2. i. Will the Jacobi iteration converge for this matrix? ii. Will the Gauss-Seidel iteration converge for this matrix? 12. An alternative conjugate gradient algorithm for solving Ax = b may be based on the error functional Ek xk = xk − x, xk − x where · denotes inner product. The solution is given as xk+1 = xk + αk σk Using σ1 = −AT r0 and σk+1 = −AT rk + βk σk , derive this conjugate gradient algorithm. 7 using this conjugate gradient algorithm. 13. Write a subroutine with two inputs (A, ﬂag) that will generate for any non-singular matrix A, the outputs (Q, P ) such that if • ﬂag=0, A = LU, P + I • ﬂag=1, P A = LU where ⎡ l11 ⎢ l21 ⎢ ⎢ L = ⎢ l31 ⎢ ..