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By Mariesa L. Crow

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-- IEEE energy & strength journal

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""This is nice source fabric for a graduate scholar getting ready for a qualifying exam""
-- IEEE strength & power journal

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Initialize Q to the zero matrix. Let j = 1. 2. Set the j th column of Q (j th column of L) to the j th column of the reduced matrix A(j) , where A(1) = A, and (j) qkj = akj for k = j, . . 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.

An iterative solution is one in which an initial guess (x0 ) to the solution is used to create a sequence x0 , x1 , x2 , . . that (hopefully) converges arbitrarily close to the desired solution x∗ . Three principal issues arise with the use of iterative methods, namely 1. Is the iterative process well defined? That is, can it be successively applied without numerical difficulties? 2. 1)? Is the solution the desired solution? 3. How economical is the entire solution process? The complete (or partial) answers to these issues are enough to fill several volumes, and as such cannot be discussed in complete detail in this chapter.

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. This is often called the GMRES(m) algorithm. GMRES(m) Algorithm for Solving Ax = b Initialization: Let k = 0, and r0 = Ax0 − b e1 = [1 0 0 .

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