Talk:Conjugate residual method

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Range of matrices applicable

The article claims that the CRM be applicable to any non-singular hermitian system. This is wrong, and no source for that wrong claim is available. Compare as a simple counter example, in Matlab notation:

A = [ 1 0 ; 0 -1 ]; b = [1 1]; x_0 = [0 0]

Then the initial residual is r_0 = [1 1]. However, 0 = r_0 * A * r_0', as you please check by yourself. So the initial alpha is 0, and one sees that the iteration keeps stuck at x_i = [ 0, 0 ], which is not the solution. Even worse in the complex valued case. Take

A = [ 1 0 ; 0 i ]; b = [1 1]; x_0 = [0 0]

Then p_0 = r_0 = b, while p_0 * A * A * p_0' = 0, as you please check by yourself, and the algorithm has a fatal break down. Hence the article makes a wrong claim. \qed --212.201.70.54 (talk) 16:55, 6 January 2013 (UTC)[reply]