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When soloving the linear equation $x=Ax+b$ (where $x$ is an unknown vector, $A$ is a matrix, and $b$ is a constant vector), one often use the follow iteration:

$x_{k+1}=Ax_k +b$.

Does the above $x_k$ is convergent to $x$ when the spectral radius of $A$ is less than 1? If yes, would you give a reason. Thanks!

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    Consider the situation that it is one dimension.In that case, y=Ax+b and y=x has one intersection,since A<1, it is the fixed point.Thus, under this situation, the answer is yes.2012-03-31

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