TABLE OF CONTENTS How good is the Gauss Seidel iteration (12.21) at solving the five-point equations on an m m grid? On the face of it, posing this question just after we have completed a whole chapter devoted to iterative methods is neither necessary nor appropriate. According to (12.51), the spectral radius of the iteration matrix is cos 2 [ ?/( m + 1)] ? 1 ? ? 2 m ? 2 and inspection of the third row of Fig. 12.5 will convince us that this presents a fair estimate of the behaviour of the scheme. Yet, by its very nature, the spectral radius displays the asymptotic attenuation rate of the error and it is entirely legitimate to query how well (or badly) Gauss Seidel performs before the onset of its asymptotic regime.
Figure 13.1 displays the logarithm of the Euclidean norm of the residual for m = 10, 20, 40, 80; we remind the reader that, given the equation
and a sequence of iterations 
, the residual is defined as r [ k ] = A x [ k ] ? b, k ? 0. [1] The emerging picture is startling: the norm drops dramatically in the first few iterations! Only after a while does the rate of attenuation approach the linear curve predicted by the spectral radius of 
1. Moreover, this phenomenon unlike the asymptotic rate of decay of ln...
