= Residual growth in an indefinite steepest-descent iteration
{title2=$A=\begin{pmatrix}2&3\\-5&-2\end{pmatrix}$}
For this nonsymmetric <matrix>, consider the formal residual update $r^+=r-(r^Tr)/(r^TAr)Ar$, with nonzero $r=(x,y)^T$. Put $D=2x^2-2xy-2y^2$ and $B=5x^2+4xy+3y^2$. Algebra gives $r^+=(B/D)(-y,x)^T$ when $D\ne0$. But $B+D=7(x+y/7)^2+6y^2/7>0$ and $B-D=3(x+y)^2+2y^2>0$. Hence $B>|D|$ and $\|r^+\|>\|r\|$ for every defined nonzero step. At $D=0$, the quotient is undefined. Thus the iteration cannot converge to zero <linear-system residual> from any nonzero initial residual. This concrete divergence proof distinguishes failure of an algorithm from mere failure to meet a sufficient theorem's hypotheses.
Back to article page
