 Solution

A sequence is Q-linearly convergent to $x^{\ast }$ in a specified norm if $\Vert x_{k+1}-x^{\ast }\Vert \le q\Vert x_k-x^{\ast }\Vert$ for all sufficiently large $k$, with some $q<1$. Its asymptotic factor is the limsup of consecutive error ratios.

Put $h=c^{-2}+d^{-2}$, $b_0=c^{-2}-d^{-2}$ and $t=b_0/h$. For an error $e=x-x^{\ast }$, an exact first-coordinate step sets $e_1=-t{e}_2$, and an exact second-coordinate step sets $e_2=-t{e}_1$. Once the first coordinate step has occurred, errors consequently alternate between the two linesThe next error has its free coordinate multiplied in magnitude by $|t|$, and both lines have the same Euclidean norm factor $\sqrt{1+t^2}$. Therefore, until finite termination,This is a sharp bound, attained after the first step; the objective errors have factor $t^2$ per step. If $c^2=d^2$, the problem separates and at most two exact steps suffice. When the condition number is large, the factor approaches one: the iterates zigzag slowly across a narrow tilted valley.