Exact counterexamples to R-superlinear convergence of cyclic steepest descent

Abstract

Cyclic steepest descent (CSD) recomputes the exact steepest-descent stepsize once per cycle and reuses it for $m$ updates. Dai's ICM 2022 survey describes CSD as likely to converge $R$-superlinearly on $n$-dimensional convex quadratics when $m\ge\lceil(n+1)/2\rceil$. We disprove the universal form of this assertion by two closed-form orbits at the stated threshold. First, for $n=m=2$, $A=\operatorname{diag}(1,3)$, $b=0$, and $x_0=(1,1/3)^{\mathsf{T}}$, the method follows the nonterminating balanced-zigzag orbit $x_k=2^{-k}(1,(-1)^k/3)^{\mathsf{T}}$, whose successive error norms have ratio $1/2$. Second, for $n=3$, $m=2$, and $A=\operatorname{diag}(1,2,3)$, we exhibit a full-support, nonresonant cycle-boundary projective period-two orbit with $g_{k+4}=g_k/49$ and $x_{k+4}=x_k/49$. This second construction is genuinely three-dimensional and is not a two-dimensional zigzag. Thus the universal claim fails through both the classical balanced-zigzag mechanism and a distinct non-zigzag period-two mechanism.