Finite recursive Galerkin V- and W-cycle for an SPD hierarchy

Fix a finite Galerkin hierarchy of real SPD matrices $A_j\in {\mathbb{R}}^{n_j\times n_j}$, $0\le j\le J$, with $n_j\ge 1$, and full-column-rank prolongations $P_j\in {\mathbb{R}}^{n_j\times n_{j-1}}$ satisfying $A_{j-1}=P_j^{\mathsf{T}}{A}_j{P}_j$ for $1\le j\le J$. Choose nonnegative integer sweep counts ${\nu }_1,{\nu }_2$, a positive integer cycle count $\gamma$, and matrices $K_j^{\mathrm{p}\mathrm{r}\mathrm{e}},K_j^{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{t}}\in {\mathbb{R}}^{n_j\times n_j}$ for $1\le j\le J$; ${}^{\mathsf{T}}$ denotes the transpose of The transpose $A^{\mathsf{T}}$ of a matrix. An invocation $\mathrm{Cycle}(j,x,b)$ takes $x,b\in {\mathbb{R}}^{n_j}$ and returns the vector specified by these instructions for the system $A_{j}z=b$:

1. If $j=0$, return the exact solution $A_0^{-1}b$.
2. Starting from $x$, perform ${\nu }_1$ pre-sweeps $x\leftarrow x+K_j^{\mathrm{p}\mathrm{r}\mathrm{e}}(b-A_{j}x)$ and set $r=b-A_{j}x$.
3. Set $e=0$. For $t=1,\dots ,\gamma$, replace $e$ by the recursive iterate $e\leftarrow \mathrm{Cycle}(j-1,e,P_j^{\mathsf{T}}r)\mathrm{.}$
4. Correct $x\leftarrow x+P_{j}e$, then perform ${\nu }_2$ post-sweeps $x\leftarrow x+K_j^{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{t}}(b-A_{j}x)$; return $x$.

The V-cycle is $\gamma =1$ and the W-cycle is $\gamma =2$. The definition fixes exact arithmetic and specifies an iterate; it makes no accuracy or convergence claim. Setup cost, floating-point error, and termination are separate claims.