Descent of the simplified residual under finite-precision initialization

Determine whether the simplified residual direction used by hardware, obtained by subtracting the finite-precision ZF residual error from the true residual, is a descending direction for the conditioned quadratic recovery surrogate; specifically, establish when Re{(r_true^(0))^H M^{-1} r_impl^(0)} > 0 holds.

Background

The proposed one-step Jacobi-preconditioned correction relies on an exact ZF initializer to cancel the dense Gram-matrix term in the initial residual. With finite-precision hardware, the initializer incurs an error e_0, so the true residual includes this error, whereas the simplified implementation omits it to preserve the lightweight element-wise datapath.

The paper gives an explicit necessary condition for the simplified direction to be descending: the real part of the inner product between the true residual and the preconditioned implemented residual must be positive. However, the paper states that this condition is not established by the available synthesis data, leaving unresolved whether the hardware-simplified direction guarantees descent under finite-precision initialization.

References

If hardware instead uses the simplified residual \mathbf{r}{\rm impl}{(0)}=\mathbf{r}{\rm true}{(0)}-\mathbf{e}_0, then, with \mathbf{M}=\operatorname{diag}(\mathbf{A}), its direction is descending only when \operatorname{Re}{(\mathbf{r}{\rm true}{(0)})H\mathbf{M}{-1}\mathbf{r}{\rm impl}{(0)}}>0; this condition is not established by the present synthesis data.

— Algorithm-Hardware Co-Design of a Lightweight PCG Equalizer with a Fixed Step Size for Massive MIMO  (2609.00890 - Wang et al., 1 Sep 2026) in Remark 1, “Finite-precision initializer” (Section III-A)