Towards energy-efficient scientific computing: Reversible numerical linear algebra kernels in floating-point arithmetic

Frontier scientific and AI workloads now reach $10^{19}-10^{25}$ fused multiply–add (FMA) operations per run (on the order of $2\times 10^{19}-2\times 10^{25}$ FLOPs). At today’s $\sim 10$  pJ per FMA, this corresponds to approximately $10^8-10^{14}$ joules of arithmetic energy. At this scale, energy becomes the limiting resource for continued growth in computational workloads, motivating a re-evaluation of long-standing algorithmic assumptions. It is often assumed that reversible computing only matters near the Landauer limit. Building on prior physical arguments that full energy recovery is only possible when computation preserves information, we demonstrate that this same requirement governs floating-point numerical kernels: overwriting state enforces a non-zero energy floor, even under ideal recovery. Thus, eliminating this wall in practice requires that the numerical algorithm itself be injective. We therefore present the first reversible floating-point realizations of core dense numerical kernels—matrix multiplication, LU factorization, and conjugate-gradient iteration—that retain rounding information rather than discarding it. Implemented directly in IEEE arithmetic, they achieve machine-precision forward–reverse agreement on well- and ill-conditioned problems with minimal auxiliary state. A toggle-based model with measured switching costs and realistic recovery factors predicts $10^3-10^4\times$ reductions in arithmetic energy. These results establish injective floating-point kernels as a foundation for energy-recovering numerical computation, and indicate that realizing this potential will require sustained co-design across applied mathematics, computer science, and hardware engineering.