Matrix Multiplication: Optimality Under Resource Constraints

We develop a rigorous, unified framework for matrix multiplication optimization that simultaneously considers space, time, communication, energy, and numerical stability. Rather than pursuing improvements to the algebraic complexity exponent ω, we prove standard I/O lower bounds and provide calibrated energy and stability analyses. Building on established communication-optimal linear algebra work, our framework treats computational resources as coordinates in a geometric space, enabling principled multi-resource optimization. We prove communication lower bounds of the form Ω (mnk/√M) for rectangular classical algorithms and Ω (n^ (ω0) /M^ ( (ω0/2) −1) for square Strassen-like recursions, and present algorithms that achieve these bounds. We establish backward error bounds following standard floating-point analysis and provide an empirical energy model calibrated against actual measurements. The key insight is that real-world optimality requires balancing multiple competing resources rather than optimizing any single metric. Our algorithm achieves: (1) communication complexity matching established I/O lower bounds; (2) workspace O (b²) with b = Θ (√M) ; total memory respects the Ω (mn+mk+kn) input/output requirement; (3) empirical energy efficiency characterized by roofline analysis; and (4) backward error O (ku) or O (u log k) depending on summation strategy. We demonstrate that our framework recovers Strassen’s algorithm and recent 4 × 4 fast schemes as special manifold paths, providing a unified view of fast matrix multiplication. Experimental validation across diverse computing platforms confirms theoretical predictions, with all code and data available for reproducibility.