Hardware Matrix Multiplication

Matrix multiplication background

Matrix multiplication, once dreaded by many in college math courses, is a cornerstone of linear algebra and a powerful tool for applying linear transformations to systems of equations. Though matrices have been studied for centuries, their importance has only grown with the rise of computation, driving continued innovation in their use and manipulation.

Today, matrix multiplication underpins a wide range of performance-critical applications—from 3D graphics and scientific simulations to search engine algorithms like PageRank. Most notably, it plays a central role in artificial intelligence, where it enables the massive parallel operations of training and inference within tensor processing units (TPUs).

The Computational Challenge of Matrix Multiplication

A major challenge in matrix multiplication lies in the sheer number of multiply-accumulate operations required. For an M×M matrix, each element in the result involves a full dot product, resulting in M³ operations. This computational load places a heavy burden on general-purpose CPUs, which typically use floating-point arithmetic. While floating-point formats offer great flexibility, they are not ideally suited for high-throughput multiply-accumulate pipelines, especially in performance-critical environments.

As artificial intelligence (AI) has advanced, tensors have emerged as a central mathematical structure. Neural network inference, for example, can often be reduced to large-scale matrix multiplication. Real-time AI applications—such as speech recognition or computer vision—depend on extremely fast matrix operations, and conventional CPUs, even those equipped with vector processing units (VPUs), often fall short in meeting these demands.

Dedicated hardware matrix multipliers provide a compelling solution. These devices use two-dimensional arrays of processing elements to perform many multiply-accumulate operations in parallel. An M×M hardware array yields a squared performance advantage over a linear vector unit, enabling the acceleration necessary for modern AI workloads. As a result, specialized matrix multipliers—often referred to as tensor processing units (TPUs)—have become a cornerstone of high-performance AI computation.

However, floating-point arithmetic remains a challenge for hardware implementation. To ease this burden, TPUs often use reduced-precision formats such as 16-bit floating point, which offer a reasonable trade-off between dynamic range and hardware efficiency. These narrower formats are easier to pipeline and integrate into dense processing arrays but come at the cost of reduced numerical precision. Depending on the AI application, this loss may be acceptable—or it may introduce significant limitations.

Major semiconductor companies have responded by integrating dedicated TPUs into their processors to accelerate AI workloads. Google’s custom TPU architecture and Nvidia’s multi-TPU chips are leading examples of this trend. As Moore’s Law slows and the need for domain-specific acceleration grows, the market for AI-specific hardware—now projected to surpass $50 billion annually—is driving a shift in processor design across the industry.

RNS Matrix Multiplication: Unlocking the Power of Modular Computation

During the early exploration of Residue Number Systems (RNS) for general-purpose processing—now more broadly recognized as modular computation—it became clear that product summation is one of RNS’s greatest strengths. As demand for faster matrix multiplication grows, modular computation is emerging as a compelling alternative to traditional arithmetic.

Matrix multiplication naturally lends itself to a 2D architecture, which aligns perfectly with RNS’s digit-parallel structure. Although RNS arithmetic requires conversion from floating-point values, recent advances in pipelined fixed-point conversion have dramatically improved efficiency. Crucially, the volume of data requiring conversion is proportional to O(M²), while the arithmetic workload scales as O(M³). This imbalance means that, in larger matrices, the relative overhead of conversion diminishes significantly.

Conversion costs are reduced even further in iterative algorithms, where intermediate results remain in RNS format until final output. Maitrix has pioneered several patented techniques that enable sustained iterative computation entirely within the RNS domain—something previously considered impractical. These normalization methods allow repeated RNS operations without intermediate conversion back to binary formats.

A key architectural innovation by Maitrix is the RNS digit matrix multiplier. In this design, each digit of an RNS word operates independently within its own matrix multiplier ALU. A full accumulator word is formed by the parallel operation of multiple digit-level multipliers. This separation enables arbitrary word width expansion without degrading individual multiplier performance. Each digit multiplier can be compact and efficient—often operating with just 6 to 7 bits—yielding faster and smaller matrix circuits than those based on floating-point MACs.

Beyond size and speed, RNS multiplication benefits from consistent bit widths: the result of a modular multiplication is no wider than its operands, unlike binary arithmetic where result widths grow with each multiplication and accumulation. This stability simplifies routing and resource usage inside the multiplier array. In binary designs, for example, a 16×16-bit multiply produces a 32-bit result, requiring even larger accumulators and routing buses. RNS avoids this ballooning of data width.

Modular computation also offers exceptional precision. In an RNS-based matrix multiplier, dot product results are accumulated across multiple digits and normalized only once at the end. This “wide word” accumulation ensures accuracy equivalent to a single rounding step, avoiding cumulative rounding errors that plague conventional systems.

Perhaps the most striking advantage lies in resource efficiency. RNS multipliers scale linearly with operand width—O(n)—whereas binary multipliers scale quadratically—O(n²). This can lead to a 4× or greater reduction in multiplier area on silicon, making RNS a game-changer for dense, high-performance AI accelerators.

The combined benefits of modular computation—carry-free arithmetic, digit-level parallelism, reduced routing complexity, improved accuracy, and linear scaling—result in remarkable gains. In FPGA implementations, Maitrix has demonstrated RNS-based TPUs achieving 7.5× to 9.5× performance improvements over equivalent fixed-point binary TPUs. More groundbreaking results will be published soon.

Stay tuned!