Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1710.04286

Deriving Correct High-Performance Algorithms

Devangi N. Parikh·Maggie E. Myers·Robert A. van de Geijn
View original source ↗Read source document ↗Export metadata ↓

Abstract

Dijkstra observed that verifying correctness of a program is difficult and conjectured that derivation of a program hand-in-hand with its proof of correctness was the answer. We illustrate this goal-oriented approach by applying it to the domain of dense linear algebra libraries for distributed memory parallel computers. We show that algorithms that underlie the implementation of most functionality for this domain can be systematically derived to be correct. The benefit is that an entire family of algorithms for an operation is discovered so that the best algorithm for a given architecture can be chosen. This approach is very practical: Ideas inspired by it have been used to rewrite the dense linear algebra software stack starting below the Basic Linear Algebra Subprograms (BLAS) and reaching up through the Elemental distributed memory library, and every level in between. The paper demonstrates how formal methods and rigorous mathematical techniques for correctness impact HPC.

Explore related subjects

cs.MS

Keep this discovery

BibTeXRIS

Devangi N. Parikh, Maggie E. Myers, Robert A. van de Geijn. 2017-10-11. Deriving Correct High-Performance Algorithms. https://arxiv.org/abs/1710.04286

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The Art of Closed-Formula Defaults: Search-Free Code Generation for Tensor Operators

Agentic search and automated optimization of GPU kernels are powerful tools for large language model inference. Their effectiveness, however, depends not on the sophistication of the search itself, but on the clarity of the optimization problem being solved. We provide an application-first approach that drives a hierarchical code generation tool from operator specifi cation down to GPU instructions, and show that a clearly defined computational model makes the optimization problem tractable.

cs.MS↗
arXiv2026

Ozaki 2.5: Engineering the Deconstruction Path of fp64-Emulated Dense Matrix Multiplication on FP8 Tensor Cores

FP8 Ozaki II emulates FP64 matrix multiplication by tensor-core products over a CRT residue system; converting the operands into residue planes (the deconstruction term in the Tensor-Memory Equilibrium model of the companion paper "FP8 is All You Need, Part 1") costs integer-pipe and memory resources before tensor instructions issue. This paper engineers that path; every result is a model projection pending measurement. First, a deconstruction-aware model: on the NVIDIA Rubin GPU the emulated rate reaches the arithmetic roof $P_{\rm FP8}/(3r+1)$ ($\approx 473$ TFLOPS at $r=12$) only within one thread-block cluster; larger outputs are re-split on the fly and held at a floor of $\approx 235$ TFLOPS (half the roof, a ratio of three design integers, not a fit), while real solvers' tall/skinny shapes stay near the crossover, $1.6$-$1.9\times$ over simple deconstruction today. Second, the method: convert-once residue workspaces, an exact two-limb constant-reduction GEMM on integer tensor pipes (or pure-SIMT dp4a), and conversion pipelined behind the MMAs, moving the crossover from $\approx 1211$ to $\approx 480$-$730$. Third, modulus co-design: all-byte and hybrid sets, two supply bounds and a carry-corrected E4M3 split of tail moduli. Fourth and central, the closed-form floor names its hardware escape, and the prize is Rubin's: a stream-side residue-conversion mode on the asynchronous copy path (Option C), a narrow fixed-function block sized as a bill of materials, takes plane formation off the arithmetic pipes and lifts the floor from 235 TFLOPS to the full 473-TFLOPS roof at unchanged cluster reach, about doubling HPL-class FP64 per Rubin GPU, and unbinds conversion-bound sparse kernels. The NVIDIA GB300 GPU, whose 135-TFLOPS roof sits at its own floor, gains little; floor and remedy are Rubin-scale. Application traces ground the analysis; constants are script-checked.

cs.MS↗
arXiv2026

Geometric Function Atlas: certified computing for geometric function theory in Python

We describe geometric-function-atlas, our open-source Python package for the sharp extremal problems of geometric function theory. We organise it around a catalogue of thirty-nine Ma--Minda starlike generators. From this catalogue we compute exact Taylor coefficients, closed-form Fekete--Szeg\H{o} constants, exact coefficients of the Ma--Minda extremal function, and admissibility screens. Our verifier answers membership questions for normalised polynomials at three levels of evidence: a floating-point grid screen, an exact sufficient condition decided in rational arithmetic, and a certified interval enclosure at the worst screened point. Every answer names the level at which we obtained it. We ship a checksummed artifact snapshot with three hundred and six coefficient certificates and seven hundred and two directed inclusion radii. Eight reviewed radius lanes carry certificates whose proof chains we replay symbolically, and we re-execute every coefficient certificate through our exact Schur-parameter machinery on request. We emit all results through one versioned envelope that records the method, the evidence status, the assumptions, and the artifact identifiers. Two optional laboratories apply the same discipline to cryptographic S-box metrics and to image-quality metrics. We present our design, state as propositions what each tier establishes, follow one radius lane from screen to replayed certificate, report measured timings, and place our package among symbolic-algebra, rigorous-numerics, and mathematical-database software. We release geometric-function-atlas under the MIT licence on the Python Package Index and at https://github.com/Prasanna28Devadiga/geometric-function-atlas.

cs.MS↗