GRG Speed & Hardware

GRG Speed & Hardware

Sep 22, 2025

GRG Computational Cost and Hardware

Binary sector indices and power-of-two scaling motivate investigation of digital implementations. They do not establish a performance advantage for the complete calculation.

Define the Task

Kernel evaluation, generation of radial nodes, drawing a raster boundary, filling a disk, and building a volume are different tasks. Their complexity depends on the output representation and implementation.

Count Complete Costs

Include generation or retrieval of geometric ratios, arithmetic precision, divisions and square roots, precomputation, memory access, and output generation. A final multiplication alone does not describe the complete cost.

Hardware Directions

Possible directions include fixed-point arithmetic, lookup tables, vectorisation, parallel execution, and programmable hardware. Each requires an implementation and measurements; their availability is not a claim that GRG already outperforms alternatives.

Evidence Required

Use the benchmark protocol and publish comparisons at equal achieved accuracy. Test rounding, cancellation, overflow, and refinement limits. State what is geometric identity and what is numerical approximation.

Current Status

The current formulation supplies geometric kernels and accuracy proofs. This page makes no unsupported claim of speed, stability, throughput superiority, or three-dimensional scaling.

Roadmap

Roadmap

Sep 13, 2025

Research Roadmap

This page records work to be completed rather than retaining expired milestones or unsupported delivery promises.

Documentation

Align the current formulation, area-kernel article, notation, historical methods, and references. Consolidate duplicate content and preserve versions.

Implementation

Trace dependencies in every interactive tool. Specify which operations construct geometry, evaluate measures, compute comparison references, and draw the visualisation.

Verification

Review domains, normalisations, error proofs, rounding, cancellation, and stopping conditions. Validate general shapes and three-dimensional methods independently.

Benchmarks and Releases

Publish actual scripts, datasets, environments, licences, and measured results. Assess equal-accuracy performance before reporting a speed advantage.

Publication

List manuscripts and releases with verified status and persistent links. No release date, throughput target, or completed milestone is implied until evidence is available.

Complexity

Complexity

Sep 12, 2025

Computational Complexity

Complexity must identify the task, algorithm, arithmetic precision, and output representation. This page distinguishes costs rather than assigning a single complexity to all GRG methods.

Scalar Kernel Evaluation

Given local geometric ratios, the current circle kernels require a fixed number of arithmetic operations. Obtaining the ratios has its own cost. A binary-bisection evaluation performs a number of refinement stages proportional to the binary depth; square-root cost depends on precision and implementation.

Explicit Geometry

Generating and storing all sector nodes has a cost proportional to the output count. Streaming may reduce storage, but does not remove output work.

Raster and Volume Tasks

Boundary drawing, disk filling, surface generation, and voxel filling have different output sizes. Their cost statements must specify whether every output element is generated.

Comparisons

Report setup, precomputation, precision, memory, and runtime at equal achieved accuracy under the benchmark protocol. Earlier repeated cost models have been consolidated here.