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Geometric algebra: an overview about different implementation representations and memory requirements
Oliver Rettig1, Jaroslav Hrdina2, Dietmar Hildenbrand3
1Computer Science, Baden-Württemberg Cooperative State University Karlsruhe , Karlsruhe, Baden-Württemberg, Germany.
Summary
Geometric algebra implementations are reviewed, focusing on memory reduction techniques for high-dimensional algebras. Advancements in tools like Gaalop are highlighted, crucial for quantum computing applications.
Area of Science:
- Mathematics
- Computer Science
Background:
- Geometric algebra has various software implementations, with basis blade sums common for low dimensions.
- High-dimensional algebras present memory challenges, impacting applications like quantum computing.
Purpose of the Study:
- To review geometric algebra implementation strategies.
- To classify software tools and libraries for geometric algebra.
- To highlight advancements in memory consumption reduction, specifically in the Gaalop tool.
Main Methods:
- Review of existing geometric algebra implementations.
- Classification of software tools and libraries.
- Analysis of memory reduction techniques (compression, sparse matrices, factorized representations).
Main Results:
- Basis blade sums are suitable for low-dimensional algebras; matrix representations are alternatives.
- Compression and sparse matrices reduce memory for low-dimensional algebras.
- Factorized representations reduce memory for high-dimensional algebras to O(n^2).
Conclusions:
- Memory efficiency is critical for geometric algebra, especially in quantum computing.
- The Gaalop tool shows recent advancements in memory reduction for geometric algebra.
- The study provides a classification of tools and strategies for geometric algebra implementation.
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