xPerm: fast index canonicalization for tensor computer algebra (0803.0862v1)

Abstract: We present a very fast implementation of the Butler-Portugal algorithm for index canonicalization with respect to permutation symmetries. It is called xPerm, and has been written as a combination of a Mathematica package and a C subroutine. The latter performs the most demanding parts of the computations and can be linked from any other program or computer algebra system. We demonstrate with tests and timings the effectively polynomial performance of the Butler-Portugal algorithm with respect to the number of indices, though we also show a case in which it is exponential. Our implementation handles generic tensorial expressions with several dozen indices in hundredths of a second, or one hundred indices in a few seconds, clearly outperforming all other current canonicalizers. The code has been already under intensive testing for several years and has been essential in recent investigations in large-scale tensor computer algebra.

• The paper introduces xPerm, which leverages the Butler-Portugal algorithm for efficient tensor index canonicalization.
• It implements a dual Mathematica and C framework, processing tensors with dozens to around 100 indices in seconds.
• Extensive testing demonstrates xPerm’s scalability and practical impact on advanced tensor computations in physics and engineering.

Overview of "xPerm: fast index canonicalization for tensor computer algebra"

The paper presents a comprehensive account of xPerm, a sophisticated implementation of the Butler-Portugal algorithm aimed at the efficient canonicalization of indices in tensor computer algebra. Developed as a mix of a Mathematica package and a C subroutine, xPerm provides a significant computational enhancement in managing the permutation symmetries associated with tensorial expressions.

Key Contributions

1. Implementation of the Butler-Portugal Algorithm: The primary focus of this work is on leveraging the Butler-Portugal algorithm, known for its effectiveness in index canonicalization with respect to permutation symmetries. By deploying this algorithm, xPerm demonstrates efficient handling of large group permutations pertinent to tensor computations.
2. Performance and Scalability: The paper presents evidence of xPerm's superior performance through rigorous tests. It efficiently processes tensorial expressions with several dozen indices in mere hundredths of a second and handles configurations with approximately 100 indices within a few seconds. This level of performance surpasses existing canonicalizers, showcasing effectively polynomial behavior under typical conditions.
3. Comprehensive System Design: xPerm is developed as both a Mathematica package and a faster C subroutine, which can be linked with other systems. This dual approach ensures flexibility and speed, catering to varied usage scenarios in computational tasks related to tensor algebra.
4. Broad Applicability and Testing: The package has undergone extensive testing and has already proven essential in several recent investigations involving large-scale tensor algebra. Its robustness has been affirmed through both theoretical analysis and practical application.

Implications

Practically, xPerm's ability to efficiently handle the index permutations required for canonicalization makes it a vital tool in fields such as general relativity and other areas of physics and engineering that rely on complex tensor calculus. Theoretically, this work represents a critical advancement in computational group theory applications, showing the integration of group-theory algorithms into practical software systems for scientific computation.

Future Prospects

The developments realized by xPerm open several avenues for future research and implementation in the field of artificial intelligence and computational algebra. The paradigm could be extended to explore more complex multiterm symmetries and further optimize existing tensor manipulation architectures, potentially informing developments in symbolic AI and other computational frameworks.

In conclusion, xPerm offers a significant advancement in the domain of tensor computer algebra, addressing key computational challenges and providing a foundation for future enhancements in both theory and application.