QuatIca: Advanced Numerical Linear Algebra and Optimization for Quaternionic Matrices in Python
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866914420560494592 |
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| author | Leplat, Valentin Ahmadi-Asl, Salman Pan, Junjun Ouerdane, Henni Ng, Michael |
| author_facet | Leplat, Valentin Ahmadi-Asl, Salman Pan, Junjun Ouerdane, Henni Ng, Michael |
| contents | Quaternion-valued representations provide a convenient way to model coupled multi-channel signals (e.g., RGB imagery, polarization data, vector fields, and multi-detector time series). Yet practical and numerically reliable software support remains far less mature than those based on the real/complex setting. Here, we present QuatIca, an open-source Python library for quaternion numerical linear algebra and optimization, designed for both research prototyping and reproducible experimentation. QuatIca provides core quaternion matrix operations and norms; dense decompositions and reductions (QR, LU, Q-SVD, eigendecomposition, Hessenberg/tridiagonal reduction, Cholesky decomposition, and Schur helpers); iterative solvers including quaternion GMRES (with preconditioning) and Newton-Schulz pseudoinverse schemes; and domain-focused routines for signal and image processing such as quaternion Tikhonov restoration. The library also includes OptiQ, which solves quaternion Hermitian semidefinite programs using log-det barrier Newton methods with $μ$-continuation. We highlight design choices that preserve quaternion structure, and we provide end-to-end demonstrations including quaternion image deblurring, Lorenz-attractor filtering, and quaternion image completion. QuatIca is distributed via PyPI and accompanied by open-source development on GitHub and continuously deployed documentation with runnable tutorials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_24074 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | QuatIca: Advanced Numerical Linear Algebra and Optimization for Quaternionic Matrices in Python Leplat, Valentin Ahmadi-Asl, Salman Pan, Junjun Ouerdane, Henni Ng, Michael Numerical Analysis Quaternion-valued representations provide a convenient way to model coupled multi-channel signals (e.g., RGB imagery, polarization data, vector fields, and multi-detector time series). Yet practical and numerically reliable software support remains far less mature than those based on the real/complex setting. Here, we present QuatIca, an open-source Python library for quaternion numerical linear algebra and optimization, designed for both research prototyping and reproducible experimentation. QuatIca provides core quaternion matrix operations and norms; dense decompositions and reductions (QR, LU, Q-SVD, eigendecomposition, Hessenberg/tridiagonal reduction, Cholesky decomposition, and Schur helpers); iterative solvers including quaternion GMRES (with preconditioning) and Newton-Schulz pseudoinverse schemes; and domain-focused routines for signal and image processing such as quaternion Tikhonov restoration. The library also includes OptiQ, which solves quaternion Hermitian semidefinite programs using log-det barrier Newton methods with $μ$-continuation. We highlight design choices that preserve quaternion structure, and we provide end-to-end demonstrations including quaternion image deblurring, Lorenz-attractor filtering, and quaternion image completion. QuatIca is distributed via PyPI and accompanied by open-source development on GitHub and continuously deployed documentation with runnable tutorials. |
| title | QuatIca: Advanced Numerical Linear Algebra and Optimization for Quaternionic Matrices in Python |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2603.24074 |