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| 1. Implement an optimized matrix type {{{Matrix_number_field}}} for matrices with entries in a number field. | 1. Implement an optimized matrix type {{{Matrix_dense_number_field}}} for matrices with entries in a number field. |
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| 1. Implement a class {{{Matrix_cyclotomic_field}}} that derives from the above class. | 1. Implement a class {{{Matrix_dense_cyclotomic_field}}} that derives from the above class. |
Cyclotomic Linear Algebra
This wiki page is about implementing optimized algorithms for linear algebra over cyclotomic fields.
Some specific tasks
Implement an optimized matrix type Matrix_dense_number_field for matrices with entries in a number field.
Implement a class Matrix_dense_cyclotomic_field that derives from the above class.
- Make very fast random_element methods for those matrix types. This will be needed for testing out our algorithms easily, and for tuning them.
- Implement multimodular matrix multiplication. This will reduce to doing a bunch of multiplies over GF(p) for many primes p.
- Implement p-adic solver with cyclotomic p-adic reconstruction algorithm.
- Implement echelon form using solver algorithm (just like we have for QQ).
- Maybe implement multimodular echelon form. Might as well.
- Implement decomposition.
- Sparse multimodular echelon form (this is a case where multimodular makes good sense). This will be needed for presentations of weight 2 modular symbols over cyclotomic fields.