Finite Rings

Base Ring

Libraries

Overview

Known Limitations

Tickets

\mathbb{F}_2

M4RI

Sage has the fastest known implementation. Matrix-matrix multiplication is either as fast as Magma 2.17 or slightly faster. Gaussian elimination is faster than Magma 2.17.

Some functions might fall back to their generic implementations because M4RI either does not implement it or its implementation is not wrapped

\mathbb{F}_{2^e} for 2 \leq e \leq 10

M4RIE

Sage has by far the fastest known implementation.

Many functions might fall back to their generic implementations because M4RIE either does not implement it or its implementation is not wrapped. Sage uses M4RIE's mzed_t but using mzd_slice_t would be slightly faster and save memory.

\mathbb{Z}_p, 2<p very small

LinBox

Sage achieves decent performance, but Magma 2.17 is faster for small values. Also, (experimental) specialised implementations exist which beat both by a large margin.

\mathbb{Z}_p, p<2^{22}

LinBox

Sage achieves decent performance on par with Magma 2.17. Performance differences likely come down to the BLAS used in both systems.

Some functions fall back to their generic implementations because LinBox either does not implement it, or the implementation was not wrapped yet. Also, echelonize uses way too much memory because it copies the matrix being worked on due to a function missing in LinBox.

Infinite Rings

\mathbb{Q}

LinBox, Pari, IML

Decent

\mathbb{Z}

LinBox, Pari, IML, NTL, custom code

Decent. Fastest known Hermite Normal Form.

\mathbb{R}, \mathbb{C}, 53-bit

SciPy, NumPy, CVXOPT

Rings RDF and CDF in Sage are floating point doubles. Performance comes from standard underlying libraries through Sage wrappers of SciPy/NumPy wrappers and interfaces. There there are no dedicated wrappers to CVXOPT, which sometimes offers better performance via SuiteSparse

Not every desirable function has been implemented, but coverage is steadily increasing.

\mathbb{R}, \mathbb{C}, arbitrary precision

generic

Rings RealField(prec) and ComplexField(prec) in Sage support arbitrary precision, but default to 53-bit when used as RR and CC. Implementation often (always?) defaults to exact algorithms.

Using exact algorithms for approximate entries can provide misleading results if not interpreted properly.

\mathbb{Q}(\zeta_n)

special code

Optimized multi-modular algorithms for arithmetic, charpoly, echelon form over cyclotomic fields.

Symbolic Rings

\mathbb{K}[x]

generic

\mathbb{K}[x_0,\dots,x_{n-1}]

generic,Singular

Generic matrices with some routines (det, echelon_form) calling libSingular. Poor performance, but apparently comparable to what other systems offer.

This implementation is not a serious attempt to implement matrices over \mathbb{K}[x_0,\dots,x_{n-1}].

Symbols

generic,Maxima

Generic matrices with some routines calling Maxima. Poor performance, but apparently comparable to what other systems offer.

Base Ring

Libraries

Overview

Known Limitations

Tickets

\mathbb{F}_{p^k}, p^k<255

MeatAxe fork

The fork adds Winograd-Strassen multiplication to the original MeatAxe. The computation of echelon forms is not asymptotically fast, but is faster than the generic implementation in Sage.

Patch needs review

#12103

Base Ring

Libraries

Overview

Known Limitations

Tickets

\mathbb{F}_{p^k}

Custom code

Matrices over extension fields are represented as tuples of matrices over the prime subfield. Then, using polynomial multiplication with matrix coefficients allows to multiply matrices. Other algorithms are implemented by reducing to matrix-matrix multiplication.

Only matrix-matrix multiplication is implemented so far.

#12177

\mathbb{Z_p} for p < 2^{64}

FLINT 2.3

Decent, not fully optimised.

\mathbb{Z}[x]

FLINT 2.3

Decent, not fully optimised.

\mathbb{Z_p}[x] for p < 2^{64}

FLINT 2.3

Decent, not fully optimised.