The state of dense linear algebra in Sage
Standard as of Version 5.0
Finite Rings 

Base Ring 
Libraries 
Overview 
Known Limitations 
Tickets 
\mathbb{F}_2 
Sage has the fastest known implementation. Matrixmatrix 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 
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 
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} 
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} 
Decent 



\mathbb{Z} 
Decent. Fastest known Hermite Normal Form. 



\mathbb{R}, \mathbb{C}, 53bit 
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 53bit 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 multimodular algorithms for arithmetic, charpoly, echelon form over cyclotomic fields. 


Symbolic Rings 

\mathbb{K}[x] 
generic 



\mathbb{K}[x_0,\dots,x_{n1}] 
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_{n1}]. 

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


Provided by Optional Packages
Base Ring 
Libraries 
Overview 
Known Limitations 
Tickets 
\mathbb{F}_{p^k}, p^k<255 
MeatAxe fork 
The fork adds WinogradStrassen 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 
In the Pipeline
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 matrixmatrix multiplication. 
Only matrixmatrix multiplication is implemented so far. 

\mathbb{Z_p} for p < 2^{64} 
Decent, not fully optimised. 



\mathbb{Z}[x] 
Decent, not fully optimised. 



\mathbb{Z_p}[x] for p < 2^{64} 
Decent, not fully optimised. 

