Comparison of linear algebra libraries explained

The following tables provide a comparison of linear algebra software libraries, either specialized or general purpose libraries with significant linear algebra coverage.

Dense linear algebra

General information

	Creator	Language	First public release	Latest stable version	Source code availability	License	Notes
ALGLIB^[1]	ALGLIB Project	C++, C#, Python, FreePascal	2006	4.00.0 / 05.2023		GPL/commercial	General purpose numerical analysis library with C++, C#, Python, FreePascal interfaces.
Armadillo^[2] ^[3]	NICTA	C++	2009	12.6.6 / 10.2023		Apache License 2.0	C++ template library for linear algebra; includes various decompositions and factorisations; syntax (API) is similar to MATLAB.
ATLAS	R. Clint Whaley et al.	C	2001	3.10.3 / 07.2016		BSD	Automatically tuned implementation of BLAS. Also includes LU and Cholesky decompositions.
Blaze^[4]	K. Iglberger et al.	C++	2012	3.8 / 08.2020		BSD	Blaze is an open-source, high-performance C++ math library for dense and sparse arithmetic.
Blitz++	Todd Veldhuizen	C++	?	1.0.2 / 10.2019		GPL	Blitz++ is a C++ template class library that provides high-performance multidimensional array containers for scientific computing.
Boost uBLAS	J. Walter, M. Koch	C++	2000	1.84.0 / 12.2023		Boost Software License	uBLAS is a C++ template class library that provides BLAS level 1, 2, 3 functionality for dense, packed and sparse matrices.
Dlib	Davis E. King	C++	2006	19.24.2 / 05.2023		Boost	C++ template library; binds to optimized BLAS such as the Intel MKL; Includes matrix decompositions, non-linear solvers, and machine learning tooling
Eigen	Benoît Jacob	C++	2008	3.4.0 / 08.2021		MPL2	Eigen is a C++ template library for linear algebra: matrices, vectors, numerical solvers, and related algorithms.
Fastor^[5]	R. Poya, A. J. Gil and R. Ortigosa	C++	2016	0.6.4 / 06.2023		MIT License	Fastor is a high performance tensor (fixed multi-dimensional array) library for modern C++.
GNU Scientific Library^[6]	GNU Project	C, C++	1996	2.7.1 / 11.2021		GPL	General purpose numerical analysis library. Includes some support for linear algebra.
IMSL Numerical Libraries	Rogue Wave Software	C, Java, C#, Fortran, Python	1970	many components		Proprietary	General purpose numerical analysis library.
LAPACK^[7] ^[8]		Fortran	1992	3.12.0 / 11.2023		3-clause BSD	Numerical linear algebra library with long history
librsb	Michele Martone	C, Fortran, M4	2011	1.2.0 / 09.2016		GPL	High-performance multi-threaded primitives for large sparse matrices. Support operations for iterative solvers: multiplication, triangular solve, scaling, matrix I/O, matrix rendering. Many variants: e.g.: symmetric, hermitian, complex, quadruple precision.
oneMKL	Intel	C, C++, Fortran	2003	2023.1 / 03.2023		Intel Simplified Software License	Numerical analysis library optimized for Intel CPUs and GPUs. C++ SYCL based reference API implementation available in source for free.
Math.NET Numerics	C. Rüegg, M. Cuda, et al.	C#	2009	5.0.0 / 04.2022		MIT License	C# numerical analysis library with linear algebra support
Matrix Template Library	Jeremy Siek, Peter Gottschling, Andrew Lumsdaine, et al.	C++	1998	4.0 / 2018		Boost Software License	High-performance C++ linear algebra library based on Generic programming
NAG Numerical Library	The Numerical Algorithms Group	C, Fortran	1971	many components		Proprietary	General purpose numerical analysis library.
NMath	CenterSpace Software	C#	2003	7.1 / 12.2019		Proprietary	Math and statistical libraries for the .NET Framework
SciPy^[9] ^[10] ^[11]	Enthought	Python	2001	1.11.1 / 6.2023		BSD	Based on Python
Xtensor^[12]	S. Corlay, W. Vollprecht, J. Mabille et al.	C++	2016	0.21.10 / 11.2020		3-clause BSD	Xtensor is a C++ library meant for numerical analysis with multi-dimensional array expressions, broadcasting and lazy computing.

Matrix types and operations

Matrix types (special types like bidiagonal/tridiagonal are not listed):

• Real – general (nonsymmetric) real
• Complex – general (nonsymmetric) complex
• SPD – symmetric positive definite (real)
• HPD – Hermitian positive definite (complex)
• SY – symmetric (real)
• HE – Hermitian (complex)
• BND – band

Operations:

• TF – triangular factorizations (LU, Cholesky)
• OF – orthogonal factorizations (QR, QL, generalized factorizations)
• EVP – eigenvalue problems
• SVD – singular value decomposition
• GEVP – generalized EVP
• GSVD – generalized SVD

Notes and References

1. Bochkanov, S., & Bystritsky, V. (2011). ALGLIB-a cross-platform numerical analysis and data processing library. ALGLIB Project.
2. Sanderson, C., & Curtin, R. (2016). Armadillo: a template-based C++ library for linear algebra. Journal of Open Source Software, 1(2), 26.
3. Sanderson, C. (2010). Armadillo: An open source C++ linear algebra library for fast prototyping and computationally intensive experiments (p. 84). Technical report, NICTA.
4. Web site: Bitbucket.
5. 10.1016/j.cpc.2017.02.016 . A high performance data parallel tensor contraction framework: Application to coupled electro-mechanics . Poya, Roman and Gil, Antonio J. and Ortigosa, Rogelio . 2017 . Computer Physics Communications . 216 . 35–52 . 2017CoPhC.216...35P .
6. Gough, B. (2009). GNU scientific library reference manual. Network Theory Ltd.
7. Anderson, E., Bai, Z., Bischof, C., Blackford, S., Dongarra, J., Du Croz, J., ... & Sorensen, D. (1999). LAPACK Users' guide. SIAM.
8. Anderson, E., Bai, Z., Dongarra, J., Greenbaum, A., McKenney, A., Du Croz, J., ... & Sorensen, D. (1990, November). LAPACK: A portable linear algebra library for high-performance computers. In Proceedings of the 1990 ACM/IEEE conference on Supercomputing (pp. 2–11). IEEE Computer Society Press.
9. Jones, E., Oliphant, T., & Peterson, P. (2001). SciPy: Open source scientific tools for Python.
10. Bressert, E. (2012). SciPy and NumPy: an overview for developers. " O'Reilly Media, Inc.".
11. Blanco-Silva, F. J. (2013). Learning SciPy for numerical and scientific computing. Packt Publishing Ltd.
12. Web site: Xtensor-stack/Xtensor. GitHub. 13 February 2022.