Comparison of linear algebra libraries

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	Free	GPL/commercial	General purpose numerical analysis library with C++, C#, Python, FreePascal interfaces.
Armadillo^[2]^[3]	NICTA	C++	2009	12.6.6 / 10.2023	Free	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	Free	BSD	Automatically tuned implementation of BLAS. Also includes LU and Cholesky decompositions.
Blaze^[4]	K. Iglberger et al.	C++	2012	3.8 / 08.2020	Free	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	Free	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	Free	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	Free	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	Free	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	Free	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	Free	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	Non-free	Proprietary	General purpose numerical analysis library.
LAPACK^[7]^[8]		Fortran	1992	3.12.0 / 11.2023	Free	3-clause BSD	Numerical linear algebra library with long history
librsb	Michele Martone	C, Fortran, M4	2011	1.2.0 / 09.2016	Free	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	Non-free	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	Free	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	Free	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	Non-free	Proprietary	General purpose numerical analysis library.
NMath	CenterSpace Software	C#	2003	7.1 / 12.2019	Non-free	Proprietary	Math and statistical libraries for the .NET Framework
SciPy^[9]^[10]^[11]	Enthought	Python	2001	1.11.1 / 6.2023	Free	BSD	Based on Python
Xtensor^[12]	S. Corlay, W. Vollprecht, J. Mabille et al.	C++	2016	0.21.10 / 11.2020	Free	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

	Real	Complex	SPD	HPD	SY	HE	BND	TF	OF	EVP	SVD	GEVP	GSVD
ALGLIB	Yes	Yes	Yes	Yes	No	No	No	Yes	Yes	Yes	Yes	Yes	No
ATLAS	Yes	Yes	Yes	Yes	No	No	No	Yes	No	No	No	No	No
Dlib	Yes	Yes	Yes	Yes	Yes	Yes	No	Yes	Yes	Yes	Yes	No	No
GNU Scientific Library	Yes	Yes	Yes	Yes	No	No	No	Yes	Yes	Yes	Yes	Yes	Yes
ILNumerics.Net	Yes	Yes	Yes	Yes	No	No	No	Yes	Yes	Yes	Yes	No	No
IMSL Numerical Libraries	Yes	Yes	Yes	Yes	No	No	Yes	Yes	No	Yes	Yes	Yes	No
LAPACK	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes
oneMKL	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes
NAG Numerical Library	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes
NMath	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes	No	No
SciPy (Python packages)	Yes	Yes	Yes	Yes	No	No	No	Yes	Yes	Yes	Yes	No	No
Eigen	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes	No
Armadillo	Yes	Yes	Yes	Yes	Yes	Yes	No	Yes	Yes	Yes	Yes	Yes	No

References

^ Bochkanov, S., & Bystritsky, V. (2011). ALGLIB-a cross-platform numerical analysis and data processing library. ALGLIB Project.
^ Sanderson, C., & Curtin, R. (2016). Armadillo: a template-based C++ library for linear algebra. Journal of Open Source Software, 1(2), 26.
^ Sanderson, C. (2010). Armadillo: An open source C++ linear algebra library for fast prototyping and computationally intensive experiments (p. 84). Technical report, NICTA.
^ "Bitbucket".
^ Poya, Roman and Gil, Antonio J. and Ortigosa, Rogelio (2017). "A high performance data parallel tensor contraction framework: Application to coupled electro-mechanics". Computer Physics Communications. 216: 35–52. Bibcode:2017CoPhC.216...35P. doi:10.1016/j.cpc.2017.02.016. hdl:10317/17584.{{cite journal}}: CS1 maint: multiple names: authors list (link)
^ Gough, B. (2009). GNU scientific library reference manual. Network Theory Ltd.
^ Anderson, E., Bai, Z., Bischof, C., Blackford, S., Dongarra, J., Du Croz, J., ... & Sorensen, D. (1999). LAPACK Users' guide. SIAM.
^ 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.
^ Jones, E., Oliphant, T., & Peterson, P. (2001). SciPy: Open source scientific tools for Python.
^ Bressert, E. (2012). SciPy and NumPy: an overview for developers. " O'Reilly Media, Inc.".
^ Blanco-Silva, F. J. (2013). Learning SciPy for numerical and scientific computing. Packt Publishing Ltd.
^ "Xtensor-stack/Xtensor". GitHub. 13 February 2022.

External links

[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] "Bitbucket".

[5] Poya, Roman and Gil, Antonio J. and Ortigosa, Rogelio (2017). "A high performance data parallel tensor contraction framework: Application to coupled electro-mechanics". Computer Physics Communications. 216: 35–52. Bibcode:2017CoPhC.216...35P. doi:10.1016/j.cpc.2017.02.016. hdl:10317/17584.{{cite journal}}: CS1 maint: multiple names: authors list (link)

[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] "Xtensor-stack/Xtensor". GitHub. 13 February 2022.

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v t e Linear algebra
Outline Glossary Template:Matrix classes
Linear equations	Linear equation System of linear equations Determinant Minor Cauchy–Binet formula Cramer's rule Gaussian elimination Gauss–Jordan elimination Overcompleteness Strassen algorithm
Matrices	Matrix Matrix addition Matrix multiplication Basis transformation matrix Characteristic polynomial Spectrum Trace Eigenvalue, eigenvector and eigenspace Cayley–Hamilton theorem Jordan normal form Weyr canonical form Rank Inverse, Pseudoinverse Adjugate, Transpose Dot product Symmetric matrix, Skew-symmetric matrix Orthogonal matrix, Unitary matrix Hermitian matrix, Antihermitian matrix Positive-(semi)definite Pfaffian Projection Spectral theorem Perron–Frobenius theorem Diagonal matrix, Triangular matrix, Tridiagonal matrix Block matrix Sparse matrix Hessenberg matrix, Hessian matrix Vandermonde matrix Stochastic matrix, Toeplitz matrix, Circulant matrix, Hankel matrix (0,1)-matrix List of matrices
Matrix decompositions	Cholesky decomposition LU decomposition QR decomposition Polar decomposition Spectral theorem Singular value decomposition Higher-order singular value decomposition Schur decomposition Schur complement Haynsworth inertia additivity formula Reducing subspace
Relations and computations	Matrix equivalence Matrix congruence Matrix similarity Matrix consimilarity Row equivalence Elementary row operations Householder transformation Least squares Linear least squares Gram–Schmidt process Woodbury matrix identity
Vector spaces	Vector space Linear combination Linear span Linear independence Basis, Hamel basis Change of basis Dimension theorem for vector spaces Hamel dimension Examples of vector spaces Linear map Shear mapping Squeeze mapping Linear subspace Row and column spaces, Null space Rank–nullity theorem Nullity theorem Cyclic subspace Dual space, Linear functional Category of vector spaces
Structures	Topological vector space Normed vector space Inner product space Euclidean space Orthogonality Orthogonal complement Orthogonal projection Orthogonal group Pseudo-Euclidean space Null vector Indefinite orthogonal group Orientation Improper rotation Symplectic structure
Multilinear algebra	Multilinear algebra Tensor Tensors (classical) Component-free treatment of tensors Outer product Tensor algebra Exterior algebra Symmetric algebra Clifford algebra Geometric algebra Bivector Multivector Gamas's theorem
Affine and projective	Affine space Affine transformation, Affine group, Affine geometry Affine coordinate system, Flat (geometry) Cartesian coordinate system Euclidean group Poincaré group Galilean group Projective space Projective transformation Projective geometry Projective linear group Quadric
Numerical linear algebra	Numerical linear algebra Floating-point arithmetic Numerical stability Basic Linear Algebra Subprograms Sparse matrix Comparison of linear algebra libraries
Category