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  2. Comparison of linear algebra libraries - Wikipedia

    en.wikipedia.org/wiki/Comparison_of_linear...

    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 ...

  3. Invariants of tensors - Wikipedia

    en.wikipedia.org/wiki/Invariants_of_tensors

    A real tensor in 3D (i.e., one with a 3x3 component matrix) has as many as six independent invariants, three being the invariants of its symmetric part and three characterizing the orientation of the axial vector of the skew-symmetric part relative to the principal directions of the symmetric part.

  4. Tensor (machine learning) - Wikipedia

    en.wikipedia.org/wiki/Tensor_(machine_learning)

    In machine learning, the term tensor informally refers to two different concepts (i) a way of organizing data and (ii) a multilinear (tensor) transformation. Data may be organized in a multidimensional array (M-way array), informally referred to as a "data tensor"; however, in the strict mathematical sense, a tensor is a multilinear mapping over a set of domain vector spaces to a range vector ...

  5. Tensor software - Wikipedia

    en.wikipedia.org/wiki/Tensor_software

    Xerus [52] is a C++ tensor algebra library for tensors of arbitrary dimensions and tensor decomposition into general tensor networks (focusing on matrix product states). It offers Einstein notation like syntax and optimizes the contraction order of any network of tensors at runtime so that dimensions need not be fixed at compile-time.

  6. Computational complexity of mathematical operations - Wikipedia

    en.wikipedia.org/wiki/Computational_complexity...

    Graphs of functions commonly used in the analysis of algorithms, showing the number of operations versus input size for each function. The following tables list the computational complexity of various algorithms for common mathematical operations.

  7. Matrix decomposition - Wikipedia

    en.wikipedia.org/wiki/Matrix_decomposition

    In the mathematical discipline of linear algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices. There are many different matrix decompositions; each finds use among a particular class of problems.

  8. Voigt notation - Wikipedia

    en.wikipedia.org/wiki/Voigt_notation

    Hooke's law has a symmetric fourth-order stiffness tensor with 81 components (3×3×3×3), but because the application of such a rank-4 tensor to a symmetric rank-2 tensor must yield another symmetric rank-2 tensor, not all of the 81 elements are independent. Voigt notation enables such a rank-4 tensor to be represented by a 6×6 matrix ...

  9. Raising and lowering indices - Wikipedia

    en.wikipedia.org/wiki/Raising_and_lowering_indices

    Concretely, in the case where the vector space has an inner product, in matrix notation these can be thought of as row vectors, which give a number when applied to column vectors. We denote this by V ∗ := Hom ( V , K ) {\displaystyle V^{*}:={\text{Hom}}(V,K)} , so that α ∈ V ∗ {\displaystyle \alpha \in V^{*}} is a linear map α : V → K ...