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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 ...
Also known as: UTV decomposition, ULV decomposition, URV decomposition. Applicable to: m-by-n matrix A. Decomposition: =, where T is a triangular matrix, and U and V are unitary matrices. Comment: Similar to the singular value decomposition and to the Schur decomposition.
The decomposition can be derived from the fundamental property of eigenvectors: = = =. The linearly independent eigenvectors q i with nonzero eigenvalues form a basis (not necessarily orthonormal) for all possible products Ax, for x ∈ C n, which is the same as the image (or range) of the corresponding matrix transformation, and also the ...
In linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced / ʃ ə ˈ l ɛ s k i / shə-LES-kee) is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful for efficient numerical solutions, e.g., Monte Carlo simulations.
LU decomposition; Singular value decomposition; QR decomposition; Cholesky decomposition; Versions exist for both C++ and the Java programming language. The C++ version uses the Template Numerical Toolkit for lower-level operations. The Java version provides the lower-level operations itself.
In linear algebra, the generalized singular value decomposition (GSVD) is the name of two different techniques based on the singular value decomposition (SVD).The two versions differ because one version decomposes two matrices (somewhat like the higher-order or tensor SVD) and the other version uses a set of constraints imposed on the left and right singular vectors of a single-matrix SVD.
Consider the following matrix as an example: = [] If we apply the full regular Cholesky decomposition, it yields: = [] And, by definition: = ′ However, by applying Cholesky decomposition, we observe that some zero elements in the original matrix end up being non-zero elements in the decomposed matrix, like elements (4,2), (5,2) and (5,3) in this example.
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.