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Related: the LUP decomposition is =, where L is lower triangular, U is upper triangular, and P is a permutation matrix. Existence: An LUP decomposition exists for any square matrix A. When P is an identity matrix, the LUP decomposition reduces to the LU 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 ...
PETSc, a large C library, containing many different matrix solvers for a variety of matrix storage formats. Trilinos, a large C++ library, with sub-libraries dedicated to the storage of dense and sparse matrices and solution of corresponding linear systems. Eigen3 is a C++ library that
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]
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 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.
The FETI-DP method is a domain decomposition method [1] that enforces equality of the solution at subdomain interfaces by Lagrange multipliers except at subdomain corners, which remain primal variables. The first mathematical analysis of the method was provided by Mandel and Tezaur. [2]
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