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The symmetric algebra S(V) can be built as the quotient of the tensor algebra T(V) by the two-sided ideal generated by the elements of the form x ⊗ y − y ⊗ x. All these definitions and properties extend naturally to the case where V is a module (not necessarily a free one) over a commutative ring .
Macaulay2 is built around fast implementations of algorithms useful for computation in commutative algebra and algebraic geometry. This core functionality includes arithmetic on rings, modules, and matrices, as well as algorithms for Gröbner bases, free resolutions, Hilbert series, determinants and Pfaffians, factoring, and similar.
In mathematics, the symbolic method in invariant theory is an algorithm developed by Arthur Cayley, [1] Siegfried Heinrich Aronhold, [2] Alfred Clebsch, [3] and Paul Gordan [4] in the 19th century for computing invariants of algebraic forms.
The conjugate residual method is an iterative numeric method used for solving systems of linear equations. It's a Krylov subspace method very similar to the much more popular conjugate gradient method, with similar construction and convergence properties. This method is used to solve linear equations of the form
Formally, the symmetric algebra of a vector space V over a field F is the group algebra of the dual, Sym(V) := F[V ∗], and the Weyl algebra is the group algebra of the (dual) Heisenberg group W(V) = F[H(V ∗)]. Since passing to group algebras is a contravariant functor, the central extension map H(V) → V becomes an inclusion Sym(V) → W(V).
If M is positive definite, any algorithm for solving (strictly) convex QPs can solve the LCP. Specially designed basis-exchange pivoting algorithms, such as Lemke's algorithm and a variant of the simplex algorithm of Dantzig have been used for decades. Besides having polynomial time complexity, interior-point methods are also effective in practice.
Originally described in Xu's Ph.D. thesis [9] and later published in Bramble-Pasciak-Xu, [10] the BPX-preconditioner is one of the two major multigrid approaches (the other is the classic multigrid algorithm such as V-cycle) for solving large-scale algebraic systems that arise from the discretization of models in science and engineering ...
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.
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