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  2. Symmetric algebra - Wikipedia

    en.wikipedia.org/wiki/Symmetric_algebra

    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 .

  3. Multigrid method - Wikipedia

    en.wikipedia.org/wiki/Multigrid_method

    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 ...

  4. Generalized minimal residual method - Wikipedia

    en.wikipedia.org/wiki/Generalized_minimal...

    The Arnoldi process also constructs ~, an (+)-by-upper Hessenberg matrix which satisfies = + ~ an equality which is used to simplify the calculation of (see § Solving the least squares problem). Note that, for symmetric matrices, a symmetric tri-diagonal matrix is actually achieved, resulting in the MINRES method.

  5. Symbolic method - Wikipedia

    en.wikipedia.org/wiki/Symbolic_method

    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.

  6. Linear complementarity problem - Wikipedia

    en.wikipedia.org/wiki/Linear_complementarity_problem

    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.

  7. Gauss–Seidel method - Wikipedia

    en.wikipedia.org/wiki/Gauss–Seidel_method

    algorithm Gauss–Seidel method is inputs: A, b output: φ Choose an initial guess φ to the solution repeat until convergence for i from 1 until n do σ ← 0 for j from 1 until n do if j ≠ i then σ ← σ + a ij φ j end if end (j-loop) φ i ← (b i − σ) / a ii end (i-loop) check if convergence is reached end (repeat)

  8. Rayleigh quotient iteration - Wikipedia

    en.wikipedia.org/wiki/Rayleigh_quotient_iteration

    Rayleigh quotient iteration is an eigenvalue algorithm which extends the idea of the inverse iteration by using the Rayleigh quotient to obtain increasingly accurate eigenvalue estimates. Rayleigh quotient iteration is an iterative method , that is, it delivers a sequence of approximate solutions that converges to a true solution in the limit.

  9. Conjugate gradient method - Wikipedia

    en.wikipedia.org/wiki/Conjugate_gradient_method

    The conjugate gradient method can also be used to solve unconstrained optimization problems such as energy minimization. It is commonly attributed to Magnus Hestenes and Eduard Stiefel, [1] [2] who programmed it on the Z4, [3] and extensively researched it. [4] [5] The biconjugate gradient method provides a