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  2. Conjugate gradient method - Wikipedia

    en.wikipedia.org/wiki/Conjugate_gradient_method

    Conjugate gradient, assuming exact arithmetic, converges in at most n steps, where n is the size of the matrix of the system (here n = 2). In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is positive-semidefinite.

  3. Gaussian elimination - Wikipedia

    en.wikipedia.org/wiki/Gaussian_elimination

    For example, to solve a system of n equations for n unknowns by performing row operations on the matrix until it is in echelon form, and then solving for each unknown in reverse order, requires n(n + 1)/2 divisions, (2n 3 + 3n 2 − 5n)/6 multiplications, and (2n 3 + 3n 2 − 5n)/6 subtractions, [10] for a total of approximately 2n 3 /3 operations.

  4. Conjugate gradient squared method - Wikipedia

    en.wikipedia.org/wiki/Conjugate_gradient_squared...

    A system of linear equations = consists of a known matrix and a known vector.To solve the system is to find the value of the unknown vector . [3] [5] A direct method for solving a system of linear equations is to take the inverse of the matrix , then calculate =.

  5. Nonlinear conjugate gradient method - Wikipedia

    en.wikipedia.org/wiki/Nonlinear_conjugate...

    There, both step direction and length are computed from the gradient as the solution of a linear system of equations, with the coefficient matrix being the exact Hessian matrix (for Newton's method proper) or an estimate thereof (in the quasi-Newton methods, where the observed change in the gradient during the iterations is used to update the ...

  6. Generalized minimal residual method - Wikipedia

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

    Note that ~ is an (n + 1)-by-n matrix, hence it gives an over-constrained linear system of n+1 equations for n unknowns. The minimum can be computed using a QR decomposition : find an ( n + 1)-by-( n + 1) orthogonal matrix Ω n and an ( n + 1)-by- n upper triangular matrix R ~ n {\displaystyle {\tilde {R}}_{n}} such that Ω n H ~ n = R ~ n ...

  7. Matrix-free methods - Wikipedia

    en.wikipedia.org/wiki/Matrix-free_methods

    It is generally used in solving non-linear equations like Euler's equations in computational fluid dynamics. Matrix-free conjugate gradient method has been applied in the non-linear elasto-plastic finite element solver. [7] Solving these equations requires the calculation of the Jacobian which is costly in terms of CPU time and storage. To ...

  8. Tridiagonal matrix algorithm - Wikipedia

    en.wikipedia.org/wiki/Tridiagonal_matrix_algorithm

    In other situations, the system of equations may be block tridiagonal (see block matrix), with smaller submatrices arranged as the individual elements in the above matrix system (e.g., the 2D Poisson problem). Simplified forms of Gaussian elimination have been developed for these situations. [6]

  9. Jacobi method - Wikipedia

    en.wikipedia.org/wiki/Jacobi_method

    In numerical linear algebra, the Jacobi method (a.k.a. the Jacobi iteration method) is an iterative algorithm for determining the solutions of a strictly diagonally dominant system of linear equations. Each diagonal element is solved for, and an approximate value is plugged in.