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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. The conjugate gradient method is often implemented as an iterative algorithm , applicable to sparse systems that are too large to be handled by a direct ...
The conjugate gradient method can be derived from several different perspectives, including specialization of the conjugate direction method [1] for optimization, and variation of the Arnoldi/Lanczos iteration for eigenvalue problems.
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 ...
The best known Krylov subspace methods are the Conjugate gradient, IDR(s) (Induced dimension reduction), GMRES (generalized minimum residual), BiCGSTAB (biconjugate gradient stabilized), QMR (quasi minimal residual), TFQMR (transpose-free QMR) and MINRES (minimal residual method).
As with the conjugate gradient method, biconjugate gradient method, and similar iterative methods for solving systems of linear equations, the CGS method can be used to find solutions to multi-variable optimisation problems, such as power-flow analysis, hyperparameter optimisation, and facial recognition. [8]
In optimization, a gradient method is an algorithm to solve problems of the form with the search directions defined by the gradient of the function at the current point. Examples of gradient methods are the gradient descent and the conjugate gradient.
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
In mathematics, more specifically in numerical linear algebra, the biconjugate gradient method is an algorithm to solve systems of linear equations A x = b . {\displaystyle Ax=b.\,} Unlike the conjugate gradient method , this algorithm does not require the matrix A {\displaystyle A} to be self-adjoint , but instead one needs to perform ...