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The conjugate gradient method with a trivial modification is extendable to solving, given complex-valued matrix A and vector b, the system of linear equations = for the complex-valued vector x, where A is Hermitian (i.e., A' = A) and positive-definite matrix, and the symbol ' denotes the conjugate transpose.
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. The intent of this article is to document the important steps in these derivations.
A good quality random Gaussian function with the zero mean is commonly the default in LOBPCG to generate the initial approximations. To fix the initial approximations, one can select a fixed seed for the random number generator. In contrast to the Lanczos method, LOBPCG rarely exhibits asymptotic superlinear convergence in practice.
The Barzilai-Borwein method [1] is an iterative gradient descent method for unconstrained optimization using either of two step sizes derived from the linear trend of the most recent two iterates. This method, and modifications, are globally convergent under mild conditions, [ 2 ] [ 3 ] and perform competitively with conjugate gradient methods ...
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]
Download as PDF; Printable version; ... Pages in category "Gradient methods" The following 20 pages are in this category, out of 20 total. ... Conjugate gradient method;
In optimization, a gradient method is an algorithm to solve problems of the form min x ∈ R n f ( x ) {\displaystyle \min _{x\in \mathbb {R} ^{n}}\;f(x)} with the search directions defined by the gradient of the function at the current point.
Whereas linear conjugate gradient seeks a solution to the linear equation =, the nonlinear conjugate gradient method is generally used to find the local minimum of a nonlinear function using its gradient alone. It works when the function is approximately quadratic near the minimum, which is the case when the function is twice differentiable at ...