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In abstract algebra, the conjugacy problem for a group G with a given presentation is the decision problem of determining, given two words x and y in G, whether or not they represent conjugate elements of G. That is, the problem is to determine whether there exists an element z of G such that =.
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.
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.
The subgroups can thus be divided into conjugacy classes, with two subgroups belonging to the same class if and only if they are conjugate. Conjugate subgroups are isomorphic, but isomorphic subgroups need not be conjugate. For example, an abelian group may have two different subgroups which are isomorphic, but they are never conjugate.
The next figure illustrates the maximization problem on the right hand side of the above equation. Tangents are drawn to each of the two curves such that both tangents have the same slope p . The problem is to adjust p in such a way that the two tangents are as far away from each other as possible (more precisely, such that the points where ...
A transformation A ↦ P −1 AP is called a similarity transformation or conjugation of the matrix A. In the general linear group , similarity is therefore the same as conjugacy , and similar matrices are also called conjugate ; however, in a given subgroup H of the general linear group, the notion of conjugacy may be more restrictive than ...
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 ...
In mathematics, two functions are said to be topologically conjugate if there exists a homeomorphism that will conjugate the one into the other. Topological conjugacy, and related-but-distinct § Topological equivalence of flows, are important in the study of iterated functions and more generally dynamical systems, since, if the dynamics of one iterative function can be determined, then that ...
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