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In mathematics, the method of steepest descent or saddle-point method is an extension of Laplace's method for approximating an integral, where one deforms a contour integral in the complex plane to pass near a stationary point (saddle point), in roughly the direction of steepest descent or stationary phase. The saddle-point approximation is ...
A (properly speaking) nonlinear steepest descent method was introduced by Kamvissis, K. McLaughlin and P. Miller in 2003, based on previous work of Lax, Levermore, Deift, Venakides and Zhou. As in the linear case, "steepest descent contours" solve a min-max problem.
The conjugate gradient method can be derived from several different perspectives, including specialization of the conjugate direction method for optimization, and variation of the Arnoldi/Lanczos iteration for eigenvalue problems. Despite differences in their approaches, these derivations share a common topic—proving the orthogonality of the ...
This method originates from the 19th century, and is due to George Gabriel Stokes and Lord Kelvin. [1] It is closely related to Laplace's method and the method of steepest descent , but Laplace's contribution precedes the others.
This method is not in general use. Davidon–Fletcher–Powell method. This method, a form of pseudo-Newton method, is similar to the one above but calculates the Hessian by successive approximation, to avoid having to use analytical expressions for the second derivatives. Steepest descent. Although a reduction in the sum of squares is ...
Subsequent search directions lose conjugacy requiring the search direction to be reset to the steepest descent direction at least every N iterations, or sooner if progress stops. However, resetting every iteration turns the method into steepest descent. The algorithm stops when it finds the minimum, determined when no progress is made after a ...
Gradient descent is a method for unconstrained mathematical optimization. It is a first-order iterative algorithm for minimizing a differentiable multivariate function . The idea is to take repeated steps in the opposite direction of the gradient (or approximate gradient) of the function at the current point, because this is the direction of ...
In asymptotic analysis, the method of Chester–Friedman–Ursell is a technique to find asymptotic expansions for contour integrals. It was developed as an extension of the steepest descent method for getting uniform asymptotic expansions in the case of coalescing saddle points . [ 1 ]