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The method of steepest descent is a method to approximate a complex integral of the form = () for large , where () and () are analytic functions of . Because the integrand is analytic, the contour C {\displaystyle C} can be deformed into a new contour C ′ {\displaystyle C'} without changing the integral.
Vs. the locally optimal steepest descent method [ edit ] In both the original and the preconditioned conjugate gradient methods one only needs to set β k := 0 {\displaystyle \beta _{k}:=0} in order to make them locally optimal, using the line search , steepest descent methods.
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 ...
Kantorovich in 1948 proposed calculating the smallest eigenvalue of a symmetric matrix by steepest descent using a direction = of a scaled gradient of a Rayleigh quotient = (,) / (,) in a scalar product (,) = ′, with the step size computed by minimizing the Rayleigh quotient in the linear span of the vectors and , i.e. in a locally optimal manner.
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 ...
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 ...
Now, () is a vector which points towards the steepest ascent of the cost function. To find the minimum of the cost function we need to take a step in the opposite direction of ∇ C ( n ) {\displaystyle \nabla C(n)} .
These steps can be used as shown in the MATLAB style pseudo code for 1 ... preconditioned steepest descent and flexible CG methods ... The method has been widely used ...