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The geometric interpretation of Newton's method is that at each iteration, it amounts to the fitting of a parabola to the graph of () at the trial value , having the same slope and curvature as the graph at that point, and then proceeding to the maximum or minimum of that parabola (in higher dimensions, this may also be a saddle point), see below.
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.
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.
Gradient descent can also be used to solve a system of nonlinear equations. Below is an example that shows how to use the gradient descent to solve for three unknown variables, x 1, x 2, and x 3. This example shows one iteration of the gradient descent. Consider the nonlinear system of equations
When the damping factor is large relative to ‖ ‖, inverting + is not necessary, as the update is well-approximated by the small gradient step [()]. To make the solution scale invariant Marquardt's algorithm solved a modified problem with each component of the gradient scaled according to the curvature.
A step of the Frank–Wolfe algorithm Initialization: Let , and let be any point in . Step 1. Direction-finding subproblem: Find solving Minimize () Subject to (Interpretation: Minimize the linear approximation of the problem given by the first-order Taylor approximation of around constrained to stay within .)
The basic idea is to convert a constrained problem into a form such that the derivative test of an unconstrained problem can still be applied. The relationship between the gradient of the function and gradients of the constraints rather naturally leads to a reformulation of the original problem, known as the Lagrangian function or Lagrangian. [2]
If an equation can be put into the form f(x) = x, and a solution x is an attractive fixed point of the function f, then one may begin with a point x 1 in the basin of attraction of x, and let x n+1 = f(x n) for n ≥ 1, and the sequence {x n} n ≥ 1 will converge to the solution x.