The Conjugate Gradient Method is an iterative technique for solving large sparse systems of linear equations. As a linear algebra and matrix manipulation technique, it is a useful tool in approximating 3.4 Conjugate Gradient. Conjugate gradient methods represent a kind of steepest descent approach “with a twist”. With steepest descent, we begin our minimization of a function \(f\) starting at \(x_0\) by traveling in the direction of the negative gradient \(-f^\prime(x_0)\).In subsequent steps, we continue to travel in the direction of the negative gradient evaluated at each successive [37] to extend the linear conjugate gradient method for nonlinear optimiza-tion. This work of Fletcher and Reeves in 1964 not only opened the door of nonlinear conjugate gradient fleld but greatly stimulated the study of non-linear optimization. In general, the nonlinear conjugate gradient method non-symmetric matrices. Various nonlinear conjugate gradient methods seek minima of nonlinear equations. Contents 1 Description of the method 2 The conjugate gradient method as a direct method 3 The conjugate gradient method as an iterative method 3.1 The resulting algorithm 3.1.1 Example code in GNU Octave 3.2 Numerical example 3.2.1 Solution 14. The Nonlinear Conjugate Gradient Method 42 14.1. Outline of the Nonlinear Conjugate Gradient Method 42 14.2. General Line Search 43 14.3. Preconditioning 47 A Notes 48 B Canned Algorithms 49 B1. Steepest Descent 49 B2. Conjugate Gradients 50 B3. Preconditioned Conjugate Gradients 51 i In this survey, we focus on conjugate gradient methods applied to the nonlinear unconstrained optimization problem (1.1) min ff(x) : x 2Rng; where f: Rn7!Ris a continuously di erentiable function, bounded from below. A nonlinear conjugate gradient method generates a sequence x k, k 1, starting from an initial guess x 0 2Rn, using the recurrence $ The conjugate gradient algorithm minimizes a quadratic function with a symmetric positive-definite Hessian: The algorithm is: step to the line minimum recalculate the gradient where: Eliminate to get the 3-term recurrence (Lanczos): Exact method and iterative method Orthogonality of the residuals implies that xm is equal to the solution x of Ax = b for some m ≤ n. For if xk 6= x for all k = 0,1,...,n− 1 then rk 6= 0for k = 0,1,...,n−1 is an orthogonal basis for Rn.But then rn ∈ Rn is orthogonal to all vectors in Rn so rn = 0and hence xn = x. So the conjugate gradient method finds the exact solution in at most Conjugate Gradient (CG) method have been utilised to solve nonlinear unconstrained optimization problems because of less storage locations and fewer computational cost in dealing with large-scale Abstract. Conjugate gradient methods are a class of important methods for unconstrained optimization and vary only with a scalar β k.In this chapter, we analyze general conjugate gradient method using the Wolfe line search and propose a condition on the scalar β k, which is sufficient for the global convergence.An example is constructed, showing that the condition is also necessary in some
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Video lecture on the Conjugate Gradient Method Learn the Multi-Dimensional Gradient Method of optimization via an example. Minimize an objective function with two variables (part 1 of 2). Cornell class CS4780. (Online version: https://tinyurl.com/eCornellML )
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