The conjugate gradient method is a mathematical technique that can be useful for the optimization of both linear and non-linear systems. This technique is generally used as an iterative algorithm, however, it can be used as a direct method, and it will produce a numerical solution. 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 3. Method of Conjugate Gradients (cg-Method) The present section will be devoted to a description of a method of solving a system of linear equations Ax=k. This method will be called the conjugate gradient method or, more briefly, the cg-method, for reasons which will unfold from the theory developed in later sections. For the moment, we shall The Conjugate Gradient Method is an iterative technique for solving large sparse systems of linear equations. For the following example for linearizing the one-dimensional heat equation, the Forward Di erence Method is utilized. Note that this process will work for all linear PDEs. CG method that does not directly involve any weighting matrix. Our derivation follows a hybrid approach that combines quasi-Newton and CG search directions. The conditioning matrix . employed. in our method exploits two-Issam A. R. Moghrabi, Member, IAENG A New Preconditioned Conjugate Gradient Method for Optimization C 6.1 The steps of the conjugate gradient algorithm applied to F(x;y).76 6.2 In this example, the conjugate gradient method also converges in four total steps, with much less zig-zagging than the gradient descent method or even Newton’s method.77 7.1 The steps of the DFP algorithm applied to F(x;y).84 Conjugate gradient method used for solving linear equation systems: As discussed before, if is the solution that minimizes the quadratic function , with being symmetric and positive definite, it also satisfies .In other words, the optimization problem is equivalent to the problem of solving the linear system , both can be solved by the conjugate gradient method. The conjugate gradient converges quadratically, which makes it an outstandingly fast. If someone is interested in the theory of conjugate gradient and also in the implementation details I would like to forward you to the amazing paper written by Jonathan Richard Shewchuk called An Introduction to the Conjugate Gradient Method Without the the Conjugate Gradient Method Without the Agonizing Pain Edition 11 4 Jonathan Richard Shewchuk August 4, 1994 School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213 Abstract The Conjugate Gradient Method is the most prominent iterative method for solving sparse systems of linear equations. 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
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