Conjugate gradient chooses the search directions to be -orthogonal. For this, we will For this, we will need some background: how to convert an arbitrary basis into an orthogonal basis using Conjugate gradient method will construct an A-orthogonal basis by recrusion. Start from an initial guess u0. Let p0 = r0 = r f(u0) . For k= 0;1;2;:::;n, let Vk = spanfp0;p1; ;pkgbe a subspace spanned by A-orthogonal basis, i.e. (pi;pj)A= 0 for i6= j;i;j= 0;:::;k. CG consists of three steps: (1)compute uk+1 by the A-orthogonal projection of u u0 to Vk. (2)add residual vector rk+1 to Vkto get The Conjugate Gradient Method Jason E. Hicken AerospaceDesignLab DepartmentofAeronautics&Astronautics StanfordUniversity 14 July 2011. Lecture Objectives describe when CG can be used to solve Ax= b relate CG to the method of conjugate directions describe what CG does geometrically explain each line in the CG algorithm. We are interested in solving the linear system Ax= b where x, b∈ Rn and A SolutionofAx = b Keyproperty: A1b 2Kn thisholdsevenwhenKn, Rn fromCayley–Hamiltontheorem, p„A”= An + a1An1 + + anI = 0 wherep„ ”= det„ I A”= n + a1 n1 + + an1 + an multiplyingontherightwithA1b shows A1b = 1 an An1b+ a 1A n2b+ + a n1b Conjugategradientmethod 13.4 An example The conjugate gradient method in extremal problems 109 zyxwvutsrqpo of a function of quite a small numberof variables, where the method(12) was required, is f(z)= A-2 i ([email protected] + 2Ae-W - qe-0.2jx2 - ~,[email protected])2, j=i WhenA = 1, 640 iterations were needed, in order to obtain ](CrY) - f(z*) = 0.5*10-13, l/P - x*11= 0.2 ’ IO-4 (when f(ti) - f(Y) = 0.5, 11ti- x*(1 = 1.7). At the same Conjugate Gradient Method • direct and indirect methods • positive definite linear systems • Krylov sequence • spectral analysis of Krylov sequence • preconditioning EE364b, Stanford University. Three classes of methods for linear equations methods to solve linear system Ax = b, A ∈ Rn×n • dense direct (factor-solve methods) – runtime depends only on size; independent of data definite. The conjugate gradient method is an iterative method, so it can be applied to sparse systems that are too large to be handled by direct methods such as the Cholesky decomposition. Such systems often arise when numerically solving partial differential equations. The conjugate gradient method can also be used to solve The Conjugate Gradient Method is the most prominent iterative method for solving sparse systems of linear equations. Unfortunately, many textbook treatments of the topic are written with neither illustrations nor intuition, and their victims can be found to this day babbling senselessly in the corners of dusty libraries. For this reason, a deep, geometric understanding of the method has been Conjugate gradient method as an optimal process. E. Stiefel in [12], [20] used the theory of orthogonal polynomials to discuss "best" iterative methods for solving finite systems of linear algebraic equations; these results, extended to a Hilbert space setting, give another character- ization of the CG method and lead to a much improved rate of convergence estimate. CONJUGATE GRADIENT METHOD So the conjugate gradient method finds the exact solution in at most n iterations. The convergence analysis shows that kx− xkkA typically becomes small quite rapidly and we can stop the iteration with k much smaller that n. It is this rapid convergence which makes the method interesting and in practice an iterative method. The Conjugate Gradient Method – p. 7/23. Conjugate Gradient
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