C++ is fast for scientific computing but has a cumbersome syntax. To make matrix computations easier to code, I wrote a templated matrix class in this repo. The class allows the programmer to define a matrix with matrix<double> A(3,3); and multiply two matrices with A*B, for example. To demonstrate the usefulness of the class, let’s solve the following linear regression problem: In these cases, iterative methods, such as conjugate gradient, are popular, especially when the matrix \(A\) is sparse. In direct matrix inversion methods, there are typically \(O(n)\) steps, each requiring \(O(n^2)\) computation; iterative methods aim to cut down on the running time of each of these numbers, and the performance typically depends on the spectral properties of the matrix. 12 Notes 13 External links Description of the method Suppose we want to solve the following system of linear equations Ax = b where the n-by-n matrix A is symmetric (i.e., AT = A), positive definite (i.e., xTAx > 0 for all non-zero vectors x in Rn), and real. We denote the unique solution of this system by x The conjugate gradient method as a direct method 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 ... 39 2 Conjugate Gradient Method 40 2.1 Minimizing a convex quadratic using A-conjugate vectors Let A2R n be a positive de nite matrix and b 2Rn. We would like to solve the convex, quadratic minimization problem min p2Rn 1 2 p TAp b p = min p2Rn 1 2 hp;pi A b p: Equivalently, we want to solve the system Ap = b: NOTES: 2 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 ... With this reasoning as background, one develops the conjugate gradient method for quadratic functions formed from symmetric positive definite matrices. For such quadratic functions, the conjugate gradient method converges to the unique global minimum in at most n steps, by moving along successive non-interfering directions. 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 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 solutions to linearized partial di erential equations. The fundamental concepts are introduced and 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 ...
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