MATH 5330: Computational Methods of Linear Algebra Lecture Note 6: Conjugate Gradient Method for General Systems
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MATH 5330: Computational Methods of Linear Algebra Lecture Note 4: The Conjugate Gradient Method
Thus we can apply any optimization algorithm to solve this minimization problem and obtain a method for solving (1.1). At the point, let us consider the steepest descent method and select any initial guess x0. With xk available we try to find the direction along which φ(x) decreases most rapidly starting from xk and compute the next point xk+1 by minimizing φ(x) in this direction. By Taylor ser...
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We look at more detail into the projection method, where a solution candidate x̃ in the affine space x0+K is searched for, so that the residual r=b−Ax is orthogonal to a linear subspace L. Let A be any non-singular n×n matrix (not necessarily symmetric), and let {v1,···,vm} and {w1,···,wm} be bases for K and L, respectively. Denote V =[v1 v2 ··· vm]∈R and W =[w1 w2 ··· wm]∈R, then we have the fi...
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[1] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004. [2] Carlos Fernandez-Granda, Lecture Notes of “Optimization-based Data Analysis”, available at http: //www.cims.nyu.edu/~cfgranda/pages/OBDA_spring16/notes.html, 2016. [3] Carlos Fernandez-Granda, Lecture Notes of DSGA1002, available at http://www.cims.nyu.edu/ ~cfgranda/pages/DSGA1002_fall15/notes.html, 201...
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