Article. Edition 1+1/4. Jonathan Richard Shewchuk. August 4, 1994. School of Computer Science. Carnegie Mellon University. Pittsburgh, P A 15213. pages: 58. Abstract. 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 reserved for the elite brilliant few who have painstakingly decoded the mumblings of their forebears. Nevertheless, the Conjugate Gradient Method is a composite of simple, elegant ideas that almost anyone can understand. Of course, a reader as intelligent as yourself will learn them almost effortlessly.Contents. Introduction. Notation. The Quadratic Form. The Method of Steepest Descent. Thinking with Eigenvectors and Eigenvalues. Convergence Analysis of Steepest Descent. The Method of Conjugate Directions. The Method of Conjugate Gradients. Convergence Analysis of Conjugate Gradients. Complexity. Starting and Stopping. Preconditioning. Conjugate Gradients on the Normal Equations. The Nonlinear Conjugate Gradient Method.
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