Calculus (dental)

Mathematics

Economic growth

Algorithm

Gradient method

Gradient descent

Mathematical analysis

Artificial intelligence

Conjugate

Derivation of the conjugate gradient method

Economics

Dentistry

Nonlinear conjugate gradient method

Convergence (economics)

Computer science

Applied mathematics

Conjugate residual method

Artificial neural network

Medicine

Conjugate gradient method

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 so that even their own authors would be mystified, if they bothered to read their own writing. For this reason, an 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. The idea of quadratic forms is introduced and used to derive the methods of Steepest Descent, Conjugate Directions, and Conjugate Gradients. Eigenvectors are explained and used to examine the convergence of the Jacobi Method, Steepest Descent, and Conjugate Gradients. Other topics include preconditioning and the nonlinear Conjugate Gradient Method. I have taken pains to make this article easy to read. Sixty-two illustrations are provided. Dense prose is avoided. Concepts are explained in several different ways. Most equations are coupled with an intuitive interpretation.

An Introduction to the Conjugate Gradient Method Without the Agonizing Pain