Conjugate gradient method

Conjugate gradient method

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A comparison of the convergence ofgradient descent with optimal step size (in green) and conjugate gradient (in red) for minimizing a quadratic function associated with a given linear system. Conjugate gradient, assuming exact arithmetics, converges in at most n steps where n is the size of the matrix of the system (here n=2).

In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix issymmetric and positive-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 solvingpartial differential equations.
The conjugate gradient method can also be used to solve unconstrained optimizationproblems such as energy minimization.
The biconjugate gradient method provides a generalization to non-symmetric matrices. Various nonlinear conjugate gradient methods seek minima of nonlinear equations.

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