Conjugate gradient method

In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is positive-semidefinite. The conjugate gradient method is often implemented as an iterative algorithm, applicable to sparse systems that are too large to be handled by a direct implementation or other direct methods such as the Cholesky decomposition. Large sparse systems often arise when numerically solving partial differential equations or optimization problems.

The conjugate gradient method can also be used to solve unconstrained optimization problems such as energy minimization. It is commonly attributed to Magnus Hestenes and Eduard Stiefel, who programmed it on the Z4, and extensively researched it.

The biconjugate gradient method provides a generalization to non-symmetric matrices. Various nonlinear conjugate gradient methods seek minima of nonlinear optimization problems.

Description of the problem addressed by conjugate gradients

Suppose we want to solve the system of linear equations

A

x

=

b

{\displaystyle \mathbf {A} \mathbf {x} =\mathbf {b} }

for the vector

x

{\displaystyle \mathbf {x} }

, where the known

n

×

n

{\displaystyle n\times n}

matrix

A

{\displaystyle \mathbf {A} }

is symmetric (i.e.,

A

T

=

A

{\displaystyle \mathbf {A} ^{\mathsf {T}}=\mathbf {A} }

), positive-definite (i.e.

x

T

A

x

>

0

{\displaystyle \mathbf {x} ^{\mathsf {T}}\mathbf {Ax} >0}

for all non-zero vectors

x

{\displaystyle \mathbf {x} }

in

R

n

{\displaystyle \mathbb {R} ^{n}}

), and real, and

b

{\displaystyle \mathbf {b} }

is known as well. We denote the unique solution of this system by

x

∗

{\displaystyle \mathbf {x} _{*}}

.