Editing Cholesky decomposition (section)

===Numerical solution of system of linear equations===
The Cholesky decomposition is mainly used for the numerical solution of [[system of linear equations|linear equations]] <math display=inline>\mathbf{Ax} = \mathbf{b}</math>. If {{math|'''A'''}} is symmetric and positive definite, then  <math display=inline>\mathbf{Ax} = \mathbf{b}</math> can be solved by first computing the Cholesky decomposition  <math display=inline>\mathbf{A} = \mathbf{LL}^\mathrm{*}</math>, then solving <math display=inline>\mathbf{Ly} = \mathbf{b}</math> for {{math|'''y'''}} by [[forward substitution]], and finally solving <math display=inline>\mathbf{L^*x} = \mathbf{y}</math> for {{math|'''x'''}} by [[back substitution]].

An alternative way to eliminate taking square roots in the <math display=inline>\mathbf{LL}^\mathrm{*}</math> decomposition is to compute the LDL decomposition <math display=inline>\mathbf{A} = \mathbf{LDL}^\mathrm{*}</math>, then solving <math display=inline>\mathbf{Ly} = \mathbf{b}</math> for {{math|'''y'''}}, and finally solving <math display=inline>\mathbf{DL}^\mathrm{*}\mathbf{x} = \mathbf{y}</math>.

For linear systems that can be put into symmetric form, the Cholesky decomposition (or its LDL variant) is the method of choice, for superior efficiency and numerical stability. Compared to the [[LU decomposition]], it is roughly twice as efficient.<ref name="NR"/>