In numerical linear algebra, the conjugate gradient squared method (CGS) is an iterative algorithm for solving systems of linear equations of the form ${\displaystyle A{\mathbf {x}}={\mathbf {b}}}$, particularly in cases where computing the transpose ${\displaystyle A^{T}}$ is impractical. The CGS method was developed as an improvement to the biconjugate gradient method.

A system of linear equations ${\displaystyle A{\mathbf {x}}={\mathbf {b}}}$ consists of a known matrix ${\displaystyle A}$ and a known vector ${\displaystyle {\mathbf {b}}}$. To solve the system is to find the value of the unknown vector ${\displaystyle {\mathbf {x}}}$. A direct method for solving a system of linear equations is to take the inverse of the matrix ${\displaystyle A}$, then calculate ${\displaystyle {\mathbf {x}}=A^{-1}{\mathbf {b}}}$. However, computing the inverse is computationally expensive. Hence, iterative methods are commonly used. Iterative methods begin with a guess ${\displaystyle {\mathbf {x}}^{(0)}}$, and on each iteration the guess is improved. Once the difference between successive guesses is sufficiently small, the method has converged to a solution.

As with the conjugate gradient method, biconjugate gradient method, and similar iterative methods for solving systems of linear equations, the CGS method can be used to find solutions to multi-variable optimisation problems, such as power-flow analysis, hyperparameter optimisation, and facial recognition.

The algorithm is as follows: