In numerical linear algebra, the conjugate gradient squared method (CGS) is an iterative algorithm for solving systems of linear equations of the form , particularly in cases where computing the transpose is impractical. The CGS method was developed as an improvement to the biconjugate gradient method.
Background A system of linear equations consists of a known matrix and a known vector . To solve the system is to find the value of the unknown vector . A direct method for solving a system of linear equations is to take the inverse of the matrix , then calculate . However, computing the inverse is computationally expensive. Hence, iterative methods are commonly used. Iterative methods begin with a guess , 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.
Algorithm
The algorithm is as follows:
Choose an initial guess Compute the residual Choose For do:
If , the method fails. If :
Else:
Solve , where is a pre-conditioner.
Solve
Check for convergence: if there is convergence, end the loop and return the result
See also Biconjugate gradient method Biconjugate gradient stabilized method Generalized minimal residual method
References