Conjugate gradient squared method

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:

1. Choose an initial guess
2. Compute the residual
3. Choose
4. For do:
6. If , the method fails.
7. If :
9. Else:
13. Solve , where is a pre-conditioner.
17. Solve
21. 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