Algorithm for solving matrix-vector equations
In numerical linear algebra, the conjugate gradient squared method (CGS) is an iterative algorithm for solving systems of linear equations of the form formula_1, particularly in cases where computing the transpose formula_2 is impractical. The CGS method was developed as an improvement to the biconjugate gradient method.
Background.
A system of linear equations formula_1 consists of a known matrix formula_4 and a known vector formula_5. To solve the system is to find the value of the unknown vector formula_6. A direct method for solving a system of linear equations is to take the inverse of the matrix formula_4, then calculate formula_8. However, computing the inverse is computationally expensive. Hence, iterative methods are commonly used. Iterative methods begin with a guess formula_9, 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:
References.
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