CG Conjugate Gradient Method: Iterative Solution for Matrix Equations

The CG (Conjugate Gradient) method is an efficient iterative algorithm specifically designed for solving large sparse linear systems of equations. Originally proposed by Hestenes and Stiefel in 1952, this method has gained widespread popularity due to its excellent convergence properties for symmetric positive definite matrices.

The core concept of the CG method leverages the properties of conjugate directions, where each iteration progresses along optimal directions toward the solution, thereby minimizing computational effort. Compared to traditional direct methods like Gaussian elimination, the CG method is particularly suitable for large-scale sparse matrix problems as it avoids the memory overhead associated with storing complete matrices. In code implementation, this typically requires only matrix-vector multiplication operations rather than full matrix storage.

In practical applications, the CG method is commonly employed in numerical simulations, finite element analysis, and optimization problems in machine learning. Its key advantage lies in requiring only matrix-vector multiplications without explicitly computing matrix inverses, which significantly reduces computational complexity. The algorithm features finite-step convergence - under ideal conditions without rounding errors, it can achieve exact solutions in at most n iterations (where n represents the matrix dimension). From a programming perspective, the method can be implemented using iterative updates of residual vectors and search directions.

However, it's important to note that the CG method is sensitive to the matrix's condition number. If the matrix is severely ill-conditioned, convergence speed may deteriorate significantly. To address this, preconditioning techniques (such as incomplete Cholesky factorization) are typically introduced to improve convergence performance. In code implementation, preconditioners help transform the original system into an equivalent better-conditioned system through carefully designed matrix operations.