共役勾配法
共役勾配法（CG）とは 反復アルゴリズム used for solving systems of linear equations, specifically those that are large and sparse, which are often encountered in various scientific and engineering applications. It is particularly effective for symmetric and positive-definite matrices.
Unlike direct methods that attempt to find the exact solution in a finite number of steps, the 共役勾配法 works by generating a sequence of approximations that converge toward the solution. The key idea is to minimize the quadratic function associated with the 線形システム, which can be visualized as finding the lowest point on a surface defined by that function.
こちらが一般的な動作方法です：
- 初期化： 解の初期推定値を設定し、初期残差を計算します。
- 反復的なプロセス: In each iteration, update the solution by moving in a direction that is conjugate to the previous directions, which helps ensure that the search does not revisit previous paths.
- 終了条件： The process continues until the residual is sufficiently small, indicating that the solution is close enough to the true answer.
One of the main advantages of the Conjugate Gradient method is its efficiency in terms of memory and computation, especially for large systems where direct methods would be infeasible. It typically requires far fewer iterations than the number of unknowns, making it a preferred choice in various applications, including numerical simulations, optimization problems, and 画像処理.