For a positive-definite matrix , the conjugate gradient method chooses mutually -conjugate search directions. Starting with and , it usesand with
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The residual of an approximate solution of is . It is the unbalanced right-hand side, while the solution error is . The Conjugate gradient method makes residuals orthogonal to successive Krylov subspaces.
At iteration of the conjugate gradient method,and itself lies in . These Galerkin conditions imply finite termination once the Krylov subspace becomes invariant under .
A symmetric preconditioner transforms intoThe aim is to make the transformed positive-definite matrix have clustered eigenvalues while keeping application of inexpensive.
For a positive-definite matrix , choosing gives , so conjugate gradients terminates in one step. Constructing or applying the exact inverse square root is generally as expensive as solving the original system.
In exact arithmetic, conjugate gradients reaches the exact solution in at most the number of distinct eigenvalues of , and therefore in at most steps for an -dimensional system.
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The Conjugate Gradient (CG) method is an iterative algorithm primarily used for solving systems of linear equations whose coefficient matrix is symmetric and positive-definite. It is particularly effective for large-scale problems, where direct methods (like Gaussian elimination) can be computationally expensive or infeasible due to memory requirements. ### Key Features of the Conjugate Gradient Method: 1. **Iteration**: The CG method generates a sequence of approximations to the solution.
For positive definite matrices only.
TODO application.
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The HPCG benchmark uses a preconditioned conjugate gradient (PCG) algorithm to measure the performance of HPC platforms with respect to frequently observed but challenging patterns of computing, communication, and memory access. While HPL provides an optimistic performance target for applications, HPCG can be considered as a lower bound on performance. Many of the top 500 supercomputers also provide their HPCG performance as a reference.