The Conjugate Gradient (CG) method is often used to solve a positive definite linear system Ax = b. This paper analyzes two hard cases for CG or any Krylov subspace type methods by either analytically finding the residual formulas or tightly bound the residuals from above and below, in contrast to existing results which only bound residuals from above. The analysis is based on a general framewo...

multigrid method as a preconditioner of the PCG method, is proposed. The multigrid method has inherent high parallelism and improves convergence of long wave length components, which is important in iterative methods. By using this method as a preconditioner of the PCG method, an e cient method with high parallelism and fast convergence is obtained. First, it is considered a necessary condition...

The Conjugate Gradient Method is a numerical optimization technique that finds the optimal solution by focusing on coefficient conjugate. This paper presents new coefficients conjugate gradient method for removing impulse noise from images, which based quadratic function and proven to be globally convergent. Results show it an effective image restoration.