Preconditioned conjugate gradient method (2)

Preconditioned conjugate gradient method (2)

Pre-selecting on matrix M

1. Recalls algorithm

The basic properties of the conjugate gradient of poison: conjugate gradient method vector generation remaining orthogonal to each other about the direction of decline mutually conjugate A

Thus, we can conclude:

2. The array of M pre-selected prerequisite

3. One view

4. Generalized Conjugate Gradient Method

The incomplete Cholesky decomposition

Recalling triangular decomposition of the matrix square root method

Theorem (the Cholesky decomposition) : If A is symmetric positive definite matrix of order n, then there is a real non-singular so that lower triangular matrix L $A = LL^T$ , is defined as positive when the diagonal elements, this decomposition is unique to L

In order to avoid the open square, we can also use shredded $A = LDL^T$

The core idea is to ensure that the sparse matrix after decomposition, making it easier to solve

6. incomplete Cholesky decomposition block

Of the block tridiagonal matrix A
we set





for the second pathway