2. 1) CG is a numerical method to solve a linear system of equations 2) CG is used when A is Symmetric and Positive definite matrix (SPD) 3) CG of Hestenes and Stiefel [1] is an effective popular method for solving large, sparse symmetric positive definite (SPD). [1] M. R. Hestenes and E. Stiefel. Methods of conjugate gradient for solving linear systems. Journal of Research of the Natural Bureau of Standards, 49:409-435, 1952

3. Standard inner product defined by:

4. Preconditioner Non-singular

5. Standard inner product defined by: Defined by:For any real symmetric Is an inner product The symmetric bilinear form Pos. def.

6. Self-adjoint in H-symmetric

7. Computational Fluid Dynamics Optimizations Saddle Point Problem Preconditioner Symmetric Indefinite Non-symmetric Positive definite Is H-symmetric and positive definite

8. PreconditionerInner Product USE SPD in H H H H H ^ ^ ^

9. SPD CG Symm MINRES Non-Sym GMRES

10. PreconditionerInner Product USE SPD in H H H H H

12. 2008

14. PreconditionerInner Product USE SPD in H USE Preconditioner SPD in H Can we ??exist Inner Product