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Conjugate Gradient Method for Indefinite Matrices

Conjugate Gradient Method for Indefinite Matrices. Conjugate Gradient . 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).

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Conjugate Gradient Method for Indefinite Matrices

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  1. Conjugate Gradient Method for Indefinite Matrices

  2. Conjugate Gradient 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. Conjugate Gradient Standard inner product defined by:

  4. Preconditioning Non-singular Preconditioner

  5. Non-standard Inner Product Standard inner product defined by: Definition For any real symmetric Defined by: The symmetric bilinear form Is an inner product Pos. def.

  6. Self-Adjoint Definition Self-adjoint in Self-adjoint in H-symmetric

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

  8. Bramble-Pasciak CG CG for Indefinite USE Preconditioner Inner Product H H SPD in < , >H H H

  9. Iterative Krylov Subspace Methods SPD Symm Non-Sym CG MINRES GMRES

  10. Bramble-Pasciak CG CG for Indefinite USE Preconditioner Inner Product H H SPD in < , >H H H

  11. Bramble-Pasciak CG

  12. Bramble-Pasciak CG 2008

  13. Bramble-Pasciak CG

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