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# KSPCG

The Preconditioned Conjugate Gradient (PCG) iterative method [HS52] and [MalekStrakovs14]

## Options Database Keys

- -ksp_cg_type Hermitian - (for complex matrices only) indicates the matrix is Hermitian, see `KSPCGSetType()`

- -ksp_cg_type symmetric - (for complex matrices only) indicates the matrix is symmetric

- -ksp_cg_single_reduction - performs both inner products needed in the algorithm with a single `MPI_Allreduce()` call, see `KSPCGUseSingleReduction()`

## Notes

The PCG method requires both the matrix and preconditioner to be symmetric positive (or negative) (semi) definite.

Only left preconditioning is supported; there are several ways to motivate preconditioned CG, but they all produce the same algorithm. One can interpret preconditioning A with B to mean any of the following:

       (1) Solve a left-preconditioned system BAx = Bb, using inv(B) to define an inner product in the algorithm.
       (2) Solve a right-preconditioned system ABy = b, x = By, using B to define an inner product in the algorithm.
       (3) Solve a symmetrically-preconditioned system, E^TAEy = E^Tb, x = Ey, where B = EE^T.
       (4) Solve Ax=b with CG, but use the inner product defined by B to define the method [2].
       In all cases, the resulting algorithm only requires application of B to vectors.

For complex numbers there are two different CG methods, one for Hermitian symmetric matrices and one for non-Hermitian symmetric matrices. Use `KSPCGSetType()` to indicate which type you are using.

One can use `KSPSetComputeEigenvalues()` and `KSPComputeEigenvalues()` to compute the eigenvalues of the (preconditioned) operator

## Developer Note

KSPSolve_CG() should actually query the matrix to determine if it is Hermitian symmetric or not and NOT require the user to indicate it to the `KSP` object.

## References

[HS52]

Magnus R. Hestenes and Eduard Steifel. Methods of conjugate gradients for solving linear systems. J. Research of the National Bureau of Standards, 49:409–436, 1952.

[MalekStrakovs14]

Josef Málek and Zdeněk Strakoš. Preconditioning and the conjugate gradient method in the context of solving PDEs. SIAM, 2014.

## See Also

KSP: Linear System Solvers, `KSPCreate()`, `KSPSetType()`, `KSPType`, `KSP`, `KSPSetComputeEigenvalues()`, `KSPComputeEigenvalues()` `KSPCGSetType()`, `KSPCGUseSingleReduction()`, `KSPPIPECG`, `KSPGROPPCG`

## Level

beginner

## Location

src/ksp/ksp/impls/cg/cg.c

## Examples

src/snes/tutorials/ex11.c
src/ksp/ksp/tutorials/ex59.c
src/tao/pde_constrained/tutorials/parabolic.c
src/tao/bound/tutorials/jbearing2.c
src/ksp/ksp/tutorials/ex78.c
src/ksp/ksp/tutorials/ex71.c
src/ksp/pc/tutorials/ex1.c
src/ksp/pc/tutorials/ex2.c

Index of all KSP routines
Table of Contents for all manual pages
Index of all manual pages