# KSPCGNE
Applies the preconditioned conjugate gradient method to the normal equations without explicitly forming A^t*A 
## Options Database Keys

- ***-ksp_cg_type <Hermitian or symmetric -*** (for complex matrices only) indicates the matrix is Hermitian or symmetric





## Notes
Eigenvalue computation routines including `KSPSetComputeEigenvalues()` and `KSPComputeEigenvalues()` will return information about the
spectrum of A^t*A, rather than A.

`KSPCGNE` is a general-purpose non-symmetric method. It works well when the singular values are much better behaved than
eigenvalues. A unitary matrix is a classic example where `KSPCGNE` converges in one iteration, but `KSPGMRES` and `KSPCGS` need N
iterations, see [1]. If you intend to solve least squares problems, use `KSPLSQR`.

This is NOT a different algorithm than used with `KSPCG`, it merely uses that algorithm with the
matrix defined by A^t*A and preconditioner defined by B^t*B where B is the preconditioner for A.

This method requires that one be able to apply the transpose of the preconditioner and operator
as well as the operator and preconditioner. If the transpose of the preconditioner is not available then
the preconditioner is used in its place so one ends up preconditioning A'A with B B. Seems odd?

This only supports left preconditioning.


## Reference

- ***[1] -*** Nachtigal, Reddy, and Trefethen, "How fast are nonsymmetric matrix iterations", 1992



## Developer Note
This object is subclassed off of `KSPCG`


## See Also
 [](chapter_ksp), `KSPCreate()`, `KSPSetType()`, `KSPType`, `KSP`, 'KSPCG', `KSPLSQR', 'KSPCGLS`,
`KSPCGSetType()`, `KSPBICG`, `KSPSetComputeEigenvalues()`, `KSPComputeEigenvalues()`

## Level
beginner

## Location
<A HREF="PETSC_DOC_OUT_ROOT_PLACEHOLDER/src/ksp/ksp/impls/cg/cgne/cgne.c.html#KSPCGNE">src/ksp/ksp/impls/cg/cgne/cgne.c</A>


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[Edit on GitLab](https://gitlab.com/petsc/petsc/-/edit/release/src/ksp/ksp/impls/cg/cgne/cgne.c)


[Index of all KSP routines](index.md)  
[Table of Contents for all manual pages](/docs/manualpages/index.md)  
[Index of all manual pages](/docs/manualpages/singleindex.md)  
