# KSPCGNE#

Applies the preconditioned conjugate gradient method to the normal equations without explicitly forming \(A^T*A\)

## Options Database Key#

(for complex matrices only) indicates the matrix is Hermitian or symmetric**-ksp_cg_type <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 [NRT90]. 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 \(A^T*A\) with \(B*B\). Seems odd?

This only supports left preconditioning.

## Developer Note#

This object is subclassed off of `KSPCG`

, see the source code in src/ksp/ksp/impls/cg for comments on the structure of the code

## References#

- NRT90
Noël M. Nachtigal, Satish C. Reddy, and Lloyd N. Trefethen. How fast are nonsymmetric matrix iterations? Technical Report 90-2, Massachusetts Institute of Technology, March 1990.

## See Also#

KSP: Linear System Solvers, `KSPCreate()`

, `KSPSetType()`

, `KSPType`

, `KSP`

, `KSPCG`

, `KSPLSQR`

, `KSPCGLS`

,
`KSPCGSetType()`

, `KSPBICG`

, `KSPSetComputeEigenvalues()`

, `KSPComputeEigenvalues()`

## Level#

beginner

## Location#

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

Index of all KSP routines

Table of Contents for all manual pages

Index of all manual pages