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# KSPPIPECGRR
Pipelined conjugate gradient method with automated residual replacements. [](sec_pipelineksp) 


## Notes
This method has only a single non-blocking reduction per iteration, compared to 2 blocking for standard `KSPCG`.  The
non-blocking reduction is overlapped by the matrix-vector product and preconditioner application.

`KSPPIPECGRR` improves the robustness of `KSPPIPECG` by adding an automated residual replacement strategy.
True residual and other auxiliary variables are computed explicitly in a number of dynamically determined
iterations to counteract the accumulation of rounding errors and thus attain a higher maximal final accuracy.

See also `KSPPIPECG`, which is identical to `KSPPIPECGRR` without residual replacements.
See also `KSPPIPECR`, where the reduction is only overlapped with the matrix-vector product.

MPI configuration may be necessary for reductions to make asynchronous progress, which is important for
performance of pipelined methods. See [](doc_faq_pipelined)


## Contributed by
Siegfried Cools, Universiteit Antwerpen, Dept. Mathematics & Computer Science,
European FP7 Project on EXascale Algorithms and Advanced Computational Techniques (EXA2CT) / Research Foundation Flanders (FWO)


## Reference
S. Cools, E.F. Yetkin, E. Agullo, L. Giraud, W. Vanroose, "Analyzing the effect of local rounding error
propagation on the maximal attainable accuracy of the pipelined Conjugate Gradients method",
SIAM Journal on Matrix Analysis and Applications (SIMAX), 39(1):426--450, 2018.


## See Also
 [](ch_ksp), [](doc_faq_pipelined), [](sec_pipelineksp), `KSPCreate()`, `KSPSetType()`, `KSPPIPECR`, `KSPGROPPCG`, `KSPPIPECG`, `KSPPGMRES`, `KSPCG`, `KSPPIPEBCGS`, `KSPCGUseSingleReduction()`

## Level
intermediate

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


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


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