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, seeKSPCGUseSingleReduction()
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#
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.
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#
Examples#
src/ksp/ksp/tutorials/ex71.c
src/tao/pde_constrained/tutorials/parabolic.c
src/tao/bound/tutorials/jbearing2.c
src/ksp/ksp/tutorials/ex59.c
src/ksp/ksp/tutorials/ex78.c
src/ksp/pc/tutorials/ex2.c
src/ksp/pc/tutorials/ex1.c
src/snes/tutorials/ex11.c
Index of all KSP routines
Table of Contents for all manual pages
Index of all manual pages