petscmaster 20180525
KSPCG
The Preconditioned Conjugate Gradient (PCG) iterative method
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 MPIU_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 leftpreconditioned system BAx = Bb, using inv(B) to define an inner product in the algorithm.
(2) Solve a rightpreconditioned system ABy = b, x = By, using B to define an inner product in the algorithm.
(3) Solve a symmetricallypreconditioned 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 nonHermitian symmetric matrices. Use
KSPCGSetType() to indicate which type you are using.
Developer Notes
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
1. Magnus R. Hestenes and Eduard Stiefel, Methods of Conjugate Gradients for Solving Linear Systems,
Journal of Research of the National Bureau of Standards Vol. 49, No. 6, December 1952 Research Paper 2379
2. Josef Malek and Zdenek Strakos, Preconditioning and the Conjugate Gradient Method in the Context of Solving PDEs,
SIAM, 2014.
See Also
KSPCreate(), KSPSetType(), KSPType (for list of available types), KSP,
KSPCGSetType(), KSPCGUseSingleReduction(), KSPPIPECG, KSPGROPPCG
Level
beginner
Location
src/ksp/ksp/impls/cg/cg.c
Examples
src/ksp/pc/examples/tutorials/ex1.c.html
src/ksp/pc/examples/tutorials/ex2.c.html
src/tao/bound/examples/tutorials/jbearing2.c.html
src/tao/pde_constrained/examples/tutorials/parabolic.c.html
Index of all KSP routines
Table of Contents for all manual pages
Index of all manual pages