1: static char help[] = "Test file for the PCFactorSetShiftType()\n";
  2: /*
  3:  * Test file for the PCFactorSetShiftType() routine or -pc_factor_shift_type POSITIVE_DEFINITE option.
  4:  * The test matrix is the example from Kershaw's paper [J.Comp.Phys 1978]
  5:  * of a positive definite matrix for which ILU(0) will give a negative pivot.
  6:  * This means that the CG method will break down; the Manteuffel shift
  7:  * [Math. Comp. 1980] repairs this.
  8:  *
  9:  * Run the executable twice:
 10:  * 1/ without options: the iterative method diverges because of an
 11:  *    indefinite preconditioner
 12:  * 2/ with -pc_factor_shift_positive_definite option (or comment in the PCFactorSetShiftType() line below):
 13:  *    the method will now successfully converge.
 14:  */

 16: #include <petscksp.h>

 18: int main(int argc, char **argv)
 19: {
 20:   KSP                ksp;
 21:   PC                 pc;
 22:   Mat                A, M;
 23:   Vec                X, B, D;
 24:   MPI_Comm           comm;
 25:   PetscScalar        v;
 26:   KSPConvergedReason reason;
 27:   PetscInt           i, j, its;

 29:   PetscFunctionBegin;
 30:   PetscFunctionBeginUser;
 31:   PetscCall(PetscInitialize(&argc, &argv, 0, help));
 32:   comm = MPI_COMM_SELF;

 34:   /*
 35:    * Construct the Kershaw matrix
 36:    * and a suitable rhs / initial guess
 37:    */
 38:   PetscCall(MatCreateSeqAIJ(comm, 4, 4, 4, 0, &A));
 39:   PetscCall(VecCreateSeq(comm, 4, &B));
 40:   PetscCall(VecDuplicate(B, &X));
 41:   for (i = 0; i < 4; i++) {
 42:     v = 3;
 43:     PetscCall(MatSetValues(A, 1, &i, 1, &i, &v, INSERT_VALUES));
 44:     v = 1;
 45:     PetscCall(VecSetValues(B, 1, &i, &v, INSERT_VALUES));
 46:     PetscCall(VecSetValues(X, 1, &i, &v, INSERT_VALUES));
 47:   }

 49:   i = 0;
 50:   v = 0;
 51:   PetscCall(VecSetValues(X, 1, &i, &v, INSERT_VALUES));

 53:   for (i = 0; i < 3; i++) {
 54:     v = -2;
 55:     j = i + 1;
 56:     PetscCall(MatSetValues(A, 1, &i, 1, &j, &v, INSERT_VALUES));
 57:     PetscCall(MatSetValues(A, 1, &j, 1, &i, &v, INSERT_VALUES));
 58:   }
 59:   i = 0;
 60:   j = 3;
 61:   v = 2;

 63:   PetscCall(MatSetValues(A, 1, &i, 1, &j, &v, INSERT_VALUES));
 64:   PetscCall(MatSetValues(A, 1, &j, 1, &i, &v, INSERT_VALUES));
 65:   PetscCall(MatAssemblyBegin(A, MAT_FINAL_ASSEMBLY));
 66:   PetscCall(MatAssemblyEnd(A, MAT_FINAL_ASSEMBLY));
 67:   PetscCall(VecAssemblyBegin(B));
 68:   PetscCall(VecAssemblyEnd(B));
 69:   PetscCall(PetscPrintf(PETSC_COMM_WORLD, "\nThe Kershaw matrix:\n\n"));
 70:   PetscCall(MatView(A, PETSC_VIEWER_STDOUT_WORLD));

 72:   /*
 73:    * A Conjugate Gradient method
 74:    * with ILU(0) preconditioning
 75:    */
 76:   PetscCall(KSPCreate(comm, &ksp));
 77:   PetscCall(KSPSetOperators(ksp, A, A));

 79:   PetscCall(KSPSetType(ksp, KSPCG));
 80:   PetscCall(KSPSetInitialGuessNonzero(ksp, PETSC_TRUE));

 82:   /*
 83:    * ILU preconditioner;
 84:    * The iterative method will break down unless you comment in the SetShift
 85:    * line below, or use the -pc_factor_shift_positive_definite option.
 86:    * Run the code twice: once as given to see the negative pivot and the
 87:    * divergence behaviour, then comment in the Shift line, or add the
 88:    * command line option, and see that the pivots are all positive and
 89:    * the method converges.
 90:    */
 91:   PetscCall(KSPGetPC(ksp, &pc));
 92:   PetscCall(PCSetType(pc, PCICC));
 93:   /* PetscCall(PCFactorSetShiftType(prec,MAT_SHIFT_POSITIVE_DEFINITE)); */

 95:   PetscCall(KSPSetFromOptions(ksp));
 96:   PetscCall(KSPSetUp(ksp));

 98:   /*
 99:    * Now that the factorisation is done, show the pivots;
100:    * note that the last one is negative. This in itself is not an error,
101:    * but it will make the iterative method diverge.
102:    */
103:   PetscCall(PCFactorGetMatrix(pc, &M));
104:   PetscCall(VecDuplicate(B, &D));
105:   PetscCall(MatGetDiagonal(M, D));
106:   PetscCall(PetscPrintf(PETSC_COMM_WORLD, "\nPivots:\n\n"));
107:   PetscCall(VecView(D, 0));

109:   /*
110:    * Solve the system;
111:    * without the shift this will diverge with
112:    * an indefinite preconditioner
113:    */
114:   PetscCall(KSPSolve(ksp, B, X));
115:   PetscCall(KSPGetConvergedReason(ksp, &reason));
116:   if (reason == KSP_DIVERGED_INDEFINITE_PC) {
117:     PetscCall(PetscPrintf(PETSC_COMM_WORLD, "\nDivergence because of indefinite preconditioner;\n"));
118:     PetscCall(PetscPrintf(PETSC_COMM_WORLD, "Run the executable again but with -pc_factor_shift_positive_definite option.\n"));
119:   } else if (reason < 0) {
120:     PetscCall(PetscPrintf(PETSC_COMM_WORLD, "\nOther kind of divergence: this should not happen.\n"));
121:   } else {
122:     PetscCall(KSPGetIterationNumber(ksp, &its));
123:     PetscCall(PetscPrintf(PETSC_COMM_WORLD, "\nConvergence in %" PetscInt_FMT " iterations.\n", its));
124:   }
125:   PetscCall(PetscPrintf(PETSC_COMM_WORLD, "\n"));

127:   PetscCall(KSPDestroy(&ksp));
128:   PetscCall(MatDestroy(&A));
129:   PetscCall(VecDestroy(&B));
130:   PetscCall(VecDestroy(&X));
131:   PetscCall(VecDestroy(&D));
132:   PetscCall(PetscFinalize());
133:   return 0;
134: }

136: /*TEST

138:    test:
139:      filter:  sed -e "s/in 5 iterations/in 4 iterations/g"

141: TEST*/