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;
 28:   PetscErrorCode     ierr;

 31:   PetscInitialize(&argc,&argv,0,help);if (ierr) return ierr;
 32:   comm = MPI_COMM_SELF;

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

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

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

 59:   MatSetValues(A,1,&i,1,&j,&v,INSERT_VALUES);
 60:   MatSetValues(A,1,&j,1,&i,&v,INSERT_VALUES);
 61:   MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
 62:   MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);
 63:   VecAssemblyBegin(B);
 64:   VecAssemblyEnd(B);
 65:   PetscPrintf(PETSC_COMM_WORLD,"\nThe Kershaw matrix:\n\n");
 66:   MatView(A,PETSC_VIEWER_STDOUT_WORLD);

 68:   /*
 69:    * A Conjugate Gradient method
 70:    * with ILU(0) preconditioning
 71:    */
 72:   KSPCreate(comm,&ksp);
 73:   KSPSetOperators(ksp,A,A);

 75:   KSPSetType(ksp,KSPCG);
 76:   KSPSetInitialGuessNonzero(ksp,PETSC_TRUE);

 78:   /*
 79:    * ILU preconditioner;
 80:    * The iterative method will break down unless you comment in the SetShift
 81:    * line below, or use the -pc_factor_shift_positive_definite option.
 82:    * Run the code twice: once as given to see the negative pivot and the
 83:    * divergence behaviour, then comment in the Shift line, or add the
 84:    * command line option, and see that the pivots are all positive and
 85:    * the method converges.
 86:    */
 87:   KSPGetPC(ksp,&pc);
 88:   PCSetType(pc,PCICC);
 89:   /* PCFactorSetShiftType(prec,MAT_SHIFT_POSITIVE_DEFINITE); */

 91:   KSPSetFromOptions(ksp);
 92:   KSPSetUp(ksp);

 94:   /*
 95:    * Now that the factorisation is done, show the pivots;
 96:    * note that the last one is negative. This in itself is not an error,
 97:    * but it will make the iterative method diverge.
 98:    */
 99:   PCFactorGetMatrix(pc,&M);
100:   VecDuplicate(B,&D);
101:   MatGetDiagonal(M,D);
102:   PetscPrintf(PETSC_COMM_WORLD,"\nPivots:\n\n");
103:   VecView(D,0);

105:   /*
106:    * Solve the system;
107:    * without the shift this will diverge with
108:    * an indefinite preconditioner
109:    */
110:   KSPSolve(ksp,B,X);
111:   KSPGetConvergedReason(ksp,&reason);
112:   if (reason==KSP_DIVERGED_INDEFINITE_PC) {
113:     PetscPrintf(PETSC_COMM_WORLD,"\nDivergence because of indefinite preconditioner;\n");
114:     PetscPrintf(PETSC_COMM_WORLD,"Run the executable again but with -pc_factor_shift_positive_definite option.\n");
115:   } else if (reason<0) {
116:     PetscPrintf(PETSC_COMM_WORLD,"\nOther kind of divergence: this should not happen.\n");
117:   } else {
118:     KSPGetIterationNumber(ksp,&its);
119:     PetscPrintf(PETSC_COMM_WORLD,"\nConvergence in %d iterations.\n",(int)its);
120:   }
121:   PetscPrintf(PETSC_COMM_WORLD,"\n");

123:   KSPDestroy(&ksp);
124:   MatDestroy(&A);
125:   VecDestroy(&B);
126:   VecDestroy(&X);
127:   VecDestroy(&D);
128:   PetscFinalize();
129:   return ierr;
130: }