/*
 * time_step solves for the time_dependence of the system
 * that was previously setup using the add_to_ham and add_lin
 * routines. Solver selection and parameters can be controlled via PETSc
 * command line options. Default solver is TSRK3BS
 *
 * Inputs:
 *       Vec     x:       The density matrix, with appropriate inital conditions
 *       double dt:       initial timestep. For certain explicit methods, this timestep
 *                        can be changed, as those methods have adaptive time steps
 *       double time_max: the maximum time to integrate to
 *       int steps_max:   max number of steps to take
 */
void time_step(Vec x, PetscReal init_time, PetscReal time_max,PetscReal dt,PetscInt steps_max){
  PetscViewer    mat_view;
  TS             ts; /* timestepping context */
  PetscInt       i,j,Istart,Iend,steps,row,col;
  PetscScalar    mat_tmp;
  PetscReal      tmp_real;
  Mat            AA;
  PetscInt       nevents,direction;
  PetscBool      terminate;
  operator       op;
  int            num_pop;
  double         *populations;
  Mat            solve_A,solve_stiff_A;


  PetscLogStagePop();
  PetscLogStagePush(solve_stage);
  if (_lindblad_terms) {
    if (nid==0) {
      printf("Lindblad terms found, using Lindblad solver.\n");
    }
    solve_A = full_A;
    if (_stiff_solver) {
      if(nid==0) printf("ERROR! Lindblad-stiff solver untested.");
      exit(0);
    }
  } else {
    if (nid==0) {
      printf("No Lindblad terms found, using (more efficient) Schrodinger solver.\n");
    }
    solve_A = ham_A;
    solve_stiff_A = ham_stiff_A;
    if (_num_time_dep&&_stiff_solver) {
      if(nid==0) printf("ERROR! Schrodinger-stiff + timedep solver untested.");
      exit(0);
    }
  }

  /* Possibly print dense ham. No stabilization is needed? */
  if (nid==0) {
    /* Print dense ham, if it was asked for */
    if (_print_dense_ham){
      FILE *fp_ham;
      fp_ham = fopen("ham","w");

      if (nid==0){
        for (i=0;i<total_levels;i++){
          for (j=0;j<total_levels;j++){
            fprintf(fp_ham,"%e %e ",PetscRealPart(_hamiltonian[i][j]),PetscImaginaryPart(_hamiltonian[i][j]));
          }
          fprintf(fp_ham,"\n");
        }
      }
      fclose(fp_ham);
      for (i=0;i<total_levels;i++){
        free(_hamiltonian[i]);
      }
      free(_hamiltonian);
      _print_dense_ham = 0;
    }
  }


  /* Remove stabilization if it was previously added */
  if (stab_added){
    if (nid==0) printf("Removing stabilization...\n");
    /*
     * We add 1.0 in the 0th spot and every n+1 after
     */
    if (nid==0) {
      row = 0;
      for (i=0;i<total_levels;i++){
        col = i*(total_levels+1);
        mat_tmp = -1.0 + 0.*PETSC_i;
        MatSetValue(full_A,row,col,mat_tmp,ADD_VALUES);
      }
    }
  }

  MatGetOwnershipRange(solve_A,&Istart,&Iend);
  /*
   * Explicitly add 0.0 to all diagonal elements;
   * this fixes a 'matrix in wrong state' message that PETSc
   * gives if the diagonal was never initialized.
   */
  //if (nid==0) printf("Adding 0 to diagonal elements...\n");
  for (i=Istart;i<Iend;i++){
//.........这里部分代码省略.........

//.........这里部分代码省略.........
  ierr = ISRestoreIndices(iscols,&cols);CHKERRQ(ierr);
  ierr = ISDestroy(&isrows);CHKERRQ(ierr);
  ierr = ISDestroy(&iscols);CHKERRQ(ierr);
  ierr = PetscRandomDestroy(&r);CHKERRQ(ierr);

  /* Test MatTranspose() */
  ierr = MatCreateTranspose(A,&C);CHKERRQ(ierr);
  ierr = MatTranspose(A,MAT_INITIAL_MATRIX,&B);CHKERRQ(ierr); /* B = A^T */
  ierr = MatMultEqual(C,B,10,&equal);CHKERRQ(ierr);
  if (!equal) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"A^T*x != (x^T*A)^T");
  ierr = MatTranspose(A,MAT_REUSE_MATRIX,&B);CHKERRQ(ierr); /* B = A^T */
  ierr = MatMultEqual(C,B,10,&equal);CHKERRQ(ierr);
  if (!equal) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"A^T*x != (x^T*A)^T");
  ierr = MatDestroy(&B);CHKERRQ(ierr);
  ierr = MatDuplicate(A,MAT_COPY_VALUES,&B);CHKERRQ(ierr);
  ierr = MatTranspose(B,MAT_INPLACE_MATRIX,&B);CHKERRQ(ierr);
  ierr = MatMultEqual(C,B,10,&equal);CHKERRQ(ierr);
  if (!equal) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"A^T*x != (x^T*A)^T");
  ierr = MatDestroy(&B);CHKERRQ(ierr);
  ierr = MatDestroy(&C);CHKERRQ(ierr);

  /* Test MatMatMult() */
  if (Test_MatMatMult) {
#if !defined(PETSC_HAVE_ELEMENTAL)
    if (size > 1) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"This test requires ELEMENTAL");
#endif
    ierr = MatTranspose(A,MAT_INITIAL_MATRIX,&B);CHKERRQ(ierr); /* B = A^T */
    ierr = MatMatMult(B,A,MAT_INITIAL_MATRIX,fill,&C);CHKERRQ(ierr); /* C = B*A = A^T*A */
    ierr = MatMatMult(B,A,MAT_REUSE_MATRIX,fill,&C);CHKERRQ(ierr);

    /* Test MatDuplicate for matrix product */
    ierr = MatDuplicate(C,MAT_COPY_VALUES,&D);CHKERRQ(ierr);
    ierr = MatDestroy(&D);CHKERRQ(ierr);

    /* Test B*A*x = C*x for n random vector x */
    ierr = MatMatMultEqual(B,A,C,10,&equal);CHKERRQ(ierr);
    if (!equal) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"B*A*x != C*x");
    ierr = MatDestroy(&C);CHKERRQ(ierr);

    ierr = MatMatMultSymbolic(B,A,fill,&C);CHKERRQ(ierr);
    for (i=0; i<2; i++) {
      /* Repeat the numeric product to test reuse of the previous symbolic product */
      ierr = MatMatMultNumeric(B,A,C);CHKERRQ(ierr);

      ierr = MatMatMultEqual(B,A,C,10,&equal);CHKERRQ(ierr);
      if (!equal) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"B*A*x != C*x");
    }
    ierr = MatDestroy(&C);CHKERRQ(ierr);
    ierr = MatDestroy(&B);CHKERRQ(ierr);
  }

  /* Test MatTransposeMatMult() */
  ierr = PetscObjectTypeCompare((PetscObject)A,MATELEMENTAL,&iselemental);CHKERRQ(ierr);
  if (!iselemental) {
    ierr = MatTransposeMatMult(A,A,MAT_INITIAL_MATRIX,fill,&D);CHKERRQ(ierr); /* D = A^T*A */
    ierr = MatTransposeMatMult(A,A,MAT_REUSE_MATRIX,fill,&D);CHKERRQ(ierr);

    /* Test MatDuplicate for matrix product */
    ierr = MatDuplicate(D,MAT_COPY_VALUES,&C);CHKERRQ(ierr);
    ierr = MatDestroy(&C);CHKERRQ(ierr);

    /* ierr = MatView(D,PETSC_VIEWER_STDOUT_WORLD);CHKERRQ(ierr); */
    ierr = MatTransposeMatMultEqual(A,A,D,10,&equal);CHKERRQ(ierr);
    if (!equal) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"D*x != A^T*A*x");
    ierr = MatDestroy(&D);CHKERRQ(ierr);

    /* Test D*x = A^T*C*A*x, where C is in AIJ format */
    ierr = MatGetLocalSize(A,&am,&an);CHKERRQ(ierr);
    ierr = MatCreate(PETSC_COMM_WORLD,&C);CHKERRQ(ierr);
    if (size == 1) {
      ierr = MatSetSizes(C,PETSC_DECIDE,PETSC_DECIDE,am,am);CHKERRQ(ierr);
    } else {
      ierr = MatSetSizes(C,am,am,PETSC_DECIDE,PETSC_DECIDE);CHKERRQ(ierr);
    }
    ierr = MatSetFromOptions(C);CHKERRQ(ierr);
    ierr = MatSetUp(C);CHKERRQ(ierr);
    ierr = MatGetOwnershipRange(C,&rstart,&rend);CHKERRQ(ierr);
    v[0] = 1.0;
    for (i=rstart; i<rend; i++) {
      ierr = MatSetValues(C,1,&i,1,&i,v,INSERT_VALUES);CHKERRQ(ierr);
    }
    ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
    ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);

    /* B = C*A, D = A^T*B */
    ierr = MatMatMult(C,A,MAT_INITIAL_MATRIX,1.0,&B);CHKERRQ(ierr);
    ierr = MatTransposeMatMult(A,B,MAT_INITIAL_MATRIX,fill,&D);CHKERRQ(ierr);
    ierr = MatTransposeMatMultEqual(A,B,D,10,&equal);CHKERRQ(ierr);
    if (!equal) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"D*x != A^T*B*x");

    ierr = MatDestroy(&D);CHKERRQ(ierr);
    ierr = MatDestroy(&C);CHKERRQ(ierr);
    ierr = MatDestroy(&B);CHKERRQ(ierr);
  }

  ierr = MatDestroy(&A);CHKERRQ(ierr);
  ierr = PetscFree(v);CHKERRQ(ierr);
  ierr = PetscFinalize();
  return ierr;
}

void
construct_operator(Elliptic1D* p, int levels) {
    // Start out with a simple tridiagonal matrix.
 
    // problem solves laplacian u = -1 on -1 to 1 with homogeneous
    // dirichlet boundary conditions. Code
    // implements a finite difference to approximate this
    // equation system.

    p->levels = levels;
    p->npoints = (2<<levels)-1;
    p->num_seg = 2<<levels;
    PetscScalar h = 2./p->num_seg;

    MatCreateMPIAIJ(PETSC_COMM_WORLD,
		    PETSC_DECIDE, //number of local rows
		    PETSC_DECIDE, //number of local cols
		    p->npoints, //global rows
		    p->npoints, //global cols
		    3,         // upper bound of diagonal nnz per row
		    PETSC_NULL, // array of diagonal nnz per row
		    1,         // upper bound of off-processor nnz per row
		    PETSC_NULL, // array of off-processor nnz per row
		    &p->A); // matrix

    MatSetFromOptions(p->A);

    PetscInt start;
    PetscInt end;
    MatGetOwnershipRange(p->A, &start, &end);
    int ii;
    for (ii=start; ii<end; ii++) {
	PetscInt col_index[3] = {ii-1, ii, ii+1};
	PetscInt row_index = ii;
	PetscScalar stencil[3] = {-1./(h*h), 2./(h*h), -1./(h*h)};

	// handle corner cases at beginning and end of matrix.
	if (ii+1 == p->npoints) {
	    col_index[2] = -1;
	} else if (ii == 0) {
	    col_index[0] = -1;
	}

	MatSetValues(p->A, 1, &row_index, 3, col_index, stencil, INSERT_VALUES);
    }
    MatAssemblyBegin(p->A, MAT_FINAL_ASSEMBLY);
    MatAssemblyEnd(p->A, MAT_FINAL_ASSEMBLY);

    //Create the corresponding vectors
    VecCreateMPI(PETSC_COMM_WORLD,
		 PETSC_DECIDE,
		 p->npoints,
		 &p->x
		 );
    VecSetFromOptions(p->x);
    VecDuplicate(p->x, &p->b);
    VecSet(p->b, 1);
    VecZeroEntries(p->x);
    
    //Fill in a random initial guess
    PetscInt high;
    PetscInt low;
    VecGetOwnershipRange(p->x, &low, &high);
    for (ii=low; ii<high; ii++) {
	VecSetValue(p->x, ii, frand(), INSERT_VALUES);
    }
    VecAssemblyBegin(p->x);
    VecAssemblyEnd(p->x);
}

/*@
   KSPComputeEigenvaluesExplicitly - Computes all of the eigenvalues of the
   preconditioned operator using LAPACK.

   Collective on KSP

   Input Parameter:
+  ksp - iterative context obtained from KSPCreate()
-  n - size of arrays r and c

   Output Parameters:
+  r - real part of computed eigenvalues
-  c - complex part of computed eigenvalues

   Notes:
   This approach is very slow but will generally provide accurate eigenvalue
   estimates.  This routine explicitly forms a dense matrix representing
   the preconditioned operator, and thus will run only for relatively small
   problems, say n < 500.

   Many users may just want to use the monitoring routine
   KSPMonitorSingularValue() (which can be set with option -ksp_monitor_singular_value)
   to print the singular values at each iteration of the linear solve.

   The preconditoner operator, rhs vector, solution vectors should be
   set before this routine is called. i.e use KSPSetOperators(),KSPSolve() or
   KSPSetOperators()

   Level: advanced

.keywords: KSP, compute, eigenvalues, explicitly

.seealso: KSPComputeEigenvalues(), KSPMonitorSingularValue(), KSPComputeExtremeSingularValues(), KSPSetOperators(), KSPSolve()
@*/
PetscErrorCode  KSPComputeEigenvaluesExplicitly(KSP ksp,PetscInt nmax,PetscReal *r,PetscReal *c)
{
  Mat                BA;
  PetscErrorCode     ierr;
  PetscMPIInt        size,rank;
  MPI_Comm           comm = ((PetscObject)ksp)->comm;
  PetscScalar        *array;
  Mat                A;
  PetscInt           m,row,nz,i,n,dummy;
  const PetscInt     *cols;
  const PetscScalar  *vals;

  PetscFunctionBegin;
  ierr = KSPComputeExplicitOperator(ksp,&BA);CHKERRQ(ierr);
  ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
  ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);

  ierr = MatGetSize(BA,&n,&n);CHKERRQ(ierr);
  if (size > 1) { /* assemble matrix on first processor */
    ierr = MatCreate(((PetscObject)ksp)->comm,&A);CHKERRQ(ierr);
    if (!rank) {
      ierr = MatSetSizes(A,n,n,n,n);CHKERRQ(ierr);
    } else {
      ierr = MatSetSizes(A,0,0,n,n);CHKERRQ(ierr);
    }
    ierr = MatSetType(A,MATMPIDENSE);CHKERRQ(ierr);
    ierr = MatMPIDenseSetPreallocation(A,PETSC_NULL);CHKERRQ(ierr);
    ierr = PetscLogObjectParent(BA,A);CHKERRQ(ierr);

    ierr = MatGetOwnershipRange(BA,&row,&dummy);CHKERRQ(ierr);
    ierr = MatGetLocalSize(BA,&m,&dummy);CHKERRQ(ierr);
    for (i=0; i<m; i++) {
      ierr = MatGetRow(BA,row,&nz,&cols,&vals);CHKERRQ(ierr);
      ierr = MatSetValues(A,1,&row,nz,cols,vals,INSERT_VALUES);CHKERRQ(ierr);
      ierr = MatRestoreRow(BA,row,&nz,&cols,&vals);CHKERRQ(ierr);
      row++;
    }

    ierr = MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
    ierr = MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
    ierr = MatDenseGetArray(A,&array);CHKERRQ(ierr);
  } else {
    ierr = MatDenseGetArray(BA,&array);CHKERRQ(ierr);
  }

#if defined(PETSC_HAVE_ESSL)
  /* ESSL has a different calling sequence for dgeev() and zgeev() than standard LAPACK */
  if (!rank) {
    PetscScalar  sdummy,*cwork;
    PetscReal    *work,*realpart;
    PetscBLASInt clen,idummy,lwork,bn,zero = 0;
    PetscInt *perm;

#if !defined(PETSC_USE_COMPLEX)
    clen = n;
#else
    clen = 2*n;
#endif
    ierr   = PetscMalloc(clen*sizeof(PetscScalar),&cwork);CHKERRQ(ierr);
    idummy = -1;                /* unused */
    bn = PetscBLASIntCast(n);
    lwork  = 5*n;
    ierr   = PetscMalloc(lwork*sizeof(PetscReal),&work);CHKERRQ(ierr);
    ierr   = PetscMalloc(n*sizeof(PetscReal),&realpart);CHKERRQ(ierr);
    ierr = PetscFPTrapPush(PETSC_FP_TRAP_OFF);CHKERRQ(ierr);
    LAPACKgeev_(&zero,array,&bn,cwork,&sdummy,&idummy,&idummy,&bn,work,&lwork);
//.........这里部分代码省略.........

int main( int argc, char **argv )
{
  Mat         	 A;		  /* operator matrix */
  Vec         	 x;
  EPS         	 eps;		  /* eigenproblem solver context */
  const EPSType  type;
  PetscReal   	 error, tol, re, im;
  PetscScalar 	 kr, ki;
  PetscErrorCode ierr;
  PetscInt    	 N, n=10, m, i, j, II, Istart, Iend, nev, maxit, its, nconv;
  PetscScalar 	 w;
  PetscBool   	 flag;

  SlepcInitialize(&argc,&argv,(char*)0,help);

  ierr = PetscOptionsGetInt(PETSC_NULL,"-n",&n,PETSC_NULL);CHKERRQ(ierr);
  ierr = PetscOptionsGetInt(PETSC_NULL,"-m",&m,&flag);CHKERRQ(ierr);
  if(!flag) m=n;
  N = n*m;
  ierr = PetscPrintf(PETSC_COMM_WORLD,"\nFiedler vector of a 2-D regular mesh, N=%d (%dx%d grid)\n\n",N,n,m);CHKERRQ(ierr);

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 
     Compute the operator matrix that defines the eigensystem, Ax=kx
     In this example, A = L(G), where L is the Laplacian of graph G, i.e.
     Lii = degree of node i, Lij = -1 if edge (i,j) exists in G
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

  ierr = MatCreate(PETSC_COMM_WORLD,&A);CHKERRQ(ierr);
  ierr = MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,N,N);CHKERRQ(ierr);
  ierr = MatSetFromOptions(A);CHKERRQ(ierr);
  
  ierr = MatGetOwnershipRange(A,&Istart,&Iend);CHKERRQ(ierr);
  for( II=Istart; II<Iend; II++ ) { 
    i = II/n; j = II-i*n;
    w = 0.0;
    if(i>0) { ierr = MatSetValue(A,II,II-n,-1.0,INSERT_VALUES);CHKERRQ(ierr); w=w+1.0; }
    if(i<m-1) { ierr = MatSetValue(A,II,II+n,-1.0,INSERT_VALUES);CHKERRQ(ierr); w=w+1.0; }
    if(j>0) { ierr = MatSetValue(A,II,II-1,-1.0,INSERT_VALUES);CHKERRQ(ierr); w=w+1.0; }
    if(j<n-1) { ierr = MatSetValue(A,II,II+1,-1.0,INSERT_VALUES);CHKERRQ(ierr); w=w+1.0; }
    ierr = MatSetValue(A,II,II,w,INSERT_VALUES);CHKERRQ(ierr);
  }

  ierr = MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 
                Create the eigensolver and set various options
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

  /* 
     Create eigensolver context
  */
  ierr = EPSCreate(PETSC_COMM_WORLD,&eps);CHKERRQ(ierr);

  /* 
     Set operators. In this case, it is a standard eigenvalue problem
  */
  ierr = EPSSetOperators(eps,A,PETSC_NULL);CHKERRQ(ierr);
  ierr = EPSSetProblemType(eps,EPS_HEP);CHKERRQ(ierr);
  
  /*
     Select portion of spectrum
  */
  ierr = EPSSetWhichEigenpairs(eps,EPS_SMALLEST_REAL);CHKERRQ(ierr);

  /*
     Set solver parameters at runtime
  */
  ierr = EPSSetFromOptions(eps);CHKERRQ(ierr);

  /*
     Attach deflation space: in this case, the matrix has a constant 
     nullspace, [1 1 ... 1]^T is the eigenvector of the zero eigenvalue
  */
  ierr = MatGetVecs(A,&x,PETSC_NULL);CHKERRQ(ierr);
  ierr = VecSet(x,1.0);CHKERRQ(ierr);
  ierr = EPSSetDeflationSpace(eps,1,&x);CHKERRQ(ierr);
  ierr = VecDestroy(x);

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 
                      Solve the eigensystem
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

  ierr = EPSSolve(eps);CHKERRQ(ierr);
  ierr = EPSGetIterationNumber(eps, &its);CHKERRQ(ierr);
  ierr = PetscPrintf(PETSC_COMM_WORLD," Number of iterations of the method: %d\n",its);CHKERRQ(ierr);

  /*
     Optional: Get some information from the solver and display it
  */
  ierr = EPSGetType(eps,&type);CHKERRQ(ierr);
  ierr = PetscPrintf(PETSC_COMM_WORLD," Solution method: %s\n\n",type);CHKERRQ(ierr);
  ierr = EPSGetDimensions(eps,&nev,PETSC_NULL,PETSC_NULL);CHKERRQ(ierr);
  ierr = PetscPrintf(PETSC_COMM_WORLD," Number of requested eigenvalues: %d\n",nev);CHKERRQ(ierr);
  ierr = EPSGetTolerances(eps,&tol,&maxit);CHKERRQ(ierr);
  ierr = PetscPrintf(PETSC_COMM_WORLD," Stopping condition: tol=%.4g, maxit=%d\n",tol,maxit);CHKERRQ(ierr);

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 
                    Display solution and clean up
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
//.........这里部分代码省略.........

int main(int argc,char **argv)
{
  PetscErrorCode ierr;
  PetscInt       time_steps=100,iout,NOUT=1;
  PetscMPIInt    size;
  Vec            global;
  PetscReal      dt,ftime,ftime_original;
  TS             ts;
  PetscViewer    viewfile;
  Mat            J = 0;
  Vec            x;
  Data           data;
  PetscInt       mn;
  PetscBool      flg;
  MatColoring    mc;
  ISColoring     iscoloring;
  MatFDColoring  matfdcoloring        = 0;
  PetscBool      fd_jacobian_coloring = PETSC_FALSE;
  SNES           snes;
  KSP            ksp;
  PC             pc;
  PetscViewer    viewer;
  char           pcinfo[120],tsinfo[120];
  TSType         tstype;
  PetscBool      sundials;

  ierr = PetscInitialize(&argc,&argv,(char*)0,help);CHKERRQ(ierr);
  ierr = MPI_Comm_size(PETSC_COMM_WORLD,&size);CHKERRQ(ierr);

  /* set data */
  data.m       = 9;
  data.n       = 9;
  data.a       = 1.0;
  data.epsilon = 0.1;
  data.dx      = 1.0/(data.m+1.0);
  data.dy      = 1.0/(data.n+1.0);
  mn           = (data.m)*(data.n);
  ierr         = PetscOptionsGetInt(NULL,"-time",&time_steps,NULL);CHKERRQ(ierr);

  /* set initial conditions */
  ierr = VecCreate(PETSC_COMM_WORLD,&global);CHKERRQ(ierr);
  ierr = VecSetSizes(global,PETSC_DECIDE,mn);CHKERRQ(ierr);
  ierr = VecSetFromOptions(global);CHKERRQ(ierr);
  ierr = Initial(global,&data);CHKERRQ(ierr);
  ierr = VecDuplicate(global,&x);CHKERRQ(ierr);

  /* create timestep context */
  ierr = TSCreate(PETSC_COMM_WORLD,&ts);CHKERRQ(ierr);
  ierr = TSMonitorSet(ts,Monitor,&data,NULL);CHKERRQ(ierr);
#if defined(PETSC_HAVE_SUNDIALS)
  ierr = TSSetType(ts,TSSUNDIALS);CHKERRQ(ierr);
#else
  ierr = TSSetType(ts,TSEULER);CHKERRQ(ierr);
#endif
  dt             = 0.1;
  ftime_original = data.tfinal = 1.0;

  ierr = TSSetInitialTimeStep(ts,0.0,dt);CHKERRQ(ierr);
  ierr = TSSetDuration(ts,time_steps,ftime_original);CHKERRQ(ierr);
  ierr = TSSetSolution(ts,global);CHKERRQ(ierr);

  /* set user provided RHSFunction and RHSJacobian */
  ierr = TSSetRHSFunction(ts,NULL,RHSFunction,&data);CHKERRQ(ierr);
  ierr = MatCreate(PETSC_COMM_WORLD,&J);CHKERRQ(ierr);
  ierr = MatSetSizes(J,PETSC_DECIDE,PETSC_DECIDE,mn,mn);CHKERRQ(ierr);
  ierr = MatSetFromOptions(J);CHKERRQ(ierr);
  ierr = MatSeqAIJSetPreallocation(J,5,NULL);CHKERRQ(ierr);
  ierr = MatMPIAIJSetPreallocation(J,5,NULL,5,NULL);CHKERRQ(ierr);

  ierr = PetscOptionsHasName(NULL,"-ts_fd",&flg);CHKERRQ(ierr);
  if (!flg) {
    ierr = TSSetRHSJacobian(ts,J,J,RHSJacobian,&data);CHKERRQ(ierr);
  } else {
    ierr = TSGetSNES(ts,&snes);CHKERRQ(ierr);
    ierr = PetscOptionsHasName(NULL,"-fd_color",&fd_jacobian_coloring);CHKERRQ(ierr);
    if (fd_jacobian_coloring) { /* Use finite differences with coloring */
      /* Get data structure of J */
      PetscBool pc_diagonal;
      ierr = PetscOptionsHasName(NULL,"-pc_diagonal",&pc_diagonal);CHKERRQ(ierr);
      if (pc_diagonal) { /* the preconditioner of J is a diagonal matrix */
        PetscInt    rstart,rend,i;
        PetscScalar zero=0.0;
        ierr = MatGetOwnershipRange(J,&rstart,&rend);CHKERRQ(ierr);
        for (i=rstart; i<rend; i++) {
          ierr = MatSetValues(J,1,&i,1,&i,&zero,INSERT_VALUES);CHKERRQ(ierr);
        }
        ierr = MatAssemblyBegin(J,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
        ierr = MatAssemblyEnd(J,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
      } else {
        /* Fill the structure using the expensive SNESComputeJacobianDefault. Temporarily set up the TS so we can call this function */
        ierr = TSSetType(ts,TSBEULER);CHKERRQ(ierr);
        ierr = TSSetUp(ts);CHKERRQ(ierr);
        ierr = SNESComputeJacobianDefault(snes,x,J,J,ts);CHKERRQ(ierr);
      }

      /* create coloring context */
      ierr = MatColoringCreate(J,&mc);CHKERRQ(ierr);
      ierr = MatColoringSetType(mc,MATCOLORINGSL);CHKERRQ(ierr);
      ierr = MatColoringSetFromOptions(mc);CHKERRQ(ierr);
      ierr = MatColoringApply(mc,&iscoloring);CHKERRQ(ierr);
//.........这里部分代码省略.........

    /*
     * Test that the matrix is calculated correctly on the cannonical triangle.
     * Tests against the analytical solution calculated by hand.
     */
    void TestAssembler()  throw(Exception)
    {
        QuadraticMesh<2> mesh;
        TrianglesMeshReader<2,2> mesh_reader("mesh/test/data/canonical_triangle_quadratic", 2, 2, false);
        mesh.ConstructFromMeshReader(mesh_reader);

        double mu = 2.0;
        c_vector<double,2> body_force;
        double g1 = 1.34254;
        double g2 = 75.3422;
        body_force(0) = g1;
        body_force(1) = g2;

        StokesFlowProblemDefinition<2> problem_defn(mesh);
        problem_defn.SetViscosity(mu);
        problem_defn.SetBodyForce(body_force);

        StokesFlowAssembler<2> assembler(&mesh, &problem_defn);

        // The tests below test the assembler against hand-calculated variables for
        // an OLD weak form (corresponding to different boundary conditions), not the
        // current Stokes weak form. This factor converts the assembler to use the old
        // weak form. See documentation for this variable for more details.
        assembler.mScaleFactor = 0.0;

        Vec vec = PetscTools::CreateVec(18);
        Mat mat;
        PetscTools::SetupMat(mat, 18, 18, 18);

        assembler.SetVectorToAssemble(vec, true);
        assembler.SetMatrixToAssemble(mat, true);
        assembler.Assemble();
        PetscMatTools::Finalise(mat);

        double A[6][6] = {
            {      1.0,  1.0/6.0,  1.0/6.0,      0.0, -2.0/3.0, -2.0/3.0},
            {  1.0/6.0,  1.0/2.0,      0.0,      0.0,      0.0, -2.0/3.0},
            {  1.0/6.0,      0.0,  1.0/2.0,      0.0, -2.0/3.0,      0.0},
            {      0.0,      0.0,      0.0,  8.0/3.0, -4.0/3.0, -4.0/3.0},
            { -2.0/3.0,      0.0, -2.0/3.0, -4.0/3.0,  8.0/3.0,      0.0},
            { -2.0/3.0, -2.0/3.0,      0.0, -4.0/3.0,      0.0,  8.0/3.0}
        };

        double Bx[6][3] = {
            { -1.0/6.0,      0.0,      0.0},
            {      0.0,  1.0/6.0,      0.0},
            {      0.0,      0.0,      0.0},
            {  1.0/6.0,  1.0/6.0,  1.0/3.0},
            { -1.0/6.0, -1.0/6.0, -1.0/3.0},
            {  1.0/6.0, -1.0/6.0,      0.0},
        };

        double By[6][3] = {
            { -1.0/6.0,      0.0,      0.0},
            {      0.0,      0.0,      0.0},
            {      0.0,      0.0,  1.0/6.0},
            {  1.0/6.0,  1.0/3.0,  1.0/6.0},
            {  1.0/6.0,      0.0, -1.0/6.0},
            { -1.0/6.0, -1.0/3.0, -1.0/6.0},
        };

        c_matrix<double,18,18> exact_A = zero_matrix<double>(18);

        // The diagonal 6x6 blocks
        for (unsigned i=0; i<6; i++)
        {
            for (unsigned j=0; j<6; j++)
            {
                exact_A(3*i,  3*j)   = mu*A[i][j];
                exact_A(3*i+1,3*j+1) = mu*A[i][j];
            }
        }


        // The 6x3 Blocks
        for (unsigned i=0; i<6; i++)
        {
            for (unsigned j=0; j<3; j++)
            {
                exact_A(3*i,3*j+2)   = -Bx[i][j];
                exact_A(3*i+1,3*j+2) = -By[i][j];
                //- as -Div(U)=0
                exact_A(3*j+2,3*i)   = -Bx[i][j];
                exact_A(3*j+2,3*i+1) = -By[i][j];
            }
        }

        int lo, hi;
        MatGetOwnershipRange(mat, &lo, &hi);
        for (unsigned i=lo; i<(unsigned)hi; i++)
        {
            for (unsigned j=0; j<18; j++)
            {
                TS_ASSERT_DELTA(PetscMatTools::GetElement(mat,i,j), exact_A(i,j), 1e-9);
            }
        }
//.........这里部分代码省略.........

PetscErrorCode FormTestMatrix(Mat A,PetscInt n,TestType type)
{
#if !defined(PETSC_USE_COMPLEX)
  SETERRQ(((PetscObject)A)->comm,1,"FormTestMatrix: These problems require complex numbers.");
#else

  PetscScalar    val[5];
  PetscErrorCode ierr;
  PetscInt       i,j,Ii,J,col[5],Istart,Iend;

  ierr = MatGetOwnershipRange(A,&Istart,&Iend);CHKERRQ(ierr);
  if (type == TEST_1) {
    val[0] = 1.0; val[1] = 4.0; val[2] = -2.0;
    for (i=1; i<n-1; i++) {
      col[0] = i-1; col[1] = i; col[2] = i+1;
      ierr = MatSetValues(A,1,&i,3,col,val,INSERT_VALUES);CHKERRQ(ierr);
    }
    i = n-1; col[0] = n-2; col[1] = n-1;
    ierr = MatSetValues(A,1,&i,2,col,val,INSERT_VALUES);CHKERRQ(ierr);
    i = 0; col[0] = 0; col[1] = 1; val[0] = 4.0; val[1] = -2.0;
    ierr = MatSetValues(A,1,&i,2,col,val,INSERT_VALUES);CHKERRQ(ierr);
  }
  else if (type == TEST_2) {
    val[0] = 1.0; val[1] = 0.0; val[2] = 2.0; val[3] = 1.0;
    for (i=2; i<n-1; i++) {
      col[0] = i-2; col[1] = i-1; col[2] = i; col[3] = i+1;
      ierr = MatSetValues(A,1,&i,4,col,val,INSERT_VALUES);CHKERRQ(ierr);
    }
    i = n-1; col[0] = n-3; col[1] = n-2; col[2] = n-1;
    ierr = MatSetValues(A,1,&i,3,col,val,INSERT_VALUES);CHKERRQ(ierr);
    i = 1; col[0] = 0; col[1] = 1; col[2] = 2;
    ierr = MatSetValues(A,1,&i,3,col,&val[1],INSERT_VALUES);CHKERRQ(ierr);
    i = 0;
    ierr = MatSetValues(A,1,&i,2,col,&val[2],INSERT_VALUES);CHKERRQ(ierr);
  }
  else if (type == TEST_3) {
    val[0] = PETSC_i * 2.0;
    val[1] = 4.0; val[2] = 0.0; val[3] = 1.0; val[4] = 0.7;
    for (i=1; i<n-3; i++) {
      col[0] = i-1; col[1] = i; col[2] = i+1; col[3] = i+2; col[4] = i+3;
      ierr = MatSetValues(A,1,&i,5,col,val,INSERT_VALUES);CHKERRQ(ierr);
    }
    i = n-3; col[0] = n-4; col[1] = n-3; col[2] = n-2; col[3] = n-1;
    ierr = MatSetValues(A,1,&i,4,col,val,INSERT_VALUES);CHKERRQ(ierr);
    i = n-2; col[0] = n-3; col[1] = n-2; col[2] = n-1;
    ierr = MatSetValues(A,1,&i,3,col,val,INSERT_VALUES);CHKERRQ(ierr);
    i = n-1; col[0] = n-2; col[1] = n-1;
    ierr = MatSetValues(A,1,&i,2,col,val,INSERT_VALUES);CHKERRQ(ierr);
    i = 0; col[0] = 0; col[1] = 1; col[2] = 2; col[3] = 3;
    ierr = MatSetValues(A,1,&i,4,col,&val[1],INSERT_VALUES);CHKERRQ(ierr);
  }
  else if (type == HELMHOLTZ_1) {
    /* Problem domain: unit square: (0,1) x (0,1)
       Solve Helmholtz equation:
          -delta u - sigma1*u + i*sigma2*u = f,
           where delta = Laplace operator
       Dirichlet b.c.'s on all sides
     */
    PetscRandom rctx;
    PetscReal   h2,sigma1 = 5.0;
    PetscScalar sigma2;
    ierr = PetscOptionsGetReal(PETSC_NULL,"-sigma1",&sigma1,PETSC_NULL);CHKERRQ(ierr);
    ierr = PetscRandomCreate(PETSC_COMM_WORLD,&rctx);CHKERRQ(ierr);
    ierr = PetscRandomSetFromOptions(rctx);CHKERRQ(ierr);
    ierr = PetscRandomSetInterval(rctx,0.0,PETSC_i);CHKERRQ(ierr);
    h2 = 1.0/((n+1)*(n+1));
    for (Ii=Istart; Ii<Iend; Ii++) {
      *val = -1.0; i = Ii/n; j = Ii - i*n;
      if (i>0) {
        J = Ii-n; ierr = MatSetValues(A,1,&Ii,1,&J,val,ADD_VALUES);CHKERRQ(ierr);}
      if (i<n-1) {
        J = Ii+n; ierr = MatSetValues(A,1,&Ii,1,&J,val,ADD_VALUES);CHKERRQ(ierr);}
      if (j>0) {
        J = Ii-1; ierr = MatSetValues(A,1,&Ii,1,&J,val,ADD_VALUES);CHKERRQ(ierr);}
      if (j<n-1) {
        J = Ii+1; ierr = MatSetValues(A,1,&Ii,1,&J,val,ADD_VALUES);CHKERRQ(ierr);}
      ierr = PetscRandomGetValue(rctx,&sigma2);CHKERRQ(ierr);
      *val = 4.0 - sigma1*h2 + sigma2*h2;
      ierr = MatSetValues(A,1,&Ii,1,&Ii,val,ADD_VALUES);CHKERRQ(ierr);
    }
    ierr = PetscRandomDestroy(&rctx);CHKERRQ(ierr);
  }
  else if (type == HELMHOLTZ_2) {
    /* Problem domain: unit square: (0,1) x (0,1)
       Solve Helmholtz equation:
          -delta u - sigma1*u = f,
           where delta = Laplace operator
       Dirichlet b.c.'s on 3 sides
       du/dn = i*alpha*u on (1,y), 0<y<1
     */
    PetscReal   h2,sigma1 = 200.0;
    PetscScalar alpha_h;
    ierr = PetscOptionsGetReal(PETSC_NULL,"-sigma1",&sigma1,PETSC_NULL);CHKERRQ(ierr);
    h2 = 1.0/((n+1)*(n+1));
    alpha_h = (PETSC_i * 10.0) / (PetscReal)(n+1);  /* alpha_h = alpha * h */
    for (Ii=Istart; Ii<Iend; Ii++) {
      *val = -1.0; i = Ii/n; j = Ii - i*n;
      if (i>0) {
        J = Ii-n; ierr = MatSetValues(A,1,&Ii,1,&J,val,ADD_VALUES);CHKERRQ(ierr);}
      if (i<n-1) {
//.........这里部分代码省略.........

int main(int argc,char **args)
{
  Mat            C;
  PetscScalar    v,none = -1.0;
  PetscInt       i,j,Ii,J,Istart,Iend,N,m = 4,n = 4,its,k;
  PetscErrorCode ierr;
  PetscMPIInt    size,rank;
  PetscReal      err_norm,res_norm,err_tol=1.e-7,res_tol=1.e-6;
  Vec            x,b,u,u_tmp;
  PetscRandom    r;
  PC             pc;
  KSP            ksp;

  PetscInitialize(&argc,&args,(char*)0,help);
  ierr = MPI_Comm_rank(PETSC_COMM_WORLD,&rank);CHKERRQ(ierr);
  ierr = MPI_Comm_size(PETSC_COMM_WORLD,&size);CHKERRQ(ierr);
  ierr = PetscOptionsGetInt(NULL,"-m",&m,NULL);CHKERRQ(ierr);
  ierr = PetscOptionsGetInt(NULL,"-n",&n,NULL);CHKERRQ(ierr);
  N    = m*n;


  /* Generate matrix */
  ierr = MatCreate(PETSC_COMM_WORLD,&C);CHKERRQ(ierr);
  ierr = MatSetSizes(C,PETSC_DECIDE,PETSC_DECIDE,N,N);CHKERRQ(ierr);
  ierr = MatSetFromOptions(C);CHKERRQ(ierr);
  ierr = MatSetUp(C);CHKERRQ(ierr);
  ierr = MatGetOwnershipRange(C,&Istart,&Iend);CHKERRQ(ierr);
  for (Ii=Istart; Ii<Iend; Ii++) {
    v = -1.0; i = Ii/n; j = Ii - i*n;
    if (i>0)   {J = Ii - n; ierr = MatSetValues(C,1,&Ii,1,&J,&v,ADD_VALUES);CHKERRQ(ierr);}
    if (i<m-1) {J = Ii + n; ierr = MatSetValues(C,1,&Ii,1,&J,&v,ADD_VALUES);CHKERRQ(ierr);}
    if (j>0)   {J = Ii - 1; ierr = MatSetValues(C,1,&Ii,1,&J,&v,ADD_VALUES);CHKERRQ(ierr);}
    if (j<n-1) {J = Ii + 1; ierr = MatSetValues(C,1,&Ii,1,&J,&v,ADD_VALUES);CHKERRQ(ierr);}
    v = 4.0; ierr = MatSetValues(C,1,&Ii,1,&Ii,&v,ADD_VALUES);CHKERRQ(ierr);
  }
  ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);

  /* a shift can make C indefinite. Preconditioners LU, ILU (for BAIJ format) and ICC may fail */
  /* ierr = MatShift(C,alpha);CHKERRQ(ierr); */
  /* ierr = MatView(C,PETSC_VIEWER_STDOUT_WORLD);CHKERRQ(ierr); */

  /* Setup and solve for system */
  /* Create vectors.  */
  ierr = VecCreate(PETSC_COMM_WORLD,&x);CHKERRQ(ierr);
  ierr = VecSetSizes(x,PETSC_DECIDE,N);CHKERRQ(ierr);
  ierr = VecSetFromOptions(x);CHKERRQ(ierr);
  ierr = VecDuplicate(x,&b);CHKERRQ(ierr);
  ierr = VecDuplicate(x,&u);CHKERRQ(ierr);
  ierr = VecDuplicate(x,&u_tmp);CHKERRQ(ierr);
  /* Set exact solution u; then compute right-hand-side vector b. */
  ierr = PetscRandomCreate(PETSC_COMM_SELF,&r);CHKERRQ(ierr);
  ierr = PetscRandomSetFromOptions(r);CHKERRQ(ierr);
  ierr = VecSetRandom(u,r);CHKERRQ(ierr);
  ierr = PetscRandomDestroy(&r);CHKERRQ(ierr);
  ierr = MatMult(C,u,b);CHKERRQ(ierr);

  for (k=0; k<3; k++) {
    if (k == 0) {                              /* CG  */
      ierr = KSPCreate(PETSC_COMM_WORLD,&ksp);CHKERRQ(ierr);
      ierr = KSPSetOperators(ksp,C,C);CHKERRQ(ierr);
      ierr = PetscPrintf(PETSC_COMM_WORLD,"\n CG: \n");CHKERRQ(ierr);
      ierr = KSPSetType(ksp,KSPCG);CHKERRQ(ierr);
    } else if (k == 1) {                       /* MINRES */
      ierr = KSPCreate(PETSC_COMM_WORLD,&ksp);CHKERRQ(ierr);
      ierr = KSPSetOperators(ksp,C,C);CHKERRQ(ierr);
      ierr = PetscPrintf(PETSC_COMM_WORLD,"\n MINRES: \n");CHKERRQ(ierr);
      ierr = KSPSetType(ksp,KSPMINRES);CHKERRQ(ierr);
    } else {                                 /* SYMMLQ */
      ierr = KSPCreate(PETSC_COMM_WORLD,&ksp);CHKERRQ(ierr);
      ierr = KSPSetOperators(ksp,C,C);CHKERRQ(ierr);
      ierr = PetscPrintf(PETSC_COMM_WORLD,"\n SYMMLQ: \n");CHKERRQ(ierr);
      ierr = KSPSetType(ksp,KSPSYMMLQ);CHKERRQ(ierr);
    }
    ierr = KSPGetPC(ksp,&pc);CHKERRQ(ierr);
    /* ierr = PCSetType(pc,PCICC);CHKERRQ(ierr); */
    ierr = PCSetType(pc,PCJACOBI);CHKERRQ(ierr);
    ierr = KSPSetTolerances(ksp,1.e-7,PETSC_DEFAULT,PETSC_DEFAULT,PETSC_DEFAULT);CHKERRQ(ierr);

    /*
    Set runtime options, e.g.,
        -ksp_type <type> -pc_type <type> -ksp_monitor -ksp_rtol <rtol>
    These options will override those specified above as long as
    KSPSetFromOptions() is called _after_ any other customization
    routines.
    */
    ierr = KSPSetFromOptions(ksp);CHKERRQ(ierr);

    /* Solve linear system; */
    ierr = KSPSetUp(ksp);CHKERRQ(ierr);
    ierr = KSPSolve(ksp,b,x);CHKERRQ(ierr);

    ierr = KSPGetIterationNumber(ksp,&its);CHKERRQ(ierr);
    /* Check error */
    ierr = VecCopy(u,u_tmp);CHKERRQ(ierr);
    ierr = VecAXPY(u_tmp,none,x);CHKERRQ(ierr);
    ierr = VecNorm(u_tmp,NORM_2,&err_norm);CHKERRQ(ierr);
    ierr = MatMult(C,x,u_tmp);CHKERRQ(ierr);
    ierr = VecAXPY(u_tmp,none,b);CHKERRQ(ierr);
    ierr = VecNorm(u_tmp,NORM_2,&res_norm);CHKERRQ(ierr);
//.........这里部分代码省略.........

int main(int argc,char **args)
{
  Mat               C,A;
  PetscInt          i,j,m = 3,n = 2,Ii,J,rstart,rend,nz;
  PetscMPIInt       rank,size;
  PetscErrorCode    ierr;
  const PetscInt    *idx;
  PetscScalar       v;
  const PetscScalar *values;

  PetscInitialize(&argc,&args,(char*)0,help);
  ierr = MPI_Comm_rank(PETSC_COMM_WORLD,&rank);CHKERRQ(ierr);
  ierr = MPI_Comm_size(PETSC_COMM_WORLD,&size);CHKERRQ(ierr);
  n    = 2*size;

  /* create the matrix for the five point stencil, YET AGAIN*/
  ierr = MatCreate(PETSC_COMM_WORLD,&C);CHKERRQ(ierr);
  ierr = MatSetSizes(C,PETSC_DECIDE,PETSC_DECIDE,m*n,m*n);CHKERRQ(ierr);
  ierr = MatSetFromOptions(C);CHKERRQ(ierr);
  ierr = MatMPIAIJSetPreallocation(C,5,NULL,5,NULL);CHKERRQ(ierr);
  ierr = MatSeqAIJSetPreallocation(C,5,NULL);CHKERRQ(ierr);
  for (i=0; i<m; i++) {
    for (j=2*rank; j<2*rank+2; j++) {
      v = -1.0;  Ii = j + n*i;
      if (i>0)   {J = Ii - n; ierr = MatSetValues(C,1,&Ii,1,&J,&v,INSERT_VALUES);CHKERRQ(ierr);}
      if (i<m-1) {J = Ii + n; ierr = MatSetValues(C,1,&Ii,1,&J,&v,INSERT_VALUES);CHKERRQ(ierr);}
      if (j>0)   {J = Ii - 1; ierr = MatSetValues(C,1,&Ii,1,&J,&v,INSERT_VALUES);CHKERRQ(ierr);}
      if (j<n-1) {J = Ii + 1; ierr = MatSetValues(C,1,&Ii,1,&J,&v,INSERT_VALUES);CHKERRQ(ierr);}
      v = 4.0; ierr = MatSetValues(C,1,&Ii,1,&Ii,&v,INSERT_VALUES);CHKERRQ(ierr);
    }
  }
  ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = PetscViewerPushFormat(PETSC_VIEWER_STDOUT_WORLD,PETSC_VIEWER_ASCII_INFO);CHKERRQ(ierr);
  ierr = MatView(C,PETSC_VIEWER_STDOUT_WORLD);CHKERRQ(ierr);
  ierr = PetscViewerPopFormat(PETSC_VIEWER_STDOUT_WORLD);CHKERRQ(ierr);
  ierr = MatView(C,PETSC_VIEWER_STDOUT_WORLD);CHKERRQ(ierr);

  ierr = MatGetOwnershipRange(C,&rstart,&rend);CHKERRQ(ierr);
  for (i=rstart; i<rend; i++) {
    ierr = MatGetRow(C,i,&nz,&idx,&values);CHKERRQ(ierr);
    ierr = PetscSynchronizedFPrintf(PETSC_COMM_WORLD,stdout,"[%d] get row %D: ",rank,i);CHKERRQ(ierr);
    for (j=0; j<nz; j++) {
      ierr = PetscSynchronizedFPrintf(PETSC_COMM_WORLD,stdout,"%D %G  ",idx[j],PetscRealPart(values[j]));CHKERRQ(ierr);
    }
    ierr = PetscSynchronizedFPrintf(PETSC_COMM_WORLD,stdout,"\n");CHKERRQ(ierr);
    ierr = MatRestoreRow(C,i,&nz,&idx,&values);CHKERRQ(ierr);
  }
  ierr = PetscSynchronizedFlush(PETSC_COMM_WORLD,stdout);CHKERRQ(ierr);CHKERRQ(ierr);

  ierr = MatConvert(C,MATSAME,MAT_INITIAL_MATRIX,&A);CHKERRQ(ierr);
  ierr = PetscViewerPushFormat(PETSC_VIEWER_STDOUT_WORLD,PETSC_VIEWER_ASCII_INFO);CHKERRQ(ierr);
  ierr = MatView(A,PETSC_VIEWER_STDOUT_WORLD);CHKERRQ(ierr);
  ierr = PetscViewerPopFormat(PETSC_VIEWER_STDOUT_WORLD);CHKERRQ(ierr);
  ierr = MatView(A,PETSC_VIEWER_STDOUT_WORLD);CHKERRQ(ierr);

  ierr = MatDestroy(&A);CHKERRQ(ierr);
  ierr = MatDestroy(&C);CHKERRQ(ierr);
  ierr = PetscFinalize();
  return 0;
}

/*
 * steady_state solves for the steady_state of the system
 * that was previously setup using the add_to_ham and add_lin
 * routines. Solver selection and parameterscan be controlled via PETSc
 * command line options.
 */
void steady_state(Vec x){
  PetscViewer    mat_view;
  PC             pc;
  Vec            b;
  KSP            ksp; /* linear solver context */
  PetscInt       row,col,its,j,i,Istart,Iend;
  PetscScalar    mat_tmp;
  long           dim;
  int            num_pop;
  double         *populations;
  Mat            solve_A;

  if (_lindblad_terms) {
    dim = total_levels*total_levels;
    solve_A = full_A;
    if (nid==0) {
      printf("Lindblad terms found, using Lindblad solver.");
    }
  } else {
    if (nid==0) {
      printf("Warning! Steady state not supported for Schrodinger.\n");
      printf("         Defaulting to (less efficient) Lindblad Solver\n");
      exit(0);
    }
    dim = total_levels*total_levels;
    solve_A = ham_A;
  }
  if (!stab_added){
    if (nid==0) printf("Adding stabilization...\n");
    /*
     * Add elements to the matrix to make the normalization work
     * I have no idea why this works, I am copying it from qutip
     * We add 1.0 in the 0th spot and every n+1 after
     */
    if (nid==0) {
      row = 0;
      for (i=0;i<total_levels;i++){
        col = i*(total_levels+1);
        mat_tmp = 1.0 + 0.*PETSC_i;
        MatSetValue(full_A,row,col,mat_tmp,ADD_VALUES);
      }

      /* Print dense ham, if it was asked for */
      if (_print_dense_ham){
        FILE *fp_ham;

        fp_ham = fopen("ham","w");

        if (nid==0){
          for (i=0;i<total_levels;i++){
            for (j=0;j<total_levels;j++){
              fprintf(fp_ham,"%e %e ",PetscRealPart(_hamiltonian[i][j]),PetscImaginaryPart(_hamiltonian[i][j]));
            }
            fprintf(fp_ham,"\n");
          }
        }
        fclose(fp_ham);
        for (i=0;i<total_levels;i++){
          free(_hamiltonian[i]);
        }
        free(_hamiltonian);
        _print_dense_ham = 0;
      }
    }
    stab_added = 1;
  }

  //  if (!matrix_assembled) {
    MatGetOwnershipRange(full_A,&Istart,&Iend);
    /*
     * Explicitly add 0.0 to all diagonal elements;
     * this fixes a 'matrix in wrong state' message that PETSc
     * gives if the diagonal was never initialized.
     */
    if (nid==0) printf("Adding 0 to diagonal elements...\n");
    for (i=Istart;i<Iend;i++){
      mat_tmp = 0 + 0.*PETSC_i;
      MatSetValue(full_A,i,i,mat_tmp,ADD_VALUES);
    }


    /* Tell PETSc to assemble the matrix */
    MatAssemblyBegin(full_A,MAT_FINAL_ASSEMBLY);
    MatAssemblyEnd(full_A,MAT_FINAL_ASSEMBLY);
    if (nid==0) printf("Matrix Assembled.\n");
    matrix_assembled = 1;
    //  }
  /* Print information about the matrix. */
  PetscViewerASCIIOpen(PETSC_COMM_WORLD,NULL,&mat_view);
  PetscViewerPushFormat(mat_view,PETSC_VIEWER_ASCII_INFO);
  MatView(full_A,mat_view);
  PetscViewerPopFormat(mat_view);
  PetscViewerDestroy(&mat_view);
  /*
//.........这里部分代码省略.........

int main(int argc,char **argv)
{
  Mat            M,C,K,A[3];      /* problem matrices */
  PEP            pep;             /* polynomial eigenproblem solver context */
  PetscInt       n=5,Istart,Iend,i;
  PetscReal      mu=1,tau=10,kappa=5;
  PetscBool      terse;
  PetscErrorCode ierr;
  PetscLogDouble time1,time2;

  SlepcInitialize(&argc,&argv,(char*)0,help);

  ierr = PetscOptionsGetInt(NULL,"-n",&n,NULL);CHKERRQ(ierr);
  ierr = PetscOptionsGetReal(NULL,"-mu",&mu,NULL);CHKERRQ(ierr);
  ierr = PetscOptionsGetReal(NULL,"-tau",&tau,NULL);CHKERRQ(ierr);
  ierr = PetscOptionsGetReal(NULL,"-kappa",&kappa,NULL);CHKERRQ(ierr);
  ierr = PetscPrintf(PETSC_COMM_WORLD,"\nDamped mass-spring system, n=%D mu=%g tau=%g kappa=%g\n\n",n,(double)mu,(double)tau,(double)kappa);CHKERRQ(ierr);

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 
     Compute the matrices that define the eigensystem, (k^2*M+k*C+K)x=0
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

  /* K is a tridiagonal */
  ierr = MatCreate(PETSC_COMM_WORLD,&K);CHKERRQ(ierr);
  ierr = MatSetSizes(K,PETSC_DECIDE,PETSC_DECIDE,n,n);CHKERRQ(ierr);
  ierr = MatSetFromOptions(K);CHKERRQ(ierr);
  ierr = MatSetUp(K);CHKERRQ(ierr);
  
  ierr = MatGetOwnershipRange(K,&Istart,&Iend);CHKERRQ(ierr);
  for (i=Istart;i<Iend;i++) {
    if (i>0) {
      ierr = MatSetValue(K,i,i-1,-kappa,INSERT_VALUES);CHKERRQ(ierr);
    }
    ierr = MatSetValue(K,i,i,kappa*3.0,INSERT_VALUES);CHKERRQ(ierr);
    if (i<n-1) {
      ierr = MatSetValue(K,i,i+1,-kappa,INSERT_VALUES);CHKERRQ(ierr);
    }
  }

  ierr = MatAssemblyBegin(K,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = MatAssemblyEnd(K,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);

  /* C is a tridiagonal */
  ierr = MatCreate(PETSC_COMM_WORLD,&C);CHKERRQ(ierr);
  ierr = MatSetSizes(C,PETSC_DECIDE,PETSC_DECIDE,n,n);CHKERRQ(ierr);
  ierr = MatSetFromOptions(C);CHKERRQ(ierr);
  ierr = MatSetUp(C);CHKERRQ(ierr);
  
  ierr = MatGetOwnershipRange(C,&Istart,&Iend);CHKERRQ(ierr);
  for (i=Istart;i<Iend;i++) {
    if (i>0) {
      ierr = MatSetValue(C,i,i-1,-tau,INSERT_VALUES);CHKERRQ(ierr);
    }
    ierr = MatSetValue(C,i,i,tau*3.0,INSERT_VALUES);CHKERRQ(ierr);
    if (i<n-1) {
      ierr = MatSetValue(C,i,i+1,-tau,INSERT_VALUES);CHKERRQ(ierr);
    }
  }

  ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  
  /* M is a diagonal matrix */
  ierr = MatCreate(PETSC_COMM_WORLD,&M);CHKERRQ(ierr);
  ierr = MatSetSizes(M,PETSC_DECIDE,PETSC_DECIDE,n,n);CHKERRQ(ierr);
  ierr = MatSetFromOptions(M);CHKERRQ(ierr);
  ierr = MatSetUp(M);CHKERRQ(ierr);
  ierr = MatGetOwnershipRange(M,&Istart,&Iend);CHKERRQ(ierr);
  for (i=Istart;i<Iend;i++) {
    ierr = MatSetValue(M,i,i,mu,INSERT_VALUES);CHKERRQ(ierr);
  }
  ierr = MatAssemblyBegin(M,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = MatAssemblyEnd(M,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 
                Create the eigensolver and solve the problem
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

  ierr = PEPCreate(PETSC_COMM_WORLD,&pep);CHKERRQ(ierr);
  A[0] = K; A[1] = C; A[2] = M;
  ierr = PEPSetOperators(pep,3,A);CHKERRQ(ierr);
  ierr = PEPSetFromOptions(pep);CHKERRQ(ierr);
  
  ierr = PetscTime(&time1); CHKERRQ(ierr);
  ierr = PEPSolve(pep);CHKERRQ(ierr);
  ierr = PetscTime(&time2); CHKERRQ(ierr);

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
                    Display solution and clean up
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  
  /* show detailed info unless -terse option is given by user */
  ierr = PetscOptionsHasName(NULL,"-terse",&terse);CHKERRQ(ierr);
  if (terse) {
    ierr = PEPErrorView(pep,PEP_ERROR_BACKWARD,NULL);CHKERRQ(ierr);
  } else {
    ierr = PetscViewerPushFormat(PETSC_VIEWER_STDOUT_WORLD,PETSC_VIEWER_ASCII_INFO_DETAIL);CHKERRQ(ierr);
    ierr = PEPReasonView(pep,PETSC_VIEWER_STDOUT_WORLD);CHKERRQ(ierr);
    ierr = PEPErrorView(pep,PEP_ERROR_BACKWARD,PETSC_VIEWER_STDOUT_WORLD);CHKERRQ(ierr);
    ierr = PetscViewerPopFormat(PETSC_VIEWER_STDOUT_WORLD);CHKERRQ(ierr);
//.........这里部分代码省略.........

int main(int argc,char **args)
{
  Mat            C,A;
  PetscInt       i,j,m = 3,n = 2,rstart,rend;
  PetscMPIInt    size,rank;
  PetscErrorCode ierr;
  PetscScalar    v;
  IS             isrow,iscol;
  Pe