int main(int argc,char **args)
{
  PetscMPIInt    size;
  PetscErrorCode ierr;
  Vec            x,y,b,s1,s2;
  Mat            A;                    /* linear system matrix */
  Mat            sA,sB,sC;             /* symmetric part of the matrices */
  PetscInt       n,mbs=16,bs=1,nz=3,prob=1,i,j,k1,k2,col[3],lf,block, row,Ii,J,n1,inc;
  PetscReal      norm1,norm2,rnorm,tol=PETSC_SMALL;
  PetscScalar    neg_one = -1.0,four=4.0,value[3];
  IS             perm, iscol;
  PetscRandom    rdm;
  PetscBool      doIcc=PETSC_TRUE,equal;
  MatInfo        minfo1,minfo2;
  MatFactorInfo  factinfo;
  MatType        type;

  PetscInitialize(&argc,&args,(char*)0,help);
  ierr = MPI_Comm_size(PETSC_COMM_WORLD,&size);CHKERRQ(ierr);
  if (size != 1) SETERRQ(PETSC_COMM_WORLD,PETSC_ERR_SUP,"This is a uniprocessor example only!");
  ierr = PetscOptionsGetInt(NULL,"-bs",&bs,NULL);CHKERRQ(ierr);
  ierr = PetscOptionsGetInt(NULL,"-mbs",&mbs,NULL);CHKERRQ(ierr);

  n    = mbs*bs;
  ierr = MatCreate(PETSC_COMM_SELF,&A);CHKERRQ(ierr);
  ierr = MatSetSizes(A,n,n,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
  ierr = MatSetType(A,MATSEQBAIJ);CHKERRQ(ierr);
  ierr = MatSetFromOptions(A);CHKERRQ(ierr);
  ierr = MatSeqBAIJSetPreallocation(A,bs,nz,NULL);CHKERRQ(ierr);

  ierr = MatCreate(PETSC_COMM_SELF,&sA);CHKERRQ(ierr);
  ierr = MatSetSizes(sA,n,n,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
  ierr = MatSetType(sA,MATSEQSBAIJ);CHKERRQ(ierr);
  ierr = MatSetFromOptions(sA);CHKERRQ(ierr);
  ierr = MatGetType(sA,&type);CHKERRQ(ierr);
  ierr = PetscObjectTypeCompare((PetscObject)sA,MATSEQSBAIJ,&doIcc);CHKERRQ(ierr);
  ierr = MatSeqSBAIJSetPreallocation(sA,bs,nz,NULL);CHKERRQ(ierr);
  ierr = MatSetOption(sA,MAT_IGNORE_LOWER_TRIANGULAR,PETSC_TRUE);CHKERRQ(ierr);

  /* Test MatGetOwnershipRange() */
  ierr = MatGetOwnershipRange(A,&Ii,&J);CHKERRQ(ierr);
  ierr = MatGetOwnershipRange(sA,&i,&j);CHKERRQ(ierr);
  if (i-Ii || j-J) {
    ierr = PetscPrintf(PETSC_COMM_SELF,"Error: MatGetOwnershipRange() in MatSBAIJ format\n");CHKERRQ(ierr);
  }

  /* Assemble matrix */
  if (bs == 1) {
    ierr = PetscOptionsGetInt(NULL,"-test_problem",&prob,NULL);CHKERRQ(ierr);
    if (prob == 1) { /* tridiagonal matrix */
      value[0] = -1.0; value[1] = 2.0; value[2] = -1.0;
      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,value,INSERT_VALUES);CHKERRQ(ierr);
        ierr   = MatSetValues(sA,1,&i,3,col,value,INSERT_VALUES);CHKERRQ(ierr);
      }
      i = n - 1; col[0]=0; col[1] = n - 2; col[2] = n - 1;

      value[0]= 0.1; value[1]=-1; value[2]=2;

      ierr = MatSetValues(A,1,&i,3,col,value,INSERT_VALUES);CHKERRQ(ierr);
      ierr = MatSetValues(sA,1,&i,3,col,value,INSERT_VALUES);CHKERRQ(ierr);

      i        = 0;
      col[0]   = n-1;   col[1] = 1;      col[2] = 0;
      value[0] = 0.1; value[1] = -1.0; value[2] = 2;

      ierr = MatSetValues(A,1,&i,3,col,value,INSERT_VALUES);CHKERRQ(ierr);
      ierr = MatSetValues(sA,1,&i,3,col,value,INSERT_VALUES);CHKERRQ(ierr);

    } else if (prob ==2) { /* matrix for the five point stencil */
      n1 = (PetscInt) (PetscSqrtReal((PetscReal)n) + 0.001);
      if (n1*n1 - n) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"sqrt(n) must be a positive interger!");
      for (i=0; i<n1; i++) {
        for (j=0; j<n1; j++) {
          Ii = j + n1*i;
          if (i>0) {
            J    = Ii - n1;
            ierr = MatSetValues(A,1,&Ii,1,&J,&neg_one,INSERT_VALUES);CHKERRQ(ierr);
            ierr = MatSetValues(sA,1,&Ii,1,&J,&neg_one,INSERT_VALUES);CHKERRQ(ierr);
          }
          if (i<n1-1) {
            J    = Ii + n1;
            ierr = MatSetValues(A,1,&Ii,1,&J,&neg_one,INSERT_VALUES);CHKERRQ(ierr);
            ierr = MatSetValues(sA,1,&Ii,1,&J,&neg_one,INSERT_VALUES);CHKERRQ(ierr);
          }
          if (j>0) {
            J    = Ii - 1;
            ierr = MatSetValues(A,1,&Ii,1,&J,&neg_one,INSERT_VALUES);CHKERRQ(ierr);
            ierr = MatSetValues(sA,1,&Ii,1,&J,&neg_one,INSERT_VALUES);CHKERRQ(ierr);
          }
          if (j<n1-1) {
            J    = Ii + 1;
            ierr = MatSetValues(A,1,&Ii,1,&J,&neg_one,INSERT_VALUES);CHKERRQ(ierr);
            ierr = MatSetValues(sA,1,&Ii,1,&J,&neg_one,INSERT_VALUES);CHKERRQ(ierr);
          }
          ierr = MatSetValues(A,1,&Ii,1,&Ii,&four,INSERT_VALUES);CHKERRQ(ierr);
          ierr = MatSetValues(sA,1,&Ii,1,&Ii,&four,INSERT_VALUES);CHKERRQ(ierr);
        }
      }
//.........这里部分代码省略.........

int main(int argc,char **argv)
{
  PetscErrorCode ierr;
  Mat            seqmat,mpimat;
  PetscMPIInt    rank;
  PetscScalar    value[3],*vals;
  PetscInt       i,col[3],n=5,bs=1;
  
  PetscInitialize(&argc,&argv,(char*)0,help);
  ierr = MPI_Comm_rank(PETSC_COMM_WORLD,&rank);CHKERRQ(ierr);
  ierr = PetscOptionsGetInt(NULL,"-bs",&bs,NULL);CHKERRQ(ierr);

  /* Create seqaij matrices of size (n+rank) by n */
  ierr = MatCreate(PETSC_COMM_SELF,&seqmat);CHKERRQ(ierr);
  ierr = MatSetSizes(seqmat,(n+rank)*bs,PETSC_DECIDE,PETSC_DECIDE,n*bs);CHKERRQ(ierr);
  ierr = MatSetFromOptions(seqmat);CHKERRQ(ierr);
  ierr = MatSeqAIJSetPreallocation(seqmat,3*bs,NULL);CHKERRQ(ierr);
  ierr = MatSeqBAIJSetPreallocation(seqmat,bs,3,NULL);CHKERRQ(ierr);

  if (bs == 1) {
    value[0] = -1.0; value[1] = 2.0; value[2] = -1.0;
    for (i=1; i<n-1; i++) {
      col[0] = i-1; col[1] = i; col[2] = i+1;
      ierr   = MatSetValues(seqmat,1,&i,3,col,value,INSERT_VALUES);CHKERRQ(ierr);
    }
    i = n - 1; col[0] = n - 2; col[1] = n - 1;
    ierr = MatSetValues(seqmat,1,&i,2,col,value,INSERT_VALUES);CHKERRQ(ierr);

    i = 0; col[0] = 0; col[1] = 1; value[0] = 2.0; value[1] = -1.0;
    ierr = MatSetValues(seqmat,1,&i,2,col,value,INSERT_VALUES);CHKERRQ(ierr);
  } else {
    PetscInt *rows,*cols,j;
    ierr = PetscMalloc3(bs*bs,&vals,bs,&rows,bs,&cols);CHKERRQ(ierr);
    /* diagonal blocks */
    for (i=0; i<bs*bs; i++) vals[i] = 2.0;
    for (i=0; i<n*bs; i+=bs) {
      for (j=0; j<bs; j++) {rows[j] = i+j; cols[j] = i+j;}
      ierr = MatSetValues(seqmat,bs,rows,bs,cols,vals,INSERT_VALUES);CHKERRQ(ierr);
    }
    /* off-diagonal blocks */
    for (i=0; i<bs*bs; i++) vals[i] = -1.0;
    for (i=0; i<(n-1)*bs; i+=bs) {
      for (j=0; j<bs; j++) {rows[j] = i+j; cols[j] = i+bs+j;}
      ierr = MatSetValues(seqmat,bs,rows,bs,cols,vals,INSERT_VALUES);CHKERRQ(ierr);
    }

    ierr = PetscFree3(vals,rows,cols);CHKERRQ(ierr);
  }
  ierr = MatAssemblyBegin(seqmat,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = MatAssemblyEnd(seqmat,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  if (rank == 1) {
    printf("[%d] seqmat:\n",rank);
    ierr = MatView(seqmat,PETSC_VIEWER_STDOUT_SELF);CHKERRQ(ierr);
  }

  ierr = MatCreateMPIMatConcatenateSeqMat(PETSC_COMM_WORLD,seqmat,PETSC_DECIDE,MAT_INITIAL_MATRIX,&mpimat);CHKERRQ(ierr);
  ierr = MatCreateMPIMatConcatenateSeqMat(PETSC_COMM_WORLD,seqmat,PETSC_DECIDE,MAT_REUSE_MATRIX,&mpimat);CHKERRQ(ierr);
  ierr = MatView(mpimat,PETSC_VIEWER_STDOUT_WORLD);CHKERRQ(ierr);

  ierr = MatDestroy(&seqmat);CHKERRQ(ierr);
  ierr = MatDestroy(&mpimat);CHKERRQ(ierr);
  ierr = PetscFinalize();
  return 0;
}

int main(int argc,char **args)
{
  Vec            x,b,u;      /* approx solution, RHS, exact solution */
  Mat            A;            /* linear system matrix */
  KSP            ksp;         /* KSP context */
  PetscErrorCode ierr;
  PetscInt       n = 10,its, dim,p = 1,use_random;
  PetscScalar    none = -1.0,pfive = 0.5;
  PetscReal      norm;
  PetscRandom    rctx;
  TestType       type;
  PetscBool      flg;

  PetscInitialize(&argc,&args,(char *)0,help);
  ierr = PetscOptionsGetInt(PETSC_NULL,"-n",&n,PETSC_NULL);CHKERRQ(ierr);
  ierr = PetscOptionsGetInt(PETSC_NULL,"-p",&p,PETSC_NULL);CHKERRQ(ierr);
  switch (p) {
    case 1:  type = TEST_1;      dim = n;   break;
    case 2:  type = TEST_2;      dim = n;   break;
    case 3:  type = TEST_3;      dim = n;   break;
    case 4:  type = HELMHOLTZ_1; dim = n*n; break;
    case 5:  type = HELMHOLTZ_2; dim = n*n; break;
    default: type = TEST_1;      dim = n;
  }

  /* Create vectors */
  ierr = VecCreate(PETSC_COMM_WORLD,&x);CHKERRQ(ierr);
  ierr = VecSetSizes(x,PETSC_DECIDE,dim);CHKERRQ(ierr);
  ierr = VecSetFromOptions(x);CHKERRQ(ierr);
  ierr = VecDuplicate(x,&b);CHKERRQ(ierr);
  ierr = VecDuplicate(x,&u);CHKERRQ(ierr);

  use_random = 1;
  flg        = PETSC_FALSE;
  ierr = PetscOptionsGetBool(PETSC_NULL,"-norandom",&flg,PETSC_NULL);CHKERRQ(ierr);
  if (flg) {
    use_random = 0;
    ierr = VecSet(u,pfive);CHKERRQ(ierr);
  } else {
    ierr = PetscRandomCreate(PETSC_COMM_WORLD,&rctx);CHKERRQ(ierr);
    ierr = PetscRandomSetFromOptions(rctx);CHKERRQ(ierr);
    ierr = VecSetRandom(u,rctx);CHKERRQ(ierr);
  }

  /* Create and assemble matrix */
  ierr = MatCreate(PETSC_COMM_WORLD,&A);CHKERRQ(ierr);
  ierr = MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,dim,dim);CHKERRQ(ierr);
  ierr = MatSetFromOptions(A);CHKERRQ(ierr);
  ierr = MatSetUp(A);CHKERRQ(ierr);
  ierr = FormTestMatrix(A,n,type);CHKERRQ(ierr);
  ierr = MatMult(A,u,b);CHKERRQ(ierr);
  flg  = PETSC_FALSE;
  ierr = PetscOptionsGetBool(PETSC_NULL,"-printout",&flg,PETSC_NULL);CHKERRQ(ierr);
  if (flg) {
    ierr = MatView(A,PETSC_VIEWER_STDOUT_WORLD);CHKERRQ(ierr);
    ierr = VecView(u,PETSC_VIEWER_STDOUT_WORLD);CHKERRQ(ierr);
    ierr = VecView(b,PETSC_VIEWER_STDOUT_WORLD);CHKERRQ(ierr);
  }

  /* Create KSP context; set operators and options; solve linear system */
  ierr = KSPCreate(PETSC_COMM_WORLD,&ksp);CHKERRQ(ierr);
  ierr = KSPSetOperators(ksp,A,A,DIFFERENT_NONZERO_PATTERN);CHKERRQ(ierr);
  ierr = KSPSetFromOptions(ksp);CHKERRQ(ierr);
  ierr = KSPSolve(ksp,b,x);CHKERRQ(ierr);
  ierr = KSPView(ksp,PETSC_VIEWER_STDOUT_WORLD);CHKERRQ(ierr);

  /* Check error */
  ierr = VecAXPY(x,none,u);CHKERRQ(ierr);
  ierr  = VecNorm(x,NORM_2,&norm);CHKERRQ(ierr);
  ierr = KSPGetIterationNumber(ksp,&its);CHKERRQ(ierr);
  ierr = PetscPrintf(PETSC_COMM_WORLD,"Norm of error %G,Iterations %D\n",norm,its);CHKERRQ(ierr);

  /* Free work space */
  ierr = VecDestroy(&x);CHKERRQ(ierr); ierr = VecDestroy(&u);CHKERRQ(ierr);
  ierr = VecDestroy(&b);CHKERRQ(ierr); ierr = MatDestroy(&A);CHKERRQ(ierr);
  if (use_random) {ierr = PetscRandomDestroy(&rctx);CHKERRQ(ierr);}
  ierr = KSPDestroy(&ksp);CHKERRQ(ierr);
  ierr = PetscFinalize();
  return 0;
}

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.........

Modified from the code contributed by Yaning Liu @lbl.gov \n\n";
/*
 Example:
   mpiexec -n <np> ./ex103 
   mpiexec -n <np> ./ex103 -mat_type elemental -mat_view
   mpiexec -n <np> ./ex103 -mat_type aij 
*/
    
#include <petscmat.h>

#undef __FUNCT__
#define __FUNCT__ "main"
int main(int argc, char** argv)
{
  Mat            A,A_elemental;
  PetscInt       i,j,M=10,N=5,nrows,ncols;
  PetscErrorCode ierr;
  PetscMPIInt    rank,size;
  IS             isrows,iscols;
  const PetscInt *rows,*cols;
  PetscScalar    *v;
  MatType        type;
  PetscBool      isDense,isAIJ,flg;

  ierr = PetscInitialize(&argc, &argv, (char*)0, help);if (ierr) return ierr;
#if !defined(PETSC_HAVE_ELEMENTAL)
  SETERRQ(PETSC_COMM_WORLD,1,"This example requires ELEMENTAL");
#endif
  ierr = MPI_Comm_size(PETSC_COMM_WORLD,&size);CHKERRQ(ierr);
  ierr = MPI_Comm_rank(PETSC_COMM_WORLD,&rank);CHKERRQ(ierr);

  /* Creat a matrix */
  ierr = PetscOptionsGetInt(NULL,NULL,"-M",&M,NULL);CHKERRQ(ierr);
  ierr = PetscOptionsGetInt(NULL,NULL,"-N",&N,NULL);CHKERRQ(ierr);
  ierr = MatCreate(PETSC_COMM_WORLD, &A);CHKERRQ(ierr);
  ierr = MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,M,N);CHKERRQ(ierr);
  ierr = MatSetType(A,MATDENSE);CHKERRQ(ierr);
  ierr = MatSetFromOptions(A);CHKERRQ(ierr);
  ierr = MatSetUp(A);CHKERRQ(ierr);

  /* Set local matrix entries */
  ierr = MatGetOwnershipIS(A,&isrows,&iscols);CHKERRQ(ierr);
  ierr = ISGetLocalSize(isrows,&nrows);CHKERRQ(ierr);
  ierr = ISGetIndices(isrows,&rows);CHKERRQ(ierr);
  ierr = ISGetLocalSize(iscols,&ncols);CHKERRQ(ierr);
  ierr = ISGetIndices(iscols,&cols);CHKERRQ(ierr);
  ierr = PetscMalloc1(nrows*ncols,&v);CHKERRQ(ierr);

  for (i=0; i<nrows; i++) {
    for (j=0; j<ncols; j++) {
      if (size == 1) {
        v[i*ncols+j] = (PetscScalar)(i+j);  
      } else {
        v[i*ncols+j] = (PetscScalar)rank+j*0.1;
      }
    }
  }
  ierr = MatSetValues(A,nrows,rows,ncols,cols,v,INSERT_VALUES);CHKERRQ(ierr);
  ierr = MatAssemblyBegin(A, MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = MatAssemblyEnd(A, MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  //ierr = PetscSynchronizedPrintf(PETSC_COMM_WORLD,"[%D] local nrows %D, ncols %D\n",rank,nrows,ncols);CHKERRQ(ierr);
  //ierr = PetscSynchronizedFlush(PETSC_COMM_WORLD,PETSC_STDOUT);CHKERRQ(ierr);

  /* Test MatSetValues() by converting A to A_elemental */
  ierr = MatGetType(A,&type);CHKERRQ(ierr);
  if (size == 1) {
    ierr = PetscObjectTypeCompare((PetscObject)A,MATSEQDENSE,&isDense);CHKERRQ(ierr);
    ierr = PetscObjectTypeCompare((PetscObject)A,MATSEQAIJ,&isAIJ);CHKERRQ(ierr);
  } else {
    ierr = PetscObjectTypeCompare((PetscObject)A,MATMPIDENSE,&isDense);CHKERRQ(ierr);
    ierr = PetscObjectTypeCompare((PetscObject)A,MATMPIAIJ,&isAIJ);CHKERRQ(ierr);
  }

  if (isDense || isAIJ) {
    Mat Aexplicit;
    ierr = MatConvert(A, MATELEMENTAL, MAT_INITIAL_MATRIX, &A_elemental);CHKERRQ(ierr);
    ierr = MatComputeExplicitOperator(A_elemental,&Aexplicit);CHKERRQ(ierr);
    ierr = MatMultEqual(Aexplicit,A_elemental,5,&flg);CHKERRQ(ierr); 
    if (!flg) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"Aexplicit != A_elemental.");
    ierr = MatDestroy(&Aexplicit);CHKERRQ(ierr); 

    /* Test MAT_REUSE_MATRIX which is only supported for inplace conversion */
    ierr = MatConvert(A, MATELEMENTAL, MAT_INPLACE_MATRIX, &A);CHKERRQ(ierr);
    ierr = MatMultEqual(A_elemental,A,5,&flg);CHKERRQ(ierr);
    if (!flg) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"A_elemental != A.");
    ierr = MatDestroy(&A_elemental);CHKERRQ(ierr);
  }

  ierr = ISRestoreIndices(isrows,&rows);CHKERRQ(ierr);
  ierr = ISRestoreIndices(iscols,&cols);CHKERRQ(ierr);
  ierr = ISDestroy(&isrows);CHKERRQ(ierr);
  ierr = ISDestroy(&iscols);CHKERRQ(ierr);
  ierr = PetscFree(v);CHKERRQ(ierr);
  ierr = MatDestroy(&A);CHKERRQ(ierr);
  ierr = PetscFinalize();
  return ierr;
}

int main(int argc,char **args)
{
  Mat          C;
  int          i,m = 5,rank,size,N,start,end,M;
  int          ierr,idx[4];
  PetscBool    flg;
  PetscScalar  Ke[16];
  PetscReal    h;
  Vec          u,b;
  KSP          ksp;
  MatNullSpace nullsp;

  PetscInitialize(&argc,&args,(char*)0,help);
  ierr = PetscOptionsGetInt(NULL,"-m",&m,NULL);CHKERRQ(ierr);
  N    = (m+1)*(m+1); /* dimension of matrix */
  M    = m*m; /* number of elements */
  h    = 1.0/m;    /* mesh width */
  ierr = MPI_Comm_rank(PETSC_COMM_WORLD,&rank);CHKERRQ(ierr);
  ierr = MPI_Comm_size(PETSC_COMM_WORLD,&size);CHKERRQ(ierr);

  /* Create stiffness 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);
  start = rank*(M/size) + ((M%size) < rank ? (M%size) : rank);
  end   = start + M/size + ((M%size) > rank);

  /* Assemble matrix */
  ierr = FormElementStiffness(h*h,Ke);   /* element stiffness for Laplacian */
  for (i=start; i<end; i++) {
    /* location of lower left corner of element */
    /* node numbers for the four corners of element */
    idx[0] = (m+1)*(i/m) + (i % m);
    idx[1] = idx[0]+1; idx[2] = idx[1] + m + 1; idx[3] = idx[2] - 1;
    ierr   = MatSetValues(C,4,idx,4,idx,Ke,ADD_VALUES);CHKERRQ(ierr);
  }
  ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);

  /* Create right-hand-side and solution vectors */
  ierr = VecCreate(PETSC_COMM_WORLD,&u);CHKERRQ(ierr);
  ierr = VecSetSizes(u,PETSC_DECIDE,N);CHKERRQ(ierr);
  ierr = VecSetFromOptions(u);CHKERRQ(ierr);
  ierr = PetscObjectSetName((PetscObject)u,"Approx. Solution");CHKERRQ(ierr);
  ierr = VecDuplicate(u,&b);CHKERRQ(ierr);
  ierr = PetscObjectSetName((PetscObject)b,"Right hand side");CHKERRQ(ierr);

  ierr = VecSet(u,1.0);CHKERRQ(ierr);
  ierr = MatMult(C,u,b);CHKERRQ(ierr);
  ierr = VecSet(u,0.0);CHKERRQ(ierr);

  /* Solve linear system */
  ierr = KSPCreate(PETSC_COMM_WORLD,&ksp);CHKERRQ(ierr);
  ierr = KSPSetOperators(ksp,C,C,DIFFERENT_NONZERO_PATTERN);CHKERRQ(ierr);
  ierr = KSPSetFromOptions(ksp);CHKERRQ(ierr);
  ierr = KSPSetInitialGuessNonzero(ksp,PETSC_TRUE);CHKERRQ(ierr);

  flg  = PETSC_FALSE;
  ierr = PetscOptionsGetBool(NULL,"-fixnullspace",&flg,NULL);CHKERRQ(ierr);
  if (flg) {
    ierr = MatNullSpaceCreate(PETSC_COMM_WORLD,PETSC_TRUE,0,NULL,&nullsp);CHKERRQ(ierr);
    ierr = KSPSetNullSpace(ksp,nullsp);CHKERRQ(ierr);
    ierr = MatNullSpaceDestroy(&&nullsp);CHKERRQ(ierr);
  }
  ierr = KSPSolve(ksp,b,u);CHKERRQ(ierr);


  /* Free work space */
  ierr = KSPDestroy(&ksp);CHKERRQ(ierr);
  ierr = VecDestroy(&u);CHKERRQ(ierr);
  ierr = VecDestroy(&b);CHKERRQ(ierr);
  ierr = MatDestroy(&C);CHKERRQ(ierr);
  ierr = PetscFinalize();
  return 0;
}

void g2_correlation(PetscScalar ***g2_values,Vec dm0,PetscInt n_tau,PetscReal tau_max,PetscInt n_st,PetscReal st_max,PetscInt number_of_ops,...){
  TSCtx tsctx;
  PetscReal st_dt,previous_start_time,this_start_time,dt;
  PetscReal tau_t_max,dt_tau;
  PetscInt i,j,dim,steps_max;
  Mat A_star_A,tmp_mat;
  Vec init_dm;
  va_list ap;
  /*Explicitly construct our jump matrix by adding up all of the operators
   * \rho = A \rho A^\dag
   * Vectorized:
   * \rho = (A* \cross A) \rho
   */
  dim = total_levels*total_levels; //Assumes Lindblad

  MatCreate(PETSC_COMM_WORLD,&tmp_mat);
  MatSetType(tmp_mat,MATMPIAIJ);
  MatSetSizes(tmp_mat,PETSC_DECIDE,PETSC_DECIDE,dim,dim);
  MatSetFromOptions(tmp_mat);
  MatMPIAIJSetPreallocation(tmp_mat,4,NULL,4,NULL);

  va_start(ap,number_of_ops);
  //Get A* \cross I
  vadd_ops_to_mat(tmp_mat,1,number_of_ops,ap);
  va_end(ap);


  MatCreate(PETSC_COMM_WORLD,&tsctx.I_cross_A);
  MatSetType(tsctx.I_cross_A,MATMPIAIJ);
  MatSetSizes(tsctx.I_cross_A,PETSC_DECIDE,PETSC_DECIDE,dim,dim);
  MatSetFromOptions(tsctx.I_cross_A);
  MatMPIAIJSetPreallocation(tsctx.I_cross_A,4,NULL,4,NULL);
  va_start(ap,number_of_ops);
  //Get I_cross_A
  vadd_ops_to_mat(tsctx.I_cross_A,-1,number_of_ops,ap);
  va_end(ap);


  //Get (A* \cross I) (I \cross A)
  MatMatMult(tmp_mat,tsctx.I_cross_A,MAT_INITIAL_MATRIX,PETSC_DEFAULT,&A_star_A);
  MatDestroy(&tmp_mat);
  VecDuplicate(dm0,&(tsctx.tmp_dm));
  VecDuplicate(dm0,&(tsctx.tmp_dm2));
  VecDuplicate(dm0,&init_dm);

  //Arbitrary, set this better
  steps_max = 51000;
  dt        = 0.025;

  //Allocate memory for g2
  (*g2_values) = (PetscScalar **)malloc((n_st+1)*sizeof(PetscScalar *));
  for (i=0;i<n_st+1;i++){
    (*g2_values)[i] = (PetscScalar *)malloc((n_tau+1)*sizeof(PetscScalar));
    for (j=0;j<n_tau+1;j++){
      (*g2_values)[i][j] = 0.0;
    }
  }
  tsctx.g2_values = (*g2_values);

  set_ts_monitor_ctx(_g2_ts_monitor,&tsctx);
  st_dt = st_max/n_st;
  previous_start_time = 0;
  tsctx.i_st = 0;
  tsctx.i_st = tsctx.i_st + 1; //Why?

  for (this_start_time=st_dt;this_start_time<=st_max;this_start_time+=st_dt){
    //Go from previous start time to this_start_time
    tsctx.tau_evolve = 0;
    dt = (this_start_time - previous_start_time)/500; //500 is arbitrary, should be picked better
    time_step(dm0,previous_start_time,this_start_time,dt,steps_max);

    //Timestep through taus
    tau_t_max = this_start_time + tau_max;
    dt_tau = (tau_t_max - this_start_time)/n_tau;

    /*
     * Force an 'emission' to get A \rho A^\dag terms
     * We already have A* \cross A - we just do the multiplication
     */
    tsctx.tau_evolve = 1;
    //Copy the timestepped dm into our init_dm for tau sweep
    VecCopy(dm0,init_dm);
    MatMult(A_star_A,dm0,init_dm); //init_dm = A * dm0
    tsctx.i_tau = 0;
    time_step(init_dm,this_start_time,tau_t_max,dt_tau,steps_max);

    previous_start_time = this_start_time;
    tsctx.i_st = tsctx.i_st + 1;
  }

  return;
}

//.........这里部分代码省略.........
    lsparse[i] = 0;
    gsparse[i] = 0;
    if (lcid[i] >= 0) {
      lsparse[i] = 1;
      gsparse[i] = 0;
    } else {
      ierr = MatGetRow(lA,i,&ncols,&rcol,&rval);CHKERRQ(ierr);
      for (j = 0;j < ncols;j++) {
        col = rcol[j];
        if (lcid[col] >= 0 && (PetscRealPart(rval[j]) > gamg->threshold*Amax_pos[i] || PetscRealPart(-rval[j]) > gamg->threshold*Amax_neg[i])) {
          lsparse[i] += 1;
        }
      }
      ierr = MatRestoreRow(lA,i,&ncols,&rcol,&rval);CHKERRQ(ierr);
      /* off */
      if (gA) {
        ierr = MatGetRow(gA,i,&ncols,&rcol,&rval);CHKERRQ(ierr);
        for (j = 0; j < ncols; j++) {
          col = rcol[j];
          if (gcid[col] >= 0 && (PetscRealPart(rval[j]) > gamg->threshold*Amax_pos[i] || PetscRealPart(-rval[j]) > gamg->threshold*Amax_neg[i])) {
            gsparse[i] += 1;
          }
        }
        ierr = MatRestoreRow(gA,i,&ncols,&rcol,&rval);CHKERRQ(ierr);
      }
    }
  }

  /* preallocate and create the prolongator */
  ierr = MatCreate(comm,P); CHKERRQ(ierr);
  ierr = MatGetType(G,&mtype);CHKERRQ(ierr);
  ierr = MatSetType(*P,mtype);CHKERRQ(ierr);

  ierr = MatSetSizes(*P,fn,cn,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
  ierr = MatMPIAIJSetPreallocation(*P,0,lsparse,0,gsparse);CHKERRQ(ierr);
  ierr = MatSeqAIJSetPreallocation(*P,0,lsparse);CHKERRQ(ierr);

  /* loop over local fine nodes -- get the diagonal, the sum of positive and negative strong and weak weights, and set up the row */
  for (i = 0;i < fn;i++) {
    /* determine on or off */
    row_f = i + fs;
    row_c = lcid[i];
    if (row_c >= 0) {
      pij = 1.;
      ierr = MatSetValues(*P,1,&row_f,1,&row_c,&pij,INSERT_VALUES);CHKERRQ(ierr);
    } else {
      g_pos = 0.;
      g_neg = 0.;
      a_pos = 0.;
      a_neg = 0.;
      diag = 0.;

      /* local connections */
      ierr = MatGetRow(lA,i,&ncols,&rcol,&rval);CHKERRQ(ierr);
      for (j = 0; j < ncols; j++) {
        col = rcol[j];
        if (lcid[col] >= 0 && (PetscRealPart(rval[j]) > gamg->threshold*Amax_pos[i] || PetscRealPart(-rval[j]) > gamg->threshold*Amax_neg[i])) {
          if (PetscRealPart(rval[j]) > 0.) {
            g_pos += rval[j];
          } else {
            g_neg += rval[j];
          }
        }
        if (col != i) {
          if (PetscRealPart(rval[j]) > 0.) {
            a_pos += rval[j];

PetscErrorCode PCGAMGTruncateProlongator_Private(PC pc,Mat *P)
{
  PetscInt          j,i,ps,pf,pn,pcs,pcf,pcn,idx,cmax;
  PetscErrorCode    ierr;
  const PetscScalar *pval;
  const PetscInt    *pcol;
  PetscScalar       *pnval;
  PetscInt          *pncol;
  PetscInt          ncols;
  Mat               Pnew;
  PetscInt          *lsparse,*gsparse;
  PetscReal         pmax_pos,pmax_neg,ptot_pos,ptot_neg,pthresh_pos,pthresh_neg;
  PC_MG             *mg          = (PC_MG*)pc->data;
  PC_GAMG           *pc_gamg     = (PC_GAMG*)mg->innerctx;
  PC_GAMG_Classical *cls         = (PC_GAMG_Classical*)pc_gamg->subctx;

  PetscFunctionBegin;
  /* trim and rescale with reallocation */
  ierr = MatGetOwnershipRange(*P,&ps,&pf);CHKERRQ(ierr);
  ierr = MatGetOwnershipRangeColumn(*P,&pcs,&pcf);CHKERRQ(ierr);
  pn = pf-ps;
  pcn = pcf-pcs;
  ierr = PetscMalloc(sizeof(PetscInt)*pn,&lsparse);CHKERRQ(ierr);
  ierr = PetscMalloc(sizeof(PetscInt)*pn,&gsparse);CHKERRQ(ierr);
  /* allocate */
  cmax = 0;
  for (i=ps;i<pf;i++) {
    lsparse[i-ps] = 0;
    gsparse[i-ps] = 0;
    ierr = MatGetRow(*P,i,&ncols,&pcol,&pval);CHKERRQ(ierr);
    if (ncols > cmax) {
      cmax = ncols;
    }
    pmax_pos = 0.;
    pmax_neg = 0.;
    for (j=0;j<ncols;j++) {
      if (PetscRealPart(pval[j]) > pmax_pos) {
        pmax_pos = PetscRealPart(pval[j]);
      } else if (PetscRealPart(pval[j]) < pmax_neg) {
        pmax_neg = PetscRealPart(pval[j]);
      }
    }
    for (j=0;j<ncols;j++) {
      if (PetscRealPart(pval[j]) >= pmax_pos*cls->interp_threshold || PetscRealPart(pval[j]) <= pmax_neg*cls->interp_threshold) {
        if (pcol[j] >= pcs && pcol[j] < pcf) {
          lsparse[i-ps]++;
        } else {
          gsparse[i-ps]++;
        }
      }
    }
    ierr = MatRestoreRow(*P,i,&ncols,&pcol,&pval);CHKERRQ(ierr);
  }

  ierr = PetscMalloc(sizeof(PetscScalar)*cmax,&pnval);CHKERRQ(ierr);
  ierr = PetscMalloc(sizeof(PetscInt)*cmax,&pncol);CHKERRQ(ierr);

  ierr = MatCreate(PetscObjectComm((PetscObject)*P),&Pnew);CHKERRQ(ierr);
  ierr = MatSetType(Pnew, MATAIJ);CHKERRQ(ierr);
  ierr = MatSetSizes(Pnew,pn,pcn,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
  ierr = MatSeqAIJSetPreallocation(Pnew,0,lsparse);CHKERRQ(ierr);
  ierr = MatMPIAIJSetPreallocation(Pnew,0,lsparse,0,gsparse);CHKERRQ(ierr);

  for (i=ps;i<pf;i++) {
    ierr = MatGetRow(*P,i,&ncols,&pcol,&pval);CHKERRQ(ierr);
    pmax_pos = 0.;
    pmax_neg = 0.;
    for (j=0;j<ncols;j++) {
      if (PetscRealPart(pval[j]) > pmax_pos) {
        pmax_pos = PetscRealPart(pval[j]);
      } else if (PetscRealPart(pval[j]) < pmax_neg) {
        pmax_neg = PetscRealPart(pval[j]);
      }
    }
    pthresh_pos = 0.;
    pthresh_neg = 0.;
    ptot_pos = 0.;
    ptot_neg = 0.;
    for (j=0;j<ncols;j++) {
      if (PetscRealPart(pval[j]) >= cls->interp_threshold*pmax_pos) {
        pthresh_pos += PetscRealPart(pval[j]);
      } else if (PetscRealPart(pval[j]) <= cls->interp_threshold*pmax_neg) {
        pthresh_neg += PetscRealPart(pval[j]);
      }
      if (PetscRealPart(pval[j]) > 0.) {
        ptot_pos += PetscRealPart(pval[j]);
      } else {
        ptot_neg += PetscRealPart(pval[j]);
      }
    }
    if (PetscAbsReal(pthresh_pos) > 0.) ptot_pos /= pthresh_pos;
    if (PetscAbsReal(pthresh_neg) > 0.) ptot_neg /= pthresh_neg;
    idx=0;
    for (j=0;j<ncols;j++) {
      if (PetscRealPart(pval[j]) >= pmax_pos*cls->interp_threshold) {
        pnval[idx] = ptot_pos*pval[j];
        pncol[idx] = pcol[j];
        idx++;
      } else if (PetscRealPart(pval[j]) <= pmax_neg*cls->interp_threshold) {
        pnval[idx] = ptot_neg*pval[j];
//.........这里部分代码省略.........

int main(int argc,char **argv)
{
  SNES           snes;                   /* SNES context */
  Vec            x,r,F,U;             /* vectors */
  Mat            J;                      /* Jacobian matrix */
  MonitorCtx     monP;                   /* monitoring context */
  PetscErrorCode ierr;
  PetscInt       its,n = 5,i,maxit,maxf;
  PetscMPIInt    size;
  PetscScalar    h,xp,v,none = -1.0;
  PetscReal      abstol,rtol,stol,norm;

  PetscInitialize(&argc,&argv,(char*)0,help);
  ierr = MPI_Comm_size(PETSC_COMM_WORLD,&size);CHKERRQ(ierr);
  if (size != 1) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"This is a uniprocessor example only!");
  ierr = PetscOptionsGetInt(NULL,"-n",&n,NULL);CHKERRQ(ierr);
  h    = 1.0/(n-1);

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Create nonlinear solver context
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

  ierr = SNESCreate(PETSC_COMM_WORLD,&snes);CHKERRQ(ierr);

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Create vector data structures; set function evaluation routine
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  /*
     Note that we form 1 vector from scratch and then duplicate as needed.
  */
  ierr = VecCreate(PETSC_COMM_WORLD,&x);CHKERRQ(ierr);
  ierr = VecSetSizes(x,PETSC_DECIDE,n);CHKERRQ(ierr);
  ierr = VecSetFromOptions(x);CHKERRQ(ierr);
  ierr = VecDuplicate(x,&r);CHKERRQ(ierr);
  ierr = VecDuplicate(x,&F);CHKERRQ(ierr);
  ierr = VecDuplicate(x,&U);CHKERRQ(ierr);

  /*
     Set function evaluation routine and vector
  */
  ierr = SNESSetFunction(snes,r,FormFunction,(void*)F);CHKERRQ(ierr);


  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Create matrix data structure; set Jacobian evaluation routine
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

  ierr = MatCreate(PETSC_COMM_WORLD,&J);CHKERRQ(ierr);
  ierr = MatSetSizes(J,PETSC_DECIDE,PETSC_DECIDE,n,n);CHKERRQ(ierr);
  ierr = MatSetFromOptions(J);CHKERRQ(ierr);
  ierr = MatSeqAIJSetPreallocation(J,3,NULL);CHKERRQ(ierr);

  /*
     Set Jacobian matrix data structure and default Jacobian evaluation
     routine. User can override with:
     -snes_fd : default finite differencing approximation of Jacobian
     -snes_mf : matrix-free Newton-Krylov method with no preconditioning
                (unless user explicitly sets preconditioner)
     -snes_mf_operator : form preconditioning matrix as set by the user,
                         but use matrix-free approx for Jacobian-vector
                         products within Newton-Krylov method
  */

  ierr = SNESSetJacobian(snes,J,J,FormJacobian,NULL);CHKERRQ(ierr);

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Customize nonlinear solver; set runtime options
   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

  /*
     Set an optional user-defined monitoring routine
  */
  ierr = PetscViewerDrawOpen(PETSC_COMM_WORLD,0,0,0,0,400,400,&monP.viewer);CHKERRQ(ierr);
  ierr = SNESMonitorSet(snes,Monitor,&monP,0);CHKERRQ(ierr);

  /*
     Set names for some vectors to facilitate monitoring (optional)
  */
  ierr = PetscObjectSetName((PetscObject)x,"Approximate Solution");CHKERRQ(ierr);
  ierr = PetscObjectSetName((PetscObject)U,"Exact Solution");CHKERRQ(ierr);

  /*
     Set SNES/KSP/KSP/PC runtime options, e.g.,
         -snes_view -snes_monitor -ksp_type <ksp> -pc_type <pc>
  */
  ierr = SNESSetFromOptions(snes);CHKERRQ(ierr);

  /*
     Print parameters used for convergence testing (optional) ... just
     to demonstrate this routine; this information is also printed with
     the option -snes_view
  */
  ierr = SNESGetTolerances(snes,&abstol,&rtol,&stol,&maxit,&maxf);CHKERRQ(ierr);
  ierr = PetscPrintf(PETSC_COMM_WORLD,"atol=%g, rtol=%g, stol=%g, maxit=%D, maxf=%D\n",(double)abstol,(double)rtol,(double)stol,maxit,maxf);CHKERRQ(ierr);

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Initialize application:
     Store right-hand-side of PDE and exact solution
   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

//.........这里部分代码省略.........

int main(int argc,char **argv)
{
  TS             ts;            /* nonlinear solver */
  Vec            x;             /* solution, residual vectors */
  Mat            A;             /* Jacobian matrix */
  PetscInt       steps;
  PetscReal      ftime   = 0.5;
  PetscBool      monitor = PETSC_FALSE;
  PetscScalar    *x_ptr;
  PetscMPIInt    size;
  struct _n_User user;
  PetscErrorCode ierr;

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Initialize program
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  PetscInitialize(&argc,&argv,NULL,help);

  ierr = MPI_Comm_size(PETSC_COMM_WORLD,&size);CHKERRQ(ierr);
  if (size != 1) SETERRQ(PETSC_COMM_SELF,1,"This is a uniprocessor example only!");

  ierr = RegisterMyARK2();CHKERRQ(ierr);

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    Set runtime options
    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

  user.next_output = 0.0;
  ierr = PetscOptionsGetBool(NULL,NULL,"-monitor",&monitor,NULL);CHKERRQ(ierr);

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    Create necessary matrix and vectors, solve same ODE on every process
    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  ierr = MatCreate(PETSC_COMM_WORLD,&A);CHKERRQ(ierr);
  ierr = MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,2,2);CHKERRQ(ierr);
  ierr = MatSetFromOptions(A);CHKERRQ(ierr);
  ierr = MatSetUp(A);CHKERRQ(ierr);
  ierr = MatCreateVecs(A,&x,NULL);CHKERRQ(ierr);

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Create timestepping solver context
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  ierr = TSCreate(PETSC_COMM_WORLD,&ts);CHKERRQ(ierr);
  ierr = TSSetType(ts,TSBEULER);CHKERRQ(ierr);
  ierr = TSSetIFunction(ts,NULL,IFunction,&user);CHKERRQ(ierr);
  ierr = TSSetIJacobian(ts,A,A,IJacobian,&user);CHKERRQ(ierr);
  ierr = TSSetDuration(ts,PETSC_DEFAULT,ftime);CHKERRQ(ierr);
  if (monitor) {
    ierr = TSMonitorSet(ts,Monitor,&user,NULL);CHKERRQ(ierr);
  }

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Set initial conditions
   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  ierr = VecGetArray(x,&x_ptr);CHKERRQ(ierr);
  x_ptr[0] = -2;   x_ptr[1] = -2.355301397608119909925287735864250951918;
  ierr = VecRestoreArray(x,&x_ptr);CHKERRQ(ierr);
  ierr = TSSetInitialTimeStep(ts,0.0,.001);CHKERRQ(ierr);

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Set runtime options
   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  ierr = TSSetFromOptions(ts);CHKERRQ(ierr);

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Solve nonlinear system
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  ierr = TSSolve(ts,x);CHKERRQ(ierr);
  ierr = TSGetSolveTime(ts,&ftime);CHKERRQ(ierr);
  ierr = TSGetTimeStepNumber(ts,&steps);CHKERRQ(ierr);
  ierr = PetscPrintf(PETSC_COMM_WORLD,"steps %D, ftime %g\n",steps,(double)ftime);CHKERRQ(ierr);
  ierr = VecView(x,PETSC_VIEWER_STDOUT_WORLD);CHKERRQ(ierr);

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Free work space.  All PETSc objects should be destroyed when they
     are no longer needed.
   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  ierr = MatDestroy(&A);CHKERRQ(ierr);
  ierr = VecDestroy(&x);CHKERRQ(ierr);
  ierr = TSDestroy(&ts);CHKERRQ(ierr);

  ierr = PetscFinalize();
  PetscFunctionReturn(0);
}

EXTERN_C_END
  

/*MC
  MATSOLVERUMFPACK = "umfpack" - A matrix type providing direct solvers (LU) for sequential matrices
  via the external package UMFPACK.

  ./configure --download-umfpack to install PETSc to use UMFPACK

  Consult UMFPACK documentation for more information about the Control parameters
  which correspond to the options database keys below.

  Options Database Keys:
+ -mat_umfpack_prl                     - UMFPACK print level: Control[UMFPACK_PRL]
. -mat_umfpack_strategy <AUTO>         - (choose one of) AUTO UNSYMMETRIC SYMMETRIC 2BY2
. -mat_umfpack_dense_col <alpha_c>     - UMFPACK dense column threshold: Control[UMFPACK_DENSE_COL]
. -mat_umfpack_dense_row <0.2>         - Control[UMFPACK_DENSE_ROW]
. -mat_umfpack_amd_dense <10>          - Control[UMFPACK_AMD_DENSE]
. -mat_umfpack_block_size <bs>         - UMFPACK block size for BLAS-Level 3 calls: Control[UMFPACK_BLOCK_SIZE]
. -mat_umfpack_2by2_tolerance <0.01>   - Control[UMFPACK_2BY2_TOLERANCE]
. -mat_umfpack_fixq <0>                - Control[UMFPACK_FIXQ]
. -mat_umfpack_aggressive <1>          - Control[UMFPACK_AGGRESSIVE]
. -mat_umfpack_pivot_tolerance <delta> - UMFPACK partial pivot tolerance: Control[UMFPACK_PIVOT_TOLERANCE]
.  -mat_umfpack_sym_pivot_tolerance <0.001> - Control[UMFPACK_SYM_PIVOT_TOLERANCE]
.  -mat_umfpack_scale <NONE>           - (choose one of) NONE SUM MAX
. -mat_umfpack_alloc_init <delta>      - UMFPACK factorized matrix allocation modifier: Control[UMFPACK_ALLOC_INIT]
.  -mat_umfpack_droptol <0>            - Control[UMFPACK_DROPTOL]
- -mat_umfpack_irstep <maxit>          - UMFPACK maximum number of iterative refinement steps: Control[UMFPACK_IRSTEP]

   Level: beginner

.seealso: PCLU, MATSOLVERSUPERLU, MATSOLVERMUMPS, MATSOLVERSPOOLES, PCFactorSetMatSolverPackage(), MatSolverPackage
M*/
EXTERN_C_BEGIN
#undef __FUNCT__  
#define __FUNCT__ "MatGetFactor_seqaij_umfpack"
PetscErrorCode MatGetFactor_seqaij_umfpack(Mat A,MatFactorType ftype,Mat *F) 
{
  Mat            B;
  Mat_UMFPACK    *lu;
  PetscErrorCode ierr;
  PetscInt       m=A->rmap->n,n=A->cmap->n,idx;

  const char     *strategy[]={"AUTO","UNSYMMETRIC","SYMMETRIC"},
                 *scale[]={"NONE","SUM","MAX"}; 
  PetscBool      flg;
  
  PetscFunctionBegin;
  /* Create the factorization matrix F */  
  ierr = MatCreate(((PetscObject)A)->comm,&B);CHKERRQ(ierr);
  ierr = MatSetSizes(B,PETSC_DECIDE,PETSC_DECIDE,m,n);CHKERRQ(ierr);
  ierr = MatSetType(B,((PetscObject)A)->type_name);CHKERRQ(ierr);
  ierr = MatSeqAIJSetPreallocation(B,0,PETSC_NULL);CHKERRQ(ierr);
  ierr = PetscNewLog(B,Mat_UMFPACK,&lu);CHKERRQ(ierr);
  B->spptr                 = lu;
  B->ops->lufactorsymbolic = MatLUFactorSymbolic_UMFPACK;
  B->ops->destroy          = MatDestroy_UMFPACK;
  B->ops->view             = MatView_UMFPACK;
  ierr = PetscObjectComposeFunctionDynamic((PetscObject)B,"MatFactorGetSolverPackage_C","MatFactorGetSolverPackage_seqaij_umfpack",MatFactorGetSolverPackage_seqaij_umfpack);CHKERRQ(ierr);
  B->factortype            = MAT_FACTOR_LU;
  B->assembled             = PETSC_TRUE;  /* required by -ksp_view */
  B->preallocated          = PETSC_TRUE;
  
  /* initializations */
  /* ------------------------------------------------*/
  /* get the default control parameters */
  umfpack_UMF_defaults(lu->Control);
  lu->perm_c = PETSC_NULL;  /* use defaul UMFPACK col permutation */
  lu->Control[UMFPACK_IRSTEP] = 0; /* max num of iterative refinement steps to attempt */

  ierr = PetscOptionsBegin(((PetscObject)A)->comm,((PetscObject)A)->prefix,"UMFPACK Options","Mat");CHKERRQ(ierr);
  /* Control parameters used by reporting routiones */
  ierr = PetscOptionsReal("-mat_umfpack_prl","Control[UMFPACK_PRL]","None",lu->Control[UMFPACK_PRL],&lu->Control[UMFPACK_PRL],PETSC_NULL);CHKERRQ(ierr);

  /* Control parameters for symbolic factorization */
  ierr = PetscOptionsEList("-mat_umfpack_strategy","ordering and pivoting strategy","None",strategy,3,strategy[0],&idx,&flg);CHKERRQ(ierr);
  if (flg) {
    switch (idx){
    case 0: lu->Control[UMFPACK_STRATEGY] = UMFPACK_STRATEGY_AUTO; break;
    case 1: lu->Control[UMFPACK_STRATEGY] = UMFPACK_STRATEGY_UNSYMMETRIC; break;
    case 2: lu->Control[UMFPACK_STRATEGY] = UMFPACK_STRATEGY_SYMMETRIC; break;
    }
  }
  ierr = PetscOptionsEList("-mat_umfpack_ordering","Internal ordering method","None",UmfpackOrderingTypes,sizeof UmfpackOrderingTypes/sizeof UmfpackOrderingTypes[0],UmfpackOrderingTypes[(int)lu->Control[UMFPACK_ORDERING]],&idx,&flg);CHKERRQ(ierr);
  if (flg) lu->Control[UMFPACK_ORDERING] = (int)idx;
  ierr = PetscOptionsReal("-mat_umfpack_dense_col","Control[UMFPACK_DENSE_COL]","None",lu->Control[UMFPACK_DENSE_COL],&lu->Control[UMFPACK_DENSE_COL],PETSC_NULL);CHKERRQ(ierr);
  ierr = PetscOptionsReal("-mat_umfpack_dense_row","Control[UMFPACK_DENSE_ROW]","None",lu->Control[UMFPACK_DENSE_ROW],&lu->Control[UMFPACK_DENSE_ROW],PETSC_NULL);CHKERRQ(ierr);
  ierr = PetscOptionsReal("-mat_umfpack_amd_dense","Control[UMFPACK_AMD_DENSE]","None",lu->Control[UMFPACK_AMD_DENSE],&lu->Control[UMFPACK_AMD_DENSE],PETSC_NULL);CHKERRQ(ierr);
  ierr = PetscOptionsReal("-mat_umfpack_block_size","Control[UMFPACK_BLOCK_SIZE]","None",lu->Control[UMFPACK_BLOCK_SIZE],&lu->Control[UMFPACK_BLOCK_SIZE],PETSC_NULL);CHKERRQ(ierr);
  ierr = PetscOptionsReal("-mat_umfpack_fixq","Control[UMFPACK_FIXQ]","None",lu->Control[UMFPACK_FIXQ],&lu->Control[UMFPACK_FIXQ],PETSC_NULL);CHKERRQ(ierr);
  ierr = PetscOptionsReal("-mat_umfpack_aggressive","Control[UMFPACK_AGGRESSIVE]","None",lu->Control[UMFPACK_AGGRESSIVE],&lu->Control[UMFPACK_AGGRESSIVE],PETSC_NULL);CHKERRQ(ierr);

  /* Control parameters used by numeric factorization */
  ierr = PetscOptionsReal("-mat_umfpack_pivot_tolerance","Control[UMFPACK_PIVOT_TOLERANCE]","None",lu->Control[UMFPACK_PIVOT_TOLERANCE],&lu->Control[UMFPACK_PIVOT_TOLERANCE],PETSC_NULL);CHKERRQ(ierr);
  ierr = PetscOptionsReal("-mat_umfpack_sym_pivot_tolerance","Control[UMFPACK_SYM_PIVOT_TOLERANCE]","None",lu->Control[UMFPACK_SYM_PIVOT_TOLERANCE],&lu->Control[UMFPACK_SYM_PIVOT_TOLERANCE],PETSC_NULL);CHKERRQ(ierr);
  ierr = PetscOptionsEList("-mat_umfpack_scale","Control[UMFPACK_SCALE]","None",scale,3,scale[0],&idx,&flg);CHKERRQ(ierr);
  if (flg) {
    switch (idx){
    case 0: lu->Control[UMFPACK_SCALE] = UMFPACK_SCALE_NONE; break;
    case 1: lu->Control[UMFPACK_SCALE] = UMFPACK_SCALE_SUM; break;
//.........这里部分代码省略.........