static const char help[] = "-Laplacian u = b as a nonlinear problem.\n\n";

/*
    The linear and nonlinear versions of these should give almost identical results on this problem

    Richardson
      Nonlinear:
        -snes_rtol 1.e-12 -snes_monitor -snes_type nrichardson -snes_linesearch_monitor

      Linear:
        -snes_rtol 1.e-12 -snes_monitor -ksp_rtol 1.e-12  -ksp_monitor -ksp_type richardson -pc_type none -ksp_richardson_self_scale -info

    GMRES
      Nonlinear:
       -snes_rtol 1.e-12 -snes_monitor  -snes_type ngmres

      Linear:
       -snes_rtol 1.e-12 -snes_monitor  -ksp_type gmres -ksp_monitor -ksp_rtol 1.e-12 -pc_type none

    CG
       Nonlinear:
            -snes_rtol 1.e-12 -snes_monitor  -snes_type ncg -snes_linesearch_monitor

       Linear:
             -snes_rtol 1.e-12 -snes_monitor  -ksp_type cg -ksp_monitor -ksp_rtol 1.e-12 -pc_type none

    Multigrid
       Linear:
          1 level:
            -snes_rtol 1.e-12 -snes_monitor  -pc_type mg -mg_levels_ksp_type richardson -mg_levels_pc_type none -mg_levels_ksp_monitor
            -mg_levels_ksp_richardson_self_scale -ksp_type richardson -ksp_monitor -ksp_rtol 1.e-12  -ksp_monitor_true_residual

          n levels:
            -da_refine n

       Nonlinear:
         1 level:
           -snes_rtol 1.e-12 -snes_monitor  -snes_type fas -fas_levels_snes_monitor

          n levels:
            -da_refine n  -fas_coarse_snes_type newtonls -fas_coarse_pc_type lu -fas_coarse_ksp_type preonly
*/

/*
   Include "petscdmda.h" so that we can use distributed arrays (DMDAs).
   Include "petscsnes.h" so that we can use SNES solvers.  Note that this
*/
#include <petscdm.h>
#include <petscdmda.h>
#include <petscsnes.h>

/*
   User-defined routines
*/
extern PetscErrorCode FormMatrix(DM, Mat);
extern PetscErrorCode MyComputeFunction(SNES, Vec, Vec, void *);
extern PetscErrorCode MyComputeJacobian(SNES, Vec, Mat, Mat, void *);
extern PetscErrorCode NonlinearGS(SNES, Vec);

int main(int argc, char **argv)
{
  SNES      snes;  /* nonlinear solver */
  SNES      psnes; /* nonlinear Gauss-Seidel approximate solver */
  Vec       x, b;  /* solution vector */
  PetscInt  its;   /* iterations for convergence */
  DM        da;
  PetscBool use_ngs_as_npc = PETSC_FALSE; /* use the nonlinear Gauss-Seidel approximate solver */

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Initialize program
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

  PetscFunctionBeginUser;
  PetscCall(PetscInitialize(&argc, &argv, NULL, help));

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Create nonlinear solver context
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  PetscCall(SNESCreate(PETSC_COMM_WORLD, &snes));

  PetscCall(PetscOptionsGetBool(NULL, NULL, "-use_ngs_as_npc", &use_ngs_as_npc, 0));

  if (use_ngs_as_npc) {
    PetscCall(SNESGetNPC(snes, &psnes));
    PetscCall(SNESSetType(psnes, SNESSHELL));
    PetscCall(SNESShellSetSolve(psnes, NonlinearGS));
  }

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Create distributed array (DMDA) to manage parallel grid and vectors
  - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  PetscCall(DMDACreate2d(PETSC_COMM_WORLD, DM_BOUNDARY_NONE, DM_BOUNDARY_NONE, DMDA_STENCIL_STAR, 4, 4, PETSC_DECIDE, PETSC_DECIDE, 1, 1, NULL, NULL, &da));
  PetscCall(DMSetFromOptions(da));
  PetscCall(DMSetUp(da));
  PetscCall(DMDASetUniformCoordinates(da, 0.0, 1.0, 0.0, 1.0, 0.0, 1.0));
  PetscCall(SNESSetDM(snes, da));
  if (use_ngs_as_npc) PetscCall(SNESShellSetContext(psnes, da));
  /*  - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Extract global vectors from DMDA; then duplicate for remaining
     vectors that are the same types
   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  PetscCall(DMCreateGlobalVector(da, &x));
  PetscCall(DMCreateGlobalVector(da, &b));
  PetscCall(VecSet(b, 1.0));

  PetscCall(SNESSetFunction(snes, NULL, MyComputeFunction, NULL));
  PetscCall(SNESSetJacobian(snes, NULL, NULL, MyComputeJacobian, NULL));

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Customize nonlinear solver; set runtime options
   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  PetscCall(SNESSetFromOptions(snes));

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Solve nonlinear system
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  PetscCall(SNESSolve(snes, b, x));
  PetscCall(SNESGetIterationNumber(snes, &its));

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Free work space.  All PETSc objects should be destroyed when they
     are no longer needed.
   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  PetscCall(VecDestroy(&x));
  PetscCall(VecDestroy(&b));
  PetscCall(SNESDestroy(&snes));
  PetscCall(DMDestroy(&da));
  PetscCall(PetscFinalize());
  return 0;
}

/* ------------------------------------------------------------------- */
PetscErrorCode MyComputeFunction(SNES snes, Vec x, Vec F, PetscCtx ctx)
{
  Mat J;
  DM  dm;

  PetscFunctionBeginUser;
  PetscCall(SNESGetDM(snes, &dm));
  PetscCall(PetscObjectQuery((PetscObject)dm, "_ex35_J", (PetscObject *)&J));
  if (!J) {
    PetscCall(DMSetMatType(dm, MATAIJ));
    PetscCall(DMCreateMatrix(dm, &J));
    PetscCall(MatSetDM(J, NULL));
    PetscCall(FormMatrix(dm, J));
    PetscCall(PetscObjectCompose((PetscObject)dm, "_ex35_J", (PetscObject)J));
    PetscCall(PetscObjectDereference((PetscObject)J));
  }
  PetscCall(MatMult(J, x, F));
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode MyComputeJacobian(SNES snes, Vec x, Mat J, Mat Jp, PetscCtx ctx)
{
  DM dm;

  PetscFunctionBeginUser;
  PetscCall(SNESGetDM(snes, &dm));
  PetscCall(FormMatrix(dm, Jp));
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode FormMatrix(DM da, Mat jac)
{
  PetscInt      i, j, nrows = 0;
  MatStencil    col[5], row, *rows;
  PetscScalar   v[5], hx, hy, hxdhy, hydhx;
  DMDALocalInfo info;

  PetscFunctionBeginUser;
  PetscCall(DMDAGetLocalInfo(da, &info));
  hx    = 1.0 / (PetscReal)(info.mx - 1);
  hy    = 1.0 / (PetscReal)(info.my - 1);
  hxdhy = hx / hy;
  hydhx = hy / hx;

  PetscCall(PetscMalloc1(info.ym * info.xm, &rows));
  /*
     Compute entries for the locally owned part of the Jacobian.
      - Currently, all PETSc parallel matrix formats are partitioned by
        contiguous chunks of rows across the processors.
      - Each processor needs to insert only elements that it owns
        locally (but any non-local elements will be sent to the
        appropriate processor during matrix assembly).
      - Here, we set all entries for a particular row at once.
      - We can set matrix entries either using either
        MatSetValuesLocal() or MatSetValues(), as discussed above.
  */
  for (j = info.ys; j < info.ys + info.ym; j++) {
    for (i = info.xs; i < info.xs + info.xm; i++) {
      row.j = j;
      row.i = i;
      /* boundary points */
      if (i == 0 || j == 0 || i == info.mx - 1 || j == info.my - 1) {
        v[0] = 2.0 * (hydhx + hxdhy);
        PetscCall(MatSetValuesStencil(jac, 1, &row, 1, &row, v, INSERT_VALUES));
        rows[nrows].i   = i;
        rows[nrows++].j = j;
      } else {
        /* interior grid points */
        v[0]     = -hxdhy;
        col[0].j = j - 1;
        col[0].i = i;
        v[1]     = -hydhx;
        col[1].j = j;
        col[1].i = i - 1;
        v[2]     = 2.0 * (hydhx + hxdhy);
        col[2].j = row.j;
        col[2].i = row.i;
        v[3]     = -hydhx;
        col[3].j = j;
        col[3].i = i + 1;
        v[4]     = -hxdhy;
        col[4].j = j + 1;
        col[4].i = i;
        PetscCall(MatSetValuesStencil(jac, 1, &row, 5, col, v, INSERT_VALUES));
      }
    }
  }

  /*
     Assemble matrix, using the 2-step process:
       MatAssemblyBegin(), MatAssemblyEnd().
  */
  PetscCall(MatAssemblyBegin(jac, MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(jac, MAT_FINAL_ASSEMBLY));
  PetscCall(MatZeroRowsColumnsStencil(jac, nrows, rows, 2.0 * (hydhx + hxdhy), NULL, NULL));
  PetscCall(PetscFree(rows));
  /*
     Tell the matrix we will never add a new nonzero location to the
     matrix. If we do, it will generate an error.
  */
  PetscCall(MatSetOption(jac, MAT_NEW_NONZERO_LOCATION_ERR, PETSC_TRUE));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/* ------------------------------------------------------------------- */
/*
      Applies some sweeps on nonlinear Gauss-Seidel on each process

 */
PetscErrorCode NonlinearGS(SNES snes, Vec X)
{
  PetscInt      i, j, Mx, My, xs, ys, xm, ym, its, l;
  PetscReal     hx, hy, hxdhy, hydhx;
  PetscScalar **x, F, J, u, uxx, uyy;
  DM            da;
  Vec           localX;

  PetscFunctionBeginUser;
  PetscCall(SNESGetTolerances(snes, NULL, NULL, NULL, &its, NULL));
  PetscCall(SNESShellGetContext(snes, &da));

  PetscCall(DMDAGetInfo(da, PETSC_IGNORE, &Mx, &My, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE));

  hx    = 1.0 / (PetscReal)(Mx - 1);
  hy    = 1.0 / (PetscReal)(My - 1);
  hxdhy = hx / hy;
  hydhx = hy / hx;

  PetscCall(DMGetLocalVector(da, &localX));

  for (l = 0; l < its; l++) {
    PetscCall(DMGlobalToLocalBegin(da, X, INSERT_VALUES, localX));
    PetscCall(DMGlobalToLocalEnd(da, X, INSERT_VALUES, localX));
    /*
     Get a pointer to vector data.
     - For default PETSc vectors, VecGetArray() returns a pointer to
     the data array.  Otherwise, the routine is implementation dependent.
     - You MUST call VecRestoreArray() when you no longer need access to
     the array.
     */
    PetscCall(DMDAVecGetArray(da, localX, &x));

    /*
     Get local grid boundaries (for 2-dimensional DMDA):
     xs, ys   - starting grid indices (no ghost points)
     xm, ym   - widths of local grid (no ghost points)

     */
    PetscCall(DMDAGetCorners(da, &xs, &ys, NULL, &xm, &ym, NULL));

    for (j = ys; j < ys + ym; j++) {
      for (i = xs; i < xs + xm; i++) {
        if (i == 0 || j == 0 || i == Mx - 1 || j == My - 1) {
          /* boundary conditions are all zero Dirichlet */
          x[j][i] = 0.0;
        } else {
          u   = x[j][i];
          uxx = (2.0 * u - x[j][i - 1] - x[j][i + 1]) * hydhx;
          uyy = (2.0 * u - x[j - 1][i] - x[j + 1][i]) * hxdhy;
          F   = uxx + uyy;
          J   = 2.0 * (hydhx + hxdhy);
          u   = u - F / J;

          x[j][i] = u;
        }
      }
    }

    /*
     Restore vector
     */
    PetscCall(DMDAVecRestoreArray(da, localX, &x));
    PetscCall(DMLocalToGlobalBegin(da, localX, INSERT_VALUES, X));
    PetscCall(DMLocalToGlobalEnd(da, localX, INSERT_VALUES, X));
  }
  PetscCall(DMRestoreLocalVector(da, &localX));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*TEST

   test:
      args: -snes_monitor_short -snes_type nrichardson
      requires: !single

   test:
      suffix: 2
      args: -snes_monitor_short -ksp_monitor_short -ksp_type richardson -pc_type none -ksp_richardson_self_scale
      requires: !single

   test:
      suffix: 3
      args: -snes_monitor_short -snes_type ngmres

   test:
      suffix: 4
      args: -snes_monitor_short -ksp_type gmres -ksp_monitor_short -pc_type none

   test:
      suffix: 5
      args: -snes_monitor_short -snes_type ncg

   test:
      suffix: 6
      args: -snes_monitor_short -ksp_type cg -ksp_monitor_short -pc_type none

   test:
      suffix: 7
      args: -da_refine 2 -snes_monitor_short -pc_type mg -mg_levels_ksp_type richardson -mg_levels_pc_type none -mg_levels_ksp_monitor_short -mg_levels_ksp_richardson_self_scale -ksp_type richardson -ksp_monitor_short
      requires: !single

   test:
      suffix: 8
      args: -da_refine 2 -snes_monitor_short -snes_type fas -fas_levels_snes_monitor_short -fas_coarse_snes_type newtonls -fas_coarse_pc_type lu -fas_coarse_ksp_type preonly -snes_type fas -snes_rtol 1.e-5

   test:
      suffix: 9
      args: -snes_monitor_short -ksp_type gmres -ksp_monitor_short -pc_type none -snes_type newtontrdc

   test:
      suffix: 10
      args: -snes_monitor_short -ksp_type gmres -ksp_monitor_short -pc_type none -snes_type newtontrdc -snes_trdc_use_cauchy false

TEST*/
