static char help[] = "Basic equation for generator stability analysis.\n";

/*F

\begin{eqnarray}
                 \frac{2 H}{\omega_s}\frac{d \omega}{dt} & = & P_m - \frac{EV}{X} \sin(\theta) \\
                 \frac{d \theta}{dt} = \omega - \omega_s
\end{eqnarray}

F*/

/*
   Include "petscts.h" so that we can use TS solvers.  Note that this
   file automatically includes:
     petscsys.h       - base PETSc routines   petscvec.h - vectors
     petscmat.h - matrices
     petscis.h     - index sets            petscksp.h - Krylov subspace methods
     petscviewer.h - viewers               petscpc.h  - preconditioners
     petscksp.h   - linear solvers
*/

#include <petscts.h>

typedef struct {
  PetscScalar H, omega_s, E, V, X;
  PetscRandom rand;
} AppCtx;

/*
     Defines the ODE passed to the ODE solver
*/
static PetscErrorCode IFunction(TS ts, PetscReal t, Vec U, Vec Udot, Vec F, AppCtx *ctx)
{
  PetscScalar       *f, r;
  const PetscScalar *u, *udot;
  static PetscScalar R = .4;

  PetscFunctionBegin;
  PetscCall(PetscRandomGetValue(ctx->rand, &r));
  if (r > .9) R = .5;
  if (r < .1) R = .4;
  R = .4;
  /*  The next three lines allow us to access the entries of the vectors directly */
  PetscCall(VecGetArrayRead(U, &u));
  PetscCall(VecGetArrayRead(Udot, &udot));
  PetscCall(VecGetArray(F, &f));
  f[0] = 2.0 * ctx->H * udot[0] / ctx->omega_s + ctx->E * ctx->V * PetscSinScalar(u[1]) / ctx->X - R;
  f[1] = udot[1] - u[0] + ctx->omega_s;

  PetscCall(VecRestoreArrayRead(U, &u));
  PetscCall(VecRestoreArrayRead(Udot, &udot));
  PetscCall(VecRestoreArray(F, &f));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*
     Defines the Jacobian of the ODE passed to the ODE solver. See TSSetIJacobian() for the meaning of a and the Jacobian.
*/
static PetscErrorCode IJacobian(TS ts, PetscReal t, Vec U, Vec Udot, PetscReal a, Mat A, Mat B, AppCtx *ctx)
{
  PetscInt           rowcol[] = {0, 1};
  PetscScalar        J[2][2];
  const PetscScalar *u, *udot;

  PetscFunctionBegin;
  PetscCall(VecGetArrayRead(U, &u));
  PetscCall(VecGetArrayRead(Udot, &udot));
  J[0][0] = 2.0 * ctx->H * a / ctx->omega_s;
  J[0][1] = -ctx->E * ctx->V * PetscCosScalar(u[1]) / ctx->X;
  J[1][0] = -1.0;
  J[1][1] = a;
  PetscCall(MatSetValues(B, 2, rowcol, 2, rowcol, &J[0][0], INSERT_VALUES));
  PetscCall(VecRestoreArrayRead(U, &u));
  PetscCall(VecRestoreArrayRead(Udot, &udot));

  PetscCall(MatAssemblyBegin(A, MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(A, MAT_FINAL_ASSEMBLY));
  if (A != B) {
    PetscCall(MatAssemblyBegin(B, MAT_FINAL_ASSEMBLY));
    PetscCall(MatAssemblyEnd(B, MAT_FINAL_ASSEMBLY));
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

int main(int argc, char **argv)
{
  TS           ts; /* ODE integrator */
  Vec          U;  /* solution will be stored here */
  Mat          A;  /* Jacobian matrix */
  PetscMPIInt  size;
  PetscInt     n = 2;
  AppCtx       ctx;
  PetscScalar *u;

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Initialize program
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  PetscFunctionBeginUser;
  PetscCall(PetscInitialize(&argc, &argv, NULL, help));
  PetscCallMPI(MPI_Comm_size(PETSC_COMM_WORLD, &size));
  PetscCheck(size == 1, PETSC_COMM_WORLD, PETSC_ERR_WRONG_MPI_SIZE, "Only for sequential runs");

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    Create necessary matrix and vectors
    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  PetscCall(MatCreate(PETSC_COMM_WORLD, &A));
  PetscCall(MatSetSizes(A, n, n, PETSC_DETERMINE, PETSC_DETERMINE));
  PetscCall(MatSetFromOptions(A));
  PetscCall(MatSetUp(A));

  PetscCall(MatCreateVecs(A, &U, NULL));

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    Set runtime options
    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  PetscOptionsBegin(PETSC_COMM_WORLD, NULL, "Reaction options", "");
  {
    ctx.omega_s = 1.0;
    PetscCall(PetscOptionsScalar("-omega_s", "", "", ctx.omega_s, &ctx.omega_s, NULL));
    ctx.H = 1.0;
    PetscCall(PetscOptionsScalar("-H", "", "", ctx.H, &ctx.H, NULL));
    ctx.E = 1.0;
    PetscCall(PetscOptionsScalar("-E", "", "", ctx.E, &ctx.E, NULL));
    ctx.V = 1.0;
    PetscCall(PetscOptionsScalar("-V", "", "", ctx.V, &ctx.V, NULL));
    ctx.X = 1.0;
    PetscCall(PetscOptionsScalar("-X", "", "", ctx.X, &ctx.X, NULL));

    PetscCall(VecGetArray(U, &u));
    u[0] = 1;
    u[1] = .7;
    PetscCall(VecRestoreArray(U, &u));
    PetscCall(PetscOptionsGetVec(NULL, NULL, "-initial", U, NULL));
  }
  PetscOptionsEnd();

  PetscCall(PetscRandomCreate(PETSC_COMM_WORLD, &ctx.rand));
  PetscCall(PetscRandomSetFromOptions(ctx.rand));

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Create timestepping solver context
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  PetscCall(TSCreate(PETSC_COMM_WORLD, &ts));
  PetscCall(TSSetProblemType(ts, TS_NONLINEAR));
  PetscCall(TSSetType(ts, TSROSW));
  PetscCall(TSSetIFunction(ts, NULL, (TSIFunctionFn *)IFunction, &ctx));
  PetscCall(TSSetIJacobian(ts, A, A, (TSIJacobianFn *)IJacobian, &ctx));

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Set initial conditions
   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  PetscCall(TSSetSolution(ts, U));

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Set solver options
   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  PetscCall(TSSetMaxTime(ts, 2000.0));
  PetscCall(TSSetExactFinalTime(ts, TS_EXACTFINALTIME_MATCHSTEP));
  PetscCall(TSSetTimeStep(ts, .001));
  PetscCall(TSSetFromOptions(ts));

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Solve nonlinear system
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  PetscCall(TSSolve(ts, U));

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Free work space.  All PETSc objects should be destroyed when they are no longer needed.
   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  PetscCall(MatDestroy(&A));
  PetscCall(VecDestroy(&U));
  PetscCall(TSDestroy(&ts));
  PetscCall(PetscRandomDestroy(&ctx.rand));
  PetscCall(PetscFinalize());
  return 0;
}

/*TEST

   build:
     requires: !complex !single

   test:
      args: -ksp_gmres_cgs_refinement_type refine_always -snes_type newtonls -ts_max_steps 10
      output_file: output/empty.out

   test:
      suffix: 2
      args: -ts_max_steps 10
      output_file: output/empty.out

TEST*/