text: /*-----------------------------------------------------------------
   * Programmer(s): Daniel R. Reynolds @ SMU
   *---------------------------------------------------------------
   * SUNDIALS Copyright Start
   * Copyright (c) 2002-2019, Lawrence Livermore National Security
   * and Southern Methodist University.
   * All rights reserved.
   *
   * See the top-level LICENSE and NOTICE files for details.
   *
   * SPDX-License-Identifier: BSD-3-Clause
   * SUNDIALS Copyright End
   *---------------------------------------------------------------
   * Example problem:
   *
   * The following is a simple example problem with analytical
   * solution,
   *    dy/dt = lamda*y + 1/(1+t^2) - lamda*atan(t)
   * for t in the interval [0.0, 10.0], with initial condition: y=0.
   *
   * The stiffness of the problem is directly proportional to the
   * value of "lamda".  The value of lamda should be negative to
   * result in a well-posed ODE; for values with magnitude larger
   * than 100 the problem becomes quite stiff.
   *
   * This program solves the problem with the DIRK method,
   * Newton iteration with the dense SUNLinearSolver, and a
   * user-supplied Jacobian routine.
   * Output is printed every 1.0 units of time (10 total).
   * Run statistics (optional outputs) are printed at the end.
   *-----------------------------------------------------------------*/
  
  /* Header files */
  #include <stdio.h>
  #include <math.h>
  #include <arkode/arkode_arkstep.h>         /* prototypes for ARKStep fcts., consts */
  #include <nvector/nvector_serial.h>        /* serial N_Vector types, fcts., macros */
  #include <sunmatrix/sunmatrix_dense.h>     /* access to dense SUNMatrix            */
  #include <sunlinsol/sunlinsol_dense.h>     /* access to dense SUNLinearSolver      */
  #include <sundials/sundials_types.h>       /* definition of type realtype          */
  #include <sundials/sundials_math.h>
  
  #if defined(SUNDIALS_EXTENDED_PRECISION)
  #define GSYM "Lg"
  #define ESYM "Le"
  #define FSYM "Lf"
  #else
  #define GSYM "g"
  #define ESYM "e"
  #define FSYM "f"
  #endif
  
  /* User-supplied Functions Called by the Solver */
  static int f(realtype t, N_Vector y, N_Vector ydot, void *user_data);
  static int Jac(realtype t, N_Vector y, N_Vector fy, SUNMatrix J,
                 void *user_data, N_Vector tmp1, N_Vector tmp2, N_Vector tmp3);
  
  /* Private function to check function return values */
  static int check_flag(void *flagvalue, const char *funcname, int opt);
  
  /* Private function to check computed solution */
  static int check_ans(N_Vector y, realtype t, realtype rtol, realtype atol);
  
  /* Main Program */
  int main()
  {
    /* general problem parameters */
    realtype T0 = RCONST(0.0);      /* initial time */
    realtype Tf = RCONST(10.0);     /* final time */
    realtype dTout = RCONST(1.0);   /* time between outputs */
    sunindextype NEQ = 1;           /* number of dependent vars. */
    realtype reltol = 1.0e-6;       /* tolerances */
    realtype abstol = 1.0e-10;
    realtype lamda  = -100.0;       /* stiffness parameter */
  
    /* general problem variables */
    int flag;                       /* reusable error-checking flag */
    N_Vector y = NULL;              /* empty vector for storing solution */
    SUNMatrix A = NULL;             /* empty matrix for linear solver */
    SUNLinearSolver LS = NULL;      /* empty linear solver object */
    void *arkode_mem = NULL;        /* empty ARKode memory structure */
    FILE *UFID;
    realtype t, tout;
    long int nst, nst_a, nfe, nfi, nsetups, nje, nfeLS, nni, ncfn, netf;
  
    /* Initial diagnostics output */
    printf("\nAnalytical ODE test problem:\n");
    printf("    lamda = %"GSYM"\n",    lamda);
    printf("   reltol = %.1"ESYM"\n",  reltol);
    printf("   abstol = %.1"ESYM"\n\n",abstol);
  
    /* Initialize data structures */
    y = N_VNew_Serial(NEQ);          /* Create serial vector for solution */
    if (check_flag((void *)y, "N_VNew_Serial", 0)) return 1;
    N_VConst(0.0, y);             /* Specify initial condition */
  
    /* Call ARKStepCreate to initialize the ARK timestepper module and
       specify the right-hand side function in y'=f(t,y), the inital time
       T0, and the initial dependent variable vector y.  Note: since this
       problem is fully implicit, we set f_E to NULL and f_I to f. */
    arkode_mem = ARKStepCreate(NULL, f, T0, y);
    if (check_flag((void *)arkode_mem, "ARKStepCreate", 0)) return 1;
  
    /* Set routines */
    flag = ARKStepSetUserData(arkode_mem, (void *) &lamda);  /* Pass lamda to user functions */
    if (check_flag(&flag, "ARKStepSetUserData", 1)) return 1;
    flag = ARKStepSStolerances(arkode_mem, reltol, abstol);  /* Specify tolerances */
    if (check_flag(&flag, "ARKStepSStolerances", 1)) return 1;
  
    /* Initialize dense matrix data structure and solver */
    A = SUNDenseMatrix(NEQ, NEQ);
    if (check_flag((void *)A, "SUNDenseMatrix", 0)) return 1;
    LS = SUNLinSol_Dense(y, A);
    if (check_flag((void *)LS, "SUNLinSol_Dense", 0)) return 1;
  
    /* Linear solver interface */
    flag = ARKStepSetLinearSolver(arkode_mem, LS, A);        /* Attach matrix and linear solver */
    if (check_flag(&flag, "ARKStepSetLinearSolver", 1)) return 1;
    flag = ARKStepSetJacFn(arkode_mem, Jac);                 /* Set Jacobian routine */
    if (check_flag(&flag, "ARKStepSetJacFn", 1)) return 1;
  
    /* Specify linearly implicit RHS, with non-time-dependent Jacobian */
    flag = ARKStepSetLinear(arkode_mem, 0);
    if (check_flag(&flag, "ARKStepSetLinear", 1)) return 1;
  
    /* Open output stream for results, output comment line */
    UFID = fopen("solution.txt","w");
    fprintf(UFID,"# t u\n");
  
    /* output initial condition to disk */
    fprintf(UFID," %.16"ESYM" %.16"ESYM"\n", T0, NV_Ith_S(y,0));
  
    /* Main time-stepping loop: calls ARKStepEvolve to perform the integration, then
       prints results.  Stops when the final time has been reached */
    t = T0;
    tout = T0+dTout;
    printf("        t           u\n");
    printf("   ---------------------\n");
    while (Tf - t > 1.0e-15) {
  
      flag = ARKStepEvolve(arkode_mem, tout, y, &t, ARK_NORMAL);      /* call integrator */
      if (check_flag(&flag, "ARKStepEvolve", 1)) break;
      printf("  %10.6"FSYM"  %10.6"FSYM"\n", t, NV_Ith_S(y,0));          /* access/print solution */
      fprintf(UFID," %.16"ESYM" %.16"ESYM"\n", t, NV_Ith_S(y,0));
      if (flag >= 0) {                                         /* successful solve: update time */
        tout += dTout;
        tout = (tout > Tf) ? Tf : tout;
      } else {                                                 /* unsuccessful solve: break */
        fprintf(stderr,"Solver failure, stopping integration\n");
        break;
      }
    }
    printf("   ---------------------\n");
    fclose(UFID);
  
    /* Get/print some final statistics on how the solve progressed */
    flag = ARKStepGetNumSteps(arkode_mem, &nst);
    check_flag(&flag, "ARKStepGetNumSteps", 1);
    flag = ARKStepGetNumStepAttempts(arkode_mem, &nst_a);
    check_flag(&flag, "ARKStepGetNumStepAttempts", 1);
    flag = ARKStepGetNumRhsEvals(arkode_mem, &nfe, &nfi);
    check_flag(&flag, "ARKStepGetNumRhsEvals", 1);
    flag = ARKStepGetNumLinSolvSetups(arkode_mem, &nsetups);
    check_flag(&flag, "ARKStepGetNumLinSolvSetups", 1);
    flag = ARKStepGetNumErrTestFails(arkode_mem, &netf);
    check_flag(&flag, "ARKStepGetNumErrTestFails", 1);
    flag = ARKStepGetNumNonlinSolvIters(arkode_mem, &nni);
    check_flag(&flag, "ARKStepGetNumNonlinSolvIters", 1);
    flag = ARKStepGetNumNonlinSolvConvFails(arkode_mem, &ncfn);
    check_flag(&flag, "ARKStepGetNumNonlinSolvConvFails", 1);
    flag = ARKStepGetNumJacEvals(arkode_mem, &nje);
    check_flag(&flag, "ARKStepGetNumJacEvals", 1);
    flag = ARKStepGetNumLinRhsEvals(arkode_mem, &nfeLS);
    check_flag(&flag, "ARKStepGetNumLinRhsEvals", 1);
  
    printf("\nFinal Solver Statistics:\n");
    printf("   Internal solver steps = %li (attempted = %li)\n", nst, nst_a);
    printf("   Total RHS evals:  Fe = %li,  Fi = %li\n", nfe, nfi);
    printf("   Total linear solver setups = %li\n", nsetups);
    printf("   Total RHS evals for setting up the linear system = %li\n", nfeLS);
    printf("   Total number of Jacobian evaluations = %li\n", nje);
    printf("   Total number of Newton iterations = %li\n", nni);
    printf("   Total number of linear solver convergence failures = %li\n", ncfn);
    printf("   Total number of error test failures = %li\n\n", netf);
  
    /* check the solution error */
    flag = check_ans(y, t, reltol, abstol);
  
    /* Clean up and return */
    N_VDestroy(y);            /* Free y vector */
    ARKStepFree(&arkode_mem); /* Free integrator memory */
    SUNLinSolFree(LS);        /* Free linear solver */
    SUNMatDestroy(A);         /* Free A matrix */
  
    return flag;
  }
  
  /*-------------------------------
   * Functions called by the solver
   *-------------------------------*/
  
  /* f routine to compute the ODE RHS function f(t,y). */
  static int f(realtype t, N_Vector y, N_Vector ydot, void *user_data)
  {
    realtype *rdata = (realtype *) user_data;   /* cast user_data to realtype */
    realtype lamda = rdata[0];                  /* set shortcut for stiffness parameter */
    realtype u = NV_Ith_S(y,0);                 /* access current solution value */
  
    /* fill in the RHS function: "NV_Ith_S" accesses the 0th entry of ydot */
    NV_Ith_S(ydot,0) = lamda*u + 1.0/(1.0+t*t) - lamda*atan(t);
  
    return 0;                                   /* return with success */
  }
  
  /* Jacobian routine to compute J(t,y) = df/dy. */
  static int Jac(realtype t, N_Vector y, N_Vector fy, SUNMatrix J,
                 void *user_data, N_Vector tmp1, N_Vector tmp2, N_Vector tmp3)
  {
    realtype *rdata = (realtype *) user_data;   /* cast user_data to realtype */
    realtype lamda = rdata[0];                  /* set shortcut for stiffness parameter */
    realtype *Jdata = SUNDenseMatrix_Data(J);
  
    /* Fill in Jacobian of f: set the first entry of the data array to set the (0,0) entry */
    Jdata[0] = lamda;
  
    return 0;                                   /* return with success */
  }
  
  /*-------------------------------
   * Private helper functions
   *-------------------------------*/
  
  /* Check function return value...
      opt == 0 means SUNDIALS function allocates memory so check if
               returned NULL pointer
      opt == 1 means SUNDIALS function returns a flag so check if
               flag >= 0
      opt == 2 means function allocates memory so check if returned
               NULL pointer
  */
  static int check_flag(void *flagvalue, const char *funcname, int opt)
  {
    int *errflag;
  
    /* Check if SUNDIALS function returned NULL pointer - no memory allocated */
    if (opt == 0 && flagvalue == NULL) {
      fprintf(stderr, "\nSUNDIALS_ERROR: %s() failed - returned NULL pointer\n\n",
              funcname);
      return 1; }
  
    /* Check if flag < 0 */
    else if (opt == 1) {
      errflag = (int *) flagvalue;
      if (*errflag < 0) {
        fprintf(stderr, "\nSUNDIALS_ERROR: %s() failed with flag = %d\n\n",
                funcname, *errflag);
        return 1; }}
  
    /* Check if function returned NULL pointer - no memory allocated */
    else if (opt == 2 && flagvalue == NULL) {
      fprintf(stderr, "\nMEMORY_ERROR: %s() failed - returned NULL pointer\n\n",
              funcname);
      return 1; }
  
    return 0;
  }
  
  /* check the computed solution */
  static int check_ans(N_Vector y, realtype t, realtype rtol, realtype atol)
  {
    int      passfail=0;     /* answer pass (0) or fail (1) flag     */
    realtype ans, err, ewt;  /* answer data, error, and error weight */
    realtype ONE=RCONST(1.0);
  
    /* compute solution error */
    ans  = atan(t);
    ewt = ONE / (rtol * SUNRabs(ans) + atol);
    err = ewt * SUNRabs(NV_Ith_S(y,0) - ans);
  
    /* is the solution within the tolerances? */
    passfail = (err < ONE) ? 0 : 1;
  
    if (passfail) {
      fprintf(stdout, "\nSUNDIALS_WARNING: check_ans error=%g \n\n", err);
    }
  
    return(passfail);
  }
  
  /*---- end of file ----*/
  