text: /*
   * -----------------------------------------------------------------
   * $Revision: 1.2 $
   * $Date: 2009/09/30 23:25:59 $
   * -----------------------------------------------------------------
   * Programmer(s): Allan Taylor, Alan Hindmarsh and
   *                Radu Serban @ LLNL
   * -----------------------------------------------------------------
   * Example program for IDA: Food web problem.
   *
   * This example program (serial version) uses the IDABAND linear 
   * solver, and IDACalcIC for initial condition calculation.
   *
   * The mathematical problem solved in this example is a DAE system
   * that arises from a system of partial differential equations after
   * spatial discretization. The PDE system is a food web population
   * model, with predator-prey interaction and diffusion on the unit
   * square in two dimensions. The dependent variable vector is:
   *
   *         1   2         ns
   *   c = (c , c ,  ..., c  ) , ns = 2 * np
   *
   * and the PDE's are as follows:
   *
   *     i             i      i
   *   dc /dt = d(i)*(c    + c  )  +  R (x,y,c)   (i = 1,...,np)
   *                   xx     yy       i
   *
   *              i      i
   *   0 = d(i)*(c    + c  )  +  R (x,y,c)   (i = np+1,...,ns)
   *              xx     yy       i
   *
   *   where the reaction terms R are:
   *
   *                   i             ns         j
   *   R  (x,y,c)  =  c  * (b(i)  + sum a(i,j)*c )
   *    i                           j=1
   *
   * The number of species is ns = 2 * np, with the first np being
   * prey and the last np being predators. The coefficients a(i,j),
   * b(i), d(i) are:
   *
   *  a(i,i) = -AA   (all i)
   *  a(i,j) = -GG   (i <= np , j >  np)
   *  a(i,j) =  EE   (i >  np, j <= np)
   *  all other a(i,j) = 0
   *  b(i) = BB*(1+ alpha * x*y + beta*sin(4 pi x)*sin(4 pi y)) (i <= np)
   *  b(i) =-BB*(1+ alpha * x*y + beta*sin(4 pi x)*sin(4 pi y)) (i  > np)
   *  d(i) = DPREY   (i <= np)
   *  d(i) = DPRED   (i > np)
   *
   * The various scalar parameters required are set using '#define'
   * statements or directly in routine InitUserData. In this program,
   * np = 1, ns = 2. The boundary conditions are homogeneous Neumann:
   * normal derivative = 0.
   *
   * A polynomial in x and y is used to set the initial values of the
   * first np variables (the prey variables) at each x,y location,
   * while initial values for the remaining (predator) variables are
   * set to a flat value, which is corrected by IDACalcIC.
   *
   * The PDEs are discretized by central differencing on a MX by MY
   * mesh.
   *
   * The DAE system is solved by IDA using the IDABAND linear solver.
   * Output is printed at t = 0, .001, .01, .1, .4, .7, 1.
   * -----------------------------------------------------------------
   * References:
   * [1] Peter N. Brown and Alan C. Hindmarsh,
   *     Reduced Storage Matrix Methods in Stiff ODE systems, Journal
   *     of Applied Mathematics and Computation, Vol. 31 (May 1989),
   *     pp. 40-91.
   *
   * [2] Peter N. Brown, Alan C. Hindmarsh, and Linda R. Petzold,
   *     Using Krylov Methods in the Solution of Large-Scale
   *     Differential-Algebraic Systems, SIAM J. Sci. Comput., 15
   *     (1994), pp. 1467-1488.
   *
   * [3] Peter N. Brown, Alan C. Hindmarsh, and Linda R. Petzold,
   *     Consistent Initial Condition Calculation for Differential-
   *     Algebraic Systems, SIAM J. Sci. Comput., 19 (1998),
   *     pp. 1495-1512.
   * -----------------------------------------------------------------
   */
  
  #include <stdio.h>
  #include <stdlib.h>
  #include <math.h>
  
  #include <ida/ida.h>
  #include <ida/ida_band.h>
  #include <nvector/nvector_serial.h>
  #include <sundials/sundials_dense.h>
  #include <sundials/sundials_types.h>
  
  /* Problem Constants. */
  
  #define NPREY       1              /* No. of prey (= no. of predators). */
  #define NUM_SPECIES 2*NPREY
  
  #define PI          RCONST(3.1415926535898)
  #define FOURPI      (RCONST(4.0)*PI)
  
  #define MX          20             /* MX = number of x mesh points      */
  #define MY          20             /* MY = number of y mesh points      */
  #define NSMX        (NUM_SPECIES * MX)
  #define NEQ         (NUM_SPECIES*MX*MY)
  #define AA          RCONST(1.0)    /* Coefficient in above eqns. for a  */
  #define EE          RCONST(10000.) /* Coefficient in above eqns. for a  */
  #define GG          RCONST(0.5e-6) /* Coefficient in above eqns. for a  */
  #define BB          RCONST(1.0)    /* Coefficient in above eqns. for b  */
  #define DPREY       RCONST(1.0)    /* Coefficient in above eqns. for d  */
  #define DPRED       RCONST(0.05)   /* Coefficient in above eqns. for d  */
  #define ALPHA       RCONST(50.)    /* Coefficient alpha in above eqns.  */
  #define BETA        RCONST(1000.)  /* Coefficient beta in above eqns.   */
  #define AX          RCONST(1.0)    /* Total range of x variable         */
  #define AY          RCONST(1.0)    /* Total range of y variable         */
  #define RTOL        RCONST(1.e-5)  /* Relative tolerance                */
  #define ATOL        RCONST(1.e-5)  /* Absolute tolerance                */
  #define NOUT        6              /* Number of output times            */
  #define TMULT       RCONST(10.0)   /* Multiplier for tout values        */
  #define TADD        RCONST(0.3)    /* Increment for tout values         */
  #define ZERO        RCONST(0.)     
  #define ONE         RCONST(1.0)    
  
  /* 
   * User-defined vector and accessor macro: IJ_Vptr.
   * IJ_Vptr is defined in order to express the underlying 3-D structure of 
   * the dependent variable vector from its underlying 1-D storage (an N_Vector).
   * IJ_Vptr(vv,i,j) returns a pointer to the location in vv corresponding to 
   * species index is = 0, x-index ix = i, and y-index jy = j.                
   */
  
  #define IJ_Vptr(vv,i,j) (&NV_Ith_S(vv, (i)*NUM_SPECIES + (j)*NSMX))
  
  /* Type: UserData.  Contains problem constants, etc. */
  
  typedef struct {
    long int Neq, ns, np, mx, my;
    realtype dx, dy, **acoef;
    realtype cox[NUM_SPECIES], coy[NUM_SPECIES], bcoef[NUM_SPECIES];
    N_Vector rates;
  } *UserData;
  
  /* Prototypes for functions called by the IDA Solver. */
  
  static int resweb(realtype time, N_Vector cc, N_Vector cp, N_Vector resval, 
                    void *user_data);
  
  /* Prototypes for private Helper Functions. */
  
  static void InitUserData(UserData webdata);
  static void SetInitialProfiles(N_Vector cc, N_Vector cp, N_Vector id,
                                 UserData webdata);
  static void PrintHeader(long int mu, long int ml, realtype rtol, realtype atol);
  static void PrintOutput(void *mem, N_Vector c, realtype t);
  static void PrintFinalStats(void *mem);
  static void Fweb(realtype tcalc, N_Vector cc, N_Vector crate, UserData webdata);
  static void WebRates(realtype xx, realtype yy, realtype *cxy, realtype *ratesxy, 
                       UserData webdata);
  static realtype dotprod(long int size, realtype *x1, realtype *x2);
  static int check_flag(void *flagvalue, char *funcname, int opt);
  
  /*
   *--------------------------------------------------------------------
   * MAIN PROGRAM
   *--------------------------------------------------------------------
   */
  
  int main()
  { 
    void *mem;
    UserData webdata;
    N_Vector cc, cp, id;
    int iout, retval;
    long int mu, ml;
    realtype rtol, atol, t0, tout, tret;
  
    mem = NULL;
    webdata = NULL;
    cc = cp = id = NULL;
  
    /* Allocate and initialize user data block webdata. */
  
    webdata = (UserData) malloc(sizeof *webdata);
    webdata->rates = N_VNew_Serial(NEQ);
    webdata->acoef = newDenseMat(NUM_SPECIES, NUM_SPECIES);
  
    InitUserData(webdata);
  
    /* Allocate N-vectors and initialize cc, cp, and id. */
  
    cc  = N_VNew_Serial(NEQ);
    if(check_flag((void *)cc, "N_VNew_Serial", 0)) return(1);
  
    cp  = N_VNew_Serial(NEQ);
    if(check_flag((void *)cp, "N_VNew_Serial", 0)) return(1);
  
    id  = N_VNew_Serial(NEQ);
    if(check_flag((void *)id, "N_VNew_Serial", 0)) return(1);
    
    SetInitialProfiles(cc, cp, id, webdata);
    
    /* Set remaining inputs to IDAMalloc. */
    
    t0 = ZERO;
    rtol = RTOL; 
    atol = ATOL;
  
    /* Call IDACreate and IDAMalloc to initialize IDA. */
    
    mem = IDACreate();
    if(check_flag((void *)mem, "IDACreate", 0)) return(1);
  
    retval = IDASetUserData(mem, webdata);
    if(check_flag(&retval, "IDASetUserData", 1)) return(1);
  
    retval = IDASetId(mem, id);
    if(check_flag(&retval, "IDASetId", 1)) return(1);
  
    retval = IDAInit(mem, resweb, t0, cc, cp);
    if(check_flag(&retval, "IDAInit", 1)) return(1);
  
    retval = IDASStolerances(mem, rtol, atol);
    if(check_flag(&retval, "IDASStolerances", 1)) return(1);
  
    /* Call IDABand to specify the IDA linear solver. */
  
    mu = ml = NSMX;
    retval = IDABand(mem, NEQ, mu, ml);
    if(check_flag(&retval, "IDABand", 1)) return(1);
  
    /* Call IDACalcIC (with default options) to correct the initial values. */
  
    tout = RCONST(0.001);
    retval = IDACalcIC(mem, IDA_YA_YDP_INIT, tout);
    if(check_flag(&retval, "IDACalcIC", 1)) return(1);
    
    /* Print heading, basic parameters, and initial values. */
  
    PrintHeader(mu, ml, rtol, atol);
    PrintOutput(mem, cc, ZERO);
    
    /* Loop over iout, call IDASolve (normal mode), print selected output. */
    
    for (iout = 1; iout <= NOUT; iout++) {
      
      retval = IDASolve(mem, tout, &tret, cc, cp, IDA_NORMAL);
      if(check_flag(&retval, "IDASolve", 1)) return(retval);
      
      PrintOutput(mem, cc, tret);
      
      if (iout < 3) tout *= TMULT; else tout += TADD;
      
    }
    
    /* Print final statistics and free memory. */  
    
    PrintFinalStats(mem);
  
    /* Free memory */
  
    IDAFree(&mem);
  
    N_VDestroy_Serial(cc);
    N_VDestroy_Serial(cp);
    N_VDestroy_Serial(id);
  
  
    destroyMat(webdata->acoef);
    N_VDestroy_Serial(webdata->rates);
    free(webdata);
  
    return(0);
  }
  
  /* Define lines for readability in later routines */ 
  
  #define acoef  (webdata->acoef)
  #define bcoef  (webdata->bcoef)
  #define cox    (webdata->cox)
  #define coy    (webdata->coy)
  
  /*
   *--------------------------------------------------------------------
   * FUNCTIONS CALLED BY IDA
   *--------------------------------------------------------------------
   */
  
  /* 
   * resweb: System residual function for predator-prey system.
   * This routine calls Fweb to get all the right-hand sides of the
   * equations, then loads the residual vector accordingly,
   * using cp in the case of prey species.                 
   */
  
  static int resweb(realtype tt, N_Vector cc, N_Vector cp, 
                    N_Vector res,  void *user_data)
  {
    long int jx, jy, is, yloc, loc, np;
    realtype *resv, *cpv;
    UserData webdata;
    
    webdata = (UserData)user_data;
    
    cpv = NV_DATA_S(cp);
    resv = NV_DATA_S(res);
    np = webdata->np;
    
    /* Call Fweb to set res to vector of right-hand sides. */
    Fweb(tt, cc, res, webdata);
    
    /* Loop over all grid points, setting residual values appropriately
       for differential or algebraic components.                        */
    
    for (jy = 0; jy < MY; jy++) {
      yloc = NSMX * jy;
      for (jx = 0; jx < MX; jx++) {
        loc = yloc + NUM_SPECIES * jx;
        for (is = 0; is < NUM_SPECIES; is++) {
          if (is < np)
            resv[loc+is] = cpv[loc+is] - resv[loc+is];
          else
            resv[loc+is] = -resv[loc+is];
        }
      }
    }
    
    return(0);
    
  }
  
  /*
   *--------------------------------------------------------------------
   * PRIVATE FUNCTIONS
   *--------------------------------------------------------------------
   */
  
  /*
   * InitUserData: Load problem constants in webdata (of type UserData).   
   */
  
  static void InitUserData(UserData webdata)
  {
    int i, j, np;
    realtype *a1,*a2, *a3, *a4, dx2, dy2;
  
    webdata->mx = MX;
    webdata->my = MY;
    webdata->ns = NUM_SPECIES;
    webdata->np = NPREY;
    webdata->dx = AX/(MX-1);
    webdata->dy = AY/(MY-1);
    webdata->Neq= NEQ;
    
    /* Set up the coefficients a and b, and others found in the equations. */
    np = webdata->np;
    dx2 = (webdata->dx)*(webdata->dx); dy2 = (webdata->dy)*(webdata->dy);
    
    for (i = 0; i < np; i++) {
      a1 = &(acoef[i][np]);
      a2 = &(acoef[i+np][0]);
      a3 = &(acoef[i][0]);
      a4 = &(acoef[i+np][np]);
      /*  Fill in the portion of acoef in the four quadrants, row by row. */
      for (j = 0; j < np; j++) {
        *a1++ =  -GG;
        *a2++ =   EE;
        *a3++ = ZERO;
        *a4++ = ZERO;
      }
      
      /* Reset the diagonal elements of acoef to -AA. */
      acoef[i][i] = -AA; acoef[i+np][i+np] = -AA;
      
      /* Set coefficients for b and diffusion terms. */
      bcoef[i] = BB; bcoef[i+np] = -BB;
      cox[i] = DPREY/dx2; cox[i+np] = DPRED/dx2;
      coy[i] = DPREY/dy2; coy[i+np] = DPRED/dy2;
    }
    
  }
  
  /* 
   * SetInitialProfiles: Set initial conditions in cc, cp, and id.
   * A polynomial profile is used for the prey cc values, and a constant
   * (1.0e5) is loaded as the initial guess for the predator cc values.
   * The id values are set to 1 for the prey and 0 for the predators.
   * The prey cp values are set according to the given system, and
   * the predator cp values are set to zero.                               
   */
  
  static void SetInitialProfiles(N_Vector cc, N_Vector cp, N_Vector id,
                                 UserData webdata)
  {
    long int loc, yloc, is, jx, jy, np;
    realtype xx, yy, xyfactor, fac;
    realtype *ccv, *cpv, *idv;
    
    ccv = NV_DATA_S(cc);
    cpv = NV_DATA_S(cp);
    idv = NV_DATA_S(id);
    np = webdata->np;
    
    /* Loop over grid, load cc values and id values. */
    for (jy = 0; jy < MY; jy++) {
      yy = jy * webdata->dy;
      yloc = NSMX * jy;
      for (jx = 0; jx < MX; jx++) {
        xx = jx * webdata->dx;
        xyfactor = RCONST(16.0)*xx*(ONE-xx)*yy*(ONE-yy);
        xyfactor *= xyfactor;
        loc = yloc + NUM_SPECIES*jx;
        fac = ONE + ALPHA * xx * yy + BETA * sin(FOURPI*xx) * sin(FOURPI*yy);
        
        for (is = 0; is < NUM_SPECIES; is++) {
          if (is < np) {
  	    ccv[loc+is] = RCONST(10.0) + (realtype)(is+1) * xyfactor;
            idv[loc+is] = ONE;
          }
          else {
  	  ccv[loc+is] = RCONST(1.0e5);
            idv[loc+is] = ZERO;
          }
        }
      }
    }
    
    /* Set c' for the prey by calling the function Fweb. */
    Fweb(ZERO, cc, cp, webdata);
    
    /* Set c' for predators to 0. */
    for (jy = 0; jy < MY; jy++) {
      yloc = NSMX * jy;
      for (jx = 0; jx < MX; jx++) {
        loc = yloc + NUM_SPECIES * jx;
        for (is = np; is < NUM_SPECIES; is++) {
          cpv[loc+is] = ZERO;
        }
      }
    }
  }
  
  /*
   * Print first lines of output (problem description)
   */
  
  static void PrintHeader(long int mu, long int ml, realtype rtol, realtype atol)
  {
    printf("\nidaFoodWeb_bnd: Predator-prey DAE serial example problem for IDA \n\n");
    printf("Number of species ns: %d", NUM_SPECIES);
    printf("     Mesh dimensions: %d x %d", MX, MY);
    printf("     System size: %d\n", NEQ);
  #if defined(SUNDIALS_EXTENDED_PRECISION) 
    printf("Tolerance parameters:  rtol = %Lg   atol = %Lg\n", rtol, atol);
  #elif defined(SUNDIALS_DOUBLE_PRECISION) 
    printf("Tolerance parameters:  rtol = %lg   atol = %lg\n", rtol, atol);
  #else
    printf("Tolerance parameters:  rtol = %g   atol = %g\n", rtol, atol);
  #endif
    printf("Linear solver: IDABAND,  Band parameters mu = %ld, ml = %ld\n",mu,ml);
    printf("CalcIC called to correct initial predator concentrations.\n\n");
    printf("-----------------------------------------------------------\n");
    printf("  t        bottom-left  top-right");
    printf("    | nst  k      h\n");
    printf("-----------------------------------------------------------\n\n");
    
  }
  
  /* 
   * PrintOutput: Print output values at output time t = tt.
   * Selected run statistics are printed.  Then values of the concentrations
   * are printed for the bottom left and top right grid points only.  
   */
  
  static void PrintOutput(void *mem, N_Vector c, realtype t)
  {
    int i, kused, flag;
    long int nst;
    realtype *c_bl, *c_tr, hused;
  
    flag = IDAGetLastOrder(mem, &kused);
    check_flag(&flag, "IDAGetLastOrder", 1);
    flag = IDAGetNumSteps(mem, &nst);
    check_flag(&flag, "IDAGetNumSteps", 1);
    flag = IDAGetLastStep(mem, &hused);
    check_flag(&flag, "IDAGetLastStep", 1);
    
    c_bl = IJ_Vptr(c,0,0);
    c_tr = IJ_Vptr(c,MX-1,MY-1);
  
  #if defined(SUNDIALS_EXTENDED_PRECISION) 
    printf("%8.2Le %12.4Le %12.4Le   | %3ld  %1d %12.4Le\n", 
           t, c_bl[0], c_tr[1], nst, kused, hused);
    for (i=1;i<NUM_SPECIES;i++)
      printf("         %12.4Le %12.4Le   |\n",c_bl[i],c_tr[i]);
  #elif defined(SUNDIALS_DOUBLE_PRECISION) 
    printf("%8.2le %12.4le %12.4le   | %3ld  %1d %12.4le\n", 
           t, c_bl[0], c_tr[1], nst, kused, hused);
    for (i=1;i<NUM_SPECIES;i++)
      printf("         %12.4le %12.4le   |\n",c_bl[i],c_tr[i]);
  #else
    printf("%8.2e %12.4e %12.4e   | %3ld  %1d %12.4e\n", 
           t, c_bl[0], c_tr[1], nst, kused, hused);
    for (i=1;i<NUM_SPECIES;i++)
      printf("         %12.4e %12.4e   |\n",c_bl[i],c_tr[i]);
  #endif
  
    printf("\n");
  }
  
  /* 
   * PrintFinalStats: Print final run data contained in iopt.              
   */
  
  static void PrintFinalStats(void *mem)
  { 
    long int nst, nre, nreLS, nni, nje, netf, ncfn;
    int flag;
  
    flag = IDAGetNumSteps(mem, &nst);
    check_flag(&flag, "IDAGetNumSteps", 1);
    flag = IDAGetNumNonlinSolvIters(mem, &nni);
    check_flag(&flag, "IDAGetNumNonlinSolvIters", 1);
    flag = IDAGetNumResEvals(mem, &nre);
    check_flag(&flag, "IDAGetNumResEvals", 1);
    flag = IDAGetNumErrTestFails(mem, &netf);
    check_flag(&flag, "IDAGetNumErrTestFails", 1);
    flag = IDAGetNumNonlinSolvConvFails(mem, &ncfn);
    check_flag(&flag, "IDAGetNumNonlinSolvConvFails", 1);
    flag = IDADlsGetNumJacEvals(mem, &nje);
    check_flag(&flag, "IDADlsGetNumJacEvals", 1);
    flag = IDADlsGetNumResEvals(mem, &nreLS);
    check_flag(&flag, "IDADlsGetNumResEvals", 1);
  
    printf("-----------------------------------------------------------\n");
    printf("Final run statistics: \n\n");
    printf("Number of steps                    = %ld\n", nst);
    printf("Number of residual evaluations     = %ld\n", nre+nreLS);
    printf("Number of Jacobian evaluations     = %ld\n", nje);
    printf("Number of nonlinear iterations     = %ld\n", nni);
    printf("Number of error test failures      = %ld\n", netf);
    printf("Number of nonlinear conv. failures = %ld\n", ncfn);
  
  }
  
  /* 
   * Fweb: Rate function for the food-web problem.                        
   * This routine computes the right-hand sides of the system equations,   
   * consisting of the diffusion term and interaction term.                
   * The interaction term is computed by the function WebRates.            
   */
  
  static void Fweb(realtype tcalc, N_Vector cc, N_Vector crate,  
                   UserData webdata)
  { 
    long int jx, jy, is, idyu, idyl, idxu, idxl;
    realtype xx, yy, *cxy, *ratesxy, *cratexy, dcyli, dcyui, dcxli, dcxui;
    
    /* Loop over grid points, evaluate interaction vector (length ns),
       form diffusion difference terms, and load crate.                    */
    
    for (jy = 0; jy < MY; jy++) {
      yy = (webdata->dy) * jy ;
      idyu = (jy!=MY-1) ? NSMX : -NSMX;
      idyl = (jy!= 0  ) ? NSMX : -NSMX;
      
      for (jx = 0; jx < MX; jx++) {
        xx = (webdata->dx) * jx;
        idxu = (jx!= MX-1) ?  NUM_SPECIES : -NUM_SPECIES;
        idxl = (jx!=  0  ) ?  NUM_SPECIES : -NUM_SPECIES;
        cxy = IJ_Vptr(cc,jx,jy);
        ratesxy = IJ_Vptr(webdata->rates,jx,jy);
        cratexy = IJ_Vptr(crate,jx,jy);
        
        /* Get interaction vector at this grid point. */
        WebRates(xx, yy, cxy, ratesxy, webdata);
        
        /* Loop over species, do differencing, load crate segment. */
        for (is = 0; is < NUM_SPECIES; is++) {
          
          /* Differencing in y. */
          dcyli = *(cxy+is) - *(cxy - idyl + is) ;
          dcyui = *(cxy + idyu + is) - *(cxy+is);
          
          /* Differencing in x. */
          dcxli = *(cxy+is) - *(cxy - idxl + is);
          dcxui = *(cxy + idxu +is) - *(cxy+is);
          
          /* Compute the crate values at (xx,yy). */
          cratexy[is] = coy[is] * (dcyui - dcyli) +
            cox[is] * (dcxui - dcxli) + ratesxy[is];
          
        } /* End is loop */
      } /* End of jx loop */
    } /* End of jy loop */
    
  }
  
  /* 
   * WebRates: Evaluate reaction rates at a given spatial point.
   * At a given (x,y), evaluate the array of ns reaction terms R.
   */
  
  static void WebRates(realtype xx, realtype yy, realtype *cxy, realtype *ratesxy,
                       UserData webdata)
  {
    int is;
    realtype fac;
    
    for (is = 0; is < NUM_SPECIES; is++)
      ratesxy[is] = dotprod(NUM_SPECIES, cxy, acoef[is]);
    
    fac = ONE + ALPHA*xx*yy + BETA*sin(FOURPI*xx)*sin(FOURPI*yy);
    
    for (is = 0; is < NUM_SPECIES; is++)  
      ratesxy[is] = cxy[is]*( bcoef[is]*fac + ratesxy[is] );
    
  }
  
  /*
   * dotprod: dot product routine for realtype arrays, for use by WebRates.    
   */
  
  static realtype dotprod(long int size, realtype *x1, realtype *x2)
  {
    long int i;
    realtype *xx1, *xx2, temp = ZERO;
    
    xx1 = x1; xx2 = x2;
    for (i = 0; i < size; i++) temp += (*xx1++) * (*xx2++);
    return(temp);
    
  }
  
  /*
   * 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, char *funcname, int opt)
  {
    int *errflag;
  
    if (opt == 0 && flagvalue == NULL) {
      /* Check if SUNDIALS function returned NULL pointer - no memory allocated */
      fprintf(stderr, 
              "\nSUNDIALS_ERROR: %s() failed - returned NULL pointer\n\n", 
              funcname);
      return(1);
    } else if (opt == 1) {
      /* Check if flag < 0 */
      errflag = (int *) flagvalue;
      if (*errflag < 0) {
        fprintf(stderr, 
                "\nSUNDIALS_ERROR: %s() failed with flag = %d\n\n", 
                funcname, *errflag);
        return(1); 
      }
    } else if (opt == 2 && flagvalue == NULL) {
      /* Check if function returned NULL pointer - no memory allocated */
      fprintf(stderr, 
              "\nMEMORY_ERROR: %s() failed - returned NULL pointer\n\n", 
              funcname);
      return(1); 
    }
  
    return(0);
  }
  