```/* nag_pde_dim1_parab_dae_coll (d03pjc) Example Program.
*
* Copyright 2019 Numerical Algorithms Group.
*
* Mark 27.0, 2019.
*/

#include <stdio.h>
#include <math.h>
#include <nag.h>

#ifdef __cplusplus
extern "C"
{
#endif
static void NAG_CALL pdedef(Integer, double, const double[], Integer, const
double[], const double[], Integer,
const double[], const double[], double[],
double[], double[], Integer *, Nag_Comm *);
static void NAG_CALL bndary(Integer, double, const double[], const double[],
Integer, const double[], const double[],
Integer, double[], double[], Integer *,
Nag_Comm *);
static void NAG_CALL odedef(Integer, double, Integer, const double[],
const double[], Integer, const double[],
const double[], const double[], const double[],
const double[], const double[], double[],
Integer *, Nag_Comm *);
static void NAG_CALL uvinit(Integer, Integer, const double[], double[],
Integer, double[], Nag_Comm *);
#ifdef __cplusplus
}
#endif

#define U(I, J)    u[npde*((J) -1)+(I) -1]
#define UX(I, J)   ux[npde*((J) -1)+(I) -1]
#define UCP(I, J)  ucp[npde*((J) -1)+(I) -1]
#define UCPX(I, J) ucpx[npde*((J) -1)+(I) -1]
#define P(I, J, K) p[npde*(npde*((K) -1)+(J) -1)+(I) -1]
#define Q(I, J)    q[npde*((J) -1)+(I) -1]
#define R(I, J)    r[npde*((J) -1)+(I) -1]

int main(void)
{
/* Constant scalars */
const Integer print_stat = 0;
const Integer npde = 1, ncode = 1, npoly = 3, m = 0, nbkpts = 30;
const Integer nel = nbkpts - 1, npts = nel * npoly + 1;
const Integer neqn = npde*npts + ncode;
const Integer nxi = 1, lisave = 24, npl1 = npoly + 1;
const Integer nwkres = npl1*(3*npl1 + npde*(npde + 6) + nbkpts + 1) +
8*npde + nxi*(5*npde + 1) + ncode + 3;
const Integer lenode = 11 * neqn + 50;
const Integer lrsave =  neqn*neqn + neqn + nwkres + lenode;

/* Constant arrays */
static double ruser[4] = { -1.0, -1.0, -1.0, -1.0 };

/* Scalars */
double       tout, ts;
Integer      exit_status = 0, i, ind, it, itask, itol, itrace;
Nag_Boolean  theta;

/* Arrays */
double       *algopt = 0, *atol = 0, *rsave = 0, *rtol = 0;
double       *u = 0, *x = 0, *xbkpts = 0, *xi = 0;
Integer      *isave = 0;

/* Nag Types */
NagError     fail;
Nag_Comm     comm;
Nag_D03_Save saved;

INIT_FAIL(fail);

printf(" nag_pde_dim1_parab_dae_coll (d03pjc) Example Program Results\n");

/* For communication with user-supplied functions: */
comm.user = ruser;

/* Allocate memory */

if (!(algopt = NAG_ALLOC(30, double)) ||
!(atol = NAG_ALLOC(1, double)) ||
!(rsave = NAG_ALLOC(lrsave, double)) ||
!(rtol = NAG_ALLOC(1, double)) ||
!(u = NAG_ALLOC(neqn, double)) ||
!(x = NAG_ALLOC(npts, double)) ||
!(xbkpts = NAG_ALLOC(nbkpts, double)) ||
!(xi = NAG_ALLOC(nxi, double)) ||
!(isave = NAG_ALLOC(lisave, Integer)))
{
printf("Allocation failure\n");
exit_status = 1;
goto END;
}

itrace = 0;
itol = 1;
atol[0] = 1e-5;
rtol[0] = atol[0];
printf("\n  Simple coupled PDE using BDF\n\n");
printf("   Degree of Polynomial =%4" NAG_IFMT "\n", npoly);
printf("   Number of elements   =%4" NAG_IFMT "\n", nbkpts - 1);
printf("   Accuracy requirement =%12.3e\n", atol[0]);
printf("   Number of points     =%4" NAG_IFMT "\n\n", npts);

/* Set break-points */

for (i = 0; i < nbkpts; ++i)
xbkpts[i] = i / (nbkpts - 1.0);

xi[0] = 1.0;
ind = 0;

/* Set theta = TRUE if the Theta integrator is required */

theta = Nag_FALSE;
for (i = 0; i < 30; ++i)
algopt[i] = 0.0;

if (theta) {
algopt[0] = 2.0;
}
else {
algopt[0] = 0.0;
}

/* Loop over output value of t */

ts = 1.e-4;
comm.p = (Pointer) &ts;

printf("%7s%8s%s\n", "time", "", "solution at x=0");

tout = 0.1;
for (it = 0; it < 5; ++it) {
tout = tout + tout;
/* nag_pde_dim1_parab_dae_coll (d03pjc).
* General system of parabolic PDEs, coupled DAEs, method of
* lines, Chebyshev C^0 collocation, one space variable
*/
nag_pde_dim1_parab_dae_coll(npde, m, &ts, tout, pdedef, bndary, u, nbkpts,
xbkpts, npoly, npts, x, ncode, odedef, nxi, xi,
neqn, uvinit, rtol, atol, itol, Nag_TwoNorm,
Nag_LinAlgFull, algopt, rsave, lrsave, isave,
lisave, itask, itrace, 0, &ind, &comm, &saved,
&fail);

if (fail.code != NE_NOERROR) {
printf("Error from nag_pde_dim1_parab_dae_coll (d03pjc).\n%s\n",
fail.message);
exit_status = 1;
goto END;
}

printf("%7.1f%15s%6.2f\n", ts, "", u[0]);
}
if (print_stat) {
printf(" Number of integration steps in time = %6" NAG_IFMT "\n", isave[0]);
printf(" Number of function evaluations = %6" NAG_IFMT "\n", isave[1]);
printf(" Number of Jacobian evaluations =%6" NAG_IFMT "\n", isave[2]);
printf(" Number of iterations = %6" NAG_IFMT "\n\n", isave[4]);
}
END:
NAG_FREE(algopt);
NAG_FREE(atol);
NAG_FREE(rsave);
NAG_FREE(rtol);
NAG_FREE(u);
NAG_FREE(x);
NAG_FREE(xbkpts);
NAG_FREE(xi);
NAG_FREE(isave);

return exit_status;
}

static void NAG_CALL uvinit(Integer npde, Integer npts, const double x[],
double u[], Integer ncode, double v[],
Nag_Comm *comm)
{
/* Routine for PDE initial values (start time is 0.1e-6) */

double *ts = (double *) comm->p;
Integer i;

if (comm->user[0] == -1.0) {
/* printf("(User-supplied callback uvinit, first invocation.)\n"); */
comm->user[0] = 0.0;
}
v[0] = *ts;
for (i = 1; i <= npts; ++i)
U(1, i) = exp(*ts * (1.0 - x[i - 1])) - 1.0;
return;
}

static void NAG_CALL odedef(Integer npde, double t, Integer ncode,
const double v[], const double vdot[],
Integer nxi, const double xi[],
const double ucp[], const double ucpx[],
const double rcp[], const double ucpt[],
const double ucptx[], double f[], Integer *ires,
Nag_Comm *comm)
{
if (comm->user[1] == -1.0) {
/* printf("(User-supplied callback odedef, first invocation.)\n"); */
comm->user[1] = 0.0;
}
if (*ires == 1) {
f[0] = vdot[0] - v[0] * UCP(1, 1) - UCPX(1, 1) - 1.0 - t;
}
else if (*ires == -1) {
f[0] = vdot[0];
}
return;
}

static void NAG_CALL pdedef(Integer npde, double t, const double x[],
Integer nptl, const double u[], const double ux[],
Integer ncode, const double v[],
const double vdot[], double p[], double q[],
double r[], Integer *ires, Nag_Comm *comm)
{
Integer i;
if (comm->user[2] == -1.0) {
/* printf("(User-supplied callback pdedef, first invocation.)\n"); */
comm->user[2] = 0.0;
}
for (i = 1; i <= nptl; ++i) {
P(1, 1, i) = v[0] * v[0];
R(1, i) = UX(1, i);
Q(1, i) = -x[i - 1] * UX(1, i) * v[0] * vdot[0];
}
return;
}

static void NAG_CALL bndary(Integer npde, double t, const double u[],
const double ux[], Integer ncode,
const double v[], const double vdot[],
Integer ibnd, double beta[], double gamma[],
Integer *ires, Nag_Comm *comm)
{
if (comm->user[3] == -1.0) {
/* printf("(User-supplied callback bndary, first invocation.)\n"); */
comm->user[3] = 0.0;
}
beta[0] = 1.0;
if (ibnd == 0) {
gamma[0] = -v[0] * exp(t);
}
else {
gamma[0] = -v[0] * vdot[0];
}
return;
}
```