NAG Library Manual, Mark 28.5
```/* nag_ode_ivp_bdf_zero_simple (d02ejc) Example Program.
*
* Copyright 2022 Numerical Algorithms Group.
*
* Mark 28.5, 2022.
*
*/

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

#ifdef __cplusplus
extern "C" {
#endif
static void NAG_CALL fcn(Integer neq, double x, const double y[], double f[],
Nag_User *comm);
static void NAG_CALL pederv(Integer neq, double x, const double y[],
double pw[], Nag_User *comm);
static double NAG_CALL g(Integer neq, double x, const double y[],
Nag_User *comm);
static void NAG_CALL out(Integer neq, double *tsol, const double y[],
Nag_User *comm);
#ifdef __cplusplus
}
#endif

struct user {
double xend, h;
Integer k;
Integer *use_comm;
};

#define NEQ 3
int main(void) {
static Integer use_comm[4] = {1, 1, 1, 1};
Integer exit_status = 0, j, neq;
NagError fail;
Nag_User comm;
double tol, x, *y = 0;
struct user s;

INIT_FAIL(fail);

printf("nag_ode_ivp_bdf_zero_simple (d02ejc) Example Program Results\n");

/* For communication with user-supplied functions
* assign address of user defined structure
* to comm.p.
*/
s.use_comm = use_comm;
comm.p = (Pointer)&s;

neq = NEQ;
if (neq >= 1) {
if (!(y = NAG_ALLOC(neq, double))) {
printf("Allocation failure\n");
exit_status = -1;
goto END;
}
} else {
exit_status = 1;
return exit_status;
}
s.xend = 10.0;
printf("\nCase 1: calculating Jacobian internally\n");
printf(" intermediate output, root-finding\n\n");

for (j = 3; j <= 4; ++j) {
tol = pow(10.0, -(double)j);
printf("\n Calculation with tol = %10.1e\n", tol);
x = 0.0;
y[0] = 1.0;
y[1] = 0.0;
y[2] = 0.0;
s.k = 4;
s.h = (s.xend - x) / (double)(s.k + 1);
printf("     X         Y(1)         Y(2)         Y(3)\n");
/* nag_ode_ivp_bdf_zero_simple (d02ejc).
* Ordinary differential equations solver, stiff, initial
* value problems using the Backward Differentiation
* Formulae
*/
nag_ode_ivp_bdf_zero_simple(neq, fcn, NULLFN, &x, y, s.xend, tol,
Nag_Relative, out, g, &comm, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_ode_ivp_bdf_zero_simple (d02ejc).\n%s\n",
fail.message);
exit_status = 1;
goto END;
}
printf("  Root of Y(1)-0.9 at %5.3f\n", x);
printf("  Solution is ");
printf("%7.4f %8.5f %7.4f\n", y[0], y[1], y[2]);
}
printf("\nCase 2: calculating Jacobian by pederv\n");
printf(" intermediate output, root-finding\n\n");

for (j = 3; j <= 4; ++j) {
tol = pow(10.0, -(double)j);
printf("\n Calculation with tol = %10.1e\n", tol);
x = 0.0;
y[0] = 1.0;
y[1] = 0.0;
y[2] = 0.0;
s.k = 4;
s.h = (s.xend - x) / (double)(s.k + 1);
printf("     X         Y(1)         Y(2)         Y(3)\n");
/* nag_ode_ivp_bdf_zero_simple (d02ejc), see above. */
nag_ode_ivp_bdf_zero_simple(neq, fcn, pederv, &x, y, s.xend, tol,
Nag_Relative, out, g, &comm, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_ode_ivp_bdf_zero_simple (d02ejc).\n%s\n",
fail.message);
exit_status = 1;
goto END;
}
printf("  Root of Y(1)-0.9 at %5.3f\n", x);
printf("  Solution is ");
printf("%7.4f %8.5f %7.4f\n", y[0], y[1], y[2]);
}
printf("\nCase 3: calculating Jacobian internally\n");
printf(" no intermediate output, root-finding\n\n");
for (j = 3; j <= 4; ++j) {
tol = pow(10.0, -(double)j);
printf("\n Calculation with tol = %10.1e\n", tol);
x = 0.0;
y[0] = 1.0;
y[1] = 0.0;
y[2] = 0.0;

/* nag_ode_ivp_bdf_zero_simple (d02ejc), see above. */
nag_ode_ivp_bdf_zero_simple(neq, fcn, NULLFN, &x, y, s.xend, tol,
Nag_Relative, NULLFN, g, &comm, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_ode_ivp_bdf_zero_simple (d02ejc).\n%s\n",
fail.message);
exit_status = 1;
goto END;
}

printf("  Root of Y(1)-0.9 at %5.3f\n", x);
printf("  Solution is ");
printf("%7.4f %8.5f %7.4f\n", y[0], y[1], y[2]);
}
printf("\nCase 4: calculating Jacobian internally\n");
printf(" intermediate output, no root-finding\n\n");

for (j = 3; j <= 4; ++j) {
tol = pow(10.0, -(double)j);
printf("\n Calculation with tol = %10.1e\n", tol);
x = 0.0;
y[0] = 1.0;
y[1] = 0.0;
y[2] = 0.0;
s.k = 4;
s.h = (s.xend - x) / (double)(s.k + 1);
printf("     X         Y(1)         Y(2)         Y(3)\n");
/* nag_ode_ivp_bdf_zero_simple (d02ejc), see above. */
nag_ode_ivp_bdf_zero_simple(neq, fcn, NULLFN, &x, y, s.xend, tol,
Nag_Relative, out, NULLDFN, &comm, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_ode_ivp_bdf_zero_simple (d02ejc).\n%s\n",
fail.message);
exit_status = 1;
goto END;
}
printf("%8.2f", x);
printf("%13.4f%13.5f%13.4f\n", y[0], y[1], y[2]);
}

printf("\nCase 5: calculating Jacobian internally\n");
printf(" no intermediate output, no root-finding (integrate to xend)\n\n");

for (j = 3; j <= 4; ++j) {
tol = pow(10.0, -(double)j);
printf("\n Calculation with tol = %10.1e\n", tol);
x = 0.0;
y[0] = 1.0;
y[1] = 0.0;
y[2] = 0.0;
printf("     X         Y(1)         Y(2)         Y(3)\n");
printf("%8.2f", x);
printf("%13.4f%13.5f%13.4f\n", y[0], y[1], y[2]);
/* nag_ode_ivp_bdf_zero_simple (d02ejc), see above. */
nag_ode_ivp_bdf_zero_simple(neq, fcn, NULLFN, &x, y, s.xend, tol,
Nag_Relative, NULLFN, NULLDFN, &comm, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_ode_ivp_bdf_zero_simple (d02ejc).\n%s\n",
fail.message);
exit_status = 1;
goto END;
}
printf("%8.2f", x);
printf("%13.4f%13.5f%13.4f\n", y[0], y[1], y[2]);
}
END:
NAG_FREE(y);
return exit_status;
}

static void NAG_CALL fcn(Integer neq, double x, const double y[], double f[],
Nag_User *comm) {
struct user *s = (struct user *)comm->p;

if (s->use_comm[0]) {
printf("(User-supplied callback fcn, first invocation.)\n");
s->use_comm[0] = 0;
}

f[0] = y[0] * -0.04 + y[1] * 1e4 * y[2];
f[1] = y[0] * 0.04 - y[1] * 1e4 * y[2] - y[1] * 3e7 * y[1];
f[2] = y[1] * 3e7 * y[1];
}

static void NAG_CALL pederv(Integer neq, double x, const double y[],
double pw[], Nag_User *comm) {
#define PW(I, J) pw[((I)-1) * neq + (J)-1]
struct user *s = (struct user *)comm->p;

if (s->use_comm[1]) {
printf("(User-supplied callback pederv, first invocation.)\n");
s->use_comm[1] = 0;
}

PW(1, 1) = -0.04;
PW(1, 2) = y[2] * 1e4;
PW(1, 3) = y[1] * 1e4;
PW(2, 1) = 0.04;
PW(2, 2) = y[2] * -1e4 - y[1] * 6e7;
PW(2, 3) = y[1] * -1e4;
PW(3, 1) = 0.0;
PW(3, 2) = y[1] * 6e7;
PW(3, 3) = 0.0;
}

static double NAG_CALL g(Integer neq, double x, const double y[],
Nag_User *comm) {
struct user *s = (struct user *)comm->p;

if (s->use_comm[2]) {
printf("(User-supplied callback g, first invocation.)\n");
s->use_comm[2] = 0;
}

return y[0] - 0.9;
}

static void NAG_CALL out(Integer neq, double *xsol, const double y[],
Nag_User *comm) {
struct user *s = (struct user *)comm->p;

printf("%8.2f", *xsol);
printf("%13.4f%13.5f%13.4f\n", y[0], y[1], y[2]);

*xsol = s->xend - (double)s->k * s->h;
s->k--;
}
```