NAG Library Manual, Mark 28.7
```/* nag_ode_bvp_coll_nlin_contin (d02txc) Example Program.
*
* Copyright 2022 Numerical Algorithms Group.
*
* Mark 28.7, 2022.
*/

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

#define Y(I, J) y[J * neq + I - 1]

typedef struct {
double el, en, s;
} func_data;

#ifdef __cplusplus
extern "C" {
#endif
static void NAG_CALL ffun(double x, const double y[], Integer neq,
const Integer m[], double f[], Nag_Comm *comm);
static void NAG_CALL fjac(double x, const double y[], Integer neq,
const Integer m[], double dfdy[], Nag_Comm *comm);
static void NAG_CALL gafun(const double ya[], Integer neq, const Integer m[],
Integer nlbc, double ga[], Nag_Comm *comm);
static void NAG_CALL gbfun(const double yb[], Integer neq, const Integer m[],
Integer nrbc, double gb[], Nag_Comm *comm);
static void NAG_CALL gajac(const double ya[], Integer neq, const Integer m[],
Integer nlbc, double dgady[], Nag_Comm *comm);
static void NAG_CALL gbjac(const double yb[], Integer neq, const Integer m[],
Integer nrbc, double dgbdy[], Nag_Comm *comm);
static void NAG_CALL guess(double x, Integer neq, const Integer m[], double y[],
double dym[], Nag_Comm *comm);
#ifdef __cplusplus
}
#endif

int main(void) {
static double ruser[7] = {-1.0, -1.0, -1.0, -1.0, -1.0, -1.0, -1.0};
Integer exit_status = 0;
Integer neq, mmax, nlbc, nrbc, nleft, nright;
Integer i, iermx, ijermx, j, licomm, lrcomm, mxmesh, ncol, ncont, nmesh;
double xsplit = 30.0;
double dx, el, el_init, en, ermx, s, s_init, xx;
double *mesh = 0, *rcomm = 0;
double *tol = 0, *y = 0;
Integer *ipmesh = 0, *icomm = 0, *m = 0;
func_data fd;
Nag_Comm comm;
NagError fail;

INIT_FAIL(fail);

printf("nag_ode_bvp_coll_nlin_contin (d02txc) Example Program Results\n\n");

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

/* Skip heading in data file */
scanf("%*[^\n] ");
scanf("%" NAG_IFMT "%" NAG_IFMT "", &neq, &mmax);
scanf("%" NAG_IFMT "%" NAG_IFMT "%*[^\n] ", &nlbc, &nrbc);
scanf("%" NAG_IFMT "%" NAG_IFMT "%*[^\n] ", &nleft, &nright);
scanf("%" NAG_IFMT "%" NAG_IFMT "%" NAG_IFMT "%*[^\n] ", &ncol, &nmesh,
&mxmesh);
licomm = mxmesh * (11 * neq + 6);
lrcomm = mxmesh * (109 * pow((double)neq, 2) + 78 * neq + 7);
/* Allocate memory */
if (!(tol = NAG_ALLOC(neq, double)) || !(y = NAG_ALLOC(neq * mmax, double)) ||
!(m = NAG_ALLOC(neq, Integer)) || !(mesh = NAG_ALLOC(mxmesh, double)) ||
!(rcomm = NAG_ALLOC(lrcomm, double)) ||
!(ipmesh = NAG_ALLOC(mxmesh, Integer)) ||
!(icomm = NAG_ALLOC(licomm, Integer))) {
printf("Allocation failure\n");
exit_status = -1;
goto END;
}
for (i = 0; i < neq; i++) {
scanf("%" NAG_IFMT "", &m[i]);
}
scanf("%*[^\n] ");
for (i = 0; i < neq; i++) {
scanf("%lf", &tol[i]);
}
scanf("%*[^\n] ");
/* Read problem (initial) parameters */
scanf("%lf%lf%lf%*[^\n] ", &en, &el_init, &s_init);
/* Initialize data */
el = el_init;
s = s_init;

/* Set data required for the user-supplied functions */
fd.el = el;
fd.en = en;
fd.s = s;
/* Associate the data structure with comm.p */
comm.p = (Pointer)&fd;

dx = 1.0 / (double)(nmesh - 1);
mesh[0] = 0.0;
for (i = 1; i < nmesh - 1; i++) {
mesh[i] = mesh[i - 1] + dx;
}
mesh[nmesh - 1] = 1.0;
ipmesh[0] = 1;
for (i = 1; i < nmesh - 1; i++) {
ipmesh[i] = 2;
}
ipmesh[nmesh - 1] = 1;

/* nag_ode_bvp_coll_nlin_setup (d02tvc).
* Ordinary differential equations, general nonlinear boundary value problem,
* setup for nag_ode_bvp_coll_nlin_solve (d02tlc).
*/
nag_ode_bvp_coll_nlin_setup(neq, m, nlbc, nrbc, ncol, tol, mxmesh, nmesh,
mesh, ipmesh, rcomm, lrcomm, icomm, licomm,
&fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_ode_bvp_coll_nlin_setup (d02tvc).\n%s\n",
fail.message);
exit_status = 1;
goto END;
}

/* Initialize number of continuation steps in el and s */
scanf("%" NAG_IFMT "%*[^\n] ", &ncont);

for (j = 0; j < ncont; j++) {
printf("\n Tolerance = %8.1e", tol[0]);
printf("  l = %8.3f  s =%7.4f\n", el, s);
/* Solve */

/* nag_ode_bvp_coll_nlin_solve (d02tlc).
* Ordinary differential equations, general nonlinear boundary value
* problem, collocation technique.
*/
nag_ode_bvp_coll_nlin_solve(ffun, fjac, gafun, gbfun, gajac, gbjac, guess,
rcomm, icomm, &comm, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_ode_bvp_coll_nlin_solve (d02tlc).\n%s\n",
fail.message);
exit_status = 2;
goto END;
}

/* Extract mesh */

/* nag_ode_bvp_coll_nlin_diag (d02tzc).
* Ordinary differential equations, general nonlinear boundary value
* problem, diagnostics for nag_ode_bvp_coll_nlin_solve (d02tlc).
*/
nag_ode_bvp_coll_nlin_diag(mxmesh, &nmesh, mesh, ipmesh, &ermx, &iermx,
&ijermx, rcomm, icomm, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_ode_bvp_coll_nlin_diag (d02tzc).\n%s\n",
fail.message);
exit_status = 3;
goto END;
}

printf("\n Used a mesh of %4" NAG_IFMT "  points\n", nmesh);
printf(" Maximum error = %10.2e", ermx);
printf("  in interval %4" NAG_IFMT " ", iermx);
printf(" for component %4" NAG_IFMT " \n", ijermx);
/* Print solution components on mesh */
printf("\n\n Solution on original interval:\n     x        f          g\n");
/* Left side domain [0,xsplit], evaluate at nleft+1 uniform grid points. */
dx = xsplit / (double)(nleft) / el;
xx = 0.0;
for (i = 0; i <= nleft; i++) {

/* nag_ode_bvp_coll_nlin_interp (d02tyc).
* Ordinary differential equations, general nonlinear boundary value
* problem, interpolation for nag_ode_bvp_coll_nlin_solve (d02tlc).
*/
nag_ode_bvp_coll_nlin_interp(xx, y, neq, mmax, rcomm, icomm, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_ode_bvp_coll_nlin_interp (d02tyc).\n%s\n",
fail.message);
exit_status = 5;
goto END;
}

printf("%8.2f %10.4f %10.4f \n", xx * el, Y(1, 0), Y(2, 0));
xx = xx + dx;
}
/* Right side domain (xsplit,L], evaluate at nright uniform grid points. */
dx = (el - xsplit) / (double)(nright) / el;
xx = xsplit / el;
for (i = 0; i < nright; i++) {
xx = MIN(1.0, xx + dx);

/* nag_ode_bvp_coll_nlin_interp (d02tyc).
* Ordinary differential equations, general nonlinear boundary value
* problem, interpolation for nag_ode_bvp_coll_nlin_solve (d02tlc).
*/
nag_ode_bvp_coll_nlin_interp(xx, y, neq, mmax, rcomm, icomm, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_ode_bvp_coll_nlin_interp (d02tyc).\n%s\n",
fail.message);
exit_status = 6;
goto END;
}

printf("%8.2f %10.4f %10.4f \n", xx * el, Y(1, 0), Y(2, 0));
}
/* Select mesh for continuation and update continuation parameters. */
if (j < ncont - 1) {
el = 2.0 * el;
s = 0.6 * s;
fd.el = el;
fd.s = s;
nmesh = (nmesh + 1) / 2;

/* nag_ode_bvp_coll_nlin_contin (d02txc).
* Ordinary differential equations, general nonlinear boundary value
* problem, continuation facility for
* nag_ode_bvp_coll_nlin_solve (d02tlc).
*/
nag_ode_bvp_coll_nlin_contin(mxmesh, nmesh, mesh, ipmesh, rcomm, icomm,
&fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_ode_bvp_coll_nlin_contin (d02txc).\n%s\n",
fail.message);
exit_status = 7;
goto END;
}
}
}

END:
NAG_FREE(mesh);
NAG_FREE(m);
NAG_FREE(tol);
NAG_FREE(rcomm);
NAG_FREE(y);
NAG_FREE(ipmesh);
NAG_FREE(icomm);
return exit_status;
}

static void NAG_CALL ffun(double x, const double y[], Integer neq,
const Integer m[], double f[], Nag_Comm *comm) {
func_data *fd = (func_data *)comm->p;
double el, en, s, t1, y11, y20;
double half = 0.5;
double one = 1.0;
double three = 3.0;

if (comm->user[0] == -1.0) {
printf("(User-supplied callback ffun, first invocation.)\n");
comm->user[0] = 0.0;
}
el = fd->el;
en = fd->en;
s = fd->s;
t1 = half * (three - en) * Y(1, 0);
y11 = Y(1, 1);
y20 = Y(2, 0);
f[0] = (pow(el, 3)) * (one - pow(y20, 2)) + (pow(el, 2)) * s * y11 -
el * (t1 * Y(1, 2) + en * pow(y11, 2));
f[1] = (pow(el, 2)) * s * (y20 - one) -
el * (t1 * Y(2, 1) + (en - one) * y11 * y20);
}

static void NAG_CALL fjac(double x, const double y[], Integer neq,
const Integer m[], double dfdy[], Nag_Comm *comm) {
#define DFDY(I, J, K) dfdy[I - 1 + (J - 1) * neq + K * neq * neq]
func_data *fd = (func_data *)comm->p;
double el, en, s;
double half = 0.5;
double one = 1.0;
double two = 2.0;
double three = 3.0;

if (comm->user[1] == -1.0) {
printf("(User-supplied callback fjac, first invocation.)\n");
comm->user[1] = 0.0;
}
el = fd->el;
en = fd->en;
s = fd->s;
DFDY(1, 2, 0) = -two * pow(el, 3) * Y(2, 0);
DFDY(1, 1, 0) = -el * half * (three - en) * Y(1, 2);
DFDY(1, 1, 1) = pow(el, 2) * s - el * two * en * Y(1, 1);
DFDY(1, 1, 2) = -el * half * (three - en) * Y(1, 0);
DFDY(2, 2, 0) = pow(el, 2) * s - el * (en - one) * Y(1, 1);
DFDY(2, 2, 1) = -el * half * (three - en) * Y(1, 0);
DFDY(2, 1, 0) = -el * half * (three - en) * Y(2, 1);
DFDY(2, 1, 1) = -el * (en - one) * Y(2, 0);
}

static void NAG_CALL gafun(const double ya[], Integer neq, const Integer m[],
Integer nlbc, double ga[], Nag_Comm *comm) {
#define YA(I, J) ya[J * neq + I - 1]

if (comm->user[2] == -1.0) {
printf("(User-supplied callback gafun, first invocation.)\n");
comm->user[2] = 0.0;
}
ga[0] = YA(1, 0);
ga[1] = YA(1, 1);
ga[2] = YA(2, 0);
}

static void NAG_CALL gbfun(const double yb[], Integer neq, const Integer m[],
Integer nrbc, double gb[], Nag_Comm *comm) {
#define YB(I, J) yb[J * neq + I - 1]

if (comm->user[3] == -1.0) {
printf("(User-supplied callback gbfun, first invocation.)\n");
comm->user[3] = 0.0;
}
gb[0] = YB(1, 1);
gb[1] = YB(2, 0) - 1.0;
}

static void NAG_CALL gajac(const double ya[], Integer neq, const Integer m[],
Integer nlbc, double dgady[], Nag_Comm *comm) {
#define DGADY(I, J, K) dgady[I - 1 + (J - 1) * nlbc + K * nlbc * neq]
double one = 1.0;

if (comm->user[4] == -1.0) {
printf("(User-supplied callback gajac, first invocation.)\n");
comm->user[4] = 0.0;
}
}

static void NAG_CALL gbjac(const double yb[], Integer neq, const Integer m[],
Integer nrbc, double dgbdy[], Nag_Comm *comm) {
#define DGBDY(I, J, K) dgbdy[I - 1 + (J - 1) * nrbc + K * nrbc * neq]
double one = 1.0;

if (comm->user[5] == -1.0) {
printf("(User-supplied callback gbjac, first invocation.)\n");
comm->user[5] = 0.0;
}
DGBDY(1, 1, 1) = one;
DGBDY(2, 2, 0) = one;
}

static void NAG_CALL guess(double x, Integer neq, const Integer m[], double y[],
double dym[], Nag_Comm *comm) {
func_data *fd = (func_data *)comm->p;
double ex, expmx;
double one = 1.0;
double two = 2.0;

if (comm->user[6] == -1.0) {
printf("(User-supplied callback guess, first invocation.)\n");
comm->user[6] = 0.0;
}
ex = x * fd->el;
expmx = exp(-ex);
Y(1, 0) = -pow(ex, 2) * expmx;
Y(1, 1) = (-two * ex + pow(ex, 2)) * expmx;
Y(1, 2) = (-two + 4.0 * ex - pow(ex, 2)) * expmx;
Y(2, 0) = one - expmx;
Y(2, 1) = expmx;
dym[0] = (6.0 - 6.0 * ex + pow(ex, 2)) * expmx;
dym[1] = -expmx;
}
```