NAG Library Manual, Mark 28.7
```/* nag_tsa_trans_hessenberg_observer (g13ewc) Example Program.
*
* Copyright 2022 Numerical Algorithms Group
*
* Mark 28.7, 2022.
*/

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

#define A(I, J) a[(I)*tda + J]
#define C(I, J) c[(I)*tdc + J]
#define U(I, J) u[(I)*tdu + J]
int main(void) {
Integer exit_status = 0, i, j, n, p, tda, tdc, tdu;
double *a = 0, *c = 0, one = 1.0, *u = 0, zero = 0.0;
Nag_ObserverForm reduceto;
NagError fail;

INIT_FAIL(fail);

printf(
"nag_tsa_trans_hessenberg_observer (g13ewc) Example Program Results\n");

/* Skip the heading in the data file and read the data. */
scanf("%*[^\n]");
scanf("%" NAG_IFMT "%" NAG_IFMT "", &n, &p);
if (n >= 1 || p >= 1) {
if (!(a = NAG_ALLOC(n * n, double)) || !(c = NAG_ALLOC(p * n, double)) ||
!(u = NAG_ALLOC(n * n, double))) {
printf("Allocation failure\n");
exit_status = -1;
goto END;
}
tda = n;
tdc = n;
tdu = n;
} else {
printf("Invalid n or p.\n");
exit_status = 1;
return exit_status;
}

reduceto = Nag_UH_Observer;

for (j = 0; j < n; ++j)
for (i = 0; i < n; ++i)
scanf("%lf", &A(i, j));
for (i = 0; i < p; ++i)
for (j = 0; j < n; ++j)
scanf("%lf", &C(i, j));

if (u) /* Initialize U as the identity matrix. */
for (i = 0; i < n; ++i) {
for (j = 0; j < n; ++j)
U(i, j) = zero;
U(i, i) = one;
}

/* Reduce the pair (A,C) to reduceto observer Hessenberg form. */
/* nag_tsa_trans_hessenberg_observer (g13ewc).
* Unitary state-space transformation to reduce (AC) to
* lower or upper observer Hessenberg form
*/
nag_tsa_trans_hessenberg_observer(n, p, reduceto, a, tda, c, tdc, u, tdu,
&fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_tsa_trans_hessenberg_observer (g13ewc).\n%s\n",
fail.message);
exit_status = 1;
goto END;
}

printf("\nThe transformed state transition matrix is \n\n");
for (i = 0; i < n; ++i) {
for (j = 0; j < n; ++j)
printf("%8.4f ", A(i, j));
printf("\n");
}
printf("\nThe transformed input matrix is \n\n");
for (i = 0; i < p; ++i) {
for (j = 0; j < n; ++j)
printf("%8.4f ", C(i, j));
printf("\n");
}
if (u) {
printf("\nThe transformation matrix that reduces (A,C) "
"to observer Hessenberg form is \n\n");
for (i = 0; i < n; ++i) {
for (j = 0; j < n; ++j)
printf("%8.4f ", U(i, j));
printf("\n");
}
}
END:
NAG_FREE(a);
NAG_FREE(c);
NAG_FREE(u);
return exit_status;
}
```