NAG Library Manual, Mark 30.1
```/* nag_lapackeig_dhgeqz (f08xec) Example Program.
*
* Copyright 2024 Numerical Algorithms Group.
*
* Mark 30.1, 2024.
*/

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

int main(void) {
/* Scalars */
Integer i, ihi, ilo, irows, j, n, pda, pdb;
Integer alpha_len, beta_len, scale_len, tau_len;
Integer exit_status = 0;

NagError fail;
Nag_OrderType order;
/* Arrays */
double *a = 0, *alphai = 0, *alphar = 0, *b = 0, *beta = 0;
double *lscale = 0, *q = 0, *rscale = 0, *tau = 0, *z = 0;

#ifdef NAG_COLUMN_MAJOR
#define A(I, J) a[(J - 1) * pda + I - 1]
#define B(I, J) b[(J - 1) * pdb + I - 1]
order = Nag_ColMajor;
#else
#define A(I, J) a[(I - 1) * pda + J - 1]
#define B(I, J) b[(I - 1) * pdb + J - 1]
order = Nag_RowMajor;
#endif

INIT_FAIL(fail);

printf("nag_lapackeig_dhgeqz (f08xec) Example Program Results\n\n");
/* Skip heading in data file */
scanf("%*[^\n] ");
scanf("%" NAG_IFMT "%*[^\n] ", &n);
pda = n;
pdb = n;
alpha_len = n;
beta_len = n;
scale_len = n;
tau_len = n;

/* Allocate memory */
if (!(a = NAG_ALLOC(n * n, double)) ||
!(alphai = NAG_ALLOC(alpha_len, double)) ||
!(alphar = NAG_ALLOC(alpha_len, double)) ||
!(b = NAG_ALLOC(n * n, double)) ||
!(beta = NAG_ALLOC(beta_len, double)) ||
!(lscale = NAG_ALLOC(scale_len, double)) ||
!(q = NAG_ALLOC(1 * 1, double)) ||
!(rscale = NAG_ALLOC(scale_len, double)) ||
!(tau = NAG_ALLOC(tau_len, double)) || !(z = NAG_ALLOC(1 * 1, double))) {
printf("Allocation failure\n");
exit_status = -1;
goto END;
}

/* READ matrix A from data file */
for (i = 1; i <= n; ++i) {
for (j = 1; j <= n; ++j)
scanf("%lf", &A(i, j));
}
scanf("%*[^\n] ");

/* READ matrix B from data file */
for (i = 1; i <= n; ++i) {
for (j = 1; j <= n; ++j)
scanf("%lf", &B(i, j));
}
scanf("%*[^\n] ");
/* Balance matrix pair (A,B) */
/* nag_lapackeig_dggbal (f08whc).
* Balance a pair of real general matrices
*/
nag_lapackeig_dggbal(order, Nag_DoBoth, n, a, pda, b, pdb, &ilo, &ihi, lscale,
rscale, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_lapackeig_dggbal (f08whc).\n%s\n", fail.message);
exit_status = 1;
goto END;
}

/* Matrix A after balancing */
/* nag_file_print_matrix_real_gen (x04cac).
* Print real general matrix (easy-to-use)
*/
fflush(stdout);
nag_file_print_matrix_real_gen(order, Nag_GeneralMatrix, Nag_NonUnitDiag, n,
n, a, pda, "Matrix A after balancing", 0,
&fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_file_print_matrix_real_gen (x04cac).\n%s\n",
fail.message);
exit_status = 1;
goto END;
}
printf("\n");

/* Matrix B after balancing */
/* nag_file_print_matrix_real_gen (x04cac), see above. */
fflush(stdout);
nag_file_print_matrix_real_gen(order, Nag_GeneralMatrix, Nag_NonUnitDiag, n,
n, b, pdb, "Matrix B after balancing", 0,
&fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_file_print_matrix_real_gen (x04cac).\n%s\n",
fail.message);
exit_status = 1;
goto END;
}
printf("\n");

/* Reduce B to triangular form using QR */
irows = ihi + 1 - ilo;
/* nag_lapackeig_dgeqrf (f08aec).
* QR factorization of real general rectangular matrix
*/
nag_lapackeig_dgeqrf(order, irows, irows, &B(ilo, ilo), pdb, tau, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_lapackeig_dgeqrf (f08aec).\n%s\n", fail.message);
exit_status = 1;
goto END;
}

/* Apply the orthogonal transformation to matrix A */
/* nag_lapackeig_dormqr (f08agc).
* Apply orthogonal transformation determined by nag_lapackeig_dgeqrf
* (f08aec) or nag_lapackeig_dgeqpf (f08bec)
*/
nag_lapackeig_dormqr(order, Nag_LeftSide, Nag_Trans, irows, irows, irows,
&B(ilo, ilo), pdb, tau, &A(ilo, ilo), pda, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_lapackeig_dormqr (f08agc).\n%s\n", fail.message);
exit_status = 1;
goto END;
}

/* Compute the generalized Hessenberg form of (A,B) */
/* nag_lapackeig_dgghd3 (f08wfc).
* Orthogonal reduction of a pair of real general matrices
* to generalized upper Hessenberg form
*/
nag_lapackeig_dgghd3(order, Nag_NotQ, Nag_NotZ, irows, 1, irows, &A(ilo, ilo),
pda, &B(ilo, ilo), pdb, q, 1, z, 1, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_lapackeig_dgghd3 (f08wfc).\n%s\n", fail.message);
exit_status = 1;
goto END;
}

/* Matrix A in generalized Hessenberg form */
/* nag_file_print_matrix_real_gen (x04cac), see above. */
fflush(stdout);
nag_file_print_matrix_real_gen(order, Nag_GeneralMatrix, Nag_NonUnitDiag, n,
n, a, pda, "Matrix A in Hessenberg form", 0,
&fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_file_print_matrix_real_gen (x04cac).\n%s\n",
fail.message);
exit_status = 1;
goto END;
}
printf("\n");
/* Matrix B in generalized Hessenberg form */
/* nag_file_print_matrix_real_gen (x04cac), see above. */
fflush(stdout);
nag_file_print_matrix_real_gen(order, Nag_GeneralMatrix, Nag_NonUnitDiag, n,
n, b, pdb, "Matrix B is triangular", 0, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_file_print_matrix_real_gen (x04cac).\n%s\n",
fail.message);
exit_status = 1;
goto END;
}

/* Compute the generalized Schur form */
/* nag_lapackeig_dhgeqz (f08xec).
* Eigenvalues and generalized Schur factorization of real
* generalized upper Hessenberg form reduced from a pair of
* real general matrices
*/
nag_lapackeig_dhgeqz(order, Nag_EigVals, Nag_NotQ, Nag_NotZ, n, ilo, ihi, a,
pda, b, pdb, alphar, alphai, beta, q, 1, z, 1, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_lapackeig_dhgeqz (f08xec).\n%s\n", fail.message);
exit_status = 1;
goto END;
}

/* Print the generalized eigenvalues */
printf("\n Generalized eigenvalues\n");
for (i = 1; i <= n; ++i) {
if (beta[i - 1] != 0.0) {
printf(" %4" NAG_IFMT "     (%7.3f,%7.3f)\n", i,
alphar[i - 1] / beta[i - 1], alphai[i - 1] / beta[i - 1]);
} else
printf(" %4" NAG_IFMT "Eigenvalue is infinite\n", i);
}
END:
NAG_FREE(a);
NAG_FREE(alphai);
NAG_FREE(alphar);
NAG_FREE(b);
NAG_FREE(beta);
NAG_FREE(lscale);
NAG_FREE(q);
NAG_FREE(rscale);
NAG_FREE(tau);
NAG_FREE(z);

return exit_status;
}
```