```/* nag_lapackeig_dgebal (f08nhc) Example Program.
*
* Copyright 2020 Numerical Algorithms Group.
*
* Mark 27.1, 2020.
*/

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

int main(void) {
/* Scalars */
Integer firstnz, i, ihi, ilo, j, m, n, pda, pdh, pdvr;
Integer scale_len, tau_len, wi_len, wr_len;
Integer exit_status = 0;
NagError fail;
Nag_OrderType order;
/* Arrays */
double *a = 0, *h = 0, *scale = 0, *tau = 0, *vl = 0, *vr = 0;
double *wi = 0, *wr = 0;
Nag_Boolean *select = 0;

#ifdef NAG_COLUMN_MAJOR
#define A(I, J) a[(J - 1) * pda + I - 1]
#define H(I, J) h[(J - 1) * pdh + I - 1]
#define VR(I, J) vr[(J - 1) * pdvr + I - 1]
order = Nag_ColMajor;
#else
#define A(I, J) a[(I - 1) * pda + J - 1]
#define H(I, J) h[(I - 1) * pdh + J - 1]
#define VR(I, J) vr[(I - 1) * pdvr + J - 1]
order = Nag_RowMajor;
#endif

INIT_FAIL(fail);

printf("nag_lapackeig_dgebal (f08nhc) Example Program Results\n\n");

/* Skip heading in data file */
scanf("%*[^\n] ");
scanf("%" NAG_IFMT "%*[^\n] ", &n);

pda = n;
pdh = n;
pdvr = n;
scale_len = n;
tau_len = n;
wi_len = n;
wr_len = n;

/* Allocate memory */
if (!(a = NAG_ALLOC(n * n, double)) || !(h = NAG_ALLOC(n * n, double)) ||
!(scale = NAG_ALLOC(scale_len, double)) ||
!(tau = NAG_ALLOC(tau_len, double)) || !(vl = NAG_ALLOC(1 * 1, double)) ||
!(vr = NAG_ALLOC(n * n, double)) || !(wi = NAG_ALLOC(wi_len, double)) ||
!(wr = NAG_ALLOC(wr_len, double)) ||
!(select = NAG_ALLOC(1, Nag_Boolean))) {
printf("Allocation failure\n");
exit_status = -1;
goto END;
}

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

/* Balance A */
/* nag_lapackeig_dgebal (f08nhc).
* Balance real general matrix
*/
nag_lapackeig_dgebal(order, Nag_DoBoth, n, a, pda, &ilo, &ihi, scale, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_lapackeig_dgebal (f08nhc).\n%s\n", fail.message);
exit_status = 1;
goto END;
}

/* Reduce A to upper Hessenberg form H = (Q^T)*A*Q */
/* nag_lapackeig_dgehrd (f08nec).
* Orthogonal reduction of real general matrix to upper
* Hessenberg form
*/
nag_lapackeig_dgehrd(order, n, ilo, ihi, a, pda, tau, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_lapackeig_dgehrd (f08nec).\n%s\n", fail.message);
exit_status = 1;
goto END;
}

/* Copy A to H and VR */
for (i = 1; i <= n; ++i) {
for (j = 1; j <= n; ++j) {
H(i, j) = A(i, j);
VR(i, j) = A(i, j);
}
}

/* Form Q explicitly, storing the result in VR */
/* nag_lapackeig_dorghr (f08nfc).
* Generate orthogonal transformation matrix from reduction
* to Hessenberg form determined by nag_lapackeig_dgehrd (f08nec)
*/
nag_lapackeig_dorghr(order, n, 1, n, vr, pdvr, tau, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_lapackeig_dorghr (f08nfc).\n%s\n", fail.message);
exit_status = 1;
goto END;
}

/* Calculate the eigenvalues and Schur factorization of A */
/* nag_lapackeig_dhseqr (f08pec).
* Eigenvalues and Schur factorization of real upper
* Hessenberg matrix reduced from real general matrix
*/
nag_lapackeig_dhseqr(order, Nag_Schur, Nag_UpdateZ, n, ilo, ihi, h, pdh, wr,
wi, vr, pdvr, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_lapackeig_dhseqr (f08pec).\n%s\n", fail.message);
exit_status = 1;
goto END;
}
printf(" Eigenvalues\n");
for (i = 1; i <= n; ++i)
printf("(%8.4f,%8.4f)\n", wr[i - 1], wi[i - 1]);
/* Calculate the eigenvectors of A, storing the result in VR */
/* nag_lapackeig_dtrevc (f08qkc).
* Left and right eigenvectors of real upper
* quasi-triangular matrix
*/
nag_lapackeig_dtrevc(order, Nag_RightSide, Nag_BackTransform, select, n, h,
pdh, vl, 1, vr, pdvr, n, &m, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_lapackeig_dtrevc (f08qkc).\n%s\n", fail.message);
exit_status = 1;
goto END;
}
/* nag_lapackeig_dgebak (f08njc).
* Transform eigenvectors of real balanced matrix to those
* of original matrix supplied to nag_lapackeig_dgebal (f08nhc)
*/
nag_lapackeig_dgebak(order, Nag_DoBoth, Nag_RightSide, n, ilo, ihi, scale, m,
vr, pdvr, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_lapackeig_dgebak (f08njc).\n%s\n", fail.message);
exit_status = 1;
goto END;
}

/* Normalize the left eigenvectors */
for (j = 1; j <= n; j++) {
firstnz = n;
for (i = n; i >= 1; i--) {
if (VR(i, j) != 0) {
firstnz = i;
}
}
for (i = n; i >= 1; i--) {
VR(i, j) = VR(i, j) / VR(firstnz, j);
}
}

/* Print eigenvectors */
printf("\n");
/* 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,
m, vr, pdvr, "Contents of array VR", 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;
}
END:
NAG_FREE(a);
NAG_FREE(h);
NAG_FREE(scale);
NAG_FREE(tau);
NAG_FREE(vl);
NAG_FREE(vr);
NAG_FREE(wi);
NAG_FREE(wr);
NAG_FREE(select);

return exit_status;
}
```