NAG Library Manual, Mark 27.3
```/* nag_lapackeig_dtgevc (f08ykc) Example Program.
*
* Copyright 2021 Numerical Algorithms Group.
*
* Mark 27.3, 2021.
*/

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

static Integer normalize_vectors(Nag_OrderType order, Integer n, double qz[],
double alphai[], const char *title);

int main(void) {
/* Scalars */
Integer i, icols, ihi, ilo, irows, j, m, n, pda, pdb, pdq, pdz;
Integer exit_status = 0;
Nag_Boolean ileft, iright;

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]
#define Q(I, J) q[(J - 1) * pdq + 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]
#define Q(I, J) q[(I - 1) * pdq + J - 1]
order = Nag_RowMajor;
#endif

INIT_FAIL(fail);

printf("nag_lapackeig_dtgevc (f08ykc) Example Program Results\n\n");

/* ileft  is Nag_TRUE if left  eigenvectors are required */
/* iright is Nag_TRUE if right eigenvectors are required */
ileft = Nag_TRUE;
iright = Nag_TRUE;

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

pda = n;
pdb = n;
pdq = n;
pdz = n;

/* Allocate memory */
if (!(a = NAG_ALLOC(n * n, double)) || !(b = NAG_ALLOC(n * n, double)) ||
!(q = NAG_ALLOC(n * n, double)) || !(z = NAG_ALLOC(n * n, double)) ||
!(alphai = NAG_ALLOC(n, double)) || !(alphar = NAG_ALLOC(n, double)) ||
!(beta = NAG_ALLOC(n, double)) || !(lscale = NAG_ALLOC(n, double)) ||
!(rscale = NAG_ALLOC(n, double)) || !(tau = NAG_ALLOC(n, 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 the real general matrix pair (A,B) using nag_lapackeig_dggbal
* (f08whc). */
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;
}

/* Print matrices A and B after balancing using
* nag_file_print_matrix_real_gen_comp (x04cbc).
*/
fflush(stdout);
nag_file_print_matrix_real_gen_comp(
order, Nag_GeneralMatrix, Nag_NonUnitDiag, n, n, a, pda, "%8.4f",
"Matrix A after balancing", Nag_IntegerLabels, NULL, Nag_IntegerLabels,
NULL, 80, 0, 0, &fail);
if (fail.code == NE_NOERROR) {
fflush(stdout);
nag_file_print_matrix_real_gen_comp(
order, Nag_GeneralMatrix, Nag_NonUnitDiag, n, n, b, pdb, "%8.4f",
"Matrix B after balancing", Nag_IntegerLabels, NULL, Nag_IntegerLabels,
NULL, 80, 0, 0, &fail);
}
if (fail.code != NE_NOERROR) {
printf("Error from nag_file_print_matrix_real_gen_comp (x04cbc).\n%s\n",
fail.message);
exit_status = 2;
goto END;
}
printf("\n");

/* Reduce B to triangular form using QR and multipling both sides by Q^T */
irows = ihi + 1 - ilo;
icols = n + 1 - ilo;
/* nag_lapackeig_dgeqrf (f08aec).
* QR factorization of real general rectangular matrix
*/
nag_lapackeig_dgeqrf(order, irows, icols, &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 = 3;
goto END;
}

/* Apply the Q to matrix A -  nag_lapackeig_dormqr (f08agc)
* as determined by nag_lapackeig_dgeqrf (f08aec).
*/
nag_lapackeig_dormqr(order, Nag_LeftSide, Nag_Trans, irows, icols, 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 = 4;
goto END;
}

/* Initialize Q (if left eigenvectors are required) */
if (ileft) {
/* Q = I. */
nag_blast_dge_load(order, n, n, 0.0, 1.0, q, pdq, &fail);
/* Copy B to Q using nag_blast_dge_copy (f16qfc). */
nag_blast_dge_copy(order, Nag_NoTrans, irows - 1, irows - 1,
&B(ilo + 1, ilo), pdb, &Q(ilo + 1, ilo), pdq, &fail);
/* nag_lapackeig_dorgqr (f08afc).
* Form all or part of orthogonal Q from QR factorization
* determined by nag_lapackeig_dgeqrf (f08aec) or nag_lapackeig_dgeqpf
* (f08bec)
*/
nag_lapackeig_dorgqr(order, irows, irows, irows, &Q(ilo, ilo), pdq, tau,
&fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_lapackeig_dorgqr (f08afc).\n%s\n", fail.message);
exit_status = 5;
goto END;
}
}

/* Initialize Z (if right eigenvectors are required) */
if (iright) {
/* Z = I. */
nag_blast_dge_load(order, n, n, 0.0, 1.0, z, pdz, &fail);
}

/* Compute the generalized Hessenberg form of (A,B) */
/* nag_lapackeig_dgghrd (f08wec).
* Orthogonal reduction of a pair of real general matrices
* to generalized upper Hessenberg form
*/
nag_lapackeig_dgghrd(order, Nag_UpdateSchur, Nag_UpdateZ, n, ilo, ihi, a, pda,
b, pdb, q, pdq, z, pdz, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_lapackeig_dgghrd (f08wec).\n%s\n", fail.message);
exit_status = 6;
goto END;
}

/* Matrix A in generalized Hessenberg form */
/* nag_file_print_matrix_real_gen_comp (x04cbc), see above. */
fflush(stdout);
nag_file_print_matrix_real_gen_comp(
order, Nag_GeneralMatrix, Nag_NonUnitDiag, n, n, a, pda, "%8.4f",
"Matrix A in Hessenberg form", Nag_IntegerLabels, NULL, Nag_IntegerLabels,
NULL, 80, 0, 0, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_file_print_matrix_real_gen_comp (x04cbc).\n%s\n",
fail.message);
exit_status = 7;
goto END;
}
printf("\n");

/* Matrix B in generalized Hessenberg form */
/* nag_file_print_matrix_real_gen_comp (x04cbc), see above. */
fflush(stdout);
nag_file_print_matrix_real_gen_comp(
order, Nag_GeneralMatrix, Nag_NonUnitDiag, n, n, b, pdb, "%8.4f",
"Matrix B in Hessenberg form", Nag_IntegerLabels, NULL, Nag_IntegerLabels,
NULL, 80, 0, 0, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_file_print_matrix_real_gen_comp (x04cbc).\n%s\n",
fail.message);
exit_status = 1;
goto END;
}

/* 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_Schur, Nag_AccumulateQ, Nag_AccumulateZ, n,
ilo, ihi, a, pda, b, pdb, alphar, alphai, beta, q, pdq,
z, pdz, &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 eigenvalue parameters */
printf("\n Generalized eigenvalues\n");
for (i = 0; i < n; ++i) {
if (beta[i] != 0.0) {
printf(" %4" NAG_IFMT "     (%8.4f,%8.4f)\n", i + 1, alphar[i] / beta[i],
alphai[i] / beta[i]);
} else
printf(" %4" NAG_IFMT "Eigenvalue is infinite\n", i + 1);
}
printf("\n");

/* Compute left and right generalized eigenvectors
* of the balanced matrix - nag_lapackeig_dtgevc (f08ykc).
*/
nag_lapackeig_dtgevc(order, Nag_BothSides, Nag_BackTransform, NULL, n, a, pda,
b, pdb, q, pdq, z, pdz, n, &m, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_lapackeig_dtgevc (f08ykc).\n%s\n", fail.message);
exit_status = 1;
goto END;
}
if (iright) {
/* Compute right eigenvectors of the original matrix pair
* supplied tonag_dggbal (f08whc) using nag_lapackeig_dggbak (f08wjc).
*/
nag_lapackeig_dggbak(order, Nag_DoBoth, Nag_RightSide, n, ilo, ihi, lscale,
rscale, n, z, pdz, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_lapackeig_dggbak (f08wjc).\n%s\n", fail.message);
exit_status = 1;
goto END;
}
/* Normalize and print the right eigenvectors */
exit_status = normalize_vectors(order, n, z, alphai, "Right eigenvectors");
}
printf("\n");

/* Compute left eigenvectors of the original matrix */
if (ileft) {
/* nag_lapackeig_dggbak (f08wjc), see above. */
nag_lapackeig_dggbak(order, Nag_DoBoth, Nag_LeftSide, n, ilo, ihi, lscale,
rscale, n, q, pdq, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_lapackeig_dggbak (f08wjc).\n%s\n", fail.message);
exit_status = 1;
goto END;
}
/* Normalize the left eigenvectors */
exit_status = normalize_vectors(order, n, q, alphai, "Left eigenvectors");
}
END:
NAG_FREE(a);
NAG_FREE(b);
NAG_FREE(q);
NAG_FREE(z);
NAG_FREE(alphai);
NAG_FREE(alphar);
NAG_FREE(beta);
NAG_FREE(lscale);
NAG_FREE(rscale);
NAG_FREE(tau);

return exit_status;
}

static Integer normalize_vectors(Nag_OrderType order, Integer n, double qz[],
double alphai[], const char *title) {
/* Real eigenvectors are scaled so that the maximum value of elements is 1.0;
* each complex eigenvector z[] is normalized so that the element of largest
* magnitude is scaled to be (1.0,0.0).
*/

double a, b, u, v, r, ri, rr;
Integer colinc, rowinc, i, ii, j, k, indqz, errors = 0;
NagError fail;

INIT_FAIL(fail);

if (order == Nag_ColMajor) {
rowinc = 1;
colinc = n;
} else {
rowinc = n;
colinc = 1;
}
indqz = 0;
for (j = 0; j < n; j++) {
if (alphai[j] >= 0.0) {
if (alphai[j] == 0.0) {
/* The 2-norm of Q is calculated using nag_blast_dge_norm (f16rac). */
nag_blast_dge_norm(order, Nag_FrobeniusNorm, n, 1, &qz[indqz], n, &r,
&fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_blast_dge_norm (f16rac).\n%s\n", fail.message);
errors = 1;
goto END;
}
for (i = 0; i < n * rowinc; i += rowinc) {
qz[indqz + i] = qz[indqz + i] / r;
}
} else {
/* norm of j-th complex eigenvector using nag_blast_dge_norm (f16rac),
* stored as two arrays of length n.
*/
k = 0;
rr = 0.0;
r = -1.0;
for (i = 0; i < n * rowinc; i += rowinc) {
ii = indqz + i;
ri = qz[ii] * qz[ii] + qz[ii + colinc] * qz[ii + colinc];
rr = rr + ri;
if (ri > r) {
k = i;
r = ri;
}
}
a = qz[indqz + k];
b = qz[indqz + colinc + k];
r = sqrt(r * rr);

for (i = 0; i < n * rowinc; i += rowinc) {
u = qz[indqz + i];
v = qz[indqz + colinc + i];
qz[indqz + i] = (u * a + v * b) / r;
qz[indqz + colinc + i] = (v * a - u * b) / r;
}
indqz += colinc;
}
indqz += colinc;
}
}
/* Print the normalized eigenvectors using
* nag_file_print_matrix_real_gen_comp (x04cbc)
*/
fflush(stdout);
nag_file_print_matrix_real_gen_comp(
order, Nag_GeneralMatrix, Nag_NonUnitDiag, n, n, qz, n, "%8.4f", title,
Nag_IntegerLabels, NULL, Nag_IntegerLabels, NULL, 80, 0, 0, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_file_print_matrix_real_gen_comp (x04cbc).\n%s\n",
fail.message);
errors = 1;
}
END:
return errors;
}
```