NAG Library Manual, Mark 27.2
```/* nag_lapackeig_dggev3 (f08wcc) Example Program.
*
* Copyright 2021 Numerical Algorithms Group.
*
* Mark 27.2, 2021.
*/

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

#ifdef __cplusplus
extern "C" {
#endif
static Integer NAG_CALL compare(const Nag_Pointer a, const Nag_Pointer b);
static Integer normalize_vectors(Integer n, double alphai[], double v[],
Complex w[], size_t rank[], const char *title);
static Integer sort_values(Integer n, Complex alpha[], size_t rank[],
double temp[]);
#ifdef __cplusplus
}
#endif

int main(void) {
/* Scalars */
Integer i, isinf, j, n, pda, pdb, pdvl, pdvr;
Integer exit_status = 0;

/* Arrays */
Complex *eval = 0, *evec = 0;
double *a = 0, *alphai = 0, *alphar = 0, *b = 0, *beta = 0;
double *vl = 0, *vr = 0, *ea = 0;
char nag_enum_arg[40];
size_t *rank = 0;

/* Nag Types */
NagError fail;
Nag_OrderType order;
Nag_LeftVecsType jobvl;
Nag_RightVecsType jobvr;

#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_dggev3 (f08wcc) Example Program Results\n");

/* Skip heading in data file */
scanf("%*[^\n]");
scanf("%" NAG_IFMT "%*[^\n]", &n);
if (n < 0) {
printf("Invalid n\n");
exit_status = 1;
goto END;
}
scanf(" %39s%*[^\n]", nag_enum_arg);
/* nag_enum_name_to_value (x04nac).
* Converts NAG enum member name to value
*/
jobvl = (Nag_LeftVecsType)nag_enum_name_to_value(nag_enum_arg);
scanf(" %39s%*[^\n]", nag_enum_arg);
jobvr = (Nag_RightVecsType)nag_enum_name_to_value(nag_enum_arg);
pda = n;
pdb = n;
pdvl = (jobvl == Nag_LeftVecs ? n : 1);
pdvr = (jobvr == Nag_RightVecs ? n : 1);

/* Allocate memory */
if (!(a = NAG_ALLOC(n * n, double)) || !(alphai = NAG_ALLOC(n, double)) ||
!(alphar = NAG_ALLOC(n, double)) || !(b = NAG_ALLOC(n * n, double)) ||
!(beta = NAG_ALLOC(n, double)) ||
!(vl = NAG_ALLOC(pdvl * pdvl, double)) ||
!(vr = NAG_ALLOC(pdvr * pdvr, double)) || !(ea = NAG_ALLOC(n, double)) ||
!(eval = NAG_ALLOC(n, Complex)) || !(evec = NAG_ALLOC(n * n, Complex)) ||
!(rank = NAG_ALLOC(n, size_t))) {
printf("Allocation failure\n");
exit_status = -1;
goto END;
}

/* Read in the matrices A and B */
for (i = 1; i <= n; ++i)
for (j = 1; j <= n; ++j)
scanf("%lf", &A(i, j));
scanf("%*[^\n]");
for (i = 1; i <= n; ++i)
for (j = 1; j <= n; ++j)
scanf("%lf", &B(i, j));
scanf("%*[^\n]");

/* Solve the generalized eigenvalue problem Ax = lambda Bx using the
* level 3 blocked routine nag_lapackeig_dggev3 (f08wcc) which returns:
*  - eigenvalues as (alphar[] + i*alphai[])./beta[];
*  - left and right eigenvectors in vl and vr respectively.
*/
nag_lapackeig_dggev3(order, jobvl, jobvr, n, a, pda, b, pdb, alphar, alphai,
beta, vl, pdvl, vr, pdvr, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_lapackeig_dggev3 (f08wcc).\n%s\n", fail.message);
exit_status = 2;
goto END;
}

isinf = 0;
for (j = 0; j < n; ++j) {
/* Check for infinite, real and complex eigenvalues in that order */
if (fabs(beta[j]) < x02ajc()) {
isinf = j + 1;
} else {
eval[j].re = alphar[j] / beta[j];
eval[j].im = alphai[j] / beta[j] + 10.0 * x02ajc();
}
}
if (isinf) {
printf("Eigenvalue %2" NAG_IFMT " is numerically infinite.\n", isinf);
} else {
/* Print the ordered (finite) eigenvalues. */
exit_status = sort_values(n, eval, rank, ea);
if (exit_status) {
goto END;
}
}

if (jobvl == Nag_LeftVecs) {
/* Normalize and print the left eigenvectors */
exit_status =
normalize_vectors(n, alphai, vl, evec, rank, "Left eigenvectors:");
printf("\n");
}
if (jobvr == Nag_RightVecs) {
/* Normalize and print the right eigenvectors */
exit_status =
normalize_vectors(n, alphai, vr, evec, rank, "Right eigenvectors:");
printf("\n");
}

END:
NAG_FREE(a);
NAG_FREE(alphai);
NAG_FREE(alphar);
NAG_FREE(b);
NAG_FREE(beta);
NAG_FREE(vl);
NAG_FREE(vr);
NAG_FREE(ea);
NAG_FREE(eval);
NAG_FREE(evec);
NAG_FREE(rank);

return exit_status;
}

static Integer normalize_vectors(Integer n, double alphai[], double v[],
Complex w[], size_t rank[],
const char *title) {
/* Each complex eigenvector is normalized so that the element of largest
* magnitude is scaled to be real and positive.
*/

Complex scal;
double r, rr;
Integer i, j, jj, k, errors = 0;
NagError fail;

INIT_FAIL(fail);

#ifdef NAG_COLUMN_MAJOR
#define V(I, J) v[(J - 1) * n + I - 1]
#else
#define V(I, J) v[(I - 1) * n + J - 1]
#endif
#define W(I, J) w[(I - 1) * n + J - 1]

/* Re-normalize the eigenvectors, largest absolute element real. */
k = 0;
for (i = 1; i <= n; i++) {
if (fabs(alphai[i - 1]) < x02ajc()) {
jj = 1;
r = 0.0;
for (j = 1; j <= n; j++) {
W(j, i).re = V(j, i);
W(j, i).im = 0.0;
rr = fabs(V(j, i));
if (rr > r) {
r = rr;
jj = j;
}
}
} else if (k == 0) {
jj = 1;
r = 0.0;
for (j = 1; j <= n; j++) {
W(j, i).re = V(j, i);
W(j, i).im = V(j, i + 1);
rr = sqrt(V(j, i) * V(j, i) + V(j, i + 1) * V(j, i + 1));
if (rr > r) {
r = rr;
jj = j;
}
}
k = 1;
} else {
for (j = 1; j <= n; j++) {
W(j, i) = nag_complex_conjg(W(j, i - 1));
}
k = 0;
}
scal = nag_complex_conjg(W(jj, i));
scal.re = scal.re / r;
scal.im = scal.im / r;
for (j = 1; j <= n; j++) {
/* nag_complex_multiply (a02ccc), multiply two complex numbers */
W(j, i) = nag_complex_multiply(W(j, i), scal);
}
}
for (j = 1; j <= n; j++) {
/* Sort eigenvectors by eigenvalue rank using
* nag_sort_reorder_vector (m01esc).
*/
nag_sort_reorder_vector((Pointer)&W(j, 1), (size_t)n, sizeof(Complex),
(ptrdiff_t)sizeof(Complex), rank, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_sort_reorder_vector (m01esc).\n%s\n",
fail.message);
errors = 2;
goto END;
}
}
/* Print the normalized eigenvectors using
* nag_file_print_matrix_complex_gen_comp (x04dbc)
*/
fflush(stdout);
nag_file_print_matrix_complex_gen_comp(
Nag_RowMajor, Nag_GeneralMatrix, Nag_NonUnitDiag, n, n, w, n,
Nag_BracketForm, "%7.4f", title, Nag_NoLabels, 0, Nag_IntegerLabels, 0,
80, 0, 0, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_file_print_matrix_complex_gen_comp (x04dbc).\n%s\n",
fail.message);
errors = 3;
}
END:
#undef V
#undef W
return errors;
}

static Integer sort_values(Integer n, Complex vec[], size_t rank[],
double temp[]) {
Integer i, errors = 0;
NagError fail;

INIT_FAIL(fail);

/* Accumulate eigenvalue modulii in temp. */
for (i = 0; i < n; ++i) {
/* nag_complex_abs (a02cdc) - modulus of complex number. */
temp[i] = nag_complex_abs(vec[i]);
}
/* Rank sort eigenvalues by absolute values using nag_sort_rank_sort (m01dsc).
*/
nag_sort_rank_sort((Pointer)temp, (size_t)n, (ptrdiff_t)(sizeof(double)),
compare, Nag_Descending, rank, &fail);
/* Turn ranks into indices using nag_sort_permute_invert (m01zac). */
nag_sort_permute_invert(rank, (size_t)n, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_sort_permute_invert (m01zac).\n%s\n", fail.message);
errors = 1;
goto END;
}
/* Sort eigenvalues using nag_sort_reorder_vector (m01esc). */
nag_sort_reorder_vector((Pointer)vec, (size_t)n, sizeof(Complex),
(ptrdiff_t)sizeof(Complex), rank, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_sort_reorder_vector (m01esc).\n%s\n", fail.message);
errors = 2;
goto END;
}
printf("\n Eigenvalues:\n");
for (i = 0; i < n; ++i) {
printf(" %4" NAG_IFMT "     (%7.3f,%7.3f)\n", i + 1, vec[i].re, vec[i].im);
}
printf("\n");
END:
return errors;
}

static Integer NAG_CALL compare(const Nag_Pointer a, const Nag_Pointer b) {
double x = *((const double *)a) - *((const double *)b);
return (x < 0.0 ? -1 : (x == 0.0 ? 0 : 1));
}
```