NAG Library Manual, Mark 28.7
```/* nag_lapackeig_dggesx (f08xbc) Example Program.
*
* Copyright 2022 Numerical Algorithms Group.
*
* Mark 28.7, 2022.
*/

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

#ifdef __cplusplus
extern "C" {
#endif
static Nag_Boolean NAG_CALL selctg(const double ar, const double ai,
const double b);
#ifdef __cplusplus
}
#endif

int main(void) {

/* Scalars */
double abnorm, dg_a, dg_b, eps, norma, normb, normd, norme, tol;
Integer i, j, n, sdim, pda, pdb, pdc, pdd, pde, pdvsl, pdvsr;
Integer exit_status = 0;

/* Arrays */
double *a = 0, *alphai = 0, *alphar = 0, *b = 0, *beta = 0;
double *c = 0, *d = 0, *e = 0, *vsl = 0, *vsr = 0;
double rconde[2], rcondv[2];
char nag_enum_arg[40];

/* Nag Types */
NagError fail;
Nag_OrderType order;
Nag_LeftVecsType jobvsl;
Nag_RightVecsType jobvsr;
Nag_SortEigValsType sort = Nag_SortEigVals;
Nag_RCondType sense = Nag_RCondBoth;

#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_dggesx (f08xbc) Example Program Results\n\n");

/* Skip heading in data file */
scanf("%*[^\n]");
scanf("%" NAG_IFMT "%*[^\n]", &n);
if (n < 0) {
printf("Invalid n\n");
exit_status = 1;
return exit_status;
}
scanf(" %39s%*[^\n]", nag_enum_arg);
/* nag_enum_name_to_value (x04nac).
* Converts NAG enum member name to value
*/
jobvsl = (Nag_LeftVecsType)nag_enum_name_to_value(nag_enum_arg);
scanf(" %39s%*[^\n]", nag_enum_arg);
jobvsr = (Nag_RightVecsType)nag_enum_name_to_value(nag_enum_arg);
scanf(" %39s%*[^\n]", nag_enum_arg);
sense = (Nag_RCondType)nag_enum_name_to_value(nag_enum_arg);

pdvsl = (jobvsl == Nag_LeftVecs ? n : 1);
pdvsr = (jobvsr == Nag_RightVecs ? n : 1);
pda = n;
pdb = n;
pdc = n;
pdd = n;
pde = n;
/* Allocate memory */
if (!(a = NAG_ALLOC(n * n, double)) || !(b = NAG_ALLOC(n * n, double)) ||
!(c = NAG_ALLOC(n * n, double)) || !(d = NAG_ALLOC(n * n, double)) ||
!(e = NAG_ALLOC(n * n, double)) || !(alphai = NAG_ALLOC(n, double)) ||
!(alphar = NAG_ALLOC(n, double)) || !(beta = NAG_ALLOC(n, double)) ||
!(vsl = NAG_ALLOC(pdvsl * pdvsl, double)) ||
!(vsr = NAG_ALLOC(pdvsr * pdvsr, double))) {
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]");

/* Copy matrices A and B to matrices D and E using nag_blast_dge_copy
* (f16qfc), real valued general matrix copy. The copies will be used as
* comparison against reconstructed matrices.
*/
nag_blast_dge_copy(order, Nag_NoTrans, n, n, a, pda, d, pdd, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_blast_dge_copy (f16qfc).\n%s\n", fail.message);
exit_status = 1;
goto END;
}
nag_blast_dge_copy(order, Nag_NoTrans, n, n, b, pdb, e, pde, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_blast_dge_copy (f16qfc).\n%s\n", fail.message);
exit_status = 1;
goto END;
}

/* nag_blast_dge_norm (f16rac): Find norms of input matrices A and B. */
nag_blast_dge_norm(order, Nag_FrobeniusNorm, n, n, a, pda, &norma, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_blast_dge_norm (f16rac).\n%s\n", fail.message);
exit_status = 1;
goto END;
}
nag_blast_dge_norm(order, Nag_FrobeniusNorm, n, n, b, pdb, &normb, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_blast_dge_norm (f16rac).\n%s\n", fail.message);
exit_status = 1;
goto END;
}

/* nag_file_print_matrix_real_gen (x04cac): Print Matrices A and B. */
fflush(stdout);
nag_file_print_matrix_real_gen(order, Nag_GeneralMatrix, Nag_NonUnitDiag, n,
n, a, pda, "Matrix A", 0, &fail);
printf("\n");
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;
}
fflush(stdout);
nag_file_print_matrix_real_gen(order, Nag_GeneralMatrix, Nag_NonUnitDiag, n,
n, b, pdb, "Matrix B", 0, &fail);
printf("\n");
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;
}

/* Find the generalized Schur form using nag_lapackeig_dggesx (f08xbc). */
nag_lapackeig_dggesx(order, jobvsl, jobvsr, sort, selctg, sense, n, a, pda, b,
pdb, &sdim, alphar, alphai, beta, vsl, pdvsl, vsr, pdvsr,
rconde, rcondv, &fail);

if (fail.code != NE_NOERROR && fail.code != NE_SCHUR_REORDER_SELECT) {
printf("Error from nag_lapackeig_dggesx (f08xbc).\n%s\n", fail.message);
exit_status = 1;
goto END;
}

/* Check generalized Schur Form by reconstruction of Schur vectors are
* available.
*/
if (jobvsl == Nag_NotLeftVecs || jobvsr == Nag_NotRightVecs) {
/* Cannot check factorization by reconstruction Schur vectors. */
goto END;
}

/* Reconstruct A as Q*S*Z^T and subtract from original (D) using the steps
* C = Q*S (Q in vsl, S in a) using nag_blast_dgemm (f16yac).
*         Note: not nag_blast_dtrmm since S may not be strictly triangular.
* D = D - C*Z^T (Z in vsr) using nag_blast_dgemm (f16yac).
*/
dg_a = 1.0;
dg_b = 0.0;
nag_blast_dgemm(order, Nag_NoTrans, Nag_NoTrans, n, n, n, dg_a, vsl, pdvsl, a,
pda, dg_b, c, pdc, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_blast_dgemm (f16yac).\n%s\n", fail.message);
exit_status = 1;
goto END;
}
dg_a = -1.0;
dg_b = 1.0;
nag_blast_dgemm(order, Nag_NoTrans, Nag_Trans, n, n, n, dg_a, c, pdc, vsr,
pdvsr, dg_b, d, pdd, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_blast_dgemm (f16yac).\n%s\n", fail.message);
exit_status = 1;
goto END;
}

/* Reconstruct B as Q*T*Z^T and subtract from original (E) using the steps
* C = Q*T (Q in vsl, T in b) using nag_blast_dgemm (f16yac).
* E = E - C*Z^T (Z in vsr) using nag_blast_dgemm (f16yac).
*/
dg_a = 1.0;
dg_b = 0.0;
nag_blast_dgemm(order, Nag_NoTrans, Nag_NoTrans, n, n, n, dg_a, vsl, pdvsl, b,
pdb, dg_b, c, pdc, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_blast_dgemm (f16yac).\n%s\n", fail.message);
exit_status = 1;
goto END;
}
dg_a = -1.0;
dg_b = 1.0;
nag_blast_dgemm(order, Nag_NoTrans, Nag_Trans, n, n, n, dg_a, c, pdc, vsr,
pdvsr, dg_b, e, pde, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_blast_dgemm (f16yac).\n%s\n", fail.message);
exit_status = 1;
goto END;
}

/* nag_blast_dge_norm (f16rac): Find norms of difference matrices D and E. */
nag_blast_dge_norm(order, Nag_FrobeniusNorm, n, n, d, pdd, &normd, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_blast_dge_norm (f16rac).\n%s\n", fail.message);
exit_status = 1;
goto END;
}
nag_blast_dge_norm(order, Nag_FrobeniusNorm, n, n, e, pde, &norme, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_blast_dge_norm (f16rac).\n%s\n", fail.message);
exit_status = 1;
goto END;
}

/* Get the machine precision, using nag_machine_precision (x02ajc) */
eps = nag_machine_precision;
if (MAX(normd, norme) > pow(eps, 0.8) * MAX(norma, normb)) {
printf("The norm of the error in the reconstructed matrices is greater "
"than expected.\nThe Schur factorization has failed.\n");
exit_status = 1;
goto END;
}

/* Print details on eigenvalues */
printf("Number of sorted eigenvalues = %4" NAG_IFMT "\n\n", sdim);
if (fail.code == NE_SCHUR_REORDER_SELECT) {
printf("*** Note that rounding errors mean that leading eigenvalues in the"
" generalized\n    Schur form no longer satisfy selctg = Nag_TRUE"
"\n\n");
} else {
printf("The selected eigenvalues are:\n");
for (i = 0; i < sdim; i++) {
if (beta[i] != 0.0)
printf("%3" NAG_IFMT " (%13.4e, %13.4e)\n", i + 1, alphar[i] / beta[i],
alphai[i] / beta[i]);
else
printf("%3" NAG_IFMT " Eigenvalue is infinite\n", i + 1);
}
}

abnorm = sqrt(pow(norma, 2) + pow(normb, 2));
tol = eps * abnorm;

if (sense == Nag_RCondEigVals || sense == Nag_RCondBoth) {
/* Print out the reciprocal condition number and error bound */
printf("\n");
printf("For the selected eigenvalues,\nthe reciprocals of projection "
"norms onto the deflating subspaces are\n");
printf(" for left  subspace, rcond = %10.1e\n for right subspace, rcond = "
"%10.1e\n\n",
rconde[0], rconde[1]);
printf(" asymptotic error bound    = %10.1e\n\n", tol / rconde[0]);
}
if (sense == Nag_RCondEigVecs || sense == Nag_RCondBoth) {
/* Print out the reciprocal condition numbers and error bound. */
printf("For the left and right deflating subspaces,\n");
printf("reciprocal condition numbers are:\n");
printf(" for left  subspace, rcond = %10.1e\n for right subspace, rcond = "
"%10.1e\n\n",
rcondv[0], rcondv[1]);
printf(" approximate error bound   = %10.1e\n", tol / rcondv[1]);
}

END:
NAG_FREE(a);
NAG_FREE(b);
NAG_FREE(c);
NAG_FREE(d);
NAG_FREE(e);
NAG_FREE(alphai);
NAG_FREE(alphar);
NAG_FREE(beta);
NAG_FREE(vsl);
NAG_FREE(vsr);

return exit_status;
}

static Nag_Boolean NAG_CALL selctg(const double ar, const double ai,
const double b) {
/* Boolean function selctg for use with nag_lapackeig_dggesx (f08xbc)
* Returns the value Nag_TRUE if the eigenvalue is real and positive.
*/

return (ar > 0.0 && ai == 0.0 && b != 0.0 ? Nag_TRUE : Nag_FALSE);
}
```