/* nag_lapackeig_dgeesx (f08pbc) Example Program.
*
* Copyright 2022 Numerical Algorithms Group.
*
* Mark 28.3, 2022.
*/
#include <math.h>
#include <nag.h>
#include <stdio.h>
#ifdef __cplusplus
extern "C" {
#endif
static Nag_Boolean NAG_CALL select_fun(const double wr, const double wi);
#ifdef __cplusplus
}
#endif
int main(void) {
/* Scalars */
double alpha, anorm, beta, eps, norm, rconde, rcondv;
Integer i, j, n, pda, pdc, pdd, pdvs, sdim;
Integer exit_status = 0;
/* Arrays */
double *a = 0, *c = 0, *d = 0, *vs = 0, *wi = 0, *wr = 0;
/* Nag Types */
NagError fail;
Nag_OrderType order;
#ifdef NAG_COLUMN_MAJOR
#define A(I, J) a[(J - 1) * pda + I - 1]
order = Nag_ColMajor;
#else
#define A(I, J) a[(I - 1) * pda + J - 1]
order = Nag_RowMajor;
#endif
INIT_FAIL(fail);
printf("nag_lapackeig_dgeesx (f08pbc) 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;
}
pda = n;
pdc = n;
pdd = n;
pdvs = n;
/* Allocate memory */
if (!(a = NAG_ALLOC(n * n, double)) || !(c = NAG_ALLOC(n * n, double)) ||
!(d = NAG_ALLOC(n * n, double)) || !(vs = NAG_ALLOC(n * n, double)) ||
!(wi = NAG_ALLOC(n, double)) || !(wr = NAG_ALLOC(n, double))) {
printf("Allocation failure\n");
exit_status = -1;
goto END;
}
/* Read in the matrix A */
for (i = 1; i <= n; ++i)
for (j = 1; j <= n; ++j)
scanf("%lf", &A(i, j));
scanf("%*[^\n]");
/* Copy A to D: nag_blast_dge_copy (f16qfc),
* real valued general matrix copy.
*/
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_norm (f16rac): Find norm of matrix A for use later
* in relative error test.
*/
nag_blast_dge_norm(order, Nag_OneNorm, n, n, a, pda, &anorm, &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 Matrix A. */
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;
}
/* Find the Schur factorization of A using nag_lapackeig_dgeesx (f08pbc). */
nag_lapackeig_dgeesx(order, Nag_Schur, Nag_SortEigVals, select_fun,
Nag_RCondBoth, n, a, pda, &sdim, wr, wi, vs, pdvs,
&rconde, &rcondv, &fail);
if (fail.code != NE_NOERROR && fail.code != NE_SCHUR_REORDER_SELECT) {
printf("Error from nag_lapackeig_dgeesx (f08pbc).\n%s\n", fail.message);
exit_status = 1;
goto END;
}
/* Reconstruct A from Schur Factorization Z*T*Trans(Z) where T is upper
* triangular and stored in A. This can be done using the following steps:
* i. C = Z*T (nag_blast_dgemm, f16yac),
* ii. D = D-C*trans(Z) (nag_blast_dgemm, f16yac).
*/
alpha = 1.0;
beta = 0.0;
nag_blast_dgemm(order, Nag_NoTrans, Nag_NoTrans, n, n, n, alpha, vs, pdvs, a,
pda, beta, 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;
}
/* nag_blast_dgemm (f16yac):
* Compute D = A - C*Z^T.
*/
alpha = -1.0;
beta = 1.0;
nag_blast_dgemm(order, Nag_NoTrans, Nag_Trans, n, n, n, alpha, c, pdc, vs,
pdvs, beta, 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;
}
/* nag_blast_dge_norm (f16rac): Find norm of difference matrix D and print
* warning if it is too large relative to norm of A.
*/
nag_blast_dge_norm(order, Nag_OneNorm, n, n, d, pdd, &norm, &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 (norm > pow(eps, 0.8) * MAX(anorm, 1.0)) {
printf("||A-(Z*T*Z^T)||/||A|| is larger than expected.\n"
"Schur factorization has failed.\n");
exit_status = 1;
goto END;
}
/* Print details on eigenvalues */
printf("Number of eigenvalues for which select is true = %4" NAG_IFMT "\n\n",
sdim);
if (fail.code == NE_SCHUR_REORDER_SELECT) {
printf(" ** Note that rounding errors mean that leading eigenvalues in the"
" Schur form\n no longer satisfy select(lambda) = Nag_TRUE\n\n");
} else {
printf("The selected eigenvalues are:\n");
for (i = 0; i < sdim; i++)
printf("%3" NAG_IFMT " (%13.4e, %13.4e)\n", i + 1, wr[i], wi[i]);
}
/* Print out the reciprocal condition numbers */
printf("\nReciprocal of projection norm onto the invariant subspace\n");
printf("%26sfor the selected eigenvalues rconde = %8.1e\n\n", "", rconde);
printf("Reciprocal condition number for the invariant subspace rcondv = "
"%8.1e\n\n",
rcondv);
/* Compute the approximate asymptotic error bound on the average absolute
* error of the selected eigenvalues given by eps*norm(A)/rconde.
*/
printf("Approximate asymptotic error bound for selected eigenvalues = "
"%8.1e\n\n",
eps * anorm / rconde);
/* Compute an approximate asymptotic bound on the maximum angular error in
* the computed invariant subspace given by eps*norm(A)/rcondv
*/
printf("Approximate asymptotic error bound for the invariant subspace = "
"%8.1e\n",
eps * anorm / rcondv);
END:
NAG_FREE(a);
NAG_FREE(c);
NAG_FREE(d);
NAG_FREE(vs);
NAG_FREE(wi);
NAG_FREE(wr);
return exit_status;
}
static Nag_Boolean NAG_CALL select_fun(const double ar, const double ai) {
/* Boolean function select for use with nag_lapackeig_dgees (f08pac)
* Returns the value Nag_TRUE if the eigenvalue is real and positive
*/
return (ar > 0.0 && ai == 0.0 ? Nag_TRUE : Nag_FALSE);
}