/* nag_lapackeig_ztgevc (f08yxc) 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(Nag_OrderType order, Integer n, Complex qz[],
Complex tz[], double temp[], size_t rank[],
const char *title);
static Integer sort_values(Integer n, Complex alpha[], Complex beta[],
size_t rank[], double temp[]);
#ifdef __cplusplus
}
#endif
int main(void) {
/* Scalars */
Integer i, icols, ihi, ilo, irows, j, m, n, pda, pdb, pdq, pdz;
Integer exit_status = 0, prbal = 0, prhess = 0;
Complex one, zero;
/* Arrays */
Complex *a = 0, *alpha = 0, *b = 0, *beta = 0, *q = 0, *tau = 0;
Complex *z = 0;
double *lscale = 0, *rscale = 0, *temp = 0;
size_t *rank = 0;
/* Nag Types */
Nag_Boolean ileft, iright;
NagError fail;
Nag_OrderType order;
#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_ztgevc (f08yxc) Example Program Results\n\n");
/* ileft is true if left eigenvectors are required;
* iright is true if right eigenvectors are required.
*/
ileft = Nag_TRUE;
iright = Nag_TRUE;
zero = nag_complex_create(0.0, 0.0);
one = nag_complex_create(1.0, 0.0);
/* Skip heading in data file and read matrix size. */
scanf("%*[^\n] ");
scanf("%" NAG_IFMT "%*[^\n] ", &n);
pda = n;
pdb = n;
pdq = n;
pdz = n;
/* Allocate memory */
if (!(a = NAG_ALLOC(n * n, Complex)) || !(b = NAG_ALLOC(n * n, Complex)) ||
!(q = NAG_ALLOC(n * n, Complex)) || !(z = NAG_ALLOC(n * n, Complex)) ||
!(alpha = NAG_ALLOC(n, Complex)) || !(beta = NAG_ALLOC(n, Complex)) ||
!(tau = NAG_ALLOC(n, Complex)) || !(temp = NAG_ALLOC(n, double)) ||
!(rank = NAG_ALLOC(n, size_t)) || !(lscale = NAG_ALLOC(n, double)) ||
!(rscale = 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 , %lf )", &A(i, j).re, &A(i, j).im);
scanf("%*[^\n] ");
/* READ matrix B from data file */
for (i = 1; i <= n; ++i)
for (j = 1; j <= n; ++j)
scanf(" ( %lf , %lf )", &B(i, j).re, &B(i, j).im);
scanf("%*[^\n] ");
/* Balance pair (A,B) of complex general matrices using
* nag_lapackeig_zggbal (f08wvc).
*/
nag_lapackeig_zggbal(order, Nag_DoBoth, n, a, pda, b, pdb, &ilo, &ihi, lscale,
rscale, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_lapackeig_zggbal (f08wvc).\n%s\n", fail.message);
exit_status = 1;
goto END;
}
if (prbal) {
/* Print complex general matrices A and B after balancing using
* nag_file_print_matrix_complex_gen_comp (x04dbc).
*/
fflush(stdout);
nag_file_print_matrix_complex_gen_comp(
order, Nag_GeneralMatrix, Nag_NonUnitDiag, n, n, a, pda,
Nag_BracketForm, "%7.4f", "Matrix A after balancing", Nag_IntegerLabels,
0, Nag_IntegerLabels, 0, 80, 0, 0, &fail);
if (fail.code == NE_NOERROR) {
fflush(stdout);
nag_file_print_matrix_complex_gen_comp(
order, Nag_GeneralMatrix, Nag_NonUnitDiag, n, n, b, pdb,
Nag_BracketForm, "%7.4f", "Matrix B after balancing",
Nag_IntegerLabels, 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);
exit_status = 1;
goto END;
}
printf("\n");
}
/* Reduce B to triangular form using QR and premultiply A by Q^H. */
irows = ihi + 1 - ilo;
icols = n + 1 - ilo;
/* nag_lapackeig_zgeqrf (f08asc).
* QR factorization of complex general rectangular matrix B.
*/
nag_lapackeig_zgeqrf(order, irows, icols, &B(ilo, ilo), pdb, tau, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_lapackeig_zgeqrf (f08asc).\n%s\n", fail.message);
exit_status = 1;
goto END;
}
/* Apply the orthogonal transformation Q^H to matrix A using
* nag_lapackeig_zunmqr (f08auc).
*/
nag_lapackeig_zunmqr(order, Nag_LeftSide, Nag_ConjTrans, irows, icols, irows,
&B(ilo, ilo), pdb, tau, &A(ilo, ilo), pda, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_lapackeig_zunmqr (f08auc).\n%s\n", fail.message);
exit_status = 1;
goto END;
}
/* Initialize Q (if left eigenvectors are required) */
if (ileft) {
/* Q = I */
nag_blast_zge_load(order, n, n, zero, one, q, pdq, &fail);
/* Q = B using nag_blast_zge_copy (f16tfc). */
nag_blast_zge_copy(order, Nag_NoTrans, irows - 1, irows - 1,
&B(ilo + 1, ilo), pdb, &Q(ilo + 1, ilo), pdq, &fail);
/* Form Q from QR factorization using nag_lapackeig_zungqr (f08atc). */
nag_lapackeig_zungqr(order, irows, irows, irows, &Q(ilo, ilo), pdq, tau,
&fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_lapackeig_zungqr (f08atc).\n%s\n", fail.message);
exit_status = 1;
goto END;
}
}
if (iright) {
/* Z = I. */
nag_blast_zge_load(order, n, n, zero, one, z, pdz, &fail);
}
/* Compute the generalized Hessenberg form of (A,B) by Unitary reduction
* using nag_lapackeig_zgghrd (f08wsc).
*/
nag_lapackeig_zgghrd(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_zgghrd (f08wsc).\n%s\n", fail.message);
exit_status = 1;
goto END;
}
if (prhess) {
/* Print generalized Hessenberg form of (A,B) using
* nag_file_print_matrix_complex_gen_comp (x04dbc).
*/
fflush(stdout);
nag_file_print_matrix_complex_gen_comp(
order, Nag_GeneralMatrix, Nag_NonUnitDiag, n, n, a, pda,
Nag_BracketForm, "%7.3f", "Matrix A in Hessenberg form",
Nag_IntegerLabels, 0, Nag_IntegerLabels, 0, 80, 0, 0, &fail);
if (fail.code == NE_NOERROR) {
printf("\n");
fflush(stdout);
nag_file_print_matrix_complex_gen_comp(
order, Nag_GeneralMatrix, Nag_NonUnitDiag, n, n, b, pdb,
Nag_BracketForm, "%7.3f", "Matrix B in Hessenberg form",
Nag_IntegerLabels, 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);
exit_status = 1;
goto END;
}
}
/* Compute the generalized Schur form - nag_lapackeig_zhgeqz (f08xsc).
* Eigenvalues and generalized Schur factorization of
* complex generalized upper Hessenberg form reduced from a
* pair of complex general matrices
*/
nag_lapackeig_zhgeqz(order, Nag_Schur, Nag_AccumulateQ, Nag_AccumulateZ, n,
ilo, ihi, a, pda, b, pdb, alpha, beta, q, pdq, z, pdz,
&fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_lapackeig_zhgeqz (f08xsc).\n%s\n", fail.message);
exit_status = 1;
goto END;
}
/* Print the ordered generalized eigenvalues if finite */
exit_status = sort_values(n, alpha, beta, rank, temp);
if (exit_status) {
goto END;
}
/* nag_lapackeig_ztgevc (f08yxc).
* Left and right eigenvectors of a pair of complex upper
* triangular matrices
*/
nag_lapackeig_ztgevc(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_ztgevc (f08yxc).\n%s\n", fail.message);
exit_status = 1;
goto END;
}
if (iright) {
/* nag_lapackeig_zggbak (f08wwc).
* Transform eigenvectors of a pair of complex balanced
* matrices to those of original matrix pair supplied to
* nag_lapackeig_zggbal (f08wvc)
*/
nag_lapackeig_zggbak(order, Nag_DoBoth, Nag_RightSide, n, ilo, ihi, lscale,
rscale, n, z, pdz, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_lapackeig_zggbak (f08wwc).\n%s\n", fail.message);
exit_status = 1;
goto END;
}
/* Normalize and print the right eigenvectors */
exit_status =
normalize_vectors(order, n, z, a, temp, rank, "Right eigenvectors");
printf("\n");
}
/* Compute left eigenvectors of the original matrix */
if (ileft) {
/* nag_lapackeig_zggbak (f08wwc), see above. */
nag_lapackeig_zggbak(order, Nag_DoBoth, Nag_LeftSide, n, ilo, ihi, lscale,
rscale, n, q, pdq, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_lapackeig_zggbak (f08wwc).\n%s\n", fail.message);
exit_status = 1;
goto END;
}
/* Normalize and print the left eigenvectors */
exit_status =
normalize_vectors(order, n, q, a, temp, rank, "Left eigenvectors");
}
END:
NAG_FREE(a);
NAG_FREE(b);
NAG_FREE(q);
NAG_FREE(z);
NAG_FREE(alpha);
NAG_FREE(beta);
NAG_FREE(tau);
NAG_FREE(temp);
NAG_FREE(rank);
NAG_FREE(lscale);
NAG_FREE(rscale);
return exit_status;
}
static Integer normalize_vectors(Nag_OrderType order, Integer n, Complex qz[],
Complex tz[], double temp[], 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 norm, r;
Integer i, j, k, errors = 0;
NagError fail;
INIT_FAIL(fail);
#ifdef NAG_COLUMN_MAJOR
#define Z(I, J) qz[(J - 1) * n + I - 1]
#define T(I, J) tz[(J - 1) * n + I - 1]
order = Nag_ColMajor;
#else
#define Z(I, J) qz[(I - 1) * n + J - 1]
#define T(I, J) tz[(I - 1) * n + J - 1]
order = Nag_RowMajor;
#endif
for (j = 1; j <= n; j++) {
/* Find element of eigenvector with largest absolute value using
* nag_complex_abs (a02dbc) and nag_blast_dmax_val (f16jnc).
*/
norm = 0.0;
for (i = 1; i <= n; i++) {
temp[i - 1] = nag_complex_abs(Z(i, j));
norm = norm + temp[i - 1] * temp[i - 1];
}
norm = sqrt(norm);
nag_blast_dmax_val(n, temp, 1, &k, &r, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_blast_dmax_val (f16jnc).\n%s\n", fail.message);
errors = 1;
goto END;
}
k = k + 1;
scal.re = Z(k, j).re / r / norm;
scal.im = -Z(k, j).im / r / norm;
for (i = 1; i <= n; i++)
Z(i, j) = nag_complex_multiply(Z(i, j), scal);
Z(k, j).im = 0.0;
}
for (j = 1; j <= n; j++) {
k = (Integer)rank[j - 1];
for (i = 1; i <= n; i++) {
T(i, j) = Z(i, k + 1);
}
}
/* Print the normalized eigenvectors using
* nag_file_print_matrix_complex_gen_comp (x04dbc)
*/
fflush(stdout);
nag_file_print_matrix_complex_gen_comp(
order, Nag_GeneralMatrix, Nag_NonUnitDiag, n, n, tz, n, Nag_BracketForm,
"%7.4f", title, Nag_IntegerLabels, 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:
return errors;
}
static Integer sort_values(Integer n, Complex alpha[], Complex beta[],
size_t rank[], double temp[]) {
Integer i, errors = 0, isinf = 0;
Complex e;
NagError fail;
INIT_FAIL(fail);
/* Accumulate eigenvalue modulii in temp. */
for (i = 0; i < n; ++i) {
if (beta[i].re != 0.0) {
/* nag_complex_divide (a02cdc) - Quotient of two complex numbers. */
e = nag_complex_divide(alpha[i], beta[i]);
temp[i] = nag_complex_abs(e);
alpha[i] = e;
} else {
isinf = i;
printf(" %4" NAG_IFMT "Eigenvalue is infinite\n", isinf + 1);
goto END;
}
}
/* 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)alpha, (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 Generalized eigenvalues\n");
for (i = 0; i < n; ++i) {
e = alpha[i];
printf(" %4" NAG_IFMT " (%7.3f,%7.3f)\n", i + 1, e.re, e.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));
}