NAG Library Manual, Mark 27.3
Interfaces:  FL   CL   CPP   AD 

NAG CL Interface Introduction
Example description
/* nag_lapackeig_dggesx (f08xbc) Example Program.
 *
 * Copyright 2021 Numerical Algorithms Group.
 *
 * Mark 27.3, 2021.
 */

#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);
}