NAG Library Manual, Mark 27.3
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NAG CL Interface Introduction
Example description
/* nag_lapackeig_dtgevc (f08ykc) Example Program.
 *
 * Copyright 2021 Numerical Algorithms Group.
 *
 * Mark 27.3, 2021.
 */

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

static Integer normalize_vectors(Nag_OrderType order, Integer n, double qz[],
                                 double alphai[], const char *title);

int main(void) {
  /* Scalars */
  Integer i, icols, ihi, ilo, irows, j, m, n, pda, pdb, pdq, pdz;
  Integer exit_status = 0;
  Nag_Boolean ileft, iright;

  NagError fail;
  Nag_OrderType order;
  /* Arrays */
  double *a = 0, *alphai = 0, *alphar = 0, *b = 0, *beta = 0;
  double *lscale = 0, *q = 0, *rscale = 0, *tau = 0, *z = 0;

#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_dtgevc (f08ykc) Example Program Results\n\n");

  /* ileft  is Nag_TRUE if left  eigenvectors are required */
  /* iright is Nag_TRUE if right eigenvectors are required */
  ileft = Nag_TRUE;
  iright = Nag_TRUE;

  /* Skip heading in data file */
  scanf("%*[^\n] ");
  scanf("%" NAG_IFMT " %*[^\n] ", &n);

  pda = n;
  pdb = n;
  pdq = n;
  pdz = n;

  /* Allocate memory */
  if (!(a = NAG_ALLOC(n * n, double)) || !(b = NAG_ALLOC(n * n, double)) ||
      !(q = NAG_ALLOC(n * n, double)) || !(z = NAG_ALLOC(n * n, double)) ||
      !(alphai = NAG_ALLOC(n, double)) || !(alphar = NAG_ALLOC(n, double)) ||
      !(beta = NAG_ALLOC(n, double)) || !(lscale = NAG_ALLOC(n, double)) ||
      !(rscale = NAG_ALLOC(n, double)) || !(tau = 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", &A(i, j));
  scanf("%*[^\n] ");

  /* READ matrix B from data file */
  for (i = 1; i <= n; ++i)
    for (j = 1; j <= n; ++j)
      scanf("%lf", &B(i, j));
  scanf("%*[^\n] ");

  /* Balance the real general matrix pair (A,B) using nag_lapackeig_dggbal
   * (f08whc). */
  nag_lapackeig_dggbal(order, Nag_DoBoth, n, a, pda, b, pdb, &ilo, &ihi, lscale,
                       rscale, &fail);
  if (fail.code != NE_NOERROR) {
    printf("Error from nag_lapackeig_dggbal (f08whc).\n%s\n", fail.message);
    exit_status = 1;
    goto END;
  }

  /* Print matrices A and B after balancing using
   * nag_file_print_matrix_real_gen_comp (x04cbc).
   */
  fflush(stdout);
  nag_file_print_matrix_real_gen_comp(
      order, Nag_GeneralMatrix, Nag_NonUnitDiag, n, n, a, pda, "%8.4f",
      "Matrix A after balancing", Nag_IntegerLabels, NULL, Nag_IntegerLabels,
      NULL, 80, 0, 0, &fail);
  if (fail.code == NE_NOERROR) {
    fflush(stdout);
    nag_file_print_matrix_real_gen_comp(
        order, Nag_GeneralMatrix, Nag_NonUnitDiag, n, n, b, pdb, "%8.4f",
        "Matrix B after balancing", Nag_IntegerLabels, NULL, Nag_IntegerLabels,
        NULL, 80, 0, 0, &fail);
  }
  if (fail.code != NE_NOERROR) {
    printf("Error from nag_file_print_matrix_real_gen_comp (x04cbc).\n%s\n",
           fail.message);
    exit_status = 2;
    goto END;
  }
  printf("\n");

  /* Reduce B to triangular form using QR and multipling both sides by Q^T */
  irows = ihi + 1 - ilo;
  icols = n + 1 - ilo;
  /* nag_lapackeig_dgeqrf (f08aec).
   * QR factorization of real general rectangular matrix
   */
  nag_lapackeig_dgeqrf(order, irows, icols, &B(ilo, ilo), pdb, tau, &fail);
  if (fail.code != NE_NOERROR) {
    printf("Error from nag_lapackeig_dgeqrf (f08aec).\n%s\n", fail.message);
    exit_status = 3;
    goto END;
  }

  /* Apply the Q to matrix A -  nag_lapackeig_dormqr (f08agc)
   * as determined by nag_lapackeig_dgeqrf (f08aec).
   */
  nag_lapackeig_dormqr(order, Nag_LeftSide, Nag_Trans, irows, icols, irows,
                       &B(ilo, ilo), pdb, tau, &A(ilo, ilo), pda, &fail);
  if (fail.code != NE_NOERROR) {
    printf("Error from nag_lapackeig_dormqr (f08agc).\n%s\n", fail.message);
    exit_status = 4;
    goto END;
  }

  /* Initialize Q (if left eigenvectors are required) */
  if (ileft) {
    /* Q = I. */
    nag_blast_dge_load(order, n, n, 0.0, 1.0, q, pdq, &fail);
    /* Copy B to Q using nag_blast_dge_copy (f16qfc). */
    nag_blast_dge_copy(order, Nag_NoTrans, irows - 1, irows - 1,
                       &B(ilo + 1, ilo), pdb, &Q(ilo + 1, ilo), pdq, &fail);
    /* nag_lapackeig_dorgqr (f08afc).
     * Form all or part of orthogonal Q from QR factorization
     * determined by nag_lapackeig_dgeqrf (f08aec) or nag_lapackeig_dgeqpf
     * (f08bec)
     */
    nag_lapackeig_dorgqr(order, irows, irows, irows, &Q(ilo, ilo), pdq, tau,
                         &fail);
    if (fail.code != NE_NOERROR) {
      printf("Error from nag_lapackeig_dorgqr (f08afc).\n%s\n", fail.message);
      exit_status = 5;
      goto END;
    }
  }

  /* Initialize Z (if right eigenvectors are required) */
  if (iright) {
    /* Z = I. */
    nag_blast_dge_load(order, n, n, 0.0, 1.0, z, pdz, &fail);
  }

  /* Compute the generalized Hessenberg form of (A,B) */
  /* nag_lapackeig_dgghrd (f08wec).
   * Orthogonal reduction of a pair of real general matrices
   * to generalized upper Hessenberg form
   */
  nag_lapackeig_dgghrd(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_dgghrd (f08wec).\n%s\n", fail.message);
    exit_status = 6;
    goto END;
  }

  /* Matrix A in generalized Hessenberg form */
  /* nag_file_print_matrix_real_gen_comp (x04cbc), see above. */
  fflush(stdout);
  nag_file_print_matrix_real_gen_comp(
      order, Nag_GeneralMatrix, Nag_NonUnitDiag, n, n, a, pda, "%8.4f",
      "Matrix A in Hessenberg form", Nag_IntegerLabels, NULL, Nag_IntegerLabels,
      NULL, 80, 0, 0, &fail);
  if (fail.code != NE_NOERROR) {
    printf("Error from nag_file_print_matrix_real_gen_comp (x04cbc).\n%s\n",
           fail.message);
    exit_status = 7;
    goto END;
  }
  printf("\n");

  /* Matrix B in generalized Hessenberg form */
  /* nag_file_print_matrix_real_gen_comp (x04cbc), see above. */
  fflush(stdout);
  nag_file_print_matrix_real_gen_comp(
      order, Nag_GeneralMatrix, Nag_NonUnitDiag, n, n, b, pdb, "%8.4f",
      "Matrix B in Hessenberg form", Nag_IntegerLabels, NULL, Nag_IntegerLabels,
      NULL, 80, 0, 0, &fail);
  if (fail.code != NE_NOERROR) {
    printf("Error from nag_file_print_matrix_real_gen_comp (x04cbc).\n%s\n",
           fail.message);
    exit_status = 1;
    goto END;
  }

  /* nag_lapackeig_dhgeqz (f08xec).
   * Eigenvalues and generalized Schur factorization of real
   * generalized upper Hessenberg form reduced from a pair of
   * real general matrices.
   */
  nag_lapackeig_dhgeqz(order, Nag_Schur, Nag_AccumulateQ, Nag_AccumulateZ, n,
                       ilo, ihi, a, pda, b, pdb, alphar, alphai, beta, q, pdq,
                       z, pdz, &fail);
  if (fail.code != NE_NOERROR) {
    printf("Error from nag_lapackeig_dhgeqz (f08xec).\n%s\n", fail.message);
    exit_status = 1;
    goto END;
  }

  /* Print the generalized eigenvalue parameters */
  printf("\n Generalized eigenvalues\n");
  for (i = 0; i < n; ++i) {
    if (beta[i] != 0.0) {
      printf(" %4" NAG_IFMT "     (%8.4f,%8.4f)\n", i + 1, alphar[i] / beta[i],
             alphai[i] / beta[i]);
    } else
      printf(" %4" NAG_IFMT "Eigenvalue is infinite\n", i + 1);
  }
  printf("\n");

  /* Compute left and right generalized eigenvectors
   * of the balanced matrix - nag_lapackeig_dtgevc (f08ykc).
   */
  nag_lapackeig_dtgevc(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_dtgevc (f08ykc).\n%s\n", fail.message);
    exit_status = 1;
    goto END;
  }
  if (iright) {
    /* Compute right eigenvectors of the original matrix pair
     * supplied tonag_dggbal (f08whc) using nag_lapackeig_dggbak (f08wjc).
     */
    nag_lapackeig_dggbak(order, Nag_DoBoth, Nag_RightSide, n, ilo, ihi, lscale,
                         rscale, n, z, pdz, &fail);
    if (fail.code != NE_NOERROR) {
      printf("Error from nag_lapackeig_dggbak (f08wjc).\n%s\n", fail.message);
      exit_status = 1;
      goto END;
    }
    /* Normalize and print the right eigenvectors */
    exit_status = normalize_vectors(order, n, z, alphai, "Right eigenvectors");
  }
  printf("\n");

  /* Compute left eigenvectors of the original matrix */
  if (ileft) {
    /* nag_lapackeig_dggbak (f08wjc), see above. */
    nag_lapackeig_dggbak(order, Nag_DoBoth, Nag_LeftSide, n, ilo, ihi, lscale,
                         rscale, n, q, pdq, &fail);
    if (fail.code != NE_NOERROR) {
      printf("Error from nag_lapackeig_dggbak (f08wjc).\n%s\n", fail.message);
      exit_status = 1;
      goto END;
    }
    /* Normalize the left eigenvectors */
    exit_status = normalize_vectors(order, n, q, alphai, "Left eigenvectors");
  }
END:
  NAG_FREE(a);
  NAG_FREE(b);
  NAG_FREE(q);
  NAG_FREE(z);
  NAG_FREE(alphai);
  NAG_FREE(alphar);
  NAG_FREE(beta);
  NAG_FREE(lscale);
  NAG_FREE(rscale);
  NAG_FREE(tau);

  return exit_status;
}

static Integer normalize_vectors(Nag_OrderType order, Integer n, double qz[],
                                 double alphai[], const char *title) {
  /* Real eigenvectors are scaled so that the maximum value of elements is 1.0;
   * each complex eigenvector z[] is normalized so that the element of largest
   * magnitude is scaled to be (1.0,0.0).
   */

  double a, b, u, v, r, ri, rr;
  Integer colinc, rowinc, i, ii, j, k, indqz, errors = 0;
  NagError fail;

  INIT_FAIL(fail);

  if (order == Nag_ColMajor) {
    rowinc = 1;
    colinc = n;
  } else {
    rowinc = n;
    colinc = 1;
  }
  indqz = 0;
  for (j = 0; j < n; j++) {
    if (alphai[j] >= 0.0) {
      if (alphai[j] == 0.0) {
        /* The 2-norm of Q is calculated using nag_blast_dge_norm (f16rac). */
        nag_blast_dge_norm(order, Nag_FrobeniusNorm, n, 1, &qz[indqz], n, &r,
                           &fail);
        if (fail.code != NE_NOERROR) {
          printf("Error from nag_blast_dge_norm (f16rac).\n%s\n", fail.message);
          errors = 1;
          goto END;
        }
        for (i = 0; i < n * rowinc; i += rowinc) {
          qz[indqz + i] = qz[indqz + i] / r;
        }
      } else {
        /* norm of j-th complex eigenvector using nag_blast_dge_norm (f16rac),
         * stored as two arrays of length n.
         */
        k = 0;
        rr = 0.0;
        r = -1.0;
        for (i = 0; i < n * rowinc; i += rowinc) {
          ii = indqz + i;
          ri = qz[ii] * qz[ii] + qz[ii + colinc] * qz[ii + colinc];
          rr = rr + ri;
          if (ri > r) {
            k = i;
            r = ri;
          }
        }
        a = qz[indqz + k];
        b = qz[indqz + colinc + k];
        r = sqrt(r * rr);

        for (i = 0; i < n * rowinc; i += rowinc) {
          u = qz[indqz + i];
          v = qz[indqz + colinc + i];
          qz[indqz + i] = (u * a + v * b) / r;
          qz[indqz + colinc + i] = (v * a - u * b) / r;
        }
        indqz += colinc;
      }
      indqz += colinc;
    }
  }
  /* Print the normalized eigenvectors using
   * nag_file_print_matrix_real_gen_comp (x04cbc)
   */
  fflush(stdout);
  nag_file_print_matrix_real_gen_comp(
      order, Nag_GeneralMatrix, Nag_NonUnitDiag, n, n, qz, n, "%8.4f", title,
      Nag_IntegerLabels, NULL, Nag_IntegerLabels, NULL, 80, 0, 0, &fail);
  if (fail.code != NE_NOERROR) {
    printf("Error from nag_file_print_matrix_real_gen_comp (x04cbc).\n%s\n",
           fail.message);
    errors = 1;
  }
END:
  return errors;
}