NAG Library Manual, Mark 30.2
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NAG CL Interface Introduction
Example description
/* nag_lapackeig_zgesvj (f08kwc) Example Program.
 *
 * Copyright 2024 Numerical Algorithms Group.
 *
 * Mark 30.2, 2024.
 */

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

static Integer normalize_vectors(Nag_OrderType order, Integer m, Integer n,
                                 Complex v[], Complex x[]);

int main(void) {
  /* Scalars */
  double eps, serrbd, ctol = 1.0;
  Integer exit_status = 0;
  Integer i, j, m, mv, n, n_vrows, n_vcols, pda, pdv, ranka, j1, j2;

  /* Arrays */
  Complex *a = 0, *v = 0, *x = 0;
  double *rcondu = 0, *rcondv = 0, *s = 0;
  double rwork[6];
  char nag_enum_arg[40];

  /* Nag Types */
  Nag_OrderType order;
  Nag_MatrixType joba;
  Nag_LeftVecsType jobu;
  Nag_RightVecsType jobv;
  NagError fail;

#ifdef NAG_COLUMN_MAJOR
#define A(I, J) a[(J - 1) * pda + I - 1]
#define V(I, J) v[(J - 1) * pdv + I - 1]
  order = Nag_ColMajor;
#else
#define A(I, J) a[(I - 1) * pda + J - 1]
#define V(I, J) v[(I - 1) * pdv + J - 1]
  order = Nag_RowMajor;
#endif

  INIT_FAIL(fail);

  printf("nag_lapackeig_zgesvj (f08kwc) Example Program Results\n\n");

  /* Skip heading in data file */
  scanf("%*[^\n]");
  scanf("%" NAG_IFMT "%" NAG_IFMT "%*[^\n]", &m, &n);
  if (n < 0 || m < n) {
    printf("Invalid m or n\n");
    exit_status = 1;
    goto END;
    ;
  }

  /* Read Nag type arguments by name and convert to value */
  scanf(" %39s%*[^\n]", nag_enum_arg);
  /* nag_enum_name_to_value (x04nac).
   * Converts NAG enum member name to value
   */
  joba = (Nag_MatrixType)nag_enum_name_to_value(nag_enum_arg);
  scanf(" %39s%*[^\n]", nag_enum_arg);
  jobu = (Nag_LeftVecsType)nag_enum_name_to_value(nag_enum_arg);
  scanf(" %39s%*[^\n]", nag_enum_arg);
  jobv = (Nag_RightVecsType)nag_enum_name_to_value(nag_enum_arg);
  scanf(" %39s%*[^\n]", nag_enum_arg);

  n_vcols = n;
  n_vrows = n;
  mv = 0;
  if (jobv == Nag_RightVecsMV) {
    scanf("%" NAG_IFMT, &mv);
    n_vrows = mv;
  } else if (jobv == Nag_NotRightVecs) {
    n_vrows = 1;
    n_vcols = 1;
  }
  scanf("%*[^\n]");

#ifdef NAG_COLUMN_MAJOR
  pda = m;
  pdv = n_vrows;
#else
  pda = n;
  pdv = n_vcols;
#endif

  if (!(a = NAG_ALLOC(m * n, Complex)) || !(rcondu = NAG_ALLOC(m, double)) ||
      !(rcondv = NAG_ALLOC(m, double)) || !(s = NAG_ALLOC(n, double)) ||
      !(v = NAG_ALLOC(n_vrows * n_vcols, Complex)) ||
      !(x = NAG_ALLOC(n, Complex))) {
    printf("Allocation failure\n");
    exit_status = -1;
    goto END;
  }

  /* Read the m by n matrix A from data file */
  j1 = 1;
  j2 = n;
  if (joba == Nag_UpperMatrix) {
    j1 = n;
  } else if (joba == Nag_LowerMatrix) {
    j2 = 1;
  }
  for (i = 1; i <= m; i++)
    for (j = j1; j <= j2; j++)
      scanf(" ( %lf , %lf )", &A(i, j).re, &A(i, j).im);
  scanf("%*[^\n]");
  if (jobv == Nag_RightVecsMV) {
    /* The first mv rows of v must be set. */
    for (i = 1; i <= mv; i++)
      for (j = 1; j <= n; j++)
        scanf(" ( %lf , %lf )", &V(i, j).re, &V(i, j).im);
    scanf("%*[^\n]");
  }

  /* nag_lapackeig_zgesvj (f08kwc)
   * Compute the singular values and left and right singular vectors
   * of A (A = U*S*V, m>=n).
   */
  nag_lapackeig_zgesvj(order, joba, jobu, jobv, m, n, a, pda, s, mv, v, pdv,
                       ctol, rwork, &fail);
  if (fail.code != NE_NOERROR) {
    printf("Error from nag_lapackeig_zgesvj (f08kwc).\n%s\n", fail.message);
    exit_status = 2;
    goto END;
  }

  /* Get the machine precision, eps and compute the approximate
   * error bound for the computed singular values. Note that for
   * the 2-norm, s[0] = norm(A).
   */
  eps = nag_machine_precision;
  serrbd = s[0];

  /* Print solution */
  printf("Singular values\n   ");
  for (j = 0; j < n; j++)
    printf("%8.4f", s[j]);
  printf("\n\n");
  if (fabs(rwork[0] - 1.0) > eps)
    printf("Values need scaling by factor = %13.5e\n\n", rwork[0]);

  ranka = (Integer)rwork[1];
  printf("Rank of A = %5" NAG_IFMT "\n\n", ranka);
  x[0].re = 2.0;
  if (jobu != Nag_NotLeftVecs) {
    /* Normalize left vectors so that largest element is real and positive */
    exit_status = normalize_vectors(order, m, ranka, a, x);
    if (exit_status > 0) {
      exit_status = 3;
      goto END;
    }
    /* nag_file_print_matrix_complex_gen_comp (x04dbc)
     * Print complex general matrix (comprehensive)
     */
    fflush(stdout);
    nag_file_print_matrix_complex_gen_comp(
        order, Nag_GeneralMatrix, Nag_NonUnitDiag, m, ranka, a, pda,
        Nag_BracketForm, "%7.4f", "Left spanning singular vectors",
        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 = 4;
      goto END;
    }
  }

  if (jobv == Nag_RightVecs) {
    /* Normalize V, either using x from U or from scratch */
    exit_status = normalize_vectors(order, n, n, v, x);
    if (exit_status > 0) {
      exit_status = 5;
      goto END;
    }
    printf("\n");
    fflush(stdout);
    nag_file_print_matrix_complex_gen_comp(
        order, Nag_GeneralMatrix, Nag_NonUnitDiag, n, n, v, pdv,
        Nag_BracketForm, "%7.4f", "Right singular vectors", Nag_IntegerLabels,
        0, Nag_IntegerLabels, 0, 80, 0, 0, &fail);
  } else if (jobv == Nag_RightVecsMV) {
    printf("\n");
    fflush(stdout);
    nag_file_print_matrix_complex_gen_comp(
        order, Nag_GeneralMatrix, Nag_NonUnitDiag, mv, n, v, pdv,
        Nag_BracketForm, "%7.4f", "Right sing. vectors applied to V",
        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 = 6;
    goto END;
  }

  /* nag_lapackeig_ddisna (f08flc)
   * Estimate reciprocal condition numbers for the singular vectors.
   */
  nag_lapackeig_ddisna(Nag_LeftSingVecs, m, n, s, rcondu, &fail);
  nag_lapackeig_ddisna(Nag_RightSingVecs, m, n, s, rcondv, &fail);
  if (fail.code != NE_NOERROR) {
    printf("Error from nag_lapackeig_ddisna (f08flc).\n%s\n", fail.message);
    exit_status = 7;
    goto END;
  }

  /* Print the approximate error bounds for the singular values and vectors. */
  printf("\n\nError estimates (as multiples of machine precision)\n");
  printf("\n   for the singular values\n%4.0f\n", serrbd);

  printf("\n   for left singular vectors\n");
  for (i = 0; i < n; i++)
    printf("%4.0f", serrbd / rcondu[i]);

  printf("\n\n   for right singular vectors\n");
  for (i = 0; i < n; i++)
    printf("%4.0f", serrbd / rcondv[i]);
  printf("\n");

END:
  NAG_FREE(a);
  NAG_FREE(rcondu);
  NAG_FREE(rcondv);
  NAG_FREE(s);
  NAG_FREE(v);
  NAG_FREE(x);

  return exit_status;
}

static Integer normalize_vectors(Nag_OrderType order, Integer m, Integer n,
                                 Complex v[], Complex x[]) {
  /* Each complex vector v[] is normalized so that the element of largest
   * magnitude is scaled to be real and positive
   */

  double r, rmax, scal;
  Integer colinc, rowinc, i, j, k, l, indv, errors = 0;
  Complex alpha, y[1];
  NagError fail;

  INIT_FAIL(fail);

  if (order == Nag_ColMajor) {
    rowinc = 1;
    colinc = m;
  } else {
    rowinc = n;
    colinc = 1;
  }

  scal = x[0].re;
  indv = 0;
  for (j = 0; j < n; j++) {

    if (scal > 1.5) {
      /* Scaling factors not found yet.
       * Find element of eigenvector with largest absolute value.
       */
      rmax = nag_complex_abs(v[indv]);
      l = indv;
      k = l;
      for (i = 0; i < m; i++) {
        /* nag_complex_abs (a02dbc). Modulus of a complex number.  */
        r = nag_complex_abs(v[l]);
        if (r > rmax) {
          rmax = r;
          k = l;
        }
        l += rowinc;
      }

      /* Normalization factor beta */
      x[j].re = v[k].re / rmax;
      x[j].im = -v[k].im / rmax;
    }

    /* Scale current vector v_j by factor x[j] using:
     * nag_blast_zaxpby (f16gcc) which performs y := alpha*x + beta*y.
     */
    alpha = nag_complex_create(0.0, 0.0);
    nag_blast_zaxpby(m, alpha, y, 1, x[j], &v[indv], rowinc, &fail);
    if (fail.code != NE_NOERROR) {
      printf("Error from nag_blast_zaxpby (f16gcc).\n%s\n", fail.message);
      errors = 2;
      goto END;
    }
    indv += colinc;
  }
END:
  return errors;
}