Example description
/* nag_lapackeig_dorcsd (f08rac) Example Program.
 *
 * Copyright 2019 Numerical Algorithms Group.
 *
 * Mark 27.0, 2019.
 */

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

int main(void)
{
  /* Scalars */
  Integer exit_status = 0;
  Integer pdx, pdu, pdv, pdx11, pdx12, pdx21, pdx22, pdu1, pdu2, pdv1t;
  Integer pdv2t, pdw;
  Integer i, j, m, p, q, n11, n12, n21, n22, r;
  Integer recombine = 1, reprint = 0;
  double alpha, beta;
  /* Arrays */
  double *theta = 0, *u = 0, *u1 = 0, *u2 = 0, *v = 0, *v1t = 0, *w = 0,
         *v2t = 0, *x = 0, *x11 = 0, *x12 = 0, *x21 = 0, *x22 = 0;
  /* Nag Types */
  Nag_OrderType order;
  NagError fail;

#ifdef NAG_COLUMN_MAJOR
#define X(I,J) x[(J-1)*pdx + I-1]
#define U(I,J) u[(J-1)*pdu + I-1]
#define V(I,J) v[(J-1)*pdv + I-1]
#define W(I,J) w[(J-1)*pdw + I-1]
#define X11(I,J) x11[(J-1)*pdx11 + I-1]
#define X12(I,J) x12[(J-1)*pdx12 + I-1]
#define X21(I,J) x21[(J-1)*pdx21 + I-1]
#define X22(I,J) x22[(J-1)*pdx22 + I-1]
  order = Nag_ColMajor;
#else
#define X(I,J) x[(I-1)*pdx + J-1]
#define U(I,J) u[(I-1)*pdu + J-1]
#define V(I,J) v[(I-1)*pdv + J-1]
#define W(I,J) w[(I-1)*pdw + J-1]
#define X11(I,J) x11[(I-1)*pdx11 + J-1]
#define X12(I,J) x12[(I-1)*pdx12 + J-1]
#define X21(I,J) x21[(I-1)*pdx21 + J-1]
#define X22(I,J) x22[(I-1)*pdx22 + J-1]
  order = Nag_RowMajor;
#endif

  INIT_FAIL(fail);

  printf("nag_lapackeig_dorcsd (f08rac) Example Program Results\n\n");
  fflush(stdout);

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

  r = MIN(MIN(p, q), MIN(m - p, m - q));

  if (!(x = NAG_ALLOC(m * m, double)) ||
      !(u = NAG_ALLOC(m * m, double)) ||
      !(v = NAG_ALLOC(m * m, double)) ||
      !(w = NAG_ALLOC(m * m, double)) ||
      !(theta = NAG_ALLOC(r, double)) ||
      !(x11 = NAG_ALLOC(p * q, double)) ||
      !(x12 = NAG_ALLOC(p * (m - q), double)) ||
      !(x21 = NAG_ALLOC((m - p) * q, double)) ||
      !(x22 = NAG_ALLOC((m - p) * (m - q), double)) ||
      !(u1 = NAG_ALLOC(p * p, double)) ||
      !(u2 = NAG_ALLOC((m - p) * (m - p), double)) ||
      !(v1t = NAG_ALLOC(q * q, double)) ||
      !(v2t = NAG_ALLOC((m - q) * (m - q), double)))
  {
    printf("Allocation failure\n");
    exit_status = -1;
    goto END;
  }
  pdx = m;
  pdu = m;
  pdv = m;
  pdw = m;
  pdu1 = p;
  pdu2 = m - p;
  pdv1t = q;
  pdv2t = m - q;
#ifdef NAG_COLUMN_MAJOR
  pdx11 = p;
  pdx12 = p;
  pdx21 = m - p;
  pdx22 = m - p;
#else
  pdx11 = q;
  pdx12 = m - q;
  pdx21 = q;
  pdx22 = m - q;
#endif
  /* Read and print orthogonal X from data file
   * (as, say, generated by a generalized singular value decomposition).
   */
  for (i = 1; i <= m; i++)
    for (j = 1; j <= m; j++)
      scanf("%lf", &X(i, j));
  /* nag_file_print_matrix_real_gen (x04cac).
   */
  nag_file_print_matrix_real_gen(order, Nag_GeneralMatrix, Nag_NonUnitDiag, m, m,
                         &X(1, 1), pdx, "    Orthogonal matrix X", 0, &fail);
  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;
  }
  printf("\n");
  fflush(stdout);

  /* nag_lapackeig_dorcsd (f08rac).
   * Compute the complete CS factorization of X:
   * X11 is stored in X(1:p,    1:q),  X12 is stored in X(1:p,   q+1:m)
   * X21 is stored in X(p+1:m,  1:q),  X22 is stored in X(p+1:m, q+1:m)
   * U1  is stored in U(1:p,    1:p),  U2  is stored in U(p+1:m, p+1:m)
   * V1  is stored in V(1:q,    1:q),  V2  is stored in V(q+1:m, q+1:m)
   */
  for (j = 1; j <= p; j++) {
    for (i = 1; i <= q; i++)
      X11(j, i) = X(j, i);
    for (i = 1; i <= m - q; i++)
      X12(j, i) = X(j, i + q);
  }
  for (j = 1; j <= m - p; j++) {
    for (i = 1; i <= q; i++)
      X21(j, i) = X(j + p, i);
    for (i = 1; i <= m - q; i++)
      X22(j, i) = X(j + p, i + q);
  }
  for (i = 1; i <= m; i++)
    for (j = 1; j <= m; j++) {
      U(i, j) = 0.0;
      V(i, j) = 0.0;
    }

  /* This is how you might pass partitions as sub-matrices */
  nag_lapackeig_dorcsd(order, Nag_AllU, Nag_AllU, Nag_AllVT, Nag_AllVT, Nag_UpperMinus,
             m, p, q, &X(1, 1), pdx, &X(1, q + 1), pdx, &X(p + 1, 1), pdx,
             &X(p + 1, q + 1), pdx, theta, &U(1, 1), pdu, &U(p + 1, p + 1),
             pdu, &V(1, 1), pdv, &V(q + 1, q + 1), pdv, &fail);
  if (fail.code != NE_NOERROR) {
    printf("Error from nag_lapackeig_dorcsd (f08rac).\n%s\n", fail.message);
    exit_status = 2;
    goto END;
  }
  /* Print Theta, U1, U2, V1T, V2T
   * using matrix printing routine nag_file_print_matrix_real_gen (x04cac).
   */
  printf("Components of CS factorization of X:\n");
  fflush(stdout);
  nag_file_print_matrix_real_gen(Nag_ColMajor, Nag_GeneralMatrix, Nag_NonUnitDiag,
                         r, 1, theta, r, "    Theta", 0, &fail);
  printf("\n");
  /* By changes of sign, elements 1:r of row 1 of U1 are made positive. */
  for (i = 1; i <= r; ++i) {
    if (U(1,i)<0.0) {
      for (j = 1; j <= p; ++j)
        U(j,i) = -U(j,i);
      for (j = p+1; j <= m; ++j)
        U(j,p+i) = -U(j,p+i);
      for (j = 1; j <= q; ++j)
        V(i,j) = -V(i,j);
      for (j = q+1; j <= m; ++j)
        V(q+i,j) = -V(q+i,j);
    }
  }
  fflush(stdout);
  nag_file_print_matrix_real_gen(order, Nag_GeneralMatrix, Nag_NonUnitDiag,
                         p, p, &U(1, 1), pdu, "    U1", 0, &fail);
  printf("\n");
  fflush(stdout);
  nag_file_print_matrix_real_gen(order, Nag_GeneralMatrix, Nag_NonUnitDiag,
                         m - p, m - p, &U(p + 1, p + 1), pdu, "    U2", 0,
                         &fail);
  printf("\n");
  fflush(stdout);
  nag_file_print_matrix_real_gen(order, Nag_GeneralMatrix, Nag_NonUnitDiag,
                         q, q, &V(1, 1), pdv, "    V1T", 0, &fail);
  printf("\n");
  fflush(stdout);
  nag_file_print_matrix_real_gen(order, Nag_GeneralMatrix, Nag_NonUnitDiag,
                         m - q, m - q, &V(q + 1, q + 1), pdv, "    V2T", 0,
                         &fail);
  printf("\n");
  fflush(stdout);

  /* And this is how you might pass partitions as separate matrices. */
  nag_lapackeig_dorcsd(order, Nag_AllU, Nag_AllU, Nag_AllVT, Nag_AllVT, Nag_UpperMinus,
             m, p, q,
             x11, pdx11, x12, pdx12, x21, pdx21, x22, pdx22, theta,
             u1, pdu1, u2, pdu2, v1t, pdv1t, v2t, pdv2t, &fail);
  if (fail.code != NE_NOERROR) {
    printf("Error second from nag_lapackeig_dorcsd (f08rac).\n%s\n", fail.message);
    exit_status = 3;
    goto END;
  }
  /* Print Theta, U1, U2, V1T, V2T
   * using matrix printing routine nag_file_print_matrix_real_gen (x04cac).
   */
  if (reprint != 0) {
    printf("Components of CS factorization of X:\n");
    fflush(stdout);
    nag_file_print_matrix_real_gen(Nag_ColMajor, Nag_GeneralMatrix, Nag_NonUnitDiag,
                           r, 1, theta, r, "    Theta", 0, &fail);
    nag_file_print_matrix_real_gen(order, Nag_GeneralMatrix, Nag_NonUnitDiag,
                           p, p, u1, pdu1, "    U1", 0, &fail);
    nag_file_print_matrix_real_gen(order, Nag_GeneralMatrix, Nag_NonUnitDiag,
                           m - p, m - p, u2, pdu2, "    U2", 0, &fail);
    nag_file_print_matrix_real_gen(order, Nag_GeneralMatrix, Nag_NonUnitDiag,
                           q, q, v1t, pdv1t, "    V1T", 0, &fail);
    nag_file_print_matrix_real_gen(order, Nag_GeneralMatrix, Nag_NonUnitDiag,
                           m - q, m - q, v2t, pdv2t, "    V2T", 0, &fail);
  }
  if (recombine != 0) {
    /* Recombining should return the original matrix. 
       Assemble Sigma_p into X 
     */
    for (i = 1; i <= m; i++)
      for (j = 1; j <= m; j++)
        X(i, j) = 0.0;
    n11 = MIN(p, q) - r;
    n12 = MIN(p, m - q) - r;
    n21 = MIN(m - p, q) - r;
    n22 = MIN(m - p, m - q) - r;

    /* top half */
    for (j = 1; j <= n11; j++)
      X(j, j) = 1.0;
    for (j = 1; j <= r; j++) {
      X(j + n11, j + n11) = cos(theta[j - 1]);
      X(j + n11, j + n11 + r + n21 + n22) = -sin(theta[j - 1]);
    }
    for (j = 1; j <= n12; j++)
      X(j + n11 + r, j + n11 + r + n21 + n22 + r) = -1.0;
    /* bottom half */
    for (j = 1; j <= n22; j++)
      X(p + j, q + j) = 1.0;
    for (j = 1; j <= r; j++) {
      X(p + n22 + j, j + n11) = sin(theta[j - 1]);
      X(p + n22 + j, j + r + n21 + n22) = cos(theta[j - 1]);
    }
    for (j = 1; j <= n21; j++)
      X(p + n22 + r + j, n11 + r + j) = 1.0;

    alpha = 1.0;
    beta = 0.0;
    /* multiply U * Sigma_p into w */
    nag_blast_dgemm(order, Nag_NoTrans, Nag_NoTrans, m, m, m, alpha,
              &U(1, 1), pdu, &X(1, 1), pdx, beta, &W(1, 1), pdw, &fail);
    /* form U * Sigma_p * V^T into u */
    nag_blast_dgemm(order, Nag_NoTrans, Nag_NoTrans, m, m, m, alpha,
              &W(1, 1), pdw, &V(1, 1), pdv, beta, &U(1, 1), pdu, &fail);
    nag_file_print_matrix_real_gen(order, Nag_GeneralMatrix, Nag_NonUnitDiag,
                           m, m, &U(1, 1), pdu, "    U * Sigma_p * V^T", 0,
                           &fail);
  }
END:
  NAG_FREE(x);
  NAG_FREE(u);
  NAG_FREE(v);
  NAG_FREE(w);
  NAG_FREE(theta);
  NAG_FREE(x11);
  NAG_FREE(x12);
  NAG_FREE(x21);
  NAG_FREE(x22);
  NAG_FREE(u1);
  NAG_FREE(u2);
  NAG_FREE(v1t);
  NAG_FREE(v2t);
  return exit_status;
}