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

NAG CL Interface Introduction
Example description
/* nag_lapackeig_dorcsd (f08rac) Example Program.
 *
 * Copyright 2021 Numerical Algorithms Group.
 *
 * Mark 27.3, 2021.
 */

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