NAG Library Manual, Mark 30.1
/* nag_lapackeig_dorcsd (f08rac) Example Program.
*
* Copyright 2024 Numerical Algorithms Group.
*
* Mark 30.1, 2024.
*/

#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;
}