NAG Library Manual, Mark 27.2
```/* nag_lapackeig_zuncsd (f08rnc) Example Program.
*
* Copyright 2021 Numerical Algorithms Group.
*
* Mark 27.2, 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, pdw;
Integer i, j, m, p, q, n11, n12, n21, n22, r;
Integer recombine = 0, reprint = 0;
Complex cone = {1.0, 0.0}, czero = {0.0, 0.0};
/* Arrays */
Complex *u = 0, *u1 = 0, *u2 = 0, *v = 0, *v1t = 0, *v2t = 0, *w = 0, *x = 0,
*x11 = 0, *x12 = 0, *x21 = 0, *x22 = 0;
double *theta = 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_zuncsd (f08rnc) 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, Complex)) || !(u = NAG_ALLOC(m * m, Complex)) ||
!(v = NAG_ALLOC(m * m, Complex)) || !(w = NAG_ALLOC(m * m, Complex)) ||
!(theta = NAG_ALLOC(r, double)) || !(x11 = NAG_ALLOC(p * q, Complex)) ||
!(x12 = NAG_ALLOC(p * (m - q), Complex)) ||
!(x21 = NAG_ALLOC((m - p) * q, Complex)) ||
!(x22 = NAG_ALLOC((m - p) * (m - q), Complex)) ||
!(u1 = NAG_ALLOC(p * p, Complex)) ||
!(u2 = NAG_ALLOC((m - p) * (m - p), Complex)) ||
!(v1t = NAG_ALLOC(q * q, Complex)) ||
!(v2t = NAG_ALLOC((m - q) * (m - q), Complex))) {
printf("Allocation failure\n");
exit_status = -1;
goto END;
}
pdx = m;
pdu = m;
pdv = m;
pdw = m;
#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 (by column) and print unitary 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 , %lf ) ", &X(j, i).re, &X(j, i).im);
scanf("%*[^\n] ");
}
/* Store partitions of X in separate matrices */
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);
}

/* nag_file_print_matrix_complex_gen_comp (x04dbc).
* Print least squares solutions.
*/
nag_file_print_matrix_complex_gen_comp(
order, Nag_GeneralMatrix, Nag_NonUnitDiag, m, m, x, pdx, Nag_BracketForm,
"%7.4f", "Unitary matrix X", 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 = 1;
goto END;
}
printf("\n");
fflush(stdout);

/* nag_lapackeig_zuncsd (f08rnc).
* 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)
*/

/* This is how you might pass partitions as sub-matrices */
nag_lapackeig_zuncsd(order, Nag_AllU, Nag_AllU, Nag_AllVT, Nag_AllVT,
Nag_UpperMinus, m, p, q, x, pdx, &X(1, q + 1), pdx,
&X(p + 1, 1), pdx, &X(p + 1, q + 1), pdx, theta, u, pdu,
&U(p + 1, p + 1), pdu, v, pdv, &V(q + 1, q + 1), pdv,
&fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_lapackeig_zuncsd (f08rnc).\n%s\n", fail.message);
exit_status = 2;
goto END;
}

/* Print Theta using matrix printing routine
* nag_file_print_matrix_real_gen (x04cac).
* Note: U1, U2, V1T, V2T not printed since these may differ by a sign
* change in columns of U1, U2 and corresponding rows of V1T, V2T.
*/
printf(" Component of CS factorization of X:\n\n");
fflush(stdout);
nag_file_print_matrix_real_gen(Nag_ColMajor, Nag_GeneralMatrix,
Nag_NonUnitDiag, r, 1, theta, r, "    Theta",
0, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_file_print_matrix_real_gen (x04cac).\n%s\n",
fail.message);
exit_status = 3;
goto END;
}
printf("\n");
fflush(stdout);

/* And this is how you might pass partitions as separate matrices. */
nag_lapackeig_zuncsd(order, Nag_AllU, Nag_AllU, Nag_AllVT, Nag_AllVT,
Nag_UpperMinus, m, p, q, x11, pdx11, x12, pdx12, x21,
pdx21, x22, pdx22, theta, u1, p, u2, m - p, v1t, q, v2t,
m - q, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_lapackeig_zuncsd (f08rnc).\n%s\n", fail.message);
exit_status = 4;
goto END;
}
if (reprint != 0) {
printf("Component of CS factorization of X using separate matrices:\n");
fflush(stdout);
nag_file_print_matrix_real_gen(Nag_ColMajor, Nag_GeneralMatrix,
Nag_NonUnitDiag, r, 1, theta, r, "    Theta",
0, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_file_print_matrix_real_gen (x04cac).\n%s\n",
fail.message);
exit_status = 5;
goto END;
}
printf("\n");
fflush(stdout);
}

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) = czero;
}
}
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) = cone;
}
for (j = 1; j <= r; j++) {
X(j + n11, j + n11).re = cos(theta[j - 1]);
X(j + n11, j + n11).im = 0.0;
X(j + n11, j + n11 + r + n21 + n22).re = -sin(theta[j - 1]);
X(j + n11, j + n11 + r + n21 + n22).im = 0.0;
}
for (j = 1; j <= n12; j++) {
X(j + n11 + r, j + n11 + r + n21 + n22 + r).re = -1.0;
X(j + n11 + r, j + n11 + r + n21 + n22 + r).im = 0.0;
}
/* bottom half */
for (j = 1; j <= n22; j++) {
X(p + j, q + j) = cone;
}
for (j = 1; j <= r; j++) {
X(p + n22 + j, j + n11).re = sin(theta[j - 1]);
X(p + n22 + j, j + n11).im = 0.0;
X(p + n22 + j, j + r + n21 + n22).re = cos(theta[j - 1]);
X(p + n22 + j, j + r + n21 + n22).im = 0.0;
}
for (j = 1; j <= n21; j++) {
X(p + n22 + r + j, n11 + r + j) = cone;
}

/* multiply U * Sigma_p into w */
nag_blast_zgemm(order, Nag_NoTrans, Nag_NoTrans, m, m, m, cone, &U(1, 1),
pdu, &X(1, 1), pdx, czero, &W(1, 1), pdw, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_blast_zgemm (f16zac).\n%s\n", fail.message);
exit_status = 6;
goto END;
}
/* form U * Sigma_p * V^T into u */
nag_blast_zgemm(order, Nag_NoTrans, Nag_NoTrans, m, m, m, cone, &W(1, 1),
pdw, &V(1, 1), pdv, czero, &U(1, 1), pdu, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_blast_zgemm (f16zac).\n%s\n", fail.message);
exit_status = 7;
goto END;
}
/* nag_file_print_matrix_complex_gen_comp (x04dbc).
* Print least squares solutions.
*/
nag_file_print_matrix_complex_gen_comp(
order, Nag_GeneralMatrix, Nag_NonUnitDiag, m, m, &U(1, 1), pdu,
Nag_BracketForm, "%7.4f", "    U * Sigma_p * V^T", 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 = 8;
goto END;
}
printf("\n");
fflush(stdout);
}

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;
}
```