NAG Library Manual, Mark 28.4
Interfaces:  FL   CL   CPP   AD
```/* nag_lapackeig_dtzrzf (f08bhc) Example Program.
*
* Copyright 2022 Numerical Algorithms Group.
*
* Mark 28.4, 2022.
*/

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

int main(void) {
/* Scalars */
double d, f, tol;
Integer i, j, k, m, n, nrhs, pda, pdb;
Integer exit_status = 0;
/* Arrays */
double *a = 0, *b = 0, *rnorm = 0, *tau = 0, *work = 0;
Integer *jpvt = 0;
/* Nag Types */
Nag_OrderType order;
NagError fail;

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

INIT_FAIL(fail);

printf("nag_lapackeig_dtzrzf (f08bhc) Example Program Results\n\n");

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

#ifdef NAG_COLUMN_MAJOR
pda = m;
pdb = m;
#else
pda = n;
pdb = nrhs;
#endif

/* Allocate memory */
if (!(a = NAG_ALLOC(m * n, double)) || !(b = NAG_ALLOC(m * nrhs, double)) ||
!(rnorm = NAG_ALLOC(nrhs, double)) || !(tau = NAG_ALLOC(n, double)) ||
!(work = NAG_ALLOC(n, double)) || !(jpvt = NAG_ALLOC(n, Integer))) {
printf("Allocation failure\n");
exit_status = -1;
goto END;
}

/* Read A and B from data file */
for (i = 1; i <= m; ++i)
for (j = 1; j <= n; ++j)
scanf("%lf", &A(i, j));
scanf("%*[^\n]");

for (i = 1; i <= m; ++i)
for (j = 1; j <= nrhs; ++j)
scanf("%lf", &B(i, j));
scanf("%*[^\n]");

/* nag_blast_iload (f16dbc).
* Initialize jpvt to be zero so that all columns are free.
*/
nag_blast_iload(n, 0, jpvt, 1, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_blast_iload (f16dbc).\n%s\n", fail.message);
exit_status = 1;
goto END;
}

/* nag_lapackeig_dgeqp3 (f08bfc).
* Compute the QR factorization of A with column pivoting as
* A = Q*(R11 R12)*(P^T)
*       ( 0  R22)
*/
nag_lapackeig_dgeqp3(order, m, n, a, pda, jpvt, tau, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_lapackeig_dgeqp3 (f08bfc).\n%s\n", fail.message);
exit_status = 1;
goto END;
}

/* nag_lapackeig_dormqr (f08ckc).
* Compute C = (C1) = (Q^T)*B, storing the result in b.
*             (C2)
*/
nag_lapackeig_dormqr(order, Nag_LeftSide, Nag_Trans, m, nrhs, n, a, pda, tau,
b, pdb, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_lapackeig_dormqr (f08ckc).\n%s\n", fail.message);
exit_status = 1;
goto END;
}

/* Choose tol to reflect the relative accuracy of the input data */
tol = 0.01;

/* Determine and print the rank, k, of R relative to tol */
for (k = 1; k <= n; ++k)
if ((f = A(k, k), fabs(f)) <= tol * (d = A(1, 1), fabs(d)))
break;
--k;

printf("Tolerance used to estimate the rank of A\n");
printf("%11.2e\n", tol);
printf("Estimated rank of A\n");
printf("%8" NAG_IFMT "\n\n", k);

/* nag_lapackeig_dtzrzf (f08bhc).
* Compute the RZ factorization of the k by k part of R as
* (R11 R12) = (T 0)*Z
*/
nag_lapackeig_dtzrzf(order, k, n, a, pda, tau, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_lapackeig_dtzrzf (f08bhc).\n%s\n", fail.message);
exit_status = 1;
goto END;
}

/* nag_blast_dtrsm (f16yjc).
* Compute least squares solutions of triangular problems by
* back substitution in T*Y1 = C1, storing the result in b.
*/
nag_blast_dtrsm(order, Nag_LeftSide, Nag_Upper, Nag_NoTrans, Nag_NonUnitDiag,
k, nrhs, 1.0, a, pda, b, pdb, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_blast_dtrsm (f16yjc).\n%s\n", fail.message);
exit_status = 1;
goto END;
}

/* nag_blast_dge_norm (f16rac).
* Compute estimates of the square roots of the residual sums of
* squares (2-norm of each of the columns of C2).
*/
for (j = 1; j <= nrhs; ++j) {
nag_blast_dge_norm(order, Nag_FrobeniusNorm, m - k, 1, &B(k + 1, j), pdb,
&rnorm[j - 1], &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_blast_dge_norm (f16rac).\n%s\n", fail.message);
exit_status = 1;
goto END;
}
}

/* nag_blast_dge_load (f16qhc).
* Set the remaining elements of the solutions to zero (to give
* the minimum-norm solutions), Y2 = 0.
*/
nag_blast_dge_load(order, n - k, nrhs, 0.0, 0.0, &B(k + 1, 1), pdb, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_blast_dge_load (f16qhc).\n%s\n", fail.message);
exit_status = 1;
goto END;
}

/* nag_lapackeig_dormrz (f08bkc).
* Form W = (Z^T)*Y.
*/
nag_lapackeig_dormrz(order, Nag_LeftSide, Nag_Trans, n, nrhs, k, n - k, a,
pda, tau, b, pdb, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_lapackeig_dormrz (f08bkc).\n%s\n", fail.message);
exit_status = 1;
goto END;
}

/* Permute the least squares solutions stored in B to give X = P*W */
for (j = 1; j <= nrhs; ++j) {
for (i = 1; i <= n; ++i)
work[jpvt[i - 1] - 1] = B(i, j);
for (i = 1; i <= n; ++i)
B(i, j) = work[i - 1];
}

/* nag_file_print_matrix_real_gen (x04cac).
* Print least squares solutions.
*/
fflush(stdout);
nag_file_print_matrix_real_gen(order, Nag_GeneralMatrix, Nag_NonUnitDiag, n,
nrhs, b, pdb, "Least squares solution(s)", 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;
}

/* Print the square roots of the residual sums of squares */
printf("\nSquare root(s) of the residual sum(s) of squares\n");

for (j = 0; j < nrhs; ++j)
printf("%11.2e%s", rnorm[j], j % 6 == 5 || j == nrhs - 1 ? "\n" : " ");

END:
NAG_FREE(a);
NAG_FREE(b);
NAG_FREE(rnorm);
NAG_FREE(tau);
NAG_FREE(work);
NAG_FREE(jpvt);
return exit_status;
}

#undef A
#undef B
```