NAG Library Manual, Mark 27.2
```/* F08KD_T1W_F C++ Header Example Program.
*
* Copyright 2021 Numerical Algorithms Group.
* Mark 27.2, 2021.
*/

#include <dco.hpp>
#include <iostream>
#include <nag.h>
#include <nagx04.h>
#include <stdio.h>
#include <string>
using namespace std;

int main(void)
{
int      exit_status = 0;
Integer  ifail       = 0;
NagError fail;
INIT_FAIL(fail);

cout << "F08KD_T1W_F C++ Header Example Program Results\n\n";
// Skip heading in data file
string mystr;
getline(cin, mystr);

// Read matrix dimensions and algorithmic mode
Integer m, n, mode;
cin >> m;
cin >> n;
cin >> mode;

// Allocate arrays containing A and its factorized form, B
// and the solution X.
Integer            lda = m, ldu = m, ldvt = n, lwork;
nagad_t1w_w_rtype *a = 0, *a_in = 0, *s = 0, *u = 0, *vt = 0, *work = 0;
double *           ur = 0, *vtr = 0, *dsda = 0;
Integer *          iwork = 0;
Charlen            lena  = 1;
a                        = new nagad_t1w_w_rtype[m * n];
a_in                     = new nagad_t1w_w_rtype[m * n];
u                        = new nagad_t1w_w_rtype[m * m];
vt                       = new nagad_t1w_w_rtype[n * n];
iwork                    = new Integer[8 * n];
dsda                     = new double[m * m];

// Read the matrix A, register and copy
double dd;
for (int i = 0; i < m; i++)
{
for (int j = 0; j < n; j++)
{
cin >> dd;
Integer k = i + j * m;
a_in[k]   = dd;
}
}

// Create AD configuration data object
ifail = 0;

// Use routine workspace query to get optimal workspace.
ifail = 0;
lwork = -1;
nag::ad::f08kd(ad_handle, "A", m, n, a, lda, s, u, ldu, vt, ldvt, dummy,
lwork, iwork, ifail);

lwork = (Integer)dco::value(dummy[0]) + 1;

double inc = 1.0, zero = 0.0;
for (int i = 0; i < m; ++i)
{
dco::derivative(a_in[i]) = inc;
for (int j = 0; j < n * m; j++)
{
a[j] = a_in[j];
}
//  Compute the singular values and left and right singular vectors
//  of A (A = U*S*(V**T), m < n)
nag::ad::f08kd(ad_handle, "A", m, n, a, lda, s, u, ldu, vt, ldvt, work,
lwork, iwork, ifail);

dco::derivative(a_in[i]) = zero;
for (int j = 0; j < m; j++)
{
dsda[j + i * m] = dco::derivative(s[j]);
}
}

// Print primal solution
cout.precision(4);
cout.width(12);
cout << " ";
cout << " Singular values:\n";
for (int i = 0; i < m; i++)
{
cout.width(11);
cout << dco::value(s[i]);
}

// Copy primal values to array for printing
ur  = new double[m * m];
vtr = new double[n * n];

for (int i = 0; i < m * m; i++)
{
ur[i] = dco::value(u[i]);
}
for (int j = 0; j < n; j++)
{
Integer k = j * n;
for (int i = 0; i < m; i++)
{
vtr[k] = dco::value(vt[k]);
k++;
}
}

cout << "\n\n";
x04cac(Nag_ColMajor, Nag_GeneralMatrix, Nag_NonUnitDiag, m, m, ur, ldu,
"Left singular vectors by column", 0, &fail);
cout << "\n";
x04cac(Nag_ColMajor, Nag_GeneralMatrix, Nag_NonUnitDiag, m, n, vtr, ldvt,
"Right singular vectors by row", 0, &fail);

cout << "\n\n Derivatives calculated: First order tangents\n";
cout << " Computational mode    : algorithmic\n";

cout << "\n Derivatives of Singular values w.r.t first column of A\n";
// Obtain derivatives for each singular value w.r.t first column of A
cout.setf(ios::scientific, ios::floatfield);
cout.setf(ios::right);
cout.precision(2);
for (int i = 0; i < m; i++)
{
cout << "\n Singular value " << i + 1 << endl;
// Get derivatives
cout.width(10);
cout << " ";
for (int j = 0; j < m; j++)
{
cout.width(10);
cout << dsda[i + j * m];
}
cout << endl;
}

// Remove computational data object
ifail = 0;

delete[] ur;
delete[] vtr;
delete[] work;
delete[] a;
delete[] a_in;
delete[] s;
delete[] u;
delete[] vt;
delete[] iwork;
delete[] dsda;
return exit_status;
}
```