```/* nag_rnla_randproj_dct_real (f10dac) Example Program.
*
* Copyright 2019 Numerical Algorithms Group.
*
* Mark 27.0, 2019.
*/

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

int main(void)
{

/* Scalars */
Integer       exit_status = 0, i, j, m, n, pda, lda;
/* Arrays */
double        *a = 0, *a_copy = 0;

/* Nag Types */
NagError      fail;
Nag_OrderType order;

/* Declarations for random projection */
Integer       k, *d = 0, *col_ind = 0;
double        *pa = 0;
char          nag_enum_arg1[40];
char          nag_enum_arg2[40];
Nag_TransType transa;

/* Declarations for RNG */
Integer       lstate;
Integer       *state = 0;
double        *x = 0;
Nag_BaseRNG   genid = Nag_Basic;
Integer       subid = 0;
Integer       seed[] = { 1762543 };
Integer       lseed = 1;

/* Declarations for lapack routines    */
Integer       tau_len, pdu, pdvt;
double        *tau = 0, *q = 0, *dgemm_inout = 0, *res_mat = 0, *work = 0,
*s = 0, *u = 0, *vt = 0;
double        alpha, beta, sum;

/* Declarations for error estimation   */
Integer       r;
double        xmu, var, err_bound, max_res_norm;
double        *omega = 0, *y = 0, *res = 0, *res_norm = 0;

#define A(I, J) a[(J - 1) * pda + I - 1]
order = Nag_ColMajor;

INIT_FAIL(fail);

printf("nag_rnla_randproj_dct_real (f10dac) Example Program Results\n\n");
/* Skip heading in data file */
scanf("%*[^\n]");
/* Read the problem dimensions, m, n, k */
scanf("%" NAG_IFMT "%" NAG_IFMT "%*[^\n] ", &m, &n);

if (m < 0 || n < 0) {
printf("Invalid m or n\n");
exit_status = 1;
goto END;
}

/* Read in random projection routine parameters - mode, transpose, k */
scanf("%39s %39s %" NAG_IFMT "%*[^\n] ", nag_enum_arg1, nag_enum_arg2, &k);

/* nag_enum_name_to_value (x04nac).
* Converts NAG enum member name to value
*/
transa = (Nag_TransType) nag_enum_name_to_value(nag_enum_arg2);

/* Allocate memory */
if (!(a       = NAG_ALLOC(m * n, double)) ||
!(a_copy  = NAG_ALLOC(m * n, double)) ||
!(pa      = NAG_ALLOC(m * k, double)) ||
!(d       = NAG_ALLOC(n, Integer)) ||
!(col_ind = NAG_ALLOC(k, Integer)) )
{
printf("Allocation failure\n");
exit_status = -1;
goto END;
}

#ifdef NAG_COLUMN_MAJOR
pda = m;
pdu = m;
pdvt = n;
#else
pda = n;
pdu = n;
pdvt = m;
#endif

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

/* Copy a into a_copy */
for (i = 0; i < m * n; i++)
a_copy[i] = a[i];

nag_file_print_matrix_real_gen(order, Nag_GeneralMatrix, Nag_NonUnitDiag,
m, n, a, pda, "Matrix A", 0, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_file_print_matrix_real_gen (x04cac).\n%s\n",
fail.message);
exit_status = 2;
goto END;
}

/* Random number generation */

/* Initialize the error structure */
INIT_FAIL(fail);

/* Get the length of the state array */
lstate = -1;
nag_rand_init_repeat(genid, subid, seed, lseed, state, &lstate, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_rand_init_repeat (g05kfc).\n%s\n",
fail.message);
exit_status = 3;
goto END;
}

/* Allocate arrays */
if (!(x = NAG_ALLOC(n, double)) || !(state = NAG_ALLOC(lstate, Integer)))
{
printf("Allocation failure\n");
exit_status = -2;
goto END;
}

/* Initialize the generator to a repeatable sequence */
nag_rand_init_repeat(genid, subid, seed, lseed, state, &lstate, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_rand_init_repeat (g05kfc).\n%s\n",
fail.message);
exit_status = 4;
goto END;
}
lda = m;
/* Perform random projection */
/* Copy a into a_copy */
for (i = 0; i < m * n; i++)
a_copy[i] = a[i];
nag_rnla_randproj_dct_real(transa, m, n, a_copy, lda, k,
state, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_rnla_randproj_dct_real (f10dac).\n%s\n",
fail.message);
exit_status = 5;
goto END;
}
for (i = 0; i < m * k; i++)
pa[i] = a[i];

/* Perform QR decomposition on random projection */
tau_len = MIN(m, k);

/* Allocate memory */
if (!(tau = NAG_ALLOC(tau_len, double)) )
{
printf("Allocation failure\n");
exit_status = -3;
goto END;
}

nag_lapackeig_dgeqrf(order, m, k, pa, m, tau, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_lapackeig_dgeqrf (f08aec).\n%s\n", fail.message);
exit_status = 6;
goto END;
}
q = pa;

nag_lapackeig_dorgqr(order, m, k, k, pa, m, tau, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_lapackeig_dorgqr (f08afc).\n%s\n", fail.message);
exit_status = 7;
goto END;
}

/* Calculate true error as largest singular value of (I-QQ')A */

/* Calculate (I - Q Q') */
if( !(dgemm_inout = NAG_ALLOC(m * m, double)) ||
!(res_mat = NAG_ALLOC(m*n, double)) ||
!(s = NAG_ALLOC(MIN(m, n), double)) ||
!(u = NAG_ALLOC(m * m, double)) ||
!(vt = NAG_ALLOC(m * n, double)) ||
!(work = NAG_ALLOC(MIN(m, n), double)))
{
printf("Allocation failure\n");
exit_status = -4;
goto END;
}

/* Define identity matrix */
for (i = 0; i<m; ++i) {
for (j = 0; j<n; ++j) {
dgemm_inout[i*m + j] = 0.0;
}
dgemm_inout[i*m + i] = 1.0;
}

alpha = -1.0;
beta = 1.0;
nag_blast_dgemm(order, Nag_NoTrans, Nag_Trans, m, m, k, alpha, q, m,
q, m, beta, dgemm_inout, m, &fail);

/* Calculate (I - Q Q') A */
alpha = 1.0;
beta = 0.0;
nag_blast_dgemm(order,Nag_NoTrans,Nag_NoTrans,m,n,m,alpha,dgemm_inout,m,
a,m,beta,res_mat,m, &fail);

/* Compute the singular values of res_mat */
nag_lapackeig_dgesvd(order, Nag_NotU,Nag_NotVT,m,n,res_mat,pda,
s,u,pdu,vt,pdvt,work,&fail);

/* Calculate error estimate */
r = 10;
if (!(omega = NAG_ALLOC(n, double)) ||
!(y = NAG_ALLOC(m, double)) ||
!(res = NAG_ALLOC(m, double)) ||
!(res_norm = NAG_ALLOC(r,double)) )
{
printf("Allocation failure\n");
exit_status = -5;
goto END;
}
max_res_norm = 0;
for(j = 0; j<r; ++j) {
/* Generate Gaussian random vector, omega */
xmu = 0.0;
var = 1.0;
nag_rand_dist_normal(n,xmu,var,state,omega,&fail);
/* calculate y = A omega */
nag_blast_dgemv(order, Nag_NoTrans,m,n,alpha,a,m,omega,1,beta,y,1, &fail);
/* Calculate (I - Q Q') y */
nag_blast_dgemv(order, Nag_NoTrans,m,m,alpha,dgemm_inout,m,y,1,beta,res,1,
&fail);
sum = 0;
for (i = 0; i<m; ++i)
sum = sum + pow(res[i], 2);

res_norm[j] = sqrt(sum);
if (res_norm[j] > max_res_norm) {
max_res_norm = res_norm[j];
}
}

err_bound = sqrt(200.0/nag_math_pi)*max_res_norm;

printf("\n True error \n %10.1e\n", s[0]);
printf("\n Probabilistic error bound \n %10.1e\n", err_bound);
printf("\n Maximum probability that true error exceeds error bound \n");
printf(" %10.1e\n", pow(10.0,-r));
END:

NAG_FREE(a);
NAG_FREE(a_copy);
NAG_FREE(pa);
NAG_FREE(d);
NAG_FREE(col_ind);
NAG_FREE(x);
NAG_FREE(state);
NAG_FREE(tau);
NAG_FREE(dgemm_inout);
NAG_FREE(res_mat);
NAG_FREE(s);
NAG_FREE(u);
NAG_FREE(vt);
NAG_FREE(work);
NAG_FREE(omega);
NAG_FREE(y);
NAG_FREE(res);
NAG_FREE(res_norm);

return exit_status;
}
```