```/* nag_kalman_sqrt_filt_cov_var (g13eac) Example Program.
*
* Copyright 2017 Numerical Algorithms Group
*
* Mark 26.2, 2017.
*
*/

#include <nag.h>
#include <math.h>
#include <stdio.h>
#include <string.h>
#include <nag_stdlib.h>
#include <nage04.h>
#include <nagf06.h>
#include <nagg05.h>
#include <nagg13.h>
#include <nagx02.h>

#ifdef __cplusplus
extern "C"
{
#endif
static void NAG_CALL objfun(Integer n, const double theta_phi[],
double *objf, double g[], Nag_Comm *comm);
#ifdef __cplusplus
}
#endif

static int ex1(void);
static int ex2(void);

int main(void)
{
/* Integer scalar and array declarations */
Integer exit_status_ex1 = 0;
Integer exit_status_ex2 = 0;

printf("nag_kalman_sqrt_filt_cov_var (g13eac) Example Program Results\n\n");

/* Run example 1 */
exit_status_ex1 = ex1();

/* Run example 2 */
exit_status_ex2 = ex2();

return (exit_status_ex1 == 0 && exit_status_ex2 == 0) ? 0 : 1;
}

/* Start of the first example ... */
#define A(I, J) a[(I) *tda + J]
#define B(I, J) b[(I) *tdb + J]
#define C(I, J) c[(I) *tdc + J]
#define K(I, J) k[(I) *tdk + J]
#define Q(I, J) q[(I) *tdq + J]
#define R(I, J) r[(I) *tdr + J]
#define S(I, J) s[(I) *tds + J]

static int ex1()
{
/* Integer scalar and array declarations */
Integer exit_status = 0;
Integer i, j, m, n, p, istep, tda, tdb, tdc, tdk, tdq, tdr, tds, tdh;

/* Double scalar and array declarations */
double tol;
double *a = 0, *b = 0, *c = 0, *k = 0, *q = 0, *r = 0, *s = 0, *h = 0;

/* NAG structures */
NagError fail;

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

printf("Example 1\n");

/* Skip the heading in the data file */
scanf("%*[^\n]");

/* Get the problem size */
scanf("%" NAG_IFMT "%" NAG_IFMT "%" NAG_IFMT "%lf", &n, &m, &p, &tol);
if (n < 1 || m < 1 || p < 1) {
printf("Invalid n or m or p.\n");
exit_status = 1;
goto END;
}

tda = tdc = tds = n;
tdb = tdq = m;
tdk = tdr = tdh = p;

/* Allocate arrays */
if (!(a = NAG_ALLOC(n * tda, double)) ||
!(b = NAG_ALLOC(n * tdb, double)) ||
!(c = NAG_ALLOC(p * tdc, double)) ||
!(k = NAG_ALLOC(n * tdk, double)) ||
!(q = NAG_ALLOC(m * tdq, double)) ||
!(r = NAG_ALLOC(p * tdr, double)) ||
!(s = NAG_ALLOC(n * tds, double)) || !(h = NAG_ALLOC(n * tdh, double)))
{
printf("Allocation failure\n");
exit_status = -1;
goto END;
}

for (i = 0; i < n; ++i)
for (j = 0; j < n; ++j)
scanf("%lf", &S(i, j));
for (i = 0; i < n; ++i)
for (j = 0; j < n; ++j)
scanf("%lf", &A(i, j));
for (i = 0; i < n; ++i)
for (j = 0; j < m; ++j)
scanf("%lf", &B(i, j));
if (q) {
for (i = 0; i < m; ++i)
for (j = 0; j < m; ++j)
scanf("%lf", &Q(i, j));
}
for (i = 0; i < p; ++i)
for (j = 0; j < n; ++j)
scanf("%lf", &C(i, j));
for (i = 0; i < p; ++i)
for (j = 0; j < p; ++j)
scanf("%lf", &R(i, j));

/* Perform three iterations of the Kalman filter recursion */
for (istep = 1; istep <= 3; ++istep) {
/* Run the kalman filter */
nag_kalman_sqrt_filt_cov_var(n, m, p, s, tds, a, tda, b, tdb, q, tdq,
c, tdc, r, tdr, k, tdk, h, tdh, tol, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_kalman_sqrt_filt_cov_var (g13eac).\n%s\n",
fail.message);
exit_status = 1;
goto END;
}
}

/* Print the results */
printf("\nThe square root of the state covariance matrix is\n\n");
for (i = 0; i < n; ++i) {
for (j = 0; j < n; ++j)
printf("%8.4f ", S(i, j));
printf("\n");
}
if (k) {
printf("\nThe matrix AK (the product of the Kalman gain\n");
printf("matrix with the state transition matrix) is\n\n");
for (i = 0; i < n; ++i) {
for (j = 0; j < p; ++j)
printf("%8.4f ", K(i, j));
printf("\n");
}
}

END:
NAG_FREE(a);
NAG_FREE(b);
NAG_FREE(c);
NAG_FREE(k);
NAG_FREE(q);
NAG_FREE(r);
NAG_FREE(s);
NAG_FREE(h);

return exit_status;
}

/* ... end of the first example */

/* Start of the second example ... */
#define NY 2000

/* This illustrates the use of the kalman filter to estimate the
parameters of an ARMA(1,1) time series model.
Note : theta_phi[0] contains theta (moving average coefficient), and
theta_phi[1] contains phi (autoregressive coefficient)
*/
static int ex2(void)
{
/* Integer scalar and array declarations */
Integer exit_status = 0;
Integer n, lr;
Integer lstate;
Integer *state = 0;

/* Double scalar and array declarations */
double a[1], sy[NY];
double *theta_phi = 0, *g = 0, *bl = 0, *bu = 0, *r = 0;
double objf, var;

/* NAG structures and data types */
Nag_Comm comm;
Nag_E04_Opt options;
NagError fail;
Nag_ModeRNG mode;

/* Choose the base generator */
Nag_BaseRNG genid = Nag_Basic;
Integer subid = 0;

/* Set the seed */
Integer seed[] = { 1762543 };
Integer lseed = 1;

/* Set the autoregressive coefficients for the (randomly generated)
time series */
Integer ip = 1;
double sphi[] = { 0.4 };

/* Set the moving average coefficients for the (randomly generated)
time series */
Integer iq = 1;
double stheta[] = { 0.9 };

/* Number of terms in the (randomly generated) time series */
Integer nterms = NY;

/* Mean and variance of the (randomly generated) time series */
double mean = 0.0, vara = 1.0;

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

printf("\n\nExample 2\n\n");

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

/* Calculate the size of the reference vector */
lr = (ip > iq + 1) ? ip : iq + 1;
lr += ip + iq + 6;

/* Allocate arrays */
if (!(state = NAG_ALLOC(lstate, Integer)) || !(r = NAG_ALLOC(lr, double)))
{
printf("Allocation failure\n");
exit_status = -1;
goto END;
}

/* Initialize the generator to a repeatable sequence */
nag_rand_init_repeatable(genid, subid, seed, lseed, state, &lstate, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_rand_init_repeatable (g05kfc).\n%s\n",
fail.message);
exit_status = 1;
goto END;
}

/* Generate a time series with nterms terms */
mode = Nag_InitializeAndGenerate;
nag_rand_arma(mode, nterms, mean, ip, sphi, iq, stheta, vara, r, lr, state,
&var, sy, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_rand_arma (g05phc).\n%s\n", fail.message);
exit_status = 1;
goto END;
}

/* Number of parameters to estimate */
n = ip + iq;

/* Allocate arrays */
if (!(theta_phi = NAG_ALLOC(n, double)) ||
!(g = NAG_ALLOC(n, double)) ||
!(bl = NAG_ALLOC(n, double)) || !(bu = NAG_ALLOC(n, double)))
{
printf("Allocation failure\n");
exit_status = -1;
goto END;
}

/* Make an initial guess of the parameters */
theta_phi[0] = 0.6;
theta_phi[1] = 0.4;

/* Set the bounds */
bl[0] = -1.0;
bu[0] = 1.0;
bl[1] = -1.0;
bu[1] = 1.0;
comm.user = &sy[0];

/* nag_opt_init (e04xxc).
* Initialization function for option setting
*/
nag_opt_init(&options);
strcpy(options.outfile, "stdout");
options.print_level = Nag_NoPrint;
options.list = Nag_FALSE;

options.obj_deriv = Nag_FALSE;
options.con_deriv = Nag_FALSE;
options.print_deriv = Nag_D_NoPrint;

/* nag_opt_nlp (e04ucc).
* Minimization with nonlinear constraints using a sequential QP method.
*/
nag_opt_nlp(n, 0, 0, a, 1, bl, bu, objfun, NULLFN, theta_phi, &objf,
g, &options, &comm, &fail);
if (fail.code != NE_NOERROR && fail.code != NW_KT_CONDITIONS) {
printf("Error from nag_opt_nlp (e04ucc).\n%s\n", fail.message);
exit_status = 1;
goto END;
}

/* nag_opt_free (e04xzc).
* Memory freeing function for use with option setting
*/
nag_opt_free(&options, "all", &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_opt_free (e04xzc).\n%s\n", fail.message);
exit_status = 1;
goto END;
}

/* Display the results */
printf("The estimates are :  theta = %7.3f, phi = %7.3f \n",
theta_phi[0], theta_phi[1]);

END:
NAG_FREE(state);
NAG_FREE(theta_phi);
NAG_FREE(g);
NAG_FREE(bl);
NAG_FREE(bu);
NAG_FREE(r);

return exit_status;
}

/* Define  objective function used in the non-linear optimization routine ... */

static void NAG_CALL objfun(Integer n, const double theta_phi[], double *objf,
double g[], Nag_Comm *comm)
{
/* Routine to evaluate objective function. */

Integer ione = 1, itwo = 2, k, m1 = 1, n1 = 2, nsteps = NY, nsum = 0, p1 =
1;
double a[2][2], ak[2][1], b[2][1], c[1][2], h[1][1], hs, k11, logdet = 0.0;
double phi, q[1][1], r[1][1];
double s[2][2], ss = 0.0, temp1, temp2, theta, tol = 0.0;
double v, xp[2], *y;
NagError fail;

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

y = comm->user;
/* The expectation of the mean of an ARMA(1,1) is 0.0 */
xp[0] = 0.0;
xp[1] = 0.0;
q[0][0] = 1.0;

c[0][0] = 1.0;
c[0][1] = 0.0;

/* There is no measurement noise */
r[0][0] = 0.0;
theta = theta_phi[0];
phi = theta_phi[1];

b[0][0] = 1.0;
b[1][0] = -theta;

a[0][0] = phi;
a[1][0] = 0.0;
a[0][1] = 1.0;
a[1][1] = 0.0;

/* set value for Cholesky factor of state covariance matrix  */
temp1 = 1.0 + (theta * theta) - (2.0 * theta * phi);
temp2 = 1.0 - (phi * phi);
k11 = temp1 / temp2;
s[0][0] = sqrt(k11);
s[0][1] = 0.0;
s[1][0] = -theta / s[0][0];
s[1][1] = theta * sqrt(1.0 - (1.0 / k11));

/* iterate kalman filter for number of observations  */
for (k = 1; k <= nsteps; ++k) {
/* nag_kalman_sqrt_filt_cov_var (g13eac).
* One iteration step of the time-varying Kalman filter
* recursion using the square root covariance implementation
*/
nag_kalman_sqrt_filt_cov_var(n1, m1, p1, &s[0][0], itwo, &a[0][0], itwo,
&b[0][0], ione, &q[0][0], ione, &c[0][0],
itwo, &r[0][0], ione, &ak[0][0], ione,
&h[0][0], ione, tol, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from  nag_kalman_sqrt_filt_cov_var (g13eac).\n%s\n",
fail.message);
*objf = 0.0;
return;
}

v = y[k - 1] - c[0][0] * xp[0];
hs = h[0][0] * h[0][0];
logdet = logdet + log(hs);
ss = ss + (v * v / hs);
nsum = nsum + 1;

xp[0] = a[0][0] * xp[0] + a[0][1] * xp[1] + ak[0][0] * v;
xp[1] = ak[1][0] * v;
}
*objf = nsum * log(ss / nsum) + logdet;
}

/* ... end of the objective function definition */
/* ... end of the second example */
```