NAG Library Manual, Mark 28.7
```/* nag_tsa_multi_kalman_sqrt_invar (g13ebc) Example Program.
*
* Copyright 2022 Numerical Algorithms Group.
*
* Mark 28.7, 2022.
*/

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

typedef enum { read, print } ioflag;

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

int main(void) {
Integer exit_status_ex1 = 0;
Integer exit_status_ex2 = 0;

printf("nag_tsa_multi_kalman_sqrt_invar (g13ebc) Example Program "
"Results\n\n");

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

exit_status_ex1 = ex1();
exit_status_ex2 = ex2();

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

#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() { /* simple example (matrices A and C are supplied in lower
observer Hessenberg form) */
Integer exit_status = 0, i, istep, j, m, n, p, tda, tdb, tdc, tdh, tdk, tdq;
Integer tdr, tds;
NagError fail;
double *a = 0, *b = 0, *c = 0, *h = 0, *k = 0, *q = 0, *r = 0, *s = 0, tol;

INIT_FAIL(fail);

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

printf("Example 1\n");
scanf("%" NAG_IFMT "%" NAG_IFMT "%" NAG_IFMT "%lf", &n, &m, &p, &tol);
if (n >= 1 && m >= 1 && p >= 1) {
if (!(a = NAG_ALLOC(n * n, double)) || !(b = NAG_ALLOC(n * m, double)) ||
!(c = NAG_ALLOC(p * n, double)) || !(k = NAG_ALLOC(n * p, double)) ||
!(q = NAG_ALLOC(m * m, double)) || !(r = NAG_ALLOC(p * p, double)) ||
!(s = NAG_ALLOC(n * n, double)) || !(h = NAG_ALLOC(n * p, double))) {
printf("Allocation failure\n");
exit_status = -1;
goto END;
}
tda = n;
tdb = m;
tdc = n;
tdk = p;
tdq = m;
tdr = p;
tds = n;
tdh = p;
} else {
printf("Invalid n or m or p.\n");
exit_status = 1;
return exit_status;
}

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)
/* nag_tsa_multi_kalman_sqrt_invar (g13ebc).
* One iteration step of the time-invariant Kalman filter
* recursion using the square root covariance implementation
* with (AC) in lower observer Hessenberg form
*/
nag_tsa_multi_kalman_sqrt_invar(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_tsa_multi_kalman_sqrt_invar (g13ebc).\n%s\n",
fail.message);
exit_status = 1;
goto END;
}

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;
}

static void mat_io(Integer n, Integer m, double mat[], Integer tdmat,
ioflag flag, const char *message);

#define UB(I, J) ub[(I)*tdub + J]
#define SF(I, J) sf[(I)*tdsf + J]
#define SE(I, J) se[(I)*tdse + J]
#define PF(I, J) pf[(I)*tdpf + J]
#define UAUT(I, J) uaut[(I)*tduaut + J]
#define CUT(I, J) cut[(I)*tdcut + J]
#define U(I, J) u[(I)*tdu + J]

static int
ex2() { /* more general example which requires the data to be transformed. The
results produced by nag_tsa_multi_kalman_sqrt_var (g13eac) and
nag_tsa_multi_kalman_sqrt_invar (g13ebc) are compared */
Integer exit_status = 0, i, istep, j, m, n, p, tda, tdb;
Integer tdc, tdcut, tdh, tdke, tdkf, tdpe, tdpf, tdq, tdr, tdrwork, tdse;
Integer tdsf, tdu, tduaut, tdub;
NagError fail;
Nag_ObserverForm reduceto = Nag_LH_Observer;
double *a = 0, *b = 0, *c = 0, *cut = 0, *h = 0;
double *ke = 0, *kf = 0, one = 1.0, *pe = 0, *pf = 0, *q = 0;
double *r = 0, *rwork = 0, *se = 0, *sf = 0, tol, *u = 0;
double *uaut = 0, *ub = 0, zero = 0.0;

INIT_FAIL(fail);

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

/* skip the heading in the data file */
scanf(" %*[^\n]");
scanf("%" NAG_IFMT "%" NAG_IFMT "%" NAG_IFMT "%lf", &n, &m, &p, &tol);
if (n >= 1 && m >= 1 && p >= 1) {
if (!(a = NAG_ALLOC(n * n, double)) || !(b = NAG_ALLOC(n * m, double)) ||
!(c = NAG_ALLOC(p * n, double)) || !(ke = NAG_ALLOC(n * p, double)) ||
!(kf = NAG_ALLOC(n * p, double)) || !(ub = NAG_ALLOC(n * m, double)) ||
!(q = NAG_ALLOC(m * m, double)) || !(r = NAG_ALLOC(p * p, double)) ||
!(rwork = NAG_ALLOC(n * n, double)) ||
!(sf = NAG_ALLOC(n * n, double)) || !(se = NAG_ALLOC(n * n, double)) ||
!(h = NAG_ALLOC(n * p, double)) || !(pf = NAG_ALLOC(n * n, double)) ||
!(pe = NAG_ALLOC(n * n, double)) ||
!(uaut = NAG_ALLOC(n * n, double)) ||
!(cut = NAG_ALLOC(p * n, double)) || !(u = NAG_ALLOC(n * n, double))) {
printf("Allocation failure\n");
exit_status = -1;
goto END;
}
tda = n;
tdb = m;
tdc = n;
tdke = p;
tdkf = p;
tdub = m;
tdq = m;
tdr = p;
tdrwork = n;
tdsf = n;
tdse = n;
tdh = p;
tdpf = n;
tdpe = n;
tduaut = n;
tdcut = n;
tdu = n;
} else {
printf("Invalid n or m or p.\n");
exit_status = 1;
return exit_status;
}
mat_io(n, n, se, tdse, read, "");
mat_io(n, n, a, tda, read, "");
mat_io(n, m, b, tdb, read, "");
if (q)
mat_io(m, m, q, tdq, read, "");
mat_io(p, n, c, tdc, read, "");
mat_io(p, p, r, tdr, read, "");
for (i = 0; i < n; ++i) {
for (j = 0; j < n; ++j) {
if (i < p)
CUT(i, j) = C(i, j);
SF(i, j) = SE(i, j);
UAUT(i, j) = A(i, j);
U(i, j) = zero;
}
U(i, i) = one;
}
/* Set up the matrix pair (A,C) in the lower observer hessenberg form */
/* nag_tsa_trans_hessenberg_observer (g13ewc).
* Unitary state-space transformation to reduce (AC) to
* lower or upper observer Hessenberg form
*/
nag_tsa_trans_hessenberg_observer(n, p, reduceto, uaut, tduaut, cut, tdcut, u,
tdu, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_tsa_trans_hessenberg_observer (g13ewc).\n%s\n",
fail.message);
exit_status = 1;
goto END;
}
for (j = 0; j < m; ++j)
for (i = 0; i < n; ++i)
nag_blast_ddot(Nag_NoConj, n, 1.0, &U(i, 0), 1, 0.0, &B(0, j), tdb,
&UB(i, j), &fail);

/* Generate noise covariance matrices PE and PF = U * PE * U' */
nag_blast_dgemm(Nag_RowMajor, Nag_NoTrans, Nag_Trans, n, n, n, one, se, tdse,
se, tdse, zero, pe, tdpe, &fail);
nag_blast_dgemm(Nag_RowMajor, Nag_NoTrans, Nag_Trans, n, n, n, one, pe, tdpe,
u, tdu, zero, rwork, tdrwork, &fail);
nag_blast_dgemm(Nag_RowMajor, Nag_NoTrans, Nag_NoTrans, n, n, n, one, u, tdu,
rwork, tdrwork, zero, pf, tdpf, &fail);

/* Now find the lower triangular (left) Cholesky factor of PF. */
/* nag_lapacklin_dpotrf (f07fdc).
* Cholesky factorization of real symmetric positive definite matrix.
*/
nag_lapacklin_dpotrf(Nag_RowMajor, Nag_Lower, n, pf, tdpf, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_lapacklin_dpotrf (f07fdc).\n%s\n", fail.message);
exit_status = 1;
goto END;
}
for (i = 0; i < n; ++i)
for (j = 0; j <= i; ++j)
SF(i, j) = PF(i, j);
/* Perform three steps of the Kalman filter recursion */
for (istep = 1; istep <= 3; ++istep) {
/* nag_tsa_multi_kalman_sqrt_var (g13eac).
* One iteration step of the time-varying Kalman filter
* recursion using the square root covariance implementation
*/
nag_tsa_multi_kalman_sqrt_var(n, m, p, se, tdse, a, tda, b, tdb, q, tdq, c,
tdc, r, tdr, ke, tdke, h, tdh, tol, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_tsa_multi_kalman_sqrt_var (g13eac).\n%s\n",
fail.message);
exit_status = 1;
goto END;
}
/* nag_tsa_multi_kalman_sqrt_invar (g13ebc), see above. */
nag_tsa_multi_kalman_sqrt_invar(n, m, p, sf, tdsf, uaut, tduaut, ub, tdub,
q, tdq, cut, tdcut, r, tdr, kf, tdkf, h,
tdh, tol, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_tsa_multi_kalman_sqrt_invar (g13ebc).\n%s\n",
fail.message);
exit_status = 1;
goto END;
}
}
nag_blast_dgemm(Nag_RowMajor, Nag_NoTrans, Nag_Trans, n, n, n, one, se, tdse,
se, tdse, zero, pe, tdpe, &fail);
nag_blast_dgemm(Nag_RowMajor, Nag_NoTrans, Nag_Trans, n, n, n, one, sf, tdsf,
sf, tdsf, zero, pf, tdpf, &fail);
mat_io(n, n, pe, tdpe, print,
"Covariance matrix PE from "
"nag_tsa_multi_kalman_sqrt_var (g13eac) is\n");
mat_io(n, n, pf, tdpf, print,
"Covariance matrix PF from "
"nag_tsa_multi_kalman_sqrt_invar (g13ebc) is\n");

/* Calculate PF = U' * PF * U */
nag_blast_dgemm(Nag_RowMajor, Nag_NoTrans, Nag_NoTrans, n, n, n, one, pf,
tdpf, u, tdu, zero, rwork, tdrwork, &fail);
nag_blast_dgemm(Nag_RowMajor, Nag_Trans, Nag_NoTrans, n, n, n, one, u, tdu,
rwork, tdrwork, zero, pf, tdpf, &fail);
mat_io(n, n, pf, tdpf, print, "Matrix U' * PF * U is \n");
mat_io(n, p, ke, tdke, print,
"The matrix KE from nag_tsa_multi_kalman_sqrt_var (g13eac) is\n");
mat_io(n, p, kf, tdkf, print,
"The matrix KF from nag_tsa_multi_kalman_sqrt_invar (g13ebc) is\n");

/* calculate U' * K */
nag_blast_dgemm(Nag_RowMajor, Nag_Trans, Nag_NoTrans, n, p, n, one, u, tdu,
kf, tdkf, zero, rwork, tdrwork, &fail);
mat_io(n, p, rwork, tdrwork, print, "U' * KF is\n");

END:
NAG_FREE(a);
NAG_FREE(b);
NAG_FREE(c);
NAG_FREE(ke);
NAG_FREE(kf);
NAG_FREE(ub);
NAG_FREE(q);
NAG_FREE(r);
NAG_FREE(rwork);
NAG_FREE(sf);
NAG_FREE(se);
NAG_FREE(h);
NAG_FREE(pf);
NAG_FREE(pe);
NAG_FREE(uaut);
NAG_FREE(cut);
NAG_FREE(u);

return exit_status;
}

static void mat_io(Integer n, Integer m, double mat[], Integer tdmat,
ioflag flag, const char *message) {
Integer i, j;
#define MAT(I, J) mat[((I)-1) * tdmat + (J)-1]
if (flag == print)
printf("%s \n", message);
for (i = 1; i <= n; ++i) {
for (j = 1; j <= m; ++j) {