NAG Library Manual, Mark 27.2
```/* nag_sparseig_real_monit (f12aec) Example Program.
*
* Copyright 2021 Numerical Algorithms Group.
*
* Mark 27.2, 2021.
*/

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

static void mv(Integer, double *, double *);
static void av(Integer, double *, double *);
static int ytax(Integer, double *, double *, double *);
static int ytmx(Integer, double *, double *, double *);
static void my_zgttrf(Integer, Complex *, Complex *, Complex *, Complex *,
Integer *, Integer *);
static void my_zgttrs(Integer, Complex *, Complex *, Complex *, Complex *,
Integer *, Complex *);

int main(void) {
/* Constants */
Integer licomm = 140, imon = 1;
/* Scalars */
Complex c1, c2, c3, eigv, num, den;
double estnrm, deni, denr, i2, numi, numr, r2;
double sigmai, sigmar;
Integer exit_status, info, irevcm, j, k, lcomm, n;
Integer nconv, ncv, nev, niter, nshift;
/* Nag types */
Nag_Boolean first;
NagError fail;

/* Arrays */
Complex *cdd = 0, *cdl = 0, *cdu = 0, *cdu2 = 0, *ctemp = 0;
double *comm = 0, *eigvr = 0, *eigvi = 0, *eigest = 0;
double *resid = 0, *v = 0;
Integer *icomm = 0, *ipiv = 0;
/* Pointers */
double *mx = 0, *x = 0, *y = 0;

exit_status = 0;
INIT_FAIL(fail);

printf("nag_sparseig_real_monit (f12aec) Example Program "
"Results\n");
/* Skip heading in data file */
scanf("%*[^\n] ");

/* Read problem parameter values from data file. */
scanf("%" NAG_IFMT "%" NAG_IFMT "%" NAG_IFMT "%lf%lf%*[^\n] ", &n, &nev, &ncv,
&sigmar, &sigmai);
/* Allocate memory */
lcomm = 3 * n + 3 * ncv * ncv + 6 * ncv + 60;
if (!(cdd = NAG_ALLOC(n, Complex)) || !(cdl = NAG_ALLOC(n, Complex)) ||
!(cdu = NAG_ALLOC(n, Complex)) || !(cdu2 = NAG_ALLOC(n, Complex)) ||
!(ctemp = NAG_ALLOC(n, Complex)) || !(comm = NAG_ALLOC(lcomm, double)) ||
!(eigvr = NAG_ALLOC(ncv, double)) || !(eigvi = NAG_ALLOC(ncv, double)) ||
!(eigest = NAG_ALLOC(ncv, double)) || !(resid = NAG_ALLOC(n, double)) ||
!(v = NAG_ALLOC(n * ncv, double)) ||
!(icomm = NAG_ALLOC(licomm, Integer)) ||
!(ipiv = NAG_ALLOC(n, Integer))) {
printf("Allocation failure\n");
exit_status = -1;
goto END;
}

/* Initialize communication arrays for problem using
nag_sparseig_real_init (f12aac). */
nag_sparseig_real_init(n, nev, ncv, icomm, licomm, comm, lcomm, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_sparseig_real_init (f12aac).\n%s\n", fail.message);
exit_status = 1;
goto END;
}
/* Select the required spectrum using
nag_sparseig_real_option("SHIFTED REAL", icomm, comm, &fail);
/* Select the problem type using
nag_sparseig_real_option("GENERALIZED", icomm, comm, &fail);
/* Solve A*x = lambda*B*x in shift-invert mode. */
/* The shift, sigma, is a complex number (sigmar, sigmai). */
/* OP = Real_Part{inv[A-(sigmar,sigmai)*M]*M and  B = M. */
c1 = nag_complex_create(-2. - sigmar, -sigmai);
c2 = nag_complex_create(2. - sigmar * 4., sigmai * -4.);
c3 = nag_complex_create(3. - sigmar, -sigmai);

for (j = 0; j <= n - 2; ++j) {
cdl[j] = c1;
cdd[j] = c2;
cdu[j] = c3;
}
cdd[n - 1] = c2;

my_zgttrf(n, cdl, cdd, cdu, cdu2, ipiv, &info);

irevcm = 0;
REVCOMLOOP:
/* repeated calls to reverse communication routine
nag_sparseig_real_iter (f12abc). */
nag_sparseig_real_iter(&irevcm, resid, v, &x, &y, &mx, &nshift, comm, icomm,
&fail);
if (irevcm != 5) {
if (irevcm == -1) {
/* Perform  x <--- OP*x = inv[A-SIGMA*M]*M*x */
mv(n, x, y);
for (j = 0; j <= n - 1; ++j) {
ctemp[j].re = y[j], ctemp[j].im = 0.;
}
my_zgttrs(n, cdl, cdd, cdu, cdu2, ipiv, ctemp);
for (j = 0; j <= n - 1; ++j) {
y[j] = ctemp[j].re;
}
} else if (irevcm == 1) {
/* Perform  x <--- OP*x = inv[A-SIGMA*M]*M*x, */
/* M*X stored in MX. */
for (j = 0; j <= n - 1; ++j) {
ctemp[j].re = mx[j], ctemp[j].im = 0.;
}
my_zgttrs(n, cdl, cdd, cdu, cdu2, ipiv, ctemp);
for (j = 0; j <= n - 1; ++j) {
y[j] = ctemp[j].re;
}
} else if (irevcm == 2) {
/* Perform  y <--- M*x */
mv(n, x, y);
} else if (irevcm == 4 && imon == 1) {
/* If imon=1, get monitoring information using
nag_sparseig_real_monit (f12aec). */
nag_sparseig_real_monit(&niter, &nconv, eigvr, eigvi, eigest, icomm,
comm);
/* Compute 2-norm of Ritz estimates using
nag_blast_dge_norm (f16rac). */
nag_blast_dge_norm(Nag_ColMajor, Nag_FrobeniusNorm, nev, 1, eigest, nev,
&estnrm, &fail);
printf("Iteration %3" NAG_IFMT ", ", niter);
printf(" No. converged = %3" NAG_IFMT ",", nconv);
printf(" norm of estimates = %17.8e\n", estnrm);
}
goto REVCOMLOOP;
}
if (fail.code == NE_NOERROR) {
/* Post-Process using nag_sparseig_real_proc
(f12acc) to compute eigenvalues/vectors. */
nag_sparseig_real_proc(&nconv, eigvr, eigvi, v, sigmar, sigmai, resid, v,
comm, icomm, &fail);
first = Nag_TRUE;
k = 0;
for (j = 0; j <= nconv - 1; ++j) {
/* Use Rayleigh Quotient to recover eigenvalues of the */
/* original problem. */
if (eigvi[j] == 0.) {
/* Ritz value is real. */
/* Numerator = Vj . AVj where Vj is j-th Ritz vector */
if (ytax(n, &v[k], &v[k], &numr)) {
goto END;
}
/* Denominator = Vj . MVj */
if (ytmx(n, &v[k], &v[k], &denr)) {
goto END;
}
eigvr[j] = numr / denr;
} else if (first) {
/* Ritz value is complex: (x,y). */
/* Compute x'(Ax)  and y'(Ax). */
if (ytax(n, &v[k], &v[k], &numr)) {
goto END;
}
if (ytax(n, &v[k], &v[k + n], &numi)) {
goto END;
}
/* Compute y'(Ay)  and x'(Ay). */
if (ytax(n, &v[k + n], &v[k + n], &r2)) {
goto END;
}
if (ytax(n, &v[k + n], &v[k], &i2)) {
goto END;
}
numr += r2;
numi = i2 - numi;
/* Assign to Complex type using nag_complex_create (a02bac). */
num = nag_complex_create(numr, numi);
/* Compute x'(Mx)  and y'(Mx). */
if (ytmx(n, &v[k], &v[k], &denr)) {
goto END;
}
if (ytmx(n, &v[k], &v[k + n], &deni)) {
goto END;
}
/* Compute y'(Ay)  and x'(Ay). */
if (ytmx(n, &v[k + n], &v[k + n], &r2)) {
goto END;
}
if (ytmx(n, &v[k + n], &v[k], &i2)) {
goto END;
}
denr += r2;
deni = i2 - deni;
/* Assign to Complex type using nag_complex_create (a02bac). */
den = nag_complex_create(denr, deni);
/* eigv = x'(Ax)/x'(Mx) */
/* Compute Complex division using nag_complex_divide
(a02cdc). */
eigv = nag_complex_divide(num, den);
eigvr[j] = eigv.re;
eigvi[j] = eigv.im;
first = Nag_FALSE;
} else {
/* Second of complex conjugate pair. */
eigvr[j] = eigvr[j - 1];
eigvi[j] = -eigvi[j - 1];
first = Nag_TRUE;
}
k = k + n;
}
/* Print computed eigenvalues. */
printf("\n The %4" NAG_IFMT " generalized Ritz values closest", nconv);
printf(" to ( %8.4f ,  %8.4f ) are:\n\n", sigmar, sigmai);
for (j = 0; j <= nconv - 1; ++j) {
printf("%8" NAG_IFMT "%5s( %7.4f, %7.4f )\n", j + 1, "", eigvr[j],
eigvi[j]);
}
} else {
printf(" Error from nag_sparseig_real_iter (f12abc).\n%s\n", fail.message);
exit_status = 1;
goto END;
}
END:
NAG_FREE(cdd);
NAG_FREE(cdl);
NAG_FREE(cdu);
NAG_FREE(cdu2);
NAG_FREE(ctemp);
NAG_FREE(comm);
NAG_FREE(eigvr);
NAG_FREE(eigvi);
NAG_FREE(eigest);
NAG_FREE(resid);
NAG_FREE(v);
NAG_FREE(icomm);
NAG_FREE(ipiv);

return exit_status;
}

static void mv(Integer n, double *v, double *y) {
/* Compute the matrix vector multiplication y<---M*x, */
/* where M is mass matrix formed by using piecewise linear elements */
/* on [0,1]. */

/* Scalars */
Integer j;

/* Function Body */
y[0] = v[0] * 4. + v[1];
for (j = 1; j <= n - 2; ++j) {
y[j] = v[j - 1] + v[j] * 4. + v[j + 1];
}
y[n - 1] = v[n - 2] + v[n - 1] * 4.;
return;
} /* mv */

static void av(Integer n, double *v, double *w) {
/* Scalars */
Integer j;

/* Function Body */
w[0] = v[0] * 2. + v[1] * 3.;
for (j = 1; j <= n - 2; ++j) {
w[j] = v[j - 1] * -2. + v[j] * 2. + v[j + 1] * 3.;
}
w[n - 1] = v[n - 2] * -2. + v[n - 1] * 2.;
return;
} /* av */

static int ytax(Integer n, double x[], double y[], double *r) {
/* Given the vectors x and y, Performs the operation */
/* y'Ax and returns the scalar value. */

/* Scalars */
Integer exit_status, j;
/* Arrays */
double *ax = 0;

/* Function Body */
exit_status = 0;
/* Allocate memory */
if (!(ax = NAG_ALLOC(n, double))) {
printf("Allocation failure\n");
exit_status = -1;
goto YTAXEND;
}
av(n, x, ax);
*r = 0.0;
for (j = 0; j <= n - 1; ++j) {
*r = *r + y[j] * ax[j];
}
YTAXEND:
NAG_FREE(ax);
return exit_status;
} /* ytax */

static int ytmx(Integer n, double x[], double y[], double *r) {
/* Given the vectors x and y, Performs the operation */
/* y'Mx and returns the scalar value. */

/* Scalars */
Integer exit_status, j;
/* Arrays */
double *mx = 0;

/* Function Body */
exit_status = 0;
/* Allocate memory */
if (!(mx = NAG_ALLOC(n, double))) {
printf("Allocation failure\n");
exit_status = -1;
goto YTMXEND;
}
mv(n, x, mx);
*r = 0.0;
for (j = 0; j <= n - 1; ++j) {
*r = *r + y[j] * mx[j];
}
YTMXEND:
NAG_FREE(mx);
return exit_status;
} /* ytmx */

static void my_zgttrf(Integer n, Complex dl[], Complex d[], Complex du[],
Complex du2[], Integer ipiv[], Integer *info) {
/* A simple C version of the Lapack routine zgttrf with argument
checking removed */
/* Scalars */
Complex temp, fact, z1;
Integer i;
/* Function Body */
*info = 0;
for (i = 0; i < n; ++i) {
ipiv[i] = i;
}
for (i = 0; i < n - 2; ++i) {
du2[i] = nag_complex_create(0.0, 0.0);
}
for (i = 0; i < n - 2; ++i) {
if (fabs(d[i].re) + fabs(d[i].im) >= fabs(dl[i].re) + fabs(dl[i].im)) {
/* No row interchange required, eliminate dl[i]. */
if (fabs(d[i].re) + fabs(d[i].im) != 0.0) {
/* Compute Complex division using nag_complex_divide
(a02cdc). */
fact = nag_complex_divide(dl[i], d[i]);
dl[i] = fact;
/* Compute Complex multiply using nag_complex_multiply
(a02ccc). */
fact = nag_complex_multiply(fact, du[i]);
/* Compute Complex subtraction using
nag_complex_subtract (a02cbc). */
d[i + 1] = nag_complex_subtract(d[i + 1], fact);
}
} else {
/* Interchange rows I and I+1, eliminate dl[I] */
/* Compute Complex division using nag_complex_divide
(a02cdc). */
fact = nag_complex_divide(d[i], dl[i]);
d[i] = dl[i];
dl[i] = fact;
temp = du[i];
du[i] = d[i + 1];
/* Compute Complex multiply using nag_complex_multiply
(a02ccc). */
z1 = nag_complex_multiply(fact, d[i + 1]);
/* Compute Complex subtraction using nag_complex_subtract
(a02cbc). */
d[i + 1] = nag_complex_subtract(temp, z1);
du2[i] = du[i + 1];
/* Compute Complex multiply using nag_complex_multiply
(a02ccc). */
du[i + 1] = nag_complex_multiply(fact, du[i + 1]);
/* Perform Complex negation using nag_complex_negate
(a02cec). */
du[i + 1] = nag_complex_negate(du[i + 1]);
ipiv[i] = i + 1;
}
}
if (n > 1) {
i = n - 2;
if (fabs(d[i].re) + fabs(d[i].im) >= fabs(dl[i].re) + fabs(dl[i].im)) {
if (fabs(d[i].re) + fabs(d[i].im) != 0.0) {
/* Compute Complex division using nag_complex_divide
(a02cdc). */
fact = nag_complex_divide(dl[i], d[i]);
dl[i] = fact;
/* Compute Complex multiply using nag_complex_multiply
(a02ccc). */
fact = nag_complex_multiply(fact, du[i]);
/* Compute Complex subtraction using
nag_complex_subtract (a02cbc). */
d[i + 1] = nag_complex_subtract(d[i + 1], fact);
}
} else {
/* Compute Complex division using nag_complex_divide
(a02cdc). */
fact = nag_complex_divide(d[i], dl[i]);
d[i] = dl[i];
dl[i] = fact;
temp = du[i];
du[i] = d[i + 1];
/* Compute Complex multiply using nag_complex_multiply
(a02ccc). */
z1 = nag_complex_multiply(fact, d[i + 1]);
/* Compute Complex subtraction using nag_complex_subtract
(a02cbc). */
d[i + 1] = nag_complex_subtract(temp, z1);
ipiv[i] = i + 1;
}
}
/* Check for a zero on the diagonal of U. */
for (i = 0; i < n; ++i) {
if (fabs(d[i].re) + fabs(d[i].im) == 0.0) {
*info = i;
goto END;
}
}
END:
return;
}

static void my_zgttrs(Integer n, Complex dl[], Complex d[], Complex du[],
Complex du2[], Integer ipiv[], Complex b[]) {
/* A simple C version of the Lapack routine zgttrs with argument
checking removed, the number of right-hand-sides=1, Trans='N' */
/* Scalars */
Complex temp, z1;
Integer i;
/* Solve L*x = b. */
for (i = 0; i < n - 1; ++i) {
if (ipiv[i] == i) {
/* b[i+1] = b[i+1] - dl[i]*b[i] */
/* Compute Complex multiply using nag_complex_multiply
(a02ccc). */
temp = nag_complex_multiply(dl[i], b[i]);
/* Compute Complex subtraction using nag_complex_subtract
(a02cbc). */
b[i + 1] = nag_complex_subtract(b[i + 1], temp);
} else {
temp = b[i];
b[i] = b[i + 1];
/* Compute Complex multiply using nag_complex_multiply
(a02ccc). */
z1 = nag_complex_multiply(dl[i], b[i]);
/* Compute Complex subtraction using nag_complex_subtract
(a02cbc). */
b[i + 1] = nag_complex_subtract(temp, z1);
}
}
/* Solve U*x = b. */
/* Compute Complex division using nag_complex_divide (a02cdc). */
b[n - 1] = nag_complex_divide(b[n - 1], d[n - 1]);
if (n > 1) {
/* Compute Complex multiply using nag_complex_multiply
(a02ccc). */
temp = nag_complex_multiply(du[n - 2], b[n - 1]);
/* Compute Complex subtraction using nag_complex_subtract
(a02cbc). */
z1 = nag_complex_subtract(b[n - 2], temp);
/* Compute Complex division using nag_complex_divide (a02cdc). */
b[n - 2] = nag_complex_divide(z1, d[n - 2]);
}
for (i = n - 3; i >= 0; --i) {
/* b[i] = (b[i]-du[i]*b[i+1]-du2[i]*b[i+2])/d[i]; */
/* Compute Complex multiply using nag_complex_multiply
(a02ccc). */
temp = nag_complex_multiply(du[i], b[i + 1]);
z1 = nag_complex_multiply(du2[i], b[i + 2]);
(a02cac). */
/* Compute Complex subtraction using nag_complex_subtract
(a02cbc). */
z1 = nag_complex_subtract(b[i], temp);
/* Compute Complex division using nag_complex_divide
(a02cdc). */
b[i] = nag_complex_divide(z1, d[i]);
}
return;
}
```