bool IterativeSolvers::pcg(const IRCMatrix &A,
Vector &x,
const Vector &b,
const Preconditioner &M) {
/*!
Solves Ax=b using the preconditioned conjugate gradient method.
*/
const idx N = x.getLength();
real resid(100.0);
Vector p(N), z(N), q(N);
real alpha;
real normr(0);
real normb = norm(b);
real rho(0), rho_1(0), beta(0);
Vector r = b - A * x;
if (normb == 0.0)
normb = 1;
resid = norm(r) / normb;
if (resid <= IterativeSolvers::toler) {
IterativeSolvers::toler = resid;
IterativeSolvers::maxIter = 0;
return true;
}
// MAIN LOOP
idx i = 1;
for (; i <= IterativeSolvers::maxIter; i++) {
M.solveMxb(z, r);
rho = dot(r, z);
if (i == 1)
p = z;
else {
beta = rho / rho_1;
aypx(beta, p, z); // p = beta*p + z;
}
// CALCULATES q = A*p AND dp = dot(q,p)
real dp = multiply_dot(A, p, q);
alpha = rho / dp;
normr = 0;
#ifdef USES_OPENMP
#pragma omp parallel for reduction(+:normr)
#endif
for (idx j = 0 ; j < N ; ++j) {
x[j] += alpha * p[j]; // x + alpha(0) * p;
r[j] -= alpha * q[j]; // r - alpha(0) * q;
normr += r[j] * r[j];
}
normr = sqrt(normr);
resid = normr / normb;
if (resid <= IterativeSolvers::toler) {
IterativeSolvers::toler = resid;
IterativeSolvers::maxIter = i;
return true;
}
rho_1 = rho;
}
IterativeSolvers::toler = resid;
return false;
}
bool IterativeSolvers::gmres(const IRCMatrix &A,
Vector &x,
const Vector &b,
const Preconditioner &M) {
const idx N = x.getLength();
idx i, j = 1, k;
Vector s(maxInnerIter + 1);
Vector cs(maxInnerIter + 1);
Vector sn(maxInnerIter + 1);
Vector w(N);
real normb = norm(M.solve(b));
Vector r = M.solve(b - A * x);
real beta = norm(r);
if (normb == 0.0)
normb = 1;
real res(norm(r) / normb);
if (res <= toler) {
toler = res;
maxIter = 0;
return true;
}
Vector *v = new Vector[maxInnerIter + 1];
for (idx id = 0; id < maxInnerIter + 1; ++id)
v[id] = Vector(N);
// CREATE HESSENBERG MATRIX NEEDED TO STORE INTERMEDIATES
DenseMatrix H(maxInnerIter + 1, maxInnerIter);
Vector temp(N);
Vector temp2(maxInnerIter + 1);
// MAIN LOOP
while (j <= maxIter) {
v[0] = r * (1.0 / beta);
s = 0.0;
s(0) = beta;
// INNER ITERATIONS
for (i = 0; i < maxInnerIter && j <= maxIter; i++, j++) {
// CALCULATE w = M^{-1}(A*v[i])
multiply(A, v[i], temp);
M.solveMxb(w, temp);
// PRE-CALCULATE DOT PRODUCTS IN PARALLEL
// H(k,i) = dot( v[k], w)
#ifdef USES_OPENMP
#pragma omp parallel for
#endif
for (k = 0; k <= i ; ++k) {
register real dp(0);
for (idx id = 0 ; id < N ; ++id)
dp += w[id] * v[k][id];
H(k, i) = dp; //dot(w,v[k]);
}
for (k = 0; k <= i; ++k) {
// w -= v[k]*H(k,i) without temporaries
register real tempr = H(k, i);
#ifdef USES_OPENMP
#pragma omp parallel for // why is this loop so critical??
#endif
for (idx id = 0 ; id < N ; ++id)
w[id] -= v[k][id] * tempr;
}
// BELOW PARALLEL REGION CALCULATES:
// H(i+1,i) = norm(w);
// v[i+1] = w * (1.0 / H(i+1, i));
H(i + 1, i) = 0;
real tempr(0);
#ifdef USES_OPENMP
#pragma omp parallel shared(tempr)
#endif
{
#ifdef USES_OPENMP
#pragma omp for reduction(+:tempr)
#endif
for (idx id = 0 ; id < N ; ++id)
tempr += w[id] * w[id]; //norm(w);
#ifdef USES_OPENMP
#pragma omp single
#endif
{
H(i + 1, i) = sqrt(tempr);
tempr = (1.0 / H(i + 1, i));
}
#ifdef USES_OPENMP
#pragma omp for
#endif
for (idx id = 0 ; id < N ; ++id)
v[i + 1][id] = w[id] * tempr;
}// end for omp parallel
for (k = 0; k < i; k++)
ApplyPlaneRotation(H(k, i), H(k + 1, i), cs(k), sn(k));
GeneratePlaneRotation(H(i, i), H(i + 1, i), cs(i), sn(i));
ApplyPlaneRotation(H(i, i), H(i + 1, i), cs(i), sn(i));
ApplyPlaneRotation(s(i), s(i + 1), cs(i), sn(i));
res = fabs(s(i + 1)) / normb;
if (res < toler) {
// COPY S INTO temp WITHOUT RESIZING
for (idx id = 0 ; id < maxInnerIter + 1 ; ++id)
temp2[id] = s[id];
Update(x, i, H, temp2, v);
toler = res;
maxIter = j;
delete [] v;
return true;
}
}// end for i IINNER ITERATIONS
// COPY S INTO temp WITHOUT RESIZING
for (idx id = 0 ; id < maxInnerIter + 1 ; ++id)
temp2[id] = s[id];
Update(x, maxInnerIter - 1, H, temp2, v);
//multiply(A, x, temp); //r = M.solve(b - A * x);
M.solveMxb(r, b - A * x);
beta = norm(r);
res = beta / normb;
if (res < toler) {
toler = res;
maxIter = j;
delete [] v;
return true;
}
}
toler = res;
delete [] v;
return false;
}