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//*****************************************************************
// Iterative template routine -- GMRES
//
// GMRES solves the unsymmetric linear system Ax = b using the
// Generalized Minimum Residual method
//
// GMRES follows the algorithm described on p. 20 of the
// SIAM Templates book.
//
// The return value indicates convergence within max_iter (input)
// iterations (0), or no convergence within max_iter iterations (1).
//
// Upon successful return, output arguments have the following values:
//
// x -- approximate solution to Ax = b
// max_iter -- the number of iterations performed before the
// tolerance was reached
// tol -- the residual after the final iteration
//
//*****************************************************************
#include <cmath>
template<class Real>
void ApplyPlaneRotation(Real &dx, Real &dy, Real &cs, Real &sn);
template<class Real>
void GeneratePlaneRotation(Real &dx, Real &dy, Real &cs, Real &sn);
template < class Matrix, class Vector >
void
Update(Vector &x, int k, Matrix &h, Vector &s, Vector v[])
{
Vector y(s);
// Backsolve:
for (int i = k; i >= 0; i--) {
y(i) /= h(i,i);
for (int j = i - 1; j >= 0; j--)
y(j) -= h(j,i) * y(i);
}
for (int j = 0; j <= k; j++)
x += v[j] * y(j);
}
template < class Real >
Real
abs(Real x)
{
return (x > 0 ? x : -x);
}
template < class Operator, class Vector, class Preconditioner,
class Matrix, class Real >
int
GMRES(const Operator &A, Vector &x, const Vector &b,
const Preconditioner &M, Matrix &H, int &m, int &max_iter,
Real &tol)
{
Real resid;
int i, j = 1, k;
Vector s(m+1), cs(m+1), sn(m+1), w;
Vector p = inv(M)*b;
Real normb = p.norm();
Vector r = inv(M) * (b - A * x);
Real beta = r.norm();
if (normb == 0.0)
normb = 1;
if ((resid = r.norm() / normb) <= tol) {
tol = resid;
max_iter = 0;
return 0;
}
Vector *v = new Vector[m+1];
while (j <= max_iter) {
v[0] = r * (1.0 / beta); // ??? r / beta
s = 0.0;
s(0) = beta;
for (i = 0; i < m && j <= max_iter; i++, j++) {
w = inv(M) * (A * v[i]);
for (k = 0; k <= i; k++) {
H(k, i) = w.dot(v[k]);
w -= H(k, i) * v[k];
}
H(i+1, i) = w.norm();
v[i+1] = w * (1.0 / H(i+1, i)); // ??? w / H(i+1, i)
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));
if ((resid = abs(s(i+1)) / normb) < tol) {
Update(x, i, H, s, v);
tol = resid;
max_iter = j;
delete [] v;
return 0;
}
}
Update(x, m - 1, H, s, v);
r = inv(M) * (b - A * x);
beta = r.norm();
if ((resid = beta / normb) < tol) {
tol = resid;
max_iter = j;
delete [] v;
return 0;
}
}
tol = resid;
delete [] v;
return 1;
}
template<class Real>
void GeneratePlaneRotation(Real &dx, Real &dy, Real &cs, Real &sn)
{
if (dy == 0.0) {
cs = 1.0;
sn = 0.0;
} else if (abs(dy) > abs(dx)) {
Real temp = dx / dy;
sn = 1.0 / sqrt( 1.0 + temp*temp );
cs = temp * sn;
} else {
Real temp = dy / dx;
cs = 1.0 / sqrt( 1.0 + temp*temp );
sn = temp * cs;
}
}
template<class Real>
void ApplyPlaneRotation(Real &dx, Real &dy, Real &cs, Real &sn)
{
Real temp = cs * dx + sn * dy;
dy = -sn * dx + cs * dy;
dx = temp;
}
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