Provided by: librheolef-dev_7.0-3_amd64

#### NAME

```       uzawa -- Uzawa algorithm.

```

#### SYNOPSIS

```         template <class Matrix, class Vector, class Preconditioner, class Real>
int uzawa (const Matrix &A, Vector &x, const Vector &b, const Preconditioner &M,
const solver_option& sopt)

```

#### EXAMPLE

```       The simplest call to 'uzawa' has the folling form:

solver_option sopt;
sopt.max_iter = 100;
sopt.tol = 1e-7;
int status = uzawa(A, x, b, eye(), sopt);

```

#### DESCRIPTION

```       uzawa solves the linear system A*x=b using the Uzawa method. The Uzawa method is a descent
method in the direction opposite to the  gradient, with a constant step length 'rho'.  The
convergence  is  assured  when  the  step  length  'rho'  is small enough.  If matrix A is
symmetric positive definite, please uses 'cg'  that  computes  automatically  the  optimal
descdnt  step  length  at  each  iteration.  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

sopt.n_iter
the number of iterations performed before the tolerance was reached

sopt.residue

```

#### IMPLEMENTATION

```       template<class Matrix, class Vector, class Preconditioner, class Real2>
int uzawa (const Matrix &A, Vector &x, const Vector &Mb, const Preconditioner &M,
const Real2& rho, const solver_option& sopt = solver_option())
{
typedef typename Vector::size_type  Size;
typedef typename Vector::float_type Real;
std::string label = (sopt.label != "" ? sopt.label : "uzawa");
Vector b = M.solve(Mb);
Real norm2_b = dot(Mb,b);
Real norm2_r = norm2_b;
if (sopt.absolute_stopping || norm2_b == Real(0)) norm2_b = 1;
if (sopt.p_err) (*sopt.p_err) << "[" << label << "] #iteration residue" << std::endl;
for (sopt.n_iter = 0; sopt.n_iter <= sopt.max_iter; sopt.n_iter++) {
Vector Mr = A*x - Mb;
Vector r = M.solve(Mr);
norm2_r = dot(Mr, r);
sopt.residue = sqrt(norm2_r/norm2_b);
if (sopt.p_err) (*sopt.p_err) << "[" << label << "] " << sopt.n_iter << " " << sopt.residue << std::endl;
if (sopt.residue <= sopt.tol) return 0;
x  -= rho*r;
}
return 1;
}

```

#### SEEALSO

```       solver_option(2)

```

```       Copyright  (C) 2000-2018 Pierre Saramito <Pierre.Saramito@imag.fr> GPLv3+: GNU GPL version