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#### NAME

```       puzawa -- Uzawa algorithm.
```

#### SYNOPSIS

```           template <class Matrix, class Vector, class Preconditioner, class Real>
int puzawa (const Matrix &A, Vector &x, const Vector &b, const Preconditioner &M,
int &max_iter, Real &tol, const Real& rho, odiststream *p_derr=0);
```

#### EXAMPLE

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

size_t max_iter = 100;
double tol = 1e-7;
int status = puzawa(A, x, b, EYE, max_iter, tol, 1.0, &derr);
```

#### DESCRIPTION

```       puzawa 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  'pcg'  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

max_iter
the number of iterations  performed  before  the  tolerance  was
reached

tol    the residual after the final iteration
```

#### IMPLEMENTATION

```       template < class Matrix, class Vector, class Preconditioner, class Real, class Size>
int puzawa(const Matrix &A, Vector &x, const Vector &Mb, const Preconditioner &M,
Size &max_iter, Real &tol, const Real& rho,
odiststream *p_derr, std::string label)
{
Vector b = M.solve(Mb);
Real norm2_b = dot(Mb,b);
Real norm2_r = norm2_b;
if (norm2_b == Real(0)) norm2_b = 1;
if (p_derr) (*p_derr) << "[" << label << "] #iteration residue" << std::endl;
for (Size n = 0; n <= max_iter; n++) {
Vector Mr = A*x - Mb;
Vector r = M.solve(Mr);
norm2_r = dot(Mr, r);
if (p_derr) (*p_derr) << "[" << label << "] " << n << " " << sqrt(norm2_r/norm2_b) << std::endl;
if (norm2_r <= sqr(tol)*norm2_b) {
tol = sqrt(norm2_r/norm2_b);
max_iter = n;
return 0;
}
x  -= rho*r;
}
tol = sqrt(norm2_r/norm2_b);
return 1;
}```

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