---
name: "uzawa"
section: "4rheolef"
platform: "linux"
package: "librheolef-dev"
version: "7.0-2+b1"
release: "Debian 10 (Buster)"
platforms: ["linux"]
url: "https://new.carta.tech/uzawa"
---

# uzawa(4rheolef)

## 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**

  the residual after the final iteration See also the solver\_option(2).

## 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;
}

```

## SEE ALSO

solver\_option(2)

## COPYRIGHT

Copyright (C) 2000-2018 Pierre Saramito \<Pierre.Saramito@imag.fr\> GPLv3+: GNU GPL version 3 or later \<http://gnu.org/licenses/gpl.html\>. This is free software: you are free to change and redistribute it.  There is NO WARRANTY, to the extent permitted by law.