---
name: "minres"
section: "5rheolef"
platform: "linux"
package: "librheolef-dev"
version: "7.2-4"
release: "Debian 13 (Trixie)"
platforms: ["linux"]
url: "https://new.carta.tech/minres"
---

# minres(5rheolef)

minimum residual algorithm (rheolef-7.2)

## NAME

minres - minimum residual algorithm (rheolef-7.2)

## SYNOPSIS

```
template <class Matrix, class Vector, class Preconditioner>
int minres (const Matrix &A, Vector &x, const Vector &Mb, const Preconditioner &M,
  const solver_option& sopt = solver_option())
```

## DESCRIPTION

This function solves the symmetric positive but possibly *singular* linear system `A*x=b` with the minimal residual method. The `minres` function follows the algorithm described in

```
    C. C. Paige and M. A. Saunders,
    Solution of sparse indefinite systems of linear equations',
    SIAM J. Numer. Anal., 12(4), 1975.
```

 For more, see http://www.stanford.edu/group/SOL/software.html and also at page 60 of the PhD report:

```
   S.-C. T. Choi,
   Iterative methods for singular linear equations and least-squares problems,
   Stanford University, 2006,
   http://www.stanford.edu/group/SOL/dissertations/sou-cheng-choi-thesis.pdf
```

## EXAMPLE

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

 The fourth argument of `minres` is a preconditionner: here, the **eye(5)** one is a do-nothing preconditionner, for simplicity. Finally, the **solver\_option(4)** variable `sopt` transmits the stopping criterion with `sopt.tol` and `sopt.max_iter`.

On return, the `sopt.residue` and `sopt.n_iter` indicate the reached residue and the number of iterations effectively performed. The return `status` is zero when the prescribed tolerance `tol` has been obtained, and non-zero otherwise. Also, the `x` variable contains the approximate solution. See also the **solver\_option(4)** for more controls upon the stopping criterion.

## IMPLEMENTATION

This documentation has been generated from file linalg/lib/minres.h

The present template implementation is inspired from the `IML++ 1.2` iterative method library, http://math.nist.gov/iml++

```
template <class Matrix, class Vector, class Preconditioner>
int minres (const Matrix &A, Vector &x, const Vector &Mb, const Preconditioner &M,
  const solver_option& sopt = solver_option())
```

```
{
  // Size &max_iter, Real &tol, odiststream *p_derr = 0
  typedef typename Vector::size_type  Size;
  typedef typename Vector::float_type Real;
  std::string label = (sopt.label != "" ? sopt.label : "minres");
  Vector b = M.solve(Mb);
  Real norm_b = sqrt(fabs(dot(Mb,b)));
  if (sopt.absolute_stopping || norm_b == Real(0.)) norm_b = 1;
  Vector Mr = Mb - A*x;
  Vector z = M.solve(Mr);
  Real beta2 = dot(Mr, z);
  Real norm_r = sqrt(fabs(beta2));
  sopt.residue = norm_r/norm_b;
  if (sopt.p_err) (*sopt.p_err) << "[" << label << "] #iteration residue" << std::endl
                                << "[" << label << "] 0 " << sopt.residue << std::endl;
  if (beta2 < 0 || sopt.residue <= sopt.tol) {
    return 0;
  }
  Real beta = sqrt(beta2);
  Real eta = beta;
  Vector Mv = Mr/beta;
  Vector  u =  z/beta;
  Real c_old = 1.;
  Real s_old = 0.;
  Real c = 1.;
  Real s = 0.;
  Vector u_old  (x.ownership(), 0.);
  Vector Mv_old (x.ownership(), 0.);
  Vector w      (x.ownership(), 0.);
  Vector w_old  (x.ownership(), 0.);
  Vector w_old2 (x.ownership(), 0.);
  for (sopt.n_iter = 1; sopt.n_iter <= sopt.max_iter; sopt.n_iter++) {
    // Lanczos
    Mr = A*u;
    z = M.solve(Mr);
    Real alpha = dot(Mr, u);
    Mr = Mr - alpha*Mv - beta*Mv_old;
    z  =  z - alpha*u  - beta*u_old;
    beta2 = dot(Mr, z);
    if (beta2 < 0) {
        dis_warning_macro ("minres: machine precision problem");
        sopt.residue = norm_r/norm_b;
        return 2;
    }
    Real beta_old = beta;
    beta = sqrt(beta2);
    // QR factorisation
    Real c_old2 = c_old;
    Real s_old2 = s_old;
    c_old = c;
    s_old = s;
    Real rho0 = c_old*alpha - c_old2*s_old*beta_old;
    Real rho2 = s_old*alpha + c_old2*c_old*beta_old;
    Real rho1 = sqrt(sqr(rho0) + sqr(beta));
    Real rho3 = s_old2 * beta_old;
    // Givens rotation
    c = rho0 / rho1;
    s = beta / rho1;
    // update
    w_old2 = w_old;
    w_old  = w;
    w = (u - rho2*w_old - rho3*w_old2)/rho1;
    x += c*eta*w;
    eta = -s*eta;
    Mv_old = Mv;
     u_old = u;
    Mv = Mr/beta;
     u =  z/beta;
    // check residue
    norm_r *= s;
    sopt.residue = norm_r/norm_b;
    if (sopt.p_err) (*sopt.p_err) << "[" << label << "] " << sopt.n_iter << " " << sopt.residue << std::endl;
    if (sopt.residue <= sopt.tol) return 0;
  }
  return 1;
}
```

## AUTHOR

Pierre  Saramito  \<Pierre.Saramito@imag.fr\>

## 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.