## damped_system_pseudocode.txt

function damped_system ( f, fp, x, xtol, ftol, itmax )

it = 0
DX = 0
alpha = 0

Begin loop

  it = it + 1

  Set Xold to X
  Set DXold to DX
  Solve for DX: DF(X) * DX = - F(X)
  
  beta = 1.0
  begin loop
    Set X = Xold + beta * DX
    if ( || F(X) || <= || F(Xold) ||
      print it, beta, ||F(X)||
      break from this damping loop
    if ( beta < 1.0 / 1024 )
      return with failure
    beta = beta / 2.0
  end loop

  Set alpha_old to alpha
  if ||DX_old|| is not 0, set alpha to ||DX|| / ||DX_old|| 
    if alpha_old is not 0, set r = log ( alpha ) / log ( alpha_old )

  if ||DX|| is less than xtol and |F(X)| is less than ftol, return success
  if it > itmax, return with failure

End loop

return updated X, and it