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lmmin

subroutine lmmin(func,x,y,iy,exitcode,x0,itermax,damp,tol,shock,usebro,iprint)
  implicit none
  external                             :: func        ! user-supplied function to be minimize
  real(kind=8) , intent(out)           :: x(:)        ! values of "x" at minimum
  real(kind=8) , intent(out)           :: y(:)        ! valuf of "func" at "x"
  integer      , intent(out)           :: iy          ! number of function evaluations
  integer      , intent(out)           :: exitcode    ! indicator of convergence
  real(kind=8) , intent(in)            :: x0(:)       ! initial guess
  real(kind=8) , intent(in) , optional :: shock       ! shock to parameter values (as %)
  real(kind=8) , intent(in) , optional :: damp        ! damping factor
  real(kind=8) , intent(in) , optional :: tol         ! tolerance level
  integer      , intent(in) , optional :: itermax     ! max number of functione valuations
  integer      , intent(in) , optional :: iprint      ! indicator for printing behaviour
  integer      , intent(in) , optional :: usebro      ! indicator for the use of Broyden method to update Jacobian

This subroutine applies the Levenberg–Marquardt algorithm (click here for more information) which minimizes the sum of squared errors of a (possibly nonlinear) system of multivariate equations.

This subroutine applies the Nelder-Mead algorithm (click here for more information).

The user should input the function func and the initial guess x0. The function func must be of the form:

  function func(xvec) result(resid)
    real(kind=8)               :: xvec(:)
    real(kind=8) , allocatable :: resid(:)
    ...
  end function func

The subroutine returns the value(s) of x that makes func smaller than tol (close enoughs to zero), the number of function evaluations (iy), and an indicator, exitcode:

• exitcode = 0: the algorithm found a root
• exitcode = 1: the jacobian of the system is close to singular
• exitcode = 2: the step in x is closed to xero
• exitcode = 9: maximum number of iterations reached

Optionally, the user can also supply:

• shock (double precision): shock to parameter values (as %), default = 0.05
• damp (double precision): damping factor, default = 1.0
• tol (double precision): tolerance level, default = 1.0d-8
• itermax (integer): max number of functione valuations, default = 500
• iprint (integer): indicator for printing behaviour
  • iprint = 0: don’t print anything (default)
  • iprint = 1: print main results
  • iprint = 2: print main results and each iteration
• usebro (integer): indicator for the use of Broyden method to update Jacobian
  • usebro = 0: re-compute the Jacobian numerically (default)
  • usebro = 1: update the existing Jacobian using the Broyden method

Note: for constrained optimization problems, one can make use of the normalize and denormalize subroutines.

Internal dependencies: inverse, broyden