!
    ! This program may be freely redistributed under the 
    ! condition that the copyright notices (including this 
    ! entire header) are not removed, and no compensation 
    ! is received through use of the software.  Private, 
    ! research, and institutional use is free.  You may 
    ! distribute modified versions of this code UNDER THE 
    ! CONDITION THAT THIS CODE AND ANY MODIFICATIONS MADE 
    ! TO IT IN THE SAME FILE REMAIN UNDER COPYRIGHT OF THE 
    ! ORIGINAL AUTHOR, BOTH SOURCE AND OBJECT CODE ARE 
    ! MADE FREELY AVAILABLE WITHOUT CHARGE, AND CLEAR 
    ! NOTICE IS GIVEN OF THE MODIFICATIONS.  Distribution 
    ! of this code as part of a commercial system is 
    ! permissible ONLY BY DIRECT ARRANGEMENT WITH THE 
    ! AUTHOR.  (If you are not directly supplying this 
    ! code to a customer, and you are instead telling them 
    ! how they can obtain it for free, then you are not 
    ! required to make any arrangement with me.) 
    !
    ! Disclaimer:  Neither I nor: Columbia University, the 
    ! National Aeronautics and Space Administration, nor 
    ! the Massachusetts Institute of Technology warrant 
    ! or certify this code in any way whatsoever.  This 
    ! code is provided "as-is" to be used at your own risk.
    !
    !

    !    
    ! UTIL1D.f90: util. func. for 1-dim. grid manipulation.
    !
    ! Darren Engwirda 
    ! 31-Mar-2019
    ! de2363 [at] columbia [dot] edu
    !
    !

    subroutine linspace(xxll,xxuu,npos,xpos)

    ! 
    ! XXLL  lower-bound grid position.
    ! NNEW  upper-bound grid position.
    ! NPOS  no. edges in the grid.
    ! XPOS  array of grid edges. XPOS has length NPOS .
    !

        implicit none

        real*8 , intent(in)  :: xxll,xxuu
        integer, intent(in)  :: npos
        real*8 , intent(out) :: xpos(:)

        integer   :: ipos
        real*8    :: xdel

        xpos(   1) = xxll
        xpos(npos) = xxuu

        xdel = (xxuu-xxll) / (npos - 1)

        do  ipos = +2, npos-1

            xpos(ipos) = (ipos-1) * xdel         

        end do

        return
    
    end  subroutine

    subroutine rndspace(xxll,xxuu,npos,xpos, &
        &               frac)

    ! 
    ! XXLL  lower-bound grid position.
    ! NNEW  upper-bound grid position.
    ! NPOS  no. edges in the grid.
    ! XPOS  array of grid edges. XPOS has length NPOS .
    ! FRAC  fractional perturbation of cell, OPTIONAL .
    !

        implicit none

        real*8 , intent(in)  :: xxll,xxuu
        integer, intent(in)  :: npos
        real*8 , intent(out) :: xpos(:)
        real*8 , intent(in), optional :: frac

        integer   :: ipos
        real*8    :: xdel,rand,move

        if (present(frac)) then
            move = +frac
        else
            move = 0.33d0
        end if

        xpos(   1) = xxll
        xpos(npos) = xxuu

        xdel = (xxuu-xxll) / (npos - 1)

        do  ipos = +2, npos-1

            xpos(ipos) = (ipos-1) * xdel         

        end do

        do ipos = +2, npos-1

            call random_number (rand)

            rand = 2.d0 * (rand-.5d0)
            
            move = rand *  move

            xpos(ipos) = &
        &       xpos(ipos) + move * xdel         

        end do

        return

    end  subroutine