brenth Subroutine

    subroutine brenth(me,ax,bx,fax,fbx,xzero,fzero,iflag)

    implicit none

    class(brenth_solver),intent(inout) :: me
    real(wp),intent(in)  :: ax      !! left endpoint of initial interval
    real(wp),intent(in)  :: bx      !! right endpoint of initial interval
    real(wp),intent(in)  :: fax     !! `f(ax)`
    real(wp),intent(in)  :: fbx     !! `f(ax)`
    real(wp),intent(out) :: xzero   !! abscissa approximating a zero of `f` in the interval `ax`,`bx`
    real(wp),intent(out) :: fzero   !! value of `f` at the root (`f(xzero)`)
    integer,intent(out)  :: iflag   !! status flag (`0`=root found, `-2`=max iterations reached)

    real(wp) :: xpre,xcur,xblk,fpre,fcur,fblk,spre,&
                scur,sbis,delta,stry,dpre,dblk,xdelta
    integer :: i !! iteration counter

    iflag = 0
    xpre = ax
    xcur = bx
    fpre = fax
    fcur = fbx

    do i = 1, me%maxiter

        if (fpre*fcur < 0.0_wp) then
            xblk = xpre
            fblk = fpre
            scur = xcur - xpre
            spre = scur
        end if
        if (abs(fblk) < abs(fcur)) then
            xpre = xcur
            xcur = xblk
            xblk = xpre
            fpre = fcur
            fcur = fblk
            fblk = fpre
        end if

        delta = (me%atol + me%rtol*abs(xcur))/2.0_wp
        sbis = (xblk - xcur)/2.0_wp
        if (abs(fcur) <= me%ftol .or. abs(sbis) < delta) exit ! converged

        if (abs(spre) > delta .and. abs(fcur) < abs(fpre)) then
            if (xpre == xblk) then
                ! interpolate
                stry = -fcur*(xcur - xpre)/(fcur - fpre)
            else
                ! extrapolate
                dpre = (fpre - fcur)/(xpre - xcur)
                dblk = (fblk - fcur)/(xblk - xcur)
                stry = -fcur*(fblk - fpre)/(fblk*dpre - fpre*dblk)  ! only difference from brentq
            end if

            if (2.0_wp*abs(stry) < min(abs(spre), 3.0_wp*abs(sbis) - delta)) then
                ! accept step
                spre = scur
                scur = stry
            else
                ! bisect
                spre = sbis
                scur = sbis
            end if
        else
            ! bisect
            spre = sbis
            scur = sbis
        end if

        xpre = xcur
        fpre = fcur
        if (abs(scur) > delta) then
            xcur = xcur + scur
        else
            if (sbis > 0.0_wp) then
                xdelta = delta
            else
                xdelta = -delta
            end if
            xcur = xcur + xdelta
        end if

        fcur = me%f(xcur)
        if (abs(fcur) <= me%ftol) exit ! converged
        if (i == me%maxiter) iflag = -2 ! max iterations reached

    end do

    xzero = xcur
    fzero = fcur

    end subroutine brenth