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! Hessian - or its inverse - update

! We use the Murtagh-Sargent update, also called Symmetric Rank 1

!

! input:

!  - nfree: number of coordinates

!  - ds: GeomOld-Geom (expressed in the optimization coord.)

!  - dgrad: GradOld-Grad  (expressed in the optimization coord.)

!

!  input/output:

!  - Hess: old Hessian in input, replaced by updated Hessian on output

!             (or inverse Hessian)

!

!

!!!!!!!!!!!!!!!!!!!!!!

! The update formula for the Hessian is:

! (following Fletcher, approximate Hessian is denoted by B)

! B(k+1)=B+[(y-Bs){(y-Bs)T}]/[{(y-Bs)T}s]

!

! it is self dual, ie for the inverse Hessian, denote by H:

! H(k+1)=H+[(s-Hy){(s-Hy)T}]/[{(s-Hy)T}y]

!

!

! We decompose the computation in two steps:

! first we compute the update, and then we add it to Hess to get the

! new Hessian.

! This allows us to introduce a scaling factor if needed later.

!

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

SUBROUTINE Hupdate_MS(Nfree,ds,dgrad,Hess)

  IMPLICIT NONE

  INTEGER, PARAMETER :: KINT=KIND(1)

  INTEGER, PARAMETER :: KREAL=KIND(1.D0)

  INTEGER(KINT), INTENT(IN) :: NFREE

  REAL(KREAL), INTENT(INOUT) :: Hess(Nfree,Nfree)

  REAL(KREAL), INTENT(IN) :: ds(Nfree)

  REAL(KREAL), INTENT(IN) :: dGrad(Nfree)

  REAL(KREAL), ALLOCATABLE  :: HessOld(:,:) , HupMS(:,:)

  REAL(KREAL), ALLOCATABLE  :: Hup(:,:)

  REAL(KREAL), ALLOCATABLE  :: HupInt(:)

  REAL(KREAL), ALLOCATABLE  :: HS(:), Num(:,:)

  REAL(KREAL) :: Phi, Num3, Den, Den2,NormDS

  CHARACTER(120) :: fmt,fmt2,Line

  INTEGER(KINT) :: i,j, idx

  LOGICAL :: debug

  interface

     function valid(string) result (isValid)

       logical                  :: isValid

       character(*), intent(in) :: string

     end function valid

  end interface

  debug=valid("Hupdate").OR.Valid("Hupdate_MS")

  if (debug) Call Header("Entering Hupdate_MS")

  ALLOCATE(HessOld(nfree,nfree),  &

       Hup(nfree,nfree), &

       HupInt(nfree), HS(nfree),  &

       Num(nfree,nfree))

  HessOld=Hess

  if (debug) THEN

     WRITE(*,*) 'DBG Hupdate HessOld'

     Idx=min(12,Nfree)

     DO I=1,NFree

        WRITE(*,'(12(1X,F8.4))') HessOld(I,1:Idx)

     END DO

     WRITE(*,*) 'DBG Hupdate ds'

     WRITE(*,'(12(1X,F8.4))') ds(1:Idx)

     WRITE(*,*) 'DBG Hupdate dGrad'

     WRITE(*,'(12(1X,F8.4))') dGrad(1:Idx)

  END IF

  NormDS=sqrt(DOT_PRODUCT(DS,DS))

  if (NormDS>=1e-6) THEN

     HS=0

     DO I=1,nfree

        DO J=1,nfree

           HS(I)=HS(I)+HessOld(i,j)*Ds(j)

        END DO

     END DO

     if (debug) WRITE(*,*) "HS:",HS

     HupInt=DGrad-HS

     ! WE calculate the MS update

     Num=0.

     Den=0.

     DO I=1,nfree

        DO J=1,nfree

           Num(I,J)=HUpInt(i)*HupInt(j)

        END DO

        Den=Den+Hupint(i)*Ds(i)

     END DO

     if (debug) WRITE(*,*) "Den:",Den,1./Den

     Hup=Num/Den

     Hess=HessOld+Hup

  ELSE

     Hess=HessOld

  END IF

  !     WRITE(*,*) 'New hess'

  !     DO I=1,nfree

  !        WRITE(*,fmt) Hess(i,:)

  !     END DO

  DEALLOCATE(HessOld,Hup, HupInt, HS, Num )

  if (debug) Call Header("Hupdate_MS Over")

END SUBROUTINE Hupdate_MS