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mod_lorenz63_lv.f90
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mod_lorenz63_lv.f90
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!===============================================================================
! program name: m_lv
!
! programmer: da,cheng org: umd ([email protected]) date: 2018-03-23
!
! Description:
! a module related to Lyapunov vector calculations
!
!-------------------------------------------------------------------------------
module mod_lorenz63_lv
use mod_math, only : eye, qr
implicit none
private
integer,parameter :: r_kind = kind(0.d0)
public :: le_blv, le_flv
contains
!
! Lyapunov exponents calculated from backward Lyapunov vectors
!
subroutine le_blv(nx,nt,dt,M,LE,Q0)
implicit none
integer,intent(in) :: nx
integer,intent(in) :: nt
real(r_kind),intent(in) :: dt
real(r_kind),intent(in) :: M(nx,nx,nt) ! M(:,:,1)
real(r_kind),intent(out) :: LE(nx)
real(r_kind),intent(in),optional :: Q0(nx,nx)
real(r_kind) :: Q(nx,nx,nt), R(nx,nx,nt)
integer :: n
call eye(nx,Q(:,:,1))
if(present(Q0)) Q(:,:,1) = Q0
call fwd_qr(nx,nt,M,Q,R,Q0=Q(:,:,1))
do n = 1, nx
LE(n) = sum(log(abs( R(n,n,1:nt) )))
enddo
LE(:)=LE(:)/(nt*dt)
endsubroutine
!
! Lyapunov exponents calculated from forward Lyapunov vectors
!
subroutine le_flv(nx,nt,dt,Mt,LE,Qn)
implicit none
integer,intent(in) :: nx
integer,intent(in) :: nt
real(r_kind),intent(in) :: dt
real(r_kind),intent(in) :: Mt(nx,nx,nt) ! M(:,:,i) for TLM from step i to step i+1
real(r_kind),intent(out) :: LE(nx)
real(r_kind),intent(in),optional :: Qn(nx,nx)
real(r_kind) :: Q(nx,nx,nt) ! Q(:,:,i) for step i
real(r_kind) :: R(nx,nx,nt) ! R(:,:,i) from step i to step i+1
real(r_kind) :: bufQ(nx,nx)
integer :: n
call eye(nx,bufQ)
if(present(Qn)) bufQ = Qn
call bwd_qr(nx,nt,Mt,Q,R,Qn=bufQ)
do n = 1, nx
LE(n) = sum(log(abs( R(n,n,1:nt) )))
enddo
LE(:)=LE(:)/(nt*dt)
endsubroutine
!
!
!
subroutine fwd_qr(nx,nt,M,Q,R,Q0)
implicit none
integer,intent(in) :: nx
integer,intent(in) :: nt
real(r_kind),intent(in) :: M(nx,nx,nt)
real(r_kind),intent(out) :: Q(nx,nx,nt)
real(r_kind),intent(out) :: R(nx,nx,nt)
real(r_kind),intent(in),optional :: Q0(nx,nx)
real(r_kind) :: bufQ(nx,nx), bufM(nx,nx)
integer :: n
call eye(nx,Q(:,:,1)) ! identiy matrix
if (present(Q0)) Q(:,:,1)=Q0
do n = 1, nt
bufM = matmul(M(:,:,n),Q(:,:,n)) ! evolve from step i -> step i+1
call qr(nx,bufM,bufQ,R(:,:,n)) ! get Q at step i+1, & R from step i->step i+1
if (n<nt) Q(:,:,n+1) = bufQ
enddo
endsubroutine
!
!
!
subroutine bwd_qr(nx,nt,Mt,Q,R,Qn)
implicit none
integer,intent(in) :: nx
integer,intent(in) :: nt
real(r_kind),intent(in) :: Mt(nx,nx,nt)
real(r_kind),intent(out) :: Q(nx,nx,nt)
real(r_kind),intent(out) :: R(nx,nx,nt)
real(r_kind),intent(in),optional :: Qn(nx,nx)
real(r_kind) :: bufQ(nx,nx), bufMt(nx,nx)
integer :: n
call eye(nx,bufQ) ! identiy matrix
if (present(Qn)) bufQ=Qn
do n = nt, 1, -1
bufMt = matmul(Mt(:,:,n),bufQ) ! evolve from step i+1 -> step i
call qr(nx,bufMt,bufQ,R(:,:,n)) ! get Q at step i, & R from step i+1 ->step i
Q(:,:,n) = bufQ
enddo
endsubroutine
endmodule