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ConjugateGradients.f90
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Module Vars
Implicit Real(8) (A-H,O-Z)
PARAMETER (NMAX=50)
Real(8) X0(nmax),P(nmax)
Integer(4) n,ifunc/0/,igrad/0/,iFuncType
End module
!********************************************
program ConjugateGradients
Use Vars, Only: n,nmax,iFuncType,ifunc,igrad
Implicit Real(8) (A-H,O-Z)
Real(8) X(nmax),G(nmax)
Open(6,File='ConGrad.out')
iFuncType=3 ! 1-Quadratic, 2-Rosenbrock, 3-Powell
n=2
If (iFuncType==3) n=4
X(1)= 3.d0
X(2)=-1.d0
X(3)= 0.d0
X(4)= 1.d0
Method=0
Call ConGrad(Method,X,n,iter,F,G,gnorm) ! Conjugate gradients (Mode: 0 - Polak-Ribiere version, 1 - Fletcher-Reeves version, -1 - Steepest Descent)
Write(6,'(//'' Optimization finished with Niter ='',i6,'' Nfunc ='',i8,'' Ngrad ='',i8)')iter,ifunc,igrad
Write(6,'('' Final F ='',f12.6,'' Final Gnorm ='',f12.6)')F,Gnorm
Write(6,'('' X ='',<n>f10.5)')(X(j),j=1,n)
Write(6,'('' G ='',<n>f10.5)')(G(j),j=1,n)
Close(6)
end program
!*********************************************************************
Subroutine CompFG(n,XX,F,G)
Use Vars, Only: iFuncType
Implicit Real(8) (A-H,O-Z)
Real(8) XX(n),G(n)
If (iFuncType==1) Then ! Quadratic F=x**2+y**2
x=XX(1)
y=XX(2)
F=x*x+y*y
G(1)=2*x
G(2)=2*y
Elseif (iFuncType==2) Then ! Rosenbrock function (Xmin=(a, a**2), Fmin=0), hard starting point (-1.2,1)
x=XX(1)
y=XX(2)
a=1.d0
b=100.d0
F=(a-x)**2 + b*(y-x*x)**2
G(1)=-2.d0*(a-x)-4.d0*b*x*(y-x*x)
G(2)=2.d0*b*(y-x*x)
ElseIf (iFuncType==3) Then !Powell function (xmin=(0,0,0,0) F=0, Hard case X0=(3,-1,0,1))
x1=XX(1)
x2=XX(2)
x3=XX(3)
x4=XX(4)
F= (x1+10.d0*x2)**2 + 5.d0*(x3-x4)**2 + (x2-2.d0*x3)**4 + 10.d0*(x1-x4)**4
G(1)=2.d0*(x1 + 10.d0*x2) + 40.d0*(x1 - x4)**3
G(2)=20.d0*(x1 + 10.d0*x2) + 4.d0*(x2 - 2*x3)**3
G(3)=-8.d0*(x2 - 2.d0*x3)**3 + 10.d0*(x3 - x4)
G(4)=-40.d0*(x1 - x4)**3 - 10.d0*(x3 - x4)
Endif
End
!**********************************************************************
SUBROUTINE ConGrad(Mode,X,n,iter,fret,G,gnorm)
Use Vars, Only: nmax,ifunc
Implicit Real(8) (A-H,O-Z)
EXTERNAL func
PARAMETER (ITMAX=10000,EPS=1.d-10)
Real(8) X(n),P(n),H(n),G(n)
! Conjugate gradients / steepest descent unconstrained optimization program
! Mode: 0 - Conjugate gradients (Polak-Ribiere)
! 1 - Conjugate gradients (Fletcher-Reeves)
! -1 - Steepest descent
! Stopping criteria
ftol=1.d-6
gtol=1.d-6
! F/G calculation at the initial point
fp=func(X)
call dfunc(X,G)
! Initialize search direction and gradients
P(1:n)=-G(1:n)
H(1:n)=P(1:n)
G(1:n)=H(1:n)
do iter=1,ITMAX ! Main loop
! Linear search
call linmin(X,G,n,fret,xmin)
! F update and G calculation at the point located during linear search
fchange=2.d0*Dabs(fret-fx)/(Dabs(fret)+Dabs(fx)+EPS)
fp=fret
call dfunc(X,G)
Write(6,'('' Iter:'',i6,'' FCN ='',i8,'' F ='',f12.5,'' X ='',<n>f8.4,'' G ='',<n>f8.4)')iter,ifunc,fret,(X(j),j=1,n),(G(j),j=1,n)
! Check for stopping criterion on F
if (fchange<=FTOL) Then
Write(6,'(/'' Stopping criteria on Delta-F satisfied in CG program: Relative-Delta-F ='',4f20.12)')fchange
return
Endif
! Prepare gradient corrections
gg=0.d0
dgg=0.d0
do j=1,n
If (Mode==-1) Then
gg=gg+G(j)**2 ! Steepest descent
G(j)=-G(j)
Cycle
Endif
gg=gg+P(j)**2
If (Mode==0) Then
dgg=dgg+(G(j)+P(j))*G(j) ! CG Polak-Ribiere formula
ElseIf (Mode==1) Then
dgg=dgg+G(j)**2 ! CG Fletcher-Reeves formula
Endif
enddo
gnorm=Dsqrt(gg)
! Check for stopping criterion on G
if (gg<gtol) Then
Write(6,'(/'' Stopping criteria on Gnorm satisfied in CG program. Gnorm ='',4f20.12)')gnorm
Return
Endif
If (Mode==-1) Cycle
! Update search direction and store gradients for CG
gam=dgg/gg
do j=1,n
P(j)=-G(j)
H(j)=P(j)+gam*H(j)
G(j)=H(j)
enddo
enddo
Write(6,'(''CG maximum iterations exceeded: '',i8)')ItMax
End
!**********************************************************
Function Func(X)
Use Vars, Only: n,ifunc
Implicit Real(8) (A-H,O-Z)
Real(8) X(n),G(n)
Call CompFG(n,X,Func,G)
ifunc=ifunc+1
Write(6,'(i5,f12.5,<n>f10.4)')ifunc,Func,(X(j),j=1,n)
End
!**********************************************************************
Subroutine DFunc(X,G)
Use Vars, only: n,igrad
Implicit Real(8) (A-H,O-Z)
Real(8) X(n),G(n)
Call CompFG(n,X,F,G)
igrad=igrad+1
End
!**********************************************************************
Function f1dim(x)
Use Vars
Implicit Real(8) (A-H,O-Z)
Real(8) X1(n)
X1(1:n)=X0(1:n)+x*P(1:n)
f1dim=func(X1)
END
!**********************************************************
Function df1dim(x)
Use Vars
Implicit Real(8) (A-H,O-Z)
Real(8) dF(n),X1(n)
X1(1:n)=X0(1:n)+x*P(1:n)
call dfunc(X1,dF)
df1dim=0.d0
do i=1,n
df1dim=df1dim+dF(i)*P(i) !/pnorm**2
enddo
End
!**********************************************************
Subroutine linmin(X0,P,n,fret,xmin)
Use Vars, Only: X0com=>X0,Pcom=>P,nmax
Implicit Real(8) (A-H,O-Z)
Real(8) X0(n),P(n)
PARAMETER (TOL=1.d-5) !Maximum anticipated n, and TOL passed to brent.
EXTERNAL f1dim,df1dim
ncom=n !Set up the common block.
do i=1,n
X0com(i)=X0(i)
Pcom(i)=P(i)
enddo
ax=0.d0 !Initial guess for brackets.
xx=0.01d0
call mnbrak(ax,xx,bx,fa,fx,fb,f1dim) ! Bracketing the interval with a minimum
fret=dbrent(ax,xx,bx,f1dim,df1dim,tol,xmin) ! Find 1D-minimum at [ax,bx]
do i=1,n !Construct the vector results to return.
P(i)=xmin*P(i)
X0(i)=X0(i)+P(i)
enddo
End
!**********************************************************
Subroutine mnbrak(ax,bx,cx,fa,fb,fc,func)
Implicit Real(8) (A-H,O-Z)
EXTERNAL func
PARAMETER (GOLD=1.618034d0, GLIMIT=100.d0, TINY=1.d-20)
fa=func(ax)
fb=func(bx)
if(fb.gt.fa)then !Switch roles of a and b so that we can go downhill in the
dum=ax !direction from a to b.
ax=bx
bx=dum
dum=fb
fb=fa
fa=dum
endif
cx=bx+GOLD*(bx-ax) !First guess for c.
fc=func(cx)
1 if(fb.ge.fc)then !\do while": keep returning here until we bracket.
r=(bx-ax)*(fb-fc) !Compute u by parabolic extrapolation from a; b; c. TINY
q=(bx-cx)*(fb-fa) !is used to prevent any possible division by zero.
u=bx-((bx-cx)*q-(bx-ax)*r)/(2.d0*dsign(dmax1(dabs(q-r),TINY),q-r))
ulim=bx+GLIMIT*(cx-bx) !We won't go farther than this. Test various possibilities:
if((bx-u)*(u-cx).gt.0.d0)then !Parabolic u is between b and c: try it.
fu=func(u)
if(fu.lt.fc)then !Got a minimum between b and c.
ax=bx
fa=fb
bx=u
fb=fu
return
else if(fu.gt.fb)then !Got a minimum between between a and u.
cx=u
fc=fu
return
endif
u=cx+GOLD*(cx-bx) !Parabolic t was no use. Use default magni cation.
fu=func(u)
else if((cx-u)*(u-ulim).gt.0.d0)then !Parabolic t is between c and its allowed
fu=func(u) !limit.
if(fu.lt.fc)then
bx=cx
cx=u
u=cx+GOLD*(cx-bx)
fb=fc
fc=fu
fu=func(u)
endif
else if((u-ulim)*(ulim-cx).ge.0.d0)then !Limit parabolic u to maximum allowed value.
u=ulim
fu=func(u)
else !Reject parabolic u, use default magni cation.
u=cx+GOLD*(cx-bx)
fu=func(u)
endif
ax=bx !Eliminate oldest point and continue.
bx=cx
cx=u
fa=fb
fb=fc
fc=fu
goto 1
endif
End
!********************************************************
FUNCTION dbrent(ax,bx,cx,f,df,tol,xmin)
Implicit Real(8) (A-H,O-Z)
EXTERNAL df,f
PARAMETER (ITMAX=100,ZEPS=1.0d-6)
LOGICAL ok1,ok2 !Will be used as flags for whether proposed steps are acceptable or not
a=dmin1(ax,cx)
b=dmax1(ax,cx)
v=bx
w=v
x=v
e=0.d0
fx=f(x)
fv=fx
fw=fx
dx=df(x)
dv=dx
dw=dx
do iter=1,ITMAX
xm=0.5d0*(a+b)
tol1=tol*Dabs(x)+ZEPS
tol2=2.d0*tol1
if(dabs(x-xm).le.(tol2-0.5d0*(b-a))) goto 3
if(dabs(e).gt.tol1) then
d1=2.d0*(b-a) !Initialize these d's to an out-of-bracket value.
d2=d1
if(dw.ne.dx) d1=(w-x)*dx/(dx-dw) !Secant method with one point.
if(dv.ne.dx) d2=(v-x)*dx/(dx-dv) !And the other.
u1=x+d1
u2=x+d2
ok1=((a-u1)*(u1-b).gt.0.d0).and.(dx*d1.le.0.d0)
ok2=((a-u2)*(u2-b).gt.0.d0).and.(dx*d2.le.0.d0)
olde=e !Movement on the step before last.
e=d
if(.not.(ok1.or.ok2))then !Take only an acceptable d, and if both
goto 1
else if (ok1.and.ok2)then
if(dabs(d1).lt.dabs(d2))then
d=d1
else
d=d2
endif
else if (ok1) then
d=d1
else
d=d2
endif
if(dabs(d).gt.dabs(0.5d0*olde))goto 1
u=x+d
if( (u-a).lt.tol2 .or. (b-u).lt.tol2 ) d=dsign(tol1,xm-x)
goto 2
endif
1 if(dx.ge.0.d0) then !Decide which segment by the sign of the derivative.
e=a-x
else
e=b-x
endif
d=0.5d0*e !Bisect, not golden section.
2 if(dabs(d).ge.tol1) then
u=x+d
fu=f(u)
else
u=x+dsign(tol1,d)
fu=f(u)
if(fu.gt.fx)goto 3 !If the minimum step in the downhill direction takes us uphill,
Endif
du=df(u) !Now all the housekeeping, sigh.
if (fu.le.fx) then
if(u.ge.x) then
a=x
else
b=x
endif
v=w
fv=fw
dv=dw
w=x
fw=fx
dw=dx
x=u
fx=fu
dx=du
else
if(u.lt.x) then
a=u
else
b=u
endif
if(fu.le.fw .or. w.eq.x) then
v=w
fv=fw
dv=dw
w=u
fw=fu
dw=du
else if(fu.le.fv .or. v.eq.x .or. v.eq.w) then
v=u
fv=fu
dv=du
endif
endif
enddo
Write(6,'(''dbrent exceeded maximum iterations'')')
3 xmin=x
dbrent=fx
End