https://github.com/cran/nleqslv
Tip revision: 29fbf340091cecba11cd5ffd76935f5c7c6a6e70 authored by Berend Hasselman on 26 September 2010, 00:00:00 UTC
version 1.8
version 1.8
Tip revision: 29fbf34
nwbrdn.f
subroutine brsolv(ldr,xc,n,scalex,maxit,
* jacflg,xtol,ftol,btol,global,xscalm,
* stepmx,dlt,sigma,
* rjac,wrk1,wrk2,wrk3,wrk4,fc,fq,dn,d,qtf,
* rcdwrk,icdwrk,qrwork,qrwsiz,
* epsm,fjac,fvec,outopt,xp,fp,gp,njcnt,nfcnt,
* termcd)
integer ldr,n,termcd,njcnt,nfcnt
integer maxit,jacflg,global,xscalm,qrwsiz
integer outopt(*)
double precision xtol,ftol,btol,stepmx,dlt,sigma,fpnorm,epsm
double precision rjac(ldr,*)
double precision xc(*),fc(*),xp(*),fp(*),dn(*),d(*)
double precision wrk1(*),wrk2(*),wrk3(*),wrk4(*)
double precision qtf(*),gp(*),fq(*)
double precision scalex(*)
double precision rcdwrk(*),qrwork(*)
integer icdwrk(*)
external fjac,fvec
c-----------------------------------------------------------------------
c
c Solve system of nonlinear equations with Broyden and global strategy
c
c
c Arguments
c
c In ldr Integer leading dimension of rjac
c In xc Real(*) initial estimate of solution
c In n Integer dimensions of problem
c Inout scalex Real(*) scaling factors x(*)
c In maxit Integer maximum number of allowable iterations
c In jacflg Integer jacobian flag
c 1 if analytic jacobian supplied
c 0 if analytic jacobian not supplied
c In xtol Real tolerance at which successive iterates x()
c are considered close enough to
c terminate algorithm
c In ftol Real tolerance at which function values f()
c are considered close enough to zero
c Inout btol Real x tolerance for backtracking
c In global Integer global strategy to use
c 0 quadratic line search
c 1 geometric line search
c 2 double dogleg
c 3 single dogleg
c In xscalm Integer x scaling method
c 1 from column norms of first jacobian
c increased if needed after first iteration
c 0 scaling user supplied
c In stepmx Real maximum allowable step size
c In dlt Real trust region radius
c In sigma Real reduction factor geometric linesearch
c Inout rjac Real(ldr,*) jacobian (2*n columns)
c Wk wrk1 Real(*) workspace
c Wk wrk2 Real(*) workspace
c Wk wrk3 Real(*) workspace
c Wk wrk4 Real(*) workspace
c Inout fc Real(*) function values f(xc)
c Wk fq Real(*) workspace
c Wk dn Real(*) workspace
c Wk d Real(*) workspace
c Wk qtf Real(*) workspace
c Wk rcdwrk Real(*) workspace
c Wk icdwrk Integer(*) workspace
c In qrwork Real(*) workspace for Lapack QR routines (call nwqmem)
c In qrwsiz Integer size of qrwork
c In epsm Real machine precision
c In fjac Name name of routine to calculate jacobian
c (optional)
c In fvec Name name of routine to calculate f()
c In outopt Integer(*) output options
c Out xp Real(*) final x()
c Out fp Real(*) final f(xp)
c Out gp Real(*) gradient at xp()
c Out njcnt Integer number of jacobian evaluations
c Out nfcnt Integer number of function evaluations
c Out termcd Integer termination code
c
c-----------------------------------------------------------------------
integer iter,gcnt,retcd,ierr
double precision dum(2),dlt0,fcnorm,cond
logical mxtake
logical jacevl,jacupd
integer priter
integer idamax
double precision Rzero, Rone
parameter(Rzero=0.0d0, Rone=1.0d0)
c initialization
retcd = 0
iter = 0
njcnt = 0
nfcnt = 0
ierr = 0
dum(1) = 0
dlt0 = dlt
if( outopt(1) .eq. 1 ) then
priter = 1
else
priter = -1
endif
c evaluate function
call vscal(n,xc,scalex)
call nwfvec(xc,n,scalex,fvec,fc,fcnorm,wrk1)
c evaluate analytic or finite difference jacobian and check analytic
c jacobian, if requested
if(jacflg .eq. 1) then
if( outopt(2) .eq. 1 ) then
call nwfjac(xc,scalex,fc,fq,n,epsm,jacflg,fvec,fjac,rjac,
* ldr,wrk1)
call chkjac(rjac,ldr,xc,fc,n,epsm,scalex,
* fq,wrk1,fvec,termcd)
if(termcd .lt. 0) return
endif
endif
c check stopping criteria for input xc
call nwtcvg(xc,fc,xc,xtol,retcd,ftol,iter,maxit,n,ierr,termcd)
if(termcd .gt. 0) then
call dcopy(n,xc,1,xp,1)
call dcopy(n,fc,1,fp,1)
fpnorm = fcnorm
return
endif
if( priter .gt. 0 ) then
dum(1) = fcnorm
dum(2) = abs(fc(idamax(n,fc,1)))
if( global .le. 1 ) then
call nwlsot(iter,-1,dum)
elseif( global .eq. 2 ) then
call nwdgot(iter,-1,dum)
elseif( global .eq. 3 ) then
call nwpwot(iter,-1,dum)
endif
endif
jacevl = .true.
do while( termcd .eq. 0 )
iter = iter+1
if( jacevl ) then
c - evaluate the jacobian at the current iterate xc
c - evaluate the gradient at the current iterate xc
call nwfjac(xc,scalex,fc,fq,n,epsm,jacflg,fvec,fjac,rjac,
* ldr,wrk1)
njcnt = njcnt + 1
c - if requested calculate x scale from jacobian column norms a la Minpack
if( xscalm .eq. 1 ) then
call vunsc(n,xc,scalex)
call nwcpsx(n,rjac,ldr,scalex,epsm,iter)
call vscal(n,xc,scalex)
endif
call nwscjac(n,rjac,ldr,scalex)
c gp = trans(Rjac) * fc
call dgemv('T',n,n,Rone,rjac,ldr,fc,1,Rzero,gp,1)
c - get newton step
c - form Q from the QR decomposition (taur/qraux in wrk1) (simple Lapack routine)
call dcopy(n,fc,1,fq,1)
call nwndir(rjac,ldr,rjac(1,n+1),fq,n,epsm,jacflg,
* wrk1,wrk2,wrk3,wrk4,dn,qtf,ierr,cond,
* rcdwrk,icdwrk,qrwork,qrwsiz)
call nwsnot(0,ierr,cond)
if( ierr .eq. 0 ) then
call liqrqq(rjac,ldr,wrk1,n,qrwork,qrwsiz,ierr)
endif
c now Rjac(* ,1..n) holds expanded Q
c now Rjac(n+1,1..n) holds full upper triangle R
else
c - get broyden step
c - calculate approximate gradient
call dcopy(n,fc,1,fq,1)
call brodir(rjac,ldr,rjac(1,n+1),fq,n,epsm,jacflg,
* wrk1,wrk2,wrk3,dn,qtf,ierr,cond,
* rcdwrk,icdwrk)
call nwsnot(1,ierr,cond)
if( ierr .eq. 0 ) then
call dcopy(n,qtf,1,gp,1)
call dtrmv('U','T','N',n,rjac(1,n+1),ldr,gp,1)
endif
endif
c - choose the next iterate xp by a global strategy
if( ierr .gt. 0 ) then
c jacobian singular or too ill-conditioned
call dcopy(n,xc,1,xp,1)
call dcopy(n,fc,1,fp,1)
fpnorm = fcnorm
gcnt = 0
if( priter .gt. 0 ) then
call nwjerr(iter)
endif
elseif(global .eq. 0) then
call nwqlsh(n,xc,fcnorm,dn,gp,stepmx,btol,scalex,
* fvec,xp,fp,fpnorm,wrk1,mxtake,retcd,gcnt,
* priter,iter)
elseif(global .eq. 1) then
call nwglsh(n,xc,fcnorm,dn,gp,sigma,stepmx,btol,scalex,
* fvec,xp,fp,fpnorm,wrk1,mxtake,retcd,gcnt,
* priter,iter)
elseif(global .eq. 2) then
call nwddlg(n,rjac(1,n+1),ldr,dn,gp,xc,fcnorm,stepmx,
* btol,mxtake,dlt,qtf,scalex,
* fvec,d,fq,wrk1,wrk2,wrk3,wrk4,
* xp,fp,fpnorm,retcd,gcnt,priter,iter)
elseif(global .eq. 3) then
call nwpdlg(n,rjac(1,n+1),ldr,dn,gp,xc,fcnorm,stepmx,
* btol,mxtake,dlt,qtf,scalex,
* fvec,d,fq,wrk1,wrk2,wrk3,wrk4,
* xp,fp,fpnorm,retcd,gcnt,priter,iter)
endif
nfcnt = nfcnt + gcnt
c - check stopping criteria for the new iterate xp
call nwtcvg(xp,fp,xc,xtol,retcd,ftol,iter,maxit,n,ierr,termcd)
if( termcd .eq. 3 .and. .not. jacevl ) then
c global strategy failed but jacobian is out of date
c try again with proper jacobian
c reset trust region radius
jacevl = .true.
jacupd = .false.
dlt = dlt0
termcd = 0
elseif(termcd .gt. 0) then
jacupd = .false.
else
jacupd = .true.
jacevl = .false.
endif
if( jacupd ) then
c perform Broyden update of current jacobian
c update xc, fc, and fcnorm
call brupdt(n,rjac,rjac(1,n+1),ldr,xc,xp,fc,fp,epsm,
* scalex,wrk1,wrk2,wrk3)
call dcopy(n,xp,1,xc,1)
call dcopy(n,fp,1,fc,1)
fcnorm = fpnorm
endif
enddo
call vunsc(n,xp,scalex)
return
end
c-----------------------------------------------------------------------
subroutine brupdt(n,q,r,ldr,xc,xp,fc,fp,epsm,scalex,dx,df,wa)
integer n,ldr
double precision q(ldr,*),r(ldr,*)
double precision xc(*),xp(*),fc(*),fp(*),dx(*),df(*),wa(*)
double precision scalex(*)
double precision epsm
c-----------------------------------------------------------------------
c
c Calculate new Q and R from rank-1 update with xp-xc and fp-fc
c using Broyden method
c
c Arguments
c
c In n Integer size of xc() etc.
c Inout Q Real(ldr,n) orthogonal matrix Q from QR
c On output updated Q
c Inout R Real(ldr,n) upper triangular R from QR
c On output updated R
c In ldr Integer leading dimension of Q and R
c In xc Real(*) current x() values
c In xp Real(*) new x() values
c In fc Real(*) current f(xc)
c In fp Real(*) new f(xp)
c In epsm Real machine precision
c Wk dx Real(*) workspace
c Wk df Real(*) workspace
c Wk wa Real(*) workspace
c
c-----------------------------------------------------------------------
integer i
double precision eta,sts
double precision dnrm2
logical doupdt
double precision Rzero, Rone, Rtwo, Rhund
parameter(Rzero=0.0d0, Rone=1.0d0, Rtwo=2.0d0, Rhund=100d0)
eta = Rhund * Rtwo * epsm
doupdt = .false.
do i=1,n
dx(i) = xp(i) - xc(i)
df(i) = fp(i) - fc(i)
enddo
c clear lower triangle
do i=1,n-1
call nuzero(n-i,r(i+1,i))
enddo
c calculate df - B*dx = df - Q*R*dx
c wa = R*dx
c df = df - Q*(R*dx) (!not really needed if qrupdt were to be changed)
c do not update with noise
call dcopy(n,dx,1,wa,1)
call dtrmv('U','N','N',n,r,ldr,wa,1)
call dgemv('N',n,n,-Rone,q,ldr,wa,1,Rone,df,1)
do i=1,n
if( abs(df(i)) .gt. eta*( abs(fp(i)) + abs(fc(i)) ) ) then
doupdt = .true.
else
df(i) = Rzero
endif
enddo
if( doupdt ) then
c equation 8.3.1 from Dennis and Schnabel (page 187)(Siam edition)
sts = dnrm2(n,dx,1)
call dscal(n,Rone/sts,dx,1)
call dscal(n,Rone/sts,df,1)
call liqrup(q,ldr,n,r,ldr,df,dx,wa)
endif
return
end
c-----------------------------------------------------------------------
subroutine brodir(q,ldr,r,fn,n,epsm,jacflg,y,w,wa,dn,qtf,
* ierr,rcond,rcdwrk,icdwrk)
integer ldr,n,ierr,jacflg
double precision epsm,q(ldr,*),r(ldr,*),fn(*)
double precision wa(*),dn(*),y(*),w(*),qtf(*)
double precision rcdwrk(*)
integer icdwrk(*)
double precision rcond
c-----------------------------------------------------------------------
c
c Calculate the approximate newton direction
c
c Arguments
c
c Inout Q Real(ldr,*) Q part from QR at current iterate
c In ldr Integer leading dimension of Q and R
c In R Real(ldr,*) upper triangular R from QR decomposition
c In fn Real(*) function values at current iterate
c In n Integer dimension of problem
c In epsm Real machine precision
c In jacflg Integer jacobian flag
c 1 for analytic
c 0 for numeric
c used for condition estimate
c Wk y Real(*) workspace
c Wk w Real(*) workspace
c Wk wa Real(*) workspace
c Out dn Real(*) Newton direction
c Out qtf Real(*) trans(Q)*f()
c Out ierr Integer 0 indicating Jacobian not ill-conditioned or singular
c 1 indicating Jacobian ill-conditioned
c 2 indicating Jacobian completely singular
c Out rcond Real inverse condition matrix
c Wk rcdwrk Real(*) workspace
c Wk icdwrk Integer(*) workspace
c
c QR decomposition with no pivoting.
c
c-----------------------------------------------------------------------
double precision Rzero, Rone
parameter(Rzero=0.0d0, Rone=1.0d0)
c check for singularity or ill conditioning
call cndjac(n,r,ldr,epsm,rcond,y,rcdwrk,icdwrk,ierr)
if( ierr .ne. 0 ) then
return
endif
c form qtf = trans(Q) * fn
call dgemv('T',n,n,Rone,q,ldr,fn,1,Rzero,qtf,1)
c solve rjac*dn = -fn
c ==> R*dn = - qtf
call dcopy(n,qtf,1,dn,1)
call dtrsv('U','N','N',n,r,ldr,dn,1)
call dscal(n, -Rone, dn, 1)
return
end
