mirror of https://gitlab.com/QEF/q-e.git
453 lines
15 KiB
Fortran
453 lines
15 KiB
Fortran
subroutine lmder(fcn,m,n,x,fvec,fjac,ldfjac,ftol,xtol,gtol,
|
|
* maxfev,diag,mode,factor,nprint,info,nfev,njev,
|
|
* ipvt,qtf,wa1,wa2,wa3,wa4)
|
|
integer m,n,ldfjac,maxfev,mode,nprint,info,nfev,njev
|
|
integer ipvt(n)
|
|
double precision ftol,xtol,gtol,factor
|
|
double precision x(n),fvec(m),fjac(ldfjac,n),diag(n),qtf(n),
|
|
* wa1(n),wa2(n),wa3(n),wa4(m)
|
|
c **********
|
|
c
|
|
c subroutine lmder
|
|
c
|
|
c the purpose of lmder is to minimize the sum of the squares of
|
|
c m nonlinear functions in n variables by a modification of
|
|
c the levenberg-marquardt algorithm. the user must provide a
|
|
c subroutine which calculates the functions and the jacobian.
|
|
c
|
|
c the subroutine statement is
|
|
c
|
|
c subroutine lmder(fcn,m,n,x,fvec,fjac,ldfjac,ftol,xtol,gtol,
|
|
c maxfev,diag,mode,factor,nprint,info,nfev,
|
|
c njev,ipvt,qtf,wa1,wa2,wa3,wa4)
|
|
c
|
|
c where
|
|
c
|
|
c fcn is the name of the user-supplied subroutine which
|
|
c calculates the functions and the jacobian. fcn must
|
|
c be declared in an external statement in the user
|
|
c calling program, and should be written as follows.
|
|
c
|
|
c subroutine fcn(m,n,x,fvec,fjac,ldfjac,iflag)
|
|
c integer m,n,ldfjac,iflag
|
|
c double precision x(n),fvec(m),fjac(ldfjac,n)
|
|
c ----------
|
|
c if iflag = 1 calculate the functions at x and
|
|
c return this vector in fvec. do not alter fjac.
|
|
c if iflag = 2 calculate the jacobian at x and
|
|
c return this matrix in fjac. do not alter fvec.
|
|
c ----------
|
|
c return
|
|
c end
|
|
c
|
|
c the value of iflag should not be changed by fcn unless
|
|
c the user wants to terminate execution of lmder.
|
|
c in this case set iflag to a negative integer.
|
|
c
|
|
c m is a positive integer input variable set to the number
|
|
c of functions.
|
|
c
|
|
c n is a positive integer input variable set to the number
|
|
c of variables. n must not exceed m.
|
|
c
|
|
c x is an array of length n. on input x must contain
|
|
c an initial estimate of the solution vector. on output x
|
|
c contains the final estimate of the solution vector.
|
|
c
|
|
c fvec is an output array of length m which contains
|
|
c the functions evaluated at the output x.
|
|
c
|
|
c fjac is an output m by n array. the upper n by n submatrix
|
|
c of fjac contains an upper triangular matrix r with
|
|
c diagonal elements of nonincreasing magnitude such that
|
|
c
|
|
c t t t
|
|
c p *(jac *jac)*p = r *r,
|
|
c
|
|
c where p is a permutation matrix and jac is the final
|
|
c calculated jacobian. column j of p is column ipvt(j)
|
|
c (see below) of the identity matrix. the lower trapezoidal
|
|
c part of fjac contains information generated during
|
|
c the computation of r.
|
|
c
|
|
c ldfjac is a positive integer input variable not less than m
|
|
c which specifies the leading dimension of the array fjac.
|
|
c
|
|
c ftol is a nonnegative input variable. termination
|
|
c occurs when both the actual and predicted relative
|
|
c reductions in the sum of squares are at most ftol.
|
|
c therefore, ftol measures the relative error desired
|
|
c in the sum of squares.
|
|
c
|
|
c xtol is a nonnegative input variable. termination
|
|
c occurs when the relative error between two consecutive
|
|
c iterates is at most xtol. therefore, xtol measures the
|
|
c relative error desired in the approximate solution.
|
|
c
|
|
c gtol is a nonnegative input variable. termination
|
|
c occurs when the cosine of the angle between fvec and
|
|
c any column of the jacobian is at most gtol in absolute
|
|
c value. therefore, gtol measures the orthogonality
|
|
c desired between the function vector and the columns
|
|
c of the jacobian.
|
|
c
|
|
c maxfev is a positive integer input variable. termination
|
|
c occurs when the number of calls to fcn with iflag = 1
|
|
c has reached maxfev.
|
|
c
|
|
c diag is an array of length n. if mode = 1 (see
|
|
c below), diag is internally set. if mode = 2, diag
|
|
c must contain positive entries that serve as
|
|
c multiplicative scale factors for the variables.
|
|
c
|
|
c mode is an integer input variable. if mode = 1, the
|
|
c variables will be scaled internally. if mode = 2,
|
|
c the scaling is specified by the input diag. other
|
|
c values of mode are equivalent to mode = 1.
|
|
c
|
|
c factor is a positive input variable used in determining the
|
|
c initial step bound. this bound is set to the product of
|
|
c factor and the euclidean norm of diag*x if nonzero, or else
|
|
c to factor itself. in most cases factor should lie in the
|
|
c interval (.1,100.).100. is a generally recommended value.
|
|
c
|
|
c nprint is an integer input variable that enables controlled
|
|
c printing of iterates if it is positive. in this case,
|
|
c fcn is called with iflag = 0 at the beginning of the first
|
|
c iteration and every nprint iterations thereafter and
|
|
c immediately prior to return, with x, fvec, and fjac
|
|
c available for printing. fvec and fjac should not be
|
|
c altered. if nprint is not positive, no special calls
|
|
c of fcn with iflag = 0 are made.
|
|
c
|
|
c info is an integer output variable. if the user has
|
|
c terminated execution, info is set to the (negative)
|
|
c value of iflag. see description of fcn. otherwise,
|
|
c info is set as follows.
|
|
c
|
|
c info = 0 improper input parameters.
|
|
c
|
|
c info = 1 both actual and predicted relative reductions
|
|
c in the sum of squares are at most ftol.
|
|
c
|
|
c info = 2 relative error between two consecutive iterates
|
|
c is at most xtol.
|
|
c
|
|
c info = 3 conditions for info = 1 and info = 2 both hold.
|
|
c
|
|
c info = 4 the cosine of the angle between fvec and any
|
|
c column of the jacobian is at most gtol in
|
|
c absolute value.
|
|
c
|
|
c info = 5 number of calls to fcn with iflag = 1 has
|
|
c reached maxfev.
|
|
c
|
|
c info = 6 ftol is too small. no further reduction in
|
|
c the sum of squares is possible.
|
|
c
|
|
c info = 7 xtol is too small. no further improvement in
|
|
c the approximate solution x is possible.
|
|
c
|
|
c info = 8 gtol is too small. fvec is orthogonal to the
|
|
c columns of the jacobian to machine precision.
|
|
c
|
|
c nfev is an integer output variable set to the number of
|
|
c calls to fcn with iflag = 1.
|
|
c
|
|
c njev is an integer output variable set to the number of
|
|
c calls to fcn with iflag = 2.
|
|
c
|
|
c ipvt is an integer output array of length n. ipvt
|
|
c defines a permutation matrix p such that jac*p = q*r,
|
|
c where jac is the final calculated jacobian, q is
|
|
c orthogonal (not stored), and r is upper triangular
|
|
c with diagonal elements of nonincreasing magnitude.
|
|
c column j of p is column ipvt(j) of the identity matrix.
|
|
c
|
|
c qtf is an output array of length n which contains
|
|
c the first n elements of the vector (q transpose)*fvec.
|
|
c
|
|
c wa1, wa2, and wa3 are work arrays of length n.
|
|
c
|
|
c wa4 is a work array of length m.
|
|
c
|
|
c subprograms called
|
|
c
|
|
c user-supplied ...... fcn
|
|
c
|
|
c minpack-supplied ... dpmpar,enorm,lmpar,qrfac
|
|
c
|
|
c fortran-supplied ... dabs,dmax1,dmin1,dsqrt,mod
|
|
c
|
|
c argonne national laboratory. minpack project. march 1980.
|
|
c burton s. garbow, kenneth e. hillstrom, jorge j. more
|
|
c
|
|
c **********
|
|
integer i,iflag,iter,j,l
|
|
double precision actred,delta,dirder,epsmch,fnorm,fnorm1,gnorm,
|
|
* one,par,pnorm,prered,p1,p5,p25,p75,p0001,ratio,
|
|
* sum,temp,temp1,temp2,xnorm,zero
|
|
double precision dpmpar,enorm
|
|
data one,p1,p5,p25,p75,p0001,zero
|
|
* /1.0d0,1.0d-1,5.0d-1,2.5d-1,7.5d-1,1.0d-4,0.0d0/
|
|
c
|
|
c epsmch is the machine precision.
|
|
c
|
|
epsmch = dpmpar(1)
|
|
c
|
|
info = 0
|
|
iflag = 0
|
|
nfev = 0
|
|
njev = 0
|
|
c
|
|
c check the input parameters for errors.
|
|
c
|
|
if (n .le. 0 .or. m .lt. n .or. ldfjac .lt. m
|
|
* .or. ftol .lt. zero .or. xtol .lt. zero .or. gtol .lt. zero
|
|
* .or. maxfev .le. 0 .or. factor .le. zero) go to 300
|
|
if (mode .ne. 2) go to 20
|
|
do 10 j = 1, n
|
|
if (diag(j) .le. zero) go to 300
|
|
10 continue
|
|
20 continue
|
|
c
|
|
c evaluate the function at the starting point
|
|
c and calculate its norm.
|
|
c
|
|
iflag = 1
|
|
call fcn(m,n,x,fvec,fjac,ldfjac,iflag)
|
|
nfev = 1
|
|
if (iflag .lt. 0) go to 300
|
|
fnorm = enorm(m,fvec)
|
|
c
|
|
c initialize levenberg-marquardt parameter and iteration counter.
|
|
c
|
|
par = zero
|
|
iter = 1
|
|
c
|
|
c beginning of the outer loop.
|
|
c
|
|
30 continue
|
|
c
|
|
c calculate the jacobian matrix.
|
|
c
|
|
iflag = 2
|
|
call fcn(m,n,x,fvec,fjac,ldfjac,iflag)
|
|
njev = njev + 1
|
|
if (iflag .lt. 0) go to 300
|
|
c
|
|
c if requested, call fcn to enable printing of iterates.
|
|
c
|
|
if (nprint .le. 0) go to 40
|
|
iflag = 0
|
|
if (mod(iter-1,nprint) .eq. 0)
|
|
* call fcn(m,n,x,fvec,fjac,ldfjac,iflag)
|
|
if (iflag .lt. 0) go to 300
|
|
40 continue
|
|
c
|
|
c compute the qr factorization of the jacobian.
|
|
c
|
|
call qrfac(m,n,fjac,ldfjac,.true.,ipvt,n,wa1,wa2,wa3)
|
|
c
|
|
c on the first iteration and if mode is 1, scale according
|
|
c to the norms of the columns of the initial jacobian.
|
|
c
|
|
if (iter .ne. 1) go to 80
|
|
if (mode .eq. 2) go to 60
|
|
do 50 j = 1, n
|
|
diag(j) = wa2(j)
|
|
if (wa2(j) .eq. zero) diag(j) = one
|
|
50 continue
|
|
60 continue
|
|
c
|
|
c on the first iteration, calculate the norm of the scaled x
|
|
c and initialize the step bound delta.
|
|
c
|
|
do 70 j = 1, n
|
|
wa3(j) = diag(j)*x(j)
|
|
70 continue
|
|
xnorm = enorm(n,wa3)
|
|
delta = factor*xnorm
|
|
if (delta .eq. zero) delta = factor
|
|
80 continue
|
|
c
|
|
c form (q transpose)*fvec and store the first n components in
|
|
c qtf.
|
|
c
|
|
do 90 i = 1, m
|
|
wa4(i) = fvec(i)
|
|
90 continue
|
|
do 130 j = 1, n
|
|
if (fjac(j,j) .eq. zero) go to 120
|
|
sum = zero
|
|
do 100 i = j, m
|
|
sum = sum + fjac(i,j)*wa4(i)
|
|
100 continue
|
|
temp = -sum/fjac(j,j)
|
|
do 110 i = j, m
|
|
wa4(i) = wa4(i) + fjac(i,j)*temp
|
|
110 continue
|
|
120 continue
|
|
fjac(j,j) = wa1(j)
|
|
qtf(j) = wa4(j)
|
|
130 continue
|
|
c
|
|
c compute the norm of the scaled gradient.
|
|
c
|
|
gnorm = zero
|
|
if (fnorm .eq. zero) go to 170
|
|
do 160 j = 1, n
|
|
l = ipvt(j)
|
|
if (wa2(l) .eq. zero) go to 150
|
|
sum = zero
|
|
do 140 i = 1, j
|
|
sum = sum + fjac(i,j)*(qtf(i)/fnorm)
|
|
140 continue
|
|
gnorm = dmax1(gnorm,dabs(sum/wa2(l)))
|
|
150 continue
|
|
160 continue
|
|
170 continue
|
|
c
|
|
c test for convergence of the gradient norm.
|
|
c
|
|
if (gnorm .le. gtol) info = 4
|
|
if (info .ne. 0) go to 300
|
|
c
|
|
c rescale if necessary.
|
|
c
|
|
if (mode .eq. 2) go to 190
|
|
do 180 j = 1, n
|
|
diag(j) = dmax1(diag(j),wa2(j))
|
|
180 continue
|
|
190 continue
|
|
c
|
|
c beginning of the inner loop.
|
|
c
|
|
200 continue
|
|
c
|
|
c determine the levenberg-marquardt parameter.
|
|
c
|
|
call lmpar(n,fjac,ldfjac,ipvt,diag,qtf,delta,par,wa1,wa2,
|
|
* wa3,wa4)
|
|
c
|
|
c store the direction p and x + p. calculate the norm of p.
|
|
c
|
|
do 210 j = 1, n
|
|
wa1(j) = -wa1(j)
|
|
wa2(j) = x(j) + wa1(j)
|
|
wa3(j) = diag(j)*wa1(j)
|
|
210 continue
|
|
pnorm = enorm(n,wa3)
|
|
c
|
|
c on the first iteration, adjust the initial step bound.
|
|
c
|
|
if (iter .eq. 1) delta = dmin1(delta,pnorm)
|
|
c
|
|
c evaluate the function at x + p and calculate its norm.
|
|
c
|
|
iflag = 1
|
|
call fcn(m,n,wa2,wa4,fjac,ldfjac,iflag)
|
|
nfev = nfev + 1
|
|
if (iflag .lt. 0) go to 300
|
|
fnorm1 = enorm(m,wa4)
|
|
c
|
|
c compute the scaled actual reduction.
|
|
c
|
|
actred = -one
|
|
if (p1*fnorm1 .lt. fnorm) actred = one - (fnorm1/fnorm)**2
|
|
c
|
|
c compute the scaled predicted reduction and
|
|
c the scaled directional derivative.
|
|
c
|
|
do 230 j = 1, n
|
|
wa3(j) = zero
|
|
l = ipvt(j)
|
|
temp = wa1(l)
|
|
do 220 i = 1, j
|
|
wa3(i) = wa3(i) + fjac(i,j)*temp
|
|
220 continue
|
|
230 continue
|
|
temp1 = enorm(n,wa3)/fnorm
|
|
temp2 = (dsqrt(par)*pnorm)/fnorm
|
|
prered = temp1**2 + temp2**2/p5
|
|
dirder = -(temp1**2 + temp2**2)
|
|
c
|
|
c compute the ratio of the actual to the predicted
|
|
c reduction.
|
|
c
|
|
ratio = zero
|
|
if (prered .ne. zero) ratio = actred/prered
|
|
c
|
|
c update the step bound.
|
|
c
|
|
if (ratio .gt. p25) go to 240
|
|
if (actred .ge. zero) temp = p5
|
|
if (actred .lt. zero)
|
|
* temp = p5*dirder/(dirder + p5*actred)
|
|
if (p1*fnorm1 .ge. fnorm .or. temp .lt. p1) temp = p1
|
|
delta = temp*dmin1(delta,pnorm/p1)
|
|
par = par/temp
|
|
go to 260
|
|
240 continue
|
|
if (par .ne. zero .and. ratio .lt. p75) go to 250
|
|
delta = pnorm/p5
|
|
par = p5*par
|
|
250 continue
|
|
260 continue
|
|
c
|
|
c test for successful iteration.
|
|
c
|
|
if (ratio .lt. p0001) go to 290
|
|
c
|
|
c successful iteration. update x, fvec, and their norms.
|
|
c
|
|
do 270 j = 1, n
|
|
x(j) = wa2(j)
|
|
wa2(j) = diag(j)*x(j)
|
|
270 continue
|
|
do 280 i = 1, m
|
|
fvec(i) = wa4(i)
|
|
280 continue
|
|
xnorm = enorm(n,wa2)
|
|
fnorm = fnorm1
|
|
iter = iter + 1
|
|
290 continue
|
|
c
|
|
c tests for convergence.
|
|
c
|
|
if (dabs(actred) .le. ftol .and. prered .le. ftol
|
|
* .and. p5*ratio .le. one) info = 1
|
|
if (delta .le. xtol*xnorm) info = 2
|
|
if (dabs(actred) .le. ftol .and. prered .le. ftol
|
|
* .and. p5*ratio .le. one .and. info .eq. 2) info = 3
|
|
if (info .ne. 0) go to 300
|
|
c
|
|
c tests for termination and stringent tolerances.
|
|
c
|
|
if (nfev .ge. maxfev) info = 5
|
|
if (dabs(actred) .le. epsmch .and. prered .le. epsmch
|
|
* .and. p5*ratio .le. one) info = 6
|
|
if (delta .le. epsmch*xnorm) info = 7
|
|
if (gnorm .le. epsmch) info = 8
|
|
if (info .ne. 0) go to 300
|
|
c
|
|
c end of the inner loop. repeat if iteration unsuccessful.
|
|
c
|
|
if (ratio .lt. p0001) go to 200
|
|
c
|
|
c end of the outer loop.
|
|
c
|
|
go to 30
|
|
300 continue
|
|
c
|
|
c termination, either normal or user imposed.
|
|
c
|
|
if (iflag .lt. 0) info = iflag
|
|
iflag = 0
|
|
if (nprint .gt. 0) call fcn(m,n,x,fvec,fjac,ldfjac,iflag)
|
|
return
|
|
c
|
|
c last card of subroutine lmder.
|
|
c
|
|
end
|