subroutine lmdif(fcn,m,n,x,fvec,ftol,xtol,gtol,maxfev,epsfcn,
* diag,mode,factor,nprint,info,nfev,fjac,ldfjac,
* ipvt,qtf,wa1,wa2,wa3,wa4)
integer m,n,maxfev,mode,nprint,info,nfev,ldfjac
integer ipvt(n)
double precision ftol,xtol,gtol,epsfcn,factor
double precision x(n),fvec(m),diag(n),fjac(ldfjac,n),qtf(n),
* wa1(n),wa2(n),wa3(n),wa4(m)
external fcn
c **********
c
c subroutine lmdif
c
c the purpose of lmdif 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. the jacobian is
c then calculated by a forward-difference approximation.
c
c the subroutine statement is
c
c subroutine lmdif(fcn,m,n,x,fvec,ftol,xtol,gtol,maxfev,epsfcn,
c diag,mode,factor,nprint,info,nfev,fjac,
c ldfjac,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. fcn must be declared
c in an external statement in the user calling
c program, and should be written as follows.
c
c subroutine fcn(m,n,x,fvec,iflag)
c integer m,n,iflag
c double precision x(n),fvec(m)
c ----------
c calculate the functions at x and
c return this vector in 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 lmdif.
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 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 is at least
c maxfev by the end of an iteration.
c
c epsfcn is an input variable used in determining a suitable
c step length for the forward-difference approximation. this
c approximation assumes that the relative errors in the
c functions are of the order of epsfcn. if epsfcn is less
c than the machine precision, it is assumed that the relative
c errors in the functions are of the order of the machine
c precision.
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 and fvec available
c for printing. 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 has reached or
c exceeded 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.
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 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,fdjac2,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
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,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 fdjac2(fcn,m,n,x,fvec,fjac,ldfjac,iflag,epsfcn,wa4)
nfev = nfev + n
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,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,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,iflag)
return
c
c last card of subroutine lmdif.
c
end