is a positive integer input variable set to the number of functions.
is a positive integer input variable set to the number of variables. n must not exceed m.
is an array of length n. on input x must contain an initial estimate of the solution vector. on output x contains the final estimate of the solution vector.
is an output array of length m which contains the functions evaluated at the output x.
is a nonnegative input variable. termination occurs when both the actual and predicted relative reductions in the sum of squares are at most ftol. therefore, ftol measures the relative error desired in the sum of squares.
is a nonnegative input variable. termination occurs when the relative error between two consecutive iterates is at most xtol. therefore, xtol measures the relative error desired in the approximate solution.
is a nonnegative input variable. termination occurs when the cosine of the angle between fvec and any column of the jacobian is at most gtol in absolute value. therefore, gtol measures the orthogonality desired between the function vector and the columns of the jacobian.
is a positive integer input variable. termination occurs when the number of calls to fcn is at least maxfev by the end of an iteration.
is an input variable used in determining a suitable step length for the forward-difference approximation. this approximation assumes that the relative errors in the functions are of the order of epsfcn. if epsfcn is less than the machine precision, it is assumed that the relative errors in the functions are of the order of the machine precision.
is an array of length n. if mode = 1 (see below), diag is internally set. if mode = 2, diag must contain positive entries that serve as multiplicative scale factors for the variables.
is an integer input variable. if mode = 1, the variables will be scaled internally. if mode = 2, the scaling is specified by the input diag. other values of mode are equivalent to mode = 1.
is a positive input variable used in determining the initial step bound. this bound is set to the product of factor and the euclidean norm of diag*x if nonzero, or else to factor itself. in most cases factor should lie in the interval (.1,100.). 100. is a generally recommended value.
is an integer input variable that enables controlled printing of iterates if it is positive. in this case, fcn is called with iflag = 0 at the beginning of the first iteration and every nprint iterations thereafter and immediately prior to return, with x and fvec available for printing. if nprint is not positive, no special calls of fcn with iflag = 0 are made.
is an output m by n array. the upper n by n submatrix of fjac contains an upper triangular matrix r with diagonal elements of nonincreasing magnitude such that
t t t p (jac *jac)p = r *r,
where p is a permutation matrix and jac is the final calculated jacobian. column j of p is column ipvt(j) (see below) of the identity matrix. the lower trapezoidal part of fjac contains information generated during the computation of r.
is a positive integer input variable not less than m which specifies the leading dimension of the array fjac.
is an integer output array of length n. ipvt defines a permutation matrix p such that jacp = qr, where jac is the final calculated jacobian, q is orthogonal (not stored), and r is upper triangular with diagonal elements of nonincreasing magnitude. column j of p is column ipvt(j) of the identity matrix.
is an output array of length n which contains the first n elements of the vector (q transpose)*fvec.
is a work array of length m.
subprograms called
user-supplied ...... fcn, jacFcn
minpack-supplied ... dpmpar,enorm,fdjac2,lmpar,qrfac
fortran-supplied ... dabs,dmax1,dmin1,dsqrt,mod
argonne national laboratory. minpack project. march 1980. burton s. garbow, kenneth e. hillstrom, jorge j. more
is the name of the user-supplied subroutine which calculates the functions. fcn must be declared in an external statement in the user calling program, and should be written as follows.
subroutine fcn(m,n,x,fvec,iflag) integer m,n,iflag double precision x(n),fvec(m)
calculate the functions at x and return this vector in fvec.
return end
the value of iflag should not be changed by fcn unless the user wants to terminate execution of lmdif. in this case set iflag to a negative integer.
subroutine lmdif
the purpose of lmdif is to minimize the sum of the squares of m nonlinear functions in n variables by a modification of the levenberg-marquardt algorithm. the user must provide a subroutine which calculates the functions. the jacobian is then calculated by a forward-difference approximation.
the subroutine statement is
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)
where