LMDER_(3)

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💖 LMDER_(3) 💖

okay bestie, so we're in section 3 of the Library Functions Manual and today's main character is lmder_ (plus its glow-up little sibling lmder1_). buckle up 🎀

✨ NAME ✨

lmder_, lmder1_ — minimize the sum of squares of m nonlinear functions, with user supplied Jacobian

🦋 SYNOPSIS 🦋

include <minpack.h>

void lmder1_ ( void (*fcn) (int *m, int *n, double *x, double *fvec,
                            double *fjac,
                            int *ldfjac, int *iflag),
               int *m, int * n, double *x, double *fvec, double *fjac,
               int *ldfjac,
               double *tol, int *info, int *iwa, double *wa, int *lwa);

void lmder_ ( void (*fcn)( int *m, int *n, double *x, double *fvec,
                           double *fjac,
                           int *ldfjac, int *iflag),
              int *m, int *n, double *x, double *fvec, double *fjac, int
              *ldfjac,
              double *ftol, double *xtol, double *gtol, int *maxfev,
              double *diag, int *mode,
              double *factor, int *nprint, int *info,
              int *nfev, int *njev, int *ipvt,
              double *qtf, double *wa1, double *wa2, double *wa3, double
              *wa4 );

💅 DESCRIPTION 💅

so the whole point of lmder_ is to minimize the sum of the squares of m nonlinear functions in n variables — and it does that with a modification of the Levenberg-Marquardt algorithm. slay. 🧚 the catch? you have to hand it a function that computes both the functions and the Jacobian. no shortcuts, bestie, you're doing the math.

lmder1_ does the exact same thing as lmder_ but with a simplified calling sequence — like the low-effort version but same results. 💖

lmstr and lmstr1 also perform the same function but use minimal storage. (storage girlies, this one's for you.)

🌈 Language notes 🌈

these functions are written in FORTRAN. so if you're calling from C, keep these points in mind, fr fr:

Name mangling. with gfortran, all the function names end in an underscore. hence the trailing _ in lmder_. iconic.

Compile with gfortran. even if your program is all C code, you should link with gfortran so it will pull in the FORTRAN libraries automatically. it's easiest just to use gfortran to do all the compiling. (it handles C just fine.) no cap.

Call by reference. all function parameters must be pointers. every single one. don't forget the &. 💀

Column-major arrays. suppose a function returns an array with 5 rows and 3 columns in an array z and in the call you have declared a leading dimension of 7. the FORTRAN and equivalent C references are:

     z(1,1)         z[0]
     z(2,1)         z[1]
     z(5,1)         z[4]
     z(1,2)         z[7]
     z(1,3)         z[14]
     z(i,j)         z[(i-1) + (j-1)*7]

yes it's giving chaos but it's consistent chaos. 💖

🎀 User-supplied Function 🎀

fcn is the name of the user-supplied subroutine which calculates the functions. in FORTRAN, 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,fjac,ldfjac,iflag)
integer m,n,iflag
double precision x(n),fvec(m),fjac(ldfjac,n)
----------
if iflag = 1 calculate the functions at x and
return this vector in fvec. do not alter fjac.
if iflag = 2 calculate the jacobian at x and
return this matrix in fjac. do not alter fvec.
----------
return
end

in C, fcn should be written as follows:

void fcn(int *m, int *n, double *x, double *fvec, double *fjac,
         int *ldfjac, int *iflag)
{
  /* if iflag = 1 calculate the functions at x and return this
     vector in fvec[0] through fvec[m-1]. do not alter fjac.
     if iflag = 2 calculate the jacobian at x and return this
     matrix in fjac. do not alter fvec. */
}

the value of iflag should not be changed by fcn unless the user wants to terminate execution of lmder_ (or lmder1_). in this case set iflag to a negative integer. that's your emergency brake, bestie. 🛑

💜 Parameters for both lmder_ and lmder1_ 💜

  • m — a positive integer input variable set to the number of functions.
  • n — a positive integer input variable set to the number of variables. n must not exceed m. that's the rule, no exceptions.
  • x — 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. glow-up arc. ✨
  • fvec — an output array of length m which contains the functions evaluated at the output x.
  • fjac — 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.

  • ldfjac — a positive integer input variable not less than m which specifies the leading dimension of the array fjac.

🌸 Parameters for lmder1_ 🌸

  • tol — a nonnegative input variable. termination occurs when the algorithm estimates either that the relative error in the sum of squares is at most tol or that the relative error between x and the solution is at most tol.
  • info — an integer output variable. if the user has terminated execution, info is set to the (negative) value of iflag. see description of fcn. otherwise, info is set as follows.
  info = 0  improper input parameters.

  info = 1  algorithm estimates that the relative error in the sum of
squares is at most tol.

  info = 2  algorithm estimates that the relative error between x and
the solution is at most tol.

  info = 3  conditions for info = 1 and info = 2 both hold.

  info = 4  fvec is orthogonal to the columns of the Jacobian to machine
precision.

  info = 5  number of calls to fcn has reached or exceeded 200*(n+1).

  info = 6  tol is too small. no further reduction in the sum of squares
is possible.

  info = 7  tol is too small. no further improvement in the approximate
solution x is possible.
  • iwa — an integer work array of length n.
  • wa — a work array of length lwa.
  • lwa — an integer input variable not less than m*n + 5*n + m for lmder1_.

🔥 Parameters for lmder_ 🔥

  • ftol — 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.
  • xtol — 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.
  • gtol — 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.
  • maxfev — a positive integer input variable. termination occurs when the number of calls to fcn is at least maxfev by the end of an iteration.
  • diag — 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.
  • mode — 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.
  • factor — 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 the latter is nonzero, or else to factor itself. in most cases factor should lie in the interval (.1,100.). 100. is a generally recommended value. 👑
  • nprint — 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.
  • info — an integer output variable. if the user has terminated execution, info is set to the (negative) value of iflag. see description of fcn. otherwise, info is set as follows.
  info = 0  improper input parameters.

  info = 1  both actual and predicted relative reductions in the sum of
squares are at most ftol.

  info = 2  relative error between two consecutive iterates is at most
xtol.

  info = 3  conditions for info = 1 and info = 2 both hold.

  info = 4  the cosine of the angle between fvec and any column of the
Jacobian is at most gtol in absolute value.

  info = 5  number of calls to fcn has reached or exceeded maxfev.

  info = 6  ftol is too small. No further reduction in the sum of
squares is possible.

  info = 7  xtol is too small. No further improvement in the approximate
solution x is possible.

  info = 8 gtol is too small. fvec is orthogonal to the columns of the
Jacobian to machine precision.
  • nfev — an integer output variable set to the number of calls to fcn with iflag = 1.
  • njev — an integer output variable set to the number of calls to fcn with iflag = 2.
  • ipvt — an integer output array of length n. ipvt defines a permutation matrix p such that jac*p = q*r, 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.
  • qtf — an output array of length n which contains the first n elements of the vector (q transpose)*fvec.
  • wa1, wa2, and wa3 — work arrays of length n.
  • wa4 — a work array of length m.

🫶 SEE ALSO 🫶

lmdif(3), lmdif1(3), lmstr(3), lmstr1(3).

👑 AUTHORS 👑

Jorge More', Burt Garbow, and Ken Hillstrom at Argonne National Laboratory. this manual page was written by Jim Van Zandt <jrv@debian.org>, for the Debian GNU/Linux system (but may be used by others). 🫶

and that's the whole page, bestie 💖 this one's from Minpack, dated March 8, 2002, and it lives under LMDER_(3) — go minimize something iconic ✨🎀🦋

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