LMDIF_(3)

💖 LMDIF_(3) 💖

okay bestie, this is the lmdif_ page from the Library Functions Manual, and it's literally the whole nonlinear-least-squares squad in one place 🫶 grab a snack, we're doing this together.

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✨ NAME ✨

lmdif_, lmdif1_ — minimize the sum of squares of m nonlinear functions

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🦋 SYNOPSIS 🦋

include <minpack.h>

void lmdif1_ ( void (*fcn)(int *m, int *n, double *x, double *fvec, int
               *iflag),
  int *m, int * n, double *x, double *fvec,
  double *tol, int *info, int *iwa, double *wa, int *lwa);

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

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🌸 DESCRIPTION 🌸

ok so lmdif_'s whole thing is minimizing the sum of the squares of m nonlinear functions in n variables, by a modification of the Levenberg-Marquardt algorithm. You, the user, have to hand it a subroutine that actually computes the functions. Then the Jacobian gets calculated by a forward-difference approximation. no cap.

lmdif1_ is giving the exact same purpose, just with a simplified calling sequence — the low-effort, high-reward bestie of the two 💅

💅 Language notes 💅

these functions are written in FORTRAN. 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.

🔥 Compile with gfortran

even if your program is 100% C code, you should link with gfortran so it'll pull in the FORTRAN libraries automatically. honestly it's easiest to just use gfortran to do all the compiling — it handles C just fine, it's giving versatile 👑

🫶 Call by reference

all function parameters must be pointers. every single one. periodt.

🌈 Column-major arrays

suppose a function returns an array with 5 rows and 3 columns in an array z, and in the call you've 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]

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,iflag)
integer m,n,iflag
double precision x(n),fvec(m)
----------
calculate the functions at x and
return this vector in fvec.
----------
return
end

in C, fcn should be written as follows:

void fcn(int *m, int *n, double *x, double *fvec, int *iflag)
{
       /* calculate the functions at x and return
          the values in fvec[0] through fvec[m-1] */
}

the value of iflag should not be changed by fcn unless the user wants to terminate execution of lmdif_ (or lmdif1_). in that case, set iflag to a negative integer. that's the abort button, bestie 🛑

⭐ Parameters for both lmdif_ and lmdif1_ ⭐

• 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. (the math said no to anything else 🙈)
• 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.

🍓 Parameters for lmdif1_ 🍓

• 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. double slay.
  • 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.

💜 Parameters for lmdif_ 💜

• 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.
• epsfcn — 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.
• 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. iconic.
  • 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.
• 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.
• 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.

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🧚 SEE ALSO 🧚

lmder(3), lmder1(3), lmstr(3), lmstr1(3).

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👑 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). give them their flowers 💐

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and that's the whole page, bestie — no cap. it came from Minpack, dated March 8, 2002, and it's the LMDIF_(3) manual page in all its glory. go forth and least-squares, you absolute main character 🦋💖