Gold Medal Software 3/Gold Medal Software - Volume 3 (Gold Medal) (1994).iso/stats/lsqrft15.arj/FIT.HLP

Text File | 1994-02-11 | 32KB | 869 lines

INSTALLATION

 Unzip the zip file in a suitable directory.  Move
 fit2.exe somewhere in your path.  If you want the online
 help to work, set the environment variable "fithelp" to
 point to the file fit.hlp.  For example:

 set fithelp=d:\docs\fit.hlp

 Under Unix, you should use setenv, something like:

 setenv FITHELP /u/docs/fit.hlp

 Adding the line to your config.sys (under OS/2) or
 autoexec.bat (under DOS) or .cshrc would keep you from
 having to type it in every time you reboot or login.
 The gnuplot.exe program should be somewhere in your path.
 The fit program requires you have gplt33b2.zip or later
 under OS/2.  Earlier versions work under Unix. This file
 is available from ftp.cdrom.com and includes complete
 documentation.  As of 10/1/93 the most recent version of
 gnuplot is 3.5.

 Under OS/2 2.x,  you should move all the *.DLL files to
 a directory in your LIBPATH.

 For use on platforms other than OS/2 2.x or DOS, you will
 have to compile. Have a look at the makefile and make it
 suitable for your compiler. You might want to make the
 executable fit2 instead of fit2.exe. You should also use
 the compiler option -DUNIX (instead of -DOS2), or whatever
 option will define UNIX for your compiler.  I recommend
 using the GCC compiler for the most painless compilation
 on any UNIX platform, but other compilers have been shown
 to work as well.  You might get some warnings.

 Under DOS, you should use the DJGPP port of GCC.  Borland
 C++ will not work.  To run in a DOS box under Windows, you
 must use version 1.10 or higher of DJGPP.  Specify the
 -DDOS option when you compile under DOS.  Plotting is
 disabled in the DOS version.  You will have to write the
 fit to a temporary file, and plot with the plotting
 package of your choice.  DJGPP is not the most usable DOS
 compiler, but it is free, has a 32 bit DOS extender, and
 is good for porting UNIX programs.  Consult the DJGPP
 documentation and the *.msdos.programmer newsgroups for
 more info on DJGPP.  You must have the file go32.exe in
 your path for the DOS version to work.

 -DUNIX affects the command used to open gnuplot.  Gnuplot
 writes some error messages to stderr and some of these
 make it to the screen.  If -DUNIX is specified, stderr is
 redirected to the file gnuout.  If you want to see the
 error messages, do not use -DUNIX.  Additionally, some
 gnuplot terminal types require stderr to go to the screen.
 If you have one of these, do not specify -DUNIX; you will
 have to live with the gnuplot error messages.  This is the
 case for vt100 terminals.  If the -DUNIX option is not
 specified, temporary files will be written to the current
 directory rather than /tmp.

END

NOTE

 FIT2  a non-linear fitting program for OS/2 2.x.  If you
 are using UNIX, you might try OS/2 2.1, where this program
 was developed and intended for use.

 Version 1.5 beta

 Written by Michael Courtney, servant of the Kingdom of God.

 Jesus Christ can take away your fear.

 I hope you find this program useful, but there are no warranties
 and no guarantees.

 Send bug reports, comments and suggestions to [e-mail]
END

GENERAL

 Fit2 is a non-linear least-squares fitting program. It uses
 the Levenberg-Marquardt method to attempt to minimize the
 least-squares error between a data file and a model
 function.  See the books, DATA REDUCTION AND ERROR ANALYSIS
 IN THE PHYSICAL SCIENCES by P. R. Bevington and NUMERICAL
 RECIPES IN C by W. H. Press et. al. for more information
 on this method.

 Due to the generality of the fitting function allowed,
 and the possible weirdness of the error function in
 parameter space, convergance to the true minimum is not
 guaranteed.  Therefore, it is important to have good
 initial guesses for the parameters.  It is a good idea to
 graph your data and the fitting function with your
 initial guesses, and tweak the initial guesses until the
 fitting function is close to the data.  Most functions
 with three or four parameters will converge quickly if the
 initial guesses are in the right ballpark.  If you have
 more than four parameters, you may have to tell the program
 to vary only four at a time (the sp command).  When the
 parameters you are varying affect a subset of the data
 much more than the rest, you should window the data
 accordingly (the wi command). If these tricks do not work,
 check and make sure that the function is computing the
 proper value of the function and derivatives with respect
 to the parameters.  You have to check your source for this.
 A sign error is disasterous.  It will cause the guess for
 the parameter to be further from the correct value than
 the current value.

 The Levenberg-Marquardt method will work for functions
 which are linear in the parameters.  It is not as fast as
 linear regression, and it is not guaranteed to converge to
 an absolute minimum.  However, it will do in a pinch and
 it is actually a little more robust if your basis functions
 are nearly linearly dependent.  Of course, if your basis
 functions are nearly linearly independent then your
 parameters will be strongly correlated, and you will have
 to sort what is meaningful and what is not.

 This program was written by Michael Courtney.  I consider
 this a beta version of the program.  I retain all rights
 to the program. You may use the program and alter the
 source to fit your needs. You may not distribute altered
 source or executable unless you obtain my permission.  You
 may port this program to other platforms if you wish.  It
 should compile and run almost unaltered on the NeXT
 platform and most Unix platforms.  You may not sell the
 program or source or receive any money for its
 distribition.  The program is provided "as is" and is
 without warranty of guarantee of any kind.  The author
 assumes no responsibility of any kind for anything the
 program does. If you use this program regularly, it is
 required that you send me email and let me know you are
 using it.  No payment is required.

 Please report bugs and suggestions to Michael Courtney
 at [e-mail].  I will consider bugs to have
 a higher priority than suggestions, although suggestions
 are welcome.

 I am considering the following future enhancements to the
 program:

 1. Multiple data sets -- the ability to simultaneously fit
 multiple data sets to different functions which share
 common parameters.  Currently, you can effectively
 accomplish this, see Multiple Data Sets.

 2. Ability to handle binary files.

 3. Command line history.

 Please let me know if any of these enhancements are of
 interest to you.
END

IMPROVEMENTS and CHANGES

 This is the 1.5 beta version of the program.

 The improvements over the 1.4 beta are the following:

 1. Ability to take commands from a file.  See the do command.

 2. fit2 can now be compiled with Cset++ under OS/2 2.x.

 3. More bug fixes.

 4. The noset command.

 The improvements over the 1.3 beta are the following:

 1. New plotting routine under OS/2 allows viewing
 of a plot of every attempted fit.  See the fp command.

 2. Minor bug fixes.

 3. A simple way to estimate errors in the parameters.
 See the command er.

 The improvements over the 1.2 beta are the following:

 1. Linear Least-Squares Fitting

 2. Minor bug fixes.

 3. Can change one parameter at a time.  Se the cp command.

END

COMMANDS

        The following can be executed at the "fit>" prompt:

        name     meaning

        help
        quit
        fn       function
        ip       initialize parameters
        cp       change one parameter
        sp       select parameters (to vary)
        gd       get data
        fi       fit
        pp       print parameters
        wp       write parameters (to a file)
        wf       write fit (data to a file)
        rp       read parameters (from a file)
        wt       weight (for fitting)
        co       covariant matrix
        order    order (of data colums in file)
        ad       allocate data array
        lf       lists functions available
        gr       graphing flag
        plot     plot data and fit
        load     load a gnuplot command file
        set      set something in gnuplot
        noset    stops fit2 from setting things in gnuplot
        wi       windows data
        ve       changes verbosity
        pr       plot residual errors
        sh       shows current settings
        md       makes data
        de       debug
        ru       run a program
        gn       sends a command to gnuplot
        li       linear least squares fit
        pa       pauses for a number of seconds
        fp       turns the use of fitplot.exe on and off
        er       estimate errors in the parameters
        do       takes commands from a file
END

FILES

        fit2 opens and stores data in several files.
 fit.tmp is used if the function is plotted to a file.
 fit.dat is used as a default filename if none is specified
 with the wf command. The wp command stores the parameters
 in the file a.dat. In addition, the text output of gnuplot
 which is sent to stderr is written in the file gnuout.
 On non-unix machines, fit2 does not erase any of these
 files, unless it overwites them.  On UNIX machines fit.tmp
 and gnuout are written in the /tmp directory.  They are
 erased ewhen the program exits, unless debugging is on.
 If the debug flag is non-zero, lamda is written to a file
 called lamda.sts on every successful iteration.  lamda
 controls how much the paramerers are changed in attempting
 to reduce chisqr.
END

fn     Usage: fn name

        fn chooses a function for fitting.

        Example: "fn gauss" chooses to fit to a gaussian.
END

ip     Usage: ip a0 a1 a2 ...

        ip initializes the fitting parameters.  If the number
 of parameters is changed, the parameters are initialized to 5
 when the fn command is  executed.  ip can be used at any time to
 change the values of the parameters. If you want to leave some
 parameters unchanged,enter a * in place of the number.  You must
 enter either a number or a * for all of the parameters.

        Example: "ip 13.7 * 12" changes a0 to 13.7, leaves
 a1 the same, and changes a2 to 12 for a three parameter fit.
END

cp        Usage: cp # #

        cp changes the parameter specified by the first
 number to the value specified by the second number.  The
 first number must be in the range of parameters ( >= 0 and
 less than the number of parameters).

        Example "cp 3 3.141" changes a3 to 3.141 and leaves
 all the other parameters unchanged.
END

sp     Usage: sp # # #

        sp selects which parameters to vary.  By default,
 all parameters are selected to be varied when the fn
 command is executed.  If you only wanted to vary
 parameters a0 and a2 of a three parameter fit, you would
 issue the command:

        Example: "sp 0 2"
END

gd     Usage: gd filename

        gd geta the data from a file.  Data should be in an
 ascii file with between two and 20 columns. For fitting
 functions of more than one independent variable, select a
 function with the number of independent variables that you
 want to fit to before reading in data.  This will allow the
 program to default to interpreting columns 1 through n
 (where n is the number of independent variables of the
 current function) as representing the independent variables
 in order.  By default, the (n+1)th column is interpreted to
 be y(x0 ... xn) and the (n+2)th column is interpreted as
 the experimental error in the measured y.  If you wish to
 assign the columns in your data file differently, or if you
 read in the data before choosing a function (for more than
 1 independent variable), you will have to use the order
 command.

 By default, if no function has been selected, or if a
 function of one independent variable has been selected,
 the first column is x, the second is y, and if there is a
 third, it is assumed to be the error in y.  If you wish to
 assign different columns to be x, y, and delta y, use the
 order command.  By default, the program can handle a file
 with five columns and 1024 rows.  The ad command can be
 used to allocate the data array differently and handle up
 to 23 columns and as many rows as memory allows.

 Lines in a data file beginning with the '#' character are taken
 to be comments.

END

fi     Usage: fi [iterations]

        fi fits the data using the non-linear fitting
 algorithm. Iterations is the  number of iterations between
 plots.  If no number it given, it defaults the last number
 used.  If none was ever entered, it defaults to 20. After
 this number of iterations, the data and fit are plotted, if
 graphing if selected.  (It is by default.  See the gr
 command.)  You are then given the option to quit iterating
 (q enter), continue iterating and turn on graphing (g enter),
 continue iterating and turn off graphing
 (n enter), or continue with the same graphing status.

 If the fi command is executed from a command file, the
 fit stops after the selected number of iterations and
 does not prompt for user input.  This is to facilitate
 batch processing.

 There is no way in general to be sure chisqr is at an
 absolute minimum. It might just be at a local minimum.
 Starting the fit with several sets of values of initial
 parameters and having every fit that converges find the
 same set of final values and same chisqr gives some
 confidence. Looking at the graph at least tells us we're
 not too far away.

 It is possible to estimate the errors in the parameters
 once chisqr is minimized.  Rather than build this into the
 program for people to use with more confidence than
 warranted, I merely make all the information you need
 available. You should look at the references and figure it
 out.  This way you will have an idea of how unreliable
 error estimates on the parameters can be.
END

li      Usage: li

        li fits the data using a linear fitting algorithm.
 No iteration is required since the best-fit parameters are
 uniquely determined by the data and basis functions.
 The linear fitting algorithm used is the simplest possible.
 A matrix equation is set up and solved by Gauss-Jordan
 elimination.  A singular matrix will cause a failure of the
 method and probably indicates a linearly dependent basis set.
 A near-singular matrix will cause inaccuracies in parameter
 values.

 li will usually work very well for less than 6 or so basis
 functions and reasonably well for up to about 10 basis
 functions. If you have problems with singular matrices,
 doubt the accuracy of your parameters, or want to
 incorporate errors in the independent parameters, I suggest
 that you use the li to obtain initial guesses and then use
 fi for to find the final parameter values.  fi will usually
 converge quickly if you start with initial guesses obtained
 from li.

 You should be aware that fitting to a subset of parameters
 with the sp command has a different effect with li than it
 does with fi.  With fi, the selected parameters are varied
 while the unselected parameters are held fixed.  With li,
 the selected parameters really define which basis functions
 are incorporated in the fit.  These are the only parameters
 found,  the other parameters are set to zero.  This may be
 inconvenient in certain situations, but it reflects a basic
 difference between linear and non-linear fitting.
END

pp     Usage: pp

        pp prints the current value of the parameters to the
 screen.
END

wp     Usage: wp [filename] [mode]

        wp writes the parameters to a file.  If no filename
 is given, parameters are written to the file "a.dat".  mode
 should be "a" (for append) or "w" for overwrite.  The
 default mode is "w".

        Examples:
 "wp" writes the parameters to the file "a.dat", overwriting
 the file if it exists.

 "wp params.dat a" writes the parameters to the file
 params.dat, appending then on to the end of the file if  it
 already exists.
END

wf     Usage: wf [filename]

        wf writes the fitted data to a file, overwriting the
 file if it exists.  If no filename is given, data is
 written to the file "fit.dat".  The fitted data is the
 value of the fitting function evaluated at the value of x
 of the current data set using the current parameters.
END

md      Usage: md filename xmin0 xmax0 xstep0 xmin1 xmax1 xstep1

        md makes a trial data set.  It's primary use is in
 testing new fitting functions.  At this time, it is only
 implemented for functions of one or two independent
 variables.  The trial data set is created using the current
 parameter values and the current fitting function.

 Examples:

 fit2> fn gauss        /* defines fitting function as gauss */
 fit2> ip 5 1 1        /* initialized parameters */
 fit2> md test 0 10 1  /* makes data for 0 <= x0 <= 10*/
 fit2> gd test         /* reads in data */
 fit2> ip 5.1 2 1      /* changes parameters */
 fit2> fi              /* test to see if fit finds real parameters */
 fit2> fn xyquad       /* defines fitting function as xyquad */
 fit2> ip 5 5 5 5 5 5  /* initialized parameters */
 fit2> md test 0 10 1 0 10 1
                      /* makes data for 0 <= x0 <=10 and 0 <= x1 <= 10*/
 fit2> gd test         /* reads in data */
 fit2> ip 1 1 1 1 1 1  /* changes parameters */
 fit2> fi              /* test to see if fit finds real parameters */
END

rp     Usage: rp [filename]

        rp reads the parameters from a file.  If no file is
 given, it attempts to read the parameters from the file
 "a.dat".
END

de      Usage: de flag

        de changes the debugging status of the program.
 Flag is 0 for no debugging, 1 for light debugging, and 2
 for heavy debugging. Only certain sections of code have
 been set up for this.  Other sections will be set up as I
 need to debug them.  This option is intended mainly for
 use of someone altering the program. If the debugging level
 is 1, all of the commands sent to gnuplot are printed,
 along with other information.
END

sh

        sh shows the value of current settings.
END

wt     Usage: wt [none stat inst other]

        The fitting function tries to minimize chisqr, where
 chisqr = sum( ((yi - f(xi))/sigmai)**2 ).  That is, chisqr
 is the sum of the squared error of the difference between
 the data and the fitting function, except that each term in
 the sum is weighted by sigmai**2.  The "wt" command chooses
 the type of weighting.  It chooses what to use for sigma.
 Errors in the independent parameters can also be used for
 non-linear fitting.  See the order command for details.
        The following weighting options are available:

 none..........no weighting, all sigma's are equal to one.
 statistical...sigmai = sqrt(yi)
 instrumental..sigmai = deltayi, as in data file
 other.........sigmai = yi

        Example: "wt s" selects statistical weighting.
END

co     Usage: co [filename]

        co prints the covariant matrix.  It will overwrite
 the file if a file of the same name exists.  If no filename
 is given, the covariant matrix is written to the screen.
 The covariant matrix can be used to estimate the error of
 the fit parameters.
END

er      Usage: er factor

        er provides a simple way to estimate the errors in the
 parameters.  The algorithm is very simple, and the answer it
 gives is intuitively satisfying, but not statistically rigorous.
 The algorithm varies one parameter at a time and finds two
 values of the parameter where chisqr is increased by the given
 factor.  An increase of 10-20% in chisqr should give a reasonable
 estimate of the errors in the parameter, so I suggest a factor
 of 1.1 or 1.2.  For factors larger than 2.0, the algorithm may
 not converge, and you answers would not make sense as error
 estimates on the parameters.  If the output is NAN or INF, try
 using a factor closer to 1.

END

order   Usage: order x0col ... xncol ycol sigycol sigx0col ... sigxncol

        order assigns significance to the columns in the
 internal data matrix.  When a data file is read with the gd
 command, the first column in the file is stored in the
 data[0].  (The 0th column of an 2-D array).  The second
 column is stored in data[1], etc.  By default, these rows
 are interpteted in a certain way (See the gd command).  The
 order command is used to change the default behavior.

 Errors in the independent variables can be used in the fit
 by using techniques of error propagation to calclate a
 corresponding error in y.  That is:

 sigy = sqrt( sigy*sigy + sigx*sigx*(dy/dx)*(dy/dx) ).

 Each data point has its own value of sigmax's and sigmay.
 Each sigx must be in a column in the data file.  The
 weighting for sigy should be statistical, instrumental or
 other if sigx is considered in the fit.  You should not
 select no weighting if you want to use sigx.  If the order
 command selects a column to be sigx, then this information
 is used for the fit.  If you wish to not use this
 information, assign sigxcol to be -1 (the default value).
        If the fitting algorithm has a problem converging
 when using sigx, try the fit without using sigx.  Then take
 those parameters as the initial parameters for the fit
 considering sigx. Note that errors in the independent
 variables are not considered during linear least squares
 fitting.

        Examples for one independent variable:

 "order 2 1 0"  x0 is the third column in the file, y is the
 second column, sigy is the first column.

 "order 1 4" x0 is the second column, y is the fifth column,
 sigy is not specified.

 "order 0 1 2 3" x0 is the first column in the file, y is
 the second column, sigy is the third column, sigx0 is the
 fourth column and is used in the fit.

 "order 0 1 2 -1" x0 is the first column in the file, y is
 the second column, sigy is the third column, sigx0 is not
 used.

        Examples for two independent variables:

 "order 2 1 0 3"  x0 is the third column in the file, x1 is
 the second column, y is the first column, sigy is the
 third column.

 "order 1 4 2" x0 is the second column, x1 is the fifth
 column, y is the third column, sigy is not specified.

 "order 0 1 2 3 4 5" x0 is the first column in the file, x1
 is the second column, y is the third column, sigy is the
 fourth, sigx0 is in the fifth column and is used in the
 fit, sigx1 is in the sixth column and is used in the fit.

 "order 0 1 2 3 -1 -1" x0 is the first column in the file,
 x1 is the second column, y is the third column, sigy is the
 fourth, sigx0 and sigx1 are not used in the fit.
END

ad     Usage: ad columns rows

        ad re-allocates the internal data matrix.  It is 8
 columns and 1024 rows by default.  You can have up to 23
 columns and as many rows as memory allows.  Three rows are
 used internally, and up to 20 rows can be used for data
 from your input file.  The ad command erases any data
 currently in memory.

        Example: ad 11 4096
END

lf     Usage: lf

        lf lists the functions available for fitting.
END

gr     Usage: gr flag

        The gr command turns the graphing on and off.  "gr 0"
 turns graphing off.  "gr 1" turns graphing on.  Graphing is
 on by default.
END

fp     Usage: fp flag

        The fp command turns the use of the fitplot.exe
 program on and off.  The fitplot.exe program is a fast
 plotting program which is used to plot the data and
 attempted fit every iteration.  This is useful if you want
 to see what the Levenberg-Marquardt algorithm is doing.
 The use of fitplot.exe is off by default, although if there
 are any command line arguments to fit2.exe, the use of
 fitplot.exe is turned on.

 Some people have found that use of this plotting program
 will hang the message queue on slower machines.  I have not
 been able to reproduce the problem.  If this occurs, I suggest
 you stick to gnuplot for plotting.
END

plot   Usage: plot

        plot plots the data and fitting function with the
 current parameters.
END

load   Usage: load filename

        load loads and executes a gnuplot command file.
END

set    Usage: set gnuplot stuff

        The entire set command is sent to gnuplot.
END

gnuplot

        Gnuplot is is an interactive plotting program. To
 use the graphing capability of the fit program, you must
 have the gnuplot program correctly installed in your system
 and it must be in a directory in the current path.
END

wi     Usage: wi [one or more numbers]

        wi selects data windowing.  It is used if you want
 to fit the data in a certain range of x values.  If wi has
 one argument, it turns windowing on and off.  "wi 1" turns
 windowing on.  "wi 0" turns windowing off.  If wi has more
 than two arguments, windowing is turned on, and the
 arguments are the minimum and maximum x values used in
 fitting.  In other words, chisqr is computed only for
 xmin <= x <= xmax.

        Examples:

 "wi 30 100" tells the fi command to perform the fit using
 only x values between 30 and 100.

 "wi 30 100 2 10" tells the fi command to perform the fit
 using only x0 values between 30 and 100 and x1 values
 between 2 and 10.
END

ve     Usage: ve flag

        If flag = 0, the fit command gives little output. If
 flag = 1, the fit command only gives the paramters every
 iteration.  If flag = 2, the output is very verbose.
END

noset  Usage: noset flag

        If flag = 0, then fit2 issues the set command to
 gnuplot often for setting ranges, labels, titles, etc.
 If flag = 1, fit2 only sends gnuplot a set command when
 absolutely necessary for performing the requested operation.
END

pr      Usage: pr flag

        pr plots residual errors.  If there is no flag or
 flag = 1, yfit vs. y is plotted.  If flag = 2, (y-yfit) vs.
 x is plotted.
END

run     Usage: run command line

        run sends a command line to the default command
 processor. It is used to run a program from within fit2.

        Example: run dir *.dat
END

gn      Usage: gn gnuplot command

        gn sends a command to the gnuplot command processor.
 It is used to execute gnuplot commands from within fit2.
 You need to send a correct gnuplot command.  You will get
 no error message if you do not.
END

pa      Usage: pa seconds

 If seconds = -1, the program is paused until enter is hit.

END

do      Usage: do file

 do executes commands from a command file.  Lines beginning with # are
 comments.  Blank lines are skipped.  All other lines are executed.
END

functions

        There are several built in functions.  Other
 functions can be added by adding them to the file
 funclib2.c and recompiling.  See that file for details.
 The built in functions are: gauss, gaussc, ngauss, lorenz,
 2lorenz, line, poly, nexp, and xyquad.
END

gauss

        The function gauss is a gaussian of the form:
 f(x) = a2*exp(-((x-a0)/a1)**2).  Fitting to this function
 converges easily if your initial guesses are even close.
 For gauss and most peak-type functions, your initial
 guesses will probably converge if your function peaks are a
 little shorter and wider than you data peaks.
END

gaussc

        The function gauss is a gaussian plus a constant.
 f(x) = a2*exp(-((x-a0)/a1)**2) + a3.  Fitting to this
 function converges easily.
END

ngauss

        This function is a sum of gaussians plus a constant.
 It uses a variable number of parameters.  The number you
 want must be chosen when you select the function.  The
 function has the form:
 f(x) = sum( a[i+2]*exp(-((x-ai)/a[i+1])**2) ) + a[n-1].
 You must choose 3*n+1 parameters, where n is the number of
 gaussians you want. This function converges slowly for more
 than 2 gaussians, but it usually does converge.  If the
 peaks in your data are separated by several widths, you
 should window the data and fit the peaks separately first,
 and then do a final fit on the whole data set.
END

lorenz

        This is a lorenzian function of the form:
 f(x) = a2/(4*(x-a0)**2 + a1).  I choose not to use the
 usual form since, I thought making the form simple in
 regards to the parameters might make for more robust
 fitting.  You have to think to eyeball the initial
 paremeters though.  If your data is a lorenzian, it will
 converge without much trouble.  If your data is not really
 a lorenzian you may have trouble.  The gaussian is a
 better choice to fit peaks that are not well characterized
 by a particular function.
END

2lorenz

        This is the sum of two lorenzians and it has the
 form: f(x) = a2/(4*(x-a0)**2 + a1) + a5/(4*(x-a3)**2 + a4).
 It sometimes takes some effort to get a fit to this
 function to converge.
END

line

        f(x) = a0 + a1*x.  Piece of cake.
END

poly

        f(x) = sum(ai*(x**i)).  Decent initial guesses and
 fourth order or less converge easily and quickly.  Higher
 orders might take some work.  Using li to obtain initial
 guesses and fi for final parameter values is best for
 polynomials higher than fourth order.
END

nexp

        This function is the sum of decaying exponentials.
 It has  the form:
 f(x) = sum( step(a[i])*a[i+2]*exp((x-a[i])/a[i+1]) )
 + a[n-1], where step(a) = 0 if x < a.  step(a) = 1 if
 x >= a.  You need to tell the program how many parameters
 you want.  4 parameters gives you one exponential, 7 gives
 you two, etc.  If you know the offset, you should
 initialize properly and not vary that parameter.  For
 example, most decaying exponentials are triggered by some
 event. If you know the time of the event, don't vary that
 parameter.
END

xyquad

        f(u,v) = a0 + a1*u + a2*v + a3*u*u + a4*v*v + a5*u*v.
 u and v are used instead of x and y or x0 and x1, because
 gnuplot needs to plot the data parametrically.  The linear
 fitting  algorithm works pretty well, but I recommend the
 result be checked with a few iterations of the L-M
 algorithm, for example, issue the command, "fi 5" after the
 "li" command.
END

sincos

        f(x) = a0 + a[i]*sin(a[i+1]*x)
 + a[i+2]*cos(a[i+3]*x).
 You need good initial guess to get this fitting function to
 converge quickly, if at all.  You will do well to get the
 frequencies close.  If the zeros of your fitting function
 with your initial parameters are close to the zeroes of
 your data, you have a good chance of converging.  For
 special cases like Fourier series, you probably want a
 specialized function.  You should take a Fourier transform
 if that is what you reeally want.
END

conic

        conic does not work very well and I do not reccomend
 its use.
END

Multiple Data Sets

        Let's say that you have several different data sets
 that you want to fit to functions which share some common
 parameters. You can simulate the functionality of multiple
 data sets.

 Put all of your data in the same file , with the same
 columns being used for x0, x1 . . . xn, y, any sigma's etc.
 Add an extra row to the file containing a number to
 uniquely identify the data set.  Define your function for
 n+1 independent variables, and assign the extra row to be
 x(n+1).  The (n+1)th independent variable will be passed to
 your fitting function.  Use it to decide which data set you
 are using and return the appropriate function value and
 derivatives.

 Sure, it's a kludge, but it's better than nothing for now.
End