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SIMPLEX
     simplex, simpfit  -  Nelder-Mead  simplex  minimization  for
     parameter estimates with standard deviations

SYNOPSIS
     xxxxfit [ -s {a|c|l|n} ] [ -d {a|c|l|n} ] [ [ -i ] inputfile
     ] [ [ -o ] diskfile ]

DESCRIPTION
     A model function explicitly relating a dependent variable to
     one or more independent variables and variable parameters is
     fit to a set of data points (observed  values  of  dependent
     and  independent  variables).  The  Nelder-Mead algorithm is
     used to locate the least squares minimum by a simplex search
     procedure,  giving  estimates  of the best-fit values of the
     parameters of the model function.   Standard  deviations  of
     the  parameters are estimated by use of a quadratic approxi-
     mation of the least squares surface near  the  minimum.  The
     quadratic  approximation,  equivalent to minimization by use
     of a Taylor's series approximation, returns improved parame-
     ter estimates.  Thus the quadratic approximation can be used
     as part of the minimization procedure.

     Alternation of a cycle of simplex  minimization,  consisting
     of  perhaps 30 iterations, with a cycle of quadratic minimi-
     zation is an effective strategy.  The simplex  search  effi-
     ciently  defines the region of parameter space that contains
     the minimum; bounds are easily incorporated; because deriva-
     tives  are  not  used, ill-behaved functions can be handled.
     The quadratic minimization, on the other  hand,  efficiently
     finds  the  least  squares  minimum, but it is reliable only
     when near the minimum.  If the quadratic minimization fails,
     as  it often will in the early stages of the search, consid-
     erable time is wasted, to reduce  which,  the  next  one  or
     several  cycles  of  quadratic  minimization  can be skipped
     after a failure.

     The files  input.doc  and  output.doc  should  be  read  for
     details of the entry of data and interpretation of output.

     Several routines have to be recoded for each new model func-
     tion.   (1)  func() calculates for a simplex vertex (for the
     set of parameter values passed to it): (a) the value of  the
     dependent  variable  for  each  data point and (b) the least
     squares function (the weighted  sum  of  residuals  squared)
     over  all  data  points.  Bounds on parameters can be imple-
     mented easily, by a test that returns an  excessively  large
     least  squares  function value (e.g., 1.E38). Code should be
     efficient; most of the minimization is spent in func().  (2)
     The  print  statement of fdatprint(), which lists calculated
     and input values for the set of data points, may have to  be
     recoded  after  change in the function or data.  fdatprint()



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     calls fpointprint(),  which  gives  additional  customizable
     data-related output.  (3) fspecial() controls a customizable
     display printed after the standard display  at  the  end  of
     each iteration of the simplex minimization.

  COMMAND LINE OPTIONS
     -s {a|c|l|n}  --  control output to terminal; default is all
     possible output ( a ):

          a = ALL  possible  output;  information  tracking  each
          iteration of the simplex minimization is displayed.

          c = summary output each CYCLE of perhaps 30 iterations;
          any  output  from a quadratic minimization is displayed
          at this point.

          l = summary output on LAST cycle only;

          n = NO output;

     -d {a|c|l|n}  --  control output to diskfile,  with  options
     as for control of terminal output; default is summary output
     ( c ):

     [ -i ] inputfile --  specify input file; default is stdin.

     [  -o  ]  diskfile  --   specify  output  file;  default  is
     /dev/null, i.e., no output is saved unless requested.

     The input file has the following structure:

          one-line title, with no tabs or ctrl characters;

          lines following give control  variables,  the  starting
          simplex, and the data array;

          the  order  of  entry  is  fixed  by   the   order   in
          read_data();  see  sample  data file for structure, and
          also the description in input.doc;

          the one-word descriptors must be present  in  the  data
          file; the data format is otherwise free-form;

          comments at the end of the file are  not  read  by  the
          program

FILES
     The files linefit.c and ldhfit.c are examples of coding  for
     specific functions.

     The files linefit.dat and ldhfit.dat are sample input files,
     with typical values of the control variables.



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     There are several documentation files: Simplex.doc describes
     the Nelder-Mead algorithm and the mechanics of the quadratic
     fit.  Output.doc describes the full ( a option)  output  for
     the minimization.  Input.doc is a template file, showing the
     style of data entry, with comments.  Data_redn.doc  contains
     low-level  comments  on  data reduction and a description of
     the model coded in ldhfit.c.

AUTHOR
     J. A. Rupley, Tucson, Arizona
     rupley!local@megaron.arizona.edu

BUGS
     There are limits, set in simpdefs.h, on the number of param-
     eters, data points, and variables for each data point.








































Sun Release 4.1           Last change:                          3