mpfit procs

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List of Routines


Routine Descriptions

GAUSS1

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 NAME:
   GAUSS1

 AUTHOR:
   Craig B. Markwardt, NASA/GSFC Code 662, Greenbelt, MD 20770
   craigm@lheamail.gsfc.nasa.gov

 PURPOSE:
   Compute Gaussian curve given the mean, sigma and area.

 MAJOR TOPICS:
   Curve and Surface Fitting

 CALLING SEQUENCE:
   YVALS = GAUSS1(XVALS, [MEAN, SIGMA, AREA], SKEW=skew)

 DESCRIPTION:

  This routine computes the values of a Gaussian function whose
  X-values, mean, sigma, and total area are given.  It is meant to be
  a demonstration for curve-fitting.

  XVALS can be an array of X-values, in which case the returned
  Y-values are an array as well.  The second parameter to GAUSS1
  should be an array containing the MEAN, SIGMA, and total AREA, in
  that order.

 INPUTS:
   X - Array of X-values.

   [MEAN, SIGMA, AREA] - the mean, sigma and total area of the 
                         desired Gaussian curve.

 INPUT KEYWORD PARAMETERS:

   SKEW - You may specify a skew value.  Default is no skew.

   PEAK - if set then AREA is interpreted as the peak value rather
          than the area under the peak.

 RETURNS:

   Returns the array of Y-values.

 EXAMPLE:

   p = [2.2D, 1.4D, 3000.D]
   x = dindgen(200)*0.1 - 10.
   y = gauss1(x, p)

   Computes the values of the Gaussian at equispaced intervals
   (spacing is 0.1).  The gaussian has a mean of 2.2, standard
   deviation of 1.4, and total area of 3000.

 REFERENCES:

 MODIFICATION HISTORY:
   Written, Jul 1998, CM
   Correct bug in normalization, CM, 01 Nov 1999
   Optimized for speed, CM, 02 Nov 1999
   Added copyright notice, 25 Mar 2001, CM
   Added PEAK keyword, 30 Sep 2001, CM

  $Id: gauss1.pro,v 1.4 2001/10/13 17:41:48 craigm Exp $

(See /dzd2/heiles/idl/gen/mpfit/gauss1.pro)


GAUSS1P

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 NAME:
   GAUSS1P

 AUTHOR:
   Craig B. Markwardt, NASA/GSFC Code 662, Greenbelt, MD 20770
   craigm@lheamail.gsfc.nasa.gov

 PURPOSE:
   Compute Gaussian curve given the mean, sigma and area (procedure).

 MAJOR TOPICS:
   Curve and Surface Fitting

 CALLING SEQUENCE:
   GAUSS1, XVALS, [MEAN, SIGMA, AREA], YVALS, SKEW=skew

 DESCRIPTION:

  This routine computes the values of a Gaussian function whose
  X-values, mean, sigma, and total area are given.  It is meant to be
  a demonstration for curve-fitting.

  XVALS can be an array of X-values, in which case the returned
  Y-values are an array as well.  The second parameter to GAUSS1
  should be an array containing the MEAN, SIGMA, and total AREA, in
  that order.

 INPUTS:
   X - Array of X-values.

   [MEAN, SIGMA, AREA] - the mean, sigma and total area of the 
                         desired Gaussian curve.

   YVALS - returns the array of Y-values.


 KEYWORD PARAMETERS:

   SKEW - You may specify a skew value.  Default is no skew.

 EXAMPLE:

   p = [2.2D, 1.4D, 3000.D]
   x = dindgen(200)*0.1 - 10.
   gauss1p, x, p, y

   Computes the values of the Gaussian at equispaced intervals
   (spacing is 0.1).  The gaussian has a mean of 2.2, standard
   deviation of 1.4, and total area of 3000.

 REFERENCES:

 MODIFICATION HISTORY:
   Transcribed from GAUSS1, 13 Dec 1999, CM
   Added copyright notice, 25 Mar 2001, CM

  $Id: gauss1p.pro,v 1.2 2001/03/25 18:55:12 craigm Exp $

(See /dzd2/heiles/idl/gen/mpfit/gauss1p.pro)


GAUSS2

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 NAME:
   GAUSS2

 AUTHOR:
   Craig B. Markwardt, NASA/GSFC Code 662, Greenbelt, MD 20770
   craigm@lheamail.gsfc.nasa.gov

 PURPOSE:
   Compute Gaussian curve given the mean, sigma and area.

 MAJOR TOPICS:
   Curve and Surface Fitting

 CALLING SEQUENCE:
   YVALS = GAUSS2(X, Y, [XCENT, YCENT, SIGMA, PEAK])

 DESCRIPTION:

  This routine computes the values of a Gaussian function whose
  X-values, mean, sigma, and total area are given.  It is meant to be
  a demonstration for curve-fitting.

  XVALS can be an array of X-values, in which case the returned
  Y-values are an array as well.  The second parameter to GAUSS1
  should be an array containing the MEAN, SIGMA, and total AREA, in
  that order.

 INPUTS:
   X - 2-dimensional array of "X"-values.
   Y - 2-dimensional array of "Y"-values.

   XCENT - X-position of gaussian centroid.
   YCENT - Y-position of gaussian centroid.

   SIGMA - sigma of the curve (X and Y widths are the same).

   PEAK - the peak value of the gaussian function.

 RETURNS:

   Returns the array of Y-values.

 EXAMPLE:

   p = [2.2D, -0.7D, 1.4D, 3000.D]
   x = (dindgen(200)*0.1 - 10.) # (dblarr(200) + 1)
   y = (dblarr(200) + 1) # (dindgen(200)*0.1 - 10.)
   z = gauss2(x, y, p)

   Computes the values of the Gaussian at equispaced intervals in X
   and Y (spacing is 0.1).  The gaussian has a centroid position of
   (2.2, -0.7), standard deviation of 1.4, and peak value of 3000.

 REFERENCES:

 MODIFICATION HISTORY:
   Written, 02 Oct 1999, CM
   Added copyright notice, 25 Mar 2001, CM

  $Id: gauss2.pro,v 1.2 2001/03/25 18:55:13 craigm Exp $

(See /dzd2/heiles/idl/gen/mpfit/gauss2.pro)


MPCHILIM

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 NAME:
   MPCHILIM

 AUTHOR:
   Craig B. Markwardt, NASA/GSFC Code 662, Greenbelt, MD 20770
   craigm@lheamail.gsfc.nasa.gov
   UPDATED VERSIONs can be found on my WEB PAGE: 
      http://cow.physics.wisc.edu/~craigm/idl/idl.html

 PURPOSE:
   Compute confidence limits for chi-square statistic

 MAJOR TOPICS:
   Curve and Surface Fitting, Statistics

 CALLING SEQUENCE:
   DELCHI = MPCHILIM(PROB, DOF, [/SIGMA, /CLEVEL, /SLEVEL ])

 DESCRIPTION:

  The function MPCHILIM() computes confidence limits of the
  chi-square statistic for a desired probability level.  The returned
  values, DELCHI, are the limiting chi-squared values: a chi-squared
  value of greater than DELCHI will occur by chance with probability
  PROB:

    P_CHI(CHI > DELCHI; DOF) = PROB

  In specifying the probability level the user has three choices:

    * give the confidence level (default);

    * give the significance level (i.e., 1 - confidence level) and
      pass the /SLEVEL keyword; OR

    * give the "sigma" of the probability (i.e., compute the
      probability based on the normal distribution) and pass the
      /SIGMA keyword.

  Note that /SLEVEL, /CLEVEL and /SIGMA are mutually exclusive.

 INPUTS:

   PROB - scalar or vector number, giving the desired probability
          level as described above.

   DOF - scalar or vector number, giving the number of degrees of
         freedom in the chi-square distribution.

 RETURNS:

  Returns a scalar or vector of chi-square confidence limits.

 KEYWORD PARAMETERS:

   SLEVEL - if set, then PROB describes the significance level.

   CLEVEL - if set, then PROB describes the confidence level
            (default).

   SIGMA - if set, then PROB is the number of "sigma" away from the
           mean in the normal distribution.

 EXAMPLES:

   print, mpchilim(0.99d, 2d, /clevel)

   Print the 99% confidence limit for a chi-squared of 2 degrees of
   freedom.

   print, mpchilim(5d, 2d, /sigma)

   Print the "5 sigma" confidence limit for a chi-squared of 2
   degrees of freedom.  Here "5 sigma" indicates the gaussian
   probability of a 5 sigma event or greater. 
       P_GAUSS(5D) = 1D - 5.7330314e-07

 REFERENCES:

   Algorithms taken from CEPHES special function library, by Stephen
   Moshier. (http://www.netlib.org/cephes/)

 MODIFICATION HISTORY:
   Completed, 1999, CM
   Documented, 16 Nov 2001, CM
   Reduced obtrusiveness of common block and math error handling, 18
     Nov 2001, CM

  $Id: mpchilim.pro,v 1.4 2001/11/18 12:59:16 craigm Exp $

(See /dzd2/heiles/idl/gen/mpfit/mpchilim.pro)


MPCHITEST

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 NAME:
   MPCHITEST

 AUTHOR:
   Craig B. Markwardt, NASA/GSFC Code 662, Greenbelt, MD 20770
   craigm@lheamail.gsfc.nasa.gov
   UPDATED VERSIONs can be found on my WEB PAGE: 
      http://cow.physics.wisc.edu/~craigm/idl/idl.html

 PURPOSE:
   Compute the probability of a given chi-squared value

 MAJOR TOPICS:
   Curve and Surface Fitting, Statistics

 CALLING SEQUENCE:
   PROB = MPCHITEST(CHI, DOF, [/SIGMA, /CLEVEL, /SLEVEL ])

 DESCRIPTION:

  The function MPCHITEST() computes the probability for a value drawn
  from the chi-square distribution to equal or exceed the given value
  CHI.  This can be used for confidence testing of a measured value
  obeying the chi-squared distribution.

    P_CHI(X > CHI; DOF) = PROB

  In specifying the returned probability level the user has three
  choices:

    * return the confidence level when the /CLEVEL keyword is passed;
      OR

    * return the significance level (i.e., 1 - confidence level) when
      the /SLEVEL keyword is passed (default); OR

    * return the "sigma" of the probability (i.e., compute the
      probability based on the normal distribution) when the /SIGMA
      keyword is passed.

  Note that /SLEVEL, /CLEVEL and /SIGMA are mutually exclusive.

 INPUTS:

   CHI - chi-squared value to be tested.

   DOF - scalar or vector number, giving the number of degrees of
         freedom in the chi-square distribution.

 RETURNS:

  Returns a scalar or vector of probabilities, as described above,
  and according to the /SLEVEL, /CLEVEL and /SIGMA keywords.

 KEYWORD PARAMETERS:

   SLEVEL - if set, then PROB describes the significance level
            (default).

   CLEVEL - if set, then PROB describes the confidence level.

   SIGMA - if set, then PROB is the number of "sigma" away from the
           mean in the normal distribution.

 EXAMPLES:

   print, mpchitest(1300d,1252d)

   Print the probability for a chi-squared value with 1252 degrees of
   freedom to exceed a value of 1300, as a confidence level.

 REFERENCES:

   Algorithms taken from CEPHES special function library, by Stephen
   Moshier. (http://www.netlib.org/cephes/)

 MODIFICATION HISTORY:
   Completed, 1999, CM
   Documented, 16 Nov 2001, CM
   Reduced obtrusiveness of common block and math error handling, 18
     Nov 2001, CM

  $Id: mpchitest.pro,v 1.5 2001/11/18 12:59:16 craigm Exp $

(See /dzd2/heiles/idl/gen/mpfit/mpchitest.pro)


MPCURVEFIT

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 NAME:
   MPCURVEFIT

 AUTHOR:
   Craig B. Markwardt, NASA/GSFC Code 662, Greenbelt, MD 20770
   craigm@lheamail.gsfc.nasa.gov
   UPDATED VERSIONs can be found on my WEB PAGE: 
      http://cow.physics.wisc.edu/~craigm/idl/idl.html

 PURPOSE:
   Perform Levenberg-Marquardt least-squares fit (replaces CURVEFIT)

 MAJOR TOPICS:
   Curve and Surface Fitting

 CALLING SEQUENCE:
   YFIT = MPCURVEFIT(X, Y, WEIGHTS, P, [SIGMA,] FUNCTION_NAME=FUNC, 
                     ITER=iter, ITMAX=itmax, 
                     CHISQ=chisq, NFREE=nfree, DOF=dof, 
                     NFEV=nfev, COVAR=covar, [/NOCOVAR, ] [/NODERIVATIVE, ]
                     FUNCTARGS=functargs, PARINFO=parinfo,
                     FTOL=ftol, XTOL=xtol, GTOL=gtol, TOL=tol,
                     ITERPROC=iterproc, ITERARGS=iterargs,
                     NPRINT=nprint, QUIET=quiet, 
                     ERRMSG=errmsg, STATUS=status)

 DESCRIPTION:

  MPCURVEFIT fits a user-supplied model -- in the form of an IDL
  function -- to a set of user-supplied data.  MPCURVEFIT calls
  MPFIT, the MINPACK-1 least-squares minimizer, to do the main
  work.

  Given the data and their uncertainties, MPCURVEFIT finds the best
  set of model parameters which match the data (in a least-squares
  sense) and returns them in the parameter P.  

  MPCURVEFIT returns the best fit function.
  
  The user must supply the following items:
   - An array of independent variable values ("X").
   - An array of "measured" *dependent* variable values ("Y").
   - An array of weighting values ("WEIGHTS").
   - The name of an IDL function which computes Y given X ("FUNC").
   - Starting guesses for all of the parameters ("P").

  There are very few restrictions placed on X, Y or FUNCT.  Simply
  put, FUNCT must map the "X" values into "Y" values given the
  model parameters.  The "X" values may represent any independent
  variable (not just Cartesian X), and indeed may be multidimensional
  themselves.  For example, in the application of image fitting, X
  may be a 2xN array of image positions.

  MPCURVEFIT carefully avoids passing large arrays where possible to
  improve performance.

  See below for an example of usage.
   
 USER FUNCTION

  The user must define a function which returns the model value.  For
  applications which use finite-difference derivatives -- the default
  -- the user function should be declared in the following way:

    PRO MYFUNCT, X, P, YMOD
     ; The independent variable is X
     ; Parameter values are passed in "P"
     YMOD = ... computed model values at X ...
    END

  The returned array YMOD must have the same dimensions and type as
  the "measured" Y values.

  User functions may also indicate a fatal error condition
  using the ERROR_CODE common block variable, as described
  below under the MPFIT_ERROR common block definition.

  See the discussion under "ANALYTIC DERIVATIVES" and AUTODERIVATIVE
  in MPFIT.PRO if you wish to compute the derivatives for yourself.
  AUTODERIVATIVE is accepted and passed directly to MPFIT.  The user
  function must accept one additional parameter, DP, which contains
  the derivative of the user function with respect to each parameter
  at each data point, as described in MPFIT.PRO.

 CONSTRAINING PARAMETER VALUES WITH THE PARINFO KEYWORD

  The behavior of MPFIT can be modified with respect to each
  parameter to be fitted.  A parameter value can be fixed; simple
  boundary constraints can be imposed; limitations on the parameter
  changes can be imposed; properties of the automatic derivative can
  be modified; and parameters can be tied to one another.

  These properties are governed by the PARINFO structure, which is
  passed as a keyword parameter to MPFIT.

  PARINFO should be an array of structures, one for each parameter.
  Each parameter is associated with one element of the array, in
  numerical order.  The structure can have the following entries
  (none are required):
  
     .VALUE - the starting parameter value (but see the START_PARAMS
              parameter for more information).
  
     .FIXED - a boolean value, whether the parameter is to be held
              fixed or not.  Fixed parameters are not varied by
              MPFIT, but are passed on to MYFUNCT for evaluation.
  
     .LIMITED - a two-element boolean array.  If the first/second
                element is set, then the parameter is bounded on the
                lower/upper side.  A parameter can be bounded on both
                sides.  Both LIMITED and LIMITS must be given
                together.
  
     .LIMITS - a two-element float or double array.  Gives the
               parameter limits on the lower and upper sides,
               respectively.  Zero, one or two of these values can be
               set, depending on the values of LIMITED.  Both LIMITED
               and LIMITS must be given together.
  
     .PARNAME - a string, giving the name of the parameter.  The
                fitting code of MPFIT does not use this tag in any
                way.  However, the default ITERPROC will print the
                parameter name if available.
  
     .STEP - the step size to be used in calculating the numerical
             derivatives.  If set to zero, then the step size is
             computed automatically.  Ignored when AUTODERIVATIVE=0.
             This value is superceded by the RELSTEP value.

     .RELSTEP - the *relative* step size to be used in calculating
                the numerical derivatives.  This number is the
                fractional size of the step, compared to the
                parameter value.  This value supercedes the STEP
                setting.  If the parameter is zero, then a default
                step size is chosen.

     .MPSIDE - the sidedness of the finite difference when computing
               numerical derivatives.  This field can take four
               values:

                  0 - one-sided derivative computed automatically
                  1 - one-sided derivative (f(x+h) - f(x)  )/h
                 -1 - one-sided derivative (f(x)   - f(x-h))/h
                  2 - two-sided derivative (f(x+h) - f(x-h))/(2*h)

              Where H is the STEP parameter described above.  The
              "automatic" one-sided derivative method will chose a
              direction for the finite difference which does not
              violate any constraints.  The other methods do not
              perform this check.  The two-sided method is in
              principle more precise, but requires twice as many
              function evaluations.  Default: 0.

     .MPMAXSTEP - the maximum change to be made in the parameter
                  value.  During the fitting process, the parameter
                  will never be changed by more than this value in
                  one iteration.

                  A value of 0 indicates no maximum.  Default: 0.
  
     .TIED - a string expression which "ties" the parameter to other
             free or fixed parameters.  Any expression involving
             constants and the parameter array P are permitted.
             Example: if parameter 2 is always to be twice parameter
             1 then use the following: parinfo(2).tied = '2 * P(1)'.
             Since they are totally constrained, tied parameters are
             considered to be fixed; no errors are computed for them.
             [ NOTE: the PARNAME can't be used in expressions. ]

     .MPPRINT - if set to 1, then the default ITERPROC will print the
                parameter value.  If set to 0, the parameter value
                will not be printed.  This tag can be used to
                selectively print only a few parameter values out of
                many.  Default: 1 (all parameters printed)

  
  Future modifications to the PARINFO structure, if any, will involve
  adding structure tags beginning with the two letters "MP".
  Therefore programmers are urged to avoid using tags starting with
  the same letters; otherwise they are free to include their own
  fields within the PARINFO structure, and they will be ignored.
  
  PARINFO Example:
  parinfo = replicate({value:0.D, fixed:0, limited:[0,0], $
                       limits:[0.D,0]}, 5)
  parinfo(0).fixed = 1
  parinfo(4).limited(0) = 1
  parinfo(4).limits(0)  = 50.D
  parinfo(*).value = [5.7D, 2.2, 500., 1.5, 2000.]
  
  A total of 5 parameters, with starting values of 5.7,
  2.2, 500, 1.5, and 2000 are given.  The first parameter
  is fixed at a value of 5.7, and the last parameter is
  constrained to be above 50.

 INPUTS:
   X - Array of independent variable values.

   Y - Array of "measured" dependent variable values.  Y should have
       the same data type as X.  The function FUNCT should map
       X->Y.

   WEIGHTS - Array of weights to be used in calculating the
             chi-squared value.  If WEIGHTS is specified then the ERR
             parameter is ignored.  The chi-squared value is computed
             as follows:

                CHISQ = TOTAL( (Y-FUNCT(X,P))^2 * ABS(WEIGHTS) )

             Here are common values of WEIGHTS:

                1D/ERR^2 - Normal weighting (ERR is the measurement error)
                1D/Y     - Poisson weighting (counting statistics)
                1D       - Unweighted

   P - An array of starting values for each of the parameters of the
       model.  The number of parameters should be fewer than the
       number of measurements.  Also, the parameters should have the
       same data type as the measurements (double is preferred).

       Upon successful completion the new parameter values are
       returned in P.

       If both START_PARAMS and PARINFO are passed, then the starting
       *value* is taken from START_PARAMS, but the *constraints* are
       taken from PARINFO.
 
   SIGMA - The formal 1-sigma errors in each parameter, computed from
           the covariance matrix.  If a parameter is held fixed, or
           if it touches a boundary, then the error is reported as
           zero.

           If the fit is unweighted (i.e. no errors were given, or
           the weights were uniformly set to unity), then SIGMA will
           probably not represent the true parameter uncertainties.

           *If* you can assume that the true reduced chi-squared
           value is unity -- meaning that the fit is implicitly
           assumed to be of good quality -- then the estimated
           parameter uncertainties can be computed by scaling SIGMA
           by the measured chi-squared value.

              DOF     = N_ELEMENTS(X) - N_ELEMENTS(P) ; deg of freedom
              CSIGMA  = SIGMA * SQRT(CHISQ / DOF)     ; scaled uncertainties

 RETURNS:

   Returns the array containing the best-fitting function.

 KEYWORD PARAMETERS:

   CHISQ - the value of the summed, squared, weighted residuals for
           the returned parameter values, i.e. the chi-square value.

   COVAR - the covariance matrix for the set of parameters returned
           by MPFIT.  The matrix is NxN where N is the number of
           parameters.  The square root of the diagonal elements
           gives the formal 1-sigma statistical errors on the
           parameters IF errors were treated "properly" in MYFUNC.
           Parameter errors are also returned in PERROR.

           To compute the correlation matrix, PCOR, use this:
           IDL> PCOR = COV * 0
           IDL> FOR i = 0, n-1 DO FOR j = 0, n-1 DO $
                PCOR(i,j) = COV(i,j)/sqrt(COV(i,i)*COV(j,j))

           If NOCOVAR is set or MPFIT terminated abnormally, then
           COVAR is set to a scalar with value !VALUES.D_NAN.

   DOF - number of degrees of freedom, computed as
             DOF = N_ELEMENTS(DEVIATES) - NFREE
         Note that this doesn't account for pegged parameters (see
         NPEGGED).

   ERRMSG - a string error or warning message is returned.

   FTOL - a nonnegative input variable. Termination occurs when both
          the actual and predicted relative reductions in the sum of
          squares are at most FTOL (and STATUS is accordingly set to
          1 or 3).  Therefore, FTOL measures the relative error
          desired in the sum of squares.  Default: 1D-10

   FUNCTION_NAME - a scalar string containing the name of an IDL
                   procedure to compute the user model values, as
                   described above in the "USER MODEL" section.

   FUNCTARGS - A structure which contains the parameters to be passed
               to the user-supplied function specified by FUNCT via
               the _EXTRA mechanism.  This is the way you can pass
               additional data to your user-supplied function without
               using common blocks.

               By default, no extra parameters are passed to the
               user-supplied function.

   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 (and STATUS is
          accordingly set to 4). Therefore, GTOL measures the
          orthogonality desired between the function vector and the
          columns of the jacobian.  Default: 1D-10

   ITER - the number of iterations completed.

   ITERARGS - The keyword arguments to be passed to ITERPROC via the
              _EXTRA mechanism.  This should be a structure, and is
              similar in operation to FUNCTARGS.
              Default: no arguments are passed.

   ITERPROC - The name of a procedure to be called upon each NPRINT
              iteration of the MPFIT routine.  It should be declared
              in the following way:

              PRO ITERPROC, FUNCT, p, iter, fnorm, FUNCTARGS=fcnargs, $
                PARINFO=parinfo, QUIET=quiet, ...
                ; perform custom iteration update
              END
         
              ITERPROC must either accept all three keyword
              parameters (FUNCTARGS, PARINFO and QUIET), or at least
              accept them via the _EXTRA keyword.
          
              FUNCT is the user-supplied function to be minimized,
              P is the current set of model parameters, ITER is the
              iteration number, and FUNCTARGS are the arguments to be
              passed to FUNCT.  FNORM should be the
              chi-squared value.  QUIET is set when no textual output
              should be printed.  See below for documentation of
              PARINFO.

              In implementation, ITERPROC can perform updates to the
              terminal or graphical user interface, to provide
              feedback while the fit proceeds.  If the fit is to be
              stopped for any reason, then ITERPROC should set the
              common block variable ERROR_CODE to negative value (see
              MPFIT_ERROR common block below).  In principle,
              ITERPROC should probably not modify the parameter
              values, because it may interfere with the algorithm's
              stability.  In practice it is allowed.

              Default: an internal routine is used to print the
                       parameter values.

   ITMAX - The maximum number of iterations to perform.  If the
             number is exceeded, then the STATUS value is set to 5
             and MPFIT returns.
             Default: 200 iterations

   NFEV - the number of FUNCT function evaluations performed.

   NFREE - the number of free parameters in the fit.  This includes
           parameters which are not FIXED and not TIED, but it does
           include parameters which are pegged at LIMITS.

   NOCOVAR - set this keyword to prevent the calculation of the
             covariance matrix before returning (see COVAR)

   NODERIVATIVE - if set, then the user function will not be queried
                  for analytical derivatives, and instead the
                  derivatives will be computed by finite differences
                  (and according to the PARINFO derivative settings;
                  see above for a description).

   NPRINT - The frequency with which ITERPROC is called.  A value of
            1 indicates that ITERPROC is called with every iteration,
            while 2 indicates every other iteration, etc.  Note that
            several Levenberg-Marquardt attempts can be made in a
            single iteration.
            Default value: 1

   PARINFO - Provides a mechanism for more sophisticated constraints
             to be placed on parameter values.  When PARINFO is not
             passed, then it is assumed that all parameters are free
             and unconstrained.  Values in PARINFO are never 
             modified during a call to MPFIT.

             See description above for the structure of PARINFO.

             Default value:  all parameters are free and unconstrained.

   QUIET - set this keyword when no textual output should be printed
           by MPFIT

   STATUS - an integer status code is returned.  All values other
            than zero can represent success.  It can have one of the
            following values:

	   0  improper input parameters.
         
	   1  both actual and predicted relative reductions
	      in the sum of squares are at most FTOL.
         
	   2  relative error between two consecutive iterates
	      is at most XTOL
         
	   3  conditions for STATUS = 1 and STATUS = 2 both hold.
         
	   4  the cosine of the angle between fvec and any
	      column of the jacobian is at most GTOL in
	      absolute value.
         
	   5  the maximum number of iterations has been reached
         
	   6  FTOL is too small. no further reduction in
	      the sum of squares is possible.
         
	   7  XTOL is too small. no further improvement in
	      the approximate solution x is possible.
         
	   8  GTOL is too small. fvec is orthogonal to the
	      columns of the jacobian to machine precision.

   TOL - synonym for FTOL.  Use FTOL instead.

   XTOL - a nonnegative input variable. Termination occurs when the
          relative error between two consecutive iterates is at most
          XTOL (and STATUS is accordingly set to 2 or 3).  Therefore,
          XTOL measures the relative error desired in the approximate
          solution.  Default: 1D-10

   YERROR - upon return, the root-mean-square variance of the
            residuals.


 EXAMPLE:

   ; First, generate some synthetic data
   npts = 200
   x  = dindgen(npts) * 0.1 - 10.                  ; Independent variable 
   yi = gauss1(x, [2.2D, 1.4, 3000.])              ; "Ideal" Y variable
   y  = yi + randomn(seed, npts) * sqrt(1000. + yi); Measured, w/ noise
   sy = sqrt(1000.D + y)                           ; Poisson errors

   ; Now fit a Gaussian to see how well we can recover
   p0 = [1.D, 1., 1000.]                           ; Initial guess
   yfit = mpcurvefit(x, y, 1/sy^2, p0, $
                   FUNCTION_NAME='GAUSS1P')        ; Fit a function
   print, p

   Generates a synthetic data set with a Gaussian peak, and Poisson
   statistical uncertainty.  Then the same function is fitted to the
   data to see how close we can get.  GAUSS1 and GAUSS1P are
   available from the same web page.


 COMMON BLOCKS:

   COMMON MPFIT_ERROR, ERROR_CODE

     User routines may stop the fitting process at any time by
     setting an error condition.  This condition may be set in either
     the user's model computation routine (MYFUNCT), or in the
     iteration procedure (ITERPROC).

     To stop the fitting, the above common block must be declared,
     and ERROR_CODE must be set to a negative number.  After the user
     procedure or function returns, MPFIT checks the value of this
     common block variable and exits immediately if the error
     condition has been set.  By default the value of ERROR_CODE is
     zero, indicating a successful function/procedure call.

 REFERENCES:

   MINPACK-1, Jorge More', available from netlib (www.netlib.org).
   "Optimization Software Guide," Jorge More' and Stephen Wright, 
     SIAM, *Frontiers in Applied Mathematics*, Number 14.

 MODIFICATION HISTORY:
   Translated from MPFITFUN, 25 Sep 1999, CM
   Alphabetized documented keywords, 02 Oct 1999, CM
   Added QUERY keyword and query checking of MPFIT, 29 Oct 1999, CM
   Check to be sure that X and Y are present, 02 Nov 1999, CM
   Documented SIGMA for unweighted fits, 03 Nov 1999, CM
   Changed to ERROR_CODE for error condition, 28 Jan 2000, CM
   Copying permission terms have been liberalized, 26 Mar 2000, CM
   Propagated improvements from MPFIT, 17 Dec 2000, CM
   Corrected behavior of NODERIVATIVE, 13 May 2002, CM
   Documented RELSTEP field of PARINFO (!!), CM, 25 Oct 2002
   Make more consistent with comparable IDL routines, 30 Jun 2003, CM
   Minor documentation adjustment, 03 Feb 2004, CM

  $Id: mpcurvefit.pro,v 1.6 2004/02/26 04:18:16 craigm Exp $

(See /dzd2/heiles/idl/gen/mpfit/mpcurvefit.pro)


MPFIT

[Previous Routine] [Next Routine] [List of Routines]
 NAME:
   MPFIT

 AUTHOR:
   Craig B. Markwardt, NASA/GSFC Code 662, Greenbelt, MD 20770
   craigm@lheamail.gsfc.nasa.gov
   UPDATED VERSIONs can be found on my WEB PAGE: 
      http://cow.physics.wisc.edu/~craigm/idl/idl.html

 PURPOSE:
   Perform Levenberg-Marquardt least-squares minimization (MINPACK-1)

 MAJOR TOPICS:
   Curve and Surface Fitting

 CALLING SEQUENCE:
   parms = MPFIT(MYFUNCT, start_parms, FUNCTARGS=fcnargs, NFEV=nfev,
                 MAXITER=maxiter, ERRMSG=errmsg, NPRINT=nprint, QUIET=quiet, 
                 FTOL=ftol, XTOL=xtol, GTOL=gtol, NITER=niter, 
                 STATUS=status, ITERPROC=iterproc, ITERARGS=iterargs,
                 COVAR=covar, PERROR=perror, BESTNORM=bestnorm,
                 PARINFO=parinfo)

 DESCRIPTION:

  MPFIT uses the Levenberg-Marquardt technique to solve the
  least-squares problem.  In its typical use, MPFIT will be used to
  fit a user-supplied function (the "model") to user-supplied data
  points (the "data") by adjusting a set of parameters.  MPFIT is
  based upon MINPACK-1 (LMDIF.F) by More' and collaborators.

  For example, a researcher may think that a set of observed data
  points is best modelled with a Gaussian curve.  A Gaussian curve is
  parameterized by its mean, standard deviation and normalization.
  MPFIT will, within certain constraints, find the set of parameters
  which best fits the data.  The fit is "best" in the least-squares
  sense; that is, the sum of the weighted squared differences between
  the model and data is minimized.

  The Levenberg-Marquardt technique is a particular strategy for
  iteratively searching for the best fit.  This particular
  implementation is drawn from MINPACK-1 (see NETLIB), and seems to
  be more robust than routines provided with IDL.  This version
  allows upper and lower bounding constraints to be placed on each
  parameter, or the parameter can be held fixed.

  The IDL user-supplied function should return an array of weighted
  deviations between model and data.  In a typical scientific problem
  the residuals should be weighted so that each deviate has a
  gaussian sigma of 1.0.  If X represents values of the independent
  variable, Y represents a measurement for each value of X, and ERR
  represents the error in the measurements, then the deviates could
  be calculated as follows:

    DEVIATES = (Y - F(X)) / ERR

  where F is the analytical function representing the model.  You are
  recommended to use the convenience functions MPFITFUN and
  MPFITEXPR, which are driver functions that calculate the deviates
  for you.  If ERR are the 1-sigma uncertainties in Y, then

    TOTAL( DEVIATES^2 ) 

  will be the total chi-squared value.  MPFIT will minimize the
  chi-square value.  The values of X, Y and ERR are passed through
  MPFIT to the user-supplied function via the FUNCTARGS keyword.

  Simple constraints can be placed on parameter values by using the
  PARINFO keyword to MPFIT.  See below for a description of this
  keyword.

  MPFIT does not perform more general optimization tasks.  See TNMIN
  instead.  MPFIT is customized, based on MINPACK-1, to the
  least-squares minimization problem.

 USER FUNCTION

  The user must define a function which returns the appropriate
  values as specified above.  The function should return the weighted
  deviations between the model and the data.  For applications which
  use finite-difference derivatives -- the default -- the user
  function should be declared in the following way:

    FUNCTION MYFUNCT, p, X=x, Y=y, ERR=err
     ; Parameter values are passed in "p"
     model = F(x, p)
     return, (y-model)/err
    END

  See below for applications with analytical derivatives.

  The keyword parameters X, Y, and ERR in the example above are
  suggestive but not required.  Any parameters can be passed to
  MYFUNCT by using the FUNCTARGS keyword to MPFIT.  Use MPFITFUN and
  MPFITEXPR if you need ideas on how to do that.  The function *must*
  accept a parameter list, P.
  
  In general there are no restrictions on the number of dimensions in
  X, Y or ERR.  However the deviates *must* be returned in a
  one-dimensional array, and must have the same type (float or
  double) as the input arrays.

  User functions may also indicate a fatal error condition using the
  ERROR_CODE common block variable, as described below under the
  MPFIT_ERROR common block definition (by setting ERROR_CODE to a
  number between -15 and -1).

 ANALYTIC DERIVATIVES
 
  In the search for the best-fit solution, MPFIT by default
  calculates derivatives numerically via a finite difference
  approximation.  The user-supplied function need not calculate the
  derivatives explicitly.  However, if you desire to compute them
  analytically, then the AUTODERIVATIVE=0 keyword must be passed.  As
  a practical matter, it is often sufficient and even faster to allow
  MPFIT to calculate the derivatives numerically, and so
  AUTODERIVATIVE=0 is not necessary.

  Also, the user function must be declared with one additional
  parameter, as follows:

    FUNCTION MYFUNCT, p, dp, X=x, Y=y, ERR=err
     model = F(x, p)
     
     if n_params() GT 1 then begin
       ; Compute derivatives
       dp = make_array(n_elements(x), n_elements(p), value=x(0)*0)
       for i = 0, n_elements(p)-1 do $
         dp(*,i) = FGRAD(x, p, i)
     endif
    
     return, (y-model)/err
    END

  where FGRAD(x, p, i) is a user function which must compute the
  derivative of the model with respect to parameter P(i) at X.  When
  finite differencing is used for computing derivatives (ie, when
  AUTODERIVATIVE=1), the parameter DP is not passed.  Therefore
  functions can use N_PARAMS() to indicate whether they must compute
  the derivatives or not.

  Derivatives should be returned in the DP array. DP should be an m x
  n array, where m is the number of data points and n is the number
  of parameters.  dp(i,j) is the derivative at the ith point with
  respect to the jth parameter.  
  
  The derivatives with respect to fixed parameters are ignored; zero
  is an appropriate value to insert for those derivatives.  Upon
  input to the user function, DP is set to a vector with the same
  length as P, with a value of 1 for a parameter which is free, and a
  value of zero for a parameter which is fixed (and hence no
  derivative needs to be calculated).  This input vector may be
  overwritten as needed.

  If the data is higher than one dimensional, then the *last*
  dimension should be the parameter dimension.  Example: fitting a
  50x50 image, "dp" should be 50x50xNPAR.
  
 CONSTRAINING PARAMETER VALUES WITH THE PARINFO KEYWORD

  The behavior of MPFIT can be modified with respect to each
  parameter to be fitted.  A parameter value can be fixed; simple
  boundary constraints can be imposed; limitations on the parameter
  changes can be imposed; properties of the automatic derivative can
  be modified; and parameters can be tied to one another.

  These properties are governed by the PARINFO structure, which is
  passed as a keyword parameter to MPFIT.

  PARINFO should be an array of structures, one for each parameter.
  Each parameter is associated with one element of the array, in
  numerical order.  The structure can have the following entries
  (none are required):
  
     .VALUE - the starting parameter value (but see the START_PARAMS
              parameter for more information).
  
     .FIXED - a boolean value, whether the parameter is to be held
              fixed or not.  Fixed parameters are not varied by
              MPFIT, but are passed on to MYFUNCT for evaluation.
  
     .LIMITED - a two-element boolean array.  If the first/second
                element is set, then the parameter is bounded on the
                lower/upper side.  A parameter can be bounded on both
                sides.  Both LIMITED and LIMITS must be given
                together.
  
     .LIMITS - a two-element float or double array.  Gives the
               parameter limits on the lower and upper sides,
               respectively.  Zero, one or two of these values can be
               set, depending on the values of LIMITED.  Both LIMITED
               and LIMITS must be given together.
  
     .PARNAME - a string, giving the name of the parameter.  The
                fitting code of MPFIT does not use this tag in any
                way.  However, the default ITERPROC will print the
                parameter name if available.
  
     .STEP - the step size to be used in calculating the numerical
             derivatives.  If set to zero, then the step size is
             computed automatically.  Ignored when AUTODERIVATIVE=0.
             This value is superceded by the RELSTEP value.

     .RELSTEP - the *relative* step size to be used in calculating
                the numerical derivatives.  This number is the
                fractional size of the step, compared to the
                parameter value.  This value supercedes the STEP
                setting.  If the parameter is zero, then a default
                step size is chosen.

     .MPSIDE - the sidedness of the finite difference when computing
               numerical derivatives.  This field can take four
               values:

                  0 - one-sided derivative computed automatically
                  1 - one-sided derivative (f(x+h) - f(x)  )/h
                 -1 - one-sided derivative (f(x)   - f(x-h))/h
                  2 - two-sided derivative (f(x+h) - f(x-h))/(2*h)

              Where H is the STEP parameter described above.  The
              "automatic" one-sided derivative method will chose a
              direction for the finite difference which does not
              violate any constraints.  The other methods do not
              perform this check.  The two-sided method is in
              principle more precise, but requires twice as many
              function evaluations.  Default: 0.

     .MPMAXSTEP - the maximum change to be made in the parameter
                  value.  During the fitting process, the parameter
                  will never be changed by more than this value in
                  one iteration.

                  A value of 0 indicates no maximum.  Default: 0.
  
     .TIED - a string expression which "ties" the parameter to other
             free or fixed parameters.  Any expression involving
             constants and the parameter array P are permitted.
             Example: if parameter 2 is always to be twice parameter
             1 then use the following: parinfo(2).tied = '2 * P(1)'.
             Since they are totally constrained, tied parameters are
             considered to be fixed; no errors are computed for them.
             [ NOTE: the PARNAME can't be used in expressions. ]

     .MPPRINT - if set to 1, then the default ITERPROC will print the
                parameter value.  If set to 0, the parameter value
                will not be printed.  This tag can be used to
                selectively print only a few parameter values out of
                many.  Default: 1 (all parameters printed)

     .MPFORMAT - IDL format string to print the parameter within
                 ITERPROC.  Default: '(G20.6)' An empty string will
                 also use the default.

  
  Future modifications to the PARINFO structure, if any, will involve
  adding structure tags beginning with the two letters "MP".
  Therefore programmers are urged to avoid using tags starting with
  the same letters; otherwise they are free to include their own
  fields within the PARINFO structure, and they will be ignored.
  
  PARINFO Example:
  parinfo = replicate({value:0.D, fixed:0, limited:[0,0], $
                       limits:[0.D,0]}, 5)
  parinfo(0).fixed = 1
  parinfo(4).limited(0) = 1
  parinfo(4).limits(0)  = 50.D
  parinfo(*).value = [5.7D, 2.2, 500., 1.5, 2000.]
  
  A total of 5 parameters, with starting values of 5.7,
  2.2, 500, 1.5, and 2000 are given.  The first parameter
  is fixed at a value of 5.7, and the last parameter is
  constrained to be above 50.


 HARD-TO-COMPUTE FUNCTIONS: "EXTERNAL" EVALUATION

  The normal mode of operation for MPFIT is for the user to pass a
  function name, and MPFIT will call the user function multiple times
  as it iterates toward a solution.

  Some user functions are particularly hard to compute using the
  standard model of MPFIT.  Usually these are functions that depend
  on a large amount of external data, and so it is not feasible, or
  at least highly impractical, to have MPFIT call it.  In those cases
  it may be possible to use the "(EXTERNAL)" evaluation option.

  In this case the user is responsible for making all function *and
  derivative* evaluations.  The function and Jacobian data are passed
  in through the EXTERNAL_FVEC and EXTERNAL_FJAC keywords,
  respectively.  The user indicates the selection of this option by
  specifying a function name (MYFUNCT) of "(EXTERNAL)".  No
  user-function calls are made when EXTERNAL evaluation is being
  used.

  At the end of each iteration, control returns to the user, who must
  reevaluate the function at its new parameter values.  Users should
  check the return value of the STATUS keyword, where a value of 9
  indicates the user should supply more data for the next iteration,
  and re-call MPFIT.  The user may refrain from calling MPFIT
  further; as usual, STATUS will indicate when the solution has
  converged and no more iterations are required.

  Because MPFIT must maintain its own data structures between calls,
  the user must also pass a named variable to the EXTERNAL_STATE
  keyword.  This variable must be maintained by the user, but not
  changed, throughout the fitting process.  When no more iterations
  are desired, the named variable may be discarded.


 INPUTS:
   MYFUNCT - a string variable containing the name of the function to
             be minimized.  The function should return the weighted
             deviations between the model and the data, as described
             above.

             For EXTERNAL evaluation of functions, this parameter
             should be set to a value of "(EXTERNAL)".

   START_PARAMS - An array of starting values for each of the
                  parameters of the model.  The number of parameters
                  should be fewer than the number of measurements.
                  Also, the parameters should have the same data type
                  as the measurements (double is preferred).

                  This parameter is optional if the PARINFO keyword
                  is used (but see PARINFO).  The PARINFO keyword
                  provides a mechanism to fix or constrain individual
                  parameters.  If both START_PARAMS and PARINFO are
                  passed, then the starting *value* is taken from
                  START_PARAMS, but the *constraints* are taken from
                  PARINFO.
 
 RETURNS:

   Returns the array of best-fit parameters.


 KEYWORD PARAMETERS:

   AUTODERIVATIVE - If this is set, derivatives of the function will
                    be computed automatically via a finite
                    differencing procedure.  If not set, then MYFUNCT
                    must provide the (analytical) derivatives.
                    Default: set (=1) 
                    NOTE: to supply your own analytical derivatives,
                      explicitly pass AUTODERIVATIVE=0

   BESTNORM - the value of the summed squared weighted residuals for
              the returned parameter values, i.e. TOTAL(DEVIATES^2).

   COVAR - the covariance matrix for the set of parameters returned
           by MPFIT.  The matrix is NxN where N is the number of
           parameters.  The square root of the diagonal elements
           gives the formal 1-sigma statistical errors on the
           parameters IF errors were treated "properly" in MYFUNC.
           Parameter errors are also returned in PERROR.

           To compute the correlation matrix, PCOR, use this example:
           IDL> PCOR = COV * 0
           IDL> FOR i = 0, n-1 DO FOR j = 0, n-1 DO $
                PCOR(i,j) = COV(i,j)/sqrt(COV(i,i)*COV(j,j))

           If NOCOVAR is set or MPFIT terminated abnormally, then
           COVAR is set to a scalar with value !VALUES.D_NAN.

   DOF - number of degrees of freedom, computed as
             DOF = N_ELEMENTS(DEVIATES) - NFREE
         Note that this doesn't account for pegged parameters (see
         NPEGGED).

   ERRMSG - a string error or warning message is returned.

   EXTERNAL_FVEC - upon input, the function values, evaluated at
                   START_PARAMS.  This should be an M-vector, where M
                   is the number of data points.

   EXTERNAL_FJAC - upon input, the Jacobian array of partial
                   derivative values.  This should be a M x N array,
                   where M is the number of data points and N is the
                   number of parameters.  NOTE: that all FIXED or
                   TIED parameters must *not* be included in this
                   array.

   EXTERNAL_STATE - a named variable to store MPFIT-related state
                    information between iterations (used in input and
                    output to MPFIT).  The user must not manipulate
                    or discard this data until the final iteration is
                    performed.

   FASTNORM - set this keyword to select a faster algorithm to
              compute sum-of-square values internally.  For systems
              with large numbers of data points, the standard
              algorithm can become prohibitively slow because it
              cannot be vectorized well.  By setting this keyword,
              MPFIT will run faster, but it will be more prone to
              floating point overflows and underflows.  Thus, setting
              this keyword may sacrifice some stability in the
              fitting process.
              
   FTOL - a nonnegative input variable. Termination occurs when both
          the actual and predicted relative reductions in the sum of
          squares are at most FTOL (and STATUS is accordingly set to
          1 or 3).  Therefore, FTOL measures the relative error
          desired in the sum of squares.  Default: 1D-10

   FUNCTARGS - A structure which contains the parameters to be passed
               to the user-supplied function specified by MYFUNCT via
               the _EXTRA mechanism.  This is the way you can pass
               additional data to your user-supplied function without
               using common blocks.

               Consider the following example:
                if FUNCTARGS = { XVAL:[1.D,2,3], YVAL:[1.D,4,9],
                                 ERRVAL:[1.D,1,1] }
                then the user supplied function should be declared
                like this:
                FUNCTION MYFUNCT, P, XVAL=x, YVAL=y, ERRVAL=err

               By default, no extra parameters are passed to the
               user-supplied function, but your function should
               accept *at least* one keyword parameter.  [ This is to
               accomodate a limitation in IDL's _EXTRA
               parameter-passing mechanism. ]

   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 (and STATUS is
          accordingly set to 4). Therefore, GTOL measures the
          orthogonality desired between the function vector and the
          columns of the jacobian.  Default: 1D-10

   ITERARGS - The keyword arguments to be passed to ITERPROC via the
              _EXTRA mechanism.  This should be a structure, and is
              similar in operation to FUNCTARGS.
              Default: no arguments are passed.

   ITERPROC - The name of a procedure to be called upon each NPRINT
              iteration of the MPFIT routine.  ITERPROC is always
              called in the final iteration.  It should be declared
              in the following way:

              PRO ITERPROC, MYFUNCT, p, iter, fnorm, FUNCTARGS=fcnargs, $
                PARINFO=parinfo, QUIET=quiet, DOF=dof, ...
                ; perform custom iteration update
              END
         
              ITERPROC must either accept all three keyword
              parameters (FUNCTARGS, PARINFO and QUIET), or at least
              accept them via the _EXTRA keyword.
          
              MYFUNCT is the user-supplied function to be minimized,
              P is the current set of model parameters, ITER is the
              iteration number, and FUNCTARGS are the arguments to be
              passed to MYFUNCT.  FNORM should be the chi-squared
              value.  QUIET is set when no textual output should be
              printed.  DOF is the number of degrees of freedom,
              normally the number of points less the number of free
              parameters.  See below for documentation of PARINFO.

              In implementation, ITERPROC can perform updates to the
              terminal or graphical user interface, to provide
              feedback while the fit proceeds.  If the fit is to be
              stopped for any reason, then ITERPROC should set the
              common block variable ERROR_CODE to negative value
              between -15 and -1 (see MPFIT_ERROR common block
              below).  In principle, ITERPROC should probably not
              modify the parameter values, because it may interfere
              with the algorithm's stability.  In practice it is
              allowed.

              Default: an internal routine is used to print the
                       parameter values.

   ITERSTOP - Set this keyword if you wish to be able to stop the
              fitting by hitting the predefined ITERKEYSTOP key on
              the keyboard.  This only works if you use the default
              ITERPROC.

   ITERKEYSTOP - A keyboard key which will halt the fit (and if
                 ITERSTOP is set and the default ITERPROC is used).
                 ITERSTOPKEY may either be a one-character string
                 with the desired key, or a scalar integer giving the
                 ASCII code of the desired key.  
                 Default: 7b (control-g)

                 NOTE: the default value of ASCI 7 (control-G) cannot
                 be read in some windowing environments, so you must
                 change to a printable character like 'q'.

   MAXITER - The maximum number of iterations to perform.  If the
             number is exceeded, then the STATUS value is set to 5
             and MPFIT returns.
             Default: 200 iterations

   NFEV - the number of MYFUNCT function evaluations performed.

   NFREE - the number of free parameters in the fit.  This includes
           parameters which are not FIXED and not TIED, but it does
           include parameters which are pegged at LIMITS.

   NITER - the number of iterations completed.

   NOCOVAR - set this keyword to prevent the calculation of the
             covariance matrix before returning (see COVAR)

   NPEGGED - the number of free parameters which are pegged at a
             LIMIT.

   NPRINT - The frequency with which ITERPROC is called.  A value of
            1 indicates that ITERPROC is called with every iteration,
            while 2 indicates every other iteration, etc.  Be aware
            that several Levenberg-Marquardt attempts can be made in
            a single iteration.  Also, the ITERPROC is *always*
            called for the final iteration, regardless of the
            iteration number.
            Default value: 1

   PARINFO - Provides a mechanism for more sophisticated constraints
             to be placed on parameter values.  When PARINFO is not
             passed, then it is assumed that all parameters are free
             and unconstrained.  Values in PARINFO are never 
             modified during a call to MPFIT.

             See description above for the structure of PARINFO.

             Default value:  all parameters are free and unconstrained.

   PERROR - The formal 1-sigma errors in each parameter, computed
            from the covariance matrix.  If a parameter is held
            fixed, or if it touches a boundary, then the error is
            reported as zero.

            If the fit is unweighted (i.e. no errors were given, or
            the weights were uniformly set to unity), then PERROR
            will probably not represent the true parameter
            uncertainties.  

            *If* you can assume that the true reduced chi-squared
            value is unity -- meaning that the fit is implicitly
            assumed to be of good quality -- then the estimated
            parameter uncertainties can be computed by scaling PERROR
            by the measured chi-squared value.

              DOF     = N_ELEMENTS(X) - N_ELEMENTS(PARMS) ; deg of freedom
              PCERROR = PERROR * SQRT(BESTNORM / DOF)   ; scaled uncertainties

   QUIET - set this keyword when no textual output should be printed
           by MPFIT

   RESDAMP - a scalar number, indicating the cut-off value of
             residuals where "damping" will occur.  Residuals with
             magnitudes greater than this number will be replaced by
             their logarithm.  This partially mitigates the so-called
             large residual problem inherent in least-squares solvers
             (as for the test problem CURVI, http://www.maxthis.com/-
             curviex.htm).  A value of 0 indicates no damping.
             Default: 0

             Note: RESDAMP doesn't work with AUTODERIV=0

   STATUS - an integer status code is returned.  All values greater
            than zero can represent success (however STATUS EQ 5 may
            indicate failure to converge).  It can have one of the
            following values:

        -16  a parameter or function value has become infinite or an
             undefined number.  This is usually a consequence of
             numerical overflow in the user's model function, which
             must be avoided.

        -15 to -1 
             these are error codes that either MYFUNCT or ITERPROC
             may return to terminate the fitting process (see
             description of MPFIT_ERROR common below).  If either
             MYFUNCT or ITERPROC set ERROR_CODE to a negative number,
             then that number is returned in STATUS.  Values from -15
             to -1 are reserved for the user functions and will not
             clash with MPFIT.

	   0  improper input parameters.
         
	   1  both actual and predicted relative reductions
	      in the sum of squares are at most FTOL.
         
	   2  relative error between two consecutive iterates
	      is at most XTOL
         
	   3  conditions for STATUS = 1 and STATUS = 2 both hold.
         
	   4  the cosine of the angle between fvec and any
	      column of the jacobian is at most GTOL in
	      absolute value.
         
	   5  the maximum number of iterations has been reached
         
	   6  FTOL is too small. no further reduction in
	      the sum of squares is possible.
         
	   7  XTOL is too small. no further improvement in
	      the approximate solution x is possible.
         
	   8  GTOL is too small. fvec is orthogonal to the
	      columns of the jacobian to machine precision.

          9  A successful single iteration has been completed, and
             the user must supply another "EXTERNAL" evaluation of
             the function and its derivatives.  This status indicator
             is neither an error nor a convergence indicator.

   XTOL - a nonnegative input variable. Termination occurs when the
          relative error between two consecutive iterates is at most
          XTOL (and STATUS is accordingly set to 2 or 3).  Therefore,
          XTOL measures the relative error desired in the approximate
          solution.  Default: 1D-10


 EXAMPLE:

   p0 = [5.7D, 2.2, 500., 1.5, 2000.]
   fa = {X:x, Y:y, ERR:err}
   p = mpfit('MYFUNCT', p0, functargs=fa)

   Minimizes sum of squares of MYFUNCT.  MYFUNCT is called with the X,
   Y, and ERR keyword parameters that are given by FUNCTARGS.  The
   resulting parameter values are returned in p.


 COMMON BLOCKS:

   COMMON MPFIT_ERROR, ERROR_CODE

     User routines may stop the fitting process at any time by
     setting an error condition.  This condition may be set in either
     the user's model computation routine (MYFUNCT), or in the
     iteration procedure (ITERPROC).

     To stop the fitting, the above common block must be declared,
     and ERROR_CODE must be set to a negative number.  After the user
     procedure or function returns, MPFIT checks the value of this
     common block variable and exits immediately if the error
     condition has been set.  This value is also returned in the
     STATUS keyword: values of -1 through -15 are reserved error
     codes for the user routines.  By default the value of ERROR_CODE
     is zero, indicating a successful function/procedure call.

   COMMON MPFIT_PROFILE
   COMMON MPFIT_MACHAR
   COMMON MPFIT_CONFIG

     These are undocumented common blocks are used internally by
     MPFIT and may change in future implementations.

 THEORY OF OPERATION:

   There are many specific strategies for function minimization.  One
   very popular technique is to use function gradient information to
   realize the local structure of the function.  Near a local minimum
   the function value can be taylor expanded about x0 as follows:

      f(x) = f(x0) + f'(x0) . (x-x0) + (1/2) (x-x0) . f''(x0) . (x-x0)
             -----   ---------------   -------------------------------  (1)
     Order    0th          1st                      2nd

   Here f'(x) is the gradient vector of f at x, and f''(x) is the
   Hessian matrix of second derivatives of f at x.  The vector x is
   the set of function parameters, not the measured data vector.  One
   can find the minimum of f, f(xm) using Newton's method, and
   arrives at the following linear equation:

      f''(x0) . (xm-x0) = - f'(x0)                            (2)

   If an inverse can be found for f''(x0) then one can solve for
   (xm-x0), the step vector from the current position x0 to the new
   projected minimum.  Here the problem has been linearized (ie, the
   gradient information is known to first order).  f''(x0) is
   symmetric n x n matrix, and should be positive definite.

   The Levenberg - Marquardt technique is a variation on this theme.
   It adds an additional diagonal term to the equation which may aid the
   convergence properties:

      (f''(x0) + nu I) . (xm-x0) = -f'(x0)                  (2a)

   where I is the identity matrix.  When nu is large, the overall
   matrix is diagonally dominant, and the iterations follow steepest
   descent.  When nu is small, the iterations are quadratically
   convergent.

   In principle, if f''(x0) and f'(x0) are known then xm-x0 can be
   determined.  However the Hessian matrix is often difficult or
   impossible to compute.  The gradient f'(x0) may be easier to
   compute, if even by finite difference techniques.  So-called
   quasi-Newton techniques attempt to successively estimate f''(x0)
   by building up gradient information as the iterations proceed.

   In the least squares problem there are further simplifications
   which assist in solving eqn (2).  The function to be minimized is
   a sum of squares:

       f = Sum(hi^2)                                         (3)

   where hi is the ith residual out of m residuals as described
   above.  This can be substituted back into eqn (2) after computing
   the derivatives:

       f'  = 2 Sum(hi  hi')     
       f'' = 2 Sum(hi' hj') + 2 Sum(hi hi'')                (4)

   If one assumes that the parameters are already close enough to a
   minimum, then one typically finds that the second term in f'' is
   negligible [or, in any case, is too difficult to compute].  Thus,
   equation (2) can be solved, at least approximately, using only
   gradient information.

   In matrix notation, the combination of eqns (2) and (4) becomes:

        hT' . h' . dx = - hT' . h                          (5)

   Where h is the residual vector (length m), hT is its transpose, h'
   is the Jacobian matrix (dimensions n x m), and dx is (xm-x0).  The
   user function supplies the residual vector h, and in some cases h'
   when it is not found by finite differences (see MPFIT_FDJAC2,
   which finds h and hT').  Even if dx is not the best absolute step
   to take, it does provide a good estimate of the best *direction*,
   so often a line minimization will occur along the dx vector
   direction.

   The method of solution employed by MINPACK is to form the Q . R
   factorization of h', where Q is an orthogonal matrix such that QT .
   Q = I, and R is upper right triangular.  Using h' = Q . R and the
   ortogonality of Q, eqn (5) becomes

        (RT . QT) . (Q . R) . dx = - (RT . QT) . h
                     RT . R . dx = - RT . QT . h         (6)
                          R . dx = - QT . h

   where the last statement follows because R is upper triangular.
   Here, R, QT and h are known so this is a matter of solving for dx.
   The routine MPFIT_QRFAC provides the QR factorization of h, with
   pivoting, and MPFIT_QRSOLV provides the solution for dx.
   
 REFERENCES:

   MINPACK-1, Jorge More', available from netlib (www.netlib.org).
   "Optimization Software Guide," Jorge More' and Stephen Wright, 
     SIAM, *Frontiers in Applied Mathematics*, Number 14.
   More', Jorge J., "The Levenberg-Marquardt Algorithm:
     Implementation and Theory," in *Numerical Analysis*, ed. Watson,
     G. A., Lecture Notes in Mathematics 630, Springer-Verlag, 1977.

 MODIFICATION HISTORY:
   Translated from MINPACK-1 in FORTRAN, Apr-Jul 1998, CM
   Fixed bug in parameter limits (x vs xnew), 04 Aug 1998, CM
   Added PERROR keyword, 04 Aug 1998, CM
   Added COVAR keyword, 20 Aug 1998, CM
   Added NITER output keyword, 05 Oct 1998
      D.L Windt, Bell Labs, windt@bell-labs.com;
   Made each PARINFO component optional, 05 Oct 1998 CM
   Analytical derivatives allowed via AUTODERIVATIVE keyword, 09 Nov 1998
   Parameter values can be tied to others, 09 Nov 1998
   Fixed small bugs (Wayne Landsman), 24 Nov 1998
   Added better exception error reporting, 24 Nov 1998 CM
   Cosmetic documentation changes, 02 Jan 1999 CM
   Changed definition of ITERPROC to be consistent with TNMIN, 19 Jan 1999 CM
   Fixed bug when AUTDERIVATIVE=0.  Incorrect sign, 02 Feb 1999 CM
   Added keyboard stop to MPFIT_DEFITER, 28 Feb 1999 CM
   Cosmetic documentation changes, 14 May 1999 CM
   IDL optimizations for speed & FASTNORM keyword, 15 May 1999 CM
   Tried a faster version of mpfit_enorm, 30 May 1999 CM
   Changed web address to cow.physics.wisc.edu, 14 Jun 1999 CM
   Found malformation of FDJAC in MPFIT for 1 parm, 03 Aug 1999 CM
   Factored out user-function call into MPFIT_CALL.  It is possible,
     but currently disabled, to call procedures.  The calling format
     is similar to CURVEFIT, 25 Sep 1999, CM
   Slightly changed mpfit_tie to be less intrusive, 25 Sep 1999, CM
   Fixed some bugs associated with tied parameters in mpfit_fdjac, 25
     Sep 1999, CM
   Reordered documentation; now alphabetical, 02 Oct 1999, CM
   Added QUERY keyword for more robust error detection in drivers, 29
     Oct 1999, CM
   Documented PERROR for unweighted fits, 03 Nov 1999, CM
   Split out MPFIT_RESETPROF to aid in profiling, 03 Nov 1999, CM
   Some profiling and speed optimization, 03 Nov 1999, CM
     Worst offenders, in order: fdjac2, qrfac, qrsolv, enorm.
     fdjac2 depends on user function, qrfac and enorm seem to be
     fully optimized.  qrsolv probably could be tweaked a little, but
     is still <10% of total compute time.
   Made sure that !err was set to 0 in MPFIT_DEFITER, 10 Jan 2000, CM
   Fixed small inconsistency in setting of QANYLIM, 28 Jan 2000, CM
   Added PARINFO field RELSTEP, 28 Jan 2000, CM
   Converted to MPFIT_ERROR common block for indicating error
     conditions, 28 Jan 2000, CM
   Corrected scope of MPFIT_ERROR common block, CM, 07 Mar 2000
   Minor speed improvement in MPFIT_ENORM, CM 26 Mar 2000
   Corrected case where ITERPROC changed parameter values and
     parameter values were TIED, CM 26 Mar 2000
   Changed MPFIT_CALL to modify NFEV automatically, and to support
     user procedures more, CM 26 Mar 2000
   Copying permission terms have been liberalized, 26 Mar 2000, CM
   Catch zero value of zero a(j,lj) in MPFIT_QRFAC, 20 Jul 2000, CM
      (thanks to David Schlegel )
   MPFIT_SETMACHAR is called only once at init; only one common block
     is created (MPFIT_MACHAR); it is now a structure; removed almost
     all CHECK_MATH calls for compatibility with IDL5 and !EXCEPT;
     profiling data is now in a structure too; noted some
     mathematical discrepancies in Linux IDL5.0, 17 Nov 2000, CM
   Some significant changes.  New PARINFO fields: MPSIDE, MPMINSTEP,
     MPMAXSTEP.  Improved documentation.  Now PTIED constraints are
     maintained in the MPCONFIG common block.  A new procedure to
     parse PARINFO fields.  FDJAC2 now computes a larger variety of
     one-sided and two-sided finite difference derivatives.  NFEV is
     stored in the MPCONFIG common now.  17 Dec 2000, CM
   Added check that PARINFO and XALL have same size, 29 Dec 2000 CM
   Don't call function in TERMINATE when there is an error, 05 Jan
     2000
   Check for float vs. double discrepancies; corrected implementation
     of MIN/MAXSTEP, which I still am not sure of, but now at least
     the correct behavior occurs *without* it, CM 08 Jan 2001
   Added SCALE_FCN keyword, to allow for scaling, as for the CASH
     statistic; added documentation about the theory of operation,
     and under the QR factorization; slowly I'm beginning to
     understand the bowels of this algorithm, CM 10 Jan 2001
   Remove MPMINSTEP field of PARINFO, for now at least, CM 11 Jan
     2001
   Added RESDAMP keyword, CM, 14 Jan 2001
   Tried to improve the DAMP handling a little, CM, 13 Mar 2001
   Corrected .PARNAME behavior in _DEFITER, CM, 19 Mar 2001
   Added checks for parameter and function overflow; a new STATUS
     value to reflect this; STATUS values of -15 to -1 are reserved
     for user function errors, CM, 03 Apr 2001
   DAMP keyword is now a TANH, CM, 03 Apr 2001
   Added more error checking of float vs. double, CM, 07 Apr 2001
   Fixed bug in handling of parameter lower limits; moved overflow
     checking to end of loop, CM, 20 Apr 2001
   Failure using GOTO, TERMINATE more graceful if FNORM1 not defined,
     CM, 13 Aug 2001
   Add MPPRINT tag to PARINFO, CM, 19 Nov 2001
   Add DOF keyword to DEFITER procedure, and print degrees of
     freedom, CM, 28 Nov 2001
   Add check to be sure MYFUNCT is a scalar string, CM, 14 Jan 2002
   Addition of EXTERNAL_FJAC, EXTERNAL_FVEC keywords; ability to save
     fitter's state from one call to the next; allow '(EXTERNAL)'
     function name, which implies that user will supply function and
     Jacobian at each iteration, CM, 10 Mar 2002
   Documented EXTERNAL evaluation code, CM, 10 Mar 2002
   Corrected signficant bug in the way that the STEP parameter, and
     FIXED parameters interacted (Thanks Andrew Steffl), CM, 02 Apr
     2002
   Allow COVAR and PERROR keywords to be computed, even in case of
     '(EXTERNAL)' function, 26 May 2002
   Add NFREE and NPEGGED keywords; compute NPEGGED; compute DOF using
     NFREE instead of n_elements(X), thanks to Kristian Kjaer, CM 11
     Sep 2002
   Hopefully PERROR is all positive now, CM 13 Sep 2002
   Documented RELSTEP field of PARINFO (!!), CM, 25 Oct 2002
   Error checking to detect missing start pars, CM 12 Apr 2003
   Add DOF keyword to return degrees of freedom, CM, 30 June 2003
   Always call ITERPROC in the final iteration; add ITERKEYSTOP
     keyword, CM, 30 June 2003
   Correct bug in MPFIT_LMPAR of singularity handling, which might
     likely be fatal for one-parameter fits, CM, 21 Nov 2003
     (with thanks to Peter Tuthill for the proper test case)
   Minor documentation adjustment, 03 Feb 2004, CM
   Correct small error in QR factorization when pivoting; document
     the return values of QRFAC when pivoting, 21 May 2004, CM
   Add MPFORMAT field to PARINFO, and correct behavior of interaction
     between MPPRINT and PARNAME in MPFIT_DEFITERPROC (thanks to Tim
     Robishaw), 23 May 2004, CM

  $Id: mpfit.pro,v 1.33 2004/05/23 07:36:21 craigm Exp $

(See /dzd2/heiles/idl/gen/mpfit/mpfit.pro)


MPFIT2DFUN

[Previous Routine] [Next Routine] [List of Routines]
 NAME:
   MPFIT2DFUN

 AUTHOR:
   Craig B. Markwardt, NASA/GSFC Code 662, Greenbelt, MD 20770
   craigm@lheamail.gsfc.nasa.gov
   UPDATED VERSIONs can be found on my WEB PAGE: 
      http://cow.physics.wisc.edu/~craigm/idl/idl.html

 PURPOSE:
   Perform Levenberg-Marquardt least-squares fit to a 2-D IDL function

 MAJOR TOPICS:
   Curve and Surface Fitting

 CALLING SEQUENCE:
   parms = MPFIT2DFUN(MYFUNCT, X, Y, Z, ERR, start_parms, ...)

 DESCRIPTION:

  MPFIT2DFUN fits a user-supplied model -- in the form of an IDL
  function -- to a set of user-supplied data.  MPFIT2DFUN calls
  MPFIT, the MINPACK-1 least-squares minimizer, to do the main
  work.  MPFIT2DFUN is a specialized version for two-dimensional 
  data.

  Given the data and their uncertainties, MPFIT2DFUN finds the best set
  of model parameters which match the data (in a least-squares
  sense) and returns them in an array.
  
  The user must supply the following items:
   - Two arrays of independent variable values ("X", "Y").
   - An array of "measured" *dependent* variable values ("Z").
   - An array of "measured" 1-sigma uncertainty values ("ERR").
   - The name of an IDL function which computes Z given (X,Y) ("MYFUNCT").
   - Starting guesses for all of the parameters ("START_PARAMS").

  There are very few restrictions placed on X, Y, Z, or MYFUNCT.
  Simply put, MYFUNCT must map the (X,Y) values into Z values given
  the model parameters.  The (X,Y) values are usually the independent
  X and Y coordinate positions in the two dimensional plane, but need
  not be.

  MPFIT2DFUN carefully avoids passing large arrays where possible to
  improve performance.

  See below for an example of usage.
   
 USER FUNCTION

  The user must define a function which returns the model value.  For
  applications which use finite-difference derivatives -- the default
  -- the user function should be declared in the following way:

    FUNCTION MYFUNCT, X, Y, P
     ; The independent variables are X and Y
     ; Parameter values are passed in "P"
     ZMOD = ... computed model values at (X,Y) ...
     return, ZMOD
    END

  The returned array YMOD must have the same dimensions and type as
  the "measured" Z values.

  User functions may also indicate a fatal error condition
  using the ERROR_CODE common block variable, as described
  below under the MPFIT_ERROR common block definition.

  See the discussion under "ANALYTIC DERIVATIVES" and AUTODERIVATIVE
  in MPFIT.PRO if you wish to compute the derivatives for yourself.
  AUTODERIVATIVE is accepted and passed directly to MPFIT.  The user
  function must accept one additional parameter, DP, which contains
  the derivative of the user function with respect to each parameter
  at each data point, as described in MPFIT.PRO.

 CREATING APPROPRIATELY DIMENSIONED INDEPENDENT VARIABLES

  The user must supply appropriate independent variables to
  MPFIT2DFUN.  For image fitting applications, this variable should
  be two-dimensional *arrays* describing the X and Y positions of
  every *pixel*.  [ Thus any two dimensional sampling is permitted,
  including irregular sampling. ]
  
  If the sampling is regular, then the x coordinates are the same for
  each row, and the y coordinates are the same for each column.  Call
  the x-row and y-column coordinates XR and YC respectively.  You can
  then compute X and Y as follows:
  
      X = XR # (YC*0 + 1)             eqn. 1
      Y = (XR*0 + 1) # YC             eqn. 2
  
  For example, if XR and YC have the following values:
  
    XR = [  1, 2, 3, 4, 5,]  ;; X positions of one row of pixels
    YC = [ 15,16,17 ]        ;; Y positions of one column of
                                pixels
  
  Then using equations 1 and 2 above will give these values to X and
  Y:
  
     X :  1  2  3  4  5       ;; X positions of all pixels
          1  2  3  4  5
          1  2  3  4  5
  
     Y : 15 15 15 15 15       ;; Y positions of all pixels
         16 16 16 16 16
         17 17 17 17 17
  
  Using the above technique is suggested, but *not* required.  You
  can do anything you wish with the X and Y values.  This technique
  only makes it easier to compute your model function values.

 CONSTRAINING PARAMETER VALUES WITH THE PARINFO KEYWORD

  The behavior of MPFIT can be modified with respect to each
  parameter to be fitted.  A parameter value can be fixed; simple
  boundary constraints can be imposed; limitations on the parameter
  changes can be imposed; properties of the automatic derivative can
  be modified; and parameters can be tied to one another.

  These properties are governed by the PARINFO structure, which is
  passed as a keyword parameter to MPFIT.

  PARINFO should be an array of structures, one for each parameter.
  Each parameter is associated with one element of the array, in
  numerical order.  The structure can have the following entries
  (none are required):
  
     .VALUE - the starting parameter value (but see the START_PARAMS
              parameter for more information).
  
     .FIXED - a boolean value, whether the parameter is to be held
              fixed or not.  Fixed parameters are not varied by
              MPFIT, but are passed on to MYFUNCT for evaluation.
  
     .LIMITED - a two-element boolean array.  If the first/second
                element is set, then the parameter is bounded on the
                lower/upper side.  A parameter can be bounded on both
                sides.  Both LIMITED and LIMITS must be given
                together.
  
     .LIMITS - a two-element float or double array.  Gives the
               parameter limits on the lower and upper sides,
               respectively.  Zero, one or two of these values can be
               set, depending on the values of LIMITED.  Both LIMITED
               and LIMITS must be given together.
  
     .PARNAME - a string, giving the name of the parameter.  The
                fitting code of MPFIT does not use this tag in any
                way.  However, the default ITERPROC will print the
                parameter name if available.
  
     .STEP - the step size to be used in calculating the numerical
             derivatives.  If set to zero, then the step size is
             computed automatically.  Ignored when AUTODERIVATIVE=0.
             This value is superceded by the RELSTEP value.

     .RELSTEP - the *relative* step size to be used in calculating
                the numerical derivatives.  This number is the
                fractional size of the step, compared to the
                parameter value.  This value supercedes the STEP
                setting.  If the parameter is zero, then a default
                step size is chosen.

     .MPSIDE - the sidedness of the finite difference when computing
               numerical derivatives.  This field can take four
               values:

                  0 - one-sided derivative computed automatically
                  1 - one-sided derivative (f(x+h) - f(x)  )/h
                 -1 - one-sided derivative (f(x)   - f(x-h))/h
                  2 - two-sided derivative (f(x+h) - f(x-h))/(2*h)

              Where H is the STEP parameter described above.  The
              "automatic" one-sided derivative method will chose a
              direction for the finite difference which does not
              violate any constraints.  The other methods do not
              perform this check.  The two-sided method is in
              principle more precise, but requires twice as many
              function evaluations.  Default: 0.

     .MPMINSTEP - the minimum change to be made in the parameter
                  value.  During the fitting process, the parameter
                  will be changed by multiples of this value.  The
                  actual step is computed as:

                     DELTA1 = MPMINSTEP*ROUND(DELTA0/MPMINSTEP)

                  where DELTA0 and DELTA1 are the estimated parameter
                  changes before and after this constraint is
                  applied.  Note that this constraint should be used
                  with care since it may cause non-converging,
                  oscillating solutions.

                  A value of 0 indicates no minimum.  Default: 0.

     .MPMAXSTEP - the maximum change to be made in the parameter
                  value.  During the fitting process, the parameter
                  will never be changed by more than this value.

                  A value of 0 indicates no maximum.  Default: 0.
  
     .TIED - a string expression which "ties" the parameter to other
             free or fixed parameters.  Any expression involving
             constants and the parameter array P are permitted.
             Example: if parameter 2 is always to be twice parameter
             1 then use the following: parinfo(2).tied = '2 * P(1)'.
             Since they are totally constrained, tied parameters are
             considered to be fixed; no errors are computed for them.
             [ NOTE: the PARNAME can't be used in expressions. ]
  
  Future modifications to the PARINFO structure, if any, will involve
  adding structure tags beginning with the two letters "MP".
  Therefore programmers are urged to avoid using tags starting with
  the same letters; otherwise they are free to include their own
  fields within the PARINFO structure, and they will be ignored.
  
  PARINFO Example:
  parinfo = replicate({value:0.D, fixed:0, limited:[0,0], $
                       limits:[0.D,0]}, 5)
  parinfo(0).fixed = 1
  parinfo(4).limited(0) = 1
  parinfo(4).limits(0)  = 50.D
  parinfo(*).value = [5.7D, 2.2, 500., 1.5, 2000.]
  
  A total of 5 parameters, with starting values of 5.7,
  2.2, 500, 1.5, and 2000 are given.  The first parameter
  is fixed at a value of 5.7, and the last parameter is
  constrained to be above 50.

 INPUTS:
   MYFUNCT - a string variable containing the name of an IDL
             function.  This function computes the "model" Z values
             given the X,Y values and model parameters, as described above.

   X - Array of "X" independent variable values, as described above.
       These values are passed directly to the fitting function
       unmodified.

   Y - Array of "Y" independent variable values, as described
       above. X and Y should have the same data type.

   Z - Array of "measured" dependent variable values.  Z should have
       the same data type as X and Y.  The function MYFUNCT should
       map (X,Y)->Z.

   ERR - Array of "measured" 1-sigma uncertainties.  ERR should have
         the same data type as Z.  ERR is ignored if the WEIGHTS
         keyword is specified.

   START_PARAMS - An array of starting values for each of the
                  parameters of the model.  The number of parameters
                  should be fewer than the number of measurements.
                  Also, the parameters should have the same data type
                  as the measurements (double is preferred).

                  This parameter is optional if the PARINFO keyword
                  is used (see MPFIT).  The PARINFO keyword provides
                  a mechanism to fix or constrain individual
                  parameters.  If both START_PARAMS and PARINFO are
                  passed, then the starting *value* is taken from
                  START_PARAMS, but the *constraints* are taken from
                  PARINFO.
 
 RETURNS:

   Returns the array of best-fit parameters.

 KEYWORD PARAMETERS:

   BESTNORM - the value of the summed, squared, weighted residuals
              for the returned parameter values, i.e. the chi-square value.

   COVAR - the covariance matrix for the set of parameters returned
           by MPFIT.  The matrix is NxN where N is the number of
           parameters.  The square root of the diagonal elements
           gives the formal 1-sigma statistical errors on the
           parameters IF errors were treated "properly" in MYFUNC.
           Parameter errors are also returned in PERROR.

           To compute the correlation matrix, PCOR, use this:
           IDL> PCOR = COV * 0
           IDL> FOR i = 0, n-1 DO FOR j = 0, n-1 DO $
                PCOR(i,j) = COV(i,j)/sqrt(COV(i,i)*COV(j,j))

           If NOCOVAR is set or MPFIT terminated abnormally, then
           COVAR is set to a scalar with value !VALUES.D_NAN.

   DOF - number of degrees of freedom, computed as
             DOF = N_ELEMENTS(DEVIATES) - NFREE
         Note that this doesn't account for pegged parameters (see
         NPEGGED).

   ERRMSG - a string error or warning message is returned.

   FTOL - a nonnegative input variable. Termination occurs when both
          the actual and predicted relative reductions in the sum of
          squares are at most FTOL (and STATUS is accordingly set to
          1 or 3).  Therefore, FTOL measures the relative error
          desired in the sum of squares.  Default: 1D-10

   FUNCTARGS - A structure which contains the parameters to be passed
               to the user-supplied function specified by MYFUNCT via
               the _EXTRA mechanism.  This is the way you can pass
               additional data to your user-supplied function without
               using common blocks.

               By default, no extra parameters are passed to the
               user-supplied function.

   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 (and STATUS is
          accordingly set to 4). Therefore, GTOL measures the
          orthogonality desired between the function vector and the
          columns of the jacobian.  Default: 1D-10

   ITERARGS - The keyword arguments to be passed to ITERPROC via the
              _EXTRA mechanism.  This should be a structure, and is
              similar in operation to FUNCTARGS.
              Default: no arguments are passed.

   ITERPROC - The name of a procedure to be called upon each NPRINT
              iteration of the MPFIT routine.  It should be declared
              in the following way:

              PRO ITERPROC, MYFUNCT, p, iter, fnorm, FUNCTARGS=fcnargs, $
                PARINFO=parinfo, QUIET=quiet, ...
                ; perform custom iteration update
              END
         
              ITERPROC must either accept all three keyword
              parameters (FUNCTARGS, PARINFO and QUIET), or at least
              accept them via the _EXTRA keyword.
          
              MYFUNCT is the user-supplied function to be minimized,
              P is the current set of model parameters, ITER is the
              iteration number, and FUNCTARGS are the arguments to be
              passed to MYFUNCT.  FNORM should be the
              chi-squared value.  QUIET is set when no textual output
              should be printed.  See below for documentation of
              PARINFO.

              In implementation, ITERPROC can perform updates to the
              terminal or graphical user interface, to provide
              feedback while the fit proceeds.  If the fit is to be
              stopped for any reason, then ITERPROC should set the
              common block variable ERROR_CODE to negative value (see
              MPFIT_ERROR common block below).  In principle,
              ITERPROC should probably not modify the parameter
              values, because it may interfere with the algorithm's
              stability.  In practice it is allowed.

              Default: an internal routine is used to print the
                       parameter values.

   MAXITER - The maximum number of iterations to perform.  If the
             number is exceeded, then the STATUS value is set to 5
             and MPFIT returns.
             Default: 200 iterations

   NFEV - the number of MYFUNCT function evaluations performed.

   NITER - the number of iterations completed.

   NOCOVAR - set this keyword to prevent the calculation of the
             covariance matrix before returning (see COVAR)

   NPRINT - The frequency with which ITERPROC is called.  A value of
            1 indicates that ITERPROC is called with every iteration,
            while 2 indicates every other iteration, etc.  Note that
            several Levenberg-Marquardt attempts can be made in a
            single iteration.
            Default value: 1

   PARINFO - Provides a mechanism for more sophisticated constraints
             to be placed on parameter values.  When PARINFO is not
             passed, then it is assumed that all parameters are free
             and unconstrained.  Values in PARINFO are never 
             modified during a call to MPFIT.

             See description above for the structure of PARINFO.

             Default value:  all parameters are free and unconstrained.

   PERROR - The formal 1-sigma errors in each parameter, computed
            from the covariance matrix.  If a parameter is held
            fixed, or if it touches a boundary, then the error is
            reported as zero.

            If the fit is unweighted (i.e. no errors were given, or
            the weights were uniformly set to unity), then PERROR
            will probably not represent the true parameter
            uncertainties.  *If* you can assume that the true reduced
            chi-squared value is unity -- meaning that the fit is
            implicitly assumed to be of good quality -- then the
            estimated parameter uncertainties can be computed by
            scaling PERROR by the measured chi-squared value.

              DOF     = N_ELEMENTS(Z) - N_ELEMENTS(PARMS) ; deg of freedom
              PCERROR = PERROR * SQRT(BESTNORM / DOF)   ; scaled uncertainties

   QUIET - set this keyword when no textual output should be printed
           by MPFIT

   STATUS - an integer status code is returned.  All values other
            than zero can represent success.  It can have one of the
            following values:

	   0  improper input parameters.
         
	   1  both actual and predicted relative reductions
	      in the sum of squares are at most FTOL.
         
	   2  relative error between two consecutive iterates
	      is at most XTOL
         
	   3  conditions for STATUS = 1 and STATUS = 2 both hold.
         
	   4  the cosine of the angle between fvec and any
	      column of the jacobian is at most GTOL in
	      absolute value.
         
	   5  the maximum number of iterations has been reached
         
	   6  FTOL is too small. no further reduction in
	      the sum of squares is possible.
         
	   7  XTOL is too small. no further improvement in
	      the approximate solution x is possible.
         
	   8  GTOL is too small. fvec is orthogonal to the
	      columns of the jacobian to machine precision.

   WEIGHTS - Array of weights to be used in calculating the
             chi-squared value.  If WEIGHTS is specified then the ERR
             parameter is ignored.  The chi-squared value is computed
             as follows:

                CHISQ = TOTAL( (Z-MYFUNCT(X,Y,P))^2 * ABS(WEIGHTS) )

             Here are common values of WEIGHTS:

                1D/ERR^2 - Normal weighting (ERR is the measurement error)
                1D/Z     - Poisson weighting (counting statistics)
                1D       - Unweighted

   XTOL - a nonnegative input variable. Termination occurs when the
          relative error between two consecutive iterates is at most
          XTOL (and STATUS is accordingly set to 2 or 3).  Therefore,
          XTOL measures the relative error desired in the approximate
          solution.  Default: 1D-10

   YFIT - the best-fit model function, as returned by MYFUNCT.

 EXAMPLE:

   p  = [2.2D, -0.7D, 1.4D, 3000.D]
   x  = (dindgen(200)*0.1 - 10.) # (dblarr(200) + 1)
   y  = (dblarr(200) + 1) # (dindgen(200)*0.1 - 10.)
   zi = gauss2(x, y, p)
   sz = sqrt(zi)
   z  = zi + randomn(seed, 200, 200) * sz

   p0 = [0D, 0D, 1D, 10D]
   p = mpfit2dfun('GAUSS2', x, y, z, sz, p0)
   
   Generates a synthetic data set with a Gaussian peak, and Poisson
   statistical uncertainty.  Then the same function (but different
   starting parameters) is fitted to the data to see how close we can
   get.

   It is especially worthy to notice that the X and Y values are
   created as full images, so that a coordinate is attached to each
   pixel independently.  This is the format that GAUSS2 accepts, and
   the easiest for you to use in your own functions.


 COMMON BLOCKS:

   COMMON MPFIT_ERROR, ERROR_CODE

     User routines may stop the fitting process at any time by
     setting an error condition.  This condition may be set in either
     the user's model computation routine (MYFUNCT), or in the
     iteration procedure (ITERPROC).

     To stop the fitting, the above common block must be declared,
     and ERROR_CODE must be set to a negative number.  After the user
     procedure or function returns, MPFIT checks the value of this
     common block variable and exits immediately if the error
     condition has been set.  By default the value of ERROR_CODE is
     zero, indicating a successful function/procedure call.


 REFERENCES:

   MINPACK-1, Jorge More', available from netlib (www.netlib.org).
   "Optimization Software Guide," Jorge More' and Stephen Wright, 
     SIAM, *Frontiers in Applied Mathematics*, Number 14.

 MODIFICATION HISTORY:
   Written, transformed from MPFITFUN, 26 Sep 1999, CM
   Alphabetized documented keywords, 02 Oct 1999, CM
   Added example, 02 Oct 1999, CM
   Tried to clarify definitions of X and Y, 29 Oct 1999, CM
   Added QUERY keyword and query checking of MPFIT, 29 Oct 1999, CM
   Check to be sure that X, Y and Z are present, 02 Nov 1999, CM
   Documented PERROR for unweighted fits, 03 Nov 1999, CM
   Changed to ERROR_CODE for error condition, 28 Jan 2000, CM
   Copying permission terms have been liberalized, 26 Mar 2000, CM
   Propagated improvements from MPFIT, 17 Dec 2000, CM
   Documented RELSTEP field of PARINFO (!!), CM, 25 Oct 2002
   Add DOF keyword to return degrees of freedom, CM, 23 June 2003
   Minor documentation adjustment, 03 Feb 2004, CM

  $Id: mpfit2dfun.pro,v 1.5 2004/02/26 04:18:16 craigm Exp $

(See /dzd2/heiles/idl/gen/mpfit/mpfit2dfun.pro)


MPFIT2DPEAK

[Previous Routine] [Next Routine] [List of Routines]
 NAME:
   MPFIT2DPEAK

 AUTHOR:
   Craig B. Markwardt, NASA/GSFC Code 662, Greenbelt, MD 20770
   craigm@lheamail.gsfc.nasa.gov
   UPDATED VERSIONs can be found on my WEB PAGE: 
      http://cow.physics.wisc.edu/~craigm/idl/idl.html

 PURPOSE:
   Fit a gaussian, lorentzian or Moffat model to data

 MAJOR TOPICS:
   Curve and Surface Fitting

 CALLING SEQUENCE:
   yfit = MPFIT2DPEAK(Z, A [, X, Y, /TILT ...] )

 DESCRIPTION:

   MPFIT2DPEAK fits a gaussian, lorentzian or Moffat model using the
   non-linear least squares fitter MPFIT.  MPFIT2DPEAK is meant to be
   a drop-in replacement for IDL's GAUSS2DFIT function (and requires
   MPFIT and MPFIT2DFUN).

   The choice of the fitting function is determined by the keywords
   GAUSSIAN, LORENTZIAN and MOFFAT.  By default the gaussian model
   function is used.  [ The Moffat function is a modified Lorentzian
   with variable power law index. ]  The two-dimensional peak has
   independent semimajor and semiminor axes, with an optional
   rotation term activated by setting the TILT keyword.  The baseline
   is assumed to be a constant.

      GAUSSIAN      A(0) + A(1)*exp(-0.5*u)
      LORENTZIAN    A(0) + A(1)/(u + 1)
      MOFFAT        A(0) + A(1)/(u + 1)^A(7)

      u = ( (x-A(4))/A(2) )^2 + ( (y-A(5))/A(3) )^2

         where x and y are cartesian coordinates in rotated
         coordinate system if TILT keyword is set.

   The returned parameter array elements have the following meanings:

      A(0)   Constant baseline level
      A(1)   Peak value
      A(2)   Peak half-width (x) -- gaussian sigma or half-width at half-max
      A(3)   Peak half-width (y) -- gaussian sigma or half-width at half-max
      A(4)   Peak centroid (x)
      A(5)   Peak centroid (y)
      A(6)   Rotation angle (radians) if TILT keyword set
      A(7)   Moffat power law index if MOFFAT keyword set

   By default the initial starting values for the parameters A are
   estimated from the data.  However, explicit starting values can be
   supplied using the ESTIMATES keyword.  Also, error or weighting
   values can optionally be provided; otherwise the fit is
   unweighted.

 RESTRICTIONS:

   If no starting parameter ESTIMATES are provided, then MPFIT2DPEAK
   attempts to estimate them from the data.  This is not a perfect
   science; however, the author believes that the technique
   implemented here is more robust than the one used in IDL's
   GAUSS2DFIT.  The author has tested cases of strong peaks, noisy
   peaks and broad peaks, all with success.


 INPUTS:

   Z - Two dimensional array of "measured" dependent variable values.
       Z should be of the same type and dimension as (X # Y).

   X - Optional vector of x positions for a single row of Z.

          X(i) should provide the x position of Z(i,*)

       Default: X values are integer increments from 0 to NX-1

   Y - Optional vector of y positions for a single column of Z.

          Y(j) should provide the y position of Z(*,j)

       Default: Y values are integer increments from 0 to NY-1

 OUTPUTS:
   A - Upon return, an array of best fit parameter values.  See the
       table above for the meanings of each parameter element.


 RETURNS:

   Returns the best fitting model function as a 2D array.

 KEYWORDS:

   ** NOTE ** Additional keywords such as PARINFO, BESTNORM, and
              STATUS are accepted by MPFIT2DPEAK but not documented
              here.  Please see the documentation for MPFIT for the
              description of these advanced options.

   CHISQ - the value of the summed squared residuals for the
           returned parameter values.

   CIRCULAR - if set, then the peak profile is assumed to be
              azimuthally symmetric.  When set, the parameters A(2)
              and A(3) will be identical and the TILT keyword will
              have no effect.

   DOF - number of degrees of freedom, computed as
             DOF = N_ELEMENTS(DEVIATES) - NFREE
         Note that this doesn't account for pegged parameters (see
         NPEGGED).

   ERROR - upon input, the measured 1-sigma uncertainties in the "Z"
           values.  If no ERROR or WEIGHTS are given, then the fit is
           unweighted.

   ESTIMATES - Array of starting values for each parameter of the
               model.
               Default: parameter values are estimated from data.

   GAUSSIAN - if set, fit a gaussian model function.  The Default.
   LORENTZIAN - if set, fit a lorentzian model function.
   MOFFAT - if set, fit a Moffat model function.

   MEASURE_ERRORS - synonym for ERRORS, for consistency with built-in
                    IDL fitting routines.

   NEGATIVE - if set, and ESTIMATES is not provided, then MPFIT2DPEAK
              will assume that a negative peak is present -- a
              valley.  Specifying this keyword is not normally
              required, since MPFIT2DPEAK can determine this
              automatically.

   NFREE - the number of free parameters in the fit.  This includes
           parameters which are not FIXED and not TIED, but it does
           include parameters which are pegged at LIMITS.

   PERROR - upon return, the 1-sigma uncertainties of the parameter
            values A.  These values are only meaningful if the ERRORS
            or WEIGHTS keywords are specified properly.

            If the fit is unweighted (i.e. no errors were given, or
            the weights were uniformly set to unity), then PERROR
            will probably not represent the true parameter
            uncertainties.  

            *If* you can assume that the true reduced chi-squared
            value is unity -- meaning that the fit is implicitly
            assumed to be of good quality -- then the estimated
            parameter uncertainties can be computed by scaling PERROR
            by the measured chi-squared value.

              DOF     = N_ELEMENTS(Z) - N_ELEMENTS(A)   ; deg of freedom
              PCERROR = PERROR * SQRT(BESTNORM / DOF)   ; scaled uncertainties

   QUIET - if set then diagnostic fitting messages are suppressed.
           Default: QUIET=1 (i.e., no diagnostics)

   SIGMA - synonym for PERROR (1-sigma parameter uncertainties), for
           compatibility with GAUSSFIT.  Do not confuse this with the
           Gaussian "sigma" width parameter.

   TILT - if set, then the major and minor axes of the peak profile
          are allowed to rotate with respect to the image axes.
          Parameter A(6) will be set to the clockwise rotation angle
          of the A(2) axis in radians.

   WEIGHTS - Array of weights to be used in calculating the
             chi-squared value.  If WEIGHTS is specified then the ERR
             parameter is ignored.  The chi-squared value is computed
             as follows:

                CHISQ = TOTAL( (Z-MYFUNCT(X,Y,P))^2 * ABS(WEIGHTS) )

             Here are common values of WEIGHTS:

                1D/ERR^2 - Normal weighting (ERR is the measurement error)
                1D/Y     - Poisson weighting (counting statistics)
                1D       - Unweighted

             The ERROR keyword takes precedence over any WEIGHTS
             keyword values.  If no ERROR or WEIGHTS are given, then
             the fit is unweighted.


 EXAMPLE:

 ; Construct a sample gaussian surface in range [-5,5] centered at [2,-3]
   x = findgen(100)*0.1 - 5. & y = x
   xx = x # (y*0 + 1)
   yy = (x*0 + 1) # y
   rr = sqrt((xx-2.)^2 + (yy+3.)^2)

 ; Gaussian surface with sigma=0.5, peak value of 3, noise with sigma=0.2
   z = 3.*exp(-(rr/0.5)^2) + randomn(seed,100,100)*.2

 ; Fit gaussian parameters A
   zfit = mpfit2dpeak(z, a, x, y)

 REFERENCES:

   MINPACK-1, Jorge More', available from netlib (www.netlib.org).
   "Optimization Software Guide," Jorge More' and Stephen Wright, 
     SIAM, *Frontiers in Applied Mathematics*, Number 14.

 MODIFICATION HISTORY:

   New algorithm for estimating starting values, CM, 31 Oct 1999
   Documented, 02 Nov 1999
   Small documentation fixes, 02 Nov 1999
   Documented PERROR for unweighted fits, 03 Nov 1999, CM
   Copying permission terms have been liberalized, 26 Mar 2000, CM
   Small cosmetic changes, 21 Sep 2000, CM
   Corrected bug introduced by cosmetic changes, 11 Oct 2000, CM :-)
   Added POSITIVE keyword, 17 Nov 2000, CM
   Removed TILT in common, in favor of FUNCTARGS approach, 23 Nov
     2000, CM
   Added SYMMETRIC keyword, documentation for TILT, and an example,
     24 Nov 2000, CM
   Changed SYMMETRIC to CIRCULAR, 17 Dec 2000, CM
   Really change SYMMETRIC to CIRCULAR!, 13 Sep 2002, CM
   Add error messages for SYMMETRIC and CIRCLE, 08 Nov 2002, CM
   Make more consistent with comparable IDL routines, 30 Jun 2003, CM

  $Id: mpfit2dpeak.pro,v 1.5 2003/06/30 21:48:01 craigm Exp $

(See /dzd2/heiles/idl/gen/mpfit/mpfit2dpeak.pro)


MPFITELLIPSE

[Previous Routine] [Next Routine] [List of Routines]
 NAME:
   MPFITELLIPSE

 AUTHOR:
   Craig B. Markwardt, NASA/GSFC Code 662, Greenbelt, MD 20770
   craigm@lheamail.gsfc.nasa.gov
   UPDATED VERSIONs can be found on my WEB PAGE: 
      http://cow.physics.wisc.edu/~craigm/idl/idl.html

 PURPOSE:
   Approximate fit to points forming an ellipse

 MAJOR TOPICS:
   Curve and Surface Fitting

 CALLING SEQUENCE:
   parms = MPFITELLIPSE(X, Y, start_parms, [/TILT, WEIGHTS=wts, ...])

 DESCRIPTION:

   MPFITELLIPSE fits a closed elliptical or circular curve to a two
   dimensional set of data points.  The user specifies the X and Y
   positions of the points, and an optional set of weights.  The
   ellipse may also be tilted at an arbitrary angle.

   The best fitting ellipse parameters are returned from by
   MPFITELLIPSE as a vector, whose values are:

      P(0)   Ellipse semi axis 1
      P(1)   Ellipse semi axis 2   ( = P(0) if CIRCLE keyword set)
      P(2)   Ellipse center - x value
      P(3)   Ellipse center - y value
      P(4)   Ellipse rotation angle (radians) if TILT keyword set

   The user may specify an initial set of trial parameters, but by
   default MPFITELLIPSE will estimate the parameters automatically.

   Users should be aware that in the presence of large amounts of
   noise, namely when the measurement error becomes significant
   compared to the ellipse axis length, then the estimated parameters
   become unreliable.  Generally speaking the computed axes will
   overestimate the true axes.  For example when (SIGMA_R/R) becomes
   0.5, the radius of the ellipse is overestimated by about 40%.

   Users can weight their data as they see appropriate.  However, the
   following prescription for the weighting appears to be the correct
   one, and produces values comparable to the typical chi-squared
   value.

     WEIGHTS = 0.75/(SIGMA_X^2 + SIGMA_Y^2)

   where SIGMA_X and SIGMA_Y are the measurement error vectors in the
   X and Y directions respectively.  However, it should be pointed
   out that this weighting is only appropriate for a set of points
   whose measurement errors are comparable.  If a more robust
   estimation of the parameter values is needed, the so-called
   orthogonal distance regression package should be used (ODRPACK,
   available in FORTRAN at www.netlib.org).

 INPUTS:

   X - measured X positions of the points in the ellipse.
   Y - measured Y positions of the points in the ellipse.

   START_PARAMS - an array of starting values for the ellipse
                  parameters, as described above.  This parameter is
                  optional; if not specified by the user, then the
                  ellipse parameters are estimated automatically from
                  the properties of the data.

 RETURNS:

   Returns the best fitting model ellipse parameters.

 KEYWORDS:

   ** NOTE ** Additional keywords such as PARINFO, BESTNORM, and
              STATUS are accepted by MPFITELLIPSE but not documented
              here.  Please see the documentation for MPFIT for the
              description of these advanced options.

   PERROR - upon return, the 1-sigma uncertainties of the returned
            ellipse parameter values.  These values are only
            meaningful if the WEIGHTS keyword is specified properly.

            If the fit is unweighted (i.e. no errors were given, or
            the weights were uniformly set to unity), then PERROR
            will probably not represent the true parameter
            uncertainties.  

   QUIET - if set then diagnostic fitting messages are suppressed.
           Default: QUIET=1 (i.e., no diagnostics)

   CIRCULAR - if set, then the curve is assumed to be a circle
              instead of ellipse.  When set, the parameters P(0) and
              P(1) will be identical and the TILT keyword will have
              no effect.

   TILT - if set, then the major and minor axes of the ellipse
          are allowed to rotate with respect to the data axes.
          Parameter P(4) will be set to the clockwise rotation angle
          of the P(0) axis in radians.

   WEIGHTS - Array of weights to be used in calculating the
             chi-squared value.  If WEIGHTS is specified then the ERR
             parameter is ignored.  The chi-squared value is computed
             as follows:

                CHISQ = TOTAL( (Z-MYFUNCT(X,Y,P))^2 * ABS(WEIGHTS)^2 )

             Users may wish to follow the guidelines for WEIGHTS
             described above.


 EXAMPLE:

 ; Construct a set of points on an ellipse, with some noise
   ph0 = 2*!pi*randomu(seed,50)
   x =  50. + 32.*cos(ph0) + 4.0*randomn(seed, 50)
   y = -75. + 65.*sin(ph0) + 0.1*randomn(seed, 50)

 ; Compute weights function
   weights = 0.75/(4.0^2 + 0.1^2)

 ; Fit ellipse and plot result
   p = mpfitellipse(x, y)
   plot, x, y, psym=1
   phi = dindgen(101)*2D*!dpi/100
   oplot, p(2)+p(0)*cos(phi), p(3)+p(1)*sin(phi)

 REFERENCES:

   MINPACK-1, Jorge More', available from netlib (www.netlib.org).
   "Optimization Software Guide," Jorge More' and Stephen Wright, 
     SIAM, *Frontiers in Applied Mathematics*, Number 14.

 MODIFICATION HISTORY:

   Ported from MPFIT2DPEAK, 17 Dec 2000, CM
   More documentation, 11 Jan 2001, CM
   Example corrected, 18 Nov 2001, CM
   Change CIRCLE keyword to the correct CIRCULAR keyword, 13 Sep
      2002, CM
   Add error messages for SYMMETRIC and CIRCLE, 08 Nov 2002, CM
   Found small error in computation of _EVAL (when CIRCULAR) was set;
      sanity check when CIRCULAR is set, 21 Jan 2003, CM

  $Id: mpfitellipse.pro,v 1.8 2003/01/24 04:04:30 craigm Exp $

(See /dzd2/heiles/idl/gen/mpfit/mpfitellipse.pro)


MPFITEXPR

[Previous Routine] [Next Routine] [List of Routines]
 NAME:
   MPFITEXPR

 AUTHOR:
   Craig B. Markwardt, NASA/GSFC Code 662, Greenbelt, MD 20770
   craigm@lheamail.gsfc.nasa.gov
   UPDATED VERSIONs can be found on my WEB PAGE: 
      http://cow.physics.wisc.edu/~craigm/idl/idl.html

 PURPOSE:
   Perform Levenberg-Marquardt least-squares fit to arbitrary expression

 MAJOR TOPICS:
   Curve and Surface Fitting

 CALLING SEQUENCE:
   MYFUNCT = 'X*(1-X)+3'
   parms = MPFITEXPR(MYFUNCT, XVAL, YVAL, ERR, start_parms, ...)

 DESCRIPTION:

  MPFITEXPR fits a user-supplied model -- in the form of an arbitrary IDL
  expression -- to a set of user-supplied data.  MPFITEXPR calls
  MPFIT, the MINPACK-1 least-squares minimizer, to do the main
  work.

  Given the data and their uncertainties, MPFITEXPR finds the best set
  of model parameters which match the data (in a least-squares
  sense) and returns them in an array.
  
  The user must supply the following items:
   - An array of independent variable values ("X").
   - An array of "measured" *dependent* variable values ("Y").
   - An array of "measured" 1-sigma uncertainty values ("ERR").
   - A text IDL expression which computes Y given X.
   - Starting guesses for all of the parameters ("START_PARAMS").

  There are very few restrictions placed on X, Y or the expression of
  the model.  Simply put, the expression must map the "X" values into
  "Y" values given the model parameters.  The "X" values may
  represent any independent variable (not just Cartesian X), and
  indeed may be multidimensional themselves.  For example, in the
  application of image fitting, X may be a 2xN array of image
  positions.

  Some rules must be obeyed in constructing the expression.  First,
  the independent variable name *MUST* be "X" in the expression,
  regardless of the name of the variable being passed to MPFITEXPR.
  This is demonstrated in the above calling sequence, where the X
  variable passed in is called "XVAL" but the expression still refers
  to "X".  Second, parameter values must be referred to as an array
  named "P".

  If you do not pass in starting values for the model parameters,
  MPFITEXPR will attempt to determine the number of parameters you
  intend to have (it does this by looking for references to the array
  variable named "P").  When no starting values are passed in, the
  values are assumed to start at zero.

  MPFITEXPR carefully avoids passing large arrays where possible to
  improve performance.

  See below for an example of usage.

 EVALUATING EXPRESSIONS

  This source module also provides a function called MPEVALEXPR.  You
  can use this function to evaluate your expression, given a list of
  parameters.  This is one of the easier ways to compute the model
  once the best-fit parameters have been found.  Here is an example:

       YMOD = MPEVALEXPR(MYFUNCT, XVAL, PARMS)

  where MYFUNCT is the expression (see MYFUNCT below), XVAL is the
  list of "X" values, and PARMS is an array of parameters.  The
  returned array YMOD contains the expression MYFUNCT evaluated at
  each point in XVAL.

 PASSING PRIVATE DATA TO AN EXPRESSION

  The most complicated optimization problems typically involve other
  external parameters, in addition to the fitted parameters.  While
  it is extremely easy to rewrite an expression dynamically, in case
  one of the external parameters changes, the other possibility is to
  use the PRIVATE data structure.

  The user merely passes a structure to the FUNCTARGS keyword.  The
  user expression receives this value as the variable PRIVATE.
  MPFITEXPR nevers accesses this variable so it can contain any
  desired values.  Usually it would be an IDL structure so that any
  named external parameters can be passed to the expression.


 CONSTRAINING PARAMETER VALUES WITH THE PARINFO KEYWORD

  The behavior of MPFIT can be modified with respect to each
  parameter to be fitted.  A parameter value can be fixed; simple
  boundary constraints can be imposed; limitations on the parameter
  changes can be imposed; properties of the automatic derivative can
  be modified; and parameters can be tied to one another.

  These properties are governed by the PARINFO structure, which is
  passed as a keyword parameter to MPFIT.

  PARINFO should be an array of structures, one for each parameter.
  Each parameter is associated with one element of the array, in
  numerical order.  The structure can have the following entries
  (none are required):
  
     .VALUE - the starting parameter value (but see the START_PARAMS
              parameter for more information).
  
     .FIXED - a boolean value, whether the parameter is to be held
              fixed or not.  Fixed parameters are not varied by
              MPFIT, but are passed on to MYFUNCT for evaluation.
  
     .LIMITED - a two-element boolean array.  If the first/second
                element is set, then the parameter is bounded on the
                lower/upper side.  A parameter can be bounded on both
                sides.  Both LIMITED and LIMITS must be given
                together.
  
     .LIMITS - a two-element float or double array.  Gives the
               parameter limits on the lower and upper sides,
               respectively.  Zero, one or two of these values can be
               set, depending on the values of LIMITED.  Both LIMITED
               and LIMITS must be given together.
  
     .PARNAME - a string, giving the name of the parameter.  The
                fitting code of MPFIT does not use this tag in any
                way.  However, the default ITERPROC will print the
                parameter name if available.
  
     .STEP - the step size to be used in calculating the numerical
             derivatives.  If set to zero, then the step size is
             computed automatically.  Ignored when AUTODERIVATIVE=0.
             This value is superceded by the RELSTEP value.

     .RELSTEP - the *relative* step size to be used in calculating
                the numerical derivatives.  This number is the
                fractional size of the step, compared to the
                parameter value.  This value supercedes the STEP
                setting.  If the parameter is zero, then a default
                step size is chosen.

     .MPSIDE - the sidedness of the finite difference when computing
               numerical derivatives.  This field can take four
               values:

                  0 - one-sided derivative computed automatically
                  1 - one-sided derivative (f(x+h) - f(x)  )/h
                 -1 - one-sided derivative (f(x)   - f(x-h))/h
                  2 - two-sided derivative (f(x+h) - f(x-h))/(2*h)

              Where H is the STEP parameter described above.  The
              "automatic" one-sided derivative method will chose a
              direction for the finite difference which does not
              violate any constraints.  The other methods do not
              perform this check.  The two-sided method is in
              principle more precise, but requires twice as many
              function evaluations.  Default: 0.

     .MPMINSTEP - the minimum change to be made in the parameter
                  value.  During the fitting process, the parameter
                  will be changed by multiples of this value.  The
                  actual step is computed as:

                     DELTA1 = MPMINSTEP*ROUND(DELTA0/MPMINSTEP)

                  where DELTA0 and DELTA1 are the estimated parameter
                  changes before and after this constraint is
                  applied.  Note that this constraint should be used
                  with care since it may cause non-converging,
                  oscillating solutions.

                  A value of 0 indicates no minimum.  Default: 0.

     .MPMAXSTEP - the maximum change to be made in the parameter
                  value.  During the fitting process, the parameter
                  will never be changed by more than this value.

                  A value of 0 indicates no maximum.  Default: 0.
  
     .TIED - a string expression which "ties" the parameter to other
             free or fixed parameters.  Any expression involving
             constants and the parameter array P are permitted.
             Example: if parameter 2 is always to be twice parameter
             1 then use the following: parinfo(2).tied = '2 * P(1)'.
             Since they are totally constrained, tied parameters are
             considered to be fixed; no errors are computed for them.
             [ NOTE: the PARNAME can't be used in expressions. ]
  
  Future modifications to the PARINFO structure, if any, will involve
  adding structure tags beginning with the two letters "MP".
  Therefore programmers are urged to avoid using tags starting with
  the same letters; otherwise they are free to include their own
  fields within the PARINFO structure, and they will be ignored.
  
  PARINFO Example:
  parinfo = replicate({value:0.D, fixed:0, limited:[0,0], $
                       limits:[0.D,0]}, 5)
  parinfo(0).fixed = 1
  parinfo(4).limited(0) = 1
  parinfo(4).limits(0)  = 50.D
  parinfo(*).value = [5.7D, 2.2, 500., 1.5, 2000.]
  
  A total of 5 parameters, with starting values of 5.7,
  2.2, 500, 1.5, and 2000 are given.  The first parameter
  is fixed at a value of 5.7, and the last parameter is
  constrained to be above 50.

 INPUTS:
   MYFUNCT - a string variable containing an IDL expression.  The
             only restriction is that the independent variable *must*
             be referred to as "X" and model parameters *must* be
             referred to as an array called "P".  Do not use symbol
             names beginning with the underscore, "_".

             The expression should calculate "model" Y values given
             the X values and model parameters.  Using the vector
             notation of IDL, this can be quite easy to do.  If your
             expression gets complicated, you may wish to make an IDL
             function which will improve performance and readibility.

             The resulting array should be of the same size and
             dimensions as the "measured" Y values.

   X - Array of independent variable values.

   Y - Array of "measured" dependent variable values.  Y should have
       the same data type as X.  The function MYFUNCT should map
       X->Y.

   ERR - Array of "measured" 1-sigma uncertainties.  ERR should have
         the same data type as Y.  ERR is ignored if the WEIGHTS
         keyword is specified.

   START_PARAMS - An array of starting values for each of the
                  parameters of the model.  The number of parameters
                  should be fewer than the number of measurements.
                  Also, the parameters should have the same data type
                  as the measurements (double is preferred).

                  This parameter is optional if the PARINFO keyword
                  is used (see MPFIT).  The PARINFO keyword provides
                  a mechanism to fix or constrain individual
                  parameters.  If both START_PARAMS and PARINFO are
                  passed, then the starting *value* is taken from
                  START_PARAMS, but the *constraints* are taken from
                  PARINFO.

                  If no parameters are given, then MPFITEXPR attempts
                  to determine the number of parameters by scanning
                  the expression.  Parameters determined this way are
                  initialized to zero.  This technique is not fully
                  reliable, so users are advised to pass explicit
                  parameter starting values.
 

 RETURNS:

   Returns the array of best-fit parameters.


 KEYWORD PARAMETERS:

   BESTNORM - the value of the summed, squared, weighted residuals
              for the returned parameter values, i.e. the chi-square value.

   COVAR - the covariance matrix for the set of parameters returned
           by MPFIT.  The matrix is NxN where N is the number of
           parameters.  The square root of the diagonal elements
           gives the formal 1-sigma statistical errors on the
           parameters IF errors were treated "properly" in MYFUNC.
           Parameter errors are also returned in PERROR.

           To compute the correlation matrix, PCOR, use this:
           IDL> PCOR = COV * 0
           IDL> FOR i = 0, n-1 DO FOR j = 0, n-1 DO $
                PCOR(i,j) = COV(i,j)/sqrt(COV(i,i)*COV(j,j))

           If NOCOVAR is set or MPFIT terminated abnormally, then
           COVAR is set to a scalar with value !VALUES.D_NAN.

   DOF - number of degrees of freedom, computed as
             DOF = N_ELEMENTS(DEVIATES) - NFREE
         Note that this doesn't account for pegged parameters (see
         NPEGGED).

   ERRMSG - a string error or warning message is returned.

   FTOL - a nonnegative input variable. Termination occurs when both
          the actual and predicted relative reductions in the sum of
          squares are at most FTOL (and STATUS is accordingly set to
          1 or 3).  Therefore, FTOL measures the relative error
          desired in the sum of squares.  Default: 1D-10

   FUNCTARGS - passed directly to the expression as the variable
               PRIVATE.  Any user-private data can be communicated to
               the user expression using this keyword.
               Default: PRIVATE is undefined in user expression

   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 (and STATUS is
          accordingly set to 4). Therefore, GTOL measures the
          orthogonality desired between the function vector and the
          columns of the jacobian.  Default: 1D-10

   ITERARGS - The keyword arguments to be passed to ITERPROC via the
              _EXTRA mechanism.  This should be a structure, and is
              similar in operation to FUNCTARGS.
              Default: no arguments are passed.

   ITERPROC - The name of a procedure to be called upon each NPRINT
              iteration of the MPFIT routine.  It should be declared
              in the following way:

              PRO ITERPROC, MYFUNCT, p, iter, fnorm, FUNCTARGS=fcnargs, $
                PARINFO=parinfo, QUIET=quiet, ...
                ; perform custom iteration update
              END
         
              ITERPROC must either accept all three keyword
              parameters (FUNCTARGS, PARINFO and QUIET), or at least
              accept them via the _EXTRA keyword.
          
              MYFUNCT is the user-supplied function to be minimized,
              P is the current set of model parameters, ITER is the
              iteration number, and FUNCTARGS are the arguments to be
              passed to MYFUNCT.  FNORM should be the
              chi-squared value.  QUIET is set when no textual output
              should be printed.  See below for documentation of
              PARINFO.

              In implementation, ITERPROC can perform updates to the
              terminal or graphical user interface, to provide
              feedback while the fit proceeds.  If the fit is to be
              stopped for any reason, then ITERPROC should set the
              common block variable ERROR_CODE to negative value (see
              MPFIT_ERROR common block below).  In principle,
              ITERPROC should probably not modify the parameter
              values, because it may interfere with the algorithm's
              stability.  In practice it is allowed.

              Default: an internal routine is used to print the
                       parameter values.

   MAXITER - The maximum number of iterations to perform.  If the
             number is exceeded, then the STATUS value is set to 5
             and MPFIT returns.
             Default: 200 iterations

   NFEV - the number of MYFUNCT function evaluations performed.

   NFREE - the number of free parameters in the fit.  This includes
           parameters which are not FIXED and not TIED, but it does
           include parameters which are pegged at LIMITS.

   NITER - the number of iterations completed.

   NOCOVAR - set this keyword to prevent the calculation of the
             covariance matrix before returning (see COVAR)

   NPEGGED - the number of free parameters which are pegged at a
             LIMIT.

   NPRINT - The frequency with which ITERPROC is called.  A value of
            1 indicates that ITERPROC is called with every iteration,
            while 2 indicates every other iteration, etc.  Note that
            several Levenberg-Marquardt attempts can be made in a
            single iteration.
            Default value: 1

   PARINFO - Provides a mechanism for more sophisticated constraints
             to be placed on parameter values.  When PARINFO is not
             passed, then it is assumed that all parameters are free
             and unconstrained.  Values in PARINFO are never 
             modified during a call to MPFIT.

             See description above for the structure of PARINFO.

             Default value:  all parameters are free and unconstrained.

   PERROR - The formal 1-sigma errors in each parameter, computed
            from the covariance matrix.  If a parameter is held
            fixed, or if it touches a boundary, then the error is
            reported as zero.

            If the fit is unweighted (i.e. no errors were given, or
            the weights were uniformly set to unity), then PERROR
            will probably not represent the true parameter
            uncertainties.  

            *If* you can assume that the true reduced chi-squared
            value is unity -- meaning that the fit is implicitly
            assumed to be of good quality -- then the estimated
            parameter uncertainties can be computed by scaling PERROR
            by the measured chi-squared value.

              DOF     = N_ELEMENTS(X) - N_ELEMENTS(PARMS) ; deg of freedom
              PCERROR = PERROR * SQRT(BESTNORM / DOF)   ; scaled uncertainties

   QUIET - set this keyword when no textual output should be printed
           by MPFIT

   STATUS - an integer status code is returned.  All values other
            than zero can represent success.  It can have one of the
            following values:

	   0  improper input parameters.
         
	   1  both actual and predicted relative reductions
	      in the sum of squares are at most FTOL.
         
	   2  relative error between two consecutive iterates
	      is at most XTOL
         
	   3  conditions for STATUS = 1 and STATUS = 2 both hold.
         
	   4  the cosine of the angle between fvec and any
	      column of the jacobian is at most GTOL in
	      absolute value.
         
	   5  the maximum number of iterations has been reached
         
	   6  FTOL is too small. no further reduction in
	      the sum of squares is possible.
         
	   7  XTOL is too small. no further improvement in
	      the approximate solution x is possible.
         
	   8  GTOL is too small. fvec is orthogonal to the
	      columns of the jacobian to machine precision.

   WEIGHTS - Array of weights to be used in calculating the
             chi-squared value.  If WEIGHTS is specified then the ERR
             parameter is ignored.  The chi-squared value is computed
             as follows:

                CHISQ = TOTAL( (Y-MYFUNCT(X,P))^2 * ABS(WEIGHTS) )

             Here are common values of WEIGHTS:

                1D/ERR^2 - Normal weighting (ERR is the measurement error)
                1D/Y     - Poisson weighting (counting statistics)
                1D       - Unweighted

   XTOL - a nonnegative input variable. Termination occurs when the
          relative error between two consecutive iterates is at most
          XTOL (and STATUS is accordingly set to 2 or 3).  Therefore,
          XTOL measures the relative error desired in the approximate
          solution.  Default: 1D-10

   YFIT - the best-fit model function, as returned by MYFUNCT.


 EXAMPLE:

   ; First, generate some synthetic data
   x  = dindgen(200) * 0.1 - 10.                   ; Independent variable 
   yi = gauss1(x, [2.2D, 1.4, 3000.]) + 1000       ; "Ideal" Y variable
   y  = yi + randomn(seed, 200) * sqrt(yi)         ; Measured, w/ noise
   sy = sqrt(y)                                    ; Poisson errors

   ; Now fit a Gaussian to see how well we can recover
   expr = 'P(0) + GAUSS1(X, P(1:3))'               ; fitting function
   p0 = [800.D, 1.D, 1., 500.]                     ; Initial guess
   p = mpfitexpr(expr, x, y, sy, p0)               ; Fit the expression
   print, p

   plot, x, y                                      ; Plot data
   oplot, x, mpevalexpr(expr, x, p)                ; Plot model

   Generates a synthetic data set with a Gaussian peak, and Poisson
   statistical uncertainty.  Then a model consisting of a constant
   plus Gaussian is fit to the data.


 COMMON BLOCKS:

   COMMON MPFIT_ERROR, ERROR_CODE

     User routines may stop the fitting process at any time by
     setting an error condition.  This condition may be set in either
     the user's model computation routine (MYFUNCT), or in the
     iteration procedure (ITERPROC).

     To stop the fitting, the above common block must be declared,
     and ERROR_CODE must be set to a negative number.  After the user
     procedure or function returns, MPFIT checks the value of this
     common block variable and exits immediately if the error
     condition has been set.  By default the value of ERROR_CODE is
     zero, indicating a successful function/procedure call.


 REFERENCES:

   MINPACK-1, Jorge More', available from netlib (www.netlib.org).
   "Optimization Software Guide," Jorge More' and Stephen Wright, 
     SIAM, *Frontiers in Applied Mathematics*, Number 14.

 MODIFICATION HISTORY:
   Written, Apr-Jul 1998, CM
   Added PERROR keyword, 04 Aug 1998, CM
   Added COVAR keyword, 20 Aug 1998, CM
   Added NITER output keyword, 05 Oct 1998
      D.L Windt, Bell Labs, windt@bell-labs.com;
   Added ability to return model function in YFIT, 09 Nov 1998
   Parameter values can be tied to others, 09 Nov 1998
   Added MPEVALEXPR utility function, 09 Dec 1998
   Cosmetic documentation updates, 16 Apr 1999, CM
   More cosmetic documentation updates, 14 May 1999, CM
   Made sure to update STATUS value, 25 Sep 1999, CM
   Added WEIGHTS keyword, 25 Sep 1999, CM
   Changed from handles to common blocks, 25 Sep 1999, CM
     - commons seem much cleaner and more logical in this case.
   Alphabetized documented keywords, 02 Oct 1999, CM
   Added QUERY keyword and query checking of MPFIT, 29 Oct 1999, CM
   Check to be sure that X and Y are present, 02 Nov 1999, CM
   Documented PERROR for unweighted fits, 03 Nov 1999, CM
   Removed ITERPROC='' when quiet EQ 1, 10 Jan 2000, CM
   Changed to ERROR_CODE for error condition, 28 Jan 2000, CM
   Updated the EXAMPLE, 26 Mar 2000, CM
   Copying permission terms have been liberalized, 26 Mar 2000, CM
   Propagated improvements from MPFIT, 17 Dec 2000, CM
   Correct reference to _WTS in MPFITEXPR_EVAL, 25 May 2001, CM
      (thanks to Mark Fardal)
   Added useful FUNCTARGS behavior (as yet undocumented), 04 Jul
      2002, CM
   Documented FUNCTARGS/PRIVATE behavior, 22 Jul 2002, CM
   Added NFREE and NPEGGED keywords, 13 Sep 2002, CM
   Documented RELSTEP field of PARINFO (!!), CM, 25 Oct 2002
   Add DOF keyword, CM, 31 Jul 2003
   Add FUNCTARGS keyword to MPEVALEXPR, CM 08 Aug 2003
   Minor documentation adjustment, 03 Feb 2004, CM

  $Id: mpfitexpr.pro,v 1.11 2004/02/26 04:18:16 craigm Exp $

(See /dzd2/heiles/idl/gen/mpfit/mpfitexpr.pro)


MPFITFUN

[Previous Routine] [Next Routine] [List of Routines]
 NAME:
   MPFITFUN

 AUTHOR:
   Craig B. Markwardt, NASA/GSFC Code 662, Greenbelt, MD 20770
   craigm@lheamail.gsfc.nasa.gov
   UPDATED VERSIONs can be found on my WEB PAGE: 
      http://cow.physics.wisc.edu/~craigm/idl/idl.html

 PURPOSE:
   Perform Levenberg-Marquardt least-squares fit to IDL function

 MAJOR TOPICS:
   Curve and Surface Fitting

 CALLING SEQUENCE:
   parms = MPFITFUN(MYFUNCT, X, Y, ERR, start_params, ...)

 DESCRIPTION:

  MPFITFUN fits a user-supplied model -- in the form of an IDL
  function -- to a set of user-supplied data.  MPFITFUN calls
  MPFIT, the MINPACK-1 least-squares minimizer, to do the main
  work.

  Given the data and their uncertainties, MPFITFUN finds the best set
  of model parameters which match the data (in a least-squares
  sense) and returns them in an array.
  
  The user must supply the following items:
   - An array of independent variable values ("X").
   - An array of "measured" *dependent* variable values ("Y").
   - An array of "measured" 1-sigma uncertainty values ("ERR").
   - The name of an IDL function which computes Y given X ("MYFUNCT").
   - Starting guesses for all of the parameters ("START_PARAMS").

  There are very few restrictions placed on X, Y or MYFUNCT.  Simply
  put, MYFUNCT must map the "X" values into "Y" values given the
  model parameters.  The "X" values may represent any independent
  variable (not just Cartesian X), and indeed may be multidimensional
  themselves.  For example, in the application of image fitting, X
  may be a 2xN array of image positions.

  MPFITFUN carefully avoids passing large arrays where possible to
  improve performance.

  See below for an example of usage.

 USER FUNCTION

  The user must define a function which returns the model value.  For
  applications which use finite-difference derivatives -- the default
  -- the user function should be declared in the following way:

    FUNCTION MYFUNCT, X, P
     ; The independent variable is X
     ; Parameter values are passed in "P"
     YMOD = ... computed model values at X ...
     return, YMOD
    END

  The returned array YMOD must have the same dimensions and type as
  the "measured" Y values.

  User functions may also indicate a fatal error condition
  using the ERROR_CODE common block variable, as described
  below under the MPFIT_ERROR common block definition.

  See the discussion under "ANALYTIC DERIVATIVES" and AUTODERIVATIVE
  in MPFIT.PRO if you wish to compute the derivatives for yourself.
  AUTODERIVATIVE is accepted by MPFITFUN and passed directly to
  MPFIT.  The user function must accept one additional parameter, DP,
  which contains the derivative of the user function with respect to
  each parameter at each data point, as described in MPFIT.PRO.

 CONSTRAINING PARAMETER VALUES WITH THE PARINFO KEYWORD

  The behavior of MPFIT can be modified with respect to each
  parameter to be fitted.  A parameter value can be fixed; simple
  boundary constraints can be imposed; limitations on the parameter
  changes can be imposed; properties of the automatic derivative can
  be modified; and parameters can be tied to one another.

  These properties are governed by the PARINFO structure, which is
  passed as a keyword parameter to MPFIT.

  PARINFO should be an array of structures, one for each parameter.
  Each parameter is associated with one element of the array, in
  numerical order.  The structure can have the following entries
  (none are required):
  
     .VALUE - the starting parameter value (but see the START_PARAMS
              parameter for more information).
  
     .FIXED - a boolean value, whether the parameter is to be held
              fixed or not.  Fixed parameters are not varied by
              MPFIT, but are passed on to MYFUNCT for evaluation.
  
     .LIMITED - a two-element boolean array.  If the first/second
                element is set, then the parameter is bounded on the
                lower/upper side.  A parameter can be bounded on both
                sides.  Both LIMITED and LIMITS must be given
                together.
  
     .LIMITS - a two-element float or double array.  Gives the
               parameter limits on the lower and upper sides,
               respectively.  Zero, one or two of these values can be
               set, depending on the values of LIMITED.  Both LIMITED
               and LIMITS must be given together.
  
     .PARNAME - a string, giving the name of the parameter.  The
                fitting code of MPFIT does not use this tag in any
                way.  However, the default ITERPROC will print the
                parameter name if available.
  
     .STEP - the step size to be used in calculating the numerical
             derivatives.  If set to zero, then the step size is
             computed automatically.  Ignored when AUTODERIVATIVE=0.
             This value is superceded by the RELSTEP value.

     .RELSTEP - the *relative* step size to be used in calculating
                the numerical derivatives.  This number is the
                fractional size of the step, compared to the
                parameter value.  This value supercedes the STEP
                setting.  If the parameter is zero, then a default
                step size is chosen.

     .MPSIDE - the sidedness of the finite difference when computing
               numerical derivatives.  This field can take four
               values:

                  0 - one-sided derivative computed automatically
                  1 - one-sided derivative (f(x+h) - f(x)  )/h
                 -1 - one-sided derivative (f(x)   - f(x-h))/h
                  2 - two-sided derivative (f(x+h) - f(x-h))/(2*h)

              Where H is the STEP parameter described above.  The
              "automatic" one-sided derivative method will chose a
              direction for the finite difference which does not
              violate any constraints.  The other methods do not
              perform this check.  The two-sided method is in
              principle more precise, but requires twice as many
              function evaluations.  Default: 0.

     .MPMINSTEP - the minimum change to be made in the parameter
                  value.  During the fitting process, the parameter
                  will be changed by multiples of this value.  The
                  actual step is computed as:

                     DELTA1 = MPMINSTEP*ROUND(DELTA0/MPMINSTEP)

                  where DELTA0 and DELTA1 are the estimated parameter
                  changes before and after this constraint is
                  applied.  Note that this constraint should be used
                  with care since it may cause non-converging,
                  oscillating solutions.

                  A value of 0 indicates no minimum.  Default: 0.

     .MPMAXSTEP - the maximum change to be made in the parameter
                  value.  During the fitting process, the parameter
                  will never be changed by more than this value.

                  A value of 0 indicates no maximum.  Default: 0.
  
     .TIED - a string expression which "ties" the parameter to other
             free or fixed parameters.  Any expression involving
             constants and the parameter array P are permitted.
             Example: if parameter 2 is always to be twice parameter
             1 then use the following: parinfo(2).tied = '2 * P(1)'.
             Since they are totally constrained, tied parameters are
             considered to be fixed; no errors are computed for them.
             [ NOTE: the PARNAME can't be used in expressions. ]
  
  Future modifications to the PARINFO structure, if any, will involve
  adding structure tags beginning with the two letters "MP".
  Therefore programmers are urged to avoid using tags starting with
  the same letters; otherwise they are free to include their own
  fields within the PARINFO structure, and they will be ignored.
  
  PARINFO Example:
  parinfo = replicate({value:0.D, fixed:0, limited:[0,0], $
                       limits:[0.D,0]}, 5)
  parinfo(0).fixed = 1
  parinfo(4).limited(0) = 1
  parinfo(4).limits(0)  = 50.D
  parinfo(*).value = [5.7D, 2.2, 500., 1.5, 2000.]
  
  A total of 5 parameters, with starting values of 5.7,
  2.2, 500, 1.5, and 2000 are given.  The first parameter
  is fixed at a value of 5.7, and the last parameter is
  constrained to be above 50.

 INPUTS:
   MYFUNCT - a string variable containing the name of an IDL function.
             This function computes the "model" Y values given the
             X values and model parameters, as desribed above.

   X - Array of independent variable values.

   Y - Array of "measured" dependent variable values.  Y should have
       the same data type as X.  The function MYFUNCT should map
       X->Y.

   ERR - Array of "measured" 1-sigma uncertainties.  ERR should have
         the same data type as Y.  ERR is ignored if the WEIGHTS
         keyword is specified.

   START_PARAMS - An array of starting values for each of the
                  parameters of the model.  The number of parameters
                  should be fewer than the number of measurements.
                  Also, the parameters should have the same data type
                  as the measurements (double is preferred).

                  This parameter is optional if the PARINFO keyword
                  is used (see MPFIT).  The PARINFO keyword provides
                  a mechanism to fix or constrain individual
                  parameters.  If both START_PARAMS and PARINFO are
                  passed, then the starting *value* is taken from
                  START_PARAMS, but the *constraints* are taken from
                  PARINFO.
 

 RETURNS:

   Returns the array of best-fit parameters.


 KEYWORD PARAMETERS:

   BESTNORM - the value of the summed squared residuals for the
              returned parameter values.

   COVAR - the covariance matrix for the set of parameters returned
           by MPFIT.  The matrix is NxN where N is the number of
           parameters.  The square root of the diagonal elements
           gives the formal 1-sigma statistical errors on the
           parameters IF errors were treated "properly" in MYFUNC.
           Parameter errors are also returned in PERROR.

           To compute the correlation matrix, PCOR, use this:
           IDL> PCOR = COV * 0
           IDL> FOR i = 0, n-1 DO FOR j = 0, n-1 DO $
                PCOR(i,j) = COV(i,j)/sqrt(COV(i,i)*COV(j,j))

           If NOCOVAR is set or MPFIT terminated abnormally, then
           COVAR is set to a scalar with value !VALUES.D_NAN.

   CASH - when set, the fit statistic is changed to a derivative of
          the CASH statistic.  The model function must be strictly
          positive.

   DOF - number of degrees of freedom, computed as
             DOF = N_ELEMENTS(DEVIATES) - NFREE
         Note that this doesn't account for pegged parameters (see
         NPEGGED).

   ERRMSG - a string error or warning message is returned.

   FTOL - a nonnegative input variable. Termination occurs when both
          the actual and predicted relative reductions in the sum of
          squares are at most FTOL (and STATUS is accordingly set to
          1 or 3).  Therefore, FTOL measures the relative error
          desired in the sum of squares.  Default: 1D-10

   FUNCTARGS - A structure which contains the parameters to be passed
               to the user-supplied function specified by MYFUNCT via
               the _EXTRA mechanism.  This is the way you can pass
               additional data to your user-supplied function without
               using common blocks.

               By default, no extra parameters are passed to the
               user-supplied function.

   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 (and STATUS is
          accordingly set to 4). Therefore, GTOL measures the
          orthogonality desired between the function vector and the
          columns of the jacobian.  Default: 1D-10

   ITERARGS - The keyword arguments to be passed to ITERPROC via the
              _EXTRA mechanism.  This should be a structure, and is
              similar in operation to FUNCTARGS.
              Default: no arguments are passed.

   ITERPROC - The name of a procedure to be called upon each NPRINT
              iteration of the MPFIT routine.  It should be declared
              in the following way:

              PRO ITERPROC, MYFUNCT, p, iter, fnorm, FUNCTARGS=fcnargs, $
                PARINFO=parinfo, QUIET=quiet, ...
                ; perform custom iteration update
              END
         
              ITERPROC must either accept all three keyword
              parameters (FUNCTARGS, PARINFO and QUIET), or at least
              accept them via the _EXTRA keyword.
          
              MYFUNCT is the user-supplied function to be minimized,
              P is the current set of model parameters, ITER is the
              iteration number, and FUNCTARGS are the arguments to be
              passed to MYFUNCT.  FNORM should be the
              chi-squared value.  QUIET is set when no textual output
              should be printed.  See below for documentation of
              PARINFO.

              In implementation, ITERPROC can perform updates to the
              terminal or graphical user interface, to provide
              feedback while the fit proceeds.  If the fit is to be
              stopped for any reason, then ITERPROC should set the
              common block variable ERROR_CODE to negative value (see
              MPFIT_ERROR common block below).  In principle,
              ITERPROC should probably not modify the parameter
              values, because it may interfere with the algorithm's
              stability.  In practice it is allowed.

              Default: an internal routine is used to print the
                       parameter values.

   MAXITER - The maximum number of iterations to perform.  If the
             number is exceeded, then the STATUS value is set to 5
             and MPFIT returns.
             Default: 200 iterations

   NFEV - the number of MYFUNCT function evaluations performed.

   NFREE - the number of free parameters in the fit.  This includes
           parameters which are not FIXED and not TIED, but it does
           include parameters which are pegged at LIMITS.

   NITER - the number of iterations completed.

   NOCOVAR - set this keyword to prevent the calculation of the
             covariance matrix before returning (see COVAR)

   NPEGGED - the number of free parameters which are pegged at a
             LIMIT.

   NPRINT - The frequency with which ITERPROC is called.  A value of
            1 indicates that ITERPROC is called with every iteration,
            while 2 indicates every other iteration, etc.  Note that
            several Levenberg-Marquardt attempts can be made in a
            single iteration.
            Default value: 1

   PARINFO - Provides a mechanism for more sophisticated constraints
             to be placed on parameter values.  When PARINFO is not
             passed, then it is assumed that all parameters are free
             and unconstrained.  Values in PARINFO are never 
             modified during a call to MPFIT.

             See description above for the structure of PARINFO.

             Default value:  all parameters are free and unconstrained.

   PERROR - The formal 1-sigma errors in each parameter, computed
            from the covariance matrix.  If a parameter is held
            fixed, or if it touches a boundary, then the error is
            reported as zero.

            If the fit is unweighted (i.e. no errors were given, or
            the weights were uniformly set to unity), then PERROR
            will probably not represent the true parameter
            uncertainties.  

            *If* you can assume that the true reduced chi-squared
            value is unity -- meaning that the fit is implicitly
            assumed to be of good quality -- then the estimated
            parameter uncertainties can be computed by scaling PERROR
            by the measured chi-squared value.

              DOF     = N_ELEMENTS(X) - N_ELEMENTS(PARMS) ; deg of freedom
              PCERROR = PERROR * SQRT(BESTNORM / DOF)   ; scaled uncertainties

   QUIET - set this keyword when no textual output should be printed
           by MPFIT

   STATUS - an integer status code is returned.  All values other
            than zero can represent success.  It can have one of the
            following values:

	   0  improper input parameters.
         
	   1  both actual and predicted relative reductions
	      in the sum of squares are at most FTOL.
         
	   2  relative error between two consecutive iterates
	      is at most XTOL
         
	   3  conditions for STATUS = 1 and STATUS = 2 both hold.
         
	   4  the cosine of the angle between fvec and any
	      column of the jacobian is at most GTOL in
	      absolute value.
         
	   5  the maximum number of iterations has been reached
         
	   6  FTOL is too small. no further reduction in
	      the sum of squares is possible.
         
	   7  XTOL is too small. no further improvement in
	      the approximate solution x is possible.
         
	   8  GTOL is too small. fvec is orthogonal to the
	      columns of the jacobian to machine precision.

   WEIGHTS - Array of weights to be used in calculating the
             chi-squared value.  If WEIGHTS is specified then the ERR
             parameter is ignored.  The chi-squared value is computed
             as follows:

                CHISQ = TOTAL( (Y-MYFUNCT(X,P))^2 * ABS(WEIGHTS) )

             Here are common values of WEIGHTS:

                1D/ERR^2 - Normal weighting (ERR is the measurement error)
                1D/Y     - Poisson weighting (counting statistics)
                1D       - Unweighted

   XTOL - a nonnegative input variable. Termination occurs when the
          relative error between two consecutive iterates is at most
          XTOL (and STATUS is accordingly set to 2 or 3).  Therefore,
          XTOL measures the relative error desired in the approximate
          solution.  Default: 1D-10

   YFIT - the best-fit model function, as returned by MYFUNCT.

   
 EXAMPLE:

   ; First, generate some synthetic data
   npts = 200
   x  = dindgen(npts) * 0.1 - 10.                  ; Independent variable 
   yi = gauss1(x, [2.2D, 1.4, 3000.])              ; "Ideal" Y variable
   y  = yi + randomn(seed, npts) * sqrt(1000. + yi); Measured, w/ noise
   sy = sqrt(1000.D + y)                           ; Poisson errors

   ; Now fit a Gaussian to see how well we can recover
   p0 = [1.D, 1., 1000.]                   ; Initial guess (cent, width, area)
   p = mpfitfun('GAUSS1', x, y, sy, p0)    ; Fit a function
   print, p

   Generates a synthetic data set with a Gaussian peak, and Poisson
   statistical uncertainty.  Then the same function is fitted to the
   data (with different starting parameters) to see how close we can
   get.


 COMMON BLOCKS:

   COMMON MPFIT_ERROR, ERROR_CODE

     User routines may stop the fitting process at any time by
     setting an error condition.  This condition may be set in either
     the user's model computation routine (MYFUNCT), or in the
     iteration procedure (ITERPROC).

     To stop the fitting, the above common block must be declared,
     and ERROR_CODE must be set to a negative number.  After the user
     procedure or function returns, MPFIT checks the value of this
     common block variable and exits immediately if the error
     condition has been set.  By default the value of ERROR_CODE is
     zero, indicating a successful function/procedure call.

 REFERENCES:

   MINPACK-1, Jorge More', available from netlib (www.netlib.org).
   "Optimization Software Guide," Jorge More' and Stephen Wright, 
     SIAM, *Frontiers in Applied Mathematics*, Number 14.

 MODIFICATION HISTORY:
   Written, Apr-Jul 1998, CM
   Added PERROR keyword, 04 Aug 1998, CM
   Added COVAR keyword, 20 Aug 1998, CM
   Added ITER output keyword, 05 Oct 1998
      D.L Windt, Bell Labs, windt@bell-labs.com;
   Added ability to return model function in YFIT, 09 Nov 1998
   Analytical derivatives allowed via AUTODERIVATIVE keyword, 09 Nov 1998
   Parameter values can be tied to others, 09 Nov 1998
   Cosmetic documentation updates, 16 Apr 1999, CM
   More cosmetic documentation updates, 14 May 1999, CM
   Made sure to update STATUS, 25 Sep 1999, CM
   Added WEIGHTS keyword, 25 Sep 1999, CM
   Changed from handles to common blocks, 25 Sep 1999, CM
     - commons seem much cleaner and more logical in this case.
   Alphabetized documented keywords, 02 Oct 1999, CM
   Added QUERY keyword and query checking of MPFIT, 29 Oct 1999, CM
   Corrected EXAMPLE (offset of 1000), 30 Oct 1999, CM
   Check to be sure that X and Y are present, 02 Nov 1999, CM
   Documented PERROR for unweighted fits, 03 Nov 1999, CM
   Changed to ERROR_CODE for error condition, 28 Jan 2000, CM
   Corrected errors in EXAMPLE, 26 Mar 2000, CM
   Copying permission terms have been liberalized, 26 Mar 2000, CM
   Propagated improvements from MPFIT, 17 Dec 2000, CM
   Added CASH statistic, 10 Jan 2001
   Added NFREE and NPEGGED keywords, 11 Sep 2002, CM
   Documented RELSTEP field of PARINFO (!!), CM, 25 Oct 2002
   Add DOF keyword to return degrees of freedom, CM, 23 June 2003

  $Id: mpfitfun.pro,v 1.6 2003/06/30 21:47:36 craigm Exp $

(See /dzd2/heiles/idl/gen/mpfit/mpfitfun.pro)


MPFITPEAK

[Previous Routine] [Next Routine] [List of Routines]
 NAME:
   MPFITPEAK

 AUTHOR:
   Craig B. Markwardt, NASA/GSFC Code 662, Greenbelt, MD 20770
   craigm@lheamail.gsfc.nasa.gov
   UPDATED VERSIONs can be found on my WEB PAGE: 
      http://cow.physics.wisc.edu/~craigm/idl/idl.html

 PURPOSE:
   Fit a gaussian, lorentzian or Moffat model to data

 MAJOR TOPICS:
   Curve and Surface Fitting

 CALLING SEQUENCE:
   yfit = MPFITPEAK(X, Y, A, NTERMS=nterms, ...)

 DESCRIPTION:

   MPFITPEAK fits a gaussian, lorentzian or Moffat model using the
   non-linear least squares fitter MPFIT.  MPFITPEAK is meant to be a
   drop-in replacement for IDL's GAUSSFIT function (and requires
   MPFIT and MPFITFUN).

   The choice of the fitting function is determined by the keywords
   GAUSSIAN, LORENTZIAN and MOFFAT.  By default the gaussian model
   function is used.  [ The Moffat function is a modified Lorentzian
   with variable power law index. (Moffat, A. F. J. 1969, Astronomy &
   Astrophysics, v. 3, p. 455-461) ]

   The functional form of the baseline is determined by NTERMS and
   the function to be fitted.  NTERMS represents the total number of
   parameters, A, to be fitted.  The functional forms and the
   meanings of the parameters are described in this table:

                 GAUSSIAN#          Lorentzian#         Moffat#

   Model     A(0)*exp(-0.5*u^2)   A(0)/(u^2 + 1)   A(0)/(u^2 + 1)^A(3)

   A(0)         Peak Value          Peak Value        Peak Value
   A(1)        Peak Centroid       Peak Centroid     Peak Centroid
   A(2)       Gaussian Sigma           HWHM%             HWHM%
   A(3)         + A(3)    *          + A(3)   *      Moffat Index
   A(4)         + A(4)*x  *          + A(4)*x *         + A(4)   *
   A(5)                                                 + A(5)*x *

   Notes: # u = (x - A(1))/A(2)
          % Half-width at half maximum
          * Optional depending on NTERMS

   By default the initial starting values for the parameters A are
   estimated from the data.  However, explicit starting values can be
   supplied using the ESTIMATES keyword.  Also, error or weighting
   values can optionally be provided; otherwise the fit is
   unweighted.

   MPFITPEAK fits the peak value of the curve.  The area under a
   gaussian peak is A(0)*A(2)*SQRT(2*!DPI); the area under a
   lorentzian peak is A(0)*A(2)*!DPI.

 RESTRICTIONS:

   If no starting parameter ESTIMATES are provided, then MPFITPEAK
   attempts to estimate them from the data.  This is not a perfect
   science; however, the author believes that the technique
   implemented here is more robust than the one used in IDL's
   GAUSSFIT.  The author has tested cases of strong peaks, noisy
   peaks and broad peaks, all with success.

   Users should be aware that if the baseline term contains a strong
   linear component then the automatic estimation may fail.  For
   automatic estimation to work the peak amplitude should dominate
   over the the maximum baseline.

 INPUTS:
   X - Array of independent variable values, whose values should
       monotonically increase.

   Y - Array of "measured" dependent variable values.  Y should have
       the same data type and dimension as X.


 OUTPUTS:
   A - Upon return, an array of NTERMS best fit parameter values.
       See the table above for the meanings of each parameter
       element.


 RETURNS:

   Returns the best fitting model function.

 KEYWORDS:

   ** NOTE ** Additional keywords such as PARINFO, BESTNORM, and
              STATUS are accepted by MPFITPEAK but not documented
              here.  Please see the documentation for MPFIT for the
              description of these advanced options.

   CHISQ - the value of the summed squared residuals for the
           returned parameter values.

   DOF - number of degrees of freedom, computed as
             DOF = N_ELEMENTS(DEVIATES) - NFREE
         Note that this doesn't account for pegged parameters (see
         NPEGGED).

   ERROR - upon input, the measured 1-sigma uncertainties in the "Y"
           values.  If no ERROR or WEIGHTS are given, then the fit is
           unweighted.

   ESTIMATES - Array of starting values for each parameter of the
               model.  The number of parameters should at least be
               three (four for Moffat), and if less than NTERMS, will
               be extended with zeroes.
               Default: parameter values are estimated from data.

   GAUSSIAN - if set, fit a gaussian model function.  The Default.
   LORENTZIAN - if set, fit a lorentzian model function.
   MOFFAT - if set, fit a Moffat model function.

   MEASURE_ERRORS - synonym for ERRORS, for consistency with built-in
                    IDL fitting routines.

   NEGATIVE / POSITIVE - if set, and ESTIMATES is not provided, then
                         MPFITPEAK will assume that a
                         negative/positive peak is present.
                         Default: determined automatically

   NFREE - the number of free parameters in the fit.  This includes
           parameters which are not FIXED and not TIED, but it does
           include parameters which are pegged at LIMITS.

   NTERMS - An integer describing the number of fitting terms.
            NTERMS must have a minimum value, but can optionally be
            larger depending on the desired baseline.  

            For gaussian and lorentzian models, NTERMS must be three
            (zero baseline), four (constant baseline) or five (linear
            baseline).  Default: 4

            For the Moffat model, NTERMS must be four (zero
            baseline), five (constant baseline), or six (linear
            baseline).  Default: 5

   PERROR - upon return, the 1-sigma uncertainties of the parameter
            values A.  These values are only meaningful if the ERRORS
            or WEIGHTS keywords are specified properly.

            If the fit is unweighted (i.e. no errors were given, or
            the weights were uniformly set to unity), then PERROR
            will probably not represent the true parameter
            uncertainties.  

            *If* you can assume that the true reduced chi-squared
            value is unity -- meaning that the fit is implicitly
            assumed to be of good quality -- then the estimated
            parameter uncertainties can be computed by scaling PERROR
            by the measured chi-squared value.

              DOF     = N_ELEMENTS(X) - N_ELEMENTS(PARMS) ; deg of freedom
              PCERROR = PERROR * SQRT(BESTNORM / DOF)   ; scaled uncertainties

   QUIET - if set then diagnostic fitting messages are suppressed.
           Default: QUIET=1 (i.e., no diagnostics)

   SIGMA - synonym for PERROR (1-sigma parameter uncertainties), for
           compatibility with GAUSSFIT.  Do not confuse this with the
           Gaussian "sigma" width parameter.

   WEIGHTS - Array of weights to be used in calculating the
             chi-squared value.  If WEIGHTS is specified then the ERROR
             keyword is ignored.  The chi-squared value is computed
             as follows:

                CHISQ = TOTAL( (Y-MYFUNCT(X,P))^2 * ABS(WEIGHTS) )

             Here are common values of WEIGHTS:

                1D/ERR^2 - Normal weighting (ERR is the measurement error)
                1D/Y     - Poisson weighting (counting statistics)
                1D       - Unweighted

             The ERROR keyword takes precedence over any WEIGHTS
             keyword values.  If no ERROR or WEIGHTS are given, then
             the fit is unweighted.

   YERROR - upon return, the root-mean-square variance of the
            residuals.


 EXAMPLE:

   ; First, generate some synthetic data
   npts = 200
   x  = dindgen(npts) * 0.1 - 10.                  ; Independent variable 
   yi = gauss1(x, [2.2D, 1.4, 3000.]) + 1000       ; "Ideal" Y variable
   y  = yi + randomn(seed, npts) * sqrt(1000. + yi); Measured, w/ noise
   sy = sqrt(1000.D + y)                           ; Poisson errors

   ; Now fit a Gaussian to see how well we can recover the original
   yfit = mpfitpeak(x, y, a, error=sy)
   print, p

   Generates a synthetic data set with a Gaussian peak, and Poisson
   statistical uncertainty.  Then the same function is fitted to the
   data.

 REFERENCES:

   MINPACK-1, Jorge More', available from netlib (www.netlib.org).
   "Optimization Software Guide," Jorge More' and Stephen Wright, 
     SIAM, *Frontiers in Applied Mathematics*, Number 14.

 MODIFICATION HISTORY:

   New algorithm for estimating starting values, CM, 31 Oct 1999
   Documented, 02 Nov 1999
   Small documentation fixes, 02 Nov 1999
   Slight correction to calculation of dx, CM, 02 Nov 1999
   Documented PERROR for unweighted fits, 03 Nov 1999, CM
   Copying permission terms have been liberalized, 26 Mar 2000, CM
   Change requirements on # elements in X and Y, 20 Jul 2000, CM
      (thanks to David Schlegel )
   Added documentation on area under curve, 29 Aug 2000, CM
   Added POSITIVE and NEGATIVE keywords, 17 Nov 2000, CM
   Added reference to Moffat paper, 10 Jan 2001, CM
   Added usage message, 26 Jul 2001, CM
   Documentation clarification, 05 Sep 2001, CM
   Make more consistent with comparable IDL routines, 30 Jun 2003, CM
   Assumption of sorted data was removed, CM, 06 Sep 2003, CM

  $Id: mpfitpeak.pro,v 1.7 2003/09/06 16:29:44 craigm Exp $

(See /dzd2/heiles/idl/gen/mpfit/mpfitpeak.pro)


MPFTEST

[Previous Routine] [Next Routine] [List of Routines]
 NAME:
   MPFTEST

 AUTHOR:
   Craig B. Markwardt, NASA/GSFC Code 662, Greenbelt, MD 20770
   craigm@lheamail.gsfc.nasa.gov
   UPDATED VERSIONs can be found on my WEB PAGE: 
      http://cow.physics.wisc.edu/~craigm/idl/idl.html

 PURPOSE:
   Compute the probability of a given F value

 MAJOR TOPICS:
   Curve and Surface Fitting, Statistics

 CALLING SEQUENCE:
   PROB = MPFTEST(F, DOF1, DOF2, [/SIGMA, /CLEVEL, /SLEVEL ])

 DESCRIPTION:

  The function MPFTEST() computes the probability for a value drawn
  from the F-distribution to equal or exceed the given value of F.
  This can be used for confidence testing of a measured value obeying
  the F-distribution (i.e., for testing the ratio of variances, or
  equivalently for the addition of parameters to a fitted model).

    P_F(X > F; DOF1, DOF2) = PROB

  In specifying the returned probability level the user has three
  choices:

    * return the confidence level when the /CLEVEL keyword is passed;
      OR

    * return the significance level (i.e., 1 - confidence level) when
      the /SLEVEL keyword is passed (default); OR

    * return the "sigma" of the probability (i.e., compute the
      probability based on the normal distribution) when the /SIGMA
      keyword is passed.

  Note that /SLEVEL, /CLEVEL and /SIGMA are mutually exclusive.

  For the ratio of variance test, the two variances, VAR1 and VAR2,
  should be distributed according to the chi-squared distribution
  with degrees of freedom DOF1 and DOF2 respectively.  The F-value is
  computed as:

     F = (VAR1/DOF1) / (VAR2/DOF2)

  and then the probability is computed as:

     PROB = MPFTEST(F, DOF1, DOF2, ... )


  For the test of additional parameters in least squares fitting, the
  user should perform two separate fits, and have two chi-squared
  values.  One fit should be the "original" fit with no additional
  parameters, and one fit should be the "new" fit with M additional
  parameters.

    CHI1 - chi-squared value for original fit

    DOF1 - number of degrees of freedom of CHI1 (number of data
           points minus number of original parameters)

    CHI2 - chi-squared value for new fit

    DOF2 - number of degrees of freedom of CHI2

  Note that according to this formalism, the number of degrees of
  freedom in the "new" fit, DOF2, should be less than the number of
  degrees of freedom in the "original" fit, DOF1 (DOF2 < DOF1); and
  also CHI2 < CHI1.

  With the above definition, the F value is computed as:

    F = ( (CHI1-CHI2)/(DOF1-DOF2) )   /  (CHI2/DOF2)

  where DOF1-DOF2 is equal to M, and then the F-test probability is
  computed as:

     PROB = MPFTEST(F, DOF1-DOF2, DOF2, ... )

  Note that this formalism assumes that the addition of the M
  parameters is a small peturbation to the overall fit.  If the
  additional parameters dramatically changes the character of the
  model, then the first "ratio of variance" test is more appropriate,
  where F = (CHI1/DOF1) / (CHI2/DOF2).

 INPUTS:

   F - ratio of variances as defined above.

   DOF1 - number of degrees of freedom in first variance component.

   DOF2 - number of degrees of freedom in second variance component.


 RETURNS:

  Returns a scalar or vector of probabilities, as described above,
  and according to the /SLEVEL, /CLEVEL and /SIGMA keywords.

 KEYWORD PARAMETERS:

   SLEVEL - if set, then PROB describes the significance level
            (default).

   CLEVEL - if set, then PROB describes the confidence level.

   SIGMA - if set, then PROB is the number of "sigma" away from the
           mean in the normal distribution.

 EXAMPLE:

   chi1 = 62.3D & dof1 = 42d
   chi2 = 54.6D & dof2 = 40d

   f = ((chi1-chi2)/(dof1-dof2)) / (chi2/dof2)
   print, mpftest(f, dof1-dof2, dof2)

   This is a test for addition of parameters.  The "original"
   chi-squared value was 62.3 with 42 degrees of freedom, and the
   "new" chi-squared value was 54.6 with 40 degrees of freedom.
   These values reflect the addition of 2 parameters and the
   reduction of the chi-squared value by 7.7.  The significance of
   this set of circumstances is 0.071464757.

 REFERENCES:

   Algorithms taken from CEPHES special function library, by Stephen
   Moshier. (http://www.netlib.org/cephes/)

 MODIFICATION HISTORY:
   Completed, 1999, CM
   Documented, 16 Nov 2001, CM
   Reduced obtrusiveness of common block and math error handling, 18
     Nov 2001, CM
   Added documentation, 30 Dec 2001, CM
   Documentation corrections (thanks W. Landsman), 17 Jan 2002, CM
   Example docs were corrected (Thanks M. Perez-Torres), 17 Feb 2002,
     CM
   Example corrected again (sigh...), 13 Feb 2003, CM

  $Id: mpftest.pro,v 1.7 2003/02/13 23:41:16 craigm Exp $

(See /dzd2/heiles/idl/gen/mpfit/mpftest.pro)


MPNORMLIM

[Previous Routine] [Next Routine] [List of Routines]
 NAME:
   MPNORMLIM

 AUTHOR:
   Craig B. Markwardt, NASA/GSFC Code 662, Greenbelt, MD 20770
   craigm@lheamail.gsfc.nasa.gov
   UPDATED VERSIONs can be found on my WEB PAGE: 
      http://cow.physics.wisc.edu/~craigm/idl/idl.html

 PURPOSE:
   Compute confidence limits for normally distributed variable

 MAJOR TOPICS:
   Curve and Surface Fitting, Statistics

 CALLING SEQUENCE:
   Z = MPNORMLIM(PROB, [/CLEVEL, /SLEVEL ])

 DESCRIPTION:

  The function MPNORMLIM() computes confidence limits of a normally
  distributed variable (with zero mean and unit variance), for a
  desired probability level.  The returned values, Z, are the
  limiting values: a the magnitude of a normally distributed value
  is greater than Z by chance with a probability PROB:

    P_NORM(ABS(X) > Z) = PROB

  In specifying the probability level the user has two choices:

    * give the confidence level (default);

    * give the significance level (i.e., 1 - confidence level) and
      pass the /SLEVEL keyword; OR

  Note that /SLEVEL and /CLEVEL are mutually exclusive.

 INPUTS:

   PROB - scalar or vector number, giving the desired probability
          level as described above.

 RETURNS:

  Returns a scalar or vector of normal confidence limits.

 KEYWORD PARAMETERS:

   SLEVEL - if set, then PROB describes the significance level.

   CLEVEL - if set, then PROB describes the confidence level
            (default).

 EXAMPLE:

   print, mpnormlim(0.99d, /clevel)

   Print the 99% confidence limit for a normally distributed
   variable.  In this case it is about 2.58 sigma.

 REFERENCES:

   Algorithms taken from CEPHES special function library, by Stephen
   Moshier. (http://www.netlib.org/cephes/)

 MODIFICATION HISTORY:
   Completed, 1999, CM
   Documented, 16 Nov 2001, CM
   Reduced obtrusiveness of common block and math error handling, 18
     Nov 2001, CM

  $Id: mpnormlim.pro,v 1.3 2001/11/18 12:59:17 craigm Exp $

(See /dzd2/heiles/idl/gen/mpfit/mpnormlim.pro)


MPNORMTEST

[Previous Routine] [Next Routine] [List of Routines]
 NAME:
   MPNORMTEST

 AUTHOR:
   Craig B. Markwardt, NASA/GSFC Code 662, Greenbelt, MD 20770
   craigm@lheamail.gsfc.nasa.gov
   UPDATED VERSIONs can be found on my WEB PAGE: 
      http://cow.physics.wisc.edu/~craigm/idl/idl.html

 PURPOSE:
   Compute the probability of a given normally distributed Z value

 MAJOR TOPICS:
   Curve and Surface Fitting, Statistics

 CALLING SEQUENCE:
   PROB = MPNORMTEST(Z, [/CLEVEL, /SLEVEL ])

 DESCRIPTION:

  The function MPNORMTEST() computes the probability for the
  magnitude of a value drawn from the normal distribution to equal or
  exceed the given value Z.  This can be used for confidence testing
  of a measured value obeying the normal distribution.

    P_NORM(ABS(X) > Z) = PROB

  In specifying the returned probability level the user has two
  choices:

    * return the confidence level when the /CLEVEL keyword is passed;
      OR

    * return the significance level (i.e., 1 - confidence level) when
      the /SLEVEL keyword is passed (default).

  Note that /SLEVEL and /CLEVEL are mutually exclusive.

 INPUTS:

   Z - the value to best tested.  Z should be drawn from a normal
       distribution with zero mean and unit variance.  If a given
       quantity Y has mean MU and standard deviation STD, then Z can
       be computed as Z = (Y-MU)/STD.

 RETURNS:

  Returns a scalar or vector of probabilities, as described above,
  and according to the /SLEVEL and /CLEVEL keywords.

 KEYWORD PARAMETERS:

   SLEVEL - if set, then PROB describes the significance level
            (default).

   CLEVEL - if set, then PROB describes the confidence level.

 EXAMPLES:

   print, mpnormtest(5d, /slevel)

   Print the probability for the magnitude of a randomly distributed
   variable with zero mean and unit variance to exceed 5, as a
   significance level.

 REFERENCES:

   Algorithms taken from CEPHES special function library, by Stephen
   Moshier. (http://www.netlib.org/cephes/)

 MODIFICATION HISTORY:
   Completed, 1999, CM
   Documented, 16 Nov 2001, CM
   Reduced obtrusiveness of common block and math error handling, 18
     Nov 2001, CM
   Corrected error in handling of CLEVEL keyword, 05 Sep 2003

  $Id: mpnormtest.pro,v 1.5 2003/09/06 16:30:02 craigm Exp $

(See /dzd2/heiles/idl/gen/mpfit/mpnormtest.pro)


SETFITPARM.PRO

[Previous Routine] [Next Routine] [List of Routines]
 NAME:
   SetFitParm.pro

 AUTHOR:
	F.Bringezu, denet - Internetservice, Halle Germany,
	bringezu@denet.de

 PURPOSE:
   Provide a widget interface for creating a parinfo structure.
   This parinfo structure can by used by mpfit routines of Craig B. Markwardt.

 MAJOR TOPICS:
   Widget, mpfit.

 CALLING SEQUENCE:
   parinfo=SetFitParm(used_parinfo)

 DESCRIPTION:

   SetFitParm creates PARINFO using a widget interface.
   PARINFO provides constraints for paramters used by the mpfit routines.

   PARINFO is an array of structures, one for each parameter.

   A detailed description can be found in the documentation of mpcurvefit.pro
   This routine creates an array that contains a structure for each element.
   The structure has the following entries.

   - VALUE (DOUBLE): The starting parameter
   - FIXED (BOOLEAN): 1 fix the parameter, 0 don't fix it at the
     point given in VALUE.
   - LIMITS (DBLARRAY(2)): Set upper and lower limit.
   - LIMITED (BOOLEAN ARRAY 2):  Fix the limit.


   The parameter OLDPARINFO is optional. OLDPARINFO is used to set
   the default values in the widget.

   You can simply run:
   test=SetFitParm() to create the array for the first time.
   Once the array is created it can be used to set the default values
   in the widget by calling

   test2=SetFitParm(test)

 INPUTS:


 OPTIONAL INPUTS:

   OLDFITPARM - The default values of the new array

 INPUT KEYWORD PARAMETERS:

   PARENT - if this widget is to be a child, set this keyword to the
            parent widget ID.

 OUTPUT KEYWORD PARAMETERS:

   CANCEL - if the user selected the cancel button on the SETFITPARM
            widget, then this keyword will be set upon exit.

 OUTPUTS:
   PARINFO array of structures

 SEE ALSO:
   mpcurvefit

 MODIFICATION HISTORY:
   Written, FB, 12/1999
   Documented, FB, Jan 2000
   Generalized positioning code, CM 01 Feb 2000

(See /dzd2/heiles/idl/gen/mpfit/setfitparm.pro)


TNMIN

[Previous Routine] [List of Routines]
 NAME:
   TNMIN

 AUTHOR:
   Craig B. Markwardt, NASA/GSFC Code 662, Greenbelt, MD 20770
   craigm@lheamail.gsfc.nasa.gov
   UPDATED VERSIONs can be found on my WEB PAGE: 
      http://cow.physics.wisc.edu/~craigm/idl/idl.html

 PURPOSE:
   Performs function minimization (Truncated-Newton Method)

 MAJOR TOPICS:
   Optimization and Minimization

 CALLING SEQUENCE:
   parms = TNMIN(MYFUNCT, X, FUNCTARGS=fcnargs, NFEV=nfev,
                 MAXITER=maxiter, ERRMSG=errmsg, NPRINT=nprint,
                 QUIET=quiet, XTOL=xtol, STATUS=status, 
                 FGUESS=fguess, PARINFO=parinfo, BESTMIN=bestmin,
                 ITERPROC=iterproc, ITERARGS=iterargs, niter=niter)

 DESCRIPTION:

  TNMIN uses the Truncated-Newton method to minimize an arbitrary IDL
  function with respect to a given set of free parameters.  The
  user-supplied function must compute the gradient with respect to
  each parameter.  TNMIN is based on TN.F (TNBC) by Stephen Nash.

  If you want to solve a least-squares problem, to perform *curve*
  *fitting*, then you will probably want to use the routines MPFIT,
  MPFITFUN and MPFITEXPR.  Those routines are specifically optimized
  for the least-squares problem.  TNMIN is suitable for constrained
  and unconstrained optimization problems with a medium number of
  variables.  Function *maximization* can be performed using the
  MAXIMIZE keyword.

  TNMIN is similar to MPFIT in that it allows parameters to be fixed,
  simple bounding limits to be placed on parameter values, and
  parameters to be tied to other parameters.  See PARINFO below for
  discussion and examples.

 USER FUNCTION

  The user must define an IDL function which returns the desired
  value as a single scalar.  The IDL function must accept a list of
  numerical parameters, P.  Additionally, keyword parameters may be
  used to pass more data or information to the user function, via the
  FUNCTARGS keyword.

  The user function should be declared in the following way:

     FUNCTION MYFUNCT, p, dp [, keywords permitted ]
       ; Parameter values are passed in "p"
       f  = ....   ; Compute function value
       dp = ....   ; Compute partial derivatives (optional)
       return, f
     END

  The function *must* accept at least one argument, the parameter
  list P.  The vector P is implicitly assumed to be a one-dimensional
  array.  Users may pass additional information to the function by
  composing and _EXTRA structure and passing it in the FUNCTARGS
  keyword.

  User functions may also indicate a fatal error condition using the
  ERROR_CODE common block variable, as described below under the
  TNMIN_ERROR common block definition (by setting ERROR_CODE to a
  number between -15 and -1).

  ANALYTIC vs. NUMERICAL DERIVATIVES

  By default, the user must compute gradient components analytically
  using AUTODERIVATIVE=0.  As explained below, numerical derivatives
  can also be calculated using AUTODERIVATIVE=1.  

  For analytic derivatives, the user should call TNMIN using the
  default keyword value AUTODERIVATIVE=0.  [ This is different
  behavior from MPFIT, where AUTODERIVATIVE=1 is the default. ] The
  IDL user routine should compute the gradient of the function as a
  one-dimensional array of values, one for each of the parameters.
  They are passed back to TNMIN via "dp" as shown above.
             
  The derivatives with respect to fixed parameters are ignored; zero
  is an appropriate value to insert for those derivatives.  Upon
  input to the user function, DP is set to a vector with the same
  length as P, with a value of 1 for a parameter which is free, and a
  value of zero for a parameter which is fixed (and hence no
  derivative needs to be calculated).  This input vector may be
  overwritten as needed.

  For numerical derivatives, a finite differencing approximation is
  used to estimate the gradient values.  Users can activate this
  feature by passing the keyword AUTODERIVATIVE=1.  Fine control over
  this behavior can be achieved using the STEP, RELSTEP and TNSIDE
  fields of the PARINFO structure.

  When finite differencing is used for computing derivatives (ie,
  when AUTODERIVATIVE=1), the parameter DP is not passed.  Therefore
  functions can use N_PARAMS() to indicate whether they must compute
  the derivatives or not.  However there is no penalty (other than
  computation time) for computing the gradient values and users may
  switch between AUTODERIVATIVE=0 or =1 in order to test both
  scenarios.

 CONSTRAINING PARAMETER VALUES WITH THE PARINFO KEYWORD

  The behavior of TNMIN can be modified with respect to each
  parameter to be optimized.  A parameter value can be fixed; simple
  boundary constraints can be imposed; limitations on the parameter
  changes can be imposed; properties of the automatic derivative can
  be modified; and parameters can be tied to one another.

  These properties are governed by the PARINFO structure, which is
  passed as a keyword parameter to TNMIN.

  PARINFO should be an array of structures, one for each parameter.
  Each parameter is associated with one element of the array, in
  numerical order.  The structure can have the following entries
  (none are required):
  
     .VALUE - the starting parameter value (but see the START_PARAMS
              parameter for more information).
  
     .FIXED - a boolean value, whether the parameter is to be held
              fixed or not.  Fixed parameters are not varied by
              TNMIN, but are passed on to MYFUNCT for evaluation.
  
     .LIMITED - a two-element boolean array.  If the first/second
                element is set, then the parameter is bounded on the
                lower/upper side.  A parameter can be bounded on both
                sides.  Both LIMITED and LIMITS must be given
                together.
  
     .LIMITS - a two-element float or double array.  Gives the
               parameter limits on the lower and upper sides,
               respectively.  Zero, one or two of these values can be
               set, depending on the values of LIMITED.  Both LIMITED
               and LIMITS must be given together.
  
     .PARNAME - a string, giving the name of the parameter.  The
                fitting code of TNMIN does not use this tag in any
                way.
  
     .STEP - the step size to be used in calculating the numerical
             derivatives.  If set to zero, then the step size is
             computed automatically.  Ignored when AUTODERIVATIVE=0.

     .MPSIDE - the sidedness of the finite difference when computing
               numerical derivatives.  This field can take four
               values:

                  0 - one-sided derivative computed automatically
                  1 - one-sided derivative (f(x+h) - f(x)  )/h
                 -1 - one-sided derivative (f(x)   - f(x-h))/h
                  2 - two-sided derivative (f(x+h) - f(x-h))/(2*h)

              Where H is the STEP parameter described above.  The
              "automatic" one-sided derivative method will chose a
              direction for the finite difference which does not
              violate any constraints.  The other methods do not
              perform this check.  The two-sided method is in
              principle more precise, but requires twice as many
              function evaluations.  Default: 0.

     .TIED - a string expression which "ties" the parameter to other
             free or fixed parameters.  Any expression involving
             constants and the parameter array P are permitted.
             Example: if parameter 2 is always to be twice parameter
             1 then use the following: parinfo(2).tied = '2 * P(1)'.
             Since they are totally constrained, tied parameters are
             considered to be fixed; no errors are computed for them.
             [ NOTE: the PARNAME can't be used in expressions. ]
  
  Future modifications to the PARINFO structure, if any, will involve
  adding structure tags beginning with the two letters "MP" or "TN".
  Therefore programmers are urged to avoid using tags starting with
  these two combinations of letters; otherwise they are free to
  include their own fields within the PARINFO structure, and they
  will be ignored.
  
  PARINFO Example:
  parinfo = replicate({value:0.D, fixed:0, limited:[0,0], $
                       limits:[0.D,0]}, 5)
  parinfo(0).fixed = 1
  parinfo(4).limited(0) = 1
  parinfo(4).limits(0)  = 50.D
  parinfo(*).value = [5.7D, 2.2, 500., 1.5, 2000.]
  
  A total of 5 parameters, with starting values of 5.7,
  2.2, 500, 1.5, and 2000 are given.  The first parameter
  is fixed at a value of 5.7, and the last parameter is
  constrained to be above 50.


 INPUTS:

   MYFUNCT - a string variable containing the name of the function to
             be minimized (see USER FUNCTION above).  The IDL routine
             should return the value of the function and optionally
             its gradients.

   X - An array of starting values for each of the parameters of the
       model.

       This parameter is optional if the PARINFO keyword is used.
       See above.  The PARINFO keyword provides a mechanism to fix or
       constrain individual parameters.  If both X and PARINFO are
       passed, then the starting *value* is taken from X, but the
       *constraints* are taken from PARINFO.
 

 RETURNS:

   Returns the array of best-fit parameters.


 KEYWORD PARAMETERS:

   AUTODERIVATIVE - If this is set, derivatives of the function will
                    be computed automatically via a finite
                    differencing procedure.  If not set, then MYFUNCT
                    must provide the (analytical) derivatives.
                    Default: 0 (analytical derivatives required)

   BESTMIN - upon return, the best minimum function value that TNMIN
             could find.

   EPSABS - a nonnegative real variable.  Termination occurs when the
            absolute error between consecutive iterates is at most
            EPSABS.  Note that using EPSREL is normally preferable
            over EPSABS, except in cases where ABS(F) is much larger
            than 1 at the optimal point.  A value of zero indicates
            the absolute error test is not applied.  If EPSABS is
            specified, then both EPSREL and EPSABS tests are applied;
            if either succeeds then termination occurs.
            Default: 0 (i.e., only EPSREL is applied).

   EPSREL - a nonnegative input variable. Termination occurs when the
            relative error between two consecutive iterates is at
            most EPSREL.  Therefore, EPSREL measures the relative
            error desired in the function.  An additional, more
            lenient, stopping condition can be applied by specifying
            the EPSABS keyword.
            Default: 100 * Machine Precision

   ERRMSG - a string error or warning message is returned.

   FGUESS - provides the routine a guess to the minimum value.
            Default: 0

   FUNCTARGS - A structure which contains the parameters to be passed
               to the user-supplied function specified by MYFUNCT via
               the _EXTRA mechanism.  This is the way you can pass
               additional data to your user-supplied function without
               using common blocks.

               Consider the following example:
                if FUNCTARGS = { XVAL:[1.D,2,3], YVAL:[1.D,4,9]}
                then the user supplied function should be declared
                like this:
                FUNCTION MYFUNCT, P, XVAL=x, YVAL=y

               By default, no extra parameters are passed to the
               user-supplied function.

   ITERARGS - The keyword arguments to be passed to ITERPROC via the
              _EXTRA mechanism.  This should be a structure, and is
              similar in operation to FUNCTARGS.
              Default: no arguments are passed.

   ITERDERIV - Intended to print function gradient information.  If
               set, then the ITERDERIV keyword is set in each call to
               ITERPROC.  In the default ITERPROC, parameter values
               and gradient information are both printed when this
               keyword is set.

   ITERPROC - The name of a procedure to be called upon each NPRINT
              iteration of the TNMIN routine.  It should be declared
              in the following way:

              PRO ITERPROC, MYFUNCT, p, iter, fnorm, FUNCTARGS=fcnargs, $
                PARINFO=parinfo, QUIET=quiet, _EXTRA=extra
                ; perform custom iteration update
              END
         
              ITERPROC must accept the _EXTRA keyword, in case of
              future changes to the calling procedure.
          
              MYFUNCT is the user-supplied function to be minimized,
              P is the current set of model parameters, ITER is the
              iteration number, and FUNCTARGS are the arguments to be
              passed to MYFUNCT.  FNORM is should be the function
              value.  QUIET is set when no textual output should be
              printed.  See below for documentation of PARINFO.

              In implementation, ITERPROC can perform updates to the
              terminal or graphical user interface, to provide
              feedback while the fit proceeds.  If the fit is to be
              stopped for any reason, then ITERPROC should set the
              common block variable ERROR_CODE to negative value
              between -15 and -1 (see TNMIN_ERROR common block
              below).  In principle, ITERPROC should probably not
              modify the parameter values, because it may interfere
              with the algorithm's stability.  In practice it is
              allowed.

              Default: an internal routine is used to print the
                       parameter values.

   MAXITER - The maximum number of iterations to perform.  If the
             number is exceeded, then the STATUS value is set to 5
             and TNMIN returns.
             Default: 200 iterations

   MAXIMIZE - If set, the function is maximized instead of
              minimized.

   MAXNFEV - The maximum number of function evaluations to perform.
             If the number is exceeded, then the STATUS value is set
             to -17 and TNMIN returns.  A value of zero indicates no
             maximum.
             Default: 0 (no maximum)

   NFEV - upon return, the number of MYFUNCT function evaluations
          performed.

   NITER - upon return, number of iterations completed.

   NPRINT - The frequency with which ITERPROC is called.  A value of
            1 indicates that ITERPROC is called with every iteration,
            while 2 indicates every other iteration, etc.  
            Default value: 1

   PARINFO - Provides a mechanism for more sophisticated constraints
             to be placed on parameter values.  When PARINFO is not
             passed, then it is assumed that all parameters are free
             and unconstrained.  Values in PARINFO are never modified
             during a call to TNMIN.

             See description above for the structure of PARINFO.

             Default value:  all parameters are free and unconstrained.

   QUIET - set this keyword when no textual output should be printed
           by TNMIN

   STATUS - an integer status code is returned.  All values greater
            than zero can represent success (however STATUS EQ 5 may
            indicate failure to converge).  Gaps in the numbering
            system below are to maintain compatibility with MPFIT.
            Upon return, STATUS can have one of the following values:

        -18  a fatal internal error occurred during optimization.

        -17  the maximum number of function evaluations has been
             reached without convergence.

        -16  a parameter or function value has become infinite or an
             undefined number.  This is usually a consequence of
             numerical overflow in the user's function, which must be
             avoided.

        -15 to -1 
             these are error codes that either MYFUNCT or ITERPROC
             may return to terminate the fitting process (see
             description of MPFIT_ERROR common below).  If either
             MYFUNCT or ITERPROC set ERROR_CODE to a negative number,
             then that number is returned in STATUS.  Values from -15
             to -1 are reserved for the user functions and will not
             clash with MPFIT.

	   0  improper input parameters.
         
	   1  convergence was reached.

          2-4 (RESERVED)
         
	   5  the maximum number of iterations has been reached

          6-8 (RESERVED)
         

 EXAMPLE:

   FUNCTION F, X, DF, _EXTRA=extra  ;; *** MUST ACCEPT KEYWORDS
     F  = (X(0)-1)^2 + (X(1)+7)^2
     DF = [ 2D * (X(0)-1), 2D * (X(1)+7) ] ; Gradient
     RETURN, F
   END

   P = TNMIN('F', [0D, 0D], BESTMIN=F0)
   Minimizes the function F(x0,x1) = (x0-1)^2 + (x1+7)^2.


 COMMON BLOCKS:

   COMMON TNMIN_ERROR, ERROR_CODE

     User routines may stop the fitting process at any time by
     setting an error condition.  This condition may be set in either
     the user's model computation routine (MYFUNCT), or in the
     iteration procedure (ITERPROC).

     To stop the fitting, the above common block must be declared,
     and ERROR_CODE must be set to a negative number.  After the user
     procedure or function returns, TNMIN checks the value of this
     common block variable and exits immediately if the error
     condition has been set.  By default the value of ERROR_CODE is
     zero, indicating a successful function/procedure call.


 REFERENCES:

   TRUNCATED-NEWTON METHOD, TN.F
      Stephen G. Nash, Operations Research and Applied Statistics
      Department
      http://www.netlib.org/opt/tn

   Nash, S. G. 1984, "Newton-Type Minimization via the Lanczos
      Method," SIAM J. Numerical Analysis, 21, p. 770-778
      

 MODIFICATION HISTORY:
   Derived from TN.F by Stephen Nash with many changes and additions,
      to conform to MPFIT paradigm, 19 Jan 1999, CM
   Changed web address to COW, 25 Sep 1999, CM
   Alphabetized documented keyword parameters, 02 Oct 1999, CM
   Changed to ERROR_CODE for error condition, 28 Jan 2000, CM
   Continued and fairly major improvements (CM, 08 Jan 2001):
      - calling of user procedure is now concentrated in TNMIN_CALL,
        and arguments are reduced by storing a large number of them
        in common blocks;
      - finite differencing is done within TNMIN_CALL; added
        AUTODERIVATIVE=1 so that finite differencing can be enabled,
        both one and two sided;
      - a new procedure to parse PARINFO fields, borrowed from MPFIT;
        brought PARINFO keywords up to date with MPFIT;
      - go through and check for float vs. double discrepancies;
      - add explicit MAXIMIZE keyword, and support in TNMIN_CALL and
        TNMIN_DEFITER to print the correct values in that case;
        TNMIN_DEFITER now prints function gradient values if
        requested;
      - convert to common-based system of MPFIT for storing machine
        constants; revert TNMIN_ENORM to simple sum of squares, at
        least for now;
      - remove limit on number of function evaluations, at least for
        now, and until I can understand what happens when we do
        numerical derivatives.
   Further changes: more float vs double; disable TNMINSTEP for now;
     experimented with convergence test in case of function
     maximization, 11 Jan 2001, CM
   TNMINSTEP is parsed but not enabled, 13 Mar 2001
   Major code cleanups; internal docs; reduced commons, CM, 08 Apr
     2001
   Continued code cleanups; documentation; the STATUS keyword
     actually means something, CM, 10 Apr 2001
   Added reference to Nash paper, CM, 08 Feb 2002
   Fixed 16-bit loop indices, D. Schelgel, 22 Apr 2003

 TODO
  - scale derivatives semi-automatically;
  - ability to scale and offset parameters;
  
  $Id: tnmin.pro,v 1.12 2003/04/22 23:42:49 craigm Exp $

(See /dzd2/heiles/idl/gen/mpfit/tnmin.pro)