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  Craig B. Markwardt, NASA/GSFC Code 662, Greenbelt, MD 20770
  UPDATED VERSIONs can be found on my WEB PAGE:


  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, ...)


  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
  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.
  Data values of NaN or Infinity for "Y", "ERR" or "WEIGHTS" will be
  ignored as missing data if the NAN keyword is set. Otherwise, they
  may cause the fitting loop to halt with an error message. Note
  that the fit will still halt if the model function, or its
  derivatives, produces infinite or NaN values.
  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:
    ; The independent variable is X
    ; Parameter values are passed in "P"
    YMOD = ... computed model values at X ...
    return, YMOD
  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.
  MPFIT by default calculates derivatives numerically via a finite
  difference approximation. However, the user function *may*
  calculate the derivatives if desired, but only if the model
  function is declared with an additional position parameter, DP, as
  described below.
  To enable explicit derivatives for all parameters, set
  When AUTODERIVATIVE=0, the user function is responsible for
  calculating the derivatives of the user function with respect to
  each parameter. The user function should be declared as follows:
    ; MYFUNCT - example user function
    ; P - input parameter values (N-element array)
    ; DP - upon input, an N-vector indicating which parameters
    ; to compute derivatives for;
    ; upon output, the user function must return
    ; an ARRAY(M,N) of derivatives in this keyword
    ; (keywords) - any other keywords specified by FUNCTARGS
    ; RETURNS - function values
    FUNCTION MYFUNCT, x, p, dp [, (additional keywords if desired)]
    model = F(x, p) ;; Model function
    if n_params() GT 2 then begin
      ; Create derivative and compute derivative array
      requested = dp ; Save original value of DP
      dp = make_array(n_elements(x), n_elements(p), value=x[0]*0)
      ; Compute derivative if requested by caller
      for i = 0, n_elements(p)-1 do if requested(i) NE 0 then $
        dp(*,i) = FGRAD(x, p, i)
    return, resid
  where FGRAD(x, p, i) is a model function which computes the
  derivative of the model F(x,p) with respect to parameter P(i) at X.
  Derivatives should be returned in the DP array. DP should be an
  ARRAY(m,n) array, where m is the number of data points and n is the
  number of parameters. DP[i,j] is the derivative of the ith
  function value with respect to the jth parameter.
  MPFIT may not always request derivatives from the user function.
  In those cases, the parameter DP is not passed. Therefore
  functions can use N_PARAMS() to indicate whether they must compute
  the derivatives or not.
  For additional information about explicit derivatives, including
  additional settings and debugging options, see the discussion under

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
    .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
                  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 as an equality constraint. 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 a TIED expression. ]
    .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
  "MP", but otherwise they are free to include their own fields
  within the PARINFO structure, which will be ignored by MPFIT.
  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.


  This function is designed to work with IDL 5.0 or greater.
  Because TIED parameters rely on the EXECUTE() function, they cannot
  be used with the free version of the IDL Virtual Machine.


  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
        NOTE: the following special cases apply:
                * if Y is NaN or Infinite, and the NAN keyword is
                  set, then the corresponding data point is ignored
  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.
        NOTE: the following special cases apply:
                * if ERR is zero, then the corresponding data point
                  is ignored
                * if ERR is NaN or Infinite, and the NAN keyword is
                  set, then the corresponding data point is ignored
                * if ERR is negative, then the absolute value of
                  ERR is used.
  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


  Returns the array of best-fit parameters.

Keyword Parameters

  BESTNORM - the value of the summed squared residuals for the
              returned parameter values.
  BEST_FJAC - upon return, BEST_FJAC contains the Jacobian, or
              partial derivative, matrix for the best-fit model.
              The values are an array,
              ARRAY(N_ELEMENTS(DEVIATES),NFREE) where NFREE is the
              number of free parameters. This array is only
              computed if /CALC_FJAC is set, otherwise BEST_FJAC is
              The returned array is such that BEST_FJAC[I,J] is the
              partial derivative of the model with respect to
              parameter PARMS[PFREE_INDEX[J]].
  BEST_RESID - upon return, an array of best-fit deviates,
                normalized by the weights or errors.
  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:
                  PCOR = COV * 0
                  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])
          or equivalently, in vector notation,
                  PCOR = COV / (PERROR # PERROR)
          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. WARNING: this option is incomplete and untested.
  DOF - number of degrees of freedom, computed as
        Note that this doesn't account for pegged parameters (see
        NPEGGED). It also does not account for data points which
        are assigned zero weight, for example if :
          * WEIGHTS[i] EQ 0, or
          * ERR[i] EQ infinity, or
          * any of the values is "undefined" and /NAN is set.
  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
              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
              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 of calculation iterations exceeds MAXITER, then
            the STATUS value is set to 5 and MPFIT returns.
            If MAXITER EQ 0, then MPFIT does not iterate to adjust
            parameter values; however, the user function is evaluated
            and parameter errors/covariance/Jacobian are estimated
            before returning.
            Default: 200 iterations
  NAN - ignore infinite or NaN values in the Y, ERR or WEIGHTS
        parameters. These values will be treated as missing data.
        However, the fit will still halt with an error condition
        if the model function becomes infinite.
  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
  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 - A one-dimensional array of structures.
            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
            *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
  PFREE_INDEX - upon return, PFREE_INDEX contains an index array
                which indicates which parameter were allowed to
                vary. I.e. of all the parameters PARMS, only
                PARMS[PFREE_INDEX] were varied.
  QUERY - if set, then MPFIT() will return immediately with one of
          the following values:
                1 - if MIN_VERSION is not set
                1 - if MIN_VERSION is set and MPFIT satisfies the minimum
                0 - if MIN_VERSION is set and MPFIT does not satisfy it
          Default: not set.
  QUIET - set this keyword when no textual output should be printed
          by MPFIT
  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:
        -18 a fatal execution error has occurred. More information
            may be available in the ERRMSG string.
        -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.
  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 for standard weightings:
                1D/ERR^2 - Normal weighting (ERR is the measurement error)
                1D/Y - Poisson weighting (counting statistics)
                1D - Unweighted
        NOTE: the following special cases apply:
                * if WEIGHTS is zero, then the corresponding data point
                  is ignored
                * if WEIGHTS is NaN or Infinite, and the NAN keyword is
                  set, then the corresponding data point is ignored
                * if WEIGHTS is negative, then the absolute value of
                  WEIGHTS is used.
  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.


  ; 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

Common Blocks

    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.


  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
  Convert to IDL 5 array syntax (!), 16 Jul 2006, CM
  Move STRICTARR compile option inside each function/procedure, 9
    Oct 2006
  Add NAN keyword, to ignore non-finite data values, 28 Oct 2006, CM
  Clarify documentation on user-function, derivatives, and PARINFO,
    27 May 2007
  Fix bug in handling of explicit derivatives with errors/weights
    (the weights were not being applied), CM, 03 Sep 2007
  Add COMPATIBILITY section, CM, 13 Dec 2007
  Add documentation about NAN behavior, CM, 30 Mar 2009
    update some documentation that had become stale, CM, 2010-10-28
  Documentation corrections, CM, 2011-08-26
  Additional documentation about explicit derivatives, CM, 2012-07-23
  $Id: mpfitfun.pro,v 1.19 2012/09/27 23:59:31 cmarkwar Exp $

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