mpfitfun.pro

非线性最小二乘-LM法 是用IDL语言编写的

;+
; NAME:
;   MPFITFUN
;
; AUTHOR:
;   Craig B. Markwardt, NASA/GSFC Code 662, Greenbelt, MD 20770
;   [e-mail]
;   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.
;
;  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 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:
;
;    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 "EXPLICIT 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.
;
;     .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.
;
; COMPATIBILITY
;
;  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.
;
;
; 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. WARNING: this option is incomplete and untested.
;
;   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
;
;   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
;             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