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MPFIT

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 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 explicit 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.
See below for error reporting mechanisms.

## Checking Status And Hanndling Errors

Upon return, MPFIT will report the status of the fitting operation
in the STATUS and ERRMSG keywords. The STATUS keyword will contain
a numerical code which indicates the success or failure status.
Generally speaking, any value 1 or greater indicates success, while
a value of 0 or less indicates a possible failure. The ERRMSG
keyword will contain a text string which should describe the error
condition more fully.
By default, MPFIT will trap fatal errors and report them to the
caller gracefully. However, during the debugging process, it is
often useful to halt execution where the error occurred. When you
set the NOCATCH keyword, MPFIT will not do any special error
trapping, and execution will stop whereever the error occurred.
MPFIT does not explicitly change the !ERROR_STATE variable
(although it may be changed implicitly if MPFIT calls MESSAGE). It
is the caller's responsibility to call MESSAGE, /RESET to ensure
that the error state is initialized before calling MPFIT.
User functions may also indicate non-fatal error conditions 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). When the user function sets an error
condition via ERROR_CODE, MPFIT will gracefully exit immediately
and report this condition to the caller. The ERROR_CODE is
returned in the STATUS keyword in that case.

## Explicit 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, 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. If the user function does not accept this additional
parameter, MPFIT will report an error. As a practical matter, it
is often sufficient and even faster to allow MPFIT to calculate the
derivatives numerically, but this option is available for users who
wish more control over the fitting process.
There are two ways to enable explicit derivatives. First, the user
can set the keyword AUTODERIVATIVE=0, which is a global switch for
all parameters. In this case, MPFIT will request explicit
derivatives for every free parameter.
Second, the user may request explicit derivatives for specifically
selected parameters using the PARINFO.MPSIDE=3 (see "CONSTRAINING
PARAMETER VALUES WITH THE PARINFO KEYWORD" below). In this
strategy, the user picks and chooses which parameter derivatives
are computed explicitly versus numerically. When PARINFO[i].MPSIDE
EQ 3, then the ith parameter derivative is computed explicitly.
The keyword setting AUTODERIVATIVE=0 always globally overrides the
individual values of PARINFO.MPSIDE. Setting AUTODERIVATIVE=0 is
equivalent to resetting PARINFO.MPSIDE=3 for all parameters.
Even if the user requests explicit derivatives for some or all
parameters, MPFIT will not always request explicit derivatives on
every user function call.

## Explicit Derivatives - Calling Interface

When AUTODERIVATIVE=0, the user function is responsible for
calculating the derivatives of the *residuals* 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 - residual values
;
FUNCTION MYFUNCT, p, dp, X=x, Y=y, ERR=err
model = F(x, p) ;; Model function
resid = (y - model)/err ;; Residual calculation (for example)

if n_params() GT 1 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)
; 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) / err
endif

return, resid
END
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.
A quirk in the implementation leaves a stray negative sign in the
definition of DP. The derivative of the *residual* should be
"-FGRAD(x,p,i) / err" because of how the residual is defined
("resid = (data - model) / err"). **HOWEVER** because of the
i.e. the opposite sign of the gradient of RESID.
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
residual with respect to the jth parameter (note the minus sign
due to the quirk described above).
As noted above, 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.

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. In the example above, the original DP
vector is saved to a variable called REQUESTED, and used as a mask
to calculate only those derivatives that are required.
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.

## Explicit Derivatives - Testing And Debugging

For reasonably complicated user functions, the calculation of
explicit derivatives of the correct sign and magnitude can be
difficult to get right. A simple sign error can cause MPFIT to be
confused. MPFIT has a derivative debugging mode which will compute
the derivatives *both* numerically and explicitly, and compare the
results.
It is expected that during production usage, derivative debugging
should be disabled for all parameters.
In order to enable derivative debugging mode, set the following
PARINFO members for the ith parameter.
PARINFO[i].MPSIDE = 3 ; Enable explicit derivatives
PARINFO[i].MPDERIV_DEBUG = 1 ; Enable derivative debugging mode
PARINFO[i].MPDERIV_RELTOL = ?? ; Relative tolerance for comparison
PARINFO[i].MPDERIV_ABSTOL = ?? ; Absolute tolerance for comparison
Note that these settings are maintained on a parameter-by-parameter
basis using PARINFO, so the user can choose which parameters
derivatives will be tested.
When .MPDERIV_DEBUG is set, then MPFIT first computes the
derivative explicitly by requesting them from the user function.
Then, it computes the derivatives numerically via finite
differencing, and compares the two values. If the difference
exceeds a tolerance threshold, then the values are printed out to
alert the user. The tolerance level threshold contains both a
relative and an absolute component, and is expressed as,
ABS(DERIV_U - DERIV_N) GE (ABSTOL + RELTOL*ABS(DERIV_U))
where DERIV_U and DERIV_N are the derivatives computed explicitly
and numerically, respectively. Appropriate values
for most users will be:
PARINFO[i].MPDERIV_RELTOL = 1d-3 ;; Suggested relative tolerance
PARINFO[i].MPDERIV_ABSTOL = 1d-7 ;; Suggested absolute tolerance
although these thresholds may have to be adjusted for a particular
problem. When the threshold is exceeded, users can expect to see a
tabular report like this one:
FJAC DEBUG BEGIN
# IPNT FUNC DERIV_U DERIV_N DIFF_ABS DIFF_REL
FJAC PARM 2
80 -0.7308 0.04233 0.04233 -5.543E-07 -1.309E-05
99 1.370 0.01417 0.01417 -5.518E-07 -3.895E-05
118 0.07187 -0.01400 -0.01400 -5.566E-07 3.977E-05
137 1.844 -0.04216 -0.04216 -5.589E-07 1.326E-05
FJAC DEBUG END
The report will be bracketed by FJAC DEBUG BEGIN/END statements.
Each parameter will be delimited by the statement FJAC PARM n,
where n is the parameter number. The columns are,
IPNT - data point number (0 ... M-1)
FUNC - function value at that point
DERIV_U - explicit derivative value at that point
DERIV_N - numerical derivative estimate at that point
DIFF_ABS - absolute difference = (DERIV_U - DERIV_N)
DIFF_REL - relative difference = (DIFF_ABS)/(DERIV_U)
When prints appear in this report, it is most important to check
that the derivatives computed in two different ways have the same
numerical sign and the same order of magnitude, since these are the
most common programming mistakes.
A line of this form may also appear
# FJAC_MASK = 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
This line indicates for which parameters explicit derivatives are
expected. A list of all-1s indicates all explicit derivatives for
all parameters are requested from the user function.

## 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

.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 - selector for type of derivative calculation. This
field can take one of five possible 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)
3 - explicit derivative used for this parameter
In the first four cases, the derivative is approximated
numerically by finite difference, with step size
H=STEP, where the STEP parameter is defined above. The
last case, MPSIDE=3, indicates to allow the user
function to compute the derivative explicitly (see
section on "EXPLICIT DERIVATIVES"). AUTODERIVATIVE=0
overrides this setting for all parameters, and is
equivalent to MPSIDE=3 for all parameters. For
MPSIDE=0, the "automatic" one-sided derivative method
will chose a direction for the finite difference which
does not violate any constraints. The other methods
(MPSIDE=-1 or MPSIDE=1) do not perform this check. The
two-sided method is in principle more precise, but
requires twice as many function evaluations. Default:
0.
.MPDERIV_DEBUG - set this value to 1 to enable debugging of
user-supplied explicit derivatives (see "TESTING and
DEBUGGING" section above). In addition, the
user must enable calculation of explicit derivatives by
either setting AUTODERIVATIVE=0, or MPSIDE=3 for the
desired parameters. When this option is enabled, a
report may be printed to the console, depending on the
MPDERIV_ABSTOL and MPDERIV_RELTOL settings.
Default: 0 (no debugging)

.MPDERIV_ABSTOL, .MPDERIV_RELTOL - tolerance settings for
print-out of debugging information, for each parameter
where debugging is enabled. See "TESTING and
DEBUGGING" section above for the meanings of these two
fields.
.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.tied = '2 * P'.
Since they are totally constrained, tied parameters are
considered to be fixed; no errors are computed for them,
and any LIMITS are not obeyed.
[ 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.fixed = 1
parinfo.limited = 1
parinfo.limits = 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.

## Recursion

Generally, recursion is not allowed. As of version 1.77, MPFIT has
recursion protection which does not allow a model function to
itself call MPFIT. Users who wish to perform multi-level
optimization should investigate the 'EXTERNAL' function evaluation
methods described below for hard-to-evaluate functions. That
method places more control in the user's hands. The user can
design a "recursive" application by taking care.
In most cases the recursion protection should be well-behaved.
However, if the user is doing debugging, it is possible for the
protection system to get "stuck." In order to reset it, run the

## Procedure

MPFIT_RESET_RECURSION
and the protection system should get "unstuck." It is save to call
this procedure at any time.

## Compatibility

This function is designed to work with IDL 5.0 or greater.

Because TIED parameters and the "(EXTERNAL)" user-model feature use
the EXECUTE() function, they cannot be used with the free version
of the IDL Virtual Machine.
DETERMINING THE VERSION OF MPFIT
MPFIT is a changing library. Users of MPFIT may also depend on a
specific version of the library being present. As of version 1.70
of MPFIT, a VERSION keyword has been added which allows the user to
query which version is present. The keyword works like this:
RESULT = MPFIT(/query, VERSION=version)
This call uses the /QUERY keyword to query the version number
without performing any computations. Users of MPFIT can call this
method to determine which version is in the IDL path before
actually using MPFIT to do any numerical work. Upon return, the
VERSION keyword contains the version number of MPFIT, expressed as
a string of the form 'X.Y' where X and Y are integers.
Users can perform their own version checking, or use the built-in
error checking of MPFIT. The MIN_VERSION keyword enforces the
requested minimum version number. For example,
RESULT = MPFIT(/query, VERSION=version, MIN_VERSION='1.70')
will check whether the accessed version is 1.70 or greater, without
performing any numerical processing.
The VERSION and MIN_VERSION keywords were added in MPFIT
version 1.70 and later. If the caller attempts to use the VERSION
or MIN_VERSION keywords, and an *older* version of the code is
present in the caller's path, then IDL will throw an 'unknown
keyword' error. Therefore, in order to be robust, the caller, must
use exception handling. Here is an example demanding at least
version 1.70.
MPFIT_OK = 0 & VERSION = '<unknown>'
CATCH, CATCHERR
IF CATCHERR EQ 0 THEN MPFIT_OK = MPFIT(/query, VERSION=version, \$
MIN_VERSION='1.70')
CATCH, /CANCEL
IF NOT MPFIT_OK THEN \$
MESSAGE, 'ERROR: you must have MPFIT version 1.70 or higher in '+\$
Of course, the caller can also do its own version number
requirements checking.
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.
** SPECIAL NOTE ** For the "(EXTERNAL)" case, the quirk noted above
does not apply. The gradient matrix, EXTERNAL_FJAC, should be
comparable to "-FGRAD(x,p)/err", which is the *opposite* sign of
the DP matrix described above. In other words, EXTERNAL_FJAC
has the same sign as the derivative of EXTERNAL_FVEC, and the
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 one-dimensional 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.
Exceptions:
* if /QUERY is set (see QUERY).

## 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 explicit derivatives.
Default: set (=1)
NOTE: to supply your own explicit derivatives,
explicitly pass AUTODERIVATIVE=0
BESTNORM - upon return, the value of the summed squared weighted
residuals for the returned parameter values,
i.e. TOTAL(DEVIATES^2).
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
undefined.
The returned array is such that BEST_FJAC[I,J] is the
partial derivative of DEVIATES[I] with respect to
parameter PARMS[PFREE_INDEX[J]]. Note that since
deviates are (data-model)*weight, the Jacobian of the
*deviates* will have the opposite sign from the
Jacobian of the *model*, and may be scaled by a
factor.
BEST_RESID - upon return, an array of best-fit deviates.
CALC_FJAC - if set, then calculate the Jacobian and return it in
BEST_FJAC. If not set, then the return value of
BEST_FJAC is undefined.
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.
DOF - number of degrees of freedom, computed as
DOF = N_ELEMENTS(DEVIATES) - NFREE
Note that this doesn't account for pegged parameters (see
NPEGGED). It also does not account for data points which
are assigned zero weight by the user function.
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
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.
ITERPRINT - The name of an IDL procedure, equivalent to PRINT,
that ITERPROC will use to render output. ITERPRINT
should be able to accept at least four positional
arguments. In addition, it should be able to accept
the standard FORMAT keyword for output formatting; and
the UNIT keyword, to redirect output to a logical file
unit (default should be UNIT=1, standard output).
These keywords are passed using the ITERARGS keyword
above. The ITERPRINT procedure must accept the _EXTRA
keyword.
NOTE: that much formatting can be handled with the
MPPRINT and MPFORMAT tags.
Default: 'MPFIT_DEFPRINT' (default internal formatter)
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, PFORMAT=pformat, \$
UNIT=unit, ...
; 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.
PFORMAT is the default parameter value format. UNIT is
passed on to the ITERPRINT procedure, and should
indicate the file unit where log output will be sent
(default: standard output).
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 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
MIN_VERSION - The minimum requested version number. This must be
a scalar string of the form returned by the VERSION
keyword. If the current version of MPFIT does not
satisfy the minimum requested version number, then,
MPFIT(/query, min_version='...') returns 0
MPFIT(...) returns NAN
Default: no version number check
NOTE: MIN_VERSION was added in MPFIT version 1.70
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.
NOCATCH - if set, then MPFIT will not perform any error trapping.
By default (not set), MPFIT will trap errors and report
them to the caller. This keyword will typically be used
for debugging.
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 - 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
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
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
The VERSION output keyword is always set upon return.
Default: not set.
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:
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
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.
VERSION - upon return, VERSION will be set to the MPFIT internal
version number. The version number will be a string of
the form "X.Y" where X is a major revision number and Y
is a minor revision number.
NOTE: the VERSION keyword was not present before
MPFIT version number 1.70, therefore, callers must
use exception handling when using this keyword.
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
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'
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_QRSOL;V provides the solution for dx.

## References

Markwardt, C. B. 2008, "Non-Linear Least Squares Fitting in IDL
with MPFIT," in proc. Astronomical Data Analysis Software and
Systems XVIII, Quebec, Canada, ASP Conference Series, Vol. XXX, eds.
D. Bohlender, P. Dowler & D. Durand (Astronomical Society of the
Pacific: San Francisco), p. 251-254 (ISBN: 978-1-58381-702-5)
http://arxiv.org/abs/0902.2850
Refer to the MPFIT website as:
http://purl.com/net/mpfit
MINPACK-1 software, by Jorge More' et al, available from netlib.
http://www.netlib.org/
"Optimization Software Guide," Jorge More' and Stephen Wright,
SIAM, *Frontiers in Applied Mathematics*, Number 14.
(ISBN: 978-0-898713-22-0)
More', J. 1978, "The Levenberg-Marquardt Algorithm: Implementation
and Theory," in Numerical Analysis, vol. 630, ed. G. A. Watson
(Springer-Verlag: Berlin), p. 105 (DOI: 10.1007/BFb0067690 )

## 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 <schlegel@astro.princeton.edu>)
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
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
Add the ITERPRINT keyword to allow redirecting output, 26 Sep
2004, CM
Correct MAXSTEP behavior in case of a negative parameter, 26 Sep
2004, CM
Fix bug in the parsing of MINSTEP/MAXSTEP, 10 Apr 2005, CM
Fix bug in the handling of upper/lower limits when the limit was
negative (the fitting code would never "stick" to the lower
limit), 29 Jun 2005, CM
Small documentation update for the TIED field, 05 Sep 2005, CM
Convert to IDL 5 array syntax (!), 16 Jul 2006, CM
If MAXITER equals zero, then do the basic parameter checking and
uncertainty analysis, but do not adjust the parameters, 15 Aug
2006, CM
Added documentation, 18 Sep 2006, CM
A few more IDL 5 array syntax changes, 25 Sep 2006, CM
Move STRICTARR compile option inside each function/procedure, 9 Oct 2006
Bug fix for case of MPMAXSTEP and fixed parameters, thanks
to Huib Intema (who found it from the Python translation!), 05 Feb 2007
Similar fix for MPFIT_FDJAC2 and the MPSIDE sidedness of
derivatives, also thanks to Huib Intema, 07 Feb 2007
Clarify documentation on user-function, derivatives, and PARINFO,
27 May 2007
Change the wording of "Analytic Derivatives" to "Explicit
Derivatives" in the documentation, CM, 03 Sep 2007
Further documentation tweaks, CM, 13 Dec 2007
2007
Document and enforce that START_PARMS and PARINFO are 1-d arrays,
CM, 29 Mar 2008
Previous change for 1-D arrays wasn't correct for
PARINFO.LIMITED/.LIMITS; now fixed, CM, 03 May 2008
Documentation adjustments, CM, 20 Aug 2008
Change some minor FOR-loop variables to type-long, CM, 03 Sep 2008
Change error handling slightly, document NOCATCH keyword,
document error handling in general, CM, 01 Oct 2008
Special case: when either LIMITS is zero, and a parameter pushes
against that limit, the coded that 'pegged' it there would not
work since it was a relative condition; now zero is handled
properly, CM, 08 Nov 2008
Documentation of how TIED interacts with LIMITS, CM, 21 Dec 2008
Better documentation of references, CM, 27 Feb 2009
If MAXITER=0, then be sure to set STATUS=5, which permits the
the covariance matrix to be computed, CM, 14 Apr 2009
Avoid numerical underflow while solving for the LM parameter,
(thanks to Sergey Koposov) CM, 14 Apr 2009
Use individual functions for all possible MPFIT_CALL permutations,
(and make sure the syntax is right) CM, 01 Sep 2009
Correct behavior of MPMAXSTEP when some parameters are frozen,
thanks to Josh Destree, CM, 22 Nov 2009
Update the references section, CM, 22 Nov 2009
1.70 - Add the VERSION and MIN_VERSION keywords, CM, 22 Nov 2009
1.71 - Store pre-calculated revision in common, CM, 23 Nov 2009
1.72-1.74 - Documented alternate method to compute correlation matrix,
CM, 05 Feb 2010
1.75 - Enforce TIED constraints when preparing to terminate the
routine, CM, 2010-06-22
1.76 - Documented input keywords now are not modified upon output,
CM, 2010-07-13
1.77 - Upon user request (/CALC_FJAC), compute Jacobian matrix and
return in BEST_FJAC; also return best residuals in
BEST_RESID; also return an index list of free parameters as
PFREE_INDEX; add a fencepost to prevent recursion
CM, 2010-10-27
1.79 - Documentation corrections. CM, 2011-08-26
1.81 - Fix bug in interaction of AUTODERIVATIVE=0 and .MPSIDE=3;