## Name

MPFIT

## Author

Craig B. Markwardt, NASA/GSFC Code 662, Greenbelt, MD 20770

craigm@lheamail.gsfc.nasa.gov

UPDATED VERSIONs can be found on my WEB PAGE:

http://cow.physics.wisc.edu/~craigm/idl/idl.html

## Purpose

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

## Major Topics

Curve and Surface Fitting

## Calling Sequence

parms = MPFIT(MYFUNCT, start_parms, FUNCTARGS=fcnargs, NFEV=nfev,

MAXITER=maxiter, ERRMSG=errmsg, NPRINT=nprint, QUIET=quiet,

FTOL=ftol, XTOL=xtol, GTOL=gtol, NITER=niter,

STATUS=status, ITERPROC=iterproc, ITERARGS=iterargs,

COVAR=covar, PERROR=perror, BESTNORM=bestnorm,

PARINFO=parinfo)

## Description

MPFIT uses the Levenberg-Marquardt technique to solve the

least-squares problem. In its typical use, MPFIT will be used to

fit a user-supplied function (the "model") to user-supplied data

points (the "data") by adjusting a set of parameters. MPFIT is

based upon MINPACK-1 (LMDIF.F) by More' and collaborators.

For example, a researcher may think that a set of observed data

points is best modelled with a Gaussian curve. A Gaussian curve is

parameterized by its mean, standard deviation and normalization.

MPFIT will, within certain constraints, find the set of parameters

which best fits the data. The fit is "best" in the least-squares

sense; that is, the sum of the weighted squared differences between

the model and data is minimized.

The Levenberg-Marquardt technique is a particular strategy for

iteratively searching for the best fit. This particular

implementation is drawn from MINPACK-1 (see NETLIB), and seems to

be more robust than routines provided with IDL. This version

allows upper and lower bounding constraints to be placed on each

parameter, or the parameter can be held fixed.

The IDL user-supplied function should return an array of weighted

deviations between model and data. In a typical scientific problem

the residuals should be weighted so that each deviate has a

gaussian sigma of 1.0. If X represents values of the independent

variable, Y represents a measurement for each value of X, and ERR

represents the error in the measurements, then the deviates could

be calculated as follows:

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

where F is the 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]*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

implementation quirk, MPFIT expects FGRAD(x,p,i)/err instead,

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

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 - 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[2].tied = '2 * P[1]'.

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[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.

## 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 '+$

'your path (found version '+version+')'

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

opposite sign of FGRAD.

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

additional data to your user-supplied function without

using common blocks.

Consider the following example:

if FUNCTARGS = { XVAL:[1.D,2,3], YVAL:[1.D,4,9],

ERRVAL:[1.D,1,1] }

then the user supplied function should be declared

like this:

FUNCTION MYFUNCT, P, XVAL=x, YVAL=y, ERRVAL=err

By default, no extra parameters are passed to the

user-supplied function, but your function should

accept *at least* one keyword parameter. [ This is to

accomodate a limitation in IDL's _EXTRA

parameter-passing mechanism. ]

GTOL - a nonnegative input variable. Termination occurs when the

cosine of the angle between fvec and any column of the

jacobian is at most GTOL in absolute value (and STATUS is

accordingly set to 4). Therefore, GTOL measures the

orthogonality desired between the function vector and the

columns of the jacobian. Default: 1D-10

ITERARGS - The keyword arguments to be passed to ITERPROC via the

_EXTRA mechanism. This should be a structure, and is

similar in operation to FUNCTARGS.

Default: no arguments are passed.

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:

-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.

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

gradient information.

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

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

Where h is the residual vector (length m), hT is its transpose, h'

is the Jacobian matrix (dimensions n x m), and dx is (xm-x0). The

user function supplies the residual vector h, and in some cases h'

when it is not found by finite differences (see MPFIT_FDJAC2,

which finds h and hT'). Even if dx is not the best absolute step

to take, it does provide a good estimate of the best *direction*,

so often a line minimization will occur along the dx vector

direction.

The method of solution employed by MINPACK is to form the Q . R

factorization of h', where Q is an orthogonal matrix such that QT .

Q = I, and R is upper right triangular. Using h' = Q . R and the

ortogonality of Q, eqn (5) becomes

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

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

R . dx = - QT . h

where the last statement follows because R is upper triangular.

Here, R, QT and h are known so this is a matter of solving for dx.

The routine MPFIT_QRFAC provides the QR factorization of h, with

pivoting, and MPFIT_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

Link to NASA ADS: http://adsabs.harvard.edu/abs/2009ASPC..411..251M

Link to ASP: http://aspbooks.org/a/volumes/table_of_contents/411

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

statistic; added documentation about the theory of operation,

and under the QR factorization; slowly I'm beginning to

understand the bowels of this algorithm, CM 10 Jan 2001

Remove MPMINSTEP field of PARINFO, for now at least, CM 11 Jan

2001

Added RESDAMP keyword, CM, 14 Jan 2001

Tried to improve the DAMP handling a little, CM, 13 Mar 2001

Corrected .PARNAME behavior in _DEFITER, CM, 19 Mar 2001

Added checks for parameter and function overflow; a new STATUS

value to reflect this; STATUS values of -15 to -1 are reserved

for user function errors, CM, 03 Apr 2001

DAMP keyword is now a TANH, CM, 03 Apr 2001

Added more error checking of float vs. double, CM, 07 Apr 2001

Fixed bug in handling of parameter lower limits; moved overflow

checking to end of loop, CM, 20 Apr 2001

Failure using GOTO, TERMINATE more graceful if FNORM1 not defined,

CM, 13 Aug 2001

Add MPPRINT tag to PARINFO, CM, 19 Nov 2001

Add DOF keyword to DEFITER procedure, and print degrees of

freedom, CM, 28 Nov 2001

Add check to be sure MYFUNCT is a scalar string, CM, 14 Jan 2002

Addition of EXTERNAL_FJAC, EXTERNAL_FVEC keywords; ability to save

fitter's state from one call to the next; allow '(EXTERNAL)'

function name, which implies that user will supply function and

Jacobian at each iteration, CM, 10 Mar 2002

Documented EXTERNAL evaluation code, CM, 10 Mar 2002

Corrected signficant bug in the way that the STEP parameter, and

FIXED parameters interacted (Thanks Andrew Steffl), CM, 02 Apr

2002

Allow COVAR and PERROR keywords to be computed, even in case of

'(EXTERNAL)' function, 26 May 2002

Add NFREE and NPEGGED keywords; compute NPEGGED; compute DOF using

NFREE instead of n_elements(X), thanks to Kristian Kjaer, CM 11

Sep 2002

Hopefully PERROR is all positive now, CM 13 Sep 2002

Documented RELSTEP field of PARINFO (!!), CM, 25 Oct 2002

Error checking to detect missing start pars, CM 12 Apr 2003

Add DOF keyword to return degrees of freedom, CM, 30 June 2003

Always call ITERPROC in the final iteration; add ITERKEYSTOP

keyword, CM, 30 June 2003

Correct bug in MPFIT_LMPAR of singularity handling, which might

likely be fatal for one-parameter fits, CM, 21 Nov 2003

(with thanks to Peter Tuthill for the proper test case)

Minor documentation adjustment, 03 Feb 2004, CM

Correct small error in QR factorization when pivoting; document

the return values of QRFAC when pivoting, 21 May 2004, CM

Add MPFORMAT field to PARINFO, and correct behavior of interaction

between MPPRINT and PARNAME in MPFIT_DEFITERPROC (thanks to Tim

Robishaw), 23 May 2004, CM

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

Add COMPATIBILITY section and add credits to copyright, CM, 13 Dec

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;

Document FJAC_MASK. CM, 2012-05-08

$Id: mpfit.pro,v 1.82 2012/09/27 23:59:44 cmarkwar Exp $