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


  Performs function minimization (Truncated-Newton Method)

Major Topics

  Optimization and Minimization

Calling Sequence

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


  TNMIN uses the Truncated-Newton method to minimize an arbitrary IDL
  function with respect to a given set of free parameters. The
  user-supplied function must compute the gradient with respect to
  each parameter. TNMIN is based on TN.F (TNBC) by Stephen Nash.
  If you want to solve a least-squares problem, to perform *curve*
  *fitting*, then you will probably want to use the routines MPFIT,
  MPFITFUN and MPFITEXPR. Those routines are specifically optimized
  for the least-squares problem. TNMIN is suitable for constrained
  and unconstrained optimization problems with a medium number of
  variables. Function *maximization* can be performed using the
  MAXIMIZE keyword.
  TNMIN is similar to MPFIT in that it allows parameters to be fixed,
  simple bounding limits to be placed on parameter values, and
  parameters to be tied to other parameters. One major difference
  between MPFIT and TNMIN is that TNMIN does not compute derivatives
  automatically by default. See PARINFO and AUTODERIVATIVE below for
  more discussion and examples.

User Function

  The user must define an IDL function which returns the desired
  value as a single scalar. The IDL function must accept a list of
  numerical parameters, P. Additionally, keyword parameters may be
  used to pass more data or information to the user function, via the
  FUNCTARGS keyword.
  The user function should be declared in the following way:
    FUNCTION MYFUNCT, p, dp [, keywords permitted ]
      ; Parameter values are passed in "p"
      f = .... ; Compute function value
      dp = .... ; Compute partial derivatives (optional)
      return, f
  The function *must* accept at least one argument, the parameter
  list P. The vector P is implicitly assumed to be a one-dimensional
  array. Users may pass additional information to the function by
  composing and _EXTRA structure and passing it in the FUNCTARGS
  User functions may also indicate a fatal error condition using the
  ERROR_CODE common block variable, as described below under the
  TNMIN_ERROR common block definition (by setting ERROR_CODE to a
  number between -15 and -1).
  By default, the user must compute the function gradient components
  explicitly using AUTODERIVATIVE=0. As explained below, numerical
  derivatives can also be calculated using AUTODERIVATIVE=1.
  For explicit derivatives, the user should call TNMIN using the
  default keyword value AUTODERIVATIVE=0. [ This is different
  behavior from MPFIT, where AUTODERIVATIVE=1 is the default. ] The
  IDL user routine should compute the gradient of the function as a
  one-dimensional array of values, one for each of the parameters.
  They are passed back to TNMIN via "dp" as shown above.
  The derivatives with respect to fixed parameters are ignored; zero
  is an appropriate value to insert for those derivatives. Upon
  input to the user function, DP is set to a vector with the same
  length as P, with a value of 1 for a parameter which is free, and a
  value of zero for a parameter which is fixed (and hence no
  derivative needs to be calculated). This input vector may be
  overwritten as needed.
  For numerical derivatives, a finite differencing approximation is
  used to estimate the gradient values. Users can activate this
  feature by passing the keyword AUTODERIVATIVE=1. Fine control over
  this behavior can be achieved using the STEP, RELSTEP and TNSIDE
  fields of the PARINFO structure.
  When finite differencing is used for computing derivatives (ie,
  when AUTODERIVATIVE=1), the parameter DP is not passed. Therefore
  functions can use N_PARAMS() to indicate whether they must compute
  the derivatives or not. However there is no penalty (other than
  computation time) for computing the gradient values and users may
  switch between AUTODERIVATIVE=0 or =1 in order to test both

Constraining Parameter Values With The Parinfo Keyword

  The behavior of TNMIN can be modified with respect to each
  parameter to be optimized. A parameter value can be fixed; simple
  boundary constraints can be imposed; limitations on the parameter
  changes can be imposed; properties of the automatic derivative can
  be modified; and parameters can be tied to one another.
  These properties are governed by the PARINFO structure, which is
  passed as a keyword parameter to TNMIN.
  PARINFO should be an array of structures, one for each parameter.
  Each parameter is associated with one element of the array, in
  numerical order. The structure can have the following entries
  (none are required):
    .VALUE - the starting parameter value (but see the START_PARAMS
              parameter for more information).
    .FIXED - a boolean value, whether the parameter is to be held
              fixed or not. Fixed parameters are not varied by
              TNMIN, but are passed on to MYFUNCT for evaluation.
    .LIMITED - a two-element boolean array. If the first/second
                element is set, then the parameter is bounded on the
                lower/upper side. A parameter can be bounded on both
                sides. Both LIMITED and LIMITS must be given
    .LIMITS - a two-element float or double array. Gives the
              parameter limits on the lower and upper sides,
              respectively. Zero, one or two of these values can be
              set, depending on the values of LIMITED. Both LIMITED
              and LIMITS must be given together.
    .PARNAME - a string, giving the name of the parameter. The
                fitting code of TNMIN does not use this tag in any
    .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.
    .TNSIDE - the sidedness of the finite difference when computing
              numerical derivatives. This field can take four
                  0 - one-sided derivative computed automatically
                  1 - one-sided derivative (f(x+h) - f(x) )/h
                -1 - one-sided derivative (f(x) - f(x-h))/h
                  2 - two-sided derivative (f(x+h) - f(x-h))/(2*h)
              Where H is the STEP parameter described above. The
              "automatic" one-sided derivative method will chose a
              direction for the finite difference which does not
              violate any constraints. The other methods do not
              perform this check. The two-sided method is in
              principle more precise, but requires twice as many
              function evaluations. Default: 0.
    .TIED - a string expression which "ties" the parameter to other
            free or fixed parameters. Any expression involving
            constants and the parameter array P are permitted.
            Example: if parameter 2 is always to be twice parameter
            1 then use the following: parinfo(2).tied = '2 * P(1)'.
            Since they are totally constrained, tied parameters are
            considered to be fixed; no errors are computed for them.
            [ NOTE: the PARNAME can't be used in expressions. ]
  Future modifications to the PARINFO structure, if any, will involve
  adding structure tags beginning with the two letters "MP" or "TN".
  Therefore programmers are urged to avoid using tags starting with
  these two combinations of letters; otherwise they are free to
  include their own fields within the PARINFO structure, and they
  will be ignored.
  PARINFO Example:
  parinfo = replicate({value:0.D, fixed:0, limited:[0,0], $
                      limits:[0.D,0]}, 5)
  parinfo(0).fixed = 1
  parinfo(4).limited(0) = 1
  parinfo(4).limits(0) = 50.D
  parinfo(*).value = [5.7D, 2.2, 500., 1.5, 2000.]
  A total of 5 parameters, with starting values of 5.7,
  2.2, 500, 1.5, and 2000 are given. The first parameter
  is fixed at a value of 5.7, and the last parameter is
  constrained to be above 50.


  MYFUNCT - a string variable containing the name of the function to
            be minimized (see USER FUNCTION above). The IDL routine
            should return the value of the function and optionally
            its gradients.
  X - An array of starting values for each of the parameters of the
      This parameter is optional if the PARINFO keyword is used.
      See above. The PARINFO keyword provides a mechanism to fix or
      constrain individual parameters. If both X and PARINFO are
      passed, then the starting *value* is taken from X, but the
      *constraints* are taken from PARINFO.


  Returns the array of best-fit parameters.

Keyword Parameters

  AUTODERIVATIVE - If this is set, derivatives of the function will
                    be computed automatically via a finite
                    differencing procedure. If not set, then MYFUNCT
                    must provide the (explicit) derivatives.
                    Default: 0 (explicit derivatives required)
  BESTMIN - upon return, the best minimum function value that TNMIN
            could find.
  EPSABS - a nonnegative real variable. Termination occurs when the
            absolute error between consecutive iterates is at most
            EPSABS. Note that using EPSREL is normally preferable
            over EPSABS, except in cases where ABS(F) is much larger
            than 1 at the optimal point. A value of zero indicates
            the absolute error test is not applied. If EPSABS is
            specified, then both EPSREL and EPSABS tests are applied;
            if either succeeds then termination occurs.
            Default: 0 (i.e., only EPSREL is applied).
  EPSREL - a nonnegative input variable. Termination occurs when the
            relative error between two consecutive iterates is at
            most EPSREL. Therefore, EPSREL measures the relative
            error desired in the function. An additional, more
            lenient, stopping condition can be applied by specifying
            the EPSABS keyword.
            Default: 100 * Machine Precision
  ERRMSG - a string error or warning message is returned.
  FGUESS - provides the routine a guess to the minimum value.
            Default: 0
  FUNCTARGS - A structure which contains the parameters to be passed
              to the user-supplied function specified by MYFUNCT via
              the _EXTRA mechanism. This is the way you can pass
              additional data to your user-supplied function without
              using common blocks.
              Consider the following example:
                if FUNCTARGS = { XVAL:[1.D,2,3], YVAL:[1.D,4,9]}
                then the user supplied function should be declared
                like this:
                FUNCTION MYFUNCT, P, XVAL=x, YVAL=y
              By default, no extra parameters are passed to the
              user-supplied function.
  ITERARGS - The keyword arguments to be passed to ITERPROC via the
              _EXTRA mechanism. This should be a structure, and is
              similar in operation to FUNCTARGS.
              Default: no arguments are passed.
  ITERDERIV - Intended to print function gradient information. If
              set, then the ITERDERIV keyword is set in each call to
              ITERPROC. In the default ITERPROC, parameter values
              and gradient information are both printed when this
              keyword is set.
  ITERPROC - The name of a procedure to be called upon each NPRINT
              iteration of the TNMIN routine. It should be declared
              in the following way:
              PRO ITERPROC, MYFUNCT, p, iter, fnorm, FUNCTARGS=fcnargs, $
                PARINFO=parinfo, QUIET=quiet, _EXTRA=extra
                ; perform custom iteration update
              ITERPROC must accept the _EXTRA keyword, in case of
              future changes to the calling procedure.
              MYFUNCT is the user-supplied function to be minimized,
              P is the current set of model parameters, ITER is the
              iteration number, and FUNCTARGS are the arguments to be
              passed to MYFUNCT. FNORM is should be the function
              value. QUIET is set when no textual output should be
              printed. See below for documentation of PARINFO.
              In implementation, ITERPROC can perform updates to the
              terminal or graphical user interface, to provide
              feedback while the fit proceeds. If the fit is to be
              stopped for any reason, then ITERPROC should set the
              common block variable ERROR_CODE to negative value
              between -15 and -1 (see TNMIN_ERROR common block
              below). In principle, ITERPROC should probably not
              modify the parameter values, because it may interfere
              with the algorithm's stability. In practice it is
              Default: an internal routine is used to print the
                      parameter values.
  MAXITER - The maximum number of iterations to perform. If the
            number is exceeded, then the STATUS value is set to 5
            and TNMIN returns.
            Default: 200 iterations
  MAXIMIZE - If set, the function is maximized instead of
  MAXNFEV - The maximum number of function evaluations to perform.
            If the number is exceeded, then the STATUS value is set
            to -17 and TNMIN returns. A value of zero indicates no
            Default: 0 (no maximum)
  NFEV - upon return, the number of MYFUNCT function evaluations
  NITER - upon return, number of iterations completed.
  NPRINT - The frequency with which ITERPROC is called. A value of
            1 indicates that ITERPROC is called with every iteration,
            while 2 indicates every other iteration, etc.
            Default value: 1
  PARINFO - Provides a mechanism for more sophisticated constraints
            to be placed on parameter values. When PARINFO is not
            passed, then it is assumed that all parameters are free
            and unconstrained. Values in PARINFO are never modified
            during a call to TNMIN.
            See description above for the structure of PARINFO.
            Default value: all parameters are free and unconstrained.
  QUIET - set this keyword when no textual output should be printed
          by TNMIN
  STATUS - an integer status code is returned. All values greater
            than zero can represent success (however STATUS EQ 5 may
            indicate failure to converge). Gaps in the numbering
            system below are to maintain compatibility with MPFIT.
            Upon return, STATUS can have one of the following values:
        -18 a fatal internal error occurred during optimization.
        -17 the maximum number of function evaluations has been
            reached without convergence.
        -16 a parameter or function value has become infinite or an
            undefined number. This is usually a consequence of
            numerical overflow in the user's function, which must be
        -15 to -1
            these are error codes that either MYFUNCT or ITERPROC
            may return to terminate the fitting process (see
            description of MPFIT_ERROR common below). If either
            MYFUNCT or ITERPROC set ERROR_CODE to a negative number,
            then that number is returned in STATUS. Values from -15
            to -1 are reserved for the user functions and will not
            clash with MPFIT.
0 improper input parameters.
1 convergence was reached.
          2-4 (RESERVED)
5 the maximum number of iterations has been reached
          6-8 (RESERVED)


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

Common Blocks

    User routines may stop the fitting process at any time by
    setting an error condition. This condition may be set in either
    the user's model computation routine (MYFUNCT), or in the
    iteration procedure (ITERPROC).
    To stop the fitting, the above common block must be declared,
    and ERROR_CODE must be set to a negative number. After the user
    procedure or function returns, TNMIN checks the value of this
    common block variable and exits immediately if the error
    condition has been set. By default the value of ERROR_CODE is
    zero, indicating a successful function/procedure call.


      Stephen G. Nash, Operations Research and Applied Statistics
  Nash, S. G. 1984, "Newton-Type Minimization via the Lanczos
      Method," SIAM J. Numerical Analysis, 21, p. 770-778

Modification History

  Derived from TN.F by Stephen Nash with many changes and additions,
      to conform to MPFIT paradigm, 19 Jan 1999, CM
  Changed web address to COW, 25 Sep 1999, CM
  Alphabetized documented keyword parameters, 02 Oct 1999, CM
  Changed to ERROR_CODE for error condition, 28 Jan 2000, CM
  Continued and fairly major improvements (CM, 08 Jan 2001):
      - calling of user procedure is now concentrated in TNMIN_CALL,
        and arguments are reduced by storing a large number of them
        in common blocks;
      - finite differencing is done within TNMIN_CALL; added
        AUTODERIVATIVE=1 so that finite differencing can be enabled,
        both one and two sided;
      - a new procedure to parse PARINFO fields, borrowed from MPFIT;
        brought PARINFO keywords up to date with MPFIT;
      - go through and check for float vs. double discrepancies;
      - add explicit MAXIMIZE keyword, and support in TNMIN_CALL and
        TNMIN_DEFITER to print the correct values in that case;
        TNMIN_DEFITER now prints function gradient values if
      - convert to common-based system of MPFIT for storing machine
        constants; revert TNMIN_ENORM to simple sum of squares, at
        least for now;
      - remove limit on number of function evaluations, at least for
        now, and until I can understand what happens when we do
        numerical derivatives.
  Further changes: more float vs double; disable TNMINSTEP for now;
    experimented with convergence test in case of function
    maximization, 11 Jan 2001, CM
  TNMINSTEP is parsed but not enabled, 13 Mar 2001
  Major code cleanups; internal docs; reduced commons, CM, 08 Apr
  Continued code cleanups; documentation; the STATUS keyword
    actually means something, CM, 10 Apr 2001
  Added reference to Nash paper, CM, 08 Feb 2002
  Fixed 16-bit loop indices, D. Schelgel, 22 Apr 2003
  Changed parens to square brackets because of conflicts with
    limits function. K. Tolbert, 23 Feb 2005
  Some documentation clarifications, CM, 09 Nov 2007
  Ensure that MY_FUNCT returns a scalar; make it more likely that
    error messages get back out to the user; fatal CATCH'd error
    now returns STATUS = -18, CM, 17 Sep 2008
  Fix TNMIN_CALL when parameters are TIEd (thanks to Alfred de
    Wijn), CM, 22 Nov 2009
  Remember to TIE the parameters before final return (thanks to
    Michael Smith), CM, 20 Jan 2010


  - scale derivatives semi-automatically;
  - ability to scale and offset parameters;

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