mpfit.UNSTABLE.py

"""
Perform Levenberg-Marquardt least-squares minimization, based on MINPACK-1.

                                   AUTHORS
                                   
  The original version of this software, called LMFIT, was written in FORTRAN
  as part of the MINPACK-1 package by Jorge More' et al, available from NETLIB 
    http://www.netlib.org/
  The algorithm is described here http://cds.cern.ch/record/126569

  Craig Markwardt converted the FORTRAN code to IDL.  
  The information for the IDL version is:
     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://purl.com/net/mpfit

  Mark Rivers created this Python version from Craig's IDL version.
    Mark Rivers, University of Chicago
    Building 434A, Argonne National Laboratory
    9700 South Cass Avenue, Argonne, IL 60439
    rivers@cars.uchicago.edu
    UPDATED VERSIONS can be found at http://cars.uchicago.edu/software/python/

 Sergey Koposov converted the Mark's Python version from Numeric to Numpy
    Sergey Koposov, University of Cambridge, Institute of Astronomy,
    Madingley road, CB3 0HA, Cambridge, UK
    koposov@ast.cam.ac.uk
    UPDATED VERSIONS can be found at https://code.google.com/p/astrolibpy/
    
  Michele Cappellari (Astrophysics, University of Oxford) made it compatible 
    with Python 3 and fixed major instabilities with degenrate Jacobians 
    already present since the original MINPACK-1 Fortran code.
    UPDATED VERSIONS can be found at http://purl.org/cappellari/software

                                 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 is much faster
 and more accurate than the version provided in the Scientific Python package
 in Scientific.Functions.LeastSquares.
 This version allows upper and lower bounding constraints to be placed on each
 parameter, or the parameter can be held fixed.

 The user-supplied Python 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 analytical 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 FUNCTKW 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.  It can also return  
 an optional partial derivative array.  For applications which
 use finite-difference derivatives -- the default -- the user
 function should be declared in the following way:

   def myfunct(p, fjac=None, x=None, y=None, err=None)
    # Parameter values are passed in "p"
    # If fjac is None then partial derivatives should not be computed.  
    # It will always be None if MPFIT is called with default flag.
    model = F(x, p)
    resid = (y - model)/err 
    return resid

 See below for applications with analytical 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 functkw 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 Numeric array of type Float.


                            ANALYTIC 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, if you desire to compute them
 analytically, then the AUTODERIVATIVE=0 keyword must be passed to MPFIT.
 As a practical matter, it is often sufficient and even faster to allow
 MPFIT to calculate the derivatives numerically, and so
 AUTODERIVATIVE=0 is not necessary.

 If AUTODERIVATIVE=0 is used then the user function must check the parameter
 FJAC, and if FJAC!=None then return the partial derivative array in the
 return list.
   def myfunct(p, fjac=None, x=None, y=None, err=None)
    # Parameter values are passed in "p"
    # If FJAC!=None then partial derivatives must be comptuer.
    # FJAC contains an array of len(p), where each entry
    # is 1 if that parameter is free and 0 if it is fixed.
    model = F(x, p)
    if (dojac):
       pderiv = zeros([len(x), len(p)], Float)
       for j in range(len(p)):
         pderiv[:,j] = FGRAD(x, p, j)
    else:
       pderiv = None
    return (y - model)/err, pderiv

 where FGRAD(x, p, i) is a user function which must compute the
 derivative of the model with respect to parameter P[i] at X.  When
 finite differencing is used for computing derivatives (ie, when
 AUTODERIVATIVE=1), or when MPFIT needs only the errors but not the
 derivatives the parameter FJAC=None.

 Derivatives should be returned in the PDERIV array. PDERIV should be an m x
 n array, where m is the number of data points and n is the number
 of parameters.  dp[i,j] is the derivative at the ith point with
 respect to the jth parameter.

 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, FJAC 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).

 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.


           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 a list of dictionaries, one list entry for each parameter.
 Each parameter is associated with one element of the array, in
 numerical order.  The dictionary can have the following keys
 (none are required, keys are case insensitive):

    '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 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 iterfunct 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.

    'mpside' - the sidedness of the finite difference when computing
              numerical derivatives.  This field can take four
              values:

                 0 - one-sided derivative computed automatically
                 1 - one-sided derivative (f(x+h) - f(x)  )/h
                -1 - one-sided derivative (f(x)   - f(x-h))/h
                 2 - two-sided derivative (f(x+h) - f(x-h))/(2*h)

             Where H is the STEP parameter described above.  The
             "automatic" one-sided derivative method will chose a
             direction for the finite difference which does not
             violate any constraints.  The other methods do not
             perform this check.  The two-sided method is in
             principle more precise, but requires twice as many
             function evaluations.  Default: 0.

    'mpmaxstep' - the maximum change to be made in the parameter
                 value.  During the fitting process, the parameter
                 will never be changed by more than this value in
                 one iteration.

                 A value of 0 indicates no maximum.  Default: 0.

    'tied' - a string expression which "ties" the parameter to other
            free or fixed parameters.  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. ]

    'mpprint' - if set to 1, then the default iterfunct 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)


 Future modifications to the PARINFO structure, if any, will involve
 adding dictionary tags beginning with the two letters "MP".
 Therefore programmers are urged to avoid using tags starting with
 the same letters; otherwise they are free to include their own
 fields within the PARINFO structure, and they will be ignored.

 PARINFO Example:
 parinfo = [{'value':0., 'fixed':0, 'limited':[0,0], 'limits':[0.,0.]}
                                                for i in range(5)]
 parinfo[0]['fixed'] = 1
 parinfo[4]['limited'][0] = 1
 parinfo[4]['limits'][0]  = 50.
 values = [5.7, 2.2, 500., 1.5, 2000.]
 for i in range(5): parinfo[i]['value']=values[i]

 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.


                                   EXAMPLE

   import mpfit
   import np.oldnumeric as Numeric
   x = arange(100, float)
   p0 = [5.7, 2.2, 500., 1.5, 2000.]
   y = ( p[0] + p[1]*[x] + p[2]*[x**2] + p[3]*sqrt(x) +
         p[4]*log(x))
   fa = {'x':x, 'y':y, 'err':err}
   m = mpfit('myfunct', p0, functkw=fa)
   print 'status = ', m.status
   if (m.status <= 0): print 'error message = ', m.errmsg
   print 'parameters = ', m.params

   Minimizes sum of squares of MYFUNCT.  MYFUNCT is called with the X,
   Y, and ERR keyword parameters that are given by FUNCTKW.  The
   results can be obtained from the returned object m.


                            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_QRSOLV provides the solution for dx.


                                 REFERENCES

   MINPACK-1, Jorge More', available from netlib (www.netlib.org).
   "Optimization Software Guide," Jorge More' and Stephen Wright,
     SIAM, *Frontiers in Applied Mathematics*, Number 14.
   More', Jorge J., "The Levenberg-Marquardt Algorithm:
     Implementation and Theory," in *Numerical Analysis*, ed. Watson,
     G. A., Lecture Notes in Mathematics 630, Springer-Verlag, 1977.


                           MODIFICATION HISTORY

 - Translated from MINPACK-1 in FORTRAN, Apr-Jul 1998, CM
 Copyright (C) 1997-2002, Craig Markwardt
 This software is provided as is without any warranty whatsoever.
 Permission to use, copy, modify, and distribute modified or
 unmodified copies is granted, provided this copyright and disclaimer
 are included unchanged.

 - Translated from MPFIT (Craig Markwardt's IDL package) to Python,
   August, 2002.  Mark Rivers
 - Converted from Numeric to numpy (Sergey Koposov, July 2008)
 

    MODIFICATION HISTORY OF THIS FORKED VERSION:
 
    V1.0.0: Included a key modification for mge_fit_sectors.
        Michele Cappellari, Oxford, 8 February 2014
    V1.1.0: Support both Python 2.6/2.7 and Python 3. MC, Oxford, 25 May 2014
    V1.1.1: Removed Scipy dependency. MC, Oxford, 13 August 2014
    V1.1.2: Replaced np.rank function with ndim attribute to avoid
        deprecation warning in Numpy 1.9. MC, Utah, 9 September 2014
    V1.2.0: Fixed *major* instabilities for degenerate Jacobians, especially 
        dramatic in double precision, by introducing thresholds to avoid 
        divisions by nearly zero diagonal elements.        
      - Made machar() always return constants for single precison numbers.
      - Consequently removed now-unnecessary change introduced in V1.0.0.
      - Removed requirement for user function to return `status` for consistency
        with most alternative optimization software.
      - Various testing and code simplifications.
        MC, Oxford, 22 May 2017
    V1.2.1: Altough I intentionally do not support outdated Numpy versions, I
        fixed a small incompatibility with Numpy 1.11 due to changes in np.full.
        This was reported by Russell Smith (Univ. of Durham).
        MC, Oxford, 16 November 2017
    V1.2.2: Dropped legacy Python 2.7 support. MC, Oxford, 12 May 2018

"""

import numpy as np

#    Original FORTRAN documentation
#    **********
#
#    subroutine lmdif
#
#    the purpose of lmdif is to minimize the sum of the squares of
#    m nonlinear functions in n variables by a modification of
#    the levenberg-marquardt algorithm. the user must provide a
#    subroutine which calculates the functions. the jacobian is
#    then calculated by a forward-difference approximation.
#
#    the subroutine statement is
#
#      subroutine lmdif(fcn,m,n,x,fvec,ftol,xtol,gtol,maxfev,epsfcn,
#                       diag,mode,factor,nprint,info,nfev,fjac,
#                       ldfjac,ipvt,qtf,wa1,wa2,wa3,wa4)
#
#    where
#
#      fcn is the name of the user-supplied subroutine which
#        calculates the functions. fcn must be declared
#        in an external statement in the user calling
#        program, and should be written as follows.
#
#        subroutine fcn(m,n,x,fvec,iflag)
#        integer m,n,iflag
#        double precision x(n),fvec(m)
#        ----------
#        calculate the functions at x and
#        return this vector in fvec.
#        ----------
#        return
#        end
#
#        the value of iflag should not be changed by fcn unless
#        the user wants to terminate execution of lmdif.
#        in this case set iflag to a negative integer.
#
#      m is a positive integer input variable set to the number
#        of functions.
#
#      n is a positive integer input variable set to the number
#        of variables. n must not exceed m.
#
#      x is an array of length n. on input x must contain
#        an initial estimate of the solution vector. on output x
#        contains the final estimate of the solution vector.
#
#      fvec is an output array of length m which contains
#        the functions evaluated at the output x.
#
#      ftol is a nonnegative input variable. termination
#        occurs when both the actual and predicted relative
#        reductions in the sum of squares are at most ftol.
#        therefore, ftol measures the relative error desired
#        in the sum of squares.
#
#      xtol is a nonnegative input variable. termination
#        occurs when the relative error between two consecutive
#        iterates is at most xtol. therefore, xtol measures the
#        relative error desired in the approximate solution.
#
#      gtol is 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. therefore, gtol measures the orthogonality
#        desired between the function vector and the columns
#        of the jacobian.
#
#      maxfev is a positive integer input variable. termination
#        occurs when the number of calls to fcn is at least
#        maxfev by the end of an iteration.
#
#      epsfcn is an input variable used in determining a suitable
#        step length for the forward-difference approximation. this
#        approximation assumes that the relative errors in the
#        functions are of the order of epsfcn. if epsfcn is less
#        than the machine precision, it is assumed that the relative
#        errors in the functions are of the order of the machine
#        precision.
#
#      diag is an array of length n. if mode = 1 (see
#        below), diag is internally set. if mode = 2, diag
#        must contain positive entries that serve as
#        multiplicative scale factors for the variables.
#
#      mode is an integer input variable. if mode = 1, the
#        variables will be scaled internally. if mode = 2,
#        the scaling is specified by the input diag. other
#        values of mode are equivalent to mode = 1.
#
#      factor is a positive input variable used in determining the
#        initial step bound. this bound is set to the product of
#        factor and the euclidean norm of diag*x if nonzero, or else
#        to factor itself. in most cases factor should lie in the
#        interval (.1,100.). 100. is a generally recommended value.
#
#      nprint is an integer input variable that enables controlled
#        printing of iterates if it is positive. in this case,
#        fcn is called with iflag = 0 at the beginning of the first
#        iteration and every nprint iterations thereafter and
#        immediately prior to return, with x and fvec available
#        for printing. if nprint is not positive, no special calls
#        of fcn with iflag = 0 are made.
#
#      info is an integer output variable. if the user has
#        terminated execution, info is set to the (negative)
#        value of iflag. see description of fcn. otherwise,
#        info is set as follows.
#
#        info = 0  improper input parameters.
#
#        info = 1  both actual and predicted relative reductions
#                  in the sum of squares are at most ftol.
#
#        info = 2  relative error between two consecutive iterates
#                  is at most xtol.
#
#        info = 3  conditions for info = 1 and info = 2 both hold.
#
#        info = 4  the cosine of the angle between fvec and any
#                  column of the jacobian is at most gtol in
#                  absolute value.
#
#        info = 5  number of calls to fcn has reached or
#                  exceeded maxfev.
#
#        info = 6  ftol is too small. no further reduction in
#                  the sum of squares is possible.
#
#        info = 7  xtol is too small. no further improvement in
#                  the approximate solution x is possible.
#
#        info = 8  gtol is too small. fvec is orthogonal to the
#                  columns of the jacobian to machine precision.
#
#      nfev is an integer output variable set to the number of
#        calls to fcn.
#
#      fjac is an output m by n array. the upper n by n submatrix
#        of fjac contains an upper triangular matrix r with
#        diagonal elements of nonincreasing magnitude such that
#
#               t    t         t
#              p *(jac *jac)*p = r *r,
#
#        where p is a permutation matrix and jac is the final
#        calculated jacobian. column j of p is column ipvt(j)
#        (see below) of the identity matrix. the lower trapezoidal
#        part of fjac contains information generated during
#        the computation of r.
#
#      ldfjac is a positive integer input variable not less than m
#        which specifies the leading dimension of the array fjac.
#
#      ipvt is an integer output array of length n. ipvt
#        defines a permutation matrix p such that jac*p = q*r,
#        where jac is the final calculated jacobian, q is
#        orthogonal (not stored), and r is upper triangular
#        with diagonal elements of nonincreasing magnitude.
#        column j of p is column ipvt(j) of the identity matrix.
#
#      qtf is an output array of length n which contains
#        the first n elements of the vector (q transpose)*fvec.
#
#      wa1, wa2, and wa3 are work arrays of length n.
#
#      wa4 is a work array of length m.
#
#    subprograms called
#
#      user-supplied ...... fcn
#
#      minpack-supplied ... dpmpar,enorm,fdjac2,,qrfac
#
#      fortran-supplied ... dabs,dmax1,dmin1,dsqrt,mod
#
#    argonne national laboratory. minpack project. march 1980.
#    burton s. garbow, kenneth e. hillstrom, jorge j. more
#
#    **********

###############################################################################

def norm(x):
    """ Euclidean norm """

    return np.sqrt(x @ x)

###############################################################################

class mpfit:

    def __init__(self, fcn, xall=None, functkw={}, parinfo=None,
                 ftol=1e-8, xtol=1e-8, gtol=1e-8, damp=0., maxiter=200,
                 factor=100., nprint=1, iterfunct='default', iterkw={},
                 nocovar=False, rescale=False, autoderivative=True,
                 quiet=False, diag=None, epsfcn=None):
        """
  Inputs:
    fcn:
       The function to be minimized.  The function should return the weighted
       deviations between the model and the data, as described above.

    xall:
       An array of starting values for each of the parameters of the model.
       The number of parameters should be fewer than the number of measurements.

       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.

  Keywords:

     autoderivative:
        If this is set, derivatives of the function will be computed
        automatically via a finite differencing procedure.  If not set, then
        fcn must provide the (analytical) derivatives.
           Default: set (=1)
           NOTE: to supply your own analytical derivatives,
                 explicitly pass autoderivative=0

     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: 1E-10

     functkw:
        A dictionary which contains the parameters to be passed to the
        user-supplied function specified by fcn via the standard Python
        keyword dictionary mechanism.  This is the way you can pass additional
        data to your user-supplied function without using global variables.

        Consider the following example:
           if functkw = {'xval':[1.,2.,3.], 'yval':[1.,4.,9.],
                         'errval':[1.,1.,1.] }
        then the user supplied function should be declared like this:
           def myfunct(p, fjac=None, xval=None, yval=None, errval=None):

        Default: {}   No extra parameters are passed to the user-supplied
                      function.

     gtol:
        A nonnegative input variable. Termination occurs when the cosine of
        the angle between fvec and any column of the jacobian is at most gtol
        in absolute value (and status is accordingly set to 4). Therefore,
        gtol measures the orthogonality desired between the function vector
        and the columns of the jacobian.
           Default: 1e-10

     iterkw:
        The keyword arguments to be passed to iterfunct via the dictionary
        keyword mechanism.  This should be a dictionary and is similar in
        operation to FUNCTKW.
           Default: {}  No arguments are passed.

     iterfunct:
        The name of a function to be called upon each NPRINT iteration of the
        MPFIT routine.  It should be declared in the following way:
           def iterfunct(myfunct, p, iter, fnorm, functkw=None,
                         parinfo=None, quiet=0, dof=None, [iterkw keywords here])
           # perform custom iteration update

        iterfunct must accept all three keyword parameters (FUNCTKW, PARINFO
        and QUIET).

        myfunct:  The user-supplied function to be minimized,
        p:      The current set of model parameters
        iter:    The iteration number
        functkw:  The arguments to be passed to myfunct.
        fnorm:  The chi-squared value.
        quiet:  Set when no textual output should be printed.
        dof:      The number of degrees of freedom, normally the number of points
                  less the number of free parameters.
        See below for documentation of parinfo.

        In implementation, iterfunct 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 iterfunct should return a
        a status value between -15 and -1.  Otherwise it should return None
        (e.g. no return statement) or 0.
        In principle, iterfunct 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.

        Set iterfunct=None if there is no user-defined routine and you don't
        want the internal default routine be called.

     maxiter:
        The maximum number of iterations to perform.  If the number is exceeded,
        then the status value is set to 5 and MPFIT returns.
        Default: 200 iterations

     nocovar:
        Set this keyword to prevent the calculation of the covariance matrix
        before returning (see COVAR)
        Default: clear (=0)  The covariance matrix is returned

     nprint:
        The frequency with which iterfunct is called.  A value of 1 indicates
        that iterfunct is called with every iteration, while 2 indicates every
        other iteration, etc.  Note that several Levenberg-Marquardt attempts
        can be made in a single iteration.
        Default value: 1

     parinfo
        Provides a mechanism for more sophisticated constraints to be placed on
        parameter values.  When parinfo is not passed, then it is assumed that
        all parameters are free and unconstrained.  Values in parinfo are never
        modified during a call to MPFIT.

        See description above for the structure of PARINFO.

        Default value: None  All parameters are free and unconstrained.

     quiet:
        Set this keyword when no textual output should be printed by MPFIT

     damp:
        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 hyperbolic tangent.  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: DAMP doesn't work with autoderivative=0

     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: 1E-10

   Outputs:

     Returns an object of type mpfit.  The results are attributes of this class,
     e.g. mpfit.status, mpfit.errmsg, mpfit.params, npfit.niter, mpfit.covar.

     .status
        An integer status code is returned.  All values greater than zero can
        represent success (however .status == 5 may indicate failure to
        converge). It can have one of the following values:

        -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 iterfunct may return to
           terminate the fitting process.  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.

     .fnorm
        The value of the summed squared residuals for the returned parameter
        values.

     .covar
        The covariance matrix for the set of parameters returned by MPFIT.
        The matrix is NxN where N is the number of  parameters.  The square root
        of the diagonal elements gives the formal 1-sigma statistical errors on
        the parameters if errors were treated "properly" in fcn.
        Parameter errors are also returned in .perror.

        To compute the correlation matrix, pcor, use this example:
           cov = mpfit.covar
           pcor = cov * 0.
           for i in range(n):
              for j in range(n):
                 pcor[i,j] = cov[i,j]/sqrt(cov[i,i]*cov[j,j])

        If nocovar is set or MPFIT terminated abnormally, then .covar is set to
        a scalar with value None.

     .errmsg
        A string error or warning message is returned.

     .nfev
        The number of calls to MYFUNCT performed.

     .niter
        The number of iterations completed.

     .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 = len(x) - len(mpfit.params) # deg of freedom
           # scaled uncertainties
           pcerror = mpfit.perror * sqrt(mpfit.fnorm / dof)

        """
        self.niter = 0
        self.params = None
        self.covar = None
        self.perror = None
        self.status = 0  # Invalid input flag set while we check inputs
        self.errmsg = ''
        self.nfev = 0
        self.damp = damp
        self.dof = 0

        if fcn is None:
            self.errmsg = "Usage: parms = mpfit('myfunt', ... )"
            return

        if iterfunct == 'default':
            iterfunct = self.defiter

        # Parameter damping doesn't work when user is providing their own
        # gradients.
        if (self.damp != 0) and (autoderivative == 0):
            self.errmsg =  'ERROR: keywords DAMP and AUTODERIVATIVE are mutually exclusive'
            return

        # Parameters can either be stored in parinfo, or x. x takes precedence if it exists
        if (xall is None) and (parinfo is None):
            self.errmsg = 'ERROR: must pass parameters in P or PARINFO'
            return

        # Be sure that PARINFO is of the right type
        if parinfo is not None:
            if not isinstance(parinfo, list):
                self.errmsg = 'ERROR: PARINFO must be a list of dictionaries.'
                return
            if not isinstance(parinfo[0], dict):
                self.errmsg = 'ERROR: PARINFO must be a list of dictionaries.'
                return
            if ((xall is not None) and (len(xall) != len(parinfo))):
                self.errmsg = 'ERROR: number of elements in PARINFO and P must agree'
                return

        # If the parameters were not specified at the command line, then
        # extract them from PARINFO
        if xall is None:
            xall = self.parinfo(parinfo, 'value')
            if xall is None:
                self.errmsg = 'ERROR: either P or PARINFO(*)["value"] must be supplied.'
                return

        # Make sure parameters are numpy arrays
        xall = np.asarray(xall, dtype=float)

        npar = len(xall)
        self.fnorm  = -1.
        fnorm1 = -1.

        # TIED parameters?
        ptied = self.parinfo(parinfo, 'tied', default='', n=npar)
        self.ptied = np.asarray([a.strip() for a in ptied])
        self.qanytied = np.any(self.ptied != '')

        # FIXED parameters ?
        pfixed = self.parinfo(parinfo, 'fixed', default=False, n=npar)
        pfixed = (pfixed == 1) | (ptied != '')   # Tied parameters are also fixed
        
        # Finite differencing step, absolute and relative, and sidedness of deriv.
        step = self.parinfo(parinfo, 'step', default=0., n=npar)
        dstep = self.parinfo(parinfo, 'relstep', default=0., n=npar)
        dside = self.parinfo(parinfo, 'mpside',  default=0, n=npar)

        # Maximum and minimum steps allowed to be taken in one iteration
        maxstep = self.parinfo(parinfo, 'mpmaxstep', default=0., n=npar)
        minstep = self.parinfo(parinfo, 'mpminstep', default=0., n=npar)
        qmin = minstep != 0
        qmin[:] = False # Remove minstep for now!!
        qmax = maxstep != 0
        if np.any(qmin & qmax & (maxstep < minstep)):
            self.errmsg = 'ERROR: MPMINSTEP is greater than MPMAXSTEP'
            return
        wh = qmin | qmax
        qminmax = wh.sum()

        # Finish up the free parameters
        ifree = np.nonzero(pfixed != 1)[0]
        nfree = len(ifree)
        if nfree == 0:
            self.errmsg = 'ERROR: no free parameters'
            return

        # Compose only VARYING parameters
        self.params = xall.copy()     # self.params is the set of parameters to be returned
        x = self.params[ifree]  # x is the set of free parameters
        
        # LIMITED parameters ?
        limited = self.parinfo(parinfo, 'limited', default=[False, False], n=npar)
        limited = limited == 1   # Ensure this is boolean
        limits = self.parinfo(parinfo, 'limits', default=[0., 0.], n=npar)
        if (limited is not None) and (limits is not None):
            # Error checking on limits in parinfo
            if np.any((limited[:, 0] & (xall < limits[:, 0]))
                    | (limited[:, 1] & (xall > limits[:, 1]))):
                self.errmsg = 'ERROR: parameters are not within PARINFO limits'
                return
            if np.any((limited[:, 0] & limited[:, 1])
                    & (limits[:, 0] >= limits[:, 1]) & (pfixed == False)):
                self.errmsg = 'ERROR: PARINFO parameter limits are not consistent'
                return

            # Transfer structure values to local variables
            qulim = limited[ifree, 1]
            qllim = limited[ifree, 0]
            ulim  = limits[ifree, 1]
            llim  = limits[ifree, 0]

            if np.any(qulim | qllim):
                qanylim = True
            else:
                qanylim = False
        else:
            # Fill in local variables with dummy values
            qulim = np.zeros(nfree)
            ulim  = x * 0.
            qllim = qulim
            llim  = x * 0.
            qanylim = 0

        n = len(x)
        # Check input parameters for errors
        if (n < 0) or (ftol <= 0) or (xtol <= 0) or (gtol <= 0) \
                    or (maxiter < 0) or (factor <= 0):
            self.errmsg = 'ERROR: input keywords are inconsistent'
            return

        if rescale:
            self.errmsg = 'ERROR: DIAG parameter scales are inconsistent'
            if diag is None:
                return
            if np.any(diag <= 0):
                return
            self.errmsg = ''

        [self.status, fvec] = self.call(fcn, self.params, functkw)

        if self.status < 0:
            self.errmsg = 'ERROR: first call to "'+str(fcn)+'" failed'
            return

        self.machar = machar()
        machep = self.machar.machep

        m = len(fvec)
        if m < n:
            self.errmsg = 'ERROR: number of parameters must not exceed data'
            return
        self.dof = m - nfree
        self.fnorm = norm(fvec)

        # Initialize Levelberg-Marquardt parameter and iteration counter

        par = 0.
        self.niter = 1
        qtf = x * 0.
        self.status = 0

        # Beginning of the outer loop

        while(1):

            # If requested, call fcn to enable printing of iterates
            self.params[ifree] = x
            if self.qanytied:
                self.params = self.tie(self.params, ptied)

            if (nprint > 0) and (iterfunct is not None):
                if ((self.niter - 1) % nprint) == 0:
                    mperr = 0
                    xnew0 = self.params.copy()

                    dof = max(len(fvec) - len(x), 0)
                    status = iterfunct(fcn, self.params, self.niter, self.fnorm**2,
                       functkw=functkw, parinfo=parinfo, quiet=quiet, dof=dof, **iterkw)
                    if status is not None:
                        self.status = status

                    # Check for user termination
                    if self.status < 0:
                        self.errmsg = 'WARNING: premature termination by ' + str(iterfunct)
                        return

                    # If parameters were changed (grrr..) then re-tie
                    if not np.array_equal(xnew0, self.params):
                        if self.qanytied:
                            self.params = self.tie(self.params, ptied)
                        x = self.params[ifree]


            # Calculate the jacobian matrix
            self.status = 2
            catch_msg = 'calling MPFIT_FDJAC2'
            fjac = self.fdjac2(fcn, x, fvec, step, qulim, ulim, dside,
                          epsfcn=epsfcn, autoderivative=autoderivative, dstep=dstep,
                          functkw=functkw, ifree=ifree, xall=self.params)
            if fjac is None:
                self.errmsg = 'WARNING: premature termination by FDJAC2'
                return

            # Determine if any of the parameters are pegged at the limits
            if qanylim:
                catch_msg = 'zeroing derivatives of pegged parameters'
                whlpeg = np.flatnonzero(qllim & (x == llim))
                nlpeg = len(whlpeg)
                whupeg = np.flatnonzero(qulim & (x == ulim))
                nupeg = len(whupeg)
                # See if any "pegged" values should keep their derivatives
                if nlpeg > 0:
                    # Total derivative of sum wrt lower pegged parameters
                    for wh in whlpeg:
                        sum0 = np.sum(fvec * fjac[:, wh])
                        if sum0 > 0:
                            fjac[:, wh] = 0
                if nupeg > 0:
                    # Total derivative of sum wrt upper pegged parameters
                    for wh in whupeg:
                        sum0 = np.sum(fvec * fjac[:, wh])
                        if sum0 < 0:
                            fjac[:, wh] = 0

            # Compute the QR factorization of the jacobian
            [fjac, ipvt, wa1, wa2] = self.qrfac(fjac, pivot=1)

            # On the first iteration if "diag" is unspecified, scale
            # according to the norms of the columns of the initial jacobian
            catch_msg = 'rescaling diagonal elements'
            if self.niter == 1:
                if (not rescale) or (diag is None):
                    diag = wa2.copy()
                    # Introduced threshold (Michele Cappellari, 19 May 2017)
                    thresh = np.max(np.abs(diag)) * self.machar.machep
                    wh = np.abs(diag) < thresh
                    diag[wh] = 1.

                # On the first iteration, calculate the norm of the scaled x
                # and initialize the step bound delta
                wa3 = diag * x
                xnorm = norm(wa3)
                delta = factor*xnorm
                if delta == 0.:
                    delta = factor

            # Form (q transpose)*fvec and store the first n components in qtf
            catch_msg = 'forming (q transpose)*fvec'
            wa4 = fvec.copy()
            for j in range(n):
                lj = ipvt[j]
                temp3 = fjac[j,lj]
                if temp3 != 0:
                    fj = fjac[j:,lj]
                    wa4[j:] -= fj * np.sum(fj*wa4[j:]) / temp3
                fjac[j,lj] = wa1[j]
                qtf[j] = wa4[j]
            # From this point on, only the square matrix, consisting of the
            # triangle of R, is needed.
            fjac = fjac[:n, :n]
            fjac = fjac[:, ipvt]

            # Compute the norm of the scaled gradient
            catch_msg = 'computing the scaled gradient'
            gnorm = 0.
            if self.fnorm != 0:
                for j in range(n):
                    l = ipvt[j]
                    if wa2[l] != 0:
                        sum0 = np.sum(fjac[:j+1, j]*qtf[:j+1])/self.fnorm
                        gnorm = max(gnorm, np.abs(sum0/wa2[l]))

            # Test for convergence of the gradient norm
            if gnorm <= gtol:
                self.status = 4
                break
            if maxiter == 0:
                self.status = 5
                break

            # Rescale if necessary
            if not rescale:
                diag = np.maximum(wa2, diag)

            # Beginning of the inner loop
            while(1):

                # Determine the levenberg-marquardt parameter
                catch_msg = 'calculating LM parameter (MPFIT_)'
                [fjac, par, wa1, wa2] = self.lmpar(fjac, ipvt, diag, qtf,
                                                   delta, wa1, wa2, par=par)

                # Store the direction p and x+p. Calculate the norm of p
                wa1 = -wa1

                if (qanylim == 0) and (qminmax == 0):
                    # No parameter limits, so just move to new position WA2
                    alpha = 1.
                    wa2 = x + wa1

                else:

                    # Respect the limits.  If a step were to go out of bounds, then
                    # we should take a step in the same direction but shorter distance.
                    # The step should take us right to the limit in that case.
                    alpha = 1.

                    if qanylim:
                        # Do not allow any steps out of bounds
                        catch_msg = 'checking for a step out of bounds'
                        if nlpeg > 0:
                            wa1[whlpeg] = np.maximum(wa1[whlpeg], 0)
                        if nupeg > 0:
                            wa1[whupeg] = np.minimum(wa1[whupeg], 0)

                        dwa1 = np.abs(wa1) > machep
                        whl = dwa1 & qllim & (x + wa1 < llim)
                        if np.any(whl):
                            t = (llim[whl] - x[whl]) / wa1[whl]
                            alpha = min(alpha, np.min(t))
                        whu = dwa1 & qulim & (x + wa1 > ulim)
                        if np.any(whu):
                            t = (ulim[whu] - x[whu]) / wa1[whu]
                            alpha = min(alpha, np.min(t))

                    # Obey any max step values.
                    if qminmax:
                        nwa1 = wa1*alpha
                        whmax = qmax & (maxstep > 0)
                        if np.any(whmax):
                            mrat = np.max(np.abs(nwa1[whmax]/maxstep[whmax]))
                            if mrat > 1:
                                alpha /= mrat

                    # Scale the resulting vector
                    wa1 *= alpha
                    wa2 = x + wa1

                    # The previous step should be feasible by design, but this
                    # may not be true to machine precision. If any value is
                    # still outside the box, we put it at the boundary.
                    wh = qulim & (wa2 > ulim)
                    wa2[wh] = ulim[wh]
                    wh = qllim & (wa2 < llim)
                    wa2[wh] = llim[wh]

                wa3 = diag*wa1
                pnorm = norm(wa3)

                # On the first iteration, adjust the initial step bound
                if self.niter == 1:
                    delta = min(delta, pnorm)

                self.params[ifree] = wa2

                # Evaluate the function at x+p and calculate its norm
                mperr = 0
                catch_msg = 'calling '+str(fcn)
                [self.status, wa4] = self.call(fcn, self.params, functkw)

                if self.status < 0:
                    self.errmsg = 'WARNING: premature termination by "'+fcn+'"'
                    return
                fnorm1 = norm(wa4)

                # Compute the scaled actual reduction
                catch_msg = 'computing convergence criteria'
                actred = -1.
                if (0.1 * fnorm1) < self.fnorm:
                    actred =  1 - (fnorm1/self.fnorm)**2

                # Compute the scaled predicted reduction and the scaled directional
                # derivative
                for j in range(n):
                    wa3[j] = 0
                    wa3[:j+1] += fjac[:j+1,j]*wa1[ipvt[j]]

                # Remember, alpha is the fraction of the full LM step actually
                # taken
                temp1 = alpha*norm(wa3)/self.fnorm
                temp2 = np.sqrt(alpha*par)*pnorm/self.fnorm
                prered = temp1**2 + 2*temp2**2
                dirder = -(temp1**2 + temp2**2)

                # Compute the ratio of the actual to the predicted reduction.
                ratio = 0.
                if prered != 0:
                    ratio = actred/prered

                # Update the step bound
                if ratio <= 0.25:
                    if actred >= 0:
                        temp = 0.5
                    else:
                        temp = 0.5*dirder/(dirder + 0.5*actred)
                    if ((0.1*fnorm1) >= self.fnorm) or (temp < 0.1):
                        temp = 0.1
                    delta = temp*min(delta, 10*pnorm)
                    par = par/temp
                else:
                    if (par == 0) or (ratio >= 0.75):
                        delta = 2*pnorm
                        par = 0.5*par

                # Test for successful iteration
                if ratio >= 0.0001:    # Successful iteration
                    x = wa2
                    wa2 = diag * x
                    fvec = wa4
                    xnorm = norm(wa2)
                    self.fnorm = fnorm1
                    self.niter += 1

                # Tests for convergence
                if (np.abs(actred) <= ftol) and (prered <= ftol) \
                     and (ratio <= 2):
                     self.status = 1
                if delta <= xtol*xnorm:
                    self.status = 2
                if (np.abs(actred) <= ftol) and (prered <= ftol) \
                     and (ratio <= 2) and (self.status == 2):
                     self.status = 3
                if self.status != 0:
                    break

                # Tests for termination and stringent tolerances
                if self.niter >= maxiter:
                    self.status = 5
                if (np.abs(actred) <= machep) and (prered <= machep) \
                    and (0.5*ratio <= 1):
                    self.status = 6
                if delta <= machep*xnorm:
                    self.status = 7
                if gnorm <= machep:
                    self.status = 8
                if self.status != 0:
                    break

                # End of inner loop. Repeat if iteration unsuccessful
                if ratio >= 0.0001:
                    break

                # Check for over/underflow
                if ~np.all(np.isfinite(wa1) & np.isfinite(wa2) & \
                            np.isfinite(x)) or ~np.isfinite(ratio):
                    self.errmsg = ('''ERROR: parameter or function value(s) have become
                        'infinite; check model function for over- 'and underflow''')
                    self.status = -16
                    break

            if self.status != 0:
                break;
        # End of outer loop.

        catch_msg = 'in the termination phase'
        # Termination, either normal or user imposed.
        if len(self.params) == 0:
            return
        if nfree == 0:
            self.params = xall.copy()
        else:
            self.params[ifree] = x
        if (nprint > 0) and (self.status > 0):
            catch_msg = 'calling ' + str(fcn)
            [status, fvec] = self.call(fcn, self.params, functkw)

            catch_msg = 'in the termination phase'
            self.fnorm = norm(fvec)

        if (self.fnorm is not None) and (fnorm1 is not None):
            self.fnorm = max(self.fnorm, fnorm1)
            self.fnorm = self.fnorm**2.

        self.covar = None
        self.perror = None
        # (very carefully) set the covariance matrix COVAR
        if (self.status > 0) and (nocovar==0) and (n is not None) \
                       and (fjac is not None) and (ipvt is not None):
            sz = fjac.shape
            if (n > 0) and (sz[0] >= n) and (sz[1] >= n) \
                and (len(ipvt) >= n):

                catch_msg = 'computing the covariance matrix'
                cv = self.calc_covar(fjac[:n,:n], ipvt[:n])
                cv.shape = [n, n]
                nn = len(xall)

                # Fill in actual covariance matrix, accounting for fixed
                # parameters.
                self.covar = np.zeros([nn, nn])
                for i in range(n):
                    self.covar[ifree,ifree[i]] = cv[:,i]

                # Compute errors in parameters
                catch_msg = 'computing parameter errors'
                self.perror = np.zeros(nn)
                d = np.diagonal(self.covar)
                wh = np.nonzero(d >= 0)[0]
                if len(wh) > 0:
                    self.perror[wh] = np.sqrt(d[wh])
        return

###############################################################################

    def __str__(self):

        return {'params': self.params,
               'niter': self.niter,
               'params': self.params,
               'covar': self.covar,
               'perror': self.perror,
               'status': self.status,
               'errmsg': self.errmsg,
               'nfev': self.nfev,
               'damp': self.damp
               }.__str__()

###############################################################################

    # Default procedure to be called every iteration.  It simply prints
    # the parameter values.
    def defiter(self, fcn, x, iter, fnorm=None, functkw=None,
                       quiet=0, iterstop=None, parinfo=None,
                       format=None, pformat='%.10g', dof=1):

        if quiet:
            return
        if fnorm is None:
            [status, fvec] = self.call(fcn, x, functkw)
            fnorm = norm(fvec)**2

        # Determine which parameters to print
        nprint = len(x)
        print("Iter ", ('%6i' % iter),"   CHI-SQUARE = ",('%.10g' % fnorm)," DOF = ", ('%i' % dof))
        for i in range(nprint):
            if (parinfo is not None) and ('parname' in parinfo[i]):
                p = '   ' + parinfo[i]['parname'] + ' = '
            else:
                p = '   P' + str(i) + ' = '
            if (parinfo is not None) and ('mpprint' in parinfo[i]):
                iprint = parinfo[i]['mpprint']
            else:
                iprint = 1
            if iprint:
                print(p + (pformat % x[i]) + '  ')
        return 0

###############################################################################

    # Procedure to parse the parameter values in PARINFO, which is a list of dictionaries
    def parinfo(self, parinfo=None, key='a', default=None, n=0):

        if parinfo is None and n == 0:
            values = default
            return values

        if key in parinfo[0]:
            values = np.array([a[key] for a in parinfo])
        else:  # The extra dtype below is only needed in Numpy < 1.12
            values = np.full(len(parinfo), default, dtype=np.array(default).dtype)

        return values

###############################################################################

    # Call user function or procedure, with derivatives or not.
    def call(self, fcn, x, functkw, fjac=None):

        if self.qanytied:
            x = self.tie(x, self.ptied)
        self.nfev += 1
        f = fcn(x, fjac=fjac, **functkw)

        # Apply the damping if requested.  This replaces the residuals
        # with their hyperbolic tangent.  Thus residuals larger than
        # DAMP are essentially clipped.
        if self.damp > 0:
            f = np.tanh(f/self.damp)

        return [0, f]

###############################################################################

    def fdjac2(self, fcn, x, fvec, step=None, ulimited=None, ulimit=None,
               dside=None, epsfcn=None, autoderivative=1, functkw=None,
               xall=None, ifree=None, dstep=None):

        m = len(fvec)
        n = len(x)
        machep = self.machar.machep

        if epsfcn is None:
            epsfcn = machep
        eps = np.sqrt(max(epsfcn, machep))
        if xall is None:
            xall = x
        if ifree is None:
            ifree = np.arange(len(xall))
        if step is None:
            step = np.full_like(x, eps)
        nall = len(xall)

        # Compute analytical derivative if requested
        if autoderivative == 0:
            fjac = np.zeros(nall)
            fjac[ifree] = 1.0  # Specify which parameters need derivatives
            self.call(fcn, xall, functkw, fjac=fjac)

            if len(fjac) != m*nall:
                print('ERROR: Derivative matrix was not computed properly.')
                return None

            # This definition is consistent with CURVEFIT
            # Sign error found (thanks Jesus Fernandez <fernande@irm.chu-caen.fr>)
            fjac.shape = [m,nall]
            fjac = -fjac

            # Select only the free parameters
            if len(ifree) < nall:
                fjac = fjac[:,ifree]
                fjac.shape = [m, n]
                return fjac

        fjac = np.zeros([m, n])
        h = eps*np.abs(x)

        # if STEP is given, use that
        # STEP includes the fixed parameters
        if step is not None:
            stepi = step[ifree]
            wh = stepi > 0
            if np.any(wh):
                h[wh] = stepi[wh]

        # if relative step is given, use that
        # DSTEP includes the fixed parameters
        if len(dstep) > 0:
            dstepi = dstep[ifree]
            wh = dstepi > 0
            if np.any(wh):
                h[wh] = np.abs(dstepi[wh]*x[wh])

        # Reverse the sign of the step if we are up against the parameter
        # limit, or if the user requested it.
        # DSIDE includes the fixed parameters (ULIMITED/ULIMIT have only
        # varying ones)
        wh = dside[ifree] == -1
        if len(ulimited) > 0 and len(ulimit) > 0:
            wh |= (ulimited != 0) & (x > ulimit - h)
            if np.any(wh):
                h[wh] = -h[wh]

        # Loop through parameters, computing the derivative for each.
        for j in range(n):
            xp = xall.copy()
            xp[ifree[j]] += h[j]
            [status, fp] = self.call(fcn, xp, functkw)

            if status < 0:
                return None

            if -1 <= dside[ifree[j]] <= 1:
                # COMPUTE THE ONE-SIDED DERIVATIVE
                fjac[:,j] = (fp - fvec)/h[j]

            else:
                # COMPUTE THE TWO-SIDED DERIVATIVE
                xp[ifree[j]] = xall[ifree[j]] - h[j]

                [status, fm] = self.call(fcn, xp, functkw)
                if status < 0:
                    return None

                fjac[:,j] = (fp - fm)/(2*h[j])

        return fjac

###############################################################################
#
#    Original FORTRAN documentation
#    **********
#
#    subroutine qrfac
#
#    this subroutine uses householder transformations with column
#    pivoting (optional) to compute a qr factorization of the
#    m by n matrix a. that is, qrfac determines an orthogonal
#    matrix q, a permutation matrix p, and an upper trapezoidal
#    matrix r with diagonal elements of nonincreasing magnitude,
#    such that a*p = q*r. the householder transformation for
#    column k, k = 1,2,...,min(m,n), is of the form
#
#                       t
#       i - (1/u(k))*u*u
#
#    where u has zeros in the first k-1 positions. the form of
#    this transformation and the method of pivoting first
#    appeared in the corresponding linpack subroutine.
#
#    the subroutine statement is
#
#   subroutine qrfac(m,n,a,lda,pivot,ipvt,lipvt,rdiag,acnorm,wa)
#
#    where
#
#   m is a positive integer input variable set to the number
#     of rows of a.
#
#   n is a positive integer input variable set to the number
#     of columns of a.
#
#   a is an m by n array. on input a contains the matrix for
#     which the qr factorization is to be computed. on output
#     the strict upper trapezoidal part of a contains the strict
#     upper trapezoidal part of r, and the lower trapezoidal
#     part of a contains a factored form of q (the non-trivial
#     elements of the u vectors described above).
#
#   lda is a positive integer input variable not less than m
#     which specifies the leading dimension of the array a.
#
#   pivot is a logical input variable. if pivot is set true,
#     then column pivoting is enforced. if pivot is set false,
#     then no column pivoting is done.
#
#   ipvt is an integer output array of length lipvt. ipvt
#     defines the permutation matrix p such that a*p = q*r.
#     column j of p is column ipvt(j) of the identity matrix.
#     if pivot is false, ipvt is not referenced.
#
#   lipvt is a positive integer input variable. if pivot is false,
#     then lipvt may be as small as 1. if pivot is true, then
#     lipvt must be at least n.
#
#   rdiag is an output array of length n which contains the
#     diagonal elements of r.
#
#   acnorm is an output array of length n which contains the
#     norms of the corresponding columns of the input matrix a.
#     if this information is not needed, then acnorm can coincide
#     with rdiag.
#
#   wa is a work array of length n. if pivot is false, then wa
#     can coincide with rdiag.
#
#    subprograms called
#
#   minpack-supplied ... dpmpar,enorm
#
#   fortran-supplied ... dmax1,dsqrt,min0
#
#    argonne national laboratory. minpack project. march 1980.
#    burton s. garbow, kenneth e. hillstrom, jorge j. more
#
#    **********
#
# PIVOTING / PERMUTING:
#
# Upon return, A(*,*) is in standard parameter order, A(*,IPVT) is in
# permuted order.
#
# RDIAG is in permuted order.
# ACNORM is in standard parameter order.
#
#
# NOTE: in IDL the factors appear slightly differently than described
# above.  The matrix A is still m x n where m >= n.
#
# The "upper" triangular matrix R is actually stored in the strict
# lower left triangle of A under the standard notation of IDL.
#
# The reflectors that generate Q are in the upper trapezoid of A upon
# output.
#
#  EXAMPLE:  decompose the matrix [[9.,2.,6.],[4.,8.,7.]]
#   aa = [[9.,2.,6.],[4.,8.,7.]]
#   mpfit_qrfac, aa, aapvt, rdiag, aanorm
#    IDL> print, aa
#         1.81818*   0.181818*   0.545455*
#        -8.54545+    1.90160*   0.432573*
#    IDL> print, rdiag
#        -11.0000+   -7.48166+
#
# The components marked with a * are the components of the
# reflectors, and those marked with a + are components of R.
#
# To reconstruct Q and R we proceed as follows.  First R.
#   r = fltarr(m, n)
#   for i = 0, n-1 do r(0:i,i) = aa(0:i,i)  # fill in lower diag
#   r(lindgen(n)*(m+1)) = rdiag
#
# Next, Q, which are composed from the reflectors.  Each reflector v
# is taken from the upper trapezoid of aa, and converted to a matrix
# via (I - 2 vT . v / (v . vT)).
#
#   hh = ident                                  # identity matrix
#   for i = 0, n-1 do begin
#   v = aa(*,i) & if i GT 0 then v(0:i-1) = 0   # extract reflector
#   hh = hh # (ident - 2*(v # v)/total(v * v))  # generate matrix
#   endfor
#
# Test the result:
#   IDL> print, hh # transpose(r)
#         9.00000     4.00000
#         2.00000     8.00000
#         6.00000     7.00000
#
# Note that it is usually never necessary to form the Q matrix
# explicitly, and MPFIT does not.

    def qrfac(self, a, pivot=0):

        machep = self.machar.machep
        sz = a.shape
        m = sz[0]
        n = sz[1]

        # Compute the initial column norms and initialize arrays
        acnorm = np.zeros(n)
        for j in range(n):
            acnorm[j] = norm(a[:,j])
        rdiag = acnorm.copy()
        wa = rdiag.copy()
        ipvt = np.arange(n)

        # Reduce a to r with householder transformations
        minmn = np.min([m,n])
        for j in range(minmn):
            if pivot != 0:
                # Bring the column of largest norm into the pivot position
                kmax = np.argmax(rdiag[j:])
                kmax += j

                # Exchange rows via the pivot only.  Avoid actually exchanging
                # the rows, in case there is lots of memory transfer.  The
                # exchange occurs later, within the body of MPFIT, after the
                # extraneous columns of the matrix have been shed.
                if kmax != j:
                    ipvt[j], ipvt[kmax] = ipvt[kmax], ipvt[j]
                    rdiag[kmax] = rdiag[j]
                    wa[kmax] = wa[j]

            # Compute the householder transformation to reduce the jth
            # column of A to a multiple of the jth unit vector
            lj = ipvt[j]
            ajj = a[j:,lj]
            ajnorm = norm(ajj)
            if ajnorm == 0:
                break
            if a[j,lj] < 0:
                ajnorm = -ajnorm

            ajj /= ajnorm
            ajj[0] += 1
            a[j:,lj] = ajj

            # Apply the transformation to the remaining columns
            # and update the norms

            # NOTE to SELF: tried to optimize this by removing the loop,
            # but it actually got slower.  Reverted to "for" loop to keep
            # it simple.
            if j+1 < n:
                for k in range(j+1, n):
                    lk = ipvt[k]
                    if a[j,lj] != 0:
                        a[j:,lk] -= ajj * np.sum(a[j:,lk]*ajj)/a[j,lj]
                        if (pivot != 0) and (rdiag[k] != 0):
                            temp = a[j,lk]/rdiag[k]
                            rdiag[k] *= np.sqrt(max(1 - temp**2, 0))
                            temp = rdiag[k]/wa[k]
                            if (0.05*temp**2) <= machep:
                                rdiag[k] = norm(a[j+1:,lk])
                                wa[k] = rdiag[k]
            rdiag[j] = -ajnorm

        return [a, ipvt, rdiag, acnorm]

###############################################################################
#
#    Original FORTRAN documentation
#    **********
#
#    subroutine qrsolv
#
#    given an m by n matrix a, an n by n diagonal matrix d,
#    and an m-vector b, the problem is to determine an x which
#    solves the system
#
#          a*x = b ,     d*x = 0 ,
#
#    in the least squares sense.
#
#    this subroutine completes the solution of the problem
#    if it is provided with the necessary information from the
#    factorization, with column pivoting, of a. that is, if
#    a*p = q*r, where p is a permutation matrix, q has orthogonal
#    columns, and r is an upper triangular matrix with diagonal
#    elements of nonincreasing magnitude, then qrsolv expects
#    the full upper triangle of r, the permutation matrix p,
#    and the first n components of (q transpose)*b. the system
#    a*x = b, d*x = 0, is then equivalent to
#
#                 t    t
#          r*z = q *b ,  p *d*p*z = 0 ,
#
#    where x = p*z. if this system does not have full rank,
#    then a least squares solution is obtained. on output qrsolv
#    also provides an upper triangular matrix s such that
#
#           t   t              t
#          p *(a *a + d*d)*p = s *s .
#
#    s is computed within qrsolv and may be of separate interest.
#
#    the subroutine statement is
#
#      subroutine qrsolv(n,r,ldr,ipvt,diag,qtb,x,sdiag,wa)
#
#    where
#
#      n is a positive integer input variable set to the order of r.
#
#      r is an n by n array. on input the full upper triangle
#        must contain the full upper triangle of the matrix r.
#        on output the full upper triangle is unaltered, and the
#        strict lower triangle contains the strict upper triangle
#        (transposed) of the upper triangular matrix s.
#
#      ldr is a positive integer input variable not less than n
#        which specifies the leading dimension of the array r.
#
#      ipvt is an integer input array of length n which defines the
#        permutation matrix p such that a*p = q*r. column j of p
#        is column ipvt(j) of the identity matrix.
#
#      diag is an input array of length n which must contain the
#        diagonal elements of the matrix d.
#
#      qtb is an input array of length n which must contain the first
#        n elements of the vector (q transpose)*b.
#
#      x is an output array of length n which contains the least
#        squares solution of the system a*x = b, d*x = 0.
#
#      sdiag is an output array of length n which contains the
#        diagonal elements of the upper triangular matrix s.
#
#      wa is a work array of length n.
#
#    subprograms called
#
#      fortran-supplied ... dabs,dsqrt
#
#    argonne national laboratory. minpack project. march 1980.
#    burton s. garbow, kenneth e. hillstrom, jorge j. more
#

    def qrsolv(self, r, ipvt, diag, qtb, sdiag):
        sz = r.shape
        m = sz[0]
        n = sz[1]

        # copy r and (q transpose)*b to preserve input and initialize s.
        # in particular, save the diagonal elements of r in x.

        for j in range(n):
            r[j:n,j] = r[j,j:n]
        x = np.diagonal(r).copy()
        wa = qtb.copy()

        # Eliminate the diagonal matrix d using a givens rotation
        for j in range(n):
            l = ipvt[j]
            if diag[l] == 0:
                break
            sdiag[j:] = 0
            sdiag[j] = diag[l]

            # The transformations to eliminate the row of d modify only a
            # single element of (q transpose)*b beyond the first n, which
            # is initially zero.

            qtbpj = 0.
            for k in range(j,n):
                if sdiag[k] == 0:
                    break
                if np.abs(r[k,k]) < np.abs(sdiag[k]):
                    cotan  = r[k,k]/sdiag[k]
                    sine   = 0.5/np.sqrt(0.25 + 0.25*cotan**2)
                    cosine = sine*cotan
                else:
                    tang   = sdiag[k]/r[k,k]
                    cosine = 0.5/np.sqrt(0.25 + 0.25*tang**2)
                    sine   = cosine*tang

                # Compute the modified diagonal element of r and the
                # modified element of ((q transpose)*b,0).
                r[k,k] = cosine*r[k,k] + sine*sdiag[k]
                temp = cosine*wa[k] + sine*qtbpj
                qtbpj = -sine*wa[k] + cosine*qtbpj
                wa[k] = temp

                # Accumulate the transformation in the row of s
                if n > k+1:
                    temp = cosine*r[k+1:n,k] + sine*sdiag[k+1:n]
                    sdiag[k+1:n] = cosine*sdiag[k+1:n] - sine*r[k+1:n,k]
                    r[k+1:n,k] = temp
            sdiag[j] = r[j,j]
            r[j,j] = x[j]

        # Solve the triangular system for z.  If the system is singular
        # then obtain a least squares solution
        nsing = n

         # Introduced threshold (Michele Cappellari, 19 May 2017)
        thresh = np.max(np.abs(sdiag)) * self.machar.machep
        wh = np.flatnonzero(np.abs(sdiag) < thresh)
        if len(wh) > 0:
            nsing = wh[0]
            wa[nsing:] = 0
        if nsing >= 1:
            wa[nsing-1] /= sdiag[nsing-1] # Degenerate case
            # *** Reverse loop ***
            for j in range(nsing-2,-1,-1):
                sum0 = np.sum(r[j+1:nsing,j]*wa[j+1:nsing])
                wa[j] = (wa[j] - sum0)/sdiag[j]

        # Permute the components of z back to components of x
        x[ipvt] = wa
        return (r, x, sdiag)

###############################################################################
#
#    Original FORTRAN documentation
#
#    subroutine lmpar
#
#    given an m by n matrix a, an n by n nonsingular diagonal
#    matrix d, an m-vector b, and a positive number delta,
#    the problem is to determine a value for the parameter
#    par such that if x solves the system
#
#       a*x = b ,    sqrt(par)*d*x = 0 ,
#
#    in the least squares sense, and dxnorm is the euclidean
#    norm of d*x, then either par is zero and
#
#       (dxnorm-delta) .le. 0.1*delta ,
#
#    or par is positive and
#
#       abs(dxnorm-delta) .le. 0.1*delta .
#
#    this subroutine completes the solution of the problem
#    if it is provided with the necessary information from the
#    qr factorization, with column pivoting, of a. that is, if
#    a*p = q*r, where p is a permutation matrix, q has orthogonal
#    columns, and r is an upper triangular matrix with diagonal
#    elements of nonincreasing magnitude, then lmpar expects
#    the full upper triangle of r, the permutation matrix p,
#    and the first n components of (q transpose)*b. on output
#    lmpar also provides an upper triangular matrix s such that
#
#        t   t                 t
#       p *(a *a + par*d*d)*p = s *s .
#
#    s is employed within lmpar and may be of separate interest.
#
#    only a few iterations are generally needed for convergence
#    of the algorithm. if, however, the limit of 10 iterations
#    is reached, then the output par will contain the best
#    value obtained so far.
#
#    the subroutine statement is
#
#   subroutine lmpar(n,r,ldr,ipvt,diag,qtb,delta,par,x,sdiag,
#                    wa1,wa2)
#
#    where
#
#   n is a positive integer input variable set to the order of r.
#
#   r is an n by n array. on input the full upper triangle
#     must contain the full upper triangle of the matrix r.
#     on output the full upper triangle is unaltered, and the
#     strict lower triangle contains the strict upper triangle
#     (transposed) of the upper triangular matrix s.
#
#   ldr is a positive integer input variable not less than n
#     which specifies the leading dimension of the array r.
#
#   ipvt is an integer input array of length n which defines the
#     permutation matrix p such that a*p = q*r. column j of p
#     is column ipvt(j) of the identity matrix.
#
#   diag is an input array of length n which must contain the
#     diagonal elements of the matrix d.
#
#   qtb is an input array of length n which must contain the first
#     n elements of the vector (q transpose)*b.
#
#   delta is a positive input variable which specifies an upper
#     bound on the euclidean norm of d*x.
#
#   par is a nonnegative variable. on input par contains an
#     initial estimate of the levenberg-marquardt parameter.
#     on output par contains the final estimate.
#
#   x is an output array of length n which contains the least
#     squares solution of the system a*x = b, sqrt(par)*d*x = 0,
#     for the output par.
#
#   sdiag is an output array of length n which contains the
#     diagonal elements of the upper triangular matrix s.
#
#   wa1 and wa2 are work arrays of length n.
#
#    subprograms called
#
#   minpack-supplied ... dpmpar,enorm,qrsolv
#
#   fortran-supplied ... dabs,dmax1,dmin1,dsqrt
#
#    argonne national laboratory. minpack project. march 1980.
#    burton s. garbow, kenneth e. hillstrom, jorge j. more
#

    def lmpar(self, r, ipvt, diag, qtb, delta, x, sdiag, par=None):

        dwarf = self.machar.minnum
        machep = self.machar.machep
        sz = r.shape
        m = sz[0]
        n = sz[1]

        # Compute and store in x the gauss-newton direction.  If the
        # jacobian is rank-deficient, obtain a least-squares solution
        nsing = n
        wa1 = qtb.copy()
        thresh = np.max(np.abs(np.diagonal(r))) * machep
        wh = np.flatnonzero(np.abs(np.diagonal(r)) < thresh)
        if len(wh) > 0:
            nsing = wh[0]
            wa1[wh[0]:] = 0
        if nsing >= 1:
            # *** Reverse loop ***
            for j in range(nsing-1,-1,-1):
                wa1[j] /= r[j,j]
                if j-1 >= 0:
                    wa1[:j] -= r[:j,j]*wa1[j]

        # Note: ipvt here is a permutation array
        x[ipvt] = wa1

        # Initialize the iteration counter.  Evaluate the function at the
        # origin, and test for acceptance of the gauss-newton direction
        iter = 0
        wa2 = diag * x
        dxnorm = norm(wa2)
        fp = dxnorm - delta
        if fp <= 0.1*delta:
            return [r, 0., x, sdiag]

        # If the jacobian is not rank deficient, the newton step provides a
        # lower bound, parl, for the zero of the function.  Otherwise set
        # this bound to zero.

        parl = 0.
        if nsing >= n:
            wa1 = diag[ipvt] * wa2[ipvt] / dxnorm
            wa1[0] /= r[0,0] # Degenerate case
            for j in range(1,n):   # Note "1" here, not zero
                sum0 = np.sum(r[:j,j]*wa1[:j])
                wa1[j] = (wa1[j] - sum0)/r[j,j]

            temp = norm(wa1)
            parl = fp/(delta*temp**2)

        # Calculate an upper bound, paru, for the zero of the function
        for j in range(n):
            sum0 = np.sum(r[:j+1,j]*qtb[:j+1])
            wa1[j] = sum0/diag[ipvt[j]]
        gnorm = norm(wa1)
        paru = gnorm/delta
        if paru == 0:
            paru = dwarf/min(delta, 0.1)

        # If the input par lies outside of the interval (parl,paru), set
        # par to the closer endpoint

        par = max(par, parl)
        par = min(par, paru)
        if par == 0:
            par = gnorm/dxnorm

        # Beginning of an interation
        while(1):
            iter = iter + 1

            # Evaluate the function at the current value of par
            if par == 0:
                par = max(dwarf, paru*0.001)
            temp = np.sqrt(par)
            wa1 = temp * diag
            [r, x, sdiag] = self.qrsolv(r, ipvt, wa1, qtb, sdiag)
            wa2 = diag*x
            dxnorm = norm(wa2)
            temp = fp
            fp = dxnorm - delta

            if (np.abs(fp) <= 0.1*delta) or \
               ((parl == 0) and (fp <= temp) and (temp < 0)) or \
               (iter == 10):
               break;

            # Compute the newton correction
            wa1 = diag[ipvt] * wa2[ipvt] / dxnorm

            for j in range(n-1):
                wa1[j] /= sdiag[j]
                wa1[j+1:n] -= r[j+1:n,j]*wa1[j]
            wa1[n-1] /= sdiag[n-1] # Degenerate case

            temp = norm(wa1)
            parc = fp/(delta*temp**2)

            # Depending on the sign of the function, update parl or paru
            if fp > 0:
                parl = max(parl, par)
            if fp < 0:
                paru = min(paru, par)

            # Compute an improved estimate for par
            par = max(parl, par + parc)

            # End of an iteration

        # Termination
        return [r, par, x, sdiag]


    # Procedure to tie one parameter to another.
    def tie(self, p, ptied=None):

        if ptied is None:
            return
        for i, pti in enumerate(ptied):
            if pti != '':
                exec('p[' + str(i) + '] = ' + pti)
        return p

###############################################################################
#
#    Original FORTRAN documentation
#    **********
#
#    subroutine covar
#
#    given an m by n matrix a, the problem is to determine
#    the covariance matrix corresponding to a, defined as
#
#                   t
#          inverse(a *a) .
#
#    this subroutine completes the solution of the problem
#    if it is provided with the necessary information from the
#    qr factorization, with column pivoting, of a. that is, if
#    a*p = q*r, where p is a permutation matrix, q has orthogonal
#    columns, and r is an upper triangular matrix with diagonal
#    elements of nonincreasing magnitude, then covar expects
#    the full upper triangle of r and the permutation matrix p.
#    the covariance matrix is then computed as
#
#                     t  t
#          p*inverse(r *r)*p  .
#
#    if a is nearly rank deficient, it may be desirable to compute
#    the covariance matrix corresponding to the linearly independent
#    columns of a. to define the numerical rank of a, covar uses
#    the tolerance tol. if l is the largest integer such that
#
#          abs(r(l,l)) .gt. tol*abs(r(1,1)) ,
#
#    then covar computes the covariance matrix corresponding to
#    the first l columns of r. for k greater than l, column
#    and row ipvt(k) of the covariance matrix are set to zero.
#
#    the subroutine statement is
#
#      subroutine covar(n,r,ldr,ipvt,tol,wa)
#
#    where
#
#      n is a positive integer input variable set to the order of r.
#
#      r is an n by n array. on input the full upper triangle must
#        contain the full upper triangle of the matrix r. on output
#        r contains the square symmetric covariance matrix.
#
#      ldr is a positive integer input variable not less than n
#        which specifies the leading dimension of the array r.
#
#      ipvt is an integer input array of length n which defines the
#        permutation matrix p such that a*p = q*r. column j of p
#        is column ipvt(j) of the identity matrix.
#
#      tol is a nonnegative input variable used to define the
#        numerical rank of a in the manner described above.
#
#      wa is a work array of length n.
#
#    subprograms called
#
#      fortran-supplied ... dabs
#
#    argonne national laboratory. minpack project. august 1980.
#    burton s. garbow, kenneth e. hillstrom, jorge j. more
#
#    **********

    def calc_covar(self, rr, ipvt=None, tol=1e-14):

        if rr.ndim != 2:
            print('ERROR: r must be a two-dimensional matrix')
            return -1
        s = rr.shape
        n = s[0]
        if s[0] != s[1]:
            print('ERROR: r must be a square matrix')
            return -1

        if ipvt is None:
            ipvt = np.arange(n)
        r = rr.copy()
        r.shape = [n,n]

        # For the inverse of r in the full upper triangle of r
        l = -1
        tolr = tol * np.abs(r[0,0])
        for k in range(n):
            if np.abs(r[k,k]) <= tolr:
                break
            r[k,k] = 1./r[k,k]
            for j in range(k):
                temp = r[k,k] * r[j,k]
                r[j,k] = 0.
                r[:j+1,k] -= temp*r[:j+1,j]
            l = k

        # Form the full upper triangle of the inverse of (r transpose)*r
        # in the full upper triangle of r
        if l >= 0:
            for k in range(l+1):
                for j in range(k):
                    temp = r[j,k]
                    r[:j+1,j] += temp*r[:j+1,k]
                temp = r[k,k]
                r[:k+1,k] *= temp

        # For the full lower triangle of the covariance matrix
        # in the strict lower triangle or and in wa
        wa = np.repeat([r[0,0]], n)
        for j in range(n):
            jj = ipvt[j]
            sing = j > l
            for i in range(j+1):
                if sing:
                    r[i,j] = 0.
                ii = ipvt[i]
                if ii > jj:
                    r[ii,jj] = r[i,j]
                if ii < jj:
                    r[jj,ii] = r[i,j]
            wa[jj] = r[j,j]

        # Symmetrize the covariance matrix in r
        for j in range(n):
            r[:j+1,j] = r[j,:j+1]
            r[j,j] = wa[j]

        return r


###############################################################################

class machar:

    def __init__(self):

        info = np.finfo(np.float32)

        self.machep = info.eps
        self.maxnum = info.max
        self.minnum = info.tiny

        self.maxlog = np.log(self.maxnum)
        self.minlog = np.log(self.minnum)
        self.rdwarf = np.sqrt(self.minnum*1.5) * 10
        self.rgiant = np.sqrt(self.maxnum) * 0.1

###############################################################################