""" 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 XXX. 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://cow.physics.wisc.edu/~craigm/idl/idl.html 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 2022-09-21 [PAK] add 'double' parameter to select machine precision for the objective function. This mainly affects numerical derivatives, forcing the step size to be large enough that x + h != x, but will also change the criteria for correcting degenerate qr decomposition and maybe stopping conditions and returned status codes. 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 should also return a status flag and 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==None then partial derivatives should not be # computed. It will always be None if MPFIT is called with default # flag. model = F(x, p) # Non-negative status value means MPFIT should continue, negative means # stop the calculation. status = 0 return([status, (y-model)/err] 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 numpy array of type float. User functions may also indicate a fatal error condition using the status return described above. If status is set to a number between -15 and -1 then MPFIT will stop the calculation and return to the caller. 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) Non-negative status value means MPFIT should continue, negative means # stop the calculation. status = 0 if (dojac): pderiv = numpy.zeros([len(x), len(p)], numpy.float64) for j in range(len(p)): pderiv[:,j] = FGRAD(x, p, j) else: pderiv = None return([status, (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.]}]*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 numpy x = numpy.arange(100, numpy.float64) p0 = [5.7, 2.2, 500., 1.5, 2000.] y = ( p[0] + p[1]*[x] + p[2]*[x**2] + p[3]*numpy.sqrt(x) + p[4]*numpy.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 """ import numpy # 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 # # ********** class mpfit: def __init__( self, fcn, xall=None, functkw={}, parinfo=None, ftol=1.0e-10, xtol=1.0e-10, gtol=1.0e-10, damp=0.0, maxiter=200, factor=100.0, nprint=1, iterfunct="default", iterkw={}, nocovar=0, fastnorm=0, rescale=0, autoderivative=1, quiet=0, diag=None, epsfcn=None, debug=0, double=1, ): """ 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 fastnorm: Set this keyword to select a faster algorithm to compute sum-of-square values internally. For systems with large numbers of data points, the standard algorithm can become prohibitively slow because it cannot be vectorized well. By setting this keyword, MPFIT will run faster, but it will be more prone to floating point overflows and underflows. Thus, setting this keyword may sacrifice some stability in the fitting process. Default: clear (=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 double: Use double=1 if fcn is double precision or double=0 for single precision. 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]/numpy.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 * numpy.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.debug = debug self.errmsg = "" self.fastnorm = fastnorm self.nfev = 0 self.damp = damp self.machar = machar(double=double) machep = self.machar.machep 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 else: 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 of type numpy.float64 xall = numpy.asarray(xall, numpy.float64) npar = len(xall) self.fnorm = -1.0 fnorm1 = -1.0 ## TIED parameters? ptied = self.parinfo(parinfo, "tied", default="", n=npar) self.qanytied = 0 for i in range(npar): ptied[i] = ptied[i].strip() if ptied[i] != "": self.qanytied = 1 self.ptied = ptied ## FIXED parameters ? pfixed = self.parinfo(parinfo, "fixed", default=0, n=npar) pfixed = pfixed == 1 for i in range(npar): pfixed[i] = pfixed[i] or (ptied[i] != "") ## Tied parameters are also effectively fixed ## Finite differencing step, absolute and relative, and sidedness of deriv. step = self.parinfo(parinfo, "step", default=0.0, n=npar) dstep = self.parinfo(parinfo, "relstep", default=0.0, 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=0.0, n=npar) minstep = self.parinfo(parinfo, "mpminstep", default=0.0, n=npar) qmin = minstep * 0 ## Remove minstep for now!! qmax = maxstep != 0 (wh,) = numpy.nonzero(((qmin != 0.0) & (qmax != 0.0)) & (maxstep < minstep)) if len(wh) > 0: self.errmsg = "ERROR: MPMINSTEP is greater than MPMAXSTEP" return (wh,) = numpy.nonzero((qmin != 0.0) & (qmax != 0.0)) qminmax = len(wh > 0) ## Finish up the free parameters (ifree,) = numpy.nonzero(pfixed != 1) nfree = len(ifree) if nfree == 0: self.errmsg = "ERROR: no free parameters" return dside = dside.take(ifree) ## Compose only VARYING parameters self.params = xall ## self.params is the set of parameters to be returned x = numpy.take(self.params, ifree) ## x is the set of free parameters ## LIMITED parameters ? limited = self.parinfo(parinfo, "limited", default=[0, 0]) limits = self.parinfo(parinfo, "limits", default=[0.0, 0.0]) if (limited is not None) and (limits is not None): ## Error checking on limits in parinfo (wh,) = numpy.nonzero((limited[:, 0] & (xall < limits[:, 0])) | (limited[:, 1] & (xall > limits[:, 1]))) if len(wh) > 0: self.errmsg = "ERROR: parameters are not within PARINFO limits" return (wh,) = numpy.nonzero((limited[:, 0] & limited[:, 1]) & (limits[:, 0] >= limits[:, 1]) & (pfixed == 0)) if len(wh) > 0: self.errmsg = "ERROR: PARINFO parameter limits are not consistent" return ## Transfer structure values to local variables qulim = numpy.take(limited[:, 1], ifree) ulim = numpy.take(limits[:, 1], ifree) qllim = numpy.take(limited[:, 0], ifree) llim = numpy.take(limits[:, 0], ifree) (wh,) = numpy.nonzero((qulim != 0.0) | (qllim != 0.0)) if len(wh) > 0: qanylim = 1 else: qanylim = 0 else: ## Fill in local variables with dummy values qulim = numpy.zeros(nfree) ulim = x * 0.0 qllim = qulim llim = x * 0.0 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 != 0: self.errmsg = "ERROR: DIAG parameter scales are inconsistent" if len(diag) < n: return (wh,) = numpy.nonzero(diag <= 0) if len(wh) > 0: return self.errmsg = "" # Make sure x is a numpy array of type numpy.float64 x = numpy.asarray(x, numpy.float64) [self.status, fvec] = self.call(fcn, self.params, functkw) if self.status < 0: self.errmsg = 'ERROR: first call to "' + str(fcn) + '" failed' return m = len(fvec) if m < n: self.errmsg = "ERROR: number of parameters must not exceed data" return self.fnorm = self.enorm(fvec) ## Initialize Levelberg-Marquardt parameter and iteration counter par = 0.0 self.niter = 1 qtf = x * 0.0 self.status = 0 ## Beginning of the outer loop while 1: ## If requested, call fcn to enable printing of iterates numpy.put(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 max(abs(xnew0 - self.params)) > 0: if self.qanytied: self.params = self.tie(self.params, ptied) x = numpy.take(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,) = numpy.nonzero(qllim & (x == llim)) nlpeg = len(whlpeg) (whupeg,) = numpy.nonzero(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 i in range(nlpeg): sum = numpy.sum(fvec * fjac[:, whlpeg[i]]) if sum > 0: fjac[:, whlpeg[i]] = 0 if nupeg > 0: ## Total derivative of sum wrt upper pegged parameters for i in range(nupeg): sum = numpy.sum(fvec * fjac[:, whupeg[i]]) if sum < 0: fjac[:, whupeg[i]] = 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 (rescale == 0) or (len(diag) < n): diag = wa2.copy() (wh,) = numpy.nonzero(diag == 0) numpy.put(diag, wh, 1.0) ## On the first iteration, calculate the norm of the scaled x ## and initialize the step bound delta wa3 = diag * x xnorm = self.enorm(wa3) delta = factor * xnorm if delta == 0.0: 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] wj = wa4[j:] ## *** optimization wa4(j:*) wa4[j:] = wj - fj * numpy.sum(fj * wj) / 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[0:n, 0:n] fjac.shape = [n, n] temp = fjac.copy() for i in range(n): temp[:, i] = fjac[:, ipvt[i]] fjac = temp.copy() ## Check for overflow. This should be a cheap test here since FJAC ## has been reduced to a (small) square matrix, and the test is ## O(N^2). # wh = where(finite(fjac) EQ 0, ct) # if ct GT 0 then goto, FAIL_OVERFLOW ## Compute the norm of the scaled gradient catch_msg = "computing the scaled gradient" gnorm = 0.0 if self.fnorm != 0: for j in range(n): l = ipvt[j] if wa2[l] != 0: sum = numpy.sum(fjac[0 : j + 1, j] * qtf[0 : j + 1]) / self.fnorm gnorm = max([gnorm, abs(sum / wa2[l])]) ## Test for convergence of the gradient norm if gnorm <= gtol: self.status = 4 return ## Rescale if necessary if rescale == 0: diag = numpy.choose(diag > wa2, (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 = 1.0 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 = 1.0 if qanylim: ## Do not allow any steps out of bounds catch_msg = "checking for a step out of bounds" if nlpeg > 0: numpy.put(wa1, whlpeg, numpy.clip(numpy.take(wa1, whlpeg), 0.0, max(wa1))) if nupeg > 0: numpy.put(wa1, whupeg, numpy.clip(numpy.take(wa1, whupeg), min(wa1), 0.0)) dwa1 = abs(wa1) > machep (whl,) = numpy.nonzero(((dwa1 != 0.0) & qllim) & ((x + wa1) < llim)) if len(whl) > 0: t = (numpy.take(llim, whl) - numpy.take(x, whl)) / numpy.take(wa1, whl) alpha = min(alpha, min(t)) (whu,) = numpy.nonzero(((dwa1 != 0.0) & qulim) & ((x + wa1) > ulim)) if len(whu) > 0: t = (numpy.take(ulim, whu) - numpy.take(x, whu)) / numpy.take(wa1, whu) alpha = min(alpha, min(t)) ## Obey any max step values. if qminmax: nwa1 = wa1 * alpha (whmax,) = numpy.nonzero((qmax != 0.0) & (maxstep > 0)) if len(whmax) > 0: mrat = max(numpy.take(nwa1, whmax) / numpy.take(maxstep, whmax)) if mrat > 1: alpha = alpha / mrat ## Scale the resulting vector wa1 = wa1 * alpha wa2 = x + wa1 ## Adjust the final output values. If the step put us exactly ## on a boundary, make sure it is exact. (wh,) = numpy.nonzero((qulim != 0.0) & (wa2 >= ulim * (1 - machep))) if len(wh) > 0: numpy.put(wa2, wh, numpy.take(ulim, wh)) (wh,) = numpy.nonzero((qllim != 0.0) & (wa2 <= llim * (1 + machep))) if len(wh) > 0: numpy.put(wa2, wh, numpy.take(llim, wh)) # endelse wa3 = diag * wa1 pnorm = self.enorm(wa3) ## On the first iteration, adjust the initial step bound if self.niter == 1: delta = min([delta, pnorm]) numpy.put(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 "' + str(fcn) + '"' return fnorm1 = self.enorm(wa4) ## Compute the scaled actual reduction catch_msg = "computing convergence criteria" actred = -1.0 if (0.1 * fnorm1) < self.fnorm: actred = -((fnorm1 / self.fnorm) ** 2) + 1.0 ## Compute the scaled predicted reduction and the scaled directional ## derivative for j in range(n): wa3[j] = 0 wa3[0 : j + 1] = wa3[0 : j + 1] + fjac[0 : j + 1, j] * wa1[ipvt[j]] ## Remember, alpha is the fraction of the full LM step actually ## taken temp1 = self.enorm(alpha * wa3) / self.fnorm temp2 = (numpy.sqrt(alpha * par) * pnorm) / self.fnorm prered = temp1 * temp1 + (temp2 * temp2) / 0.5 dirder = -(temp1 * temp1 + temp2 * temp2) ## Compute the ratio of the actual to the predicted reduction. ratio = 0.0 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, pnorm / 0.1]) par = par / temp else: if (par == 0) or (ratio >= 0.75): delta = pnorm / 0.5 par = 0.5 * par ## Test for successful iteration if ratio >= 0.0001: ## Successful iteration. Update x, fvec, and their norms x = wa2 wa2 = diag * x fvec = wa4 xnorm = self.enorm(wa2) self.fnorm = fnorm1 self.niter = self.niter + 1 ## Tests for convergence if (abs(actred) <= ftol) and (prered <= ftol) and (0.5 * ratio <= 1): self.status = 1 if delta <= xtol * xnorm: self.status = 2 if (abs(actred) <= ftol) and (prered <= ftol) and (0.5 * ratio <= 1) 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 (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 - SKIP FOR NOW ##wh = where(finite(wa1) EQ 0 OR finite(wa2) EQ 0 OR finite(x) EQ 0, ct) ##if ct GT 0 OR finite(ratio) EQ 0 then begin ## 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: numpy.put(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 = self.enorm(fvec) if (self.fnorm is not None) and (fnorm1 is not None): self.fnorm = max([self.fnorm, fnorm1]) self.fnorm = self.fnorm**2.0 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 = numpy.shape(fjac) 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[0:n, 0:n], ipvt[0:n]) cv.shape = [n, n] nn = len(xall) ## Fill in actual covariance matrix, accounting for fixed ## parameters. self.covar = numpy.zeros([nn, nn], numpy.float64) for i in range(n): indices = ifree + ifree[i] * n numpy.put(self.covar, indices, cv[:, i]) ## Compute errors in parameters catch_msg = "computing parameter errors" self.perror = numpy.zeros(nn, numpy.float64) d = numpy.diagonal(self.covar).copy() (wh,) = numpy.nonzero(d >= 0) if len(wh) > 0: numpy.put(self.perror, wh, numpy.sqrt(numpy.take(d, wh))) return ## 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 self.debug: print("Entering defiter...") if quiet: return if fnorm is None: [status, fvec] = self.call(fcn, x, functkw) fnorm = self.enorm(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 ## DO_ITERSTOP: ## if keyword_set(iterstop) then begin ## k = get_kbrd(0) ## if k EQ string(byte(7)) then begin ## message, 'WARNING: minimization not complete', /info ## print, 'Do you want to terminate this procedure? (y/n)', $ ## format='(A,$)' ## k = '' ## read, k ## if strupcase(strmid(k,0,1)) EQ 'Y' then begin ## message, 'WARNING: Procedure is terminating.', /info ## mperr = -1 ## endif ## endif ## endif ## 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 self.debug: print("Entering parinfo...") if (n == 0) and (parinfo is not None): n = len(parinfo) if n == 0: values = default return values values = [] for i in range(n): if parinfo is not None and key in parinfo[i]: values.append(parinfo[i][key]) else: values.append(default) # Convert to numeric arrays if possible test = default if isinstance(default, list): test = default[0] if isinstance(test, int): values = numpy.asarray(values, numpy.int32) elif isinstance(test, float): values = numpy.asarray(values, numpy.float64) return values ## Call user function or procedure, with _EXTRA or not, with ## derivatives or not. def call(self, fcn, x, functkw, fjac=None): if self.debug: print("Entering call...") if self.qanytied: x = self.tie(x, self.ptied) self.nfev = self.nfev + 1 if fjac is None: [status, f] = fcn(x, fjac=fjac, **functkw) if self.damp > 0: ## Apply the damping if requested. This replaces the residuals ## with their hyperbolic tangent. Thus residuals larger than ## DAMP are essentially clipped. f = numpy.tanh(f / self.damp) return [status, f] else: return fcn(x, fjac=fjac, **functkw) def enorm(self, vec): if self.debug: print("Entering enorm...") ## NOTE: it turns out that, for systems that have a lot of data ## points, this routine is a big computing bottleneck. The extended ## computations that need to be done cannot be effectively ## vectorized. The introduction of the FASTNORM configuration ## parameter allows the user to select a faster routine, which is ## based on TOTAL() alone. # Very simple-minded sum-of-squares if self.fastnorm: ans = numpy.sqrt(numpy.sum(vec * vec)) else: agiant = self.machar.rgiant / len(vec) adwarf = self.machar.rdwarf * len(vec) ## This is hopefully a compromise between speed and robustness. ## Need to do this because of the possibility of over- or underflow. mx = max(vec) mn = min(vec) mx = max(abs(mx), abs(mn)) if mx == 0: return vec[0] * 0.0 if mx > agiant or mx < adwarf: ans = mx * numpy.sqrt(numpy.sum((vec / mx) * (vec / mx))) else: ans = numpy.sqrt(numpy.sum(vec * vec)) return ans 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, ): if self.debug: print("Entering fdjac2...") machep = self.machar.machep if epsfcn is None: epsfcn = machep if xall is None: xall = x if ifree is None: ifree = numpy.arange(len(xall)) if step is None: step = x * 0.0 nall = len(xall) eps = numpy.sqrt(max([epsfcn, machep])) m = len(fvec) n = len(x) ## Compute analytical derivative if requested if autoderivative == 0: mperr = 0 fjac = numpy.zeros(nall, numpy.float64) numpy.put(fjac, ifree, 1.0) ## Specify which parameters need derivatives [status, fp, pderiv] = self.call(fcn, xall, functkw, fjac=fjac) fjac = pderiv if fjac.shape != (m, nall): print("ERROR: Derivative matrix was not computed properly.") return None ## This definition is c1onsistent with CURVEFIT ## Sign error found (thanks Jesus Fernandez ) fjac = -fjac ## Select only the free parameters if len(ifree) < nall: fjac = fjac[:, ifree] fjac.shape = [m, n] return fjac fjac = numpy.zeros([m, n], numpy.float64) h = eps * abs(x) ## if STEP is given, use that if step is not None: stepi = numpy.take(step, ifree) (wh,) = numpy.nonzero(stepi > 0) if len(wh) > 0: numpy.put(h, wh, numpy.take(stepi, wh)) ## if relative step is given, use that if len(dstep) > 0: dstepi = numpy.take(dstep, ifree) (wh,) = numpy.nonzero(dstepi > 0) if len(wh) > 0: numpy.put(h, wh, abs(numpy.take(dstepi, wh) * numpy.take(x, wh))) ## In case any of the step values are zero (wh,) = numpy.nonzero(h == 0) if len(wh) > 0: numpy.put(h, wh, eps) ## Reverse the sign of the step if we are up against the parameter ## limit, or if the user requested it. mask = dside == -1 if len(ulimited) > 0 and len(ulimit) > 0: mask = mask | (ulimited & (x > ulimit - h)) (wh,) = numpy.nonzero(mask) if len(wh) > 0: numpy.put(h, wh, -numpy.take(h, wh)) ## Loop through parameters, computing the derivative for each for j in range(n): xp = xall.copy() xp[ifree[j]] = xp[ifree[j]] + h[j] [status, fp] = self.call(fcn, xp, functkw) if status < 0: return None if abs(dside[j]) <= 1: ## COMPUTE THE ONE-SIDED DERIVATIVE ## Note optimization fjac(0:*,j) fjac[0:, j] = (fp - fvec) / h[j] else: ## COMPUTE THE TWO-SIDED DERIVATIVE xp[ifree[j]] = xall[ifree[j]] - h[j] mperr = 0 [status, fm] = self.call(fcn, xp, functkw) if status < 0: return None ## Note optimization fjac(0:*,j) fjac[0:, 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 # # ********** # 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): if self.debug: print("Entering qrfac...") machep = self.machar.machep sz = numpy.shape(a) m = sz[0] n = sz[1] ## Compute the initial column norms and initialize arrays acnorm = numpy.zeros(n, numpy.float64) for j in range(n): acnorm[j] = self.enorm(a[:, j]) rdiag = acnorm.copy() wa = rdiag.copy() ipvt = numpy.arange(n) ## Reduce a to r with householder transformations minmn = min([m, n]) for j in range(minmn): if pivot != 0: ## Bring the column of largest norm into the pivot position rmax = max(rdiag[j:]) (kmax,) = numpy.nonzero(rdiag[j:] == rmax) ct = len(kmax) kmax = kmax + j if ct > 0: kmax = kmax[0] ## 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: temp = ipvt[j] ipvt[j] = ipvt[kmax] ipvt[kmax] = temp 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 = self.enorm(ajj) if ajnorm == 0: break if a[j, j] < 0: ajnorm = -ajnorm ajj = ajj / ajnorm ajj[0] = ajj[0] + 1 ## *** Note optimization a(j:*,j) 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] ajk = a[j:, lk] ## *** Note optimization a(j:*,lk) ## (corrected 20 Jul 2000) if a[j, lj] != 0: a[j:, lk] = ajk - ajj * numpy.sum(ajk * ajj) / a[j, lj] if (pivot != 0) and (rdiag[k] != 0): temp = a[j, lk] / rdiag[k] rdiag[k] = rdiag[k] * numpy.sqrt(max((1.0 - temp**2), 0.0)) temp = rdiag[k] / wa[k] if (0.05 * temp * temp) <= machep: rdiag[k] = self.enorm(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): if self.debug: print("Entering qrsolv...") sz = numpy.shape(r) 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 = numpy.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 = 0.0 for k in range(j, n): if sdiag[k] == 0: break if abs(r[k, k]) < abs(sdiag[k]): cotan = r[k, k] / sdiag[k] sine = 0.5 / numpy.sqrt(0.25 + 0.25 * cotan * cotan) cosine = sine * cotan else: tang = sdiag[k] / r[k, k] cosine = 0.5 / numpy.sqrt(0.25 + 0.25 * tang * tang) 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] = -sine * r[k + 1 : n, k] + cosine * sdiag[k + 1 : n] 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 (wh,) = numpy.nonzero(sdiag == 0) if len(wh) > 0: nsing = wh[0] wa[nsing:] = 0 if nsing >= 1: wa[nsing - 1] = wa[nsing - 1] / sdiag[nsing - 1] ## Degenerate case ## *** Reverse loop *** for j in range(nsing - 2, -1, -1): sum = numpy.sum(r[j + 1 : nsing, j] * wa[j + 1 : nsing]) wa[j] = (wa[j] - sum) / sdiag[j] ## Permute the components of z back to components of x # if (self.debug) print("wa = ",wa, ipvt) numpy.put(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): if self.debug: print("Entering lmpar...") dwarf = self.machar.minnum sz = numpy.shape(r) 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() (wh,) = numpy.nonzero(numpy.diagonal(r) == 0) 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] = wa1[j] / r[j, j] if j - 1 >= 0: wa1[0:j] = wa1[0:j] - r[0:j, j] * wa1[j] ## Note: ipvt here is a permutation array numpy.put(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 = self.enorm(wa2) fp = dxnorm - delta if fp <= 0.1 * delta: return [r, 0.0, 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 = 0.0 if nsing >= n: wa1 = numpy.take(diag, ipvt) * numpy.take(wa2, ipvt) / dxnorm wa1[0] = wa1[0] / r[0, 0] ## Degenerate case for j in range(1, n): ## Note "1" here, not zero sum = numpy.sum(r[0:j, j] * wa1[0:j]) wa1[j] = (wa1[j] - sum) / r[j, j] temp = self.enorm(wa1) parl = ((fp / delta) / temp) / temp ## Calculate an upper bound, paru, for the zero of the function for j in range(n): sum = numpy.sum(r[0 : j + 1, j] * qtb[0 : j + 1]) wa1[j] = sum / diag[ipvt[j]] gnorm = self.enorm(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 = numpy.sqrt(par) wa1 = temp * diag [r, x, sdiag] = self.qrsolv(r, ipvt, wa1, qtb, sdiag) wa2 = diag * x dxnorm = self.enorm(wa2) temp = fp fp = dxnorm - delta if (abs(fp) <= 0.1 * delta) or ((parl == 0) and (fp <= temp) and (temp < 0)) or (iter == 10): break ## Compute the newton correction wa1 = numpy.take(diag, ipvt) * numpy.take(wa2, ipvt) / dxnorm for j in range(n - 1): wa1[j] = wa1[j] / sdiag[j] wa1[j + 1 : n] = wa1[j + 1 : n] - r[j + 1 : n, j] * wa1[j] wa1[n - 1] = wa1[n - 1] / sdiag[n - 1] ## Degenerate case temp = self.enorm(wa1) parc = ((fp / delta) / temp) / temp ## 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 self.debug: print("Entering tie...") if ptied is None: return for i in range(len(ptied)): if ptied[i] == "": continue cmd = "p[" + str(i) + "] = " + ptied[i] exec(cmd) 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=1.0e-14): if self.debug: print("Entering calc_covar...") if numpy.rank(rr) != 2: print("ERROR: r must be a two-dimensional matrix") return -1 s = numpy.shape(rr) n = s[0] if s[0] != s[1]: print("ERROR: r must be a square matrix") return -1 if ipvt is None: ipvt = numpy.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 * abs(r[0, 0]) for k in range(n): if abs(r[k, k]) <= tolr: break r[k, k] = 1.0 / r[k, k] for j in range(k): temp = r[k, k] * r[j, k] r[j, k] = 0.0 r[0 : j + 1, k] = r[0 : j + 1, k] - temp * r[0 : 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[0 : j + 1, j] = r[0 : j + 1, j] + temp * r[0 : j + 1, k] temp = r[k, k] r[0 : k + 1, k] = temp * r[0 : k + 1, k] ## For the full lower triangle of the covariance matrix ## in the strict lower triangle or and in wa wa = numpy.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] = 0.0 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[0 : j + 1, j] = r[j, 0 : j + 1] r[j, j] = wa[j] return r class machar: def __init__(self, double=1): if double == 0: self.machep = 1.19209e-007 self.maxnum = 3.40282e038 self.minnum = 1.17549e-038 self.maxgam = 171.624376956302725 else: self.machep = 2.2204460e-016 self.maxnum = 1.7976931e308 self.minnum = 2.2250739e-308 self.maxgam = 171.624376956302725 self.maxlog = numpy.log(self.maxnum) self.minlog = numpy.log(self.minnum) self.rdwarf = numpy.sqrt(self.minnum * 1.5) * 10 self.rgiant = numpy.sqrt(self.maxnum) * 0.1