Minimize the sum of squares of a set of equations.
Description:
Return the point which minimizes the sum of squares of M (non-linear) equations in N unknowns given a starting estimate, x0, using a modification of the Levenberg-Marquardt algorithm.
- x = arg min(sum(func(y)**2,axis=0))
- y
Inputs:
- func – A Python function or method which takes at least one
- (possibly length N vector) argument and returns M floating point numbers.
x0 – The starting estimate for the minimization. args – Any extra arguments to func are placed in this tuple. Dfun – A function or method to compute the Jacobian of func with
derivatives across the rows. If this is None, the Jacobian will be estimated.full_output – non-zero to return all optional outputs. col_deriv – non-zero to specify that the Jacobian function
computes derivatives down the columns (faster, because there is no transpose operation).
- warning – True to print a warning message when the call is
- unsuccessful; False to suppress the warning message.
Outputs: (x, {cov_x, infodict, mesg}, ier)
- x – the solution (or the result of the last iteration for an
- unsuccessful call.
- cov_x – uses the fjac and ipvt optional outputs to construct an
- estimate of the covariance matrix of the solution. None if a singular matrix encountered (indicates infinite covariance in some direction).
- infodict – a dictionary of optional outputs with the keys:
‘nfev’ : the number of function calls ‘fvec’ : the function evaluated at the output ‘fjac’ : A permutation of the R matrix of a QR
factorization of the final approximate Jacobian matrix, stored column wise. Together with ipvt, the covariance of the estimate can be approximated.
- ‘ipvt’ : an integer array of length N which defines
- a permutation matrix, p, such that fjac*p = q*r, where r is upper triangular with diagonal elements of nonincreasing magnitude. Column j of p is column ipvt(j) of the identity matrix.
‘qtf’ : the vector (transpose(q) * fvec).
mesg – a string message giving information about the cause of failure. ier – an integer flag. If it is equal to 1, 2, 3 or 4, the
solution was found. Otherwise, the solution was not found. In either case, the optional output variable ‘mesg’ gives more information.
Extended Inputs:
ftol – Relative error desired in the sum of squares. xtol – Relative error desired in the approximate solution. gtol – Orthogonality desired between the function vector
and the columns of the Jacobian.
- maxfev – The maximum number of calls to the function. If zero,
- then 100*(N+1) is the maximum where N is the number of elements in x0.
- epsfcn – A suitable step length for the forward-difference
- approximation of the Jacobian (for Dfun=None). 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.
- factor – A parameter determining the initial step bound
- (factor * || diag * x||). Should be in interval (0.1,100).
- diag – A sequency of N positive entries that serve as a
- scale factors for the variables.
Remarks:
“leastsq” is a wrapper around MINPACK’s lmdif and lmder algorithms.
See also:
scikits.openopt, which offers a unified syntax to call this and other solvers
- fmin, fmin_powell, fmin_cg,
- fmin_bfgs, fmin_ncg – multivariate local optimizers
- fmin_l_bfgs_b, fmin_tnc,
- fmin_cobyla – constrained multivariate optimizers
anneal, brute – global optimizers
fminbound, brent, golden, bracket – local scalar minimizers
fsolve – n-dimenstional root-finding
brentq, brenth, ridder, bisect, newton – one-dimensional root-finding
fixed_point – scalar and vector fixed-point finder