leastsq(func, x0, args=(), Dfun=None, full_output=0, col_deriv=0, ftol=1.49012e-08, xtol=1.49012e-08, gtol=0.0, maxfev=0, epsfcn=None, factor=100, diag=None)¶
Minimize the sum of squares of a set of equations.
x = arg min(sum(func(y)**2,axis=0)) y
- func : callable
should take at least one (possibly length N vector) argument and returns M floating point numbers. It must not return NaNs or fitting might fail.
- x0 : ndarray
The starting estimate for the minimization.
- args : tuple, optional
Any extra arguments to func are placed in this tuple.
- Dfun : callable, optional
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 : bool, optional
non-zero to return all optional outputs.
- col_deriv : bool, optional
non-zero to specify that the Jacobian function computes derivatives down the columns (faster, because there is no transpose operation).
- ftol : float, optional
Relative error desired in the sum of squares.
- xtol : float, optional
Relative error desired in the approximate solution.
- gtol : float, optional
Orthogonality desired between the function vector and the columns of the Jacobian.
- maxfev : int, optional
The maximum number of calls to the function. If Dfun is provided then the default maxfev is 100*(N+1) where N is the number of elements in x0, otherwise the default maxfev is 200*(N+1).
- epsfcn : float, optional
A variable used in determining a suitable step length for the forward- difference approximation of the Jacobian (for Dfun=None). Normally the actual step length will be sqrt(epsfcn)*x If epsfcn is less than the machine precision, it is assumed that the relative errors are of the order of the machine precision.
- factor : float, optional
A parameter determining the initial step bound (
factor * || diag * x||). Should be in interval
- diag : sequence, optional
N positive entries that serve as a scale factors for the variables.
- x : ndarray
The solution (or the result of the last iteration for an unsuccessful call).
- cov_x : ndarray
Uses the fjac and ipvt optional outputs to construct an estimate of the jacobian around the solution. None if a singular matrix encountered (indicates very flat curvature in some direction). This matrix must be multiplied by the residual variance to get the covariance of the parameter estimates – see curve_fit.
- infodict : dict
a dictionary of optional outputs with the key s:
The number of function calls
The function evaluated at the output
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.
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.
The vector (transpose(q) * fvec).
- mesg : str
A string message giving information about the cause of failure.
- ier : int
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.
“leastsq” is a wrapper around MINPACK’s lmdif and lmder algorithms.
cov_x is a Jacobian approximation to the Hessian of the least squares objective function. This approximation assumes that the objective function is based on the difference between some observed target data (ydata) and a (non-linear) function of the parameters f(xdata, params)
func(params) = ydata - f(xdata, params)
so that the objective function is
min sum((ydata - f(xdata, params))**2, axis=0) params
The solution, x, is always a 1D array, regardless of the shape of x0, or whether x0 is a scalar.