scipy.optimize.leastsq#
- scipy.optimize.leastsq(func, x0, args=(), Dfun=None, full_output=False, col_deriv=False, ftol=1.49012e-08, xtol=1.49012e-08, gtol=0.0, maxfev=0, epsfcn=None, factor=100, diag=None)[source]#
Minimize the sum of squares of a set of equations.
x = arg min(sum(func(y)**2,axis=0)) y
- Parameters:
- funccallable
Should take at least one (possibly length
N
vector) argument and returnsM
floating point numbers. It must not return NaNs or fitting might fail.M
must be greater than or equal toN
.- x0ndarray
The starting estimate for the minimization.
- argstuple, optional
Any extra arguments to func are placed in this tuple.
- Dfuncallable, 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_outputbool, optional
If
True
, return all optional outputs (not just x and ier).- col_derivbool, optional
If
True
, specify that the Jacobian function computes derivatives down the columns (faster, because there is no transpose operation).- ftolfloat, optional
Relative error desired in the sum of squares.
- xtolfloat, optional
Relative error desired in the approximate solution.
- gtolfloat, optional
Orthogonality desired between the function vector and the columns of the Jacobian.
- maxfevint, 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).
- epsfcnfloat, 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.
- factorfloat, optional
A parameter determining the initial step bound (
factor * || diag * x||
). Should be in interval(0.1, 100)
.- diagsequence, optional
N positive entries that serve as a scale factors for the variables.
- Returns:
- xndarray
The solution (or the result of the last iteration for an unsuccessful call).
- cov_xndarray
The inverse of the Hessian. fjac and ipvt are used to construct an estimate of the Hessian. A value of None indicates a singular matrix, which means the curvature in parameters x is numerically flat. To obtain the covariance matrix of the parameters x, cov_x must be multiplied by the variance of the residuals – see curve_fit. Only returned if full_output is
True
.- infodictdict
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).
Only returned if full_output is
True
.- mesgstr
A string message giving information about the cause of failure. Only returned if full_output is
True
.- ierint
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.
See also
least_squares
Newer interface to solve nonlinear least-squares problems with bounds on the variables. See
method='lm'
in particular.
Notes
“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 1-D array, regardless of the shape of x0, or whether x0 is a scalar.
Examples
>>> from scipy.optimize import leastsq >>> def func(x): ... return 2*(x-3)**2+1 >>> leastsq(func, 0) (array([2.99999999]), 1)