scipy.optimize.leastsq#

scipy.optimize.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)[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 returns M floating point numbers. It must not return NaNs or fitting might fail. M must be greater than or equal to N.
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: non-zero to return all optional outputs.
col_derivbool, optional: non-zero to 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.

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).

mesgstr

A string message giving information about the cause of failure.

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)

scipy.optimize.golden

scipy.optimize.fsolve