hat the
        reduced `chisq` for the optimal parameters `popt` when using the
        *scaled* `sigma` equals unity. In other words, `sigma` is scaled to
        match the sample variance of the residuals after the fit. Default is False.
        Mathematically,
        ``pcov(absolute_sigma=False) = pcov(absolute_sigma=True) * chisq(popt)/(M-N)``
    check_finite : bool, optional
        If True, check that the input arrays do not contain nans of infs,
        and raise a ValueError if they do. Setting this parameter to
        False may silently produce nonsensical results if the input arrays
        do contain nans. Default is True if `nan_policy` is not specified
        explicitly and False otherwise.
    bounds : 2-tuple of array_like or `Bounds`, optional
        Lower and upper bounds on parameters. Defaults to no bounds.
        There are two ways to specify the bounds:

            - Instance of `Bounds` class.

            - 2-tuple of array_like: Each element of the tuple must be either
              an array with the length equal to the number of parameters, or a
              scalar (in which case the bound is taken to be the same for all
              parameters). Use ``np.inf`` with an appropriate sign to disable
              bounds on all or some parameters.

    method : {'lm', 'trf', 'dogbox'}, optional
        Method to use for optimization. See `least_squares` for more details.
        Default is 'lm' for unconstrained problems and 'trf' if `bounds` are
        provided. The method 'lm' won't work when the number of observations
        is less than the number of variables, use 'trf' or 'dogbox' in this
        case.

        .. versionadded:: 0.17
    jac : callable, string or None, optional
        Function with signature ``jac(x, ...)`` which computes the Jacobian
        matrix of the model function with respect to parameters as a dense
        array_like structure. It will be scaled according to provided `sigma`.
        If None (default), the Jacobian will be estimated numerically.
        String keywords for 'trf' and 'dogbox' methods can be used to select
        a finite difference scheme, see `least_squares`.

        .. versionadded:: 0.18
    full_output : boolean, optional
        If True, this function returns additioal information: `infodict`,
        `mesg`, and `ier`.

        .. versionadded:: 1.9
    nan_policy : {'raise', 'omit', None}, optional
        Defines how to handle when input contains nan.
        The following options are available (default is None):

          * 'raise': throws an error
          * 'omit': performs the calculations ignoring nan values
          * None: no special handling of NaNs is performed
            (except what is done by check_finite); the behavior when NaNs
            are present is implementation-dependent and may change.

        Note that if this value is specified explicitly (not None),
        `check_finite` will be set as False.

        .. versionadded:: 1.11
    **kwargs
        Keyword arguments passed to `leastsq` for ``method='lm'`` or
        `least_squares` otherwise.

    Returns
    -------
    popt : array
        Optimal values for the parameters so that the sum of the squared
        residuals of ``f(xdata, *popt) - ydata`` is minimized.
    pcov : 2-D array
        The estimated approximate covariance of popt. The diagonals provide
        the variance of the parameter estimate. To compute one standard
        deviation errors on the parameters, use
        ``perr = np.sqrt(np.diag(pcov))``. Note that the relationship between
        `cov` and parameter error estimates is derived based on a linear
        approximation to the model function around the optimum [1].
        When this approximation becomes inaccurate, `cov` may not provide an
        accurate measure of uncertainty.

        How the `sigma` parameter affects the estimated covariance
        depends on `absolute_sigma` argument, as described above.

        If the Jacobian matrix at the solution doesn't have a full rank, then
        'lm' method returns a matrix filled with ``np.inf``, on the other hand
        'trf'  and 'dogbox' methods use Moore-Penrose pseudoinverse to compute
        the covariance matrix. Covariance matrices with large condition numbers
        (e.g. computed with `numpy.linalg.cond`) may indicate that results are
        unreliable.
    infodict : dict (returned only if `full_output` is True)
        a dictionary of optional outputs with the keys:

        ``nfev``
            The number of function calls. Methods 'trf' and 'dogbox' do not
            count function calls for numerical Jacobian approximation,
            as opposed to 'lm' method.
        ``fvec``
            The residual values evaluated at the solution, for a 1-D `sigma`
            this is ``(f(x, *popt) - ydata)/sigma``.
        ``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.
            Method 'lm' only provides this information.
        ``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.
            Method 'lm' only provides this information.
        ``qtf``
            The vector (transpose(q) * fvec).
            Method 'lm' only provides this information.

        .. versionadded:: 1.9
    mesg : str (returned only if `full_output` is True)
        A string message giving information about the solution.

        .. versionadded:: 1.9
    ier : int (returnned only if `full_output` is True)
        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.

        .. versionadded:: 1.9

    Raises
    ------
    ValueError
        if either `ydata` or `xdata` contain NaNs, or if incompatible options
        are used.

    RuntimeError
        if the least-squares minimization fails.

    OptimizeWarning
        if covariance of the parameters can not be estimated.

    See Also
    --------
    least_squares : Minimize the sum of squares of nonlinear functions.
    scipy.stats.linregress : Calculate a linear least squares regression for
                             two sets of measurements.

    Notes
    -----
    Users should ensure that inputs `xdata`, `ydata`, and the output of `f`
    are ``float64``, or else the optimization may return incorrect results.

    With ``method='lm'``, the algorithm uses the Levenberg-Marquardt algorithm
    through `leastsq`. Note that this algorithm can only deal with
    unconstrained problems.

    Box constraints can be handled by methods 'trf' and 'dogbox'. Refer to
    the docstring of `least_squares` for more information.

    References
    ----------
    [1] K. Vugrin et al. Confidence region estimation techniques for nonlinear
        regression in groundwater flow: Three case studies. Water Resources
        Research, Vol. 43, W03423, :doi:`10.1029/2005WR004804`

    Examples
    --------
    >>> import numpy as np
    >>> import matplotlib.pyplot as plt
    >>> from scipy.optimize import curve_fit

    >>> def func(x, a, b, c):
    ...     return a * np.exp(-b * x) + c

    Define the data to be fit with some noise:

    >>> xdata = np.linspace(0, 4, 50)
    >>> y = func(xdata, 2.5, 1.3, 0.5)
    >>> rng = np.random.default_rng()
    >>> y_noise = 0.2 * rng.normal(size=xdata.size)
    >>> ydata = y + y_noise
    >>> plt.plot(xdata, ydata, 'b-', label='data')

    Fit for the parameters a, b, c of the function `func`:

    >>> popt, pcov = curve_fit(func, xdata, ydata)
    >>> popt
    array([2.56274217, 1.37268521, 0.47427475])
    >>> plt.plot(xdata, func(xdata, *popt), 'r-',
    ...          label='fit: a=%5.3f, b=%5.3f, c=%5.3f' % tuple(popt))

    Constrain the optimization to the region of ``0 <= a <= 3``,
    ``0 <= b <= 1`` and ``0 <= c <= 0.5``:

    >>> popt, pcov = curve_fit(func, xdata, ydata, bounds=(0, [3., 1., 0.5]))
    >>> popt
    array([2.43736712, 1.        , 0.34463856])
    >>> plt.plot(xdata, func(xdata, *popt), 'g--',
    ...          label='fit: a=%5.3f, b=%5.3f, c=%5.3f' % tuple(popt))

    >>> plt.xlabel('x')
    >>> plt.ylabel('y')
    >>> plt.legend()
    >>> plt.show()

    For reliable results, the model `func` should not be overparametrized;
    redundant parameters can cause unreliable covariance matrices and, in some
    cases, poorer quality fits. As a quick check of whether the model may be
    overparameterized, calculate the condition number of the covariance matrix:

    >>> np.linalg.cond(pcov)
    34.571092161547405  # may vary

    The value is small, so it does not raise much concern. If, however, we were
    to add a fourth parameter ``d`` to `func` with the same effect as ``a``:

    >>> def func(x, a, b, c, d):
    ...     return a * d * np.exp(-b * x) + c  # a and d are redundant
    >>> popt, pcov = curve_fit(func, xdata, ydata)
    >>> np.linalg.cond(pcov)
    1.13250718925596e+32  # may vary

    Such a large value is cause for concern. The diagonal elements of the
    covariance matrix, which is related to uncertainty of the fit, gives more
    information:

    >>> np.diag(pcov)
    array([1.48814742e+29, 3.78596560e-02, 5.39253738e-03, 2.76417220e+28])  # may vary

    Note that the first and last terms are much larger than the other elements,
    suggesting that the optimal values of these parameters are ambiguous and
    that only one of these parameters is needed in the model.

    NrV