singular_values, rcond] : list These values are only returned if ``full == True`` - residuals -- sum of squared residuals of the least squares fit - rank -- the numerical rank of the scaled Vandermonde matrix - singular_values -- singular values of the scaled Vandermonde matrix - rcond -- value of `rcond`. For more details, see `numpy.linalg.lstsq`. Warns ----- RankWarning The rank of the coefficient matrix in the least-squares fit is deficient. The warning is only raised if ``full == False``. The warnings can be turned off by >>> import warnings >>> warnings.simplefilter('ignore', np.RankWarning) See Also -------- numpy.polynomial.polynomial.polyfit numpy.polynomial.chebyshev.chebfit numpy.polynomial.laguerre.lagfit numpy.polynomial.hermite.hermfit numpy.polynomial.hermite_e.hermefit legval : Evaluates a Legendre series. legvander : Vandermonde matrix of Legendre series. legweight : Legendre weight function (= 1). numpy.linalg.lstsq : Computes a least-squares fit from the matrix. scipy.interpolate.UnivariateSpline : Computes spline fits. Notes ----- The solution is the coefficients of the Legendre series `p` that minimizes the sum of the weighted squared errors .. math:: E = \sum_j w_j^2 * |y_j - p(x_j)|^2, where :math:`w_j` are the weights. This problem is solved by setting up as the (typically) overdetermined matrix equation .. math:: V(x) * c = w * y, where `V` is the weighted pseudo Vandermonde matrix of `x`, `c` are the coefficients to be solved for, `w` are the weights, and `y` are the observed values. This equation is then solved using the singular value decomposition of `V`. If some of the singular values of `V` are so small that they are neglected, then a `RankWarning` will be issued. This means that the coefficient values may be poorly determined. Using a lower order fit will usually get rid of the warning. The `rcond` parameter can also be set to a value smaller than its default, but the resulting fit may be spurious and have large contributions from roundoff error. Fits using Legendre series are usually better conditioned than fits using power series, but much can depend on the distribution of the sample points and the smoothness of the data. If the quality of the fit is inadequate splines may be a good alternative. References ---------- .. [1] Wikipedia, "Curve fitting", https://en.wikipedia.org/wiki/Curve_fitting Examples -------- )