M) or (M,M,K) Present only if ``full == False`` and ``cov == True``. The covariance matrix of the polynomial coefficient estimates. The diagonal of this matrix are the variance estimates for each coefficient. If y is a 2-D array, then the covariance matrix for the `k`-th data set are in ``V[:,:,k]`` 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 -------- polyval : Compute polynomial values. linalg.lstsq : Computes a least-squares fit. scipy.interpolate.UnivariateSpline : Computes spline fits. Notes ----- The solution minimizes the squared error .. math:: E = \sum_{j=0}^k |p(x_j) - y_j|^2 in the equations:: x[0]**n * p[0] + ... + x[0] * p[n-1] + p[n] = y[0] x[1]**n * p[0] + ... + x[1] * p[n-1] + p[n] = y[1] ... x[k]**n * p[0] + ... + x[k] * p[n-1] + p[n] = y[k] The coefficient matrix of the coefficients `p` is a Vandermonde matrix. `polyfit` issues a `RankWarning` when the least-squares fit is badly conditioned. This implies that the best fit is not well-defined due to numerical error. The results may be improved by lowering the polynomial degree or by replacing `x` by `x` - `x`.mean(). The `rcond` parameter can also be set to a value smaller than its default, but the resulting fit may be spurious: including contributions from the small singular values can add numerical noise to the result. Note that fitting polynomial coefficients is inherently badly conditioned when the degree of the polynomial is large or the interval of sample points is badly centered. The quality of the fit should always be checked in these cases. When polynomial fits are not satisfactory, splines may be a good alternative. References ---------- .. [1] Wikipedia, "Curve fitting", https://en.wikipedia.org/wiki/Curve_fitting .. [2] Wikipedia, "Polynomial interpolation", https://en.wikipedia.org/wiki/Polynomial_interpolation Examples -------- >>> import warnings >>> x = np.array([0.0, 1.0, 2.0, 3.0, 4.0, 5.0]) >>> y = np.array([0.0, 0.8, 0.9, 0.1, -0.8, -1.0]) >>> z = np.polyfit(x, y, 3) >>> z array([ 0.08703704, -0.81349206, 1.69312169, -0.03968254]) # may vary It is convenient to use `poly1d` objects for dealing with polynomials: >>> p = np.poly1d(z) >>> p(0.5) 0.6143849206349179 # may vary >>> p(3.5) -0.34732142857143039 # may vary >>> p(10) 22.579365079365115 # may vary High-order polynomials may oscillate wildly: >>> with warnings.catch_warnings(): ... warnings.simplefilter('ignore', np.RankWarning) ... p30 = np.poly1d(np.polyfit(x, y, 30)) ... >>> p30(4) -0.80000000000000204 # may vary >>> p30(5) -0.99999999999999445 # may vary >>> p30(4.5) -0.10547061179440398 # may vary Illustration: >>> import matplotlib.pyplot as plt >>> xp = np.linspace(-2, 6, 100) >>> _ = plt.plot(x, y, '.', xp, p(xp), '-', xp, p30(xp), '--') >>> plt.ylim(-2,2) (-2, 2) >>> plt.show() r/