diff options
Diffstat (limited to 'numpy/lib/polynomial.py')
| -rw-r--r-- | numpy/lib/polynomial.py | 98 |
1 files changed, 58 insertions, 40 deletions
diff --git a/numpy/lib/polynomial.py b/numpy/lib/polynomial.py index 81af185eb..1f08abf36 100644 --- a/numpy/lib/polynomial.py +++ b/numpy/lib/polynomial.py @@ -110,7 +110,7 @@ def poly(seq_of_zeros): Given a sequence of a polynomial's zeros: >>> np.poly((0, 0, 0)) # Multiple root example - array([1, 0, 0, 0]) + array([1., 0., 0., 0.]) The line above represents z**3 + 0*z**2 + 0*z + 0. @@ -119,14 +119,14 @@ def poly(seq_of_zeros): The line above represents z**3 - z/4 - >>> np.poly((np.random.random(1.)[0], 0, np.random.random(1.)[0])) - array([ 1. , -0.77086955, 0.08618131, 0. ]) #random + >>> np.poly((np.random.random(1)[0], 0, np.random.random(1)[0])) + array([ 1. , -0.77086955, 0.08618131, 0. ]) # random Given a square array object: >>> P = np.array([[0, 1./3], [-1./2, 0]]) >>> np.poly(P) - array([ 1. , 0. , 0.16666667]) + array([1. , 0. , 0.16666667]) Note how in all cases the leading coefficient is always 1. @@ -295,7 +295,7 @@ def polyint(p, m=1, k=None): >>> p = np.poly1d([1,1,1]) >>> P = np.polyint(p) >>> P - poly1d([ 0.33333333, 0.5 , 1. , 0. ]) + poly1d([ 0.33333333, 0.5 , 1. , 0. ]) # may vary >>> np.polyder(P) == p True @@ -310,7 +310,7 @@ def polyint(p, m=1, k=None): 0.0 >>> P = np.polyint(p, 3, k=[6,5,3]) >>> P - poly1d([ 0.01666667, 0.04166667, 0.16666667, 3. , 5. , 3. ]) + poly1d([ 0.01666667, 0.04166667, 0.16666667, 3. , 5. , 3. ]) # may vary Note that 3 = 6 / 2!, and that the constants are given in the order of integrations. Constant of the highest-order polynomial term comes first: @@ -404,7 +404,7 @@ def polyder(p, m=1): >>> np.polyder(p, 3) poly1d([6]) >>> np.polyder(p, 4) - poly1d([ 0.]) + poly1d([0.]) """ m = int(m) @@ -463,9 +463,14 @@ def polyfit(x, y, deg, rcond=None, full=False, w=None, cov=False): w : array_like, shape (M,), optional Weights to apply to the y-coordinates of the sample points. For gaussian uncertainties, use 1/sigma (not 1/sigma**2). - cov : bool, optional - Return the estimate and the covariance matrix of the estimate - If full is True, then cov is not returned. + cov : bool or str, optional + If given and not `False`, return not just the estimate but also its + covariance matrix. By default, the covariance are scaled by + chi2/sqrt(N-dof), i.e., the weights are presumed to be unreliable + except in a relative sense and everything is scaled such that the + reduced chi2 is unity. This scaling is omitted if ``cov='unscaled'``, + as is relevant for the case that the weights are 1/sigma**2, with + sigma known to be a reliable estimate of the uncertainty. Returns ------- @@ -543,32 +548,35 @@ def polyfit(x, y, deg, rcond=None, full=False, w=None, cov=False): 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]) + 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 + 0.6143849206349179 # may vary >>> p(3.5) - -0.34732142857143039 + -0.34732142857143039 # may vary >>> p(10) - 22.579365079365115 + 22.579365079365115 # may vary High-order polynomials may oscillate wildly: - >>> p30 = np.poly1d(np.polyfit(x, y, 30)) - /... RankWarning: Polyfit may be poorly conditioned... + >>> with warnings.catch_warnings(): + ... warnings.simplefilter('ignore', np.RankWarning) + ... p30 = np.poly1d(np.polyfit(x, y, 30)) + ... >>> p30(4) - -0.80000000000000204 + -0.80000000000000204 # may vary >>> p30(5) - -0.99999999999999445 + -0.99999999999999445 # may vary >>> p30(4.5) - -0.10547061179440398 + -0.10547061179440398 # may vary Illustration: @@ -626,21 +634,24 @@ def polyfit(x, y, deg, rcond=None, full=False, w=None, cov=False): # warn on rank reduction, which indicates an ill conditioned matrix if rank != order and not full: msg = "Polyfit may be poorly conditioned" - warnings.warn(msg, RankWarning, stacklevel=2) + warnings.warn(msg, RankWarning, stacklevel=3) if full: return c, resids, rank, s, rcond elif cov: Vbase = inv(dot(lhs.T, lhs)) Vbase /= NX.outer(scale, scale) - # Some literature ignores the extra -2.0 factor in the denominator, but - # it is included here because the covariance of Multivariate Student-T - # (which is implied by a Bayesian uncertainty analysis) includes it. - # Plus, it gives a slightly more conservative estimate of uncertainty. - if len(x) <= order + 2: - raise ValueError("the number of data points must exceed order + 2 " - "for Bayesian estimate the covariance matrix") - fac = resids / (len(x) - order - 2.0) + if cov == "unscaled": + fac = 1 + else: + if len(x) <= order: + raise ValueError("the number of data points must exceed order " + "to scale the covariance matrix") + # note, this used to be: fac = resids / (len(x) - order - 2.0) + # it was deciced that the "- 2" (originally justified by "Bayesian + # uncertainty analysis") is not was the user expects + # (see gh-11196 and gh-11197) + fac = resids / (len(x) - order) if y.ndim == 1: return c, Vbase * fac else: @@ -695,6 +706,8 @@ def polyval(p, x): for polynomials of high degree the values may be inaccurate due to rounding errors. Use carefully. + If `x` is a subtype of `ndarray` the return value will be of the same type. + References ---------- .. [1] I. N. Bronshtein, K. A. Semendyayev, and K. A. Hirsch (Eng. @@ -706,18 +719,18 @@ def polyval(p, x): >>> np.polyval([3,0,1], 5) # 3 * 5**2 + 0 * 5**1 + 1 76 >>> np.polyval([3,0,1], np.poly1d(5)) - poly1d([ 76.]) + poly1d([76.]) >>> np.polyval(np.poly1d([3,0,1]), 5) 76 >>> np.polyval(np.poly1d([3,0,1]), np.poly1d(5)) - poly1d([ 76.]) + poly1d([76.]) """ p = NX.asarray(p) if isinstance(x, poly1d): y = 0 else: - x = NX.asarray(x) + x = NX.asanyarray(x) y = NX.zeros_like(x) for i in range(len(p)): y = y * x + p[i] @@ -863,8 +876,7 @@ def polymul(a1, a2): See Also -------- poly1d : A one-dimensional polynomial class. - poly, polyadd, polyder, polydiv, polyfit, polyint, polysub, - polyval + poly, polyadd, polyder, polydiv, polyfit, polyint, polysub, polyval convolve : Array convolution. Same output as polymul, but has parameter for overlap mode. @@ -926,7 +938,7 @@ def polydiv(u, v): See Also -------- - poly, polyadd, polyder, polydiv, polyfit, polyint, polymul, polysub, + poly, polyadd, polyder, polydiv, polyfit, polyint, polymul, polysub polyval Notes @@ -943,7 +955,7 @@ def polydiv(u, v): >>> x = np.array([3.0, 5.0, 2.0]) >>> y = np.array([2.0, 1.0]) >>> np.polydiv(x, y) - (array([ 1.5 , 1.75]), array([ 0.25])) + (array([1.5 , 1.75]), array([0.25])) """ truepoly = (isinstance(u, poly1d) or isinstance(u, poly1d)) @@ -1038,7 +1050,7 @@ class poly1d(object): >>> p.r array([-1.+1.41421356j, -1.-1.41421356j]) >>> p(p.r) - array([ -4.44089210e-16+0.j, -4.44089210e-16+0.j]) + array([ -4.44089210e-16+0.j, -4.44089210e-16+0.j]) # may vary These numbers in the previous line represent (0, 0) to machine precision @@ -1065,7 +1077,7 @@ class poly1d(object): poly1d([ 1, 4, 10, 12, 9]) >>> (p**3 + 4) / p - (poly1d([ 1., 4., 10., 12., 9.]), poly1d([ 4.])) + (poly1d([ 1., 4., 10., 12., 9.]), poly1d([4.])) ``asarray(p)`` gives the coefficient array, so polynomials can be used in all functions that accept arrays: @@ -1087,7 +1099,7 @@ class poly1d(object): Construct a polynomial from its roots: >>> np.poly1d([1, 2], True) - poly1d([ 1, -3, 2]) + poly1d([ 1., -3., 2.]) This is the same polynomial as obtained by: @@ -1099,8 +1111,14 @@ class poly1d(object): @property def coeffs(self): - """ A copy of the polynomial coefficients """ - return self._coeffs.copy() + """ The polynomial coefficients """ + return self._coeffs + + @coeffs.setter + def coeffs(self, value): + # allowing this makes p.coeffs *= 2 legal + if value is not self._coeffs: + raise AttributeError("Cannot set attribute") @property def variable(self): |