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| """ | |
| ================================================= | |
| Power Series (:mod:`numpy.polynomial.polynomial`) | |
| ================================================= | |
| This module provides a number of objects (mostly functions) useful for | |
| dealing with polynomials, including a `Polynomial` class that | |
| encapsulates the usual arithmetic operations. (General information | |
| on how this module represents and works with polynomial objects is in | |
| the docstring for its "parent" sub-package, `numpy.polynomial`). | |
| Classes | |
| ------- | |
| .. autosummary:: | |
| :toctree: generated/ | |
| Polynomial | |
| Constants | |
| --------- | |
| .. autosummary:: | |
| :toctree: generated/ | |
| polydomain | |
| polyzero | |
| polyone | |
| polyx | |
| Arithmetic | |
| ---------- | |
| .. autosummary:: | |
| :toctree: generated/ | |
| polyadd | |
| polysub | |
| polymulx | |
| polymul | |
| polydiv | |
| polypow | |
| polyval | |
| polyval2d | |
| polyval3d | |
| polygrid2d | |
| polygrid3d | |
| Calculus | |
| -------- | |
| .. autosummary:: | |
| :toctree: generated/ | |
| polyder | |
| polyint | |
| Misc Functions | |
| -------------- | |
| .. autosummary:: | |
| :toctree: generated/ | |
| polyfromroots | |
| polyroots | |
| polyvalfromroots | |
| polyvander | |
| polyvander2d | |
| polyvander3d | |
| polycompanion | |
| polyfit | |
| polytrim | |
| polyline | |
| See Also | |
| -------- | |
| `numpy.polynomial` | |
| """ | |
| __all__ = [ | |
| 'polyzero', 'polyone', 'polyx', 'polydomain', 'polyline', 'polyadd', | |
| 'polysub', 'polymulx', 'polymul', 'polydiv', 'polypow', 'polyval', | |
| 'polyvalfromroots', 'polyder', 'polyint', 'polyfromroots', 'polyvander', | |
| 'polyfit', 'polytrim', 'polyroots', 'Polynomial', 'polyval2d', 'polyval3d', | |
| 'polygrid2d', 'polygrid3d', 'polyvander2d', 'polyvander3d'] | |
| import numpy as np | |
| import numpy.linalg as la | |
| from numpy.core.multiarray import normalize_axis_index | |
| from . import polyutils as pu | |
| from ._polybase import ABCPolyBase | |
| polytrim = pu.trimcoef | |
| # | |
| # These are constant arrays are of integer type so as to be compatible | |
| # with the widest range of other types, such as Decimal. | |
| # | |
| # Polynomial default domain. | |
| polydomain = np.array([-1, 1]) | |
| # Polynomial coefficients representing zero. | |
| polyzero = np.array([0]) | |
| # Polynomial coefficients representing one. | |
| polyone = np.array([1]) | |
| # Polynomial coefficients representing the identity x. | |
| polyx = np.array([0, 1]) | |
| # | |
| # Polynomial series functions | |
| # | |
| def polyline(off, scl): | |
| """ | |
| Returns an array representing a linear polynomial. | |
| Parameters | |
| ---------- | |
| off, scl : scalars | |
| The "y-intercept" and "slope" of the line, respectively. | |
| Returns | |
| ------- | |
| y : ndarray | |
| This module's representation of the linear polynomial ``off + | |
| scl*x``. | |
| See Also | |
| -------- | |
| numpy.polynomial.chebyshev.chebline | |
| numpy.polynomial.legendre.legline | |
| numpy.polynomial.laguerre.lagline | |
| numpy.polynomial.hermite.hermline | |
| numpy.polynomial.hermite_e.hermeline | |
| Examples | |
| -------- | |
| >>> from numpy.polynomial import polynomial as P | |
| >>> P.polyline(1,-1) | |
| array([ 1, -1]) | |
| >>> P.polyval(1, P.polyline(1,-1)) # should be 0 | |
| 0.0 | |
| """ | |
| if scl != 0: | |
| return np.array([off, scl]) | |
| else: | |
| return np.array([off]) | |
| def polyfromroots(roots): | |
| """ | |
| Generate a monic polynomial with given roots. | |
| Return the coefficients of the polynomial | |
| .. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n), | |
| where the ``r_n`` are the roots specified in `roots`. If a zero has | |
| multiplicity n, then it must appear in `roots` n times. For instance, | |
| if 2 is a root of multiplicity three and 3 is a root of multiplicity 2, | |
| then `roots` looks something like [2, 2, 2, 3, 3]. The roots can appear | |
| in any order. | |
| If the returned coefficients are `c`, then | |
| .. math:: p(x) = c_0 + c_1 * x + ... + x^n | |
| The coefficient of the last term is 1 for monic polynomials in this | |
| form. | |
| Parameters | |
| ---------- | |
| roots : array_like | |
| Sequence containing the roots. | |
| Returns | |
| ------- | |
| out : ndarray | |
| 1-D array of the polynomial's coefficients If all the roots are | |
| real, then `out` is also real, otherwise it is complex. (see | |
| Examples below). | |
| See Also | |
| -------- | |
| numpy.polynomial.chebyshev.chebfromroots | |
| numpy.polynomial.legendre.legfromroots | |
| numpy.polynomial.laguerre.lagfromroots | |
| numpy.polynomial.hermite.hermfromroots | |
| numpy.polynomial.hermite_e.hermefromroots | |
| Notes | |
| ----- | |
| The coefficients are determined by multiplying together linear factors | |
| of the form ``(x - r_i)``, i.e. | |
| .. math:: p(x) = (x - r_0) (x - r_1) ... (x - r_n) | |
| where ``n == len(roots) - 1``; note that this implies that ``1`` is always | |
| returned for :math:`a_n`. | |
| Examples | |
| -------- | |
| >>> from numpy.polynomial import polynomial as P | |
| >>> P.polyfromroots((-1,0,1)) # x(x - 1)(x + 1) = x^3 - x | |
| array([ 0., -1., 0., 1.]) | |
| >>> j = complex(0,1) | |
| >>> P.polyfromroots((-j,j)) # complex returned, though values are real | |
| array([1.+0.j, 0.+0.j, 1.+0.j]) | |
| """ | |
| return pu._fromroots(polyline, polymul, roots) | |
| def polyadd(c1, c2): | |
| """ | |
| Add one polynomial to another. | |
| Returns the sum of two polynomials `c1` + `c2`. The arguments are | |
| sequences of coefficients from lowest order term to highest, i.e., | |
| [1,2,3] represents the polynomial ``1 + 2*x + 3*x**2``. | |
| Parameters | |
| ---------- | |
| c1, c2 : array_like | |
| 1-D arrays of polynomial coefficients ordered from low to high. | |
| Returns | |
| ------- | |
| out : ndarray | |
| The coefficient array representing their sum. | |
| See Also | |
| -------- | |
| polysub, polymulx, polymul, polydiv, polypow | |
| Examples | |
| -------- | |
| >>> from numpy.polynomial import polynomial as P | |
| >>> c1 = (1,2,3) | |
| >>> c2 = (3,2,1) | |
| >>> sum = P.polyadd(c1,c2); sum | |
| array([4., 4., 4.]) | |
| >>> P.polyval(2, sum) # 4 + 4(2) + 4(2**2) | |
| 28.0 | |
| """ | |
| return pu._add(c1, c2) | |
| def polysub(c1, c2): | |
| """ | |
| Subtract one polynomial from another. | |
| Returns the difference of two polynomials `c1` - `c2`. The arguments | |
| are sequences of coefficients from lowest order term to highest, i.e., | |
| [1,2,3] represents the polynomial ``1 + 2*x + 3*x**2``. | |
| Parameters | |
| ---------- | |
| c1, c2 : array_like | |
| 1-D arrays of polynomial coefficients ordered from low to | |
| high. | |
| Returns | |
| ------- | |
| out : ndarray | |
| Of coefficients representing their difference. | |
| See Also | |
| -------- | |
| polyadd, polymulx, polymul, polydiv, polypow | |
| Examples | |
| -------- | |
| >>> from numpy.polynomial import polynomial as P | |
| >>> c1 = (1,2,3) | |
| >>> c2 = (3,2,1) | |
| >>> P.polysub(c1,c2) | |
| array([-2., 0., 2.]) | |
| >>> P.polysub(c2,c1) # -P.polysub(c1,c2) | |
| array([ 2., 0., -2.]) | |
| """ | |
| return pu._sub(c1, c2) | |
| def polymulx(c): | |
| """Multiply a polynomial by x. | |
| Multiply the polynomial `c` by x, where x is the independent | |
| variable. | |
| Parameters | |
| ---------- | |
| c : array_like | |
| 1-D array of polynomial coefficients ordered from low to | |
| high. | |
| Returns | |
| ------- | |
| out : ndarray | |
| Array representing the result of the multiplication. | |
| See Also | |
| -------- | |
| polyadd, polysub, polymul, polydiv, polypow | |
| Notes | |
| ----- | |
| .. versionadded:: 1.5.0 | |
| """ | |
| # c is a trimmed copy | |
| [c] = pu.as_series([c]) | |
| # The zero series needs special treatment | |
| if len(c) == 1 and c[0] == 0: | |
| return c | |
| prd = np.empty(len(c) + 1, dtype=c.dtype) | |
| prd[0] = c[0]*0 | |
| prd[1:] = c | |
| return prd | |
| def polymul(c1, c2): | |
| """ | |
| Multiply one polynomial by another. | |
| Returns the product of two polynomials `c1` * `c2`. The arguments are | |
| sequences of coefficients, from lowest order term to highest, e.g., | |
| [1,2,3] represents the polynomial ``1 + 2*x + 3*x**2.`` | |
| Parameters | |
| ---------- | |
| c1, c2 : array_like | |
| 1-D arrays of coefficients representing a polynomial, relative to the | |
| "standard" basis, and ordered from lowest order term to highest. | |
| Returns | |
| ------- | |
| out : ndarray | |
| Of the coefficients of their product. | |
| See Also | |
| -------- | |
| polyadd, polysub, polymulx, polydiv, polypow | |
| Examples | |
| -------- | |
| >>> from numpy.polynomial import polynomial as P | |
| >>> c1 = (1,2,3) | |
| >>> c2 = (3,2,1) | |
| >>> P.polymul(c1,c2) | |
| array([ 3., 8., 14., 8., 3.]) | |
| """ | |
| # c1, c2 are trimmed copies | |
| [c1, c2] = pu.as_series([c1, c2]) | |
| ret = np.convolve(c1, c2) | |
| return pu.trimseq(ret) | |
| def polydiv(c1, c2): | |
| """ | |
| Divide one polynomial by another. | |
| Returns the quotient-with-remainder of two polynomials `c1` / `c2`. | |
| The arguments are sequences of coefficients, from lowest order term | |
| to highest, e.g., [1,2,3] represents ``1 + 2*x + 3*x**2``. | |
| Parameters | |
| ---------- | |
| c1, c2 : array_like | |
| 1-D arrays of polynomial coefficients ordered from low to high. | |
| Returns | |
| ------- | |
| [quo, rem] : ndarrays | |
| Of coefficient series representing the quotient and remainder. | |
| See Also | |
| -------- | |
| polyadd, polysub, polymulx, polymul, polypow | |
| Examples | |
| -------- | |
| >>> from numpy.polynomial import polynomial as P | |
| >>> c1 = (1,2,3) | |
| >>> c2 = (3,2,1) | |
| >>> P.polydiv(c1,c2) | |
| (array([3.]), array([-8., -4.])) | |
| >>> P.polydiv(c2,c1) | |
| (array([ 0.33333333]), array([ 2.66666667, 1.33333333])) # may vary | |
| """ | |
| # c1, c2 are trimmed copies | |
| [c1, c2] = pu.as_series([c1, c2]) | |
| if c2[-1] == 0: | |
| raise ZeroDivisionError() | |
| # note: this is more efficient than `pu._div(polymul, c1, c2)` | |
| lc1 = len(c1) | |
| lc2 = len(c2) | |
| if lc1 < lc2: | |
| return c1[:1]*0, c1 | |
| elif lc2 == 1: | |
| return c1/c2[-1], c1[:1]*0 | |
| else: | |
| dlen = lc1 - lc2 | |
| scl = c2[-1] | |
| c2 = c2[:-1]/scl | |
| i = dlen | |
| j = lc1 - 1 | |
| while i >= 0: | |
| c1[i:j] -= c2*c1[j] | |
| i -= 1 | |
| j -= 1 | |
| return c1[j+1:]/scl, pu.trimseq(c1[:j+1]) | |
| def polypow(c, pow, maxpower=None): | |
| """Raise a polynomial to a power. | |
| Returns the polynomial `c` raised to the power `pow`. The argument | |
| `c` is a sequence of coefficients ordered from low to high. i.e., | |
| [1,2,3] is the series ``1 + 2*x + 3*x**2.`` | |
| Parameters | |
| ---------- | |
| c : array_like | |
| 1-D array of array of series coefficients ordered from low to | |
| high degree. | |
| pow : integer | |
| Power to which the series will be raised | |
| maxpower : integer, optional | |
| Maximum power allowed. This is mainly to limit growth of the series | |
| to unmanageable size. Default is 16 | |
| Returns | |
| ------- | |
| coef : ndarray | |
| Power series of power. | |
| See Also | |
| -------- | |
| polyadd, polysub, polymulx, polymul, polydiv | |
| Examples | |
| -------- | |
| >>> from numpy.polynomial import polynomial as P | |
| >>> P.polypow([1,2,3], 2) | |
| array([ 1., 4., 10., 12., 9.]) | |
| """ | |
| # note: this is more efficient than `pu._pow(polymul, c1, c2)`, as it | |
| # avoids calling `as_series` repeatedly | |
| return pu._pow(np.convolve, c, pow, maxpower) | |
| def polyder(c, m=1, scl=1, axis=0): | |
| """ | |
| Differentiate a polynomial. | |
| Returns the polynomial coefficients `c` differentiated `m` times along | |
| `axis`. At each iteration the result is multiplied by `scl` (the | |
| scaling factor is for use in a linear change of variable). The | |
| argument `c` is an array of coefficients from low to high degree along | |
| each axis, e.g., [1,2,3] represents the polynomial ``1 + 2*x + 3*x**2`` | |
| while [[1,2],[1,2]] represents ``1 + 1*x + 2*y + 2*x*y`` if axis=0 is | |
| ``x`` and axis=1 is ``y``. | |
| Parameters | |
| ---------- | |
| c : array_like | |
| Array of polynomial coefficients. If c is multidimensional the | |
| different axis correspond to different variables with the degree | |
| in each axis given by the corresponding index. | |
| m : int, optional | |
| Number of derivatives taken, must be non-negative. (Default: 1) | |
| scl : scalar, optional | |
| Each differentiation is multiplied by `scl`. The end result is | |
| multiplication by ``scl**m``. This is for use in a linear change | |
| of variable. (Default: 1) | |
| axis : int, optional | |
| Axis over which the derivative is taken. (Default: 0). | |
| .. versionadded:: 1.7.0 | |
| Returns | |
| ------- | |
| der : ndarray | |
| Polynomial coefficients of the derivative. | |
| See Also | |
| -------- | |
| polyint | |
| Examples | |
| -------- | |
| >>> from numpy.polynomial import polynomial as P | |
| >>> c = (1,2,3,4) # 1 + 2x + 3x**2 + 4x**3 | |
| >>> P.polyder(c) # (d/dx)(c) = 2 + 6x + 12x**2 | |
| array([ 2., 6., 12.]) | |
| >>> P.polyder(c,3) # (d**3/dx**3)(c) = 24 | |
| array([24.]) | |
| >>> P.polyder(c,scl=-1) # (d/d(-x))(c) = -2 - 6x - 12x**2 | |
| array([ -2., -6., -12.]) | |
| >>> P.polyder(c,2,-1) # (d**2/d(-x)**2)(c) = 6 + 24x | |
| array([ 6., 24.]) | |
| """ | |
| c = np.array(c, ndmin=1, copy=True) | |
| if c.dtype.char in '?bBhHiIlLqQpP': | |
| # astype fails with NA | |
| c = c + 0.0 | |
| cdt = c.dtype | |
| cnt = pu._deprecate_as_int(m, "the order of derivation") | |
| iaxis = pu._deprecate_as_int(axis, "the axis") | |
| if cnt < 0: | |
| raise ValueError("The order of derivation must be non-negative") | |
| iaxis = normalize_axis_index(iaxis, c.ndim) | |
| if cnt == 0: | |
| return c | |
| c = np.moveaxis(c, iaxis, 0) | |
| n = len(c) | |
| if cnt >= n: | |
| c = c[:1]*0 | |
| else: | |
| for i in range(cnt): | |
| n = n - 1 | |
| c *= scl | |
| der = np.empty((n,) + c.shape[1:], dtype=cdt) | |
| for j in range(n, 0, -1): | |
| der[j - 1] = j*c[j] | |
| c = der | |
| c = np.moveaxis(c, 0, iaxis) | |
| return c | |
| def polyint(c, m=1, k=[], lbnd=0, scl=1, axis=0): | |
| """ | |
| Integrate a polynomial. | |
| Returns the polynomial coefficients `c` integrated `m` times from | |
| `lbnd` along `axis`. At each iteration the resulting series is | |
| **multiplied** by `scl` and an integration constant, `k`, is added. | |
| The scaling factor is for use in a linear change of variable. ("Buyer | |
| beware": note that, depending on what one is doing, one may want `scl` | |
| to be the reciprocal of what one might expect; for more information, | |
| see the Notes section below.) The argument `c` is an array of | |
| coefficients, from low to high degree along each axis, e.g., [1,2,3] | |
| represents the polynomial ``1 + 2*x + 3*x**2`` while [[1,2],[1,2]] | |
| represents ``1 + 1*x + 2*y + 2*x*y`` if axis=0 is ``x`` and axis=1 is | |
| ``y``. | |
| Parameters | |
| ---------- | |
| c : array_like | |
| 1-D array of polynomial coefficients, ordered from low to high. | |
| m : int, optional | |
| Order of integration, must be positive. (Default: 1) | |
| k : {[], list, scalar}, optional | |
| Integration constant(s). The value of the first integral at zero | |
| is the first value in the list, the value of the second integral | |
| at zero is the second value, etc. If ``k == []`` (the default), | |
| all constants are set to zero. If ``m == 1``, a single scalar can | |
| be given instead of a list. | |
| lbnd : scalar, optional | |
| The lower bound of the integral. (Default: 0) | |
| scl : scalar, optional | |
| Following each integration the result is *multiplied* by `scl` | |
| before the integration constant is added. (Default: 1) | |
| axis : int, optional | |
| Axis over which the integral is taken. (Default: 0). | |
| .. versionadded:: 1.7.0 | |
| Returns | |
| ------- | |
| S : ndarray | |
| Coefficient array of the integral. | |
| Raises | |
| ------ | |
| ValueError | |
| If ``m < 1``, ``len(k) > m``, ``np.ndim(lbnd) != 0``, or | |
| ``np.ndim(scl) != 0``. | |
| See Also | |
| -------- | |
| polyder | |
| Notes | |
| ----- | |
| Note that the result of each integration is *multiplied* by `scl`. Why | |
| is this important to note? Say one is making a linear change of | |
| variable :math:`u = ax + b` in an integral relative to `x`. Then | |
| :math:`dx = du/a`, so one will need to set `scl` equal to | |
| :math:`1/a` - perhaps not what one would have first thought. | |
| Examples | |
| -------- | |
| >>> from numpy.polynomial import polynomial as P | |
| >>> c = (1,2,3) | |
| >>> P.polyint(c) # should return array([0, 1, 1, 1]) | |
| array([0., 1., 1., 1.]) | |
| >>> P.polyint(c,3) # should return array([0, 0, 0, 1/6, 1/12, 1/20]) | |
| array([ 0. , 0. , 0. , 0.16666667, 0.08333333, # may vary | |
| 0.05 ]) | |
| >>> P.polyint(c,k=3) # should return array([3, 1, 1, 1]) | |
| array([3., 1., 1., 1.]) | |
| >>> P.polyint(c,lbnd=-2) # should return array([6, 1, 1, 1]) | |
| array([6., 1., 1., 1.]) | |
| >>> P.polyint(c,scl=-2) # should return array([0, -2, -2, -2]) | |
| array([ 0., -2., -2., -2.]) | |
| """ | |
| c = np.array(c, ndmin=1, copy=True) | |
| if c.dtype.char in '?bBhHiIlLqQpP': | |
| # astype doesn't preserve mask attribute. | |
| c = c + 0.0 | |
| cdt = c.dtype | |
| if not np.iterable(k): | |
| k = [k] | |
| cnt = pu._deprecate_as_int(m, "the order of integration") | |
| iaxis = pu._deprecate_as_int(axis, "the axis") | |
| if cnt < 0: | |
| raise ValueError("The order of integration must be non-negative") | |
| if len(k) > cnt: | |
| raise ValueError("Too many integration constants") | |
| if np.ndim(lbnd) != 0: | |
| raise ValueError("lbnd must be a scalar.") | |
| if np.ndim(scl) != 0: | |
| raise ValueError("scl must be a scalar.") | |
| iaxis = normalize_axis_index(iaxis, c.ndim) | |
| if cnt == 0: | |
| return c | |
| k = list(k) + [0]*(cnt - len(k)) | |
| c = np.moveaxis(c, iaxis, 0) | |
| for i in range(cnt): | |
| n = len(c) | |
| c *= scl | |
| if n == 1 and np.all(c[0] == 0): | |
| c[0] += k[i] | |
| else: | |
| tmp = np.empty((n + 1,) + c.shape[1:], dtype=cdt) | |
| tmp[0] = c[0]*0 | |
| tmp[1] = c[0] | |
| for j in range(1, n): | |
| tmp[j + 1] = c[j]/(j + 1) | |
| tmp[0] += k[i] - polyval(lbnd, tmp) | |
| c = tmp | |
| c = np.moveaxis(c, 0, iaxis) | |
| return c | |
| def polyval(x, c, tensor=True): | |
| """ | |
| Evaluate a polynomial at points x. | |
| If `c` is of length `n + 1`, this function returns the value | |
| .. math:: p(x) = c_0 + c_1 * x + ... + c_n * x^n | |
| The parameter `x` is converted to an array only if it is a tuple or a | |
| list, otherwise it is treated as a scalar. In either case, either `x` | |
| or its elements must support multiplication and addition both with | |
| themselves and with the elements of `c`. | |
| If `c` is a 1-D array, then `p(x)` will have the same shape as `x`. If | |
| `c` is multidimensional, then the shape of the result depends on the | |
| value of `tensor`. If `tensor` is true the shape will be c.shape[1:] + | |
| x.shape. If `tensor` is false the shape will be c.shape[1:]. Note that | |
| scalars have shape (,). | |
| Trailing zeros in the coefficients will be used in the evaluation, so | |
| they should be avoided if efficiency is a concern. | |
| Parameters | |
| ---------- | |
| x : array_like, compatible object | |
| If `x` is a list or tuple, it is converted to an ndarray, otherwise | |
| it is left unchanged and treated as a scalar. In either case, `x` | |
| or its elements must support addition and multiplication with | |
| with themselves and with the elements of `c`. | |
| c : array_like | |
| Array of coefficients ordered so that the coefficients for terms of | |
| degree n are contained in c[n]. If `c` is multidimensional the | |
| remaining indices enumerate multiple polynomials. In the two | |
| dimensional case the coefficients may be thought of as stored in | |
| the columns of `c`. | |
| tensor : boolean, optional | |
| If True, the shape of the coefficient array is extended with ones | |
| on the right, one for each dimension of `x`. Scalars have dimension 0 | |
| for this action. The result is that every column of coefficients in | |
| `c` is evaluated for every element of `x`. If False, `x` is broadcast | |
| over the columns of `c` for the evaluation. This keyword is useful | |
| when `c` is multidimensional. The default value is True. | |
| .. versionadded:: 1.7.0 | |
| Returns | |
| ------- | |
| values : ndarray, compatible object | |
| The shape of the returned array is described above. | |
| See Also | |
| -------- | |
| polyval2d, polygrid2d, polyval3d, polygrid3d | |
| Notes | |
| ----- | |
| The evaluation uses Horner's method. | |
| Examples | |
| -------- | |
| >>> from numpy.polynomial.polynomial import polyval | |
| >>> polyval(1, [1,2,3]) | |
| 6.0 | |
| >>> a = np.arange(4).reshape(2,2) | |
| >>> a | |
| array([[0, 1], | |
| [2, 3]]) | |
| >>> polyval(a, [1,2,3]) | |
| array([[ 1., 6.], | |
| [17., 34.]]) | |
| >>> coef = np.arange(4).reshape(2,2) # multidimensional coefficients | |
| >>> coef | |
| array([[0, 1], | |
| [2, 3]]) | |
| >>> polyval([1,2], coef, tensor=True) | |
| array([[2., 4.], | |
| [4., 7.]]) | |
| >>> polyval([1,2], coef, tensor=False) | |
| array([2., 7.]) | |
| """ | |
| c = np.array(c, ndmin=1, copy=False) | |
| if c.dtype.char in '?bBhHiIlLqQpP': | |
| # astype fails with NA | |
| c = c + 0.0 | |
| if isinstance(x, (tuple, list)): | |
| x = np.asarray(x) | |
| if isinstance(x, np.ndarray) and tensor: | |
| c = c.reshape(c.shape + (1,)*x.ndim) | |
| c0 = c[-1] + x*0 | |
| for i in range(2, len(c) + 1): | |
| c0 = c[-i] + c0*x | |
| return c0 | |
| def polyvalfromroots(x, r, tensor=True): | |
| """ | |
| Evaluate a polynomial specified by its roots at points x. | |
| If `r` is of length `N`, this function returns the value | |
| .. math:: p(x) = \\prod_{n=1}^{N} (x - r_n) | |
| The parameter `x` is converted to an array only if it is a tuple or a | |
| list, otherwise it is treated as a scalar. In either case, either `x` | |
| or its elements must support multiplication and addition both with | |
| themselves and with the elements of `r`. | |
| If `r` is a 1-D array, then `p(x)` will have the same shape as `x`. If `r` | |
| is multidimensional, then the shape of the result depends on the value of | |
| `tensor`. If `tensor is ``True`` the shape will be r.shape[1:] + x.shape; | |
| that is, each polynomial is evaluated at every value of `x`. If `tensor` is | |
| ``False``, the shape will be r.shape[1:]; that is, each polynomial is | |
| evaluated only for the corresponding broadcast value of `x`. Note that | |
| scalars have shape (,). | |
| .. versionadded:: 1.12 | |
| Parameters | |
| ---------- | |
| x : array_like, compatible object | |
| If `x` is a list or tuple, it is converted to an ndarray, otherwise | |
| it is left unchanged and treated as a scalar. In either case, `x` | |
| or its elements must support addition and multiplication with | |
| with themselves and with the elements of `r`. | |
| r : array_like | |
| Array of roots. If `r` is multidimensional the first index is the | |
| root index, while the remaining indices enumerate multiple | |
| polynomials. For instance, in the two dimensional case the roots | |
| of each polynomial may be thought of as stored in the columns of `r`. | |
| tensor : boolean, optional | |
| If True, the shape of the roots array is extended with ones on the | |
| right, one for each dimension of `x`. Scalars have dimension 0 for this | |
| action. The result is that every column of coefficients in `r` is | |
| evaluated for every element of `x`. If False, `x` is broadcast over the | |
| columns of `r` for the evaluation. This keyword is useful when `r` is | |
| multidimensional. The default value is True. | |
| Returns | |
| ------- | |
| values : ndarray, compatible object | |
| The shape of the returned array is described above. | |
| See Also | |
| -------- | |
| polyroots, polyfromroots, polyval | |
| Examples | |
| -------- | |
| >>> from numpy.polynomial.polynomial import polyvalfromroots | |
| >>> polyvalfromroots(1, [1,2,3]) | |
| 0.0 | |
| >>> a = np.arange(4).reshape(2,2) | |
| >>> a | |
| array([[0, 1], | |
| [2, 3]]) | |
| >>> polyvalfromroots(a, [-1, 0, 1]) | |
| array([[-0., 0.], | |
| [ 6., 24.]]) | |
| >>> r = np.arange(-2, 2).reshape(2,2) # multidimensional coefficients | |
| >>> r # each column of r defines one polynomial | |
| array([[-2, -1], | |
| [ 0, 1]]) | |
| >>> b = [-2, 1] | |
| >>> polyvalfromroots(b, r, tensor=True) | |
| array([[-0., 3.], | |
| [ 3., 0.]]) | |
| >>> polyvalfromroots(b, r, tensor=False) | |
| array([-0., 0.]) | |
| """ | |
| r = np.array(r, ndmin=1, copy=False) | |
| if r.dtype.char in '?bBhHiIlLqQpP': | |
| r = r.astype(np.double) | |
| if isinstance(x, (tuple, list)): | |
| x = np.asarray(x) | |
| if isinstance(x, np.ndarray): | |
| if tensor: | |
| r = r.reshape(r.shape + (1,)*x.ndim) | |
| elif x.ndim >= r.ndim: | |
| raise ValueError("x.ndim must be < r.ndim when tensor == False") | |
| return np.prod(x - r, axis=0) | |
| def polyval2d(x, y, c): | |
| """ | |
| Evaluate a 2-D polynomial at points (x, y). | |
| This function returns the value | |
| .. math:: p(x,y) = \\sum_{i,j} c_{i,j} * x^i * y^j | |
| The parameters `x` and `y` are converted to arrays only if they are | |
| tuples or a lists, otherwise they are treated as a scalars and they | |
| must have the same shape after conversion. In either case, either `x` | |
| and `y` or their elements must support multiplication and addition both | |
| with themselves and with the elements of `c`. | |
| If `c` has fewer than two dimensions, ones are implicitly appended to | |
| its shape to make it 2-D. The shape of the result will be c.shape[2:] + | |
| x.shape. | |
| Parameters | |
| ---------- | |
| x, y : array_like, compatible objects | |
| The two dimensional series is evaluated at the points `(x, y)`, | |
| where `x` and `y` must have the same shape. If `x` or `y` is a list | |
| or tuple, it is first converted to an ndarray, otherwise it is left | |
| unchanged and, if it isn't an ndarray, it is treated as a scalar. | |
| c : array_like | |
| Array of coefficients ordered so that the coefficient of the term | |
| of multi-degree i,j is contained in `c[i,j]`. If `c` has | |
| dimension greater than two the remaining indices enumerate multiple | |
| sets of coefficients. | |
| Returns | |
| ------- | |
| values : ndarray, compatible object | |
| The values of the two dimensional polynomial at points formed with | |
| pairs of corresponding values from `x` and `y`. | |
| See Also | |
| -------- | |
| polyval, polygrid2d, polyval3d, polygrid3d | |
| Notes | |
| ----- | |
| .. versionadded:: 1.7.0 | |
| """ | |
| return pu._valnd(polyval, c, x, y) | |
| def polygrid2d(x, y, c): | |
| """ | |
| Evaluate a 2-D polynomial on the Cartesian product of x and y. | |
| This function returns the values: | |
| .. math:: p(a,b) = \\sum_{i,j} c_{i,j} * a^i * b^j | |
| where the points `(a, b)` consist of all pairs formed by taking | |
| `a` from `x` and `b` from `y`. The resulting points form a grid with | |
| `x` in the first dimension and `y` in the second. | |
| The parameters `x` and `y` are converted to arrays only if they are | |
| tuples or a lists, otherwise they are treated as a scalars. In either | |
| case, either `x` and `y` or their elements must support multiplication | |
| and addition both with themselves and with the elements of `c`. | |
| If `c` has fewer than two dimensions, ones are implicitly appended to | |
| its shape to make it 2-D. The shape of the result will be c.shape[2:] + | |
| x.shape + y.shape. | |
| Parameters | |
| ---------- | |
| x, y : array_like, compatible objects | |
| The two dimensional series is evaluated at the points in the | |
| Cartesian product of `x` and `y`. If `x` or `y` is a list or | |
| tuple, it is first converted to an ndarray, otherwise it is left | |
| unchanged and, if it isn't an ndarray, it is treated as a scalar. | |
| c : array_like | |
| Array of coefficients ordered so that the coefficients for terms of | |
| degree i,j are contained in ``c[i,j]``. If `c` has dimension | |
| greater than two the remaining indices enumerate multiple sets of | |
| coefficients. | |
| Returns | |
| ------- | |
| values : ndarray, compatible object | |
| The values of the two dimensional polynomial at points in the Cartesian | |
| product of `x` and `y`. | |
| See Also | |
| -------- | |
| polyval, polyval2d, polyval3d, polygrid3d | |
| Notes | |
| ----- | |
| .. versionadded:: 1.7.0 | |
| """ | |
| return pu._gridnd(polyval, c, x, y) | |
| def polyval3d(x, y, z, c): | |
| """ | |
| Evaluate a 3-D polynomial at points (x, y, z). | |
| This function returns the values: | |
| .. math:: p(x,y,z) = \\sum_{i,j,k} c_{i,j,k} * x^i * y^j * z^k | |
| The parameters `x`, `y`, and `z` are converted to arrays only if | |
| they are tuples or a lists, otherwise they are treated as a scalars and | |
| they must have the same shape after conversion. In either case, either | |
| `x`, `y`, and `z` or their elements must support multiplication and | |
| addition both with themselves and with the elements of `c`. | |
| If `c` has fewer than 3 dimensions, ones are implicitly appended to its | |
| shape to make it 3-D. The shape of the result will be c.shape[3:] + | |
| x.shape. | |
| Parameters | |
| ---------- | |
| x, y, z : array_like, compatible object | |
| The three dimensional series is evaluated at the points | |
| `(x, y, z)`, where `x`, `y`, and `z` must have the same shape. If | |
| any of `x`, `y`, or `z` is a list or tuple, it is first converted | |
| to an ndarray, otherwise it is left unchanged and if it isn't an | |
| ndarray it is treated as a scalar. | |
| c : array_like | |
| Array of coefficients ordered so that the coefficient of the term of | |
| multi-degree i,j,k is contained in ``c[i,j,k]``. If `c` has dimension | |
| greater than 3 the remaining indices enumerate multiple sets of | |
| coefficients. | |
| Returns | |
| ------- | |
| values : ndarray, compatible object | |
| The values of the multidimensional polynomial on points formed with | |
| triples of corresponding values from `x`, `y`, and `z`. | |
| See Also | |
| -------- | |
| polyval, polyval2d, polygrid2d, polygrid3d | |
| Notes | |
| ----- | |
| .. versionadded:: 1.7.0 | |
| """ | |
| return pu._valnd(polyval, c, x, y, z) | |
| def polygrid3d(x, y, z, c): | |
| """ | |
| Evaluate a 3-D polynomial on the Cartesian product of x, y and z. | |
| This function returns the values: | |
| .. math:: p(a,b,c) = \\sum_{i,j,k} c_{i,j,k} * a^i * b^j * c^k | |
| where the points `(a, b, c)` consist of all triples formed by taking | |
| `a` from `x`, `b` from `y`, and `c` from `z`. The resulting points form | |
| a grid with `x` in the first dimension, `y` in the second, and `z` in | |
| the third. | |
| The parameters `x`, `y`, and `z` are converted to arrays only if they | |
| are tuples or a lists, otherwise they are treated as a scalars. In | |
| either case, either `x`, `y`, and `z` or their elements must support | |
| multiplication and addition both with themselves and with the elements | |
| of `c`. | |
| If `c` has fewer than three dimensions, ones are implicitly appended to | |
| its shape to make it 3-D. The shape of the result will be c.shape[3:] + | |
| x.shape + y.shape + z.shape. | |
| Parameters | |
| ---------- | |
| x, y, z : array_like, compatible objects | |
| The three dimensional series is evaluated at the points in the | |
| Cartesian product of `x`, `y`, and `z`. If `x`,`y`, or `z` is a | |
| list or tuple, it is first converted to an ndarray, otherwise it is | |
| left unchanged and, if it isn't an ndarray, it is treated as a | |
| scalar. | |
| c : array_like | |
| Array of coefficients ordered so that the coefficients for terms of | |
| degree i,j are contained in ``c[i,j]``. If `c` has dimension | |
| greater than two the remaining indices enumerate multiple sets of | |
| coefficients. | |
| Returns | |
| ------- | |
| values : ndarray, compatible object | |
| The values of the two dimensional polynomial at points in the Cartesian | |
| product of `x` and `y`. | |
| See Also | |
| -------- | |
| polyval, polyval2d, polygrid2d, polyval3d | |
| Notes | |
| ----- | |
| .. versionadded:: 1.7.0 | |
| """ | |
| return pu._gridnd(polyval, c, x, y, z) | |
| def polyvander(x, deg): | |
| """Vandermonde matrix of given degree. | |
| Returns the Vandermonde matrix of degree `deg` and sample points | |
| `x`. The Vandermonde matrix is defined by | |
| .. math:: V[..., i] = x^i, | |
| where `0 <= i <= deg`. The leading indices of `V` index the elements of | |
| `x` and the last index is the power of `x`. | |
| If `c` is a 1-D array of coefficients of length `n + 1` and `V` is the | |
| matrix ``V = polyvander(x, n)``, then ``np.dot(V, c)`` and | |
| ``polyval(x, c)`` are the same up to roundoff. This equivalence is | |
| useful both for least squares fitting and for the evaluation of a large | |
| number of polynomials of the same degree and sample points. | |
| Parameters | |
| ---------- | |
| x : array_like | |
| Array of points. The dtype is converted to float64 or complex128 | |
| depending on whether any of the elements are complex. If `x` is | |
| scalar it is converted to a 1-D array. | |
| deg : int | |
| Degree of the resulting matrix. | |
| Returns | |
| ------- | |
| vander : ndarray. | |
| The Vandermonde matrix. The shape of the returned matrix is | |
| ``x.shape + (deg + 1,)``, where the last index is the power of `x`. | |
| The dtype will be the same as the converted `x`. | |
| See Also | |
| -------- | |
| polyvander2d, polyvander3d | |
| """ | |
| ideg = pu._deprecate_as_int(deg, "deg") | |
| if ideg < 0: | |
| raise ValueError("deg must be non-negative") | |
| x = np.array(x, copy=False, ndmin=1) + 0.0 | |
| dims = (ideg + 1,) + x.shape | |
| dtyp = x.dtype | |
| v = np.empty(dims, dtype=dtyp) | |
| v[0] = x*0 + 1 | |
| if ideg > 0: | |
| v[1] = x | |
| for i in range(2, ideg + 1): | |
| v[i] = v[i-1]*x | |
| return np.moveaxis(v, 0, -1) | |
| def polyvander2d(x, y, deg): | |
| """Pseudo-Vandermonde matrix of given degrees. | |
| Returns the pseudo-Vandermonde matrix of degrees `deg` and sample | |
| points `(x, y)`. The pseudo-Vandermonde matrix is defined by | |
| .. math:: V[..., (deg[1] + 1)*i + j] = x^i * y^j, | |
| where `0 <= i <= deg[0]` and `0 <= j <= deg[1]`. The leading indices of | |
| `V` index the points `(x, y)` and the last index encodes the powers of | |
| `x` and `y`. | |
| If ``V = polyvander2d(x, y, [xdeg, ydeg])``, then the columns of `V` | |
| correspond to the elements of a 2-D coefficient array `c` of shape | |
| (xdeg + 1, ydeg + 1) in the order | |
| .. math:: c_{00}, c_{01}, c_{02} ... , c_{10}, c_{11}, c_{12} ... | |
| and ``np.dot(V, c.flat)`` and ``polyval2d(x, y, c)`` will be the same | |
| up to roundoff. This equivalence is useful both for least squares | |
| fitting and for the evaluation of a large number of 2-D polynomials | |
| of the same degrees and sample points. | |
| Parameters | |
| ---------- | |
| x, y : array_like | |
| Arrays of point coordinates, all of the same shape. The dtypes | |
| will be converted to either float64 or complex128 depending on | |
| whether any of the elements are complex. Scalars are converted to | |
| 1-D arrays. | |
| deg : list of ints | |
| List of maximum degrees of the form [x_deg, y_deg]. | |
| Returns | |
| ------- | |
| vander2d : ndarray | |
| The shape of the returned matrix is ``x.shape + (order,)``, where | |
| :math:`order = (deg[0]+1)*(deg([1]+1)`. The dtype will be the same | |
| as the converted `x` and `y`. | |
| See Also | |
| -------- | |
| polyvander, polyvander3d, polyval2d, polyval3d | |
| """ | |
| return pu._vander_nd_flat((polyvander, polyvander), (x, y), deg) | |
| def polyvander3d(x, y, z, deg): | |
| """Pseudo-Vandermonde matrix of given degrees. | |
| Returns the pseudo-Vandermonde matrix of degrees `deg` and sample | |
| points `(x, y, z)`. If `l, m, n` are the given degrees in `x, y, z`, | |
| then The pseudo-Vandermonde matrix is defined by | |
| .. math:: V[..., (m+1)(n+1)i + (n+1)j + k] = x^i * y^j * z^k, | |
| where `0 <= i <= l`, `0 <= j <= m`, and `0 <= j <= n`. The leading | |
| indices of `V` index the points `(x, y, z)` and the last index encodes | |
| the powers of `x`, `y`, and `z`. | |
| If ``V = polyvander3d(x, y, z, [xdeg, ydeg, zdeg])``, then the columns | |
| of `V` correspond to the elements of a 3-D coefficient array `c` of | |
| shape (xdeg + 1, ydeg + 1, zdeg + 1) in the order | |
| .. math:: c_{000}, c_{001}, c_{002},... , c_{010}, c_{011}, c_{012},... | |
| and ``np.dot(V, c.flat)`` and ``polyval3d(x, y, z, c)`` will be the | |
| same up to roundoff. This equivalence is useful both for least squares | |
| fitting and for the evaluation of a large number of 3-D polynomials | |
| of the same degrees and sample points. | |
| Parameters | |
| ---------- | |
| x, y, z : array_like | |
| Arrays of point coordinates, all of the same shape. The dtypes will | |
| be converted to either float64 or complex128 depending on whether | |
| any of the elements are complex. Scalars are converted to 1-D | |
| arrays. | |
| deg : list of ints | |
| List of maximum degrees of the form [x_deg, y_deg, z_deg]. | |
| Returns | |
| ------- | |
| vander3d : ndarray | |
| The shape of the returned matrix is ``x.shape + (order,)``, where | |
| :math:`order = (deg[0]+1)*(deg([1]+1)*(deg[2]+1)`. The dtype will | |
| be the same as the converted `x`, `y`, and `z`. | |
| See Also | |
| -------- | |
| polyvander, polyvander3d, polyval2d, polyval3d | |
| Notes | |
| ----- | |
| .. versionadded:: 1.7.0 | |
| """ | |
| return pu._vander_nd_flat((polyvander, polyvander, polyvander), (x, y, z), deg) | |
| def polyfit(x, y, deg, rcond=None, full=False, w=None): | |
| """ | |
| Least-squares fit of a polynomial to data. | |
| Return the coefficients of a polynomial of degree `deg` that is the | |
| least squares fit to the data values `y` given at points `x`. If `y` is | |
| 1-D the returned coefficients will also be 1-D. If `y` is 2-D multiple | |
| fits are done, one for each column of `y`, and the resulting | |
| coefficients are stored in the corresponding columns of a 2-D return. | |
| The fitted polynomial(s) are in the form | |
| .. math:: p(x) = c_0 + c_1 * x + ... + c_n * x^n, | |
| where `n` is `deg`. | |
| Parameters | |
| ---------- | |
| x : array_like, shape (`M`,) | |
| x-coordinates of the `M` sample (data) points ``(x[i], y[i])``. | |
| y : array_like, shape (`M`,) or (`M`, `K`) | |
| y-coordinates of the sample points. Several sets of sample points | |
| sharing the same x-coordinates can be (independently) fit with one | |
| call to `polyfit` by passing in for `y` a 2-D array that contains | |
| one data set per column. | |
| deg : int or 1-D array_like | |
| Degree(s) of the fitting polynomials. If `deg` is a single integer | |
| all terms up to and including the `deg`'th term are included in the | |
| fit. For NumPy versions >= 1.11.0 a list of integers specifying the | |
| degrees of the terms to include may be used instead. | |
| rcond : float, optional | |
| Relative condition number of the fit. Singular values smaller | |
| than `rcond`, relative to the largest singular value, will be | |
| ignored. The default value is ``len(x)*eps``, where `eps` is the | |
| relative precision of the platform's float type, about 2e-16 in | |
| most cases. | |
| full : bool, optional | |
| Switch determining the nature of the return value. When ``False`` | |
| (the default) just the coefficients are returned; when ``True``, | |
| diagnostic information from the singular value decomposition (used | |
| to solve the fit's matrix equation) is also returned. | |
| w : array_like, shape (`M`,), optional | |
| Weights. If not None, the contribution of each point | |
| ``(x[i],y[i])`` to the fit is weighted by ``w[i]``. Ideally the | |
| weights are chosen so that the errors of the products ``w[i]*y[i]`` | |
| all have the same variance. The default value is None. | |
| .. versionadded:: 1.5.0 | |
| Returns | |
| ------- | |
| coef : ndarray, shape (`deg` + 1,) or (`deg` + 1, `K`) | |
| Polynomial coefficients ordered from low to high. If `y` was 2-D, | |
| the coefficients in column `k` of `coef` represent the polynomial | |
| fit to the data in `y`'s `k`-th column. | |
| [residuals, rank, singular_values, rcond] : list | |
| These values are only returned if `full` = True | |
| resid -- sum of squared residuals of the least squares fit | |
| rank -- the numerical rank of the scaled Vandermonde matrix | |
| sv -- singular values of the scaled Vandermonde matrix | |
| rcond -- value of `rcond`. | |
| For more details, see `numpy.linalg.lstsq`. | |
| Raises | |
| ------ | |
| RankWarning | |
| Raised if the matrix in the least-squares fit is rank 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.chebyshev.chebfit | |
| numpy.polynomial.legendre.legfit | |
| numpy.polynomial.laguerre.lagfit | |
| numpy.polynomial.hermite.hermfit | |
| numpy.polynomial.hermite_e.hermefit | |
| polyval : Evaluates a polynomial. | |
| polyvander : Vandermonde matrix for powers. | |
| 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 polynomial `p` that minimizes | |
| the sum of the weighted squared errors | |
| .. math :: E = \\sum_j w_j^2 * |y_j - p(x_j)|^2, | |
| where the :math:`w_j` are the weights. This problem is solved by | |
| setting up the (typically) over-determined 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 (and `full` == ``False``), a `RankWarning` will be raised. | |
| This means that the coefficient values may be poorly determined. | |
| Fitting to a lower order polynomial will usually get rid of the warning | |
| (but may not be what you want, of course; if you have independent | |
| reason(s) for choosing the degree which isn't working, you may have to: | |
| a) reconsider those reasons, and/or b) reconsider the quality of your | |
| data). 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. | |
| Polynomial fits using double precision tend to "fail" at about | |
| (polynomial) degree 20. Fits using Chebyshev or Legendre series are | |
| generally better conditioned, but much can still 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. | |
| Examples | |
| -------- | |
| >>> np.random.seed(123) | |
| >>> from numpy.polynomial import polynomial as P | |
| >>> x = np.linspace(-1,1,51) # x "data": [-1, -0.96, ..., 0.96, 1] | |
| >>> y = x**3 - x + np.random.randn(len(x)) # x^3 - x + N(0,1) "noise" | |
| >>> c, stats = P.polyfit(x,y,3,full=True) | |
| >>> np.random.seed(123) | |
| >>> c # c[0], c[2] should be approx. 0, c[1] approx. -1, c[3] approx. 1 | |
| array([ 0.01909725, -1.30598256, -0.00577963, 1.02644286]) # may vary | |
| >>> stats # note the large SSR, explaining the rather poor results | |
| [array([ 38.06116253]), 4, array([ 1.38446749, 1.32119158, 0.50443316, # may vary | |
| 0.28853036]), 1.1324274851176597e-014] | |
| Same thing without the added noise | |
| >>> y = x**3 - x | |
| >>> c, stats = P.polyfit(x,y,3,full=True) | |
| >>> c # c[0], c[2] should be "very close to 0", c[1] ~= -1, c[3] ~= 1 | |
| array([-6.36925336e-18, -1.00000000e+00, -4.08053781e-16, 1.00000000e+00]) | |
| >>> stats # note the minuscule SSR | |
| [array([ 7.46346754e-31]), 4, array([ 1.38446749, 1.32119158, # may vary | |
| 0.50443316, 0.28853036]), 1.1324274851176597e-014] | |
| """ | |
| return pu._fit(polyvander, x, y, deg, rcond, full, w) | |
| def polycompanion(c): | |
| """ | |
| Return the companion matrix of c. | |
| The companion matrix for power series cannot be made symmetric by | |
| scaling the basis, so this function differs from those for the | |
| orthogonal polynomials. | |
| Parameters | |
| ---------- | |
| c : array_like | |
| 1-D array of polynomial coefficients ordered from low to high | |
| degree. | |
| Returns | |
| ------- | |
| mat : ndarray | |
| Companion matrix of dimensions (deg, deg). | |
| Notes | |
| ----- | |
| .. versionadded:: 1.7.0 | |
| """ | |
| # c is a trimmed copy | |
| [c] = pu.as_series([c]) | |
| if len(c) < 2: | |
| raise ValueError('Series must have maximum degree of at least 1.') | |
| if len(c) == 2: | |
| return np.array([[-c[0]/c[1]]]) | |
| n = len(c) - 1 | |
| mat = np.zeros((n, n), dtype=c.dtype) | |
| bot = mat.reshape(-1)[n::n+1] | |
| bot[...] = 1 | |
| mat[:, -1] -= c[:-1]/c[-1] | |
| return mat | |
| def polyroots(c): | |
| """ | |
| Compute the roots of a polynomial. | |
| Return the roots (a.k.a. "zeros") of the polynomial | |
| .. math:: p(x) = \\sum_i c[i] * x^i. | |
| Parameters | |
| ---------- | |
| c : 1-D array_like | |
| 1-D array of polynomial coefficients. | |
| Returns | |
| ------- | |
| out : ndarray | |
| Array of the roots of the polynomial. If all the roots are real, | |
| then `out` is also real, otherwise it is complex. | |
| See Also | |
| -------- | |
| numpy.polynomial.chebyshev.chebroots | |
| numpy.polynomial.legendre.legroots | |
| numpy.polynomial.laguerre.lagroots | |
| numpy.polynomial.hermite.hermroots | |
| numpy.polynomial.hermite_e.hermeroots | |
| Notes | |
| ----- | |
| The root estimates are obtained as the eigenvalues of the companion | |
| matrix, Roots far from the origin of the complex plane may have large | |
| errors due to the numerical instability of the power series for such | |
| values. Roots with multiplicity greater than 1 will also show larger | |
| errors as the value of the series near such points is relatively | |
| insensitive to errors in the roots. Isolated roots near the origin can | |
| be improved by a few iterations of Newton's method. | |
| Examples | |
| -------- | |
| >>> import numpy.polynomial.polynomial as poly | |
| >>> poly.polyroots(poly.polyfromroots((-1,0,1))) | |
| array([-1., 0., 1.]) | |
| >>> poly.polyroots(poly.polyfromroots((-1,0,1))).dtype | |
| dtype('float64') | |
| >>> j = complex(0,1) | |
| >>> poly.polyroots(poly.polyfromroots((-j,0,j))) | |
| array([ 0.00000000e+00+0.j, 0.00000000e+00+1.j, 2.77555756e-17-1.j]) # may vary | |
| """ | |
| # c is a trimmed copy | |
| [c] = pu.as_series([c]) | |
| if len(c) < 2: | |
| return np.array([], dtype=c.dtype) | |
| if len(c) == 2: | |
| return np.array([-c[0]/c[1]]) | |
| # rotated companion matrix reduces error | |
| m = polycompanion(c)[::-1,::-1] | |
| r = la.eigvals(m) | |
| r.sort() | |
| return r | |
| # | |
| # polynomial class | |
| # | |
| class Polynomial(ABCPolyBase): | |
| """A power series class. | |
| The Polynomial class provides the standard Python numerical methods | |
| '+', '-', '*', '//', '%', 'divmod', '**', and '()' as well as the | |
| attributes and methods listed in the `ABCPolyBase` documentation. | |
| Parameters | |
| ---------- | |
| coef : array_like | |
| Polynomial coefficients in order of increasing degree, i.e., | |
| ``(1, 2, 3)`` give ``1 + 2*x + 3*x**2``. | |
| domain : (2,) array_like, optional | |
| Domain to use. The interval ``[domain[0], domain[1]]`` is mapped | |
| to the interval ``[window[0], window[1]]`` by shifting and scaling. | |
| The default value is [-1, 1]. | |
| window : (2,) array_like, optional | |
| Window, see `domain` for its use. The default value is [-1, 1]. | |
| .. versionadded:: 1.6.0 | |
| """ | |
| # Virtual Functions | |
| _add = staticmethod(polyadd) | |
| _sub = staticmethod(polysub) | |
| _mul = staticmethod(polymul) | |
| _div = staticmethod(polydiv) | |
| _pow = staticmethod(polypow) | |
| _val = staticmethod(polyval) | |
| _int = staticmethod(polyint) | |
| _der = staticmethod(polyder) | |
| _fit = staticmethod(polyfit) | |
| _line = staticmethod(polyline) | |
| _roots = staticmethod(polyroots) | |
| _fromroots = staticmethod(polyfromroots) | |
| # Virtual properties | |
| domain = np.array(polydomain) | |
| window = np.array(polydomain) | |
| basis_name = None | |
| def _str_term_unicode(cls, i, arg_str): | |
| return f"·{arg_str}{i.translate(cls._superscript_mapping)}" | |
| def _str_term_ascii(i, arg_str): | |
| return f" {arg_str}**{i}" | |
| def _repr_latex_term(i, arg_str, needs_parens): | |
| if needs_parens: | |
| arg_str = rf"\left({arg_str}\right)" | |
| if i == 0: | |
| return '1' | |
| elif i == 1: | |
| return arg_str | |
| else: | |
| return f"{arg_str}^{{{i}}}" | |