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""" Objects for dealing with polynomials.
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`).
Constants --------- - `polydomain` -- Polynomial default domain, [-1,1]. - `polyzero` -- (Coefficients of the) "zero polynomial." - `polyone` -- (Coefficients of the) constant polynomial 1. - `polyx` -- (Coefficients of the) identity map polynomial, ``f(x) = x``.
Arithmetic ---------- - `polyadd` -- add two polynomials. - `polysub` -- subtract one polynomial from another. - `polymulx` -- multiply a polynomial in ``P_i(x)`` by ``x``. - `polymul` -- multiply two polynomials. - `polydiv` -- divide one polynomial by another. - `polypow` -- raise a polynomial to a positive integer power. - `polyval` -- evaluate a polynomial at given points. - `polyval2d` -- evaluate a 2D polynomial at given points. - `polyval3d` -- evaluate a 3D polynomial at given points. - `polygrid2d` -- evaluate a 2D polynomial on a Cartesian product. - `polygrid3d` -- evaluate a 3D polynomial on a Cartesian product.
Calculus -------- - `polyder` -- differentiate a polynomial. - `polyint` -- integrate a polynomial.
Misc Functions -------------- - `polyfromroots` -- create a polynomial with specified roots. - `polyroots` -- find the roots of a polynomial. - `polyvalfromroots` -- evaluate a polynomial at given points from roots. - `polyvander` -- Vandermonde-like matrix for powers. - `polyvander2d` -- Vandermonde-like matrix for 2D power series. - `polyvander3d` -- Vandermonde-like matrix for 3D power series. - `polycompanion` -- companion matrix in power series form. - `polyfit` -- least-squares fit returning a polynomial. - `polytrim` -- trim leading coefficients from a polynomial. - `polyline` -- polynomial representing given straight line.
Classes ------- - `Polynomial` -- polynomial class.
See Also -------- `numpy.polynomial`
"""
'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']
# # 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.
# Polynomial coefficients representing zero.
# Polynomial coefficients representing one.
# Polynomial coefficients representing the identity x.
# # Polynomial series functions #
""" 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 -------- chebline
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])
""" 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 -------- chebfromroots, legfromroots, lagfromroots, hermfromroots 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])
""" if len(roots) == 0: return np.ones(1) else: [roots] = pu.as_series([roots], trim=False) roots.sort() p = [polyline(-r, 1) for r in roots] n = len(p) while n > 1: m, r = divmod(n, 2) tmp = [polymul(p[i], p[i+m]) for i in range(m)] if r: tmp[0] = polymul(tmp[0], p[-1]) p = tmp n = m return p[0]
""" 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
""" # c1, c2 are trimmed copies [c1, c2] = pu.as_series([c1, c2]) if len(c1) > len(c2): c1[:c2.size] += c2 ret = c1 else: c2[:c1.size] += c1 ret = c2 return pu.trimseq(ret)
""" 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.])
""" # c1, c2 are trimmed copies [c1, c2] = pu.as_series([c1, c2]) if len(c1) > len(c2): c1[:c2.size] -= c2 ret = c1 else: c2 = -c2 c2[:c1.size] += c1 ret = c2 return pu.trimseq(ret)
"""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
""" 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)
""" 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]))
""" # c1, c2 are trimmed copies [c1, c2] = pu.as_series([c1, c2]) if c2[-1] == 0: raise ZeroDivisionError()
len1 = len(c1) len2 = len(c2) if len2 == 1: return c1/c2[-1], c1[:1]*0 elif len1 < len2: return c1[:1]*0, c1 else: dlen = len1 - len2 scl = c2[-1] c2 = c2[:-1]/scl i = dlen j = len1 - 1 while i >= 0: c1[i:j] -= c2*c1[j] i -= 1 j -= 1 return c1[j+1:]/scl, pu.trimseq(c1[:j+1])
"""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.])
""" # c is a trimmed copy [c] = pu.as_series([c]) power = int(pow) if power != pow or power < 0: raise ValueError("Power must be a non-negative integer.") elif maxpower is not None and power > maxpower: raise ValueError("Power is too large") elif power == 0: return np.array([1], dtype=c.dtype) elif power == 1: return c else: # This can be made more efficient by using powers of two # in the usual way. prd = c for i in range(2, power + 1): prd = np.convolve(prd, c) return prd
""" 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=1) if c.dtype.char in '?bBhHiIlLqQpP': # astype fails with NA c = c + 0.0 cdt = c.dtype cnt, iaxis = [int(t) for t in [m, axis]]
if cnt != m: raise ValueError("The order of derivation must be integer") if cnt < 0: raise ValueError("The order of derivation must be non-negative") if iaxis != axis: raise ValueError("The axis must be integer") 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
""" 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, 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=1) 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, iaxis = [int(t) for t in [m, axis]]
if cnt != m: raise ValueError("The order of integration must be integer") 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.") if iaxis != axis: raise ValueError("The axis must be integer") 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
""" 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=0) 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
""" 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=0) 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)
""" 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
""" try: x, y = np.array((x, y), copy=0) except Exception: raise ValueError('x, y are incompatible')
c = polyval(x, c) c = polyval(y, c, tensor=False) return 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
""" c = polyval(x, c) c = polyval(y, c) return 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
""" try: x, y, z = np.array((x, y, z), copy=0) except Exception: raise ValueError('x, y, z are incompatible')
c = polyval(x, c) c = polyval(y, c, tensor=False) c = polyval(z, c, tensor=False) return 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
""" c = polyval(x, c) c = polyval(y, c) c = polyval(z, c) return c
"""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 = int(deg) if ideg != deg: raise ValueError("deg must be integer") if ideg < 0: raise ValueError("deg must be non-negative")
x = np.array(x, copy=0, 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)
"""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
""" ideg = [int(d) for d in deg] is_valid = [id == d and id >= 0 for id, d in zip(ideg, deg)] if is_valid != [1, 1]: raise ValueError("degrees must be non-negative integers") degx, degy = ideg x, y = np.array((x, y), copy=0) + 0.0
vx = polyvander(x, degx) vy = polyvander(y, degy) v = vx[..., None]*vy[..., None,:] # einsum bug #v = np.einsum("...i,...j->...ij", vx, vy) return v.reshape(v.shape[:-2] + (-1,))
"""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
""" ideg = [int(d) for d in deg] is_valid = [id == d and id >= 0 for id, d in zip(ideg, deg)] if is_valid != [1, 1, 1]: raise ValueError("degrees must be non-negative integers") degx, degy, degz = ideg x, y, z = np.array((x, y, z), copy=0) + 0.0
vx = polyvander(x, degx) vy = polyvander(y, degy) vz = polyvander(z, degz) v = vx[..., None, None]*vy[..., None,:, None]*vz[..., None, None,:] # einsum bug #v = np.einsum("...i, ...j, ...k->...ijk", vx, vy, vz) return v.reshape(v.shape[:-3] + (-1,))
""" 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 `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', RankWarning)
See Also -------- chebfit, legfit, lagfit, hermfit, hermefit polyval : Evaluates a polynomial. polyvander : Vandermonde matrix for powers. 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 -------- >>> 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) >>> 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]) >>> stats # note the large SSR, explaining the rather poor results [array([ 38.06116253]), 4, array([ 1.38446749, 1.32119158, 0.50443316, 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([ -1.73362882e-17, -1.00000000e+00, -2.67471909e-16, 1.00000000e+00]) >>> stats # note the minuscule SSR [array([ 7.46346754e-31]), 4, array([ 1.38446749, 1.32119158, 0.50443316, 0.28853036]), 1.1324274851176597e-014]
""" x = np.asarray(x) + 0.0 y = np.asarray(y) + 0.0 deg = np.asarray(deg)
# check arguments. if deg.ndim > 1 or deg.dtype.kind not in 'iu' or deg.size == 0: raise TypeError("deg must be an int or non-empty 1-D array of int") if deg.min() < 0: raise ValueError("expected deg >= 0") if x.ndim != 1: raise TypeError("expected 1D vector for x") if x.size == 0: raise TypeError("expected non-empty vector for x") if y.ndim < 1 or y.ndim > 2: raise TypeError("expected 1D or 2D array for y") if len(x) != len(y): raise TypeError("expected x and y to have same length")
if deg.ndim == 0: lmax = deg order = lmax + 1 van = polyvander(x, lmax) else: deg = np.sort(deg) lmax = deg[-1] order = len(deg) van = polyvander(x, lmax)[:, deg]
# set up the least squares matrices in transposed form lhs = van.T rhs = y.T if w is not None: w = np.asarray(w) + 0.0 if w.ndim != 1: raise TypeError("expected 1D vector for w") if len(x) != len(w): raise TypeError("expected x and w to have same length") # apply weights. Don't use inplace operations as they # can cause problems with NA. lhs = lhs * w rhs = rhs * w
# set rcond if rcond is None: rcond = len(x)*np.finfo(x.dtype).eps
# Determine the norms of the design matrix columns. if issubclass(lhs.dtype.type, np.complexfloating): scl = np.sqrt((np.square(lhs.real) + np.square(lhs.imag)).sum(1)) else: scl = np.sqrt(np.square(lhs).sum(1)) scl[scl == 0] = 1
# Solve the least squares problem. c, resids, rank, s = la.lstsq(lhs.T/scl, rhs.T, rcond) c = (c.T/scl).T
# Expand c to include non-fitted coefficients which are set to zero if deg.ndim == 1: if c.ndim == 2: cc = np.zeros((lmax + 1, c.shape[1]), dtype=c.dtype) else: cc = np.zeros(lmax + 1, dtype=c.dtype) cc[deg] = c c = cc
# warn on rank reduction if rank != order and not full: msg = "The fit may be poorly conditioned" warnings.warn(msg, pu.RankWarning, stacklevel=2)
if full: return c, [resids, rank, s, rcond] else: return 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
""" 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 -------- chebroots
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])
""" # 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]])
m = polycompanion(c) r = la.eigvals(m) r.sort() return r
# # polynomial class #
"""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
# Virtual properties
def _repr_latex_term(i, arg_str, needs_parens): if needs_parens: arg_str = r'\left({}\right)'.format(arg_str) if i == 0: return '1' elif i == 1: return arg_str else: return '{}^{{{}}}'.format(arg_str, i) |