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""" Objects for dealing with Laguerre series.
This module provides a number of objects (mostly functions) useful for dealing with Laguerre series, including a `Laguerre` class that encapsulates the usual arithmetic operations. (General information on how this module represents and works with such polynomials is in the docstring for its "parent" sub-package, `numpy.polynomial`).
Constants --------- - `lagdomain` -- Laguerre series default domain, [-1,1]. - `lagzero` -- Laguerre series that evaluates identically to 0. - `lagone` -- Laguerre series that evaluates identically to 1. - `lagx` -- Laguerre series for the identity map, ``f(x) = x``.
Arithmetic ---------- - `lagadd` -- add two Laguerre series. - `lagsub` -- subtract one Laguerre series from another. - `lagmulx` -- multiply a Laguerre series in ``P_i(x)`` by ``x``. - `lagmul` -- multiply two Laguerre series. - `lagdiv` -- divide one Laguerre series by another. - `lagpow` -- raise a Laguerre series to a positive integer power. - `lagval` -- evaluate a Laguerre series at given points. - `lagval2d` -- evaluate a 2D Laguerre series at given points. - `lagval3d` -- evaluate a 3D Laguerre series at given points. - `laggrid2d` -- evaluate a 2D Laguerre series on a Cartesian product. - `laggrid3d` -- evaluate a 3D Laguerre series on a Cartesian product.
Calculus -------- - `lagder` -- differentiate a Laguerre series. - `lagint` -- integrate a Laguerre series.
Misc Functions -------------- - `lagfromroots` -- create a Laguerre series with specified roots. - `lagroots` -- find the roots of a Laguerre series. - `lagvander` -- Vandermonde-like matrix for Laguerre polynomials. - `lagvander2d` -- Vandermonde-like matrix for 2D power series. - `lagvander3d` -- Vandermonde-like matrix for 3D power series. - `laggauss` -- Gauss-Laguerre quadrature, points and weights. - `lagweight` -- Laguerre weight function. - `lagcompanion` -- symmetrized companion matrix in Laguerre form. - `lagfit` -- least-squares fit returning a Laguerre series. - `lagtrim` -- trim leading coefficients from a Laguerre series. - `lagline` -- Laguerre series of given straight line. - `lag2poly` -- convert a Laguerre series to a polynomial. - `poly2lag` -- convert a polynomial to a Laguerre series.
Classes ------- - `Laguerre` -- A Laguerre series class.
See also -------- `numpy.polynomial`
"""
'lagzero', 'lagone', 'lagx', 'lagdomain', 'lagline', 'lagadd', 'lagsub', 'lagmulx', 'lagmul', 'lagdiv', 'lagpow', 'lagval', 'lagder', 'lagint', 'lag2poly', 'poly2lag', 'lagfromroots', 'lagvander', 'lagfit', 'lagtrim', 'lagroots', 'Laguerre', 'lagval2d', 'lagval3d', 'laggrid2d', 'laggrid3d', 'lagvander2d', 'lagvander3d', 'lagcompanion', 'laggauss', 'lagweight']
""" poly2lag(pol)
Convert a polynomial to a Laguerre series.
Convert an array representing the coefficients of a polynomial (relative to the "standard" basis) ordered from lowest degree to highest, to an array of the coefficients of the equivalent Laguerre series, ordered from lowest to highest degree.
Parameters ---------- pol : array_like 1-D array containing the polynomial coefficients
Returns ------- c : ndarray 1-D array containing the coefficients of the equivalent Laguerre series.
See Also -------- lag2poly
Notes ----- The easy way to do conversions between polynomial basis sets is to use the convert method of a class instance.
Examples -------- >>> from numpy.polynomial.laguerre import poly2lag >>> poly2lag(np.arange(4)) array([ 23., -63., 58., -18.])
""" [pol] = pu.as_series([pol]) deg = len(pol) - 1 res = 0 for i in range(deg, -1, -1): res = lagadd(lagmulx(res), pol[i]) return res
""" Convert a Laguerre series to a polynomial.
Convert an array representing the coefficients of a Laguerre series, ordered from lowest degree to highest, to an array of the coefficients of the equivalent polynomial (relative to the "standard" basis) ordered from lowest to highest degree.
Parameters ---------- c : array_like 1-D array containing the Laguerre series coefficients, ordered from lowest order term to highest.
Returns ------- pol : ndarray 1-D array containing the coefficients of the equivalent polynomial (relative to the "standard" basis) ordered from lowest order term to highest.
See Also -------- poly2lag
Notes ----- The easy way to do conversions between polynomial basis sets is to use the convert method of a class instance.
Examples -------- >>> from numpy.polynomial.laguerre import lag2poly >>> lag2poly([ 23., -63., 58., -18.]) array([ 0., 1., 2., 3.])
""" from .polynomial import polyadd, polysub, polymulx
[c] = pu.as_series([c]) n = len(c) if n == 1: return c else: c0 = c[-2] c1 = c[-1] # i is the current degree of c1 for i in range(n - 1, 1, -1): tmp = c0 c0 = polysub(c[i - 2], (c1*(i - 1))/i) c1 = polyadd(tmp, polysub((2*i - 1)*c1, polymulx(c1))/i) return polyadd(c0, polysub(c1, polymulx(c1)))
# # These are constant arrays are of integer type so as to be compatible # with the widest range of other types, such as Decimal. #
# Laguerre
# Laguerre coefficients representing zero.
# Laguerre coefficients representing one.
# Laguerre coefficients representing the identity x.
""" Laguerre series whose graph is a straight line.
Parameters ---------- off, scl : scalars The specified line is given by ``off + scl*x``.
Returns ------- y : ndarray This module's representation of the Laguerre series for ``off + scl*x``.
See Also -------- polyline, chebline
Examples -------- >>> from numpy.polynomial.laguerre import lagline, lagval >>> lagval(0,lagline(3, 2)) 3.0 >>> lagval(1,lagline(3, 2)) 5.0
""" if scl != 0: return np.array([off + scl, -scl]) else: return np.array([off])
""" Generate a Laguerre series with given roots.
The function returns the coefficients of the polynomial
.. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n),
in Laguerre form, 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 * L_1(x) + ... + c_n * L_n(x)
The coefficient of the last term is not generally 1 for monic polynomials in Laguerre form.
Parameters ---------- roots : array_like Sequence containing the roots.
Returns ------- out : ndarray 1-D array of coefficients. If all roots are real then `out` is a real array, if some of the roots are complex, then `out` is complex even if all the coefficients in the result are real (see Examples below).
See Also -------- polyfromroots, legfromroots, chebfromroots, hermfromroots, hermefromroots.
Examples -------- >>> from numpy.polynomial.laguerre import lagfromroots, lagval >>> coef = lagfromroots((-1, 0, 1)) >>> lagval((-1, 0, 1), coef) array([ 0., 0., 0.]) >>> coef = lagfromroots((-1j, 1j)) >>> lagval((-1j, 1j), coef) array([ 0.+0.j, 0.+0.j])
""" if len(roots) == 0: return np.ones(1) else: [roots] = pu.as_series([roots], trim=False) roots.sort() p = [lagline(-r, 1) for r in roots] n = len(p) while n > 1: m, r = divmod(n, 2) tmp = [lagmul(p[i], p[i+m]) for i in range(m)] if r: tmp[0] = lagmul(tmp[0], p[-1]) p = tmp n = m return p[0]
""" Add one Laguerre series to another.
Returns the sum of two Laguerre series `c1` + `c2`. The arguments are sequences of coefficients ordered from lowest order term to highest, i.e., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.
Parameters ---------- c1, c2 : array_like 1-D arrays of Laguerre series coefficients ordered from low to high.
Returns ------- out : ndarray Array representing the Laguerre series of their sum.
See Also -------- lagsub, lagmulx, lagmul, lagdiv, lagpow
Notes ----- Unlike multiplication, division, etc., the sum of two Laguerre series is a Laguerre series (without having to "reproject" the result onto the basis set) so addition, just like that of "standard" polynomials, is simply "component-wise."
Examples -------- >>> from numpy.polynomial.laguerre import lagadd >>> lagadd([1, 2, 3], [1, 2, 3, 4]) array([ 2., 4., 6., 4.])
""" # 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 Laguerre series from another.
Returns the difference of two Laguerre series `c1` - `c2`. The sequences of coefficients are from lowest order term to highest, i.e., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.
Parameters ---------- c1, c2 : array_like 1-D arrays of Laguerre series coefficients ordered from low to high.
Returns ------- out : ndarray Of Laguerre series coefficients representing their difference.
See Also -------- lagadd, lagmulx, lagmul, lagdiv, lagpow
Notes ----- Unlike multiplication, division, etc., the difference of two Laguerre series is a Laguerre series (without having to "reproject" the result onto the basis set) so subtraction, just like that of "standard" polynomials, is simply "component-wise."
Examples -------- >>> from numpy.polynomial.laguerre import lagsub >>> lagsub([1, 2, 3, 4], [1, 2, 3]) array([ 0., 0., 0., 4.])
""" # 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 Laguerre series by x.
Multiply the Laguerre series `c` by x, where x is the independent variable.
Parameters ---------- c : array_like 1-D array of Laguerre series coefficients ordered from low to high.
Returns ------- out : ndarray Array representing the result of the multiplication.
See Also -------- lagadd, lagsub, lagmul, lagdiv, lagpow
Notes ----- The multiplication uses the recursion relationship for Laguerre polynomials in the form
.. math::
xP_i(x) = (-(i + 1)*P_{i + 1}(x) + (2i + 1)P_{i}(x) - iP_{i - 1}(x))
Examples -------- >>> from numpy.polynomial.laguerre import lagmulx >>> lagmulx([1, 2, 3]) array([ -1., -1., 11., -9.])
""" # 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] prd[1] = -c[0] for i in range(1, len(c)): prd[i + 1] = -c[i]*(i + 1) prd[i] += c[i]*(2*i + 1) prd[i - 1] -= c[i]*i return prd
""" Multiply one Laguerre series by another.
Returns the product of two Laguerre series `c1` * `c2`. The arguments are sequences of coefficients, from lowest order "term" to highest, e.g., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.
Parameters ---------- c1, c2 : array_like 1-D arrays of Laguerre series coefficients ordered from low to high.
Returns ------- out : ndarray Of Laguerre series coefficients representing their product.
See Also -------- lagadd, lagsub, lagmulx, lagdiv, lagpow
Notes ----- In general, the (polynomial) product of two C-series results in terms that are not in the Laguerre polynomial basis set. Thus, to express the product as a Laguerre series, it is necessary to "reproject" the product onto said basis set, which may produce "unintuitive" (but correct) results; see Examples section below.
Examples -------- >>> from numpy.polynomial.laguerre import lagmul >>> lagmul([1, 2, 3], [0, 1, 2]) array([ 8., -13., 38., -51., 36.])
""" # s1, s2 are trimmed copies [c1, c2] = pu.as_series([c1, c2])
if len(c1) > len(c2): c = c2 xs = c1 else: c = c1 xs = c2
if len(c) == 1: c0 = c[0]*xs c1 = 0 elif len(c) == 2: c0 = c[0]*xs c1 = c[1]*xs else: nd = len(c) c0 = c[-2]*xs c1 = c[-1]*xs for i in range(3, len(c) + 1): tmp = c0 nd = nd - 1 c0 = lagsub(c[-i]*xs, (c1*(nd - 1))/nd) c1 = lagadd(tmp, lagsub((2*nd - 1)*c1, lagmulx(c1))/nd) return lagadd(c0, lagsub(c1, lagmulx(c1)))
""" Divide one Laguerre series by another.
Returns the quotient-with-remainder of two Laguerre series `c1` / `c2`. The arguments are sequences of coefficients from lowest order "term" to highest, e.g., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.
Parameters ---------- c1, c2 : array_like 1-D arrays of Laguerre series coefficients ordered from low to high.
Returns ------- [quo, rem] : ndarrays Of Laguerre series coefficients representing the quotient and remainder.
See Also -------- lagadd, lagsub, lagmulx, lagmul, lagpow
Notes ----- In general, the (polynomial) division of one Laguerre series by another results in quotient and remainder terms that are not in the Laguerre polynomial basis set. Thus, to express these results as a Laguerre series, it is necessary to "reproject" the results onto the Laguerre basis set, which may produce "unintuitive" (but correct) results; see Examples section below.
Examples -------- >>> from numpy.polynomial.laguerre import lagdiv >>> lagdiv([ 8., -13., 38., -51., 36.], [0, 1, 2]) (array([ 1., 2., 3.]), array([ 0.])) >>> lagdiv([ 9., -12., 38., -51., 36.], [0, 1, 2]) (array([ 1., 2., 3.]), array([ 1., 1.]))
""" # c1, c2 are trimmed copies [c1, c2] = pu.as_series([c1, c2]) if c2[-1] == 0: raise ZeroDivisionError()
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: quo = np.empty(lc1 - lc2 + 1, dtype=c1.dtype) rem = c1 for i in range(lc1 - lc2, - 1, -1): p = lagmul([0]*i + [1], c2) q = rem[-1]/p[-1] rem = rem[:-1] - q*p[:-1] quo[i] = q return quo, pu.trimseq(rem)
"""Raise a Laguerre series to a power.
Returns the Laguerre series `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 ``P_0 + 2*P_1 + 3*P_2.``
Parameters ---------- c : array_like 1-D array of Laguerre series coefficients ordered from low to high. 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 Laguerre series of power.
See Also -------- lagadd, lagsub, lagmulx, lagmul, lagdiv
Examples -------- >>> from numpy.polynomial.laguerre import lagpow >>> lagpow([1, 2, 3], 2) array([ 14., -16., 56., -72., 54.])
""" # 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 = lagmul(prd, c) return prd
""" Differentiate a Laguerre series.
Returns the Laguerre series 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 series ``1*L_0 + 2*L_1 + 3*L_2`` while [[1,2],[1,2]] represents ``1*L_0(x)*L_0(y) + 1*L_1(x)*L_0(y) + 2*L_0(x)*L_1(y) + 2*L_1(x)*L_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``.
Parameters ---------- c : array_like Array of Laguerre series 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 Laguerre series of the derivative.
See Also -------- lagint
Notes ----- In general, the result of differentiating a Laguerre series does not resemble the same operation on a power series. Thus the result of this function may be "unintuitive," albeit correct; see Examples section below.
Examples -------- >>> from numpy.polynomial.laguerre import lagder >>> lagder([ 1., 1., 1., -3.]) array([ 1., 2., 3.]) >>> lagder([ 1., 0., 0., -4., 3.], m=2) array([ 1., 2., 3.])
""" c = np.array(c, ndmin=1, copy=1) if c.dtype.char in '?bBhHiIlLqQpP': c = c.astype(np.double) 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=c.dtype) for j in range(n, 1, -1): der[j - 1] = -c[j] c[j - 1] += c[j] der[0] = -c[1] c = der c = np.moveaxis(c, 0, iaxis) return c
""" Integrate a Laguerre series.
Returns the Laguerre series 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 series ``L_0 + 2*L_1 + 3*L_2`` while [[1,2],[1,2]] represents ``1*L_0(x)*L_0(y) + 1*L_1(x)*L_0(y) + 2*L_0(x)*L_1(y) + 2*L_1(x)*L_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``.
Parameters ---------- c : array_like Array of Laguerre series 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 Order of integration, must be positive. (Default: 1) k : {[], list, scalar}, optional Integration constant(s). The value of the first integral at ``lbnd`` is the first value in the list, the value of the second integral at ``lbnd`` 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 Laguerre series coefficients of the integral.
Raises ------ ValueError If ``m < 0``, ``len(k) > m``, ``np.ndim(lbnd) != 0``, or ``np.ndim(scl) != 0``.
See Also -------- lagder
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.
Also note that, in general, the result of integrating a C-series needs to be "reprojected" onto the C-series basis set. Thus, typically, the result of this function is "unintuitive," albeit correct; see Examples section below.
Examples -------- >>> from numpy.polynomial.laguerre import lagint >>> lagint([1,2,3]) array([ 1., 1., 1., -3.]) >>> lagint([1,2,3], m=2) array([ 1., 0., 0., -4., 3.]) >>> lagint([1,2,3], k=1) array([ 2., 1., 1., -3.]) >>> lagint([1,2,3], lbnd=-1) array([ 11.5, 1. , 1. , -3. ]) >>> lagint([1,2], m=2, k=[1,2], lbnd=-1) array([ 11.16666667, -5. , -3. , 2. ])
""" c = np.array(c, ndmin=1, copy=1) if c.dtype.char in '?bBhHiIlLqQpP': c = c.astype(np.double) 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
c = np.moveaxis(c, iaxis, 0) k = list(k) + [0]*(cnt - len(k)) 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=c.dtype) tmp[0] = c[0] tmp[1] = -c[0] for j in range(1, n): tmp[j] += c[j] tmp[j + 1] = -c[j] tmp[0] += k[i] - lagval(lbnd, tmp) c = tmp c = np.moveaxis(c, 0, iaxis) return c
""" Evaluate a Laguerre series at points x.
If `c` is of length `n + 1`, this function returns the value:
.. math:: p(x) = c_0 * L_0(x) + c_1 * L_1(x) + ... + c_n * L_n(x)
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, algebra_like The shape of the return value is described above.
See Also -------- lagval2d, laggrid2d, lagval3d, laggrid3d
Notes ----- The evaluation uses Clenshaw recursion, aka synthetic division.
Examples -------- >>> from numpy.polynomial.laguerre import lagval >>> coef = [1,2,3] >>> lagval(1, coef) -0.5 >>> lagval([[1,2],[3,4]], coef) array([[-0.5, -4. ], [-4.5, -2. ]])
""" c = np.array(c, ndmin=1, copy=0) if c.dtype.char in '?bBhHiIlLqQpP': c = c.astype(np.double) if isinstance(x, (tuple, list)): x = np.asarray(x) if isinstance(x, np.ndarray) and tensor: c = c.reshape(c.shape + (1,)*x.ndim)
if len(c) == 1: c0 = c[0] c1 = 0 elif len(c) == 2: c0 = c[0] c1 = c[1] else: nd = len(c) c0 = c[-2] c1 = c[-1] for i in range(3, len(c) + 1): tmp = c0 nd = nd - 1 c0 = c[-i] - (c1*(nd - 1))/nd c1 = tmp + (c1*((2*nd - 1) - x))/nd return c0 + c1*(1 - x)
""" Evaluate a 2-D Laguerre series at points (x, y).
This function returns the values:
.. math:: p(x,y) = \\sum_{i,j} c_{i,j} * L_i(x) * L_j(y)
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` is a 1-D array a one is 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 -------- lagval, laggrid2d, lagval3d, laggrid3d
Notes -----
.. versionadded:: 1.7.0
""" try: x, y = np.array((x, y), copy=0) except Exception: raise ValueError('x, y are incompatible')
c = lagval(x, c) c = lagval(y, c, tensor=False) return c
""" Evaluate a 2-D Laguerre series on the Cartesian product of x and y.
This function returns the values:
.. math:: p(a,b) = \\sum_{i,j} c_{i,j} * L_i(a) * L_j(b)
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 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 Chebyshev series at points in the Cartesian product of `x` and `y`.
See Also -------- lagval, lagval2d, lagval3d, laggrid3d
Notes -----
.. versionadded:: 1.7.0
""" c = lagval(x, c) c = lagval(y, c) return c
""" Evaluate a 3-D Laguerre series at points (x, y, z).
This function returns the values:
.. math:: p(x,y,z) = \\sum_{i,j,k} c_{i,j,k} * L_i(x) * L_j(y) * L_k(z)
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 multidimension polynomial on points formed with triples of corresponding values from `x`, `y`, and `z`.
See Also -------- lagval, lagval2d, laggrid2d, laggrid3d
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 = lagval(x, c) c = lagval(y, c, tensor=False) c = lagval(z, c, tensor=False) return c
""" Evaluate a 3-D Laguerre series 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} * L_i(a) * L_j(b) * L_k(c)
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 -------- lagval, lagval2d, laggrid2d, lagval3d
Notes -----
.. versionadded:: 1.7.0
""" c = lagval(x, c) c = lagval(y, c) c = lagval(z, c) return c
"""Pseudo-Vandermonde matrix of given degree.
Returns the pseudo-Vandermonde matrix of degree `deg` and sample points `x`. The pseudo-Vandermonde matrix is defined by
.. math:: V[..., i] = L_i(x)
where `0 <= i <= deg`. The leading indices of `V` index the elements of `x` and the last index is the degree of the Laguerre polynomial.
If `c` is a 1-D array of coefficients of length `n + 1` and `V` is the array ``V = lagvander(x, n)``, then ``np.dot(V, c)`` and ``lagval(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 Laguerre series 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 pseudo-Vandermonde matrix. The shape of the returned matrix is ``x.shape + (deg + 1,)``, where The last index is the degree of the corresponding Laguerre polynomial. The dtype will be the same as the converted `x`.
Examples -------- >>> from numpy.polynomial.laguerre import lagvander >>> x = np.array([0, 1, 2]) >>> lagvander(x, 3) array([[ 1. , 1. , 1. , 1. ], [ 1. , 0. , -0.5 , -0.66666667], [ 1. , -1. , -1. , -0.33333333]])
""" 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] = 1 - x for i in range(2, ideg + 1): v[i] = (v[i-1]*(2*i - 1 - x) - v[i-2]*(i - 1))/i 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] = L_i(x) * L_j(y),
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 degrees of the Laguerre polynomials.
If ``V = lagvander2d(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 ``lagval2d(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 Laguerre series 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 -------- lagvander, lagvander3d. lagval2d, lagval3d
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]: raise ValueError("degrees must be non-negative integers") degx, degy = ideg x, y = np.array((x, y), copy=0) + 0.0
vx = lagvander(x, degx) vy = lagvander(y, degy) v = vx[..., None]*vy[..., None,:] 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] = L_i(x)*L_j(y)*L_k(z),
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 degrees of the Laguerre polynomials.
If ``V = lagvander3d(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 ``lagval3d(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 Laguerre series 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 -------- lagvander, lagvander3d. lagval2d, lagval3d
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 = lagvander(x, degx) vy = lagvander(y, degy) vz = lagvander(z, degz) v = vx[..., None, None]*vy[..., None,:, None]*vz[..., None, None,:] return v.reshape(v.shape[:-3] + (-1,))
""" Least squares fit of Laguerre series to data.
Return the coefficients of a Laguerre series 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 * L_1(x) + ... + c_n * L_n(x),
where `n` is `deg`.
Parameters ---------- x : array_like, shape (M,) x-coordinates of the M sample points ``(x[i], y[i])``. y : array_like, shape (M,) or (M, K) y-coordinates of the sample points. Several data sets of sample points sharing the same x-coordinates can be fitted at once by passing in a 2D-array that contains one dataset 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 this relative to the largest singular value will be ignored. The default value is len(x)*eps, where eps is the relative precision of the float type, about 2e-16 in most cases. full : bool, optional Switch determining nature of return value. When it is False (the default) just the coefficients are returned, when True diagnostic information from the singular value decomposition 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.
Returns ------- coef : ndarray, shape (M,) or (M, K) Laguerre coefficients ordered from low to high. If `y` was 2-D, the coefficients for the data in column k of `y` are in column `k`.
[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`.
Warns ----- RankWarning The rank of the coefficient matrix in the least-squares fit is deficient. The warning is only raised if `full` = False. The warnings can be turned off by
>>> import warnings >>> warnings.simplefilter('ignore', RankWarning)
See Also -------- chebfit, legfit, polyfit, hermfit, hermefit lagval : Evaluates a Laguerre series. lagvander : pseudo Vandermonde matrix of Laguerre series. lagweight : Laguerre weight function. linalg.lstsq : Computes a least-squares fit from the matrix. scipy.interpolate.UnivariateSpline : Computes spline fits.
Notes ----- The solution is the coefficients of the Laguerre series `p` that minimizes the sum of the weighted squared errors
.. math:: E = \\sum_j w_j^2 * |y_j - p(x_j)|^2,
where the :math:`w_j` are the weights. This problem is solved by setting up as the (typically) overdetermined matrix equation
.. math:: V(x) * c = w * y,
where `V` is the weighted pseudo Vandermonde matrix of `x`, `c` are the coefficients to be solved for, `w` are the weights, and `y` are the observed values. This equation is then solved using the singular value decomposition of `V`.
If some of the singular values of `V` are so small that they are neglected, then a `RankWarning` will be issued. This means that the coefficient values may be poorly determined. Using a lower order fit will usually get rid of the warning. The `rcond` parameter can also be set to a value smaller than its default, but the resulting fit may be spurious and have large contributions from roundoff error.
Fits using Laguerre series are probably most useful when the data can be approximated by ``sqrt(w(x)) * p(x)``, where `w(x)` is the Laguerre weight. In that case the weight ``sqrt(w(x[i])`` should be used together with data values ``y[i]/sqrt(w(x[i])``. The weight function is available as `lagweight`.
References ---------- .. [1] Wikipedia, "Curve fitting", https://en.wikipedia.org/wiki/Curve_fitting
Examples -------- >>> from numpy.polynomial.laguerre import lagfit, lagval >>> x = np.linspace(0, 10) >>> err = np.random.randn(len(x))/10 >>> y = lagval(x, [1, 2, 3]) + err >>> lagfit(x, y, 2) array([ 0.96971004, 2.00193749, 3.00288744])
""" 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 = lagvander(x, lmax) else: deg = np.sort(deg) lmax = deg[-1] order = len(deg) van = lagvander(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 > 0: 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 usual companion matrix of the Laguerre polynomials is already symmetric when `c` is a basis Laguerre polynomial, so no scaling is applied.
Parameters ---------- c : array_like 1-D array of Laguerre series 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([[1 + c[0]/c[1]]])
n = len(c) - 1 mat = np.zeros((n, n), dtype=c.dtype) top = mat.reshape(-1)[1::n+1] mid = mat.reshape(-1)[0::n+1] bot = mat.reshape(-1)[n::n+1] top[...] = -np.arange(1, n) mid[...] = 2.*np.arange(n) + 1. bot[...] = top mat[:, -1] += (c[:-1]/c[-1])*n return mat
""" Compute the roots of a Laguerre series.
Return the roots (a.k.a. "zeros") of the polynomial
.. math:: p(x) = \\sum_i c[i] * L_i(x).
Parameters ---------- c : 1-D array_like 1-D array of coefficients.
Returns ------- out : ndarray Array of the roots of the series. If all the roots are real, then `out` is also real, otherwise it is complex.
See Also -------- polyroots, legroots, chebroots, hermroots, 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 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.
The Laguerre series basis polynomials aren't powers of `x` so the results of this function may seem unintuitive.
Examples -------- >>> from numpy.polynomial.laguerre import lagroots, lagfromroots >>> coef = lagfromroots([0, 1, 2]) >>> coef array([ 2., -8., 12., -6.]) >>> lagroots(coef) array([ -4.44089210e-16, 1.00000000e+00, 2.00000000e+00])
""" # c is a trimmed copy [c] = pu.as_series([c]) if len(c) <= 1: return np.array([], dtype=c.dtype) if len(c) == 2: return np.array([1 + c[0]/c[1]])
m = lagcompanion(c) r = la.eigvals(m) r.sort() return r
""" Gauss-Laguerre quadrature.
Computes the sample points and weights for Gauss-Laguerre quadrature. These sample points and weights will correctly integrate polynomials of degree :math:`2*deg - 1` or less over the interval :math:`[0, \\inf]` with the weight function :math:`f(x) = \\exp(-x)`.
Parameters ---------- deg : int Number of sample points and weights. It must be >= 1.
Returns ------- x : ndarray 1-D ndarray containing the sample points. y : ndarray 1-D ndarray containing the weights.
Notes -----
.. versionadded:: 1.7.0
The results have only been tested up to degree 100 higher degrees may be problematic. The weights are determined by using the fact that
.. math:: w_k = c / (L'_n(x_k) * L_{n-1}(x_k))
where :math:`c` is a constant independent of :math:`k` and :math:`x_k` is the k'th root of :math:`L_n`, and then scaling the results to get the right value when integrating 1.
""" ideg = int(deg) if ideg != deg or ideg < 1: raise ValueError("deg must be a non-negative integer")
# first approximation of roots. We use the fact that the companion # matrix is symmetric in this case in order to obtain better zeros. c = np.array([0]*deg + [1]) m = lagcompanion(c) x = la.eigvalsh(m)
# improve roots by one application of Newton dy = lagval(x, c) df = lagval(x, lagder(c)) x -= dy/df
# compute the weights. We scale the factor to avoid possible numerical # overflow. fm = lagval(x, c[1:]) fm /= np.abs(fm).max() df /= np.abs(df).max() w = 1/(fm * df)
# scale w to get the right value, 1 in this case w /= w.sum()
return x, w
"""Weight function of the Laguerre polynomials.
The weight function is :math:`exp(-x)` and the interval of integration is :math:`[0, \\inf]`. The Laguerre polynomials are orthogonal, but not normalized, with respect to this weight function.
Parameters ---------- x : array_like Values at which the weight function will be computed.
Returns ------- w : ndarray The weight function at `x`.
Notes -----
.. versionadded:: 1.7.0
""" w = np.exp(-x) return w
# # Laguerre series class #
"""A Laguerre series class.
The Laguerre 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 Laguerre coefficients in order of increasing degree, i.e, ``(1, 2, 3)`` gives ``1*L_0(x) + 2*L_1(X) + 3*L_2(x)``. 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 [0, 1]. window : (2,) array_like, optional Window, see `domain` for its use. The default value is [0, 1].
.. versionadded:: 1.6.0
""" # Virtual Functions
# Virtual properties |