neulab/docprompting-conala huggingface dataset:
numpy.polynomial.legendre.leggauss   polynomial.legendre.leggauss(deg)[source]
 
Gauss-Legendre quadrature. Computes the sample points and weights for Gauss-Legendre quadrature. These sample points and weights will correctly integrate polynomials of degree \(2*deg - 1\) or less over the interval \([-1, 1]\) with the weight function \(f(x) = 1\).  Parameters 
 
degint


Number of sample points and weights. It must be >= 1.    Returns 
 
xndarray


1-D ndarray containing the sample points.  
yndarray


1-D ndarray containing the weights.     Notes  New in version 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  \[w_k = c / (L'_n(x_k) * L_{n-1}(x_k))\] where \(c\) is a constant independent of \(k\) and \(x_k\) is the k’th root of \(L_n\), and then scaling the results to get the right value when integrating 1.
numpy.polynomial.legendre.leggrid2d   polynomial.legendre.leggrid2d(x, y, c)[source]
 
Evaluate a 2-D Legendre series on the Cartesian product of x and y. This function returns the values:  \[p(a,b) = \sum_{i,j} c_{i,j} * L_i(a) * L_j(b)\] 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, yarray_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.  
carray_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 
 
valuesndarray, compatible object


The values of the two dimensional Chebyshev series at points in the Cartesian product of x and y.      See also  
legval, legval2d, legval3d, leggrid3d

  Notes  New in version 1.7.0.
numpy.polynomial.legendre.leggrid3d   polynomial.legendre.leggrid3d(x, y, z, c)[source]
 
Evaluate a 3-D Legendre series on the Cartesian product of x, y, and z. This function returns the values:  \[p(a,b,c) = \sum_{i,j,k} c_{i,j,k} * L_i(a) * L_j(b) * L_k(c)\] 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, zarray_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.  
carray_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 
 
valuesndarray, compatible object


The values of the two dimensional polynomial at points in the Cartesian product of x and y.      See also  
legval, legval2d, leggrid2d, legval3d

  Notes  New in version 1.7.0.
numpy.polynomial.legendre.legint   polynomial.legendre.legint(c, m=1, k=[], lbnd=0, scl=1, axis=0)[source]
 
Integrate a Legendre series. Returns the Legendre 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 
 
carray_like


Array of Legendre series coefficients. If c is multidimensional the different axis correspond to different variables with the degree in each axis given by the corresponding index.  
mint, 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.  
lbndscalar, optional


The lower bound of the integral. (Default: 0)  
sclscalar, optional


Following each integration the result is multiplied by scl before the integration constant is added. (Default: 1)  
axisint, optional


Axis over which the integral is taken. (Default: 0).  New in version 1.7.0.     Returns 
 
Sndarray


Legendre series coefficient array of the integral.    Raises 
 ValueError

If m < 0, len(k) > m, np.ndim(lbnd) != 0, or np.ndim(scl) != 0.      See also  legder
  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 \(u = ax + b\) in an integral relative to x. Then \(dx = du/a\), so one will need to set scl equal to \(1/a\) - 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 import legendre as L
>>> c = (1,2,3)
>>> L.legint(c)
array([ 0.33333333,  0.4       ,  0.66666667,  0.6       ]) # may vary
>>> L.legint(c, 3)
array([  1.66666667e-02,  -1.78571429e-02,   4.76190476e-02, # may vary
         -1.73472348e-18,   1.90476190e-02,   9.52380952e-03])
>>> L.legint(c, k=3)
 array([ 3.33333333,  0.4       ,  0.66666667,  0.6       ]) # may vary
>>> L.legint(c, lbnd=-2)
array([ 7.33333333,  0.4       ,  0.66666667,  0.6       ]) # may vary
>>> L.legint(c, scl=2)
array([ 0.66666667,  0.8       ,  1.33333333,  1.2       ]) # may vary
numpy.polynomial.legendre.legline   polynomial.legendre.legline(off, scl)[source]
 
Legendre series whose graph is a straight line.  Parameters 
 
off, sclscalars


The specified line is given by off + scl*x.    Returns 
 
yndarray


This module’s representation of the Legendre series for off + scl*x.      See also  numpy.polynomial.polynomial.polyline
numpy.polynomial.chebyshev.chebline
numpy.polynomial.laguerre.lagline
numpy.polynomial.hermite.hermline
numpy.polynomial.hermite_e.hermeline
  Examples >>> import numpy.polynomial.legendre as L
>>> L.legline(3,2)
array([3, 2])
>>> L.legval(-3, L.legline(3,2)) # should be -3
-3.0
numpy.polynomial.legendre.legmul   polynomial.legendre.legmul(c1, c2)[source]
 
Multiply one Legendre series by another. Returns the product of two Legendre 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, c2array_like


1-D arrays of Legendre series coefficients ordered from low to high.    Returns 
 
outndarray


Of Legendre series coefficients representing their product.      See also  
legadd, legsub, legmulx, legdiv, legpow

  Notes In general, the (polynomial) product of two C-series results in terms that are not in the Legendre polynomial basis set. Thus, to express the product as a Legendre 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 import legendre as L
>>> c1 = (1,2,3)
>>> c2 = (3,2)
>>> L.legmul(c1,c2) # multiplication requires "reprojection"
array([  4.33333333,  10.4       ,  11.66666667,   3.6       ]) # may vary
numpy.polynomial.legendre.legmulx   polynomial.legendre.legmulx(c)[source]
 
Multiply a Legendre series by x. Multiply the Legendre series c by x, where x is the independent variable.  Parameters 
 
carray_like


1-D array of Legendre series coefficients ordered from low to high.    Returns 
 
outndarray


Array representing the result of the multiplication.      See also  
legadd, legmul, legdiv, legpow

  Notes The multiplication uses the recursion relationship for Legendre polynomials in the form  \[xP_i(x) = ((i + 1)*P_{i + 1}(x) + i*P_{i - 1}(x))/(2i + 1)\] Examples >>> from numpy.polynomial import legendre as L
>>> L.legmulx([1,2,3])
array([ 0.66666667, 2.2, 1.33333333, 1.8]) # may vary
numpy.polynomial.legendre.legpow   polynomial.legendre.legpow(c, pow, maxpower=16)[source]
 
Raise a Legendre series to a power. Returns the Legendre 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 
 
carray_like


1-D array of Legendre series coefficients ordered from low to high.  
powinteger


Power to which the series will be raised  
maxpowerinteger, optional


Maximum power allowed. This is mainly to limit growth of the series to unmanageable size. Default is 16    Returns 
 
coefndarray


Legendre series of power.      See also  
legadd, legsub, legmulx, legmul, legdiv
numpy.polynomial.legendre.legroots   polynomial.legendre.legroots(c)[source]
 
Compute the roots of a Legendre series. Return the roots (a.k.a. “zeros”) of the polynomial  \[p(x) = \sum_i c[i] * L_i(x).\]  Parameters 
 
c1-D array_like


1-D array of coefficients.    Returns 
 
outndarray


Array of the roots of the series. If all the roots are real, then out is also real, otherwise it is complex.      See also  numpy.polynomial.polynomial.polyroots
numpy.polynomial.chebyshev.chebroots
numpy.polynomial.laguerre.lagroots
numpy.polynomial.hermite.hermroots
numpy.polynomial.hermite_e.hermeroots
  Notes The root estimates are obtained as the eigenvalues of the companion matrix, Roots far from the origin of the complex plane may have large errors due to the numerical instability of the 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 Legendre series basis polynomials aren’t powers of x so the results of this function may seem unintuitive. Examples >>> import numpy.polynomial.legendre as leg
>>> leg.legroots((1, 2, 3, 4)) # 4L_3 + 3L_2 + 2L_1 + 1L_0, all real roots
array([-0.85099543, -0.11407192,  0.51506735]) # may vary
numpy.polynomial.legendre.legsub   polynomial.legendre.legsub(c1, c2)[source]
 
Subtract one Legendre series from another. Returns the difference of two Legendre 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, c2array_like


1-D arrays of Legendre series coefficients ordered from low to high.    Returns 
 
outndarray


Of Legendre series coefficients representing their difference.      See also  
legadd, legmulx, legmul, legdiv, legpow

  Notes Unlike multiplication, division, etc., the difference of two Legendre series is a Legendre 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 import legendre as L
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> L.legsub(c1,c2)
array([-2.,  0.,  2.])
>>> L.legsub(c2,c1) # -C.legsub(c1,c2)
array([ 2.,  0., -2.])
numpy.polynomial.legendre.legtrim   polynomial.legendre.legtrim(c, tol=0)[source]
 
Remove “small” “trailing” coefficients from a polynomial. “Small” means “small in absolute value” and is controlled by the parameter tol; “trailing” means highest order coefficient(s), e.g., in [0, 1, 1, 0, 0] (which represents 0 + x + x**2 + 0*x**3 + 0*x**4) both the 3-rd and 4-th order coefficients would be “trimmed.”  Parameters 
 
carray_like


1-d array of coefficients, ordered from lowest order to highest.  
tolnumber, optional


Trailing (i.e., highest order) elements with absolute value less than or equal to tol (default value is zero) are removed.    Returns 
 
trimmedndarray


1-d array with trailing zeros removed. If the resulting series would be empty, a series containing a single zero is returned.    Raises 
 ValueError

If tol < 0      See also  trimseq
  Examples >>> from numpy.polynomial import polyutils as pu
>>> pu.trimcoef((0,0,3,0,5,0,0))
array([0.,  0.,  3.,  0.,  5.])
>>> pu.trimcoef((0,0,1e-3,0,1e-5,0,0),1e-3) # item == tol is trimmed
array([0.])
>>> i = complex(0,1) # works for complex
>>> pu.trimcoef((3e-4,1e-3*(1-i),5e-4,2e-5*(1+i)), 1e-3)
array([0.0003+0.j   , 0.001 -0.001j])
numpy.polynomial.legendre.legval   polynomial.legendre.legval(x, c, tensor=True)[source]
 
Evaluate a Legendre series at points x. If c is of length n + 1, this function returns the value:  \[p(x) = c_0 * L_0(x) + c_1 * L_1(x) + ... + c_n * L_n(x)\] 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 
 
xarray_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.  
carray_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.  
tensorboolean, 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.  New in version 1.7.0.     Returns 
 
valuesndarray, algebra_like


The shape of the return value is described above.      See also  
legval2d, leggrid2d, legval3d, leggrid3d

  Notes The evaluation uses Clenshaw recursion, aka synthetic division.
numpy.polynomial.legendre.legval2d   polynomial.legendre.legval2d(x, y, c)[source]
 
Evaluate a 2-D Legendre series at points (x, y). This function returns the values:  \[p(x,y) = \sum_{i,j} c_{i,j} * L_i(x) * L_j(y)\] 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, yarray_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.  
carray_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 
 
valuesndarray, compatible object


The values of the two dimensional Legendre series at points formed from pairs of corresponding values from x and y.      See also  
legval, leggrid2d, legval3d, leggrid3d

  Notes  New in version 1.7.0.
numpy.polynomial.legendre.legval3d   polynomial.legendre.legval3d(x, y, z, c)[source]
 
Evaluate a 3-D Legendre series at points (x, y, z). This function returns the values:  \[p(x,y,z) = \sum_{i,j,k} c_{i,j,k} * L_i(x) * L_j(y) * L_k(z)\] 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, zarray_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.  
carray_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 
 
valuesndarray, compatible object


The values of the multidimensional polynomial on points formed with triples of corresponding values from x, y, and z.      See also  
legval, legval2d, leggrid2d, leggrid3d

  Notes  New in version 1.7.0.
numpy.polynomial.legendre.legvander   polynomial.legendre.legvander(x, deg)[source]
 
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  \[V[..., i] = L_i(x)\] where 0 <= i <= deg. The leading indices of V index the elements of x and the last index is the degree of the Legendre polynomial. If c is a 1-D array of coefficients of length n + 1 and V is the array V = legvander(x, n), then np.dot(V, c) and legval(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 Legendre series of the same degree and sample points.  Parameters 
 
xarray_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.  
degint


Degree of the resulting matrix.    Returns 
 
vanderndarray


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 Legendre polynomial. The dtype will be the same as the converted x.
numpy.polynomial.legendre.legvander2d   polynomial.legendre.legvander2d(x, y, deg)[source]
 
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  \[V[..., (deg[1] + 1)*i + j] = L_i(x) * L_j(y),\] 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 Legendre polynomials. If V = legvander2d(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  \[c_{00}, c_{01}, c_{02} ... , c_{10}, c_{11}, c_{12} ...\] and np.dot(V, c.flat) and legval2d(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 Legendre series of the same degrees and sample points.  Parameters 
 
x, yarray_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.  
deglist of ints


List of maximum degrees of the form [x_deg, y_deg].    Returns 
 
vander2dndarray


The shape of the returned matrix is x.shape + (order,), where \(order = (deg[0]+1)*(deg[1]+1)\). The dtype will be the same as the converted x and y.      See also  
legvander, legvander3d, legval2d, legval3d

  Notes  New in version 1.7.0.
numpy.polynomial.legendre.legvander3d   polynomial.legendre.legvander3d(x, y, z, deg)[source]
 
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  \[V[..., (m+1)(n+1)i + (n+1)j + k] = L_i(x)*L_j(y)*L_k(z),\] 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 Legendre polynomials. If V = legvander3d(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  \[c_{000}, c_{001}, c_{002},... , c_{010}, c_{011}, c_{012},...\] and np.dot(V, c.flat) and legval3d(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 Legendre series of the same degrees and sample points.  Parameters 
 
x, y, zarray_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.  
deglist of ints


List of maximum degrees of the form [x_deg, y_deg, z_deg].    Returns 
 
vander3dndarray


The shape of the returned matrix is x.shape + (order,), where \(order = (deg[0]+1)*(deg[1]+1)*(deg[2]+1)\). The dtype will be the same as the converted x, y, and z.      See also  
legvander, legvander3d, legval2d, legval3d

  Notes  New in version 1.7.0.
numpy.polynomial.legendre.legweight   polynomial.legendre.legweight(x)[source]
 
Weight function of the Legendre polynomials. The weight function is \(1\) and the interval of integration is \([-1, 1]\). The Legendre polynomials are orthogonal, but not normalized, with respect to this weight function.  Parameters 
 
xarray_like


Values at which the weight function will be computed.    Returns 
 
wndarray


The weight function at x.     Notes  New in version 1.7.0.