"""Node sets for the `d`-simplex implemented by ``recursivenodes``"""
import numpy as np
from recursivenodes.utils import (npolys,
multiindex_equal,
multiindex_up_to,
coord_map)
from recursivenodes.quadrature import gaussjacobi, lobattogaussjacobi
from recursivenodes.polynomials import proriolkoornwinderdubinervandermonde
[docs]def equispaced(d, n, domain='biunit'):
'''Equispaced nodes for polynomials up to degree `n` on the `d`-simplex.
Args:
d (int): The dimension of the simplex.
n (int): The polynomial degree
domain (str, optional) -- The domain of the simplex. See
":ref:`domains`" for the choices and their formal definitions.
Returns:
ndarray: Equispaced nodes as a 2D array with one row for each of
`\\binom{n+d}{d}` nodes, and `d` columns for coordinates (or `d+1` if
``domain='barycentric'``).
Example:
.. plot::
:include-source: True
>>> import matplotlib.pyplot as plt
>>> from numpy import eye
>>> from recursivenodes.nodes import equispaced
>>> from recursivenodes.utils import coord_map
>>> nodes = equispaced(2, 7, domain='equilateral')
>>> corners = coord_map(eye(3), 'barycentric', 'equilateral')
>>> plt.plot(corners[[0,1,2,0],0], corners[[0,1,2,0],1])
[<matplotlib.lines.Line2D object at ...>]
>>> plt.scatter(nodes[:,0], nodes[:,1])
<matplotlib.collections.PathCollection object at ...>
>>> plt.gca().set_aspect('equal')
>>> plt.title('Equispaced Nodes')
Text(0.5, 1.0, 'Equispaced Nodes')
>>> plt.show()
'''
N = npolys(d, n)
x = np.ndarray((N, d))
for (k, i) in enumerate(multiindex_up_to(d, n)):
x[k, :] = np.array(i) / n
return coord_map(x, 'unit', domain)
[docs]def equispaced_interior(d, n, domain='biunit'):
'''Equispaced nodes for polynomials up to degree `n` on the `d`-simplex,
all of which are in the interior.
Args:
d (int): The dimension of the simplex.
n (int): The polynomial degree
domain (str, optional): The domain of the simplex. See
":ref:`domains`" for the choices and their formal definitions.
Returns:
ndarray: Equispaced interior nodes as a 2D array with one row for each
of `\\binom{n+d}{d}` nodes, and `d` columns for coordinates (or `d+1`
if ``domain='barycentric'``).
Example:
.. plot::
:include-source: True
>>> import matplotlib.pyplot as plt
>>> from numpy import eye
>>> from recursivenodes.nodes import equispaced_interior
>>> from recursivenodes.utils import coord_map
>>> nodes = equispaced_interior(2, 7, domain='equilateral')
>>> corners = coord_map(eye(3), 'barycentric', 'equilateral')
>>> plt.plot(corners[[0,1,2,0],0], corners[[0,1,2,0],1])
[<matplotlib.lines.Line2D object at ...>]
>>> plt.scatter(nodes[:,0], nodes[:,1])
<matplotlib.collections.PathCollection object at ...>
>>> plt.gca().set_aspect('equal')
>>> plt.title('Equispaced Interior Nodes')
Text(0.5, 1.0, 'Equispaced Interior Nodes')
>>> plt.show()
'''
N = npolys(d, n)
x = np.ndarray((N, d))
for (k, i) in enumerate(multiindex_up_to(d, n)):
x[k, :] = (np.array(i) + 1/(d+1)) / (n+1)
return coord_map(x, 'unit', domain)
[docs]def blyth_luo_pozrikidis(d, n, x=None, domain='biunit'):
'''Create Blyth-Luo-Pozrikidis nodes from a 1D node set for polynomials up
to degree `n` on the `d`-simplex.
Notes:
The Blyth-Luo-Pozrikidis rule places an interior node with multi-index
`\\boldsymbol{\\alpha}` at the barycentric point `\\boldsymbol{b}(\\boldsymbol{\\alpha})` such that
.. math::
:label: blp
b_i(\\boldsymbol{\\alpha}) = x_{n,\\boldsymbol{\\alpha}_i} + \\frac{1}{d}(1 - \\sum_{j\\neq i}
x_{n,\\boldsymbol{\\alpha}_j}).
Points on the boundary look like `(d-1)`-simplex Blyth-Luo-Pozrikidis
nodes.
Args:
d (int): The dimension of the simplex.
n (int): The polynomial degree
x (ndarray, optional): 1D node set on `[0, 1]` with `n+1` points.
Lobatto-Gauss-Legendre nodes are used if ``x=None``.
domain (str, optional): The domain of the simplex. See
":ref:`domains`" for the choices and their formal definitions.
Returns:
ndarray: Blyth-Luo-Pozrikidis nodes as a 2D array with one row for each
of `\\binom{n+d}{d}` nodes, and `d` columns for coordinates (or `d+1`
if ``domain='barycentric'``).
Example:
This plot shows the Blyth-Luo-Pozrikidis nodes with lines connecting
the Lobatto-Gauss-Legendre nodes on the edges. The definition in
:eq:`blp` is designed to place the nodes in the centroids of the
triangles created by the intersecting lines.
.. plot::
:include-source: True
>>> import matplotlib.pyplot as plt
>>> from numpy import eye
>>> from recursivenodes.nodes import blyth_luo_pozrikidis
>>> from recursivenodes.utils import coord_map, multiindex_equal
>>> nodes = blyth_luo_pozrikidis(2, 7, domain='equilateral')
>>> corners = coord_map(eye(3), 'barycentric', 'equilateral')
>>> # plot grid lines
>>> for (i, al) in enumerate(multiindex_equal(3, 7)):
... if min(al) > 0 or max(al) == sum(al): continue
... for (j, be) in enumerate(multiindex_equal(3, 7)):
... if (j <= i or min(be) > 0 or max(be) != max(al)
... or sum(be) != sum(al)):
... continue
... for d in range(3):
... if al[d] == be[d] and al[d] != 0:
... _ = plt.plot(nodes[[i,j],0], nodes[[i,j],1],
... linestyle='--', c='grey')
>>> plt.plot(corners[[0,1,2,0],0], corners[[0,1,2,0],1])
[<matplotlib.lines.Line2D object at ...>]
>>> plt.scatter(nodes[:,0], nodes[:,1])
<matplotlib.collections.PathCollection object at ...>
>>> plt.gca().set_aspect('equal')
>>> plt.title('Blyth-Luo-Pozrikidis Nodes')
Text(0.5, 1.0, 'Blyth-Luo-Pozrikidis Nodes')
>>> plt.show()
References:
:cite:`BlPo06,LuPo06`
'''
if x is None:
x = coord_map(lobattogaussjacobi(n+1)[0].reshape(n+1, 1), 'biunit', domain)
if domain != 'barycentric':
x = x[:, 0]
n = len(x) - 1
x = x.reshape(x.shape + (1,))
y = coord_map(x, domain, 'unit')
y = y[:, 0]
N = npolys(d, n)
xd = np.zeros((N, d))
for (k, i) in enumerate(multiindex_equal(d+1, n)):
numzero = sum([int(y[m] == 0.) for m in i])
weight = 0.
for j in range(d+1):
ibutj = i[0:j] + i[(j+1):]
yibutj = [y[m] for m in ibutj]
thisnumzero = sum([int(z == 0) for z in yibutj])
if thisnumzero < numzero:
continue
yj = 1. - sum(yibutj)
weight += 1
for m in range(j):
if (m < d):
xd[k, m] += y[i[m]]
if (j < d):
xd[k, j] += yj
for m in range(j+1, d+1):
if (m < d):
xd[k, m] += y[i[m]]
xd[k, :] /= weight
return coord_map(xd, 'unit', domain)
def _warburton_b(d, xt, i):
b = np.ones((xt.shape[0], 1))
tol = 1.e-8
bdry = np.zeros(b.shape)
for j in range(d+1):
if j == i:
continue
nonzero = np.abs(2. * xt[:, j] + xt[:, i]) > tol
b[nonzero, 0] *= ((2. * xt[nonzero, j]) /
(2. * xt[nonzero, j] + xt[nonzero, i]))
bdry[~nonzero, 0] += 1.
isbdry = bdry[:, 0] > 0.
if d == 2:
'''The implementation of warp and blend, evident in the Lebesgue
constants in the paper and in modepy
(https://github.com/inducer/modepy/blob/master/modepy/nodes.py)
uses the blending scaling (2 * l2) * (2 * l3) / (1. - (l2 - l3)**2)
instead of (2 * l2) * (2 * l3) / ((2 * l2 + l1)*(2 * l3 + l1)).
If l1 + l2 + l3 = 1, this is equivalent, because
(1. - (l2 - l3)**2) = (1. + (l2 - l3))*(1. - (l2 - l3))
= (l1+l2+l3 +l2-l3)*(l1+l2+l3 -l2+l3)
= (2*l2+l1)*(2.*l3+l1)
BUT, when we are blending for higher dimensions, we may be using a
subset of barycentric coordinates, i.e. l1 + l2 + l3 + l4 = 1, l4 > 0,
and we are computing a face warp for the l4 face, so we use only the
(l1,l2,l3) coordinate. In this case l1+l2+l3 != 1, so the two blending
scalings are not equivalent. I'm choosing to follow the implementation
so that I can reproduce the computed Lebesgue constants.
'''
i2 = (i + 1) % 3
i3 = (i + 2) % 3
b[~isbdry, 0] = (4. * xt[~isbdry, i2] * xt[~isbdry, i3] /
(1. - (xt[~isbdry, i2]-xt[~isbdry, i3])**2))
b[isbdry, 0] = 1. / (bdry[isbdry, 0] + 1.)
return b
def _warburton_g(d, n, xt, alpha=0., x=None):
g = np.zeros(xt.shape)
if (d == 1):
if x is None:
x, _ = lobattogaussjacobi(n+1)
xe = np.linspace(-1., 1., n+1, endpoint=True)
diff = (x - xe) / 2
V = proriolkoornwinderdubinervandermonde(1, n, xe.reshape((n+1, 1)))
P = np.linalg.solve(V, diff)
T = proriolkoornwinderdubinervandermonde(1, n, xt[:, [0]] - xt[:, [1]])
g[:, 0] = T.dot(P)
g[:, 1] = -g[:, 0]
else:
for i in range(d+1):
rem = [j for j in range(d+1) if j != i]
gi = _warburton_g(d-1, n, xt[:, rem], alpha, x)
bi = _warburton_b(d, xt, i)
g[:, rem] += (1. + (alpha * xt[:, [i]])**2) * gi * bi
return g
#: A dictionary of optimal values of the blending parameter `\alpha` computed
#: in :cite:`Warb06`. Keyed by ``(d, n)`` tuples.
#:
#: Example:
#: >>> warburton_alpha[(2,7)]
#: 1.0999
warburton_alpha = {
(2, 3): 1.4152, (3, 3): 0.0000,
(2, 4): 0.1001, (3, 4): 0.1002,
(2, 5): 0.2751, (3, 5): 1.1332,
(2, 6): 0.9808, (3, 6): 1.5608,
(2, 7): 1.0999, (3, 7): 1.3413,
(2, 8): 1.2832, (3, 8): 1.2577,
(2, 9): 1.3648, (3, 9): 1.1603,
(2, 10): 1.4773, (3, 10): 1.0153,
(2, 11): 1.4959, (3, 11): 0.6080,
(2, 12): 1.5743, (3, 12): 0.4523,
(2, 13): 1.5770, (3, 13): 0.8856,
(2, 14): 1.6223, (3, 14): 0.8717,
(2, 15): 1.6258, (3, 15): 0.9655,
}
[docs]def warburton(d, n, x=None, alpha=None, domain='biunit'):
'''Warburton *warp & blend* nodes from a 1D node set for polynomials up
to degree `n` on the `d`-simplex.
Notes:
The Warburton *warp & blend* nodes define a node's coordinates as a
displacement from equispaced coordinates by blending together
distortion maps of the `d`-simplex that warp the edge nodes
to match the 1D node set. One optimization parameter `\\boldsymbol{\\alpha}`
controls the blending in the interior of the simplex.
Args:
d (int): The dimension of the simplex.
n (int): The polynomial degree
x (ndarray, optional): 1D node set on `[0, 1]` with `n+1` points.
Lobatto-Gauss-Legendre nodes are used if ``x=None``.
alpha (float, optional): The blending parameter. If ``alpha=None``, a
precomputed optimal parameter is used if known.
domain (str, optional): The domain of the simplex. See
":ref:`domains`" for the choices and their formal definitions.
Returns:
ndarray: Warburton *warp & blend* nodes as a 2D array with one row for
each of `\\binom{n+d}{d}` nodes, and `d` columns for coordinates (or
`d+1` if ``domain='barycentric'``).
Example:
.. plot::
:include-source: True
>>> import matplotlib.pyplot as plt
>>> from numpy import eye
>>> from recursivenodes.nodes import warburton
>>> from recursivenodes.utils import coord_map
>>> nodes = warburton(2, 7, domain='equilateral')
>>> corners = coord_map(eye(3), 'barycentric', 'equilateral')
>>> plt.plot(corners[[0,1,2,0],0], corners[[0,1,2,0],1])
[<matplotlib.lines.Line2D object at ...>]
>>> plt.scatter(nodes[:,0], nodes[:,1])
<matplotlib.collections.PathCollection object at ...>
>>> plt.gca().set_aspect('equal')
>>> plt.title('Warburton Warp & Blend Nodes')
Text(0.5, 1.0, 'Warburton Warp & Blend Nodes')
>>> plt.show()
References:
:cite:`Warb06`
'''
if not (x is None):
x = coord_map(x, domain, 'biunit')
if alpha is None:
try:
alpha = warburton_alpha[(d, n)]
except KeyError:
alpha = 0.
xt = equispaced(d, n, domain='barycentric')
g = _warburton_g(d, n, xt, alpha, x)
xt += g
return coord_map(xt, 'barycentric', domain)
class NodeFamily:
'''Family of nodes on the unit interval. This class essentially is a
lazy-evaluate-and-cache dictionary: the user passes a routine to evaluate
entries for unknown keys'''
def __init__(self, f):
self._f = f
self._cache = {}
def __getitem__(self, key):
try:
return self._cache[key]
except KeyError:
value = self._f(key)
self._cache[key] = value
return value
# For each family, family[n] should be a a set of n+1 points in [0,1] in
# increasing order that is symmetric about 1/2, family[n] should be None if the
# family does not have a representative for that index
#
# We predefine:
#
# - shifted Lobatto-Gauss-Legendre (lgl_family)
# - shifted Lobatto-Gauss-Chebyshev (lgc_family)
# - shifted Gauss-Legendre (gl_family)
# - shifted Gauss-Chebyshev (gc_family)
# - equispaced, including endpoints (equi_family)
# - equispaced, interior (equi_interior_family)
lgl_family = NodeFamily(lambda n:
coord_map(lobattogaussjacobi(n+1)[0], 'biunit', 'unit')
if n > 0 else np.array([0.5]))
lgc_family = NodeFamily(lambda n:
coord_map(lobattogaussjacobi(n+1, -0.5, -0.5)[0],
'biunit', 'unit')
if n > 0 else np.array([0.5]))
gl_family = NodeFamily(lambda n:
coord_map(gaussjacobi(n+1)[0], 'biunit', 'unit'))
gc_family = NodeFamily(lambda n:
coord_map(gaussjacobi(n+1, -0.5, -0.5)[0],
'biunit', 'unit'))
equi_family = NodeFamily(lambda n:
np.linspace(0., 1., n+1) if n > 0 else
np.array([0.5]))
equi_interior_family = NodeFamily(lambda n:
np.linspace(0.5/(n+1), 1.-0.5/(n+1), n+1))
def _recursive(d, n, alpha, family):
'''The barycentric d-simplex coordinates for a
multiindex alpha with sum n, based on a 1D node family.'''
xn = family[n]
b = np.zeros((d+1,))
if d == 1:
b[:] = xn[[alpha[0], alpha[1]]]
return b
weight = 0.
for i in range(d+1):
alpha_noti = alpha[:i] + alpha[i+1:]
n_noti = n - alpha[i]
w = xn[n_noti]
br = _recursive(d-1, n_noti, alpha_noti, family)
b[:i] += w * br[:i]
b[i+1:] += w * br[i:]
weight += w
b /= weight
return b
def _decode_family(family):
if family is None:
family = lgl_family
elif isinstance(family, str):
if family == 'lgl':
family = lgl_family
elif family == 'lgc':
family = lgc_family
elif family == 'gl':
family = gl_family
elif family == 'gc':
family = gc_family
elif family == 'equi':
family = equi_family
elif family == 'equi_interior':
family = equi_interior_family
elif isinstance(family, tuple) and len(family) == 2 and family[0] == 'lgg':
a = family[1]
family = NodeFamily(lambda n:
coord_map(lobattogaussjacobi(n+1, a-0.5, a-0.5)[0],
'biunit', 'unit')
if n > 0 else np.array([0.5]))
elif isinstance(family, tuple) and len(family) == 2 and family[0] == 'gg':
a = family[1]
family = NodeFamily(lambda n:
coord_map(gaussjacobi(n+1, a-0.5, a-0.5)[0],
'biunit', 'unit'))
return family
[docs]def recursive(d, n, family='lgl', domain='barycentric'):
'''Recursively defined nodes for `\\mathcal{P}_n(\\Delta^d)`, the polynomials
with degree at most `n` on the `d`-simplex, based on a 1D node family.
Notes:
Some definitions:
- A 1D *node set* `X_k=(x_{k,0}, \\dots, x_{k,k})` is a
sorted list of `k+1` points in `[0,1]` that is symmetric about `1/2`.
- A 1D *node family* `\\boldsymbol{X} = \\{X_k\\}` is a collection of 1D
node sets for every degree `k`.
- The `barycentric triangle`_ is a canonical domain for the `d`-simplex
in `\\mathbb{R}^{d+1}`: positive coordinates
`\\boldsymbol{b} = (b_0, b_1, \\dots b_d)` such that
`\\sum_i b_i = 1`.
- For a multi-index_ `\\boldsymbol{\\alpha}`, let `\\#\\boldsymbol{\\alpha}` be its length,
`|\\boldsymbol{\\alpha}|` its sum, and `\\boldsymbol{\\alpha}_{\\backslash i}` the multi-index
created by removing `\\boldsymbol{\\alpha}_i`: nodes that define a basis of
`\\mathcal{P}_n(\\Delta^d)` can be indexed by `\\boldsymbol{\\alpha}` such that
`\\#\\boldsymbol{\\alpha} = d+1` and `|\\boldsymbol{\\alpha}| = n`.
- For a vector `\\boldsymbol{b}`, let `\\boldsymbol{b}_{+i}` be the
vector created by inserting a zero for the `i`-th coordinate.
The recursive definition of barycentric node coordinates,
`\\boldsymbol{b}_{\\boldsymbol{X}}(\\boldsymbol{\\alpha})
\\in \\mathbb{R}^{\\#\\boldsymbol{\\alpha}}` has the base case
.. math::
:label: basecase
\\boldsymbol{b}_{\\boldsymbol{X}}(\\boldsymbol{\\alpha}) =
(x_{|\\boldsymbol{\\alpha}|,\\boldsymbol{\\alpha}_0}, x_{|\\boldsymbol{\\alpha}|,\\boldsymbol{\\alpha}_1}), \\quad
\\#\\boldsymbol{\\alpha} = 2,
and the recursion
.. math::
:label: recursion
\\boldsymbol{b}_{\\boldsymbol{X}}(\\boldsymbol{\\alpha}) =
\\frac{\\sum_i x_{|\\boldsymbol{\\alpha}|,|\\boldsymbol{\\alpha}_{\\backslash i}|}
\\boldsymbol{b}_{\\boldsymbol{X}}(\\boldsymbol{\\alpha}_{\\backslash i})_{+i}}
{\\sum_i x_{|\\boldsymbol{\\alpha}|,|\\boldsymbol{\\alpha}_{\\backslash i}|}}, \\quad
\\#\\boldsymbol{\\alpha} > 2.
The full set of nodes is
.. math::
:label: fullset
R^d_{\\boldsymbol{X},n} =
\\{\\boldsymbol{b}_{\\boldsymbol{X}}(\\boldsymbol{\\alpha}): \\#\\boldsymbol{\\alpha} = d+1,
|\\boldsymbol{\\alpha}| = n\\}.
Args:
d (int): The dimension of the simplex
n (int): The maximum degree of the polynomials
family (optional): The 1D node family used to define the coordinates in
the barycentric `d`-simplex. The default ``family='lgl'``
corresponds to the shifted Lobatto-Gauss-Legendre_ nodes. See
":ref:`families`" for using other node families.
domain (str, optional): The domain for the `d`-simplex where the
returned coordinates will be defined. See ":ref:`domains`" for the
choices and their formal definitions.
Returns:
ndarray: The nodes `R^d_{\\boldsymbol{X},n}` defined in
:eq:`fullset`, as a 2D array with `\\binom{n+d}{d}` rows. If
``domain='barycentric'``, it has `d+1` columns, otherwise it has `d`
columns.
Examples:
Nodes for `\\mathcal{P}^4(\\Delta^2)` in barycentric coordinates:
>>> recursive_nodes(2, 4)
array([[0. , 0. , 1. ],
[0. , 0.17267316, 0.82732684],
[0. , 0.5 , 0.5 ],
[0. , 0.82732684, 0.17267316],
[0. , 1. , 0. ],
[0.17267316, 0. , 0.82732684],
[0.2221552 , 0.2221552 , 0.5556896 ],
[0.2221552 , 0.5556896 , 0.2221552 ],
[0.17267316, 0.82732684, 0. ],
[0.5 , 0. , 0.5 ],
[0.5556896 , 0.2221552 , 0.2221552 ],
[0.5 , 0.5 , 0. ],
[0.82732684, 0. , 0.17267316],
[0.82732684, 0.17267316, 0. ],
[1. , 0. , 0. ]])
The same nodes on the unit triangle:
>>> recursive_nodes(2, 4, domain='unit')
array([[0. , 0. ],
[0. , 0.17267316],
[0. , 0.5 ],
[0. , 0.82732684],
[0. , 1. ],
[0.17267316, 0. ],
[0.2221552 , 0.2221552 ],
[0.2221552 , 0.5556896 ],
[0.17267316, 0.82732684],
[0.5 , 0. ],
[0.5556896 , 0.2221552 ],
[0.5 , 0.5 ],
[0.82732684, 0. ],
[0.82732684, 0.17267316],
[1. , 0. ]])
If we construct the node set not from the Lobatto-Gauss-Legendre 1D
node family, but from the equispaced 1D node family, we get equispaced
2D nodes:
>>> recursive_nodes(2, 4, family='equi', domain='unit')
array([[0. , 0. ],
[0. , 0.25],
[0. , 0.5 ],
[0. , 0.75],
[0. , 1. ],
[0.25, 0. ],
[0.25, 0.25],
[0.25, 0.5 ],
[0.25, 0.75],
[0.5 , 0. ],
[0.5 , 0.25],
[0.5 , 0.5 ],
[0.75, 0. ],
[0.75, 0.25],
[1. , 0. ]])
This is what they look like mapped to the equilateral triangle:
.. plot::
:include-source: True
>>> import matplotlib.pyplot as plt
>>> from recursivenodes import recursive_nodes
>>> nodes_equi = recursive_nodes(2, 4, family='equi', domain='equilateral')
>>> nodes_lgl = recursive_nodes(2, 4, domain='equilateral')
>>> plt.scatter(nodes_lgl[:,0], nodes_lgl[:,1], marker='o', label='recursive LGL')
<matplotlib.collections.PathCollection object at ...>
>>> plt.scatter(nodes_equi[:,0], nodes_equi[:,1], marker='^', label='equispaced')
<matplotlib.collections.PathCollection object at ...>
>>> plt.gca().set_aspect('equal')
>>> plt.legend()
<matplotlib.legend.Legend object at ...>
>>> plt.show()
.. _multi-index: https://en.wikipedia.org/wiki/Multi-index_notation
.. _barycentric triangle: https://en.wikipedia.org/wiki/Barycentric_coordinate_system
.. _Lobatto-Gauss-Legendre: https://en.wikipedia.org/wiki/Gaussian_quadrature#Gauss%E2%80%93Lobatto_rules
'''
family = _decode_family(family)
N = npolys(d, n)
x = np.zeros((N, d+1))
for (k, i) in enumerate(multiindex_equal(d+1, n)):
x[k, :] = _recursive(d, n, i, family)
return coord_map(x, 'barycentric', domain)
# ---------------------------START---------------------------- #
# Rapetti, Sommariva, & Vianello Symmetric Lebesgue-minimizing #
# Lobatto-Gauss-Legendre nodes: doi:10.1016/j.cam.2011.11.023 #
_rsv_lebgls = {}
_rsv_lebgls[1] = [
[1.000000000000000e+00, 0.000000000000000e+00],
[0.000000000000000e+00, 1.000000000000000e+00],
[0.000000000000000e+00, 0.000000000000000e+00],
]
_rsv_lebgls[2] = [
[1.000000000000000e+00, 0.000000000000000e+00],
[0.000000000000000e+00, 1.000000000000000e+00],
[0.000000000000000e+00, 0.000000000000000e+00],
[5.000000000000000e-01, 5.000000000000000e-01],
[5.000000000000000e-01, 0.000000000000000e+00],
[0.000000000000000e+00, 5.000000000000000e-01],
]
_rsv_lebgls[3] = [
[3.333333333333333e-01, 3.333333333333333e-01],
[1.000000000000000e+00, 0.000000000000000e+00],
[0.000000000000000e+00, 1.000000000000000e+00],
[0.000000000000000e+00, 0.000000000000000e+00],
[2.763932022500210e-01, 7.236067977499789e-01],
[7.236067977499789e-01, 2.763932022500210e-01],
[2.763932022500210e-01, 0.000000000000000e+00],
[7.236067977499789e-01, 0.000000000000000e+00],
[0.000000000000000e+00, 2.763932022500210e-01],
[0.000000000000000e+00, 7.236067977499789e-01],
]
_rsv_lebgls[4] = [
[2.371066302371978e-01, 2.371066302371978e-01],
[2.371066302371978e-01, 5.257867395256041e-01],
[5.257867395256041e-01, 2.371066302371978e-01],
[1.000000000000000e+00, 0.000000000000000e+00],
[0.000000000000000e+00, 1.000000000000000e+00],
[0.000000000000000e+00, 0.000000000000000e+00],
[1.726731646460114e-01, 8.273268353539885e-01],
[5.000000000000000e-01, 5.000000000000000e-01],
[8.273268353539885e-01, 1.726731646460114e-01],
[1.726731646460114e-01, 0.000000000000000e+00],
[5.000000000000000e-01, 0.000000000000000e+00],
[8.273268353539885e-01, 0.000000000000000e+00],
[0.000000000000000e+00, 1.726731646460114e-01],
[0.000000000000000e+00, 5.000000000000000e-01],
[0.000000000000000e+00, 8.273268353539885e-01],
]
_rsv_lebgls[5] = [
[4.123406940197869e-01, 4.123406940197869e-01],
[1.737985196489401e-01, 1.737985196489401e-01],
[4.123406940197869e-01, 1.753186119604261e-01],
[1.737985196489401e-01, 6.524029607021195e-01],
[1.753186119604261e-01, 4.123406940197869e-01],
[6.524029607021195e-01, 1.737985196489401e-01],
[1.000000000000000e+00, 0.000000000000000e+00],
[0.000000000000000e+00, 1.000000000000000e+00],
[0.000000000000000e+00, 0.000000000000000e+00],
[1.174723380352677e-01, 8.825276619647322e-01],
[3.573842417596774e-01, 6.426157582403224e-01],
[6.426157582403225e-01, 3.573842417596774e-01],
[8.825276619647324e-01, 1.174723380352675e-01],
[1.174723380352677e-01, 0.000000000000000e+00],
[3.573842417596774e-01, 0.000000000000000e+00],
[6.426157582403225e-01, 0.000000000000000e+00],
[8.825276619647324e-01, 0.000000000000000e+00],
[0.000000000000000e+00, 1.174723380352677e-01],
[0.000000000000000e+00, 3.573842417596774e-01],
[0.000000000000000e+00, 6.426157582403225e-01],
[0.000000000000000e+00, 8.825276619647324e-01],
]
_rsv_lebgls[6] = [
[3.333333333333333e-01, 3.333333333333333e-01],
[1.236700206266602e-01, 1.236700206266602e-01],
[1.236700206266602e-01, 7.526599587466795e-01],
[7.526599587466795e-01, 1.236700206266602e-01],
[3.254637175704040e-01, 5.425665225375484e-01],
[3.254637175704040e-01, 1.319697598920475e-01],
[5.425665225375484e-01, 3.254637175704040e-01],
[5.425665225375484e-01, 1.319697598920475e-01],
[1.319697598920475e-01, 3.254637175704040e-01],
[1.319697598920475e-01, 5.425665225375484e-01],
[1.000000000000000e+00, 0.000000000000000e+00],
[0.000000000000000e+00, 1.000000000000000e+00],
[0.000000000000000e+00, 0.000000000000000e+00],
[8.488805186071646e-02, 9.151119481392835e-01],
[2.655756032646428e-01, 7.344243967353571e-01],
[5.000000000000001e-01, 4.999999999999998e-01],
[7.344243967353571e-01, 2.655756032646428e-01],
[9.151119481392833e-01, 8.488805186071668e-02],
[8.488805186071646e-02, 0.000000000000000e+00],
[2.655756032646428e-01, 0.000000000000000e+00],
[5.000000000000001e-01, 0.000000000000000e+00],
[7.344243967353571e-01, 0.000000000000000e+00],
[9.151119481392833e-01, 0.000000000000000e+00],
[0.000000000000000e+00, 8.488805186071646e-02],
[0.000000000000000e+00, 2.655756032646428e-01],
[0.000000000000000e+00, 5.000000000000001e-01],
[0.000000000000000e+00, 7.344243967353571e-01],
[0.000000000000000e+00, 9.151119481392833e-01],
]
_rsv_lebgls[7] = [
[2.741584394628903e-01, 2.741584394628903e-01],
[8.791461398121544e-02, 8.791461398121544e-02],
[2.741584394628903e-01, 4.516831210742193e-01],
[8.791461398121544e-02, 8.241707720375690e-01],
[4.516831210742193e-01, 2.741584394628903e-01],
[8.241707720375690e-01, 8.791461398121544e-02],
[4.500074460521015e-01, 4.500074460521015e-01],
[4.500074460521015e-01, 9.998510789579695e-02],
[9.998510789579695e-02, 4.500074460521015e-01],
[2.574575396908898e-01, 6.389182038566255e-01],
[2.574575396908898e-01, 1.036242564524846e-01],
[6.389182038566255e-01, 2.574575396908898e-01],
[6.389182038566255e-01, 1.036242564524846e-01],
[1.036242564524846e-01, 2.574575396908898e-01],
[1.036242564524846e-01, 6.389182038566255e-01],
[1.000000000000000e+00, 0.000000000000000e+00],
[0.000000000000000e+00, 1.000000000000000e+00],
[0.000000000000000e+00, 0.000000000000000e+00],
[6.412992574519671e-02, 9.358700742548032e-01],
[2.041499092834289e-01, 7.958500907165710e-01],
[3.953503910487606e-01, 6.046496089512394e-01],
[6.046496089512393e-01, 3.953503910487606e-01],
[7.958500907165713e-01, 2.041499092834286e-01],
[9.358700742548031e-01, 6.412992574519682e-02],
[6.412992574519671e-02, 0.000000000000000e+00],
[2.041499092834289e-01, 0.000000000000000e+00],
[3.953503910487606e-01, 0.000000000000000e+00],
[6.046496089512393e-01, 0.000000000000000e+00],
[7.958500907165713e-01, 0.000000000000000e+00],
[9.358700742548031e-01, 0.000000000000000e+00],
[0.000000000000000e+00, 6.412992574519671e-02],
[0.000000000000000e+00, 2.041499092834289e-01],
[0.000000000000000e+00, 3.953503910487606e-01],
[0.000000000000000e+00, 6.046496089512393e-01],
[0.000000000000000e+00, 7.958500907165713e-01],
[0.000000000000000e+00, 9.358700742548031e-01],
]
_rsv_lebgls[8] = [
[2.238114422946618e-01, 2.238114422946618e-01],
[3.879143834569392e-01, 3.879143834569392e-01],
[6.749944697564996e-02, 6.749944697564996e-02],
[2.238114422946618e-01, 5.523771154106762e-01],
[3.879143834569392e-01, 2.241712330861213e-01],
[6.749944697564996e-02, 8.650011060487001e-01],
[5.523771154106762e-01, 2.238114422946618e-01],
[2.241712330861213e-01, 3.879143834569392e-01],
[8.650011060487001e-01, 6.749944697564996e-02],
[2.106241399947925e-01, 7.064768674963645e-01],
[3.702480264146488e-01, 5.482896391715870e-01],
[2.106241399947925e-01, 8.289899250884291e-02],
[3.702480264146488e-01, 8.146233441376415e-02],
[7.064768674963645e-01, 2.106241399947925e-01],
[5.482896391715870e-01, 3.702480264146488e-01],
[7.064768674963645e-01, 8.289899250884291e-02],
[5.482896391715870e-01, 8.146233441376415e-02],
[8.289899250884291e-02, 2.106241399947925e-01],
[8.146233441376415e-02, 3.702480264146488e-01],
[8.289899250884291e-02, 7.064768674963645e-01],
[8.146233441376415e-02, 5.482896391715870e-01],
[1.000000000000000e+00, 0.000000000000000e+00],
[0.000000000000000e+00, 1.000000000000000e+00],
[0.000000000000000e+00, 0.000000000000000e+00],
[5.012100229426996e-02, 9.498789977057300e-01],
[1.614068602446310e-01, 8.385931397553689e-01],
[3.184412680869108e-01, 6.815587319130891e-01],
[5.000000000000001e-01, 4.999999999999998e-01],
[6.815587319130891e-01, 3.184412680869108e-01],
[8.385931397553689e-01, 1.614068602446310e-01],
[9.498789977057298e-01, 5.012100229427018e-02],
[5.012100229426996e-02, 0.000000000000000e+00],
[1.614068602446310e-01, 0.000000000000000e+00],
[3.184412680869108e-01, 0.000000000000000e+00],
[5.000000000000001e-01, 0.000000000000000e+00],
[6.815587319130891e-01, 0.000000000000000e+00],
[8.385931397553689e-01, 0.000000000000000e+00],
[9.498789977057298e-01, 0.000000000000000e+00],
[0.000000000000000e+00, 5.012100229426996e-02],
[0.000000000000000e+00, 1.614068602446310e-01],
[0.000000000000000e+00, 3.184412680869108e-01],
[0.000000000000000e+00, 5.000000000000001e-01],
[0.000000000000000e+00, 6.815587319130891e-01],
[0.000000000000000e+00, 8.385931397553689e-01],
[0.000000000000000e+00, 9.498789977057298e-01],
]
_rsv_lebgls[9] = [
[3.333333333333333e-01, 3.333333333333333e-01],
[5.208382829746331e-02, 5.208382829746331e-02],
[1.835842328020949e-01, 1.835842328020949e-01],
[5.208382829746331e-02, 8.958323434050733e-01],
[1.835842328020949e-01, 6.328315343958100e-01],
[8.958323434050733e-01, 5.208382829746331e-02],
[6.328315343958100e-01, 1.835842328020949e-01],
[4.677778353675260e-01, 4.677778353675260e-01],
[4.677778353675260e-01, 6.444432926494780e-02],
[6.444432926494780e-02, 4.677778353675260e-01],
[3.335906038202737e-01, 4.796482965082551e-01],
[1.723053623095807e-01, 7.617793422357387e-01],
[3.093824727565271e-01, 6.225829731720886e-01],
[3.335906038202737e-01, 1.867610996714711e-01],
[1.723053623095807e-01, 6.591529545468055e-02],
[3.093824727565271e-01, 6.803455407138414e-02],
[4.796482965082551e-01, 3.335906038202737e-01],
[7.617793422357387e-01, 1.723053623095807e-01],
[6.225829731720886e-01, 3.093824727565271e-01],
[4.796482965082551e-01, 1.867610996714711e-01],
[7.617793422357387e-01, 6.591529545468055e-02],
[6.225829731720886e-01, 6.803455407138414e-02],
[1.867610996714711e-01, 3.335906038202737e-01],
[6.591529545468055e-02, 1.723053623095807e-01],
[6.803455407138414e-02, 3.093824727565271e-01],
[1.867610996714711e-01, 4.796482965082551e-01],
[6.591529545468055e-02, 7.617793422357387e-01],
[6.803455407138414e-02, 6.225829731720886e-01],
[1.000000000000000e+00, 0.000000000000000e+00],
[0.000000000000000e+00, 1.000000000000000e+00],
[0.000000000000000e+00, 0.000000000000000e+00],
[4.023304591677051e-02, 9.597669540832294e-01],
[1.306130674472475e-01, 8.693869325527523e-01],
[2.610375250947778e-01, 7.389624749052221e-01],
[4.173605211668064e-01, 5.826394788331934e-01],
[5.826394788331933e-01, 4.173605211668066e-01],
[7.389624749052223e-01, 2.610375250947776e-01],
[8.693869325527523e-01, 1.306130674472476e-01],
[9.597669540832292e-01, 4.023304591677079e-02],
[4.023304591677051e-02, 0.000000000000000e+00],
[1.306130674472475e-01, 0.000000000000000e+00],
[2.610375250947778e-01, 0.000000000000000e+00],
[4.173605211668064e-01, 0.000000000000000e+00],
[5.826394788331933e-01, 0.000000000000000e+00],
[7.389624749052223e-01, 0.000000000000000e+00],
[8.693869325527523e-01, 0.000000000000000e+00],
[9.597669540832292e-01, 0.000000000000000e+00],
[0.000000000000000e+00, 4.023304591677051e-02],
[0.000000000000000e+00, 1.306130674472475e-01],
[0.000000000000000e+00, 2.610375250947778e-01],
[0.000000000000000e+00, 4.173605211668064e-01],
[0.000000000000000e+00, 5.826394788331933e-01],
[0.000000000000000e+00, 7.389624749052223e-01],
[0.000000000000000e+00, 8.693869325527523e-01],
[0.000000000000000e+00, 9.597669540832292e-01],
]
_rsv_lebgls[10] = [
[1.312733767493933e-01, 1.312733767493933e-01],
[5.386953019930516e-02, 5.386953019930516e-02],
[2.252378431088206e-01, 2.252378431088206e-01],
[1.312733767493933e-01, 7.374532465012133e-01],
[5.386953019930516e-02, 8.922609396013896e-01],
[2.252378431088206e-01, 5.495243137823585e-01],
[7.374532465012133e-01, 1.312733767493933e-01],
[8.922609396013896e-01, 5.386953019930516e-02],
[5.495243137823585e-01, 2.252378431088206e-01],
[3.824877939073025e-01, 3.824877939073025e-01],
[3.824877939073025e-01, 2.350244121853949e-01],
[2.350244121853949e-01, 3.824877939073025e-01],
[2.622175009707391e-01, 6.564279951505325e-01],
[3.776961071421849e-01, 5.223502436193665e-01],
[1.820746613840211e-01, 7.792454007990795e-01],
[3.779628077066509e-01, 5.892534012902733e-01],
[2.622175009707391e-01, 8.135450387872833e-02],
[3.776961071421849e-01, 9.995364923844851e-02],
[1.820746613840211e-01, 3.867993781689926e-02],
[3.779628077066509e-01, 3.278379100307571e-02],
[6.564279951505325e-01, 2.622175009707391e-01],
[5.223502436193665e-01, 3.776961071421849e-01],
[7.792454007990795e-01, 1.820746613840211e-01],
[5.892534012902733e-01, 3.779628077066509e-01],
[6.564279951505325e-01, 8.135450387872833e-02],
[5.223502436193665e-01, 9.995364923844851e-02],
[7.792454007990795e-01, 3.867993781689926e-02],
[5.892534012902733e-01, 3.278379100307571e-02],
[8.135450387872833e-02, 2.622175009707391e-01],
[9.995364923844851e-02, 3.776961071421849e-01],
[3.867993781689926e-02, 1.820746613840211e-01],
[3.278379100307571e-02, 3.779628077066509e-01],
[8.135450387872833e-02, 6.564279951505325e-01],
[9.995364923844851e-02, 5.223502436193665e-01],
[3.867993781689926e-02, 7.792454007990795e-01],
[3.278379100307571e-02, 5.892534012902733e-01],
[1.000000000000000e+00, 0.000000000000000e+00],
[0.000000000000000e+00, 1.000000000000000e+00],
[0.000000000000000e+00, 0.000000000000000e+00],
[3.299928479597025e-02, 9.670007152040297e-01],
[1.077582631684279e-01, 8.922417368315720e-01],
[2.173823365018974e-01, 7.826176634981025e-01],
[3.521209322065304e-01, 6.478790677934695e-01],
[5.000000000000000e-01, 5.000000000000000e-01],
[6.478790677934695e-01, 3.521209322065304e-01],
[7.826176634981024e-01, 2.173823365018975e-01],
[8.922417368315722e-01, 1.077582631684277e-01],
[9.670007152040296e-01, 3.299928479597036e-02],
[3.299928479597025e-02, 0.000000000000000e+00],
[1.077582631684279e-01, 0.000000000000000e+00],
[2.173823365018974e-01, 0.000000000000000e+00],
[3.521209322065304e-01, 0.000000000000000e+00],
[5.000000000000000e-01, 0.000000000000000e+00],
[6.478790677934695e-01, 0.000000000000000e+00],
[7.826176634981024e-01, 0.000000000000000e+00],
[8.922417368315722e-01, 0.000000000000000e+00],
[9.670007152040296e-01, 0.000000000000000e+00],
[0.000000000000000e+00, 3.299928479597025e-02],
[0.000000000000000e+00, 1.077582631684279e-01],
[0.000000000000000e+00, 2.173823365018974e-01],
[0.000000000000000e+00, 3.521209322065304e-01],
[0.000000000000000e+00, 5.000000000000000e-01],
[0.000000000000000e+00, 6.478790677934695e-01],
[0.000000000000000e+00, 7.826176634981024e-01],
[0.000000000000000e+00, 8.922417368315722e-01],
[0.000000000000000e+00, 9.670007152040296e-01],
]
_rsv_lebgls[11] = [
[1.050821229581930e-01, 1.050821229581930e-01],
[4.295650634280591e-02, 4.295650634280591e-02],
[2.195219638878813e-01, 2.195219638878813e-01],
[1.050821229581930e-01, 7.898357540836139e-01],
[4.295650634280591e-02, 9.140869873143881e-01],
[2.195219638878813e-01, 5.609560722242372e-01],
[7.898357540836139e-01, 1.050821229581930e-01],
[9.140869873143881e-01, 4.295650634280591e-02],
[5.609560722242372e-01, 2.195219638878813e-01],
[4.807601837574821e-01, 4.807601837574821e-01],
[4.286809903071824e-01, 4.286809903071824e-01],
[4.807601837574821e-01, 3.847963248503571e-02],
[4.286809903071824e-01, 1.426380193856351e-01],
[3.847963248503571e-02, 4.807601837574821e-01],
[1.426380193856351e-01, 4.286809903071824e-01],
[1.994095640340481e-01, 7.069562305387490e-01],
[1.474224961654246e-01, 8.177875722890196e-01],
[3.075960627730014e-01, 6.626688579526240e-01],
[3.278085121320318e-01, 4.251975363710968e-01],
[3.205222948140067e-01, 5.840805307965026e-01],
[1.994095640340481e-01, 9.363420542720279e-02],
[1.474224961654246e-01, 3.478993154555565e-02],
[3.075960627730014e-01, 2.973507927437446e-02],
[3.278085121320318e-01, 2.469939514968714e-01],
[3.205222948140067e-01, 9.539717438949057e-02],
[7.069562305387490e-01, 1.994095640340481e-01],
[8.177875722890196e-01, 1.474224961654246e-01],
[6.626688579526240e-01, 3.075960627730014e-01],
[4.251975363710968e-01, 3.278085121320318e-01],
[5.840805307965026e-01, 3.205222948140067e-01],
[7.069562305387490e-01, 9.363420542720279e-02],
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_rsv_lebgls[12] = [
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[2.530162527885500e-01, 2.530162527885500e-01],
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]
_rsv_lebgls[13] = [
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]
_rsv_lebgls[14] = [
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_rsv_lebgls[15] = [
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_rsv_lebgls[16] = [
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_rsv_lebgls[17] = [
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_rsv_lebgls[18] = [
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[6.635759045983397e-01, 7.202789714473578e-02],
[7.771019226176585e-01, 9.058984177461271e-02],
[5.474017498960880e-01, 1.367017859779495e-02],
[7.043665532533164e-01, 1.199612814632355e-01],
[6.538843552798158e-01, 1.757463907520784e-02],
[5.183440648484733e-01, 1.431573199596718e-01],
[8.801628839826043e-01, 1.314701330527023e-02],
[5.152283097309622e-01, 4.587916374769451e-02],
[7.356822819499346e-01, 6.516494540200501e-02],
[6.001860955307918e-01, 1.294702426870414e-01],
[6.037477082466996e-01, 4.356228799123151e-02],
[7.742157293365573e-01, 8.029643948994191e-03],
[7.226718910081446e-01, 3.093560547662865e-02],
[4.292054611356153e-01, 2.465093763482417e-01],
[2.171341263416559e-01, 2.697535482222230e-01],
[4.335907411852446e-02, 1.013663979040955e-01],
[2.790950955580528e-02, 1.577802125320718e-01],
[1.224879008153767e-02, 5.704260843925303e-02],
[9.655335587584701e-02, 3.518333725591127e-01],
[7.202789714473578e-02, 2.643961982569245e-01],
[9.058984177461271e-02, 1.323082356077286e-01],
[1.367017859779495e-02, 4.389280715061170e-01],
[1.199612814632355e-01, 1.756721652834479e-01],
[1.757463907520784e-02, 3.285410056449763e-01],
[1.431573199596718e-01, 3.384986151918548e-01],
[1.314701330527023e-02, 1.066901027121254e-01],
[4.587916374769451e-02, 4.388925265213433e-01],
[6.516494540200501e-02, 1.991527726480603e-01],
[1.294702426870414e-01, 2.703436617821667e-01],
[4.356228799123151e-02, 3.526900037620688e-01],
[8.029643948994191e-03, 2.177546267144484e-01],
[3.093560547662865e-02, 2.463925035152266e-01],
[2.465093763482417e-01, 3.242851625161429e-01],
[2.171341263416559e-01, 5.131123254361209e-01],
[4.335907411852446e-02, 8.552745279773800e-01],
[2.790950955580528e-02, 8.143102779121228e-01],
[1.224879008153767e-02, 9.307086014792093e-01],
[9.655335587584701e-02, 5.516132715650402e-01],
[7.202789714473578e-02, 6.635759045983397e-01],
[9.058984177461271e-02, 7.771019226176585e-01],
[1.367017859779495e-02, 5.474017498960880e-01],
[1.199612814632355e-01, 7.043665532533164e-01],
[1.757463907520784e-02, 6.538843552798158e-01],
[1.431573199596718e-01, 5.183440648484733e-01],
[1.314701330527023e-02, 8.801628839826043e-01],
[4.587916374769451e-02, 5.152283097309622e-01],
[6.516494540200501e-02, 7.356822819499346e-01],
[1.294702426870414e-01, 6.001860955307918e-01],
[4.356228799123151e-02, 6.037477082466996e-01],
[8.029643948994191e-03, 7.742157293365573e-01],
[3.093560547662865e-02, 7.226718910081446e-01],
[2.465093763482417e-01, 4.292054611356153e-01],
[1.000000000000000e+00, 0.000000000000000e+00],
[0.000000000000000e+00, 1.000000000000000e+00],
[0.000000000000000e+00, 0.000000000000000e+00],
[1.069411688896010e-02, 9.893058831110399e-01],
[3.554923592370679e-02, 9.644507640762931e-01],
[7.376971110167684e-02, 9.262302888983231e-01],
[1.242528987236936e-01, 8.757471012763062e-01],
[1.855459313673898e-01, 8.144540686326101e-01],
[2.558853571596433e-01, 7.441146428403566e-01],
[3.332475760877505e-01, 6.667524239122494e-01],
[4.154069882953593e-01, 5.845930117046407e-01],
[5.000000000000000e-01, 5.000000000000000e-01],
[5.845930117046408e-01, 4.154069882953591e-01],
[6.667524239122492e-01, 3.332475760877507e-01],
[7.441146428403569e-01, 2.558853571596430e-01],
[8.144540686326102e-01, 1.855459313673897e-01],
[8.757471012763063e-01, 1.242528987236936e-01],
[9.262302888983231e-01, 7.376971110167684e-02],
[9.644507640762930e-01, 3.554923592370695e-02],
[9.893058831110401e-01, 1.069411688895982e-02],
[1.069411688896010e-02, 0.000000000000000e+00],
[3.554923592370679e-02, 0.000000000000000e+00],
[7.376971110167684e-02, 0.000000000000000e+00],
[1.242528987236936e-01, 0.000000000000000e+00],
[1.855459313673898e-01, 0.000000000000000e+00],
[2.558853571596433e-01, 0.000000000000000e+00],
[3.332475760877505e-01, 0.000000000000000e+00],
[4.154069882953593e-01, 0.000000000000000e+00],
[5.000000000000000e-01, 0.000000000000000e+00],
[5.845930117046408e-01, 0.000000000000000e+00],
[6.667524239122492e-01, 0.000000000000000e+00],
[7.441146428403569e-01, 0.000000000000000e+00],
[8.144540686326102e-01, 0.000000000000000e+00],
[8.757471012763063e-01, 0.000000000000000e+00],
[9.262302888983231e-01, 0.000000000000000e+00],
[9.644507640762930e-01, 0.000000000000000e+00],
[9.893058831110401e-01, 0.000000000000000e+00],
[0.000000000000000e+00, 1.069411688896010e-02],
[0.000000000000000e+00, 3.554923592370679e-02],
[0.000000000000000e+00, 7.376971110167684e-02],
[0.000000000000000e+00, 1.242528987236936e-01],
[0.000000000000000e+00, 1.855459313673898e-01],
[0.000000000000000e+00, 2.558853571596433e-01],
[0.000000000000000e+00, 3.332475760877505e-01],
[0.000000000000000e+00, 4.154069882953593e-01],
[0.000000000000000e+00, 5.000000000000000e-01],
[0.000000000000000e+00, 5.845930117046408e-01],
[0.000000000000000e+00, 6.667524239122492e-01],
[0.000000000000000e+00, 7.441146428403569e-01],
[0.000000000000000e+00, 8.144540686326102e-01],
[0.000000000000000e+00, 8.757471012763063e-01],
[0.000000000000000e+00, 9.262302888983231e-01],
[0.000000000000000e+00, 9.644507640762930e-01],
[0.000000000000000e+00, 9.893058831110401e-01],
]
# Rapetti, Sommariva, & Vianello Symmetric Lebesgue-minimizing #
# Lobatto-Gauss-Legendre nodes: doi:10.1016/j.cam.2011.11.023 #
# ----------------------------END----------------------------- #
[docs]def rapetti_sommariva_vianello(n, domain='unit'):
'''Create Rapetti-Sommariva-Vianello nodes for the triangle.
that minimize the Lebesgue constant while remaining symmetric
and having Lobatto-Gauss-Legendre nodes on the edges.
Notes:
The Rapetti-Sommariva-Vianello nodes are implicitly defined
by minimizing the Lebesgue constanat while maintaining `S_3` symmetry
and edge nodes that are Lobatto-Gauss-Legendre nodes.
Args:
n (int): The polynomial degree, must be `\\leq 18`.
domain (str, optional): The domain of the simplex. See
":ref:`domains`" for the choices and their formal definitions.
Returns:
ndarray: Rapetti-Sommariva-Vianellow nodes as a 2D array with one row
for each of `\\binom{n+2}{2}` nodes, and 2 columns for coordinates
(or 3 if ``domain='barycentric'``).
Example:
This plot shows the Rapetti-Sommariva-Vianello nodes for `n = 10`,
which do not have the lattice-like structure of the explicitly
defined nodes in the module.
.. plot::
:include-source: True
>>> import matplotlib.pyplot as plt
>>> from numpy import eye
>>> from recursivenodes.nodes import rapetti_sommariva_vianello
>>> from recursivenodes.utils import coord_map
>>> nodes = rapetti_sommariva_vianello(10, domain='equilateral')
>>> corners = coord_map(eye(3), 'barycentric', 'equilateral')
>>> plt.plot(corners[[0,1,2,0],0], corners[[0,1,2,0],1])
[<matplotlib.lines.Line2D object at ...>]
>>> plt.scatter(nodes[:,0], nodes[:,1])
<matplotlib.collections.PathCollection object at ...>
>>> plt.gca().set_aspect('equal')
>>> plt.title('Rapetti-Sommariva-Vianello Nodes')
Text(0.5, 1.0, 'Rapetti-Sommariva-Vianello Nodes')
>>> plt.show()
References:
:cite:`RaSV12`
Warning:
These nodes cannot be used with
:func:`recursivenodes.lebesgue.lebesguemax` for `n \\geq 10`, because
they do not have lattice-like structure.
'''
nodes = np.array(_rsv_lebgls[n])
return coord_map(nodes, 'unit', domain)
def expand_to_boundary(d, n, nodes, ex_nodes, domain='biunit'):
''' given degree n nodes for the d-simplex that are logically laid out like
the equispaced nodes and are only in the interior, construct degree (n + d
+ 1) nodes for the d-simplex which are equal to nodes in the interior and
ex_nodes on the boundary. No attempt is made to make the new boundary
nodes good for interpolation: they are just there to aid in computing
lebesguemax() in lebesgue.py '''
nd = n + d + 1
tuple_to_index = {}
for (k, i) in enumerate(multiindex_equal(d+1, n)):
tuple_to_index[i] = k
ex_nodes = ex_nodes.copy()
for (k, i) in enumerate(multiindex_equal(d+1, nd)):
if min(i) > 0:
ihat = tuple([a - 1 for a in i])
j = tuple_to_index[ihat]
ex_nodes[k, :] = nodes[j, :]
return ex_nodes
def add_nodes_to_parser(parser):
parser.add_argument('-n', '--degree', type=int, nargs='?', default=7,
help=('degree of polynomial space for '
'which node set is equisolvent'))
parser.add_argument('-d', '--dimension', type=int, nargs='?', default=3,
help='spatial dimensions to node set simplex')
parser.add_argument('--nodes', type=str,
choices=['recursive',
'warburton',
'blyth_luo_pozrikidis',
'equispaced',
'equispaced_interior'],
nargs='?', default='recursive',
help='node placement algorithm')
parser.add_argument('-w', '--w-alpha', type=float, nargs='?',
help=('blending parameter for '
'Warburton warp & blend method'))
parser.add_argument('--domain', type=str,
choices=['biunit',
'unit',
'barycentric',
'equilateral'],
default='biunit', nargs='?',
help='output simlex domain')
parser.add_argument('-f', '--family', type=str,
choices=['lgl',
'lgc',
'gl',
'gc',
'equi',
'equi_interior',
'lgg',
'gg'],
default='lgl', nargs='?',
help=('base 1D node family for Blyth-Pozrikidis, '
'Warburton, and recursive nodes'))
parser.add_argument('-g', '--g-alpha', type=float, nargs='?', default=0.,
help=('Gegenbauer alpha exponent (only used with '
'--family lgg or --family gg)'))
def nodes_from_args(args, expanded=False):
d = args.dimension
n = args.degree
domain = args.domain
fam_arg = args.family
need_expanded = False
if expanded:
need_expanded = True
if (args.nodes == 'equispaced' or
(args.nodes != 'equispaced_interior'
and (args.family == 'equi' or fam_arg[0] == 'l'))):
need_expanded = False
if need_expanded:
if (args.nodes == 'equispaced_interior'
or args.family == 'equi_interior'):
ex_fam_arg = 'equi'
else:
ex_fam_arg = 'l' + fam_arg
if (fam_arg == 'lgg' or fam_arg == 'gg'):
fam_arg = (fam_arg, args.g_alpha)
family = _decode_family(fam_arg)
x = coord_map(family[n], 'unit', domain)
if need_expanded:
if (ex_fam_arg == 'lgg'):
ex_fam_arg = (ex_fam_arg, args.g_alpha)
ex_family = _decode_family(ex_fam_arg)
ex_x = coord_map(family[n+d+1], 'unit', domain)
if args.nodes == 'equispaced':
nodes = equispaced(d, n, domain=domain)
elif args.nodes == 'equispaced_interior':
nodes = equispaced_interior(d, n, domain=domain)
if need_expanded:
ex_nodes = equispaced(d, n+d+1, domain=domain)
elif args.nodes == 'blyth_luo_pozrikidis':
nodes = blyth_luo_pozrikidis(d, n, x, domain=domain)
if need_expanded:
ex_nodes = blyth_luo_pozrikidis(d, n+d+1, ex_x, domain=domain)
elif args.nodes == 'warburton':
alpha = args.w_alpha
nodes = warburton(d, n, x, alpha=alpha, domain=domain)
if need_expanded:
ex_nodes = warburton(d, n+d+1, ex_x, alpha=alpha, domain=domain)
elif args.nodes == 'recursive':
nodes = recursive(d, n, family, domain=domain)
if need_expanded:
ex_nodes = recursive(d, n+d+1, ex_family, domain=domain)
if expanded:
if need_expanded:
ex_nodes = expand_to_boundary(d, n, nodes, ex_nodes, domain=domain)
return (d, n, domain, family, nodes, ex_nodes)
else:
return (d, n, domain, family, nodes, nodes)
return (d, n, domain, family, nodes)
if __name__ == '__main__':
from argparse import ArgumentParser
parser = ArgumentParser(description=('Print out a node set '
'implemented in nodes.py'))
add_nodes_to_parser(parser)
args = parser.parse_args()
d, n, domain, family, nodes = nodes_from_args(args)
print(nodes)
