nodes

All of the node sets for the \(d\)-simplex implemented by recursivenodes.

Node sets on the simplex

recursivenodes.nodes.recursive(d, n, family='lgl', domain='barycentric')[source]

Alias of recursivenodes.recursive_nodes().

recursivenodes.nodes.equispaced(d, n, domain='biunit')[source]

Equispaced nodes for polynomials up to degree \(n\) on the \(d\)-simplex.

Parameters
  • d (int) – The dimension of the simplex.

  • n (int) – The polynomial degree

  • domain (str, optional) – “The domain optional argument” for the choices and their formal definitions.

Returns

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').

Return type

ndarray

Example

>>> 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()
_images/nodes-1.png
recursivenodes.nodes.equispaced_interior(d, n, domain='biunit')[source]

Equispaced nodes for polynomials up to degree \(n\) on the \(d\)-simplex, all of which are in the interior.

Parameters
  • d (int) – The dimension of the simplex.

  • n (int) – The polynomial degree

  • domain (str, optional) – The domain of the simplex. See “The domain optional argument” for the choices and their formal definitions.

Returns

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').

Return type

ndarray

Example

>>> 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()
_images/nodes-2.png
recursivenodes.nodes.blyth_luo_pozrikidis(d, n, x=None, domain='biunit')[source]

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

(1)\[b_i(\boldsymbol{\alpha}) = x_{n,\boldsymbol{\alpha}_i} + \frac{1}{d}(1 - \sum_{j\neq i} x_{n,\boldsymbol{\alpha}_j}).\]

Points on the boundary look like \((d-1)\)-simplex Blyth-Luo-Pozrikidis nodes.

Parameters
  • 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 “The domain optional argument” for the choices and their formal definitions.

Returns

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').

Return type

ndarray

Example

This plot shows the Blyth-Luo-Pozrikidis nodes with lines connecting the Lobatto-Gauss-Legendre nodes on the edges. The definition in (1) is designed to place the nodes in the centroids of the triangles created by the intersecting lines.

>>> 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()
_images/nodes-3.png

References

[BP06][LP06]

recursivenodes.nodes.warburton(d, n, x=None, alpha=None, domain='biunit')[source]

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.

Parameters
  • 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 “The domain optional argument” for the choices and their formal definitions.

Returns

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').

Return type

ndarray

Example

>>> 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()
_images/nodes-4.png

References

[War06]

recursivenodes.nodes.warburton_alpha

A dictionary of optimal values of the blending parameter \(\alpha\) computed in [War06]. Keyed by (d, n) tuples.

Example

>>> warburton_alpha[(2,7)]
1.0999
recursivenodes.nodes.rapetti_sommariva_vianello(n, domain='unit')[source]

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.

Parameters
  • n (int) – The polynomial degree, must be \(\leq 18\).

  • domain (str, optional) – The domain of the simplex. See “The domain optional argument” for the choices and their formal definitions.

Returns

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').

Return type

ndarray

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.

>>> 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()
_images/nodes-5.png

References

[RSV12]

Warning

These nodes cannot be used with recursivenodes.lebesgue.lebesguemax() for \(n \geq 10\), because they do not have lattice-like structure.

The family optional argument

A full definition of 1D node families is given in the documentation of recursivenodes.recursive_nodes().

Here are ways to specify 1D node families in recursivenodes:

  • Shifted Lobatto-Gauss-Jacobi nodes for different values of the weight exponents \(\boldsymbol{\alpha}\) and \(\beta\), which always include the endpoints \(0\) and \(1\):

    • 'lgl' (str) – Shifted L-G-Legendre (\(\boldsymbol{\alpha} = \beta = 0\)).

    • 'lgc' (str) – Shifted L-G-Chebyshev (\(\boldsymbol{\alpha} = \beta = -1/2\)).

    • ('lgg', a) ((str, float)) – Shifted L-G-Gegenbauer (\(\boldsymbol{\alpha} = \beta = a - 1/2\)).

  • Shifted Gauss-Jacobi nodes for different values of the weight exponents \(\boldsymbol{\alpha}\) and \(\beta\), which are in the interior away from the endpoints \(0\) and \(1\):

    • 'gl' (str) – Shifted G-Legendre (\(\boldsymbol{\alpha} = \beta = 0\)).

    • 'gc' (str) – Shifted G-Chebyshev (\(\boldsymbol{\alpha} = \beta = -1/2\)).

    • ('gg', a) ((str, float)) – Shifted G-Gegenbauer (\(\boldsymbol{\alpha} = \beta = a - 1/2\)).

  • Equispaced nodes:

    • 'equi' (str) – Equispaced nodes including the endpoints \(0\) and \(1\).

    • 'equi_interior' (str) – Equispaced nodes excluding the endpoints \(0\) and \(1\). The \(k\)-point equispaced interior nodes are a half-phase off from the \((k+1)\)-point equispaced nodes.

      >>> equispaced(1, 4, domain='unit')
      array([[0.  ],
             [0.25],
             [0.5 ],
             [0.75],
             [1.  ]])
      >>> equispaced_interior(1, 3, domain='unit')
      array([[0.125],
             [0.375],
             [0.625],
             [0.875]])
      

The domain optional argument

Four reference domains for the \(d\)-simplex are implemented in recursivenodes.

  • 'unit' (str) – In \(\mathbb{R}^d\), defined by \(x_i \geq 0, \sum_i x_i <= 1\).

  • 'biunit' (str) – In \(\mathbb{R}^d\), defined by \(x_i \geq -1, \sum_i x_i <= 2-d\).

  • 'equilateral' (str) – In \(\mathbb{R}^d\), defined by having:

    • edge length 2,

    • centroid at the origin,

    • a downward-facing facet,

    • a projection onto the first \(d-1\) coordinates that is the corresponding equilateral \(d-1\)-simplex.

    (For \(d=1\), 'equilateral' is the same as 'biunit').

  • 'barycentric' (str) – In \(\mathbb{R}^{d+1}\), defined by \(x_i \geq 0, \sum_i x_i = 1\).

    _images/nodes-6.png