# An Explicit Construction of Optimized Interpolation Points on the 4-Simplex

Trenton J. Gobel<EMAIL_ADDRESS>David M. Williams  Department of Mechanical
Engineering, The Pennsylvania State University, University Park, Pennsylvania
16802

###### Abstract

In this work, a family of symmetric interpolation points are generated on the
four-dimensional simplex (i.e. the pentatope). These points are optimized in
order to minimize the Lebesgue constant. The process of generating these
points closely follows that outlined by Warburton in “An explicit construction
of interpolation nodes on the simplex,” Journal of Engineering Mathematics,
2006. Here, Warburton generated optimal interpolation points on the triangle
and tetrahedron by formulating explicit geometric warping and blending
functions, and applying these functions to equidistant nodal distributions.
The locations of the resulting points were Lebesgue-optimized. In our work, we
extend this procedure to four dimensions, and construct interpolation points
on the pentatope up to order ten. The Lebesgue constants of our nodal sets are
calculated, and are shown to outperform those of equidistant nodal
distributions.

###### keywords:

Interpolation , space-time , four dimensions , finite element methods ,
Lebesgue constant , symmetry

###### MSC:

[2010] 76M10 , 65D05 , 65M12 , 65M60

††journal: Applied Numerical Mathematicsfn1fn1footnotetext: Distribution
Statement A: Approved for public release. Distribution is unlimited.

##  1 Introduction

The nodal polynomial approximation of a well-defined function is called a
‘nodal representation’. One may use nodal representations in order to
approximate _all_ functions for a given finite element method (FEM), including
both the solution and the geometry. Such FEM’s are referred to as nodal FEM’s.
However, alternative options exist. In particular, one may use either nodal or
modal polynomials to represent the FEM solution and geometry. With this in
mind, one may consider four different types of FEM’s: a) FEM’s which use a
nodal representation for both the solution and geometry, b) FEM’s which use a
modal representation for the solution and a nodal representation for the
geometry, c) FEM’s which use a nodal representation for the solution and a
modal representation for the geometry, and d) FEM’s which use a modal
representation for both the solution and geometry. We note that FEM’s of types
c) and d) are not very practical, as it is difficult to approximate the
geometry with a modal representation. In particular, one must obtain modal
coefficients for the geometry by solving linear systems of equations; these
systems must be customized for each element in the mesh, and the sizes of the
systems grow rapidly with the polynomial order and dimension. In contrast,
nodal representations bypass this issue, as they directly interpolate the
geometry. Consequently, there is no need to solve a customized linear system
for the nodal coefficients, for each element in the mesh. As a result, FEM’s
of types a) and b) are the most popular in practice.

We can summarize the above discussion by stating that virtually all popular
FEM’s require nodal interpolation points in some capacity.

In what follows, we will discuss the important characteristics of nodal and
modal representations, the key differences between the two, the unique
challenges associated with constructing a nodal representation, and the
importance of the Lebesgue constant.

##  2 Background

Prior to discussing the specifics of the nodal representation, it is important
to first provide some contextual background by describing the modal
representation and its properties.

In order to construct a modal representation for the solution, we multiply
modal coefficients by a set of element-spanning polynomials—i.e. modes. These
modes are not required to be orthogonal, but are often chosen as such in order
to facilitate implementation. In particular, the orthogonality of the
polynomials results in a diagonal mass matrix thereby simplifying
computations. The modal coefficients and polynomials are independent of any
interpolation node locations, and confer the advantage that changing the
polynomial order does not necessarily require changing all of the basis
functions. This latter property is guaranteed to hold if the modal basis is
hierarchical.

To construct a nodal representation, we must choose specific point locations
in the reference element at which the values of the solution will be defined
or sampled. These solution values are then multiplied by nodal basis functions
(usually Lagrange polynomials) and summed in order to return the solution on
the entire element. Many researchers consider nodal representations to be more
intuitive than modal representations, as the nodal coefficients directly
correspond to the values of the FEM solution at particular points in physical
space. Conversely, modal coefficients correspond to the amounts of energy that
are stored in particular modes. Unfortunately, nodal basis functions are
neither hierarchical nor orthogonal, which makes their
implementation/computation less straightforward than their modal counterparts.

Naturally, this raises the question of how nodal basis functions can be
generated. In one dimension, these basis functions can be calculated
analytically using the well-known Lagrange formula. The extension of this
approach to higher dimensions is straightforward on quadrilaterals, hexahedra,
tesseracts and other hypercube elements, as the tensor products of Lagrange
polynomials can be used to construct multi-dimensional nodal basis functions.
However, the construction of nodal basis functions on higher-dimensional
simplices is more difficult, owing in part to the nonexistence of an elegant,
tensor-product-based procedure. Some explicit basis function formulae
exist—such as those given by Hughes [1] for triangles and tetrahedra—but these
functions are limited to low polynomial orders, and are difficult to generate
in an automatic fashion.

Evidently, a different approach for generating nodal basis functions is
needed. Fortunately, as discussed in [2], Vandermonde matrices relate modal
representations to nodal representations. The modal basis in question is
constructed using products of Jacobi polynomials of the desired order.
Utilizing these Jacobi polynomials, we can construct an orthonormal modal
basis on the reference simplicial element. Once this is obtained, we can
multiply a vector of the modal basis functions by the inverse transpose of the
Vandermonde matrix, and thereafter determine the values of the multi-
dimensional Lagrange polynomials.

Proper behavior of the Vandermonde matrix is crucial for interpolation
accuracy and stability, especially at high polynomial orders. Therefore, one
often seeks to maximize the absolute value of the matrix determinant. The
maximization process depends on two factors. Firstly, we must ensure the
orthonormality of the modal polynomial basis. Fortunately, an equation for an
orthonormal basis in four dimensions on the pentatope has already been
obtained—see the formulation in [3]. Secondly, we must select interpolation
points for the construction of the Vandermonde matrix which are optimally
distributed throughout the simplex. It turns out that Legendre-Gauss-Lobatto
points do an excellent job of amplifying the Vandermonde matrix determinant in
one dimension. Therefore, we seek to construct generalizations of Legendre-
Gauss-Lobatto points on to higher-dimensional simplices.

In addition to maximizing the determinant of the Vandermonde matrix, there are
two qualities that bear consideration: symmetry and error minimization.
Symmetry of interpolation points on the simplex is defined as invariance of
the point locations under affine transformations of the simplex on to itself.
Error minimization is defined based on the points’ ability to minimize the
interpolation error with respect to a particular norm.

It is well-known that minimization of the pointwise error is equivalent to
minimizing the error with respect to the $L^{\infty}$ norm. In turn,
minimizing the $L^{\infty}$ norm is achieved by minimizing the Lebesgue
constant for a given nodal distribution. Finally, it is important to note that
minimizing the Lebesgue constant is directly connected to maximizing the
Vandermonde matrix determinant. Therefore, we can focus on minimizing the
Lebesgue constant, and obtain some of the numerical-stability benefits
associated with maximizing the determinant without additional effort.

As an alternative strategy, we can also minimize the amount of error
introduced during numerical quadrature procedures. We will refer to this
strategy as quadrature-error minimization. It turns out, that for low
polynomial orders, quadrature-optimized interpolation points can have
relatively small Lebesgue constants. As a result, they simultaneously maintain
good integration accuracy and $L^{\infty}$ accuracy [4].

In summary, one often seeks to construct interpolation points on the simplex
which satisfy the following objectives: i) the points are arranged in a
symmetric fashion, ii) the points maximize the determinant of the Vandermonde
matrix, and iii) the points minimize either quadrature error or $L^{\infty}$
error.

###  2.1 Literature Review

Early efforts to construct interpolation points focused on maximizing the
determinant of the Vandermonde matrix. Interpolation points constructed in
this fashion are called Fekete points [5]. In the 1980’s, Bos used this
optimization approach to generate symmetric interpolation points on triangles
up to order seven [6]. Chen and Babuška furthered this concept up to order
twenty on the line segment and order thirteen on the triangle [7]. They also
constructed symmetric interpolation points on the tetrahedron up to order nine
[8]. Thereafter, Taylor, Wingate, and Vincent generated symmetric
interpolation points up to order nineteen on the triangle [9], using the same
methodology. In [10], Heinrichs used a different approach, minimizing the
Lebesgue constant directly rather than maximizing the Vandermonde matrix
determinant. With this approach, interpolation points up to order eighteen
were generated on the triangle. Curiously, the resulting points were not
symmetric.

Another optimization method was formulated based on electrostatic principles
by Hesthaven [11]. In this work, an analogy was drawn between the electric
charge distributions of Stieltjes [12, 13] and the Legendre-Gauss-Lobatto
points in one dimension. Hesthaven extended the electrostatic analogy into two
dimensions, and constructed symmetric interpolations points on the triangle
with favorable Lebesgue constants up to order twelve.

In addition, Warburton proposed a simple optimization procedure for minimizing
the Lebesgue constant based on geometric movement (warping and blending) of
point locations [14]. In this procedure, the interpolation points are
initially distributed on an equispaced grid within the simplex, and then
allowed to move in conjunction with explicitly defined warping and blending
functions. The functions are defined on the faces and edges of the simplex,
and are parameterized by a single variable. In turn, this variable is
optimized in order to minimize the Lebesgue constant. Using this approach,
Warburton generated symmetric interpolation points on the triangle up to order
fifteen, and on the tetrahedron up to order ten.

Recently, Jameson et al. [15] and Castonguay et al. [16] proved the
effectiveness of quadrature-optimized interpolation points for FEM’s applied
to non-linear advection problems. In this work, they leveraged symmetric
quadrature rules which happened to have the correct number of points to
represent polynomials on the line segment and triangle. Thereafter, Shunn and
Ham [17], and Williams et al. [18] expanded this work, and constructed
symmetric quadrature-optimized interpolation points on the triangle and
tetrahedron up to orders ten and six, respectively. In [4], Williams and
Jameson showed that the Lebesgue constants for these points are well-behaved
for up to order five on the tetrahedron. Most recently, Williams et al. [19]
extended the work of [18], and generated quadrature-optimized, symmetric
interpolation points on the pentatope up to order five.

###  2.2 Overview of the Paper

The majority of previous work has focused on generating interpolation points
in one, two, and three dimensions. The comparatively small amount of work that
focuses on four-dimensional space has been limited to quadrature-optimized
points. In this paper, we seek to generate new, Lebesgue-optimized
interpolation points on 4-simplex elements (pentatopes). Our work is a natural
extension of Warburton’s approach [14] into four dimensions. To our knowledge,
this represents the first ever such endeavor.

In section 3 of this paper, we discuss the preliminary concepts and formulae
relevant to this method. Section 4 explains the methodology behind the warping
and blending functions used in the generation of the Lebesgue-optimized
points. The final results of our optimization process are given in section 5.
Lastly, conclusions are given in section 6 along with suggestions for future
work. Tables of the optimized interpolation points are given in an appendix
subsequent to the main text.

##  3 Preliminaries

###  3.1 Nodal Representations

A nodal solution representation in $d$-dimensional space is given by

$\displaystyle u^{h}(\bm{r})=\sum_{i=1}^{N_{p}}u_{i}^{h}l_{i}(\bm{r}),$ (3.1)

where $u^{h}$ is the finite element solution, $N_{p}$ is the number of nodes,
$l_{i}$ are the nodal basis functions, and $\bm{r}$ is the coordinate vector
in reference space. Here, $u_{i}^{h}$ represents the value of the solution at
the specific nodal locations $\bm{r}_{i}$ in reference space. For a given
dimension $d$ and polynomial order $p$, the number of nodes is calculated via

$\displaystyle N_{p}=\frac{1}{d!}\prod_{k=1}^{d}(p+k).$ (3.2)

In one dimension, this simplifies to $N_{p}=p+1$. The nodal basis functions
can be calculated for a one-dimensional mesh using Lagrange polynomials

$\displaystyle l_{j}(x)=\prod^{N_{p}}_{\stackrel{{\scriptstyle m=0}}{{m\neq
j}}}\frac{x-x_{m}}{x_{j}-x_{m}}=\frac{x-x_{0}}{x_{j}-x_{0}}\cdot\cdot\cdot\frac{x-x_{j-1}}{x_{j}-x_{j-1}}\frac{x-x_{j+1}}{x_{j}-x_{j+1}}\cdot\cdot\cdot\frac{x-x_{N_{p}}}{x_{j}-x_{N_{p}}},$
(3.3)

where each $x_{j}$ represents a specific nodal location. However, as mentioned
previously, it is difficult to construct analytical formulas for nodal basis
functions on simplex elements in $\mathbb{R}^{d}$ when $d\geq 2$. This issue
will be addressed in the next section.

###  3.2 Modal Representations

The general form for modal solution representations is very similar to nodal
representations

$\displaystyle
u_{h}(\bm{r})=\sum_{i=1}^{N_{p}}\hat{u}_{i}^{h}\psi_{i}(\bm{r}),$ (3.4)

where $\hat{u}_{i}^{h}$ are the modal coefficients, and $\psi_{i}(\bm{r})$ are
the modal basis functions. However, unlike nodal basis functions, there exist
many analytical formulas for modal basis functions. Oftentimes, these formulas
are written in terms of Jacobi polynomials. We note that the Jacobi
polynomials themselves can be formulated recursively as follows

$\displaystyle P_{0}^{(\alpha,\beta)}(x)$
$\displaystyle=\sqrt{2^{-\alpha-\beta-1}\frac{\Gamma(\alpha+\beta+2)}{\Gamma(\alpha+1)\Gamma(\beta+1)}},$
(3.5) $\displaystyle P_{1}^{(\alpha,\beta)}(x)$
$\displaystyle=\frac{1}{2}P_{0}^{(\alpha,\beta)}(x)\sqrt{\frac{\alpha+\beta+3}{(\alpha+1)(\beta+1)}}((\alpha+\beta+2)x+(\alpha-\beta)),$
(3.6) $\displaystyle P_{n+1}^{(\alpha,\beta)}(x)$
$\displaystyle=\frac{1}{a_{n+1}}\left(xP_{n}^{(\alpha,\beta)}(x)-a_{n}P_{n-1}^{(\alpha,\beta)}(x)-b_{n}P_{n}^{(\alpha,\beta)}(x)\right),$
(3.7)

where $\alpha,\beta$ and $n$ are integer parameters, $\Gamma(\cdot)$ is the
well-known Gamma function, and $a_{n}$ and $b_{n}$ are defined as

$\displaystyle a_{n}$
$\displaystyle=\frac{2}{2n+\alpha+\beta}\sqrt{\frac{n(n+\alpha+\beta)(n+\alpha)(n+\beta)}{(2n+\alpha+\beta-1)(2n+\alpha+\beta+1)}},$
(3.8) $\displaystyle b_{n}$
$\displaystyle=-\frac{\alpha^{2}-\beta^{2}}{(2n+\alpha+\beta)(2n+\alpha+\beta+2)}.$
(3.9)

Hesthaven and Warburton [2] give a Jacobi polynomial-based formulation in one
dimension for the orthonormal modal basis as follows

$\displaystyle\psi_{m}(r)=\sqrt{\frac{2m-1}{2}}P_{m-1}^{(0,0)}(r),\qquad 0\leq
m\leq p+1.$ (3.10)

Note that here the $\alpha$ and $\beta$ parameters of the Jacobi polynomial
are zero, returning the $(m-1)^{\text{th}}$ Legendre polynomial in place of
the Jacobi polynomial.

Now that we have an expression for the modal basis, we can relate it to the
nodal basis. This action is completed via the Vandermonde matrix $\mathcal{V}$

$\displaystyle\mathcal{V}_{ij}=\psi_{j}(\bm{r}_{i}),\qquad\mathcal{V}^{T}\bm{l(r)}=\tilde{\boldsymbol{\psi}}\bm{(r)},$
(3.11)

where $\bm{r}_{i}$ are the nodal point locations in reference space, (as
mentioned previously). Consequently,

$\displaystyle\bm{l(r)}=\mathcal{V}^{-T}\tilde{\boldsymbol{\psi}}\bm{(r)}.$
(3.12)

Therefore, in order to obtain a higher-dimensional nodal basis, we merely need
to construct a higher-dimensional _modal_ basis, and then use Eq. (3.12). For
the sake of completeness, in what follows we provide the modal basis formulas
for two, three, and four dimensions.

In two dimensions, the modal basis is given by

$\displaystyle\psi_{m}(\bm{r})=\sqrt{2}P_{i}^{(0,0)}(a)P_{j}^{(2i+1,0)}(b)(1-b)^{i},$
(3.13)

where $m$ is the modal index for two-dimensional polynomials, defined over $i$
and $j$ as

$\displaystyle
m=1+\left(\frac{2p+3}{2}\right)i+\left(-\frac{1}{2}\right)i^{2}+j,\quad(i,j)\geq
0,\quad i+j\leq p,$ (3.14)

and $a$ and $b$ are defined relative to the reference coordinates
$\bm{r}=(r,s)$

$\displaystyle a=2\left(\frac{1+r}{1-s}\right)-1,\quad b=s.$ (3.15)

The modal basis in three dimensions is an extension of the two-dimensional
case

$\displaystyle\psi_{m}(\bm{r})=2\sqrt{2}P_{i}^{(0,0)}(a)P_{j}^{(2i+1,0)}(b)P_{k}^{(2i+2j+2,0)}(c)(1-b)^{i}(1-c)^{i+j},$
(3.16)

where again we have a modal indexing function

$\displaystyle m$
$\displaystyle=1+\left(\frac{3p^{2}+12p+11}{6}\right)i+\left(\frac{-p-2}{2}\right)i^{2}+\left(\frac{1}{6}\right)i^{3}+\left(\frac{2p+3}{2}\right)j$
$\displaystyle+\left(-\frac{1}{2}\right)j^{2}-ij+k,\quad(i,j,k)\geq 0,\quad
i+j+k\leq p,$ (3.17)

and for $\bm{r}=(r,s,t)$

$\displaystyle a=-2\left(\frac{1+r}{s+t}\right)-1,\quad
b=2\left(\frac{1+s}{1-t}\right)-1,\quad c=t.$ (3.18)

Frontin et. al. [3] give a formulation for the four-dimensional modal basis
which follows the pattern set by its lower-dimensional predecessors

$\displaystyle\psi_{m}(\bm{r})=8P_{i}^{(0,0)}(a)P_{j}^{(2i+1,0)}(b)P_{k}^{(2i+2j+2,0)}(c)P_{q}^{(2i+2j+2k+3,0)}(e)$
$\displaystyle\times(1-b)^{i}(1-c)^{i+j}(1-e)^{i+j+k},$ (3.19)

and for $\bm{r}=(r,s,t,u)$

$\displaystyle a=-2\left(\frac{1+r}{s+t+u+1}\right)-1,\quad
b=-2\left(\frac{1+s}{t+u}\right)-1,\quad
c=2\left(\frac{1+t}{1-u}\right)-1,\quad e=u.$ (3.20)

As in the lower dimensional cases, a modal indexing formula is necessary in
four dimensions. However, no expression for $m$ has been derived in previous
works.

By observation, we realize that the indexing equation is polynomial in nature
with each term of the form $C_{n}i^{\xi}j^{\zeta}k^{\mu}q^{\nu}$, where $\xi$,
$\zeta$, $\mu$, and $\nu$ are non-negative integers. We also note that each
coefficient $C_{n}$ is either constant or a polynomial function of $p$. This
suggests the following formula for $m$ in four dimensions

$\displaystyle m$
$\displaystyle=1+C_{1}i+C_{2}i^{2}+C_{3}i^{3}+C_{4}i^{4}+C_{5}j+C_{6}ij+C_{7}i^{2}j+C_{8}j^{2}+C_{9}ij^{2}+C_{10}j^{3}$
$\displaystyle+C_{11}k+C_{12}ik+C_{13}jk+C_{14}k^{2}+C_{15}q,$
$\displaystyle(i,j,k,q)\geq 0,\quad i+j+k+q\leq p.$ (3.21)

Note that every permutation of $i,j,k,$ and $q$ is not represented in the
above formula. Certain terms were excluded based on the assumption that, like
the lower-dimensional equations before it, there exists an upper limit on the
order of a term depending on the variables appearing in that term and the
dimension $d$. For example, no term containing an $i$ can have an order above
$d$, $j$ above $d-1$, $k$ above $d-2$, and $q$ above $d-3$.

To find explicit expressions for the coefficients in Eq. (3.21), we must first
generate the correct node index ordering using a series of nested for loops.
Once proper ordering is ascertained, we carefully choose values of $i,j,k,$
and $q$ to calculate the numerical values for each variable $C_{n}$ for a
given $p$ as shown in Table LABEL:C_Table_num. The MATLAB code used to
generate Table LABEL:C_Table_num is given in Appendix B of [20].

Table 1: Four-dimensional node indexing equation coefficients for varying $p$ $p$ | 4 | 5 | 6 | 7 | 8
---|---|---|---|---|---
$C_{1}$ | $\nicefrac{{533}}{{12}}$ | $\nicefrac{{275}}{{4}}$ | $\nicefrac{{1207}}{{12}}$ | $\nicefrac{{1691}}{{12}}$ | $\nicefrac{{763}}{{4}}$
$C_{2}$ | $\nicefrac{{-251}}{{24}}$ | $\nicefrac{{-335}}{{24}}$ | $\nicefrac{{-431}}{{24}}$ | $\nicefrac{{-539}}{{24}}$ | $\nicefrac{{-659}}{{24}}$
$C_{3}$ | $\nicefrac{{13}}{{12}}$ | $\nicefrac{{5}}{{4}}$ | $\nicefrac{{17}}{{12}}$ | $\nicefrac{{19}}{{12}}$ | $\nicefrac{{7}}{{4}}$
$C_{4}$ | $\nicefrac{{-1}}{{24}}$ | $\nicefrac{{-1}}{{24}}$ | $\nicefrac{{-1}}{{24}}$ | $\nicefrac{{-1}}{{24}}$ | $\nicefrac{{-1}}{{24}}$
$C_{5}$ | $\nicefrac{{107}}{{6}}$ | $\nicefrac{{73}}{{3}}$ | $\nicefrac{{191}}{{6}}$ | $\nicefrac{{121}}{{3}}$ | $\nicefrac{{299}}{{6}}$
$C_{6}$ | -6 | -7 | -8 | -9 | -10
$C_{7}$ | $\nicefrac{{1}}{{2}}$ | $\nicefrac{{1}}{{2}}$ | $\nicefrac{{1}}{{2}}$ | $\nicefrac{{1}}{{2}}$ | $\nicefrac{{1}}{{2}}$
$C_{8}$ | -3 | $\nicefrac{{-7}}{{2}}$ | -4 | $\nicefrac{{-9}}{{2}}$ | -5
$C_{9}$ | $\nicefrac{{1}}{{2}}$ | $\nicefrac{{1}}{{2}}$ | $\nicefrac{{1}}{{2}}$ | $\nicefrac{{1}}{{2}}$ | $\nicefrac{{1}}{{2}}$
$C_{10}$ | $\nicefrac{{1}}{{6}}$ | $\nicefrac{{1}}{{6}}$ | $\nicefrac{{1}}{{6}}$ | $\nicefrac{{1}}{{6}}$ | $\nicefrac{{1}}{{6}}$
$C_{11}$ | $\nicefrac{{11}}{{2}}$ | $\nicefrac{{13}}{{2}}$ | $\nicefrac{{15}}{{2}}$ | $\nicefrac{{17}}{{2}}$ | $\nicefrac{{19}}{{2}}$
$C_{12}$ | -1 | -1 | -1 | -1 | -1
$C_{13}$ | -1 | -1 | -1 | -1 | -1
$C_{14}$ | $\nicefrac{{-1}}{{2}}$ | $\nicefrac{{-1}}{{2}}$ | $\nicefrac{{-1}}{{2}}$ | $\nicefrac{{-1}}{{2}}$ | $\nicefrac{{-1}}{{2}}$
$C_{15}$ | 1 | 1 | 1 | 1 | 1

Solving for several of the coefficients is trivial, as they remain constant
regardless of the order $p$. The remaining coefficients change with $p$;
however they can be obtained through a simple curve-fitting procedure.
Following this approach, the final nodal index formulation in four dimensions
is given by

$\displaystyle m$
$\displaystyle=1+\left(\frac{2p^{3}+15p^{2}+35p+25}{12}\right)i+\left(\frac{-6p^{2}-30p-35}{24}\right)i^{2}$
$\displaystyle+\left(\frac{2p+5}{12}\right)i^{3}-\left(\frac{1}{24}\right)i^{4}+\left(\frac{3p^{2}+12p+11}{6}\right)j+(-p-2)ij+\left(\frac{1}{2}\right)i^{2}j$
$\displaystyle+\left(\frac{-p-2}{2}\right)j^{2}+\left(\frac{1}{2}\right)ij^{2}+\left(\frac{1}{6}\right)j^{3}+\left(\frac{2p+3}{2}\right)k-ik-
jk-\left(\frac{1}{2}\right)k^{2}+q,$ $\displaystyle(i,j,k,q)\geq 0,\quad
i+j+k+q\leq p.$ (3.22)

In order to minimize computational roundoff error, the authors find it useful
to premultiply each term in Eq. (3.2) by a factor of 24 and then multiply the
result of the entire equation by $\nicefrac{{1}}{{24}}$ after summing terms.

###  3.3 Lebesgue Constant

With the indexing function derived, all the preliminary pieces are now in
place to generate the nodal basis functions on the pentatope. It remains for
us to establish a metric for assessing the validity of the interpolation point
sets. This assessment is performed by calculating the Lebesgue constant
$\Lambda$ for each point set $\left\\{\bm{r}_{i}\right\\}$

$\displaystyle\Lambda=\max_{\bm{r}}\sum_{i=1}^{N_{p}}\left|l_{i}(\bm{r})\right|.$
(3.23)

Minimization of the Lebesgue constant is proven to minimize the $L^{\infty}$
error as follows

$\displaystyle||u-u_{h}||_{\infty}=||u-u^{*}+u^{*}-u_{h}||_{\infty}\leq||u-u^{*}||_{\infty}+||u^{*}-u_{h}||_{\infty}\leq(1+\Lambda)||u-u^{*}||_{\infty},$
(3.24)

where $u^{*}$ is the best polynomial approximation of the exact solution $u$.

##  4 Methodology

Optimization of the Lebesgue constant for a given point set is performed using
warping and blending functions to adjust interpolation point locations. These
functions are generated sequentially, with the four-dimensional functions
dependent on the three-dimensional ones—which are themselves dependent on the
two-dimensional functions and so on. Here, the process of generating these
warping and blending functions is detailed, beginning with the one-dimensional
case and extending up to four dimensions.

###  4.1 The Line Segment

Consider the reference line segment, which is defined as the shortest distance
between the following two vertices

$\displaystyle\bm{v}_{1}$ $\displaystyle=-1,\qquad\bm{v}_{2}=1.$ (4.1)

By construction, the reference line segment is of length 2, and the sole edge
is given by the subset consisting of the vertices

$\displaystyle E_{1}$ $\displaystyle=\left\\{\bm{v}_{1},\bm{v}_{2}\right\\}.$

The simplest discretization of the reference line is that of equally-spaced
nodes

$\displaystyle r^{e}_{i}=-1+\frac{2i}{p},\qquad i\in[0,...,p],$

However, this discretization can result in the onset of the infamous Runge
phenomenon for large values of $p$. As such, it is often advantageous to adopt
a ‘warped’ discretization using Legendre-Gauss-Lobatto point locations, which
are optimized to maximize the determinant of the associated Vandermonde
matrix. These points will be referred to as $r_{i}^{LGL}$. These two nodal
discretizations can be connected via

$\displaystyle w(r)=\sum_{i=1}^{p+1}(r_{i}^{LGL}-r_{i}^{e})\ell_{i}^{e}(r),$

where $\ell_{i}^{e}$ are the Lagrange basis functions derived for $r_{i}^{e}$.

The warping function (above) can be modified in order to obtain

$\displaystyle\widetilde{w}(r)=\frac{w(r)}{1-r^{2}}.$ (4.2)

This function will be used frequently during the subsequent sections.

###  4.2 The Triangle

Consider the reference triangle, which is defined as the convex hull of the
following three vertices

$\displaystyle\bm{v}_{1}=\left(-1,-\frac{1}{\sqrt{3}}\right),\qquad\bm{v}_{2}=\left(1,-\frac{1}{\sqrt{3}}\right),\qquad\bm{v}_{3}=\left(0,\frac{2}{\sqrt{3}}\right).$
(4.3)

All points within the triangle interior can be represented as follows

$\displaystyle\bm{r}=\lambda^{2}\bm{v}_{1}+\lambda^{3}\bm{v}_{2}+\lambda^{1}\bm{v}_{3},$

where

$\displaystyle 0\leq\lambda^{1},\lambda^{2},\lambda^{3}\leq
1,\qquad\lambda^{1}+\lambda^{2}+\lambda^{3}=1.$

By construction, the triangle is equilateral and all of the edges are of
length 2. The edges are given by

$\displaystyle E_{1}$
$\displaystyle=\left\\{\bm{v}_{1},\bm{v}_{2}\right\\},\qquad
E_{2}=\left\\{\bm{v}_{2},\bm{v}_{3}\right\\},\qquad
E_{3}=\left\\{\bm{v}_{1},\bm{v}_{3}\right\\}.$

A warping function for each edge is derived from Eq. (4.2) as follows

$\displaystyle\bm{w}^{1}(\lambda^{1},\lambda^{2},\lambda^{3})$
$\displaystyle=\widetilde{w}(\lambda^{3}-\lambda^{2})\bm{t}^{1},$
$\displaystyle\bm{w}^{2}(\lambda^{1},\lambda^{2},\lambda^{3})$
$\displaystyle=\widetilde{w}(\lambda^{1}-\lambda^{3})\bm{t}^{2},$
$\displaystyle\bm{w}^{3}(\lambda^{1},\lambda^{2},\lambda^{3})$
$\displaystyle=\widetilde{w}(\lambda^{2}-\lambda^{1})\bm{t}^{3}.$

Here, the $\bm{t}^{f}$’s are the vectors associated with each edge

$\displaystyle\bm{t}^{1}$ $\displaystyle=\left(1,0\right),$
$\displaystyle\bm{t}^{2}$
$\displaystyle=\left(-\frac{1}{2},\frac{\sqrt{3}}{2}\right),$
$\displaystyle\bm{t}^{3}$
$\displaystyle=\left(-\frac{1}{2},-\frac{\sqrt{3}}{2}\right).$

In addition, we can define the following blending functions for each edge

$\displaystyle
b^{1}(\lambda^{1},\lambda^{2},\lambda^{3})=4\lambda^{3}\lambda^{2},$
$\displaystyle
b^{2}(\lambda^{1},\lambda^{2},\lambda^{3})=4\lambda^{3}\lambda^{1},$
$\displaystyle
b^{3}(\lambda^{1},\lambda^{2},\lambda^{3})=4\lambda^{2}\lambda^{1}.$

Next, we can define the following warping/blending function on the entire
triangle

$\displaystyle\bm{g}\left(\lambda^{1},\lambda^{2},\lambda^{3}\right)$
$\displaystyle=\left(1+\left(\alpha\lambda^{1}\right)^{2}\right)b^{1}\bm{w}^{1}+\left(1+\left(\alpha\lambda^{2}\right)^{2}\right)b^{2}\bm{w}^{2}+\left(1+\left(\alpha\lambda^{3}\right)^{2}\right)b^{3}\bm{w}^{3},$
(4.4)

where $\alpha$ is an optimization parameter. Generally speaking, our objective
is to shift the locations of equidistant points on the triangle by using the
warping/blending function
$\bm{g}\left(\lambda^{1},\lambda^{2},\lambda^{3}\right)$, and thereby obtain
good interpolation points. The locations of the equidistant points are given
by

$\displaystyle(i,j)\geq 0,\quad i+j\leq
p:\qquad\left(\lambda^{1}_{e},\lambda^{3}_{e}\right)=\left(\frac{i}{p},\frac{j}{p}\right),\qquad\lambda^{2}_{e}=1-\lambda^{1}_{e}-\lambda^{3}_{e},$

in barycentric coordinates, or

$\displaystyle\bm{r}_{e}=\lambda^{2}_{e}\bm{v}_{1}+\lambda^{3}_{e}\bm{v}_{2}+\lambda^{1}_{e}\bm{v}_{3},$

in reference coordinates. We obtain the final shifted point locations by
simply adding the warping/blending function to the equidistant point locations
as follows

$\displaystyle\bm{r}_{s}=\bm{r}_{e}+\bm{g}\left(\lambda^{1}_{e},\lambda^{2}_{e},\lambda^{3}_{e}\right).$
(4.5)

If the warping/blending function has been formulated correctly, then the
shifted points $\bm{r}_{s}$ will act as good-quality interpolation points.

###  4.3 The Tetrahedron

Consider the reference tetrahedron, which is defined as the convex hull of the
following four vertices

$\displaystyle\bm{v}_{1}$
$\displaystyle=\left(-1,-\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{6}}\right),\qquad\bm{v}_{2}=\left(1,-\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{6}}\right),$
$\displaystyle\bm{v}_{3}$
$\displaystyle=\left(0,\frac{2}{\sqrt{3}},-\frac{1}{\sqrt{6}}\right),\qquad\qquad\bm{v}_{4}=\left(0,0,\frac{3}{\sqrt{6}}\right).$
(4.6)

All points within the tetrahedron interior can be represented as follows

$\displaystyle\bm{r}=\lambda^{3}\bm{v}_{1}+\lambda^{4}\bm{v}_{2}+\lambda^{2}\bm{v}_{3}+\lambda^{1}\bm{v}_{4},$

where

$\displaystyle 0\leq\lambda^{1},\lambda^{2},\lambda^{3},\lambda^{4}\leq
1,\qquad\lambda^{1}+\lambda^{2}+\lambda^{3}+\lambda^{4}=1.$

The tetrahedron has a total of four equilateral triangular faces (each with
edge length 2), given by the following subsets of vertices

$\displaystyle F_{1}$
$\displaystyle=\left\\{\bm{v}_{1},\bm{v}_{2},\bm{v}_{3}\right\\},\qquad
F_{2}=\left\\{\bm{v}_{1},\bm{v}_{2},\bm{v}_{4}\right\\},$ $\displaystyle
F_{3}$ $\displaystyle=\left\\{\bm{v}_{2},\bm{v}_{3},\bm{v}_{4}\right\\},\qquad
F_{4}=\left\\{\bm{v}_{1},\bm{v}_{3},\bm{v}_{4}\right\\}.$

Let us now define the warping functions for each face as follows

$\displaystyle\bm{w}^{1}$
$\displaystyle=\bm{g}_{1}\left(\lambda^{2},\lambda^{3},\lambda^{4}\right)\bm{t}^{1,1}+\bm{g}_{2}\left(\lambda^{2},\lambda^{3},\lambda^{4}\right)\bm{t}^{1,2},$
$\displaystyle\bm{w}^{2}$
$\displaystyle=\bm{g}_{1}\left(\lambda^{1},\lambda^{3},\lambda^{4}\right)\bm{t}^{2,1}+\bm{g}_{2}\left(\lambda^{1},\lambda^{3},\lambda^{4}\right)\bm{t}^{2,2},$
$\displaystyle\bm{w}^{3}$
$\displaystyle=\bm{g}_{1}\left(\lambda^{1},\lambda^{4},\lambda^{2}\right)\bm{t}^{3,1}+\bm{g}_{2}\left(\lambda^{1},\lambda^{4},\lambda^{2}\right)\bm{t}^{3,2},$
$\displaystyle\bm{w}^{4}$
$\displaystyle=\bm{g}_{1}\left(\lambda^{1},\lambda^{3},\lambda^{2}\right)\bm{t}^{4,1}+\bm{g}_{2}\left(\lambda^{1},\lambda^{3},\lambda^{2}\right)\bm{t}^{4,2}.$

Here, $\bm{t}^{f,1}$ and $\bm{t}^{f,2}$ are two orthonormal vectors associated
with each face

$\displaystyle\bm{t}^{1,1}$
$\displaystyle=\left(1,0,0\right),\quad\bm{t}^{1,2}=\left(0,1,0\right),$
$\displaystyle\bm{t}^{2,1}$
$\displaystyle=\left(1,0,0\right),\quad\bm{t}^{2,2}=\left(0,\frac{1}{3},\frac{4}{3\sqrt{2}}\right),$
$\displaystyle\bm{t}^{3,1}$
$\displaystyle=\left(-\frac{1}{2},\frac{3}{2\sqrt{3}},0\right),\quad\bm{t}^{3,2}=\left(-\frac{1}{2\sqrt{3}},-\frac{1}{6},\frac{4}{3\sqrt{2}}\right),$
$\displaystyle\bm{t}^{4,1}$
$\displaystyle=\left(\frac{1}{2},\frac{3}{2\sqrt{3}},0\right),\quad\bm{t}^{4,2}=\left(\frac{1}{2\sqrt{3}},-\frac{1}{6},\frac{4}{3\sqrt{2}}\right).$

Next, we can define the following blending functions for each face

$\displaystyle b^{1}(\lambda^{1},\lambda^{2},\lambda^{3},\lambda^{4})$
$\displaystyle=\left(\frac{2\lambda^{2}}{2\lambda^{2}+\lambda^{1}}\right)\left(\frac{2\lambda^{3}}{2\lambda^{3}+\lambda^{1}}\right)\left(\frac{2\lambda^{4}}{2\lambda^{4}+\lambda^{1}}\right),$
$\displaystyle b^{2}(\lambda^{1},\lambda^{2},\lambda^{3},\lambda^{4})$
$\displaystyle=\left(\frac{2\lambda^{1}}{2\lambda^{1}+\lambda^{2}}\right)\left(\frac{2\lambda^{3}}{2\lambda^{3}+\lambda^{2}}\right)\left(\frac{2\lambda^{4}}{2\lambda^{4}+\lambda^{2}}\right),$
$\displaystyle b^{3}(\lambda^{1},\lambda^{2},\lambda^{3},\lambda^{4})$
$\displaystyle=\left(\frac{2\lambda^{1}}{2\lambda^{1}+\lambda^{3}}\right)\left(\frac{2\lambda^{2}}{2\lambda^{2}+\lambda^{3}}\right)\left(\frac{2\lambda^{4}}{2\lambda^{4}+\lambda^{3}}\right),$
$\displaystyle b^{4}(\lambda^{1},\lambda^{2},\lambda^{3},\lambda^{4})$
$\displaystyle=\left(\frac{2\lambda^{1}}{2\lambda^{1}+\lambda^{4}}\right)\left(\frac{2\lambda^{2}}{2\lambda^{2}+\lambda^{4}}\right)\left(\frac{2\lambda^{3}}{2\lambda^{3}+\lambda^{4}}\right).$

Furthermore, we can define the following warping/blending function on the
entire tetrahedron

$\displaystyle\bm{g}\left(\lambda^{1},\lambda^{2},\lambda^{3},\lambda^{4}\right)$
$\displaystyle=\left(1+\left(\beta\lambda^{1}\right)^{2}\right)b^{1}\bm{w}^{1}+\left(1+\left(\beta\lambda^{2}\right)^{2}\right)b^{2}\bm{w}^{2}$
$\displaystyle+\left(1+\left(\beta\lambda^{3}\right)^{2}\right)b^{3}\bm{w}^{3}+\left(1+\left(\beta\lambda^{4}\right)^{2}\right)b^{4}\bm{w}^{4},$
(4.7)

where $\beta$ is an optimization parameter. Figure 1 illustrates the combined
impact of the warping and blending functions on the tetrahedron.

As before, we can use the function
$\bm{g}\left(\lambda^{1},\lambda^{2},\lambda^{3},\lambda^{4}\right)$ to shift
the locations of equidistant points on the tetrahedron. The locations of the
equidistant points are given by

$\displaystyle(i,j,k)\geq 0,\quad i+j+k\leq
p:\qquad\left(\lambda^{1}_{e},\lambda^{2}_{e},\lambda^{4}_{e}\right)=\left(\frac{i}{p},\frac{j}{p},\frac{k}{p}\right),$
$\displaystyle\lambda^{3}_{e}=1-\lambda^{1}_{e}-\lambda^{2}_{e}-\lambda^{4}_{e},$

in barycentric coordinates, or

$\displaystyle\bm{r}_{e}=\lambda^{3}_{e}\bm{v}_{1}+\lambda^{4}_{e}\bm{v}_{2}+\lambda^{2}_{e}\bm{v}_{3}+\lambda^{1}_{e}\bm{v}_{4},$

in reference coordinates. We obtain the final shifted point locations by
adding the warping/blending function to the equidistant point locations as
follows

$\displaystyle\bm{r}_{s}=\bm{r}_{e}+\bm{g}\left(\lambda^{1}_{e},\lambda^{2}_{e},\lambda^{3}_{e},\lambda^{4}_{e}\right).$
(4.8)

Figure 1: Effect of warping and blending functions on points on the reference
tetrahedron. Note that points located at vertices, midpoints, and barycenters
undergo zero warping and/or blending.

###  4.4 The Pentatope

Consider the reference pentatope, which is defined as the convex hull of the
following five vertices

$\displaystyle\bm{v}_{1}$
$\displaystyle=\left(-1,-\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{6}},-\frac{1}{\sqrt{10}}\right),\qquad\bm{v}_{2}=\left(1,-\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{6}},-\frac{1}{\sqrt{10}}\right),$
$\displaystyle\bm{v}_{3}$
$\displaystyle=\left(0,\frac{2}{\sqrt{3}},-\frac{1}{\sqrt{6}},-\frac{1}{\sqrt{10}}\right),\,\;\quad\qquad\bm{v}_{4}=\left(0,0,\frac{3}{\sqrt{6}},-\frac{1}{\sqrt{10}}\right),$
$\displaystyle\bm{v}_{5}$
$\displaystyle=\left(0,0,0,\frac{4}{\sqrt{10}}\right).$ (4.9)

All points within the pentatope interior can be represented as follows

$\displaystyle\bm{r}=\lambda^{4}\bm{v}_{1}+\lambda^{5}\bm{v}_{2}+\lambda^{3}\bm{v}_{3}+\lambda^{2}\bm{v}_{4}+\lambda^{1}\bm{v}_{5},$

where

$\displaystyle
0\leq\lambda^{1},\lambda^{2},\lambda^{3},\lambda^{4},\lambda^{5}\leq
1,\qquad\lambda^{1}+\lambda^{2}+\lambda^{3}+\lambda^{4}+\lambda^{5}=1.$

By construction, the pentatope is equi-facetal, and all of the edges are of
length 2. In addition, the pentatope has a total of five tetrahedral facets,
given by the following subsets of vertices

$\displaystyle F_{1}$
$\displaystyle=\left\\{\bm{v}_{1},\bm{v}_{2},\bm{v}_{3},\bm{v}_{4}\right\\},\qquad
F_{2}=\left\\{\bm{v}_{1},\bm{v}_{2},\bm{v}_{3},\bm{v}_{5}\right\\},$
$\displaystyle F_{3}$
$\displaystyle=\left\\{\bm{v}_{1},\bm{v}_{2},\bm{v}_{4},\bm{v}_{5}\right\\},\qquad
F_{4}=\left\\{\bm{v}_{2},\bm{v}_{3},\bm{v}_{4},\bm{v}_{5}\right\\},$
$\displaystyle F_{5}$
$\displaystyle=\left\\{\bm{v}_{3},\bm{v}_{1},\bm{v}_{4},\bm{v}_{5},\right\\}.$

We can now define vector warping functions for each facet as follows

$\displaystyle\bm{w}^{1}$
$\displaystyle=\bm{g}_{1}\left(\lambda^{2},\lambda^{3},\lambda^{4},\lambda^{5}\right)\bm{t}^{1,1}+\bm{g}_{2}\left(\lambda^{2},\lambda^{3},\lambda^{4},\lambda^{5}\right)\bm{t}^{1,2}+\bm{g}_{3}\left(\lambda^{2},\lambda^{3},\lambda^{4},\lambda^{5}\right)\bm{t}^{1,3},$
$\displaystyle\bm{w}^{2}$
$\displaystyle=\bm{g}_{1}\left(\lambda^{1},\lambda^{3},\lambda^{4},\lambda^{5}\right)\bm{t}^{2,1}+\bm{g}_{2}\left(\lambda^{1},\lambda^{3},\lambda^{4},\lambda^{5}\right)\bm{t}^{2,2}+\bm{g}_{3}\left(\lambda^{1},\lambda^{3},\lambda^{4},\lambda^{5}\right)\bm{t}^{2,3},$
$\displaystyle\bm{w}^{3}$
$\displaystyle=\bm{g}_{1}\left(\lambda^{1},\lambda^{2},\lambda^{4},\lambda^{5}\right)\bm{t}^{3,1}+\bm{g}_{2}\left(\lambda^{1},\lambda^{2},\lambda^{4},\lambda^{5}\right)\bm{t}^{3,2}+\bm{g}_{3}\left(\lambda^{1},\lambda^{2},\lambda^{4},\lambda^{5}\right)\bm{t}^{3,3},$
$\displaystyle\bm{w}^{4}$
$\displaystyle=\bm{g}_{1}\left(\lambda^{1},\lambda^{2},\lambda^{5},\lambda^{3}\right)\bm{t}^{4,1}+\bm{g}_{2}\left(\lambda^{1},\lambda^{2},\lambda^{5},\lambda^{3}\right)\bm{t}^{4,2}+\bm{g}_{3}\left(\lambda^{1},\lambda^{2},\lambda^{5},\lambda^{3}\right)\bm{t}^{4,3},$
$\displaystyle\bm{w}^{5}$
$\displaystyle=\bm{g}_{1}\left(\lambda^{1},\lambda^{2},\lambda^{4},\lambda^{3}\right)\bm{t}^{5,1}+\bm{g}_{2}\left(\lambda^{1},\lambda^{2},\lambda^{4},\lambda^{3}\right)\bm{t}^{5,2}+\bm{g}_{3}\left(\lambda^{1},\lambda^{2},\lambda^{4},\lambda^{3}\right)\bm{t}^{5,3}.$

Here, $\bm{t}^{f,1}$, $\bm{t}^{f,2}$, and $\bm{t}^{f,3}$ are three orthonormal
vectors associated with each facet

$\displaystyle\bm{t}^{1,1}$
$\displaystyle=\left(1,0,0,0\right),\quad\bm{t}^{1,2}=\left(0,1,0,0\right),\quad\bm{t}^{1,3}=\left(0,0,1,0\right),$
$\displaystyle\bm{t}^{2,1}$
$\displaystyle=\left(1,0,0,0\right),\quad\bm{t}^{2,2}=\left(0,1,0,0\right),\quad\bm{t}^{2,3}=\left(0,0,\frac{1}{4},\frac{\sqrt{15}}{4}\right),$
$\displaystyle\bm{t}^{3,1}$
$\displaystyle=\left(1,0,0,0\right),\quad\bm{t}^{3,2}=\left(0,\frac{1}{3},\frac{4}{3\sqrt{2}},0\right),\quad\bm{t}^{3,3}=\left(0,\frac{1}{\sqrt{18}},-\frac{1}{12},\frac{\sqrt{15}}{4}\right),$
$\displaystyle\bm{t}^{4,1}$
$\displaystyle=\left(-\frac{1}{2},\frac{3}{2\sqrt{3}},0,0\right),\quad\bm{t}^{4,2}=\left(-\frac{1}{2\sqrt{3}},-\frac{1}{6},\frac{4}{3\sqrt{2}},0\right),\quad\bm{t}^{4,3}=\left(-\frac{\sqrt{6}}{12},-\frac{\sqrt{2}}{12},-\frac{1}{12},\frac{\sqrt{15}}{4}\right),$
$\displaystyle\bm{t}^{5,1}$
$\displaystyle=\left(\frac{1}{2},\frac{3}{2\sqrt{3}},0,0\right),\quad\bm{t}^{5,2}=\left(\frac{1}{2\sqrt{3}},-\frac{1}{6},\frac{4}{3\sqrt{2}},0\right),\quad\bm{t}^{5,3}=\left(\frac{\sqrt{6}}{12},-\frac{\sqrt{2}}{12},-\frac{1}{12},\frac{\sqrt{15}}{4}\right).$

In addition, we can define the following blending functions for each facet

$\displaystyle
b^{1}(\lambda^{1},\lambda^{2},\lambda^{3},\lambda^{4},\lambda^{5})$
$\displaystyle=\left(\frac{2\lambda^{2}}{2\lambda^{2}+\lambda^{1}}\right)\left(\frac{2\lambda^{3}}{2\lambda^{3}+\lambda^{1}}\right)\left(\frac{2\lambda^{4}}{2\lambda^{4}+\lambda^{1}}\right)\left(\frac{2\lambda^{5}}{2\lambda^{5}+\lambda^{1}}\right),$
$\displaystyle
b^{2}(\lambda^{1},\lambda^{2},\lambda^{3},\lambda^{4},\lambda^{5})$
$\displaystyle=\left(\frac{2\lambda^{1}}{2\lambda^{1}+\lambda^{2}}\right)\left(\frac{2\lambda^{3}}{2\lambda^{3}+\lambda^{2}}\right)\left(\frac{2\lambda^{4}}{2\lambda^{4}+\lambda^{2}}\right)\left(\frac{2\lambda^{5}}{2\lambda^{5}+\lambda^{2}}\right),$
$\displaystyle
b^{3}(\lambda^{1},\lambda^{2},\lambda^{3},\lambda^{4},\lambda^{5})$
$\displaystyle=\left(\frac{2\lambda^{1}}{2\lambda^{1}+\lambda^{3}}\right)\left(\frac{2\lambda^{2}}{2\lambda^{2}+\lambda^{3}}\right)\left(\frac{2\lambda^{4}}{2\lambda^{4}+\lambda^{3}}\right)\left(\frac{2\lambda^{5}}{2\lambda^{5}+\lambda^{3}}\right),$
$\displaystyle
b^{4}(\lambda^{1},\lambda^{2},\lambda^{3},\lambda^{4},\lambda^{5})$
$\displaystyle=\left(\frac{2\lambda^{1}}{2\lambda^{1}+\lambda^{4}}\right)\left(\frac{2\lambda^{2}}{2\lambda^{2}+\lambda^{4}}\right)\left(\frac{2\lambda^{3}}{2\lambda^{3}+\lambda^{4}}\right)\left(\frac{2\lambda^{5}}{2\lambda^{5}+\lambda^{4}}\right),$
$\displaystyle
b^{5}(\lambda^{1},\lambda^{2},\lambda^{3},\lambda^{4},\lambda^{5})$
$\displaystyle=\left(\frac{2\lambda^{1}}{2\lambda^{1}+\lambda^{5}}\right)\left(\frac{2\lambda^{2}}{2\lambda^{2}+\lambda^{5}}\right)\left(\frac{2\lambda^{3}}{2\lambda^{3}+\lambda^{5}}\right)\left(\frac{2\lambda^{4}}{2\lambda^{4}+\lambda^{5}}\right).$

Next, we can define the following warping/blending function on the entire
pentatope

$\displaystyle\bm{g}\left(\lambda^{1},\lambda^{2},\lambda^{3},\lambda^{4},\lambda^{5}\right)$
$\displaystyle=\left(1+\left(\gamma\lambda^{1}\right)^{2}\right)b^{1}\bm{w}^{1}+\left(1+\left(\gamma\lambda^{2}\right)^{2}\right)b^{2}\bm{w}^{2}+\left(1+\left(\gamma\lambda^{3}\right)^{2}\right)b^{3}\bm{w}^{3}$
$\displaystyle+\left(1+\left(\gamma\lambda^{4}\right)^{2}\right)b^{4}\bm{w}^{4}+\left(1+\left(\gamma\lambda^{5}\right)^{2}\right)b^{5}\bm{w}^{5},$
(4.10)

where $\gamma$ is an optimization parameter. As before, we can use the
function
$\bm{g}\left(\lambda^{1},\lambda^{2},\lambda^{3},\lambda^{4},\lambda^{5}\right)$
to shift the locations of equidistant points on the pentatope. The locations
of the equidistant points are given by

$\displaystyle(i,j,k,m)\geq 0,\quad i+j+k+m\leq p$
$\displaystyle:\qquad\left(\lambda^{1}_{e},\lambda^{2}_{e},\lambda^{3}_{e},\lambda^{5}_{e}\right)=\left(\frac{i}{p},\frac{j}{p},\frac{k}{p},\frac{m}{p}\right),$
$\displaystyle\lambda^{4}_{e}=1-\lambda^{1}_{e}$
$\displaystyle-\lambda^{2}_{e}-\lambda^{3}_{e}-\lambda^{5}_{e},$

in barycentric coordinates, or

$\displaystyle\bm{r}_{e}=\lambda^{4}_{e}\bm{v}_{1}+\lambda^{5}_{e}\bm{v}_{2}+\lambda^{3}_{e}\bm{v}_{3}+\lambda^{2}_{e}\bm{v}_{4}+\lambda^{1}_{e}\bm{v}_{5},$

in reference coordinates. We obtain the final shifted point locations by
adding the warping/blending function to the equidistant point locations as
follows

$\displaystyle\bm{r}_{s}=\bm{r}_{e}+\bm{g}\left(\lambda^{1}_{e},\lambda^{2}_{e},\lambda^{3}_{e},\lambda^{4}_{e},\lambda^{5}_{e}\right).$
(4.11)

##  5 Results

In the interest of expediting computation we assumed that our three
optimization parameters $\alpha,\beta,$ and $\gamma$ were all equivalent,
thereby leading to a single-parameter optimization scheme with
$\alpha=\beta=\gamma$. The resultant single optimization parameter is
henceforth referred to simply as $\alpha$. We note that when $\alpha=0$, we
counterintuitively still obtain nontrivial warping and blending of our
equidistant point sets, as evidenced by Eq. (4.4).

In order to find the optimal value of $\alpha$ for a given polynomial order
$p$, a simple bisection algorithm was created. The initial range for this
algorithm was $\alpha=[0,3]$, as preliminary computational results showed that
for $\alpha$ values greater than 3, the Lebesgue constant of the shifted nodes
tended to be higher than that of the equispaced nodes. The resultant optimized
values of $\alpha$ for polynomial orders $p=1$ to $p=10$ are tabulated in
Table 2, alongside the associated Lebesgue constants $\Lambda$. It is clear
from the table that in virtually all cases the optimized points outperform the
equispaced points, i.e. they have smaller Lebesgue constants. The only
exception is the $p=3$ case, for which we obtain a slightly worse Lebesgue
constant relative to the equispaced points.

The Lebesgue constants in Table 2 were obtained using a naive sampling
approach. Absolute values of each nodal basis function were calculated on a
uniform grid within the pentatope (this uniform grid was independent of the
equispaced points used for interpolation purposes). Due to rising
computational cost, higher polynomial order cases were carried out using a
progressively coarser uniform grid. The grid spacing used for each polynomial
order is given in the penultimate column of Table 2. Conveniently, previous
researchers have computed Lebesgue constants for equispaced interpolation
points up to dimension 6 and $p=10$ [21]. As expected, our equispaced results
are in close agreement with theirs for $d=4$.

In addition, the geometric arrangements of the optimized points from $p=5$ to
$p=8$ are illustrated in Figures 2—5 (compact view) as well as 6—9 (expanded
view). Finally, the barycentric coordinates for optimized point sets for $p=1$
to $p=6$ appear in A, and the remaining coordinates for $p=7$ through $p=10$
appear in [20].

Table 2: Comparison of Lebesgue constants for varying polynomial orders $p$ and node generation parameters $\alpha$ on the reference pentatope $p$ | $\alpha$ | $\Lambda$ | $\Lambda$ | $\Lambda$ | Grid | $\Lambda$
---|---|---|---|---|---|---
Optimized | $\alpha=0$ | Equidistant | Spacing | Equidistant [21]
1 | 0 | 1.0000 | 1.0000 | 1.0000 | 0.01 | * ‣ 2
2 | 0 | 2.2000 | 2.2000 | 2.2000 | 0.01 | * ‣ 2
3 | 0 | 4.2000 | 4.2000 | 3.8800 | 0.01 | * ‣ 2
4 | 0 | 6.1240 | 6.1240 | 6.2384 | 0.01 | * ‣ 2
5 | 0 | 8.6423 | 8.6423 | 10.9171 | 0.01 | ***Data not given in reference [21].
6 | 1.5000 | 12.0326 | 12.1297 | 19.1418 | 0.02 | 19.22
7 | 2.2500 | 17.1032 | 18.0971 | 33.8915 | 0.02 | 34.08
8 | 1.8890 | 23.9226 | 26.0423 | 60.5048 | 0.02 | 60.86
9 | 1.5000 | 36.1110 | 39.1431 | 109.4267 | 0.02 | 109.43
10 | 1.5469 | 53.3404 | 57.7742 | 194.8739 | 0.04 | 198.08
Figure 2: Optimized interpolation points on a projected version of the
pentatope for $p=5$. Black points appear at the vertices, green points appear
on the edges, blue points appear on the triangular faces, red points appear on
the tetrahedral facets, and yellow points appear in the interior of the
pentatope. Figure 3: Optimized interpolation points on a projected version of
the pentatope for $p=6$. Black points appear at the vertices, green points
appear on the edges, blue points appear on the triangular faces, red points
appear on the tetrahedral facets, and yellow points appear in the interior of
the pentatope. Figure 4: Optimized interpolation points on a projected version
of the pentatope for $p=7$. Black points appear at the vertices, green points
appear on the edges, blue points appear on the triangular faces, red points
appear on the tetrahedral facets, and yellow points appear in the interior of
the pentatope. Figure 5: Optimized interpolation points on a projected version
of the pentatope for $p=8$. Black points appear at the vertices, green points
appear on the edges, blue points appear on the triangular faces, red points
appear on the tetrahedral facets, and yellow points appear in the interior of
the pentatope.

Figure 6: Optimized interpolation points on a projected version of the
pentatope for $p=5$, expanded view. Pentatope interior (top left), facet 1
(top right), facet 2 (middle left), facet 3 (middle right), facet 4 (bottom
left), and facet 5 (bottom right). Black points appear at the vertices, green
points appear on the edges, blue points appear on the triangular faces, red
points appear on the tetrahedral facets, and yellow points appear in the
interior of the pentatope.

Figure 7: Optimized interpolation points on a projected version of the
pentatope for $p=6$, expanded view. Pentatope interior (top left), facet 1
(top right), facet 2 (middle left), facet 3 (middle right), facet 4 (bottom
left), and facet 5 (bottom right). Black points appear at the vertices, green
points appear on the edges, blue points appear on the triangular faces, red
points appear on the tetrahedral facets, and yellow points appear in the
interior of the pentatope.

Figure 8: Optimized interpolation points on a projected version of the
pentatope for $p=7$, expanded view. Pentatope interior (top left), facet 1
(top right), facet 2 (middle left), facet 3 (middle right), facet 4 (bottom
left), and facet 5 (bottom right). Black points appear at the vertices, green
points appear on the edges, blue points appear on the triangular faces, red
points appear on the tetrahedral facets, and yellow points appear in the
interior of the pentatope.

Figure 9: Optimized interpolation points on a projected version of the
pentatope for $p=8$, expanded view. Pentatope interior (top left), facet 1
(top right), facet 2 (middle left), facet 3 (middle right), facet 4 (bottom
left), and facet 5 (bottom right). Black points appear at the vertices, green
points appear on the edges, blue points appear on the triangular faces, red
points appear on the tetrahedral facets, and yellow points appear in the
interior of the pentatope.

##  6 Concluding Remarks

In this paper, we generated Lebesgue-optimized interpolation points on four-
dimensional simplex elements using an extension of Warburton’s approach [14]
which was originally formulated for problems in two and three dimensions. To
this end, we first gave an overview of prior interpolation research on lower
dimensional simplices. Then, we constructed the extension of Warburton’s
warping and blending functions in four dimensions, and discussed how these
functions can be applied to the equispaced interpolation point set. Using
these functions, we optimized the interpolation point sets around a single
parameter $\alpha$ and we found that for polynomial orders $p\leq 5$, the
smallest Lebesgue constants occurred for $\alpha=0$. This minimal warping and
blending—due primarily to the 1D Legendre-Gauss-Lobatto distribution used as a
starting point—produces a slightly suboptimal point distribution for $p=3$.
However the optimized points began to outperform the equispaced points
beginning with $p=4$. For values of $p>5$, the optimization parameter $\alpha$
assumed non-zero values. In addition, the gap between the Lebesgue constants
of the equispaced points and those of the optimized points rapidly grew. For
example, for $p=10$ the Lebesgue constant of the optimized points is
approximately one-fourth of that of the equispaced points.

###  6.1 Future Work

The interpolation point generation procedure detailed in this work can be
conceivably extended to higher dimensions, $d>4$. However, this would require
careful construction of the blending/warping functions. In addition, a higher-
dimensional extension of the nodal indexing equation would be advantageous.
Using the existing indexing equations along with the one derived here, some
patterns can be observed regarding coefficients and the terms upon which they
act. These patterns will prove useful should a need for the higher-dimensional
indexing equations arise.

The simplifications introduced by the present paper can also be modified or
omitted entirely. For example, it remains to be shown that equating the
optimization parameters of $\alpha,\beta,$ and $\gamma$ leads to the most
optimized interpolation point sets. However, the Lebesgue constants of the
single-parameter sets in both lower dimensions and in this work are reasonably
small, thereby bringing the necessity of further optimization into question.

Finally, in the interest of completeness, the interpolation points found in
this work can be tested independently of their Lebesgue constants. For
example, one could perform a four-dimensional collocation projection of a
transcendental function and compute the $L^{2}$ and/or $L^{\infty}$ error of
the interpolation. We anticipate that this type of practical analysis will be
investigated in future studies.

## Declaration of Competing Interests

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

## Funding

This research received funding from the United States Naval Research
Laboratory (NRL) under grant number N00173-22-2-C008. In turn, the NRL grant
itself was funded by Steven Martens, Program Officer for the Power, Propulsion
and Thermal Management Program, Code 35, in the United States Office of Naval
Research.

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##  Appendix A Interpolation Point Tables

###  A.1 Optimized Interpolation Points, p = 1

$\lambda^{1}$ | $\lambda^{2}$ | $\lambda^{3}$ | $\lambda^{4}$ | $\lambda^{5}$
---|---|---|---|---
1.000000000000000 | 0 | 0 | 0 | 0
0 | 1.000000000000000 | 0 | 0 | 0
0 | 0 | 1.000000000000000 | 0 | 0
0 | 0 | 0 | 1.000000000000000 | 0
0 | 0 | 0 | 0 | 1.000000000000000

###  A.2 Optimized Interpolation Points, p = 2

$\lambda^{1}$ | $\lambda^{2}$ | $\lambda^{3}$ | $\lambda^{4}$ | $\lambda^{5}$
---|---|---|---|---
1.000000000000000 | 0 | 0 | 0 | 0
0.500000000000000 | 0.500000000000000 | 0 | 0 | 0
0 | 1.000000000000000 | 0 | 0 | 0
0.500000000000000 | 0 | 0.500000000000000 | 0 | 0
0 | 0.500000000000000 | 0.500000000000000 | 0 | 0
0 | 0 | 1.000000000000000 | 0 | 0
0.500000000000000 | 0 | 0 | 0.500000000000000 | 0
0 | 0.500000000000000 | 0 | 0.500000000000000 | 0
0 | 0 | 0.500000000000000 | 0.500000000000000 | 0
0 | 0 | 0 | 1.000000000000000 | 0
0.500000000000000 | 0 | 0 | 0 | 0.500000000000000
0 | 0.500000000000000 | 0 | 0 | 0.500000000000000
0 | 0 | 0.500000000000000 | 0 | 0.500000000000000
0 | 0 | 0 | 0.500000000000000 | 0.500000000000000
0 | 0 | 0 | 0 | 1.000000000000000

###  A.3 Optimized Interpolation Points, p = 3

$\lambda^{1}$ | $\lambda^{2}$ | $\lambda^{3}$ | $\lambda^{4}$ | $\lambda^{5}$
---|---|---|---|---
1.000000000000000 | 0 | 0 | 0 | 0
0.723606797749979 | 0.276393202250021 | 0 | 0 | 0
0.276393202250021 | 0.723606797749979 | 0 | 0 | 0
0 | 1.000000000000000 | 0 | 0 | 0
0.723606797749979 | 0 | 0.276393202250021 | 0 | 0
0.333333333333333 | 0.333333333333333 | 0.333333333333333 | 0 | 0
0 | 0.723606797749979 | 0.276393202250021 | 0 | 0
0.276393202250021 | 0 | 0.723606797749979 | 0 | 0
0 | 0.276393202250021 | 0.723606797749979 | 0 | 0
0 | 0 | 1.000000000000000 | 0 | 0
0.723606797749979 | 0 | 0 | 0.276393202250021 | 0
0.333333333333333 | 0.333333333333333 | 0 | 0.333333333333333 | 0
0 | 0.723606797749979 | 0 | 0.276393202250021 | 0
0.333333333333333 | 0 | 0.333333333333333 | 0.333333333333333 | 0
0 | 0.333333333333333 | 0.333333333333333 | 0.333333333333333 | 0
0 | 0 | 0.723606797749979 | 0.276393202250021 | 0
0.276393202250021 | 0 | 0 | 0.723606797749979 | 0
0 | 0.276393202250021 | 0 | 0.723606797749979 | 0
0 | 0 | 0.276393202250021 | 0.723606797749979 | 0
0 | 0 | 0 | 1.000000000000000 | 0
0.723606797749979 | 0 | 0 | 0 | 0.276393202250021
0.333333333333333 | 0.333333333333333 | 0 | 0 | 0.333333333333333
0 | 0.723606797749979 | 0 | 0 | 0.276393202250021
0.333333333333333 | 0 | 0.333333333333333 | 0 | 0.333333333333333
0 | 0.333333333333333 | 0.333333333333333 | 0 | 0.333333333333333
0 | 0 | 0.723606797749979 | 0 | 0.276393202250021
0.333333333333333 | 0 | 0 | 0.333333333333333 | 0.333333333333333
0 | 0.333333333333333 | 0 | 0.333333333333333 | 0.333333333333333
0 | 0 | 0.333333333333333 | 0.333333333333333 | 0.333333333333333
0 | 0 | 0 | 0.723606797749979 | 0.276393202250021
0.276393202250021 | 0 | 0 | 0 | 0.723606797749979
0 | 0.276393202250021 | 0 | 0 | 0.723606797749979
0 | 0 | 0.276393202250021 | 0 | 0.723606797749979
0 | 0 | 0 | 0.276393202250021 | 0.723606797749979
0 | 0 | 0 | 0 | 1.000000000000000

###  A.4 Optimized Interpolation Points, p = 4

$\lambda^{1}$ | $\lambda^{2}$ | $\lambda^{3}$ | $\lambda^{4}$ | $\lambda^{5}$
---|---|---|---|---
1.000000000000000 | 0 | 0 | 0 | 0
0.827326835353989 | 0.172673164646011 | 0 | 0 | 0
0.500000000000000 | 0.500000000000000 | 0 | 0 | 0
0.172673164646011 | 0.827326835353989 | 0 | 0 | 0
0 | 1.000000000000000 | 0 | 0 | 0
0.827326835353989 | 0 | 0.172673164646011 | 0 | 0
0.551551223569326 | 0.224224388215337 | 0.224224388215337 | 0 | 0
0.224224388215337 | 0.551551223569326 | 0.224224388215337 | 0 | 0
0 | 0.827326835353989 | 0.172673164646011 | 0 | 0
0.500000000000000 | 0 | 0.500000000000000 | 0 | 0
0.224224388215337 | 0.224224388215337 | 0.551551223569326 | 0 | 0
0 | 0.500000000000000 | 0.500000000000000 | 0 | 0
0.172673164646011 | 0 | 0.827326835353989 | 0 | 0
0 | 0.172673164646011 | 0.827326835353988 | 0 | 0
0 | 0 | 1.000000000000000 | 0 | 0
0.827326835353989 | 0 | 0 | 0.172673164646011 | 0
0.551551223569326 | 0.224224388215337 | 0 | 0.224224388215337 | 0
0.224224388215337 | 0.551551223569326 | 0 | 0.224224388215337 | 0
0 | 0.827326835353989 | 0 | 0.172673164646011 | 0
0.551551223569326 | 0 | 0.224224388215337 | 0.224224388215337 | 0
0.250000000000000 | 0.250000000000000 | 0.250000000000000 | 0.250000000000000 | 0
0 | 0.551551223569326 | 0.224224388215337 | 0.224224388215337 | 0
0.224224388215337 | 0 | 0.551551223569326 | 0.224224388215337 | 0
0 | 0.224224388215337 | 0.551551223569326 | 0.224224388215337 | 0
0 | 0 | 0.827326835353989 | 0.172673164646011 | 0
0.500000000000000 | 0 | 0 | 0.500000000000000 | 0
0.224224388215337 | 0.224224388215337 | 0 | 0.551551223569326 | 0
0 | 0.500000000000000 | 0 | 0.500000000000000 | 0
0.224224388215337 | 0 | 0.224224388215337 | 0.551551223569326 | 0
0 | 0.224224388215337 | 0.224224388215337 | 0.551551223569326 | 0
0 | 0 | 0.500000000000000 | 0.500000000000000 | 0
0.172673164646011 | 0 | 0 | 0.827326835353988 | 0
0 | 0.172673164646011 | 0 | 0.827326835353988 | 0
0 | 0 | 0.172673164646011 | 0.827326835353988 | 0
0 | 0 | 0 | 1.000000000000000 | 0
0.827326835353989 | 0 | 0 | 0 | 0.172673164646011
0.551551223569326 | 0.224224388215337 | 0 | 0 | 0.224224388215337
0.224224388215337 | 0.551551223569326 | 0 | 0 | 0.224224388215337
0 | 0.827326835353989 | 0 | 0 | 0.172673164646011
0.551551223569326 | 0 | 0.224224388215337 | 0 | 0.224224388215337
0.250000000000000 | 0.250000000000000 | 0.250000000000000 | 0 | 0.250000000000000
0 | 0.551551223569326 | 0.224224388215337 | 0 | 0.224224388215337
0.224224388215337 | 0 | 0.551551223569326 | 0 | 0.224224388215337
0 | 0.224224388215337 | 0.551551223569326 | 0 | 0.224224388215337
0 | 0 | 0.827326835353989 | 0 | 0.172673164646011
0.551551223569326 | 0 | 0 | 0.224224388215337 | 0.224224388215337
0.250000000000000 | 0.250000000000000 | 0 | 0.250000000000000 | 0.250000000000000
0 | 0.551551223569326 | 0 | 0.224224388215337 | 0.224224388215337
0.250000000000000 | 0 | 0.250000000000000 | 0.250000000000000 | 0.250000000000000
0 | 0.250000000000000 | 0.250000000000000 | 0.250000000000000 | 0.250000000000000
0 | 0 | 0.551551223569326 | 0.224224388215337 | 0.224224388215337
0.224224388215337 | 0 | 0 | 0.551551223569326 | 0.224224388215337
0 | 0.224224388215337 | 0 | 0.551551223569326 | 0.224224388215337
0 | 0 | 0.224224388215337 | 0.551551223569326 | 0.224224388215337
0 | 0 | 0 | 0.827326835353989 | 0.172673164646011
0.500000000000000 | 0 | 0 | 0 | 0.500000000000000
0.224224388215337 | 0.224224388215337 | 0 | 0 | 0.551551223569326
0 | 0.500000000000000 | 0 | 0 | 0.500000000000000
0.224224388215337 | 0 | 0.224224388215337 | 0 | 0.551551223569326
0 | 0.224224388215337 | 0.224224388215337 | 0 | 0.551551223569326
0 | 0 | 0.500000000000000 | 0 | 0.500000000000000
0.224224388215337 | 0 | 0 | 0.224224388215337 | 0.551551223569326
0 | 0.224224388215337 | 0 | 0.224224388215337 | 0.551551223569326
0 | 0 | 0.224224388215337 | 0.224224388215337 | 0.551551223569326
0 | 0 | 0 | 0.500000000000000 | 0.500000000000000
0.172673164646011 | 0 | 0 | 0 | 0.827326835353989
0 | 0.172673164646011 | 0 | 0 | 0.827326835353989
0 | 0 | 0.172673164646011 | 0 | 0.827326835353989
0 | 0 | 0 | 0.172673164646011 | 0.827326835353989
0 | 0 | 0 | 0 | 1.000000000000000

###  A.5 Optimized Interpolation Points, p = 5

$\lambda^{1}$ | $\lambda^{2}$ | $\lambda^{3}$ | $\lambda^{4}$ | $\lambda^{5}$
---|---|---|---|---
1.000000000000000 | 0 | 0 | 0 | 0
0.882527661964732 | 0.117472338035268 | 0 | 0 | 0
0.642615758240323 | 0.357384241759677 | 0 | 0 | 0
0.357384241759677 | 0.642615758240323 | 0 | 0 | 0
0.117472338035268 | 0.882527661964732 | 0 | 0 | 0
0 | 1.000000000000000 | 0 | 0 | 0
0.882527661964732 | 0 | 0.117472338035268 | 0 | 0
0.684217571037178 | 0.157891214481411 | 0.157891214481411 | 0 | 0
0.414205252746774 | 0.414205252746774 | 0.171589494506452 | 0 | 0
0.157891214481411 | 0.684217571037178 | 0.157891214481411 | 0 | 0
0 | 0.882527661964732 | 0.117472338035268 | 0 | 0
0.642615758240323 | 0 | 0.357384241759677 | 0 | 0
0.414205252746774 | 0.171589494506452 | 0.414205252746774 | 0 | 0
0.171589494506452 | 0.414205252746774 | 0.414205252746774 | 0 | 0
0 | 0.642615758240323 | 0.357384241759677 | 0 | 0
0.357384241759677 | 0 | 0.642615758240323 | 0 | 0
0.157891214481411 | 0.157891214481411 | 0.684217571037178 | 0 | 0
0 | 0.357384241759677 | 0.642615758240322 | 0 | 0
0.117472338035268 | 0 | 0.882527661964732 | 0 | 0
0 | 0.117472338035268 | 0.882527661964732 | 0 | 0
0 | 0 | 1.000000000000000 | 0 | 0
0.882527661964732 | 0 | 0 | 0.117472338035268 | 0
0.684217571037178 | 0.157891214481411 | 0 | 0.157891214481411 | 0
0.414205252746774 | 0.414205252746774 | 0 | 0.171589494506452 | 0
0.157891214481411 | 0.684217571037178 | 0 | 0.157891214481411 | 0
0 | 0.882527661964732 | 0 | 0.117472338035268 | 0
0.684217571037178 | 0 | 0.157891214481411 | 0.157891214481411 | 0
0.430304539193118 | 0.189898486935627 | 0.189898486935627 | 0.189898486935627 | 0
0.189898486935627 | 0.430304539193118 | 0.189898486935627 | 0.189898486935627 | 0
0 | 0.684217571037178 | 0.157891214481411 | 0.157891214481411 | 0
0.414205252746774 | 0 | 0.414205252746774 | 0.171589494506452 | 0
0.189898486935627 | 0.189898486935627 | 0.430304539193118 | 0.189898486935627 | 0
0 | 0.414205252746774 | 0.414205252746774 | 0.171589494506452 | 0
0.157891214481411 | 0 | 0.684217571037178 | 0.157891214481411 | 0
0 | 0.157891214481411 | 0.684217571037178 | 0.157891214481411 | 0
0 | 0 | 0.882527661964732 | 0.117472338035268 | 0
0.642615758240323 | 0 | 0 | 0.357384241759677 | 0
0.414205252746774 | 0.171589494506452 | 0 | 0.414205252746774 | 0
0.171589494506452 | 0.414205252746774 | 0 | 0.414205252746774 | 0
0 | 0.642615758240323 | 0 | 0.357384241759677 | 0
0.414205252746774 | 0 | 0.171589494506452 | 0.414205252746774 | 0
0.189898486935627 | 0.189898486935627 | 0.189898486935627 | 0.430304539193118 | 0
0 | 0.414205252746774 | 0.171589494506452 | 0.414205252746774 | 0
0.171589494506452 | 0 | 0.414205252746774 | 0.414205252746774 | 0
0 | 0.171589494506452 | 0.414205252746774 | 0.414205252746774 | 0
0 | 0 | 0.642615758240323 | 0.357384241759677 | 0
0.357384241759677 | 0 | 0 | 0.642615758240323 | 0
0.157891214481411 | 0.157891214481411 | 0 | 0.684217571037178 | 0
0 | 0.357384241759677 | 0 | 0.642615758240322 | 0
0.157891214481411 | 0 | 0.157891214481411 | 0.684217571037178 | 0
0 | 0.157891214481411 | 0.157891214481411 | 0.684217571037178 | 0
0 | 0 | 0.357384241759677 | 0.642615758240323 | 0
0.117472338035268 | 0 | 0 | 0.882527661964732 | 0
0 | 0.117472338035268 | 0 | 0.882527661964732 | 0
0 | 0 | 0.117472338035268 | 0.882527661964732 | 0
0 | 0 | 0 | 1.000000000000000 | 0
0.882527661964732 | 0 | 0 | 0 | 0.117472338035268
0.684217571037178 | 0.157891214481411 | 0 | 0 | 0.157891214481411
0.414205252746774 | 0.414205252746774 | 0 | 0 | 0.171589494506452
0.157891214481411 | 0.684217571037178 | 0 | 0 | 0.157891214481411
0 | 0.882527661964732 | 0 | 0 | 0.117472338035268
0.684217571037178 | 0 | 0.157891214481411 | 0 | 0.157891214481411
0.430304539193118 | 0.189898486935627 | 0.189898486935627 | 0 | 0.189898486935627
0.189898486935627 | 0.430304539193118 | 0.189898486935627 | 0 | 0.189898486935627
0 | 0.684217571037178 | 0.157891214481411 | 0 | 0.157891214481411
0.414205252746774 | 0 | 0.414205252746774 | 0 | 0.171589494506452
0.189898486935627 | 0.189898486935627 | 0.430304539193118 | 0 | 0.189898486935627
0 | 0.414205252746774 | 0.414205252746774 | 0 | 0.171589494506452
0.157891214481411 | 0 | 0.684217571037178 | 0 | 0.157891214481411
0 | 0.157891214481411 | 0.684217571037178 | 0 | 0.157891214481411
0 | 0 | 0.882527661964732 | 0 | 0.117472338035268
0.684217571037178 | 0 | 0 | 0.157891214481411 | 0.157891214481411
0.430304539193118 | 0.189898486935627 | 0 | 0.189898486935627 | 0.189898486935627
0.189898486935627 | 0.430304539193118 | 0 | 0.189898486935627 | 0.189898486935627
0 | 0.684217571037178 | 0 | 0.157891214481411 | 0.157891214481411
0.430304539193118 | 0 | 0.189898486935627 | 0.189898486935627 | 0.189898486935627
0.200000000000000 | 0.200000000000000 | 0.200000000000000 | 0.200000000000000 | 0.200000000000000
0 | 0.430304539193118 | 0.189898486935627 | 0.189898486935627 | 0.189898486935627
0.189898486935627 | 0 | 0.430304539193118 | 0.189898486935627 | 0.189898486935627
0 | 0.189898486935627 | 0.430304539193118 | 0.189898486935627 | 0.189898486935627
0 | 0 | 0.684217571037178 | 0.157891214481411 | 0.157891214481411
0.414205252746774 | 0 | 0 | 0.414205252746774 | 0.171589494506452
0.189898486935627 | 0.189898486935627 | 0 | 0.430304539193118 | 0.189898486935627
0 | 0.414205252746774 | 0 | 0.414205252746774 | 0.171589494506452
0.189898486935627 | 0 | 0.189898486935627 | 0.430304539193118 | 0.189898486935627
0 | 0.189898486935627 | 0.189898486935627 | 0.430304539193118 | 0.189898486935627
0 | 0 | 0.414205252746774 | 0.414205252746774 | 0.171589494506452
0.157891214481411 | 0 | 0 | 0.684217571037178 | 0.157891214481411
0 | 0.157891214481411 | 0 | 0.684217571037178 | 0.157891214481411
0 | 0 | 0.157891214481411 | 0.684217571037178 | 0.157891214481411
0 | 0 | 0 | 0.882527661964732 | 0.117472338035268
0.642615758240323 | 0 | 0 | 0 | 0.357384241759677
0.414205252746774 | 0.171589494506452 | 0 | 0 | 0.414205252746774
0.171589494506452 | 0.414205252746774 | 0 | 0 | 0.414205252746774
0 | 0.642615758240323 | 0 | 0 | 0.357384241759677
0.414205252746774 | 0 | 0.171589494506452 | 0 | 0.414205252746774
0.189898486935627 | 0.189898486935627 | 0.189898486935627 | 0 | 0.430304539193118
0 | 0.414205252746774 | 0.171589494506452 | 0 | 0.414205252746774
0.171589494506452 | 0 | 0.414205252746774 | 0 | 0.414205252746774
0 | 0.171589494506452 | 0.414205252746774 | 0 | 0.414205252746774
0 | 0 | 0.642615758240323 | 0 | 0.357384241759677
0.414205252746774 | 0 | 0 | 0.171589494506452 | 0.414205252746774
0.189898486935627 | 0.189898486935627 | 0 | 0.189898486935627 | 0.430304539193118
0 | 0.414205252746774 | 0 | 0.171589494506452 | 0.414205252746774
0.189898486935627 | 0 | 0.189898486935627 | 0.189898486935627 | 0.430304539193118
0 | 0.189898486935627 | 0.189898486935627 | 0.189898486935627 | 0.430304539193118
0 | 0 | 0.414205252746774 | 0.171589494506452 | 0.414205252746774
0.171589494506452 | 0 | 0 | 0.414205252746774 | 0.414205252746774
0 | 0.171589494506452 | 0 | 0.414205252746774 | 0.414205252746774
0 | 0 | 0.171589494506452 | 0.414205252746774 | 0.414205252746774
0 | 0 | 0 | 0.642615758240323 | 0.357384241759677
0.357384241759677 | 0 | 0 | 0 | 0.642615758240322
0.157891214481411 | 0.157891214481411 | 0 | 0 | 0.684217571037178
0 | 0.357384241759677 | 0 | 0 | 0.642615758240322
0.157891214481411 | 0 | 0.157891214481411 | 0 | 0.684217571037178
0 | 0.157891214481411 | 0.157891214481411 | 0 | 0.684217571037178
0 | 0 | 0.357384241759677 | 0 | 0.642615758240322
0.157891214481411 | 0 | 0 | 0.157891214481411 | 0.684217571037178
0 | 0.157891214481411 | 0 | 0.157891214481411 | 0.684217571037178
0 | 0 | 0.157891214481411 | 0.157891214481411 | 0.684217571037178
0 | 0 | 0 | 0.357384241759677 | 0.642615758240323
0.117472338035268 | 0 | 0 | 0 | 0.882527661964732
0 | 0.117472338035268 | 0 | 0 | 0.882527661964733
0 | 0 | 0.117472338035268 | 0 | 0.882527661964733
0 | 0 | 0 | 0.117472338035268 | 0.882527661964733
0 | 0 | 0 | 0 | 1.000000000000000

###  A.6 Optimized Interpolation Points, p = 6

$\lambda^{1}$ | $\lambda^{2}$ | $\lambda^{3}$ | $\lambda^{4}$ | $\lambda^{5}$
---|---|---|---|---
1.000000000000000 | 0 | 0 | 0 | 0
0.915111948139283 | 0.084888051860717 | 0 | 0 | 0
0.734424396735357 | 0.265575603264643 | 0 | 0 | 0
0.500000000000000 | 0.500000000000000 | 0 | 0 | 0
0.265575603264643 | 0.734424396735357 | 0 | 0 | 0
0.084888051860716 | 0.915111948139284 | 0 | 0 | 0
0 | 1.000000000000000 | 0 | 0 | 0
0.915111948139283 | 0 | 0.084888051860717 | 0 | 0
0.773105070455563 | 0.113447464772218 | 0.113447464772218 | 0 | 0
0.558991228320154 | 0.319451478409572 | 0.121557293270274 | 0 | 0
0.319451478409572 | 0.558991228320154 | 0.121557293270274 | 0 | 0
0.113447464772218 | 0.773105070455563 | 0.113447464772218 | 0 | 0
0 | 0.915111948139284 | 0.084888051860717 | 0 | 0
0.734424396735357 | 0 | 0.265575603264643 | 0 | 0
0.558991228320154 | 0.121557293270274 | 0.319451478409572 | 0 | 0
0.333333333333333 | 0.333333333333333 | 0.333333333333333 | 0 | 0
0.121557293270274 | 0.558991228320154 | 0.319451478409571 | 0 | 0
0 | 0.734424396735357 | 0.265575603264643 | 0 | 0
0.500000000000000 | 0 | 0.500000000000000 | 0 | 0
0.319451478409572 | 0.121557293270274 | 0.558991228320154 | 0 | 0
0.121557293270274 | 0.319451478409572 | 0.558991228320154 | 0 | 0
0 | 0.500000000000000 | 0.500000000000000 | 0 | 0
0.265575603264643 | 0 | 0.734424396735357 | 0 | 0
0.113447464772218 | 0.113447464772218 | 0.773105070455563 | 0 | 0
0 | 0.265575603264643 | 0.734424396735357 | 0 | 0
0.084888051860716 | 0 | 0.915111948139284 | 0 | 0
0 | 0.084888051860716 | 0.915111948139284 | 0 | 0
0 | 0 | 1.000000000000000 | 0 | 0
0.915111948139283 | 0 | 0 | 0.084888051860717 | 0
0.773105070455563 | 0.113447464772218 | 0 | 0.113447464772218 | 0
0.558991228320154 | 0.319451478409572 | 0 | 0.121557293270274 | 0
0.319451478409572 | 0.558991228320154 | 0 | 0.121557293270274 | 0
0.113447464772218 | 0.773105070455563 | 0 | 0.113447464772218 | 0
0 | 0.915111948139283 | 0 | 0.084888051860717 | 0
0.773105070455564 | 0 | 0.113447464772218 | 0.113447464772218 | 0
0.565564678537450 | 0.144811773820850 | 0.144811773820850 | 0.144811773820850 | 0
0.346796952884392 | 0.346796952884392 | 0.153203047115608 | 0.153203047115608 | 0
0.144811773820850 | 0.565564678537449 | 0.144811773820850 | 0.144811773820850 | 0
0 | 0.773105070455563 | 0.113447464772218 | 0.113447464772218 | 0
0.558991228320154 | 0 | 0.319451478409572 | 0.121557293270274 | 0
0.346796952884392 | 0.153203047115608 | 0.346796952884392 | 0.153203047115608 | 0
0.153203047115608 | 0.346796952884392 | 0.346796952884392 | 0.153203047115608 | 0
0 | 0.558991228320154 | 0.319451478409571 | 0.121557293270274 | 0
0.319451478409572 | 0 | 0.558991228320154 | 0.121557293270274 | 0
0.144811773820850 | 0.144811773820850 | 0.565564678537449 | 0.144811773820850 | 0
0 | 0.319451478409572 | 0.558991228320154 | 0.121557293270274 | 0
0.113447464772218 | 0 | 0.773105070455563 | 0.113447464772218 | 0
0 | 0.113447464772218 | 0.773105070455563 | 0.113447464772218 | 0
0 | 0 | 0.915111948139284 | 0.084888051860717 | 0
0.734424396735357 | 0 | 0 | 0.265575603264643 | 0
0.558991228320154 | 0.121557293270274 | 0 | 0.319451478409572 | 0
0.333333333333333 | 0.333333333333333 | 0 | 0.333333333333333 | 0
0.121557293270274 | 0.558991228320154 | 0 | 0.319451478409571 | 0
0 | 0.734424396735357 | 0 | 0.265575603264643 | 0
0.558991228320154 | 0 | 0.121557293270274 | 0.319451478409571 | 0
0.346796952884392 | 0.153203047115608 | 0.153203047115608 | 0.346796952884392 | 0
0.153203047115608 | 0.346796952884392 | 0.153203047115608 | 0.346796952884392 | 0
0 | 0.558991228320154 | 0.121557293270274 | 0.319451478409571 | 0
0.333333333333333 | 0 | 0.333333333333333 | 0.333333333333333 | 0
0.153203047115608 | 0.153203047115608 | 0.346796952884392 | 0.346796952884392 | 0
0 | 0.333333333333333 | 0.333333333333333 | 0.333333333333333 | 0
0.121557293270274 | 0 | 0.558991228320154 | 0.319451478409572 | 0
0 | 0.121557293270274 | 0.558991228320154 | 0.319451478409572 | 0
0 | 0 | 0.734424396735357 | 0.265575603264643 | 0
0.500000000000000 | 0 | 0 | 0.500000000000000 | 0
0.319451478409572 | 0.121557293270274 | 0 | 0.558991228320154 | 0
0.121557293270274 | 0.319451478409572 | 0 | 0.558991228320154 | 0
0 | 0.500000000000000 | 0 | 0.500000000000000 | 0
0.319451478409572 | 0 | 0.121557293270274 | 0.558991228320154 | 0
0.144811773820850 | 0.144811773820850 | 0.144811773820850 | 0.565564678537449 | 0
0 | 0.319451478409572 | 0.121557293270274 | 0.558991228320154 | 0
0.121557293270274 | 0 | 0.319451478409572 | 0.558991228320154 | 0
0 | 0.121557293270274 | 0.319451478409572 | 0.558991228320154 | 0
0 | 0 | 0.500000000000000 | 0.500000000000000 | 0
0.265575603264643 | 0 | 0 | 0.734424396735357 | 0
0.113447464772218 | 0.113447464772218 | 0 | 0.773105070455563 | 0
0 | 0.265575603264643 | 0 | 0.734424396735357 | 0
0.113447464772218 | 0 | 0.113447464772218 | 0.773105070455563 | 0
0 | 0.113447464772218 | 0.113447464772218 | 0.773105070455563 | 0
0 | 0 | 0.265575603264643 | 0.734424396735357 | 0
0.084888051860716 | 0 | 0 | 0.915111948139284 | 0
0 | 0.084888051860716 | 0 | 0.915111948139284 | 0
0 | 0 | 0.084888051860716 | 0.915111948139284 | 0
0 | 0 | 0 | 1.000000000000000 | 0
0.915111948139284 | 0 | 0 | 0 | 0.084888051860716
0.773105070455563 | 0.113447464772218 | 0 | 0 | 0.113447464772218
0.558991228320154 | 0.319451478409572 | 0 | 0 | 0.121557293270274
0.319451478409572 | 0.558991228320154 | 0 | 0 | 0.121557293270274
0.113447464772218 | 0.773105070455563 | 0 | 0 | 0.113447464772218
0 | 0.915111948139284 | 0 | 0 | 0.084888051860716
0.773105070455564 | 0 | 0.113447464772218 | 0 | 0.113447464772218
0.565564678537450 | 0.144811773820850 | 0.144811773820850 | 0 | 0.144811773820850
0.346796952884392 | 0.346796952884392 | 0.153203047115608 | 0 | 0.153203047115608
0.144811773820850 | 0.565564678537449 | 0.144811773820850 | 0 | 0.144811773820850
0 | 0.773105070455563 | 0.113447464772218 | 0 | 0.113447464772218
0.558991228320154 | 0 | 0.319451478409572 | 0 | 0.121557293270274
0.346796952884392 | 0.153203047115608 | 0.346796952884392 | 0 | 0.153203047115608
0.153203047115608 | 0.346796952884392 | 0.346796952884392 | 0 | 0.153203047115608
0 | 0.558991228320154 | 0.319451478409572 | 0 | 0.121557293270274
0.319451478409572 | 0 | 0.558991228320154 | 0 | 0.121557293270274
0.144811773820850 | 0.144811773820850 | 0.565564678537449 | 0 | 0.144811773820850
0 | 0.319451478409572 | 0.558991228320154 | 0 | 0.121557293270274
0.113447464772218 | 0 | 0.773105070455563 | 0 | 0.113447464772218
0 | 0.113447464772218 | 0.773105070455563 | 0 | 0.113447464772218
0 | 0 | 0.915111948139284 | 0 | 0.084888051860716
0.773105070455563 | 0 | 0 | 0.113447464772218 | 0.113447464772218
0.565564678537450 | 0.144811773820850 | 0 | 0.144811773820850 | 0.144811773820850
0.346796952884392 | 0.346796952884392 | 0 | 0.153203047115608 | 0.153203047115608
0.144811773820850 | 0.565564678537449 | 0 | 0.144811773820850 | 0.144811773820850
0 | 0.773105070455563 | 0 | 0.113447464772218 | 0.113447464772218
0.565564678537450 | 0 | 0.144811773820850 | 0.144811773820850 | 0.144811773820850
0.354059351412100 | 0.161485162146975 | 0.161485162146975 | 0.161485162146975 | 0.161485162146975
0.161485162146975 | 0.354059351412100 | 0.161485162146975 | 0.161485162146975 | 0.161485162146975
0 | 0.565564678537450 | 0.144811773820850 | 0.144811773820850 | 0.144811773820850
0.346796952884392 | 0 | 0.346796952884392 | 0.153203047115608 | 0.153203047115608
0.161485162146975 | 0.161485162146975 | 0.354059351412100 | 0.161485162146975 | 0.161485162146975
0 | 0.346796952884392 | 0.346796952884392 | 0.153203047115608 | 0.153203047115608
0.144811773820850 | 0 | 0.565564678537449 | 0.144811773820850 | 0.144811773820850
0 | 0.144811773820850 | 0.565564678537449 | 0.144811773820850 | 0.144811773820850
0 | 0 | 0.773105070455563 | 0.113447464772218 | 0.113447464772218
0.558991228320154 | 0 | 0 | 0.319451478409572 | 0.121557293270274
0.346796952884392 | 0.153203047115608 | 0 | 0.346796952884392 | 0.153203047115608
0.153203047115608 | 0.346796952884392 | 0 | 0.346796952884392 | 0.153203047115608
0 | 0.558991228320154 | 0 | 0.319451478409572 | 0.121557293270274
0.346796952884392 | 0 | 0.153203047115608 | 0.346796952884392 | 0.153203047115608
0.161485162146975 | 0.161485162146975 | 0.161485162146975 | 0.354059351412100 | 0.161485162146975
0 | 0.346796952884392 | 0.153203047115608 | 0.346796952884392 | 0.153203047115608
0.153203047115608 | 0 | 0.346796952884392 | 0.346796952884392 | 0.153203047115608
0 | 0.153203047115608 | 0.346796952884392 | 0.346796952884392 | 0.153203047115608
0 | 0 | 0.558991228320154 | 0.319451478409572 | 0.121557293270274
0.319451478409572 | 0 | 0 | 0.558991228320154 | 0.121557293270274
0.144811773820850 | 0.144811773820850 | 0 | 0.565564678537449 | 0.144811773820850
0 | 0.319451478409572 | 0 | 0.558991228320154 | 0.121557293270274
0.144811773820850 | 0 | 0.144811773820850 | 0.565564678537449 | 0.144811773820850
0 | 0.144811773820850 | 0.144811773820850 | 0.565564678537449 | 0.144811773820850
0 | 0 | 0.319451478409572 | 0.558991228320154 | 0.121557293270274
0.113447464772218 | 0 | 0 | 0.773105070455563 | 0.113447464772218
0 | 0.113447464772218 | 0 | 0.773105070455563 | 0.113447464772218
0 | 0 | 0.113447464772218 | 0.773105070455563 | 0.113447464772218
0 | 0 | 0 | 0.915111948139283 | 0.084888051860717
0.734424396735357 | 0 | 0 | 0 | 0.265575603264643
0.558991228320154 | 0.121557293270274 | 0 | 0 | 0.319451478409572
0.333333333333333 | 0.333333333333333 | 0 | 0 | 0.333333333333333
0.121557293270274 | 0.558991228320154 | 0 | 0 | 0.319451478409572
0 | 0.734424396735357 | 0 | 0 | 0.265575603264643
0.558991228320154 | 0 | 0.121557293270274 | 0 | 0.319451478409572
0.346796952884392 | 0.153203047115608 | 0.153203047115608 | 0 | 0.346796952884392
0.153203047115608 | 0.346796952884392 | 0.153203047115608 | 0 | 0.346796952884392
0 | 0.558991228320154 | 0.121557293270274 | 0 | 0.319451478409572
0.333333333333333 | 0 | 0.333333333333333 | 0 | 0.333333333333333
0.153203047115608 | 0.153203047115608 | 0.346796952884392 | 0 | 0.346796952884392
0 | 0.333333333333333 | 0.333333333333333 | 0 | 0.333333333333333
0.121557293270274 | 0 | 0.558991228320154 | 0 | 0.319451478409572
0 | 0.121557293270274 | 0.558991228320154 | 0 | 0.319451478409572
0 | 0 | 0.734424396735357 | 0 | 0.265575603264643
0.558991228320154 | 0 | 0 | 0.121557293270274 | 0.319451478409572
0.346796952884392 | 0.153203047115608 | 0 | 0.153203047115608 | 0.346796952884392
0.153203047115608 | 0.346796952884392 | 0 | 0.153203047115608 | 0.346796952884392
0 | 0.558991228320154 | 0 | 0.121557293270274 | 0.319451478409572
0.346796952884392 | 0 | 0.153203047115608 | 0.153203047115608 | 0.346796952884392
0.161485162146975 | 0.161485162146975 | 0.161485162146975 | 0.161485162146975 | 0.354059351412100
0 | 0.346796952884392 | 0.153203047115608 | 0.153203047115608 | 0.346796952884392
0.153203047115608 | 0 | 0.346796952884392 | 0.153203047115608 | 0.346796952884392
0 | 0.153203047115608 | 0.346796952884392 | 0.153203047115608 | 0.346796952884392
0 | 0 | 0.558991228320154 | 0.121557293270274 | 0.319451478409572
0.333333333333333 | 0 | 0 | 0.333333333333333 | 0.333333333333333
0.153203047115608 | 0.153203047115608 | 0 | 0.346796952884392 | 0.346796952884392
0 | 0.333333333333333 | 0 | 0.333333333333333 | 0.333333333333333
0.153203047115608 | 0 | 0.153203047115608 | 0.346796952884392 | 0.346796952884392
0 | 0.153203047115608 | 0.153203047115608 | 0.346796952884392 | 0.346796952884392
0 | 0 | 0.333333333333333 | 0.333333333333333 | 0.333333333333333
0.121557293270274 | 0 | 0 | 0.558991228320154 | 0.319451478409572
0 | 0.121557293270274 | 0 | 0.558991228320154 | 0.319451478409572
0 | 0 | 0.121557293270274 | 0.558991228320154 | 0.319451478409572
0 | 0 | 0 | 0.734424396735357 | 0.265575603264643
0.500000000000000 | 0 | 0 | 0 | 0.500000000000000
0.319451478409572 | 0.121557293270274 | 0 | 0 | 0.558991228320154
0.121557293270274 | 0.319451478409572 | 0 | 0 | 0.558991228320154
0 | 0.500000000000000 | 0 | 0 | 0.500000000000000
0.319451478409572 | 0 | 0.121557293270274 | 0 | 0.558991228320154
0.144811773820850 | 0.144811773820850 | 0.144811773820850 | 0 | 0.565564678537449
0 | 0.319451478409572 | 0.121557293270274 | 0 | 0.558991228320154
0.121557293270274 | 0 | 0.319451478409572 | 0 | 0.558991228320154
0 | 0.121557293270274 | 0.319451478409572 | 0 | 0.558991228320154
0 | 0 | 0.500000000000000 | 0 | 0.500000000000000
0.319451478409572 | 0 | 0 | 0.121557293270274 | 0.558991228320154
0.144811773820850 | 0.144811773820850 | 0 | 0.144811773820850 | 0.565564678537449
0 | 0.319451478409572 | 0 | 0.121557293270274 | 0.558991228320154
0.144811773820850 | 0 | 0.144811773820850 | 0.144811773820850 | 0.565564678537449
0 | 0.144811773820850 | 0.144811773820850 | 0.144811773820850 | 0.565564678537449
0 | 0 | 0.319451478409572 | 0.121557293270274 | 0.558991228320154
0.121557293270274 | 0 | 0 | 0.319451478409572 | 0.558991228320154
0 | 0.121557293270274 | 0 | 0.319451478409572 | 0.558991228320154
0 | 0 | 0.121557293270274 | 0.319451478409572 | 0.558991228320154
0 | 0 | 0 | 0.500000000000000 | 0.500000000000000
0.265575603264643 | 0 | 0 | 0 | 0.734424396735357
0.113447464772218 | 0.113447464772218 | 0 | 0 | 0.773105070455563
0 | 0.265575603264643 | 0 | 0 | 0.734424396735357
0.113447464772218 | 0 | 0.113447464772218 | 0 | 0.773105070455563
0 | 0.113447464772218 | 0.113447464772218 | 0 | 0.773105070455563
0 | 0 | 0.265575603264643 | 0 | 0.734424396735357
0.113447464772218 | 0 | 0 | 0.113447464772218 | 0.773105070455563
0 | 0.113447464772218 | 0 | 0.113447464772218 | 0.773105070455563
0 | 0 | 0.113447464772218 | 0.113447464772218 | 0.773105070455563
0 | 0 | 0 | 0.265575603264643 | 0.734424396735357
0.084888051860717 | 0 | 0 | 0 | 0.915111948139283
0 | 0.084888051860717 | 0 | 0 | 0.915111948139283
0 | 0 | 0.084888051860717 | 0 | 0.915111948139284
0 | 0 | 0 | 0.084888051860716 | 0.915111948139284
0 | 0 | 0 | 0 | 1.000000000000000