text:
Socceroos' Mathew Leckie robbed of penalty against Saudi Arabia in World Cup qualifier
By WWOS staff
Socceroos coach Ange Postecoglou has questioned the decision to deny his team an obvious penalty against Saudi Arabia in a move which could have ramifications for Australia's World Cup campaign.
With the Socceroos already up 3-2, Mathew Leckie looked like he would have a chance to go further ahead after charging on to a ball in the Saudi's box.
He was brought down by defender Salman Alfaraj but the Socceroos appeal for a penalty to Uzbekistan ref Ravshan Irmatov fell on deaf ears.
"Unless you find a more obvious one than that one, it looked like a penalty to me," Postecoglou said.
"It wasn't given, so be it. In retrospect it's a pretty clear penalty."
Saudi Arabia snub minute's silence
Socceroos down Saudis in World Cup Qualifier
Fox Sports commentator Mark Bosnich later described the penalty as a "20 out of 10" when asked to rate it on a scale of 1 to 10.
"This guy is a good referee but I don't know what he was thinking ... players make mistakes and referees make mistakes," Bosnich said.
"Hopefully it won't make a difference in the end."
While the Socceroos went on to win the match, the incident could still prove costly with Australia level with Japan and Saudi Arabia on 16 points but in third position on goal difference.
If the penalty had been awarded, the Socceroos would have leapfrogged Saudi Arabia on the table with the top two teams automatically qualifying for the World Cup while the third-placed team faces a playoff.
Australia still has a chance to directly qualify with games remaining against Japan and Thailand.
Socceroos down Saudis in Cup qualifier
Neymar attempts terrifying shot at goal on Los Angeles rooftops


Ar`bi*tra"tion (?), n. [F. arbitration, L. arbitratio, fr. arbitrari.] The hearing and determination of a cause between parties in controversy, by a person or persons chosen by the parties.
Arbitration bond, a bond which obliges one to abide by the award of an arbitration. -- Arbitration of Exchange, the operation of converting the currency of one country into that of another, or determining the rate of exchange between such countries or currencies. An arbitrated rate is one determined by such arbitration through the medium of one or more intervening currencies.


Microsoft has entered a new phase of its $300 million Windows ad campaign. In a briefing this morning, the software maker said it's releasing a 60-second commercial that seems designed to directly counter Apple's popular PC-bashing Macintosh ads.
It also debuted a print ad themed "Windows vs. Walls" that covers two facing pages. Internally dubbed "Manifesto," the ad appears in both Seattle's daily papers today. It "celebrates choice and individuality," said Bill Veghte, Microsoft's senior vice-president of online services & Windows, in an interview this morning.
Veghte said Phase II of the commercial campaign omits comedian Jerry Seinfeld, who figured in the three commercials making up Phase I, but Phase II's first spot does feature a brief appearance by company co-founder and chairman Bill Gates.
The newest commercial is the first to actually address Microsoft products. Phase I was more about creating curiosity and buzz, Microsoft has said.
The commercial begins with an actor who resembles John Hodgman, one of the two characters in the Apple campaign, speaking the line, "Hello, I'm a PC — and I've been made into a stereotype." That's followed by brief, loosely linked scenes of scores of PC users in widely varied locales, doing non-stereotypical work or projects with their computers.
In the long-running Apple campaign, the PC consistently fails to equal, much less exceed, the Mac's power or cool. That's what the Microsoft campaign apparently seeks to challenge.
He said an online version of the spot will let viewers upload their own film clips demonstrating how they, too, are a PC.
A story in the New York Times today discusses the wisdom of attacking a competitor by riffing on the rival's own ad campaign.


text:
\section{Introduction}

Given data $\{f_j\}_{j=0}^n$ and distinct nodes $\{x_j\}_{j=0}^n$, the interpolation problem consists of finding a polynomial $p$ of degree $n$ that satisfies
\begin{equation}
p(x_j)=f_j
\end{equation}
for each $0\leq j\leq n$. If a basis for the space of polynomials of degree at most $n$ is chosen, then the interpolation problem can be expressed as a system of linear equations, where the coefficient matrix is a Vandermonde-like matrix \cite{gautschi1999orthogonal}. Recently, Bernstein polynomials have been considered as a tool for high-order approximation of partial differential equations via the finite element method \cite{ainsworth2011bernstein,duffy1982quadrature}, and the interpolant is often used as a polynomial approximation of initial or boundary data. The corresponding Bernstein--Vandermonde matrix is found to be highly ill-conditioned \cite{davis1975condition}, but the structure of the matrix has led to fast algorithms that avoid some of the issues that arise from the ill-conditioning. For example, Marco and Mart\'{i}nez \cite{marco2007bidiagonal} used the fact that the Bernstein--Vandermonde matrix is strictly totally positive \cite{gasca1992positivity} to obtain a bidiagonal factorization of the inverse, and Ainsworth and Sanchez \cite{ainsworth2016newton} adapted the standard divided difference algorithm for the monomial Vandermonde matrix \cite{burden2012difference} to the Bernstein basis.

In \cite{allenkirby2020mass}, we used the relationship between monomial B\'{e}zout matrices and the inverse of Hankel matrices described by Heinig and Rost \cite{heinig1984algebraic} to obtain a decomposition of the inverse of the Bernstein mass matrix in terms of Hankel, Toeplitz, and diagonal matrices. In this paper, we generalize an argument made by Kaplan \cite{kaplan2006bezout} to obtain a decomposition of the inverse of Vandermonde-like matrices in terms of their transpose, a diagonal matrix, and the corresponding B\'{e}zout matrix. When applied to the Bernstein basis, this gives a decomposition of the inverse of the Bernstein--Vandermonde matrix in terms of Hankel, Toeplitz, and diagonal matrices, which in turn leads to a fast algorithm for solving the interpolation problem. Additionally, we use the nonstandard matrix norm defined in \cite{allenkirby2020mass} to give an explanation for the relatively good performance of the Bernstein--Vandermonde matrix despite its massive condition number.

Bernstein polynomials also extend naturally to give a basis for multivariate polynomials of total degree $n$. Properties of Bernstein polynomials lead to special recursive blockwise-structure for finite element matrices \cite{kirby2017fast,kirby2012fast}. In this paper, we use this structure to obtain a block $LU$ decomposition of the Bernstein--Vandermonde matrix associated to equispaced nodes on the $d$-simplex.

\section{Inverse Formulas}
\label{sec:inverse}
\subsection{General Formulas}
\label{ssec:general}

For integers $m,n\geq 0$, let $\{ b^n_j(x) \}_{j=0}^n$ be a basis for the space of univariate polynomials of degree at most $n$, and let $\mathbf{x}\in\mathbb{R}^{m+1}$ with $\mathbf{x}_i<\mathbf{x}_{i+1}$ for each $0\leq i<m$, where boldface is used to distinguish the vector of nodes $\mathbf{x}$ from the indeterminate $x$ (a similar convention will be used to distinguish a polynomial $v$ from its vector of coefficients $\mathbf{v}$). The Vandermonde matrix $V^n(\mathbf{x})$ associated to $\{ b^n_j(x) \}_{j=0}^n$ and $\mathbf{x}$ is the $(m+1)\times(n+1)$ matrix given by
\begin{equation}
\label{eq:vandermonde}
V^n_{ij}(\mathbf{x})=b^n_j(\mathbf{x}_i).
\end{equation}
We consider the case $m=n$ so that $V^n(\mathbf{x})$ is invertible.

If $v$ and $w$ are polynomials of degree at most $n+1$ expressed in the basis $\{b^{n+1}_j(x)\}_{j=0}^{n+1}$, then the B\'{e}zout matrix generated by $v$ and $w$, denoted $\Bez(v,w)$, is the $(n+1)\times (n+1)$ matrix whose entries satisfy
\begin{equation}
\label{eq:bezout}
\frac{v(s)w(t)-v(t)w(s)}{s-t} = \sum_{i,j=0}^n \Bez_{ij}(v,w)b^n_i(s)b^n_j(t).
\end{equation}
In \cite{kaplan2006bezout}, it was shown that if the monomial basis is used, then the inverse of a B\'{e}zout matrix is a Hankel matrix. As a consequence of the arguments involved, one obtains a formula for the inverse of the monomial Vandermonde matrix in terms of its transpose, a diagonal matrix, and a particular B\'{e}zout matrix. Since none of the arguments require anything specific to the monomial basis, we can generalize the arguments to any basis, which we summarize here.

For $t\in\mathbb{R}$, let $\mathbf{b}^n(t)$ denote the column vector $(b^n_0(t),\dots,b^n_n(t))^T$. As a consequence of (\ref{eq:bezout}), if $s\neq t$, then
\begin{align*}
(\mathbf{b}^n(s))^T\Bez(v,w)\mathbf{b}^n(t)
&= \sum_{i,j=0}^n \Bez_{ij}(v,w)b^n_i(s)b^n_j(t) \\
&= \frac{v(s)w(t)-v(t)w(s)}{s-t}.
\end{align*}
This implies that
\begin{align*}
(\mathbf{b}^n(t))^T\Bez(v,w)\mathbf{b}^n(t)
&= \lim_{\varepsilon\rightarrow 0} \left[ (\mathbf{b}^n(t))^T\Bez(v,w)\mathbf{b}^n(t+\varepsilon) \right] \\
&= \lim_{\varepsilon\rightarrow 0} \frac{v(t+\varepsilon)w(t)-v(t)w(t+\varepsilon)}{\varepsilon} \\
&= v'(t)w(t)-v(t)w'(t).
\end{align*}
Therefore, if $v$ has simple zeros at $\mathbf{x}_i$ and $w(\mathbf{x}_i)\neq 0$ for each $0\leq i\leq n$, then
\begin{equation}
V^n(\mathbf{x})\Bez(v,w)\left(V^n(\mathbf{x})\right)^T = \diag(v'(\mathbf{x}_j)w(\mathbf{x}_j))_{j=0}^n,
\end{equation}
and so we have the following:

\begin{theorem}
\label{thm:bezout}
The inverse of $V^n(\mathbf{x})$ is given by
\begin{equation}
\left(V^n(\mathbf{x})\right)^{-1} = \Bez(v,w)\left(V^n(\mathbf{x})\right)^T\diag\left(\frac{1}{v'(\mathbf{x}_j)w(\mathbf{x}_j)}\right)_{j=0}^n,
\end{equation}
where $v$ and $w$ are any polynomials of degree $n+1$ satisfying $v(\mathbf{x}_i)=0$, $v'(\mathbf{x}_i)\neq 0$, and $w(\mathbf{x}_i)\neq 0$ for each $0\leq i\leq n$.
\end{theorem}

\begin{corollary}
\label{cor:bezout}
The inverse of $V^n(\mathbf{x})$ is given by
\begin{equation}
\left(V^n(\mathbf{x})\right)^{-1} = \Bez(v,1)\left(V^n(\mathbf{x})\right)^T\diag\left(\frac{1}{v'(\mathbf{x}_j)}\right)_{j=0}^n,
\end{equation}
where $v$ is any polynomial of degree $n+1$ that has simple zeros at $\mathbf{x}_i$ for each $0\leq i\leq n$.
\end{corollary}

\subsection{Bernstein--Vandermonde}
\label{ssec:BernsteinVandermonde}

We now focus on the case where $0\leq \mathbf{x}_i\leq 1$ for each $0\leq i\leq n$ and the basis consists of the polynomials $\{B^n_j(x)\}_{j=0}^n$, where
\begin{equation}
\label{eq:bernstein}
B^n_j(x) = \binom{n}{j}x^j(1-x)^{n-j}
\end{equation}
are the Bernstein polynomials of degree $n$. In order to apply Theorem~\ref{thm:bezout} or Corollary~\ref{cor:bezout}, we need a method for computing the entries of the Bernstein--B\'{e}zout matrix; in fact, they satisfy a recurrence relation involving the Bernstein coefficients of the polynomials.

\begin{theorem}
\label{thm:recursion}
If $v(t)=\sum_{i=0}^{n+1} \mathbf{v}_iB^{n+1}_i(t)$ and $w(t)=\sum_{i=0}^{n+1} \mathbf{w}_iB^{n+1}_i(t)$ are polynomials of degree at most $n+1$, then the entries $b_{ij}$ of the Bernstein--B\'{e}zout matrix $\Bez(v,w)$ generated by $v$ and $w$ satisfy
\begin{equation}
\label{eq:recursion}
b_{ij} = \frac{1}{(i+1)(n-j+1)}\left[j(n-i)b_{i+1,j-1}+(n+1)^2\left( \mathbf{v}_{i+1}\mathbf{w}_j-\mathbf{v}_j\mathbf{w}_{i+1}\right)\right].
\end{equation}
\end{theorem}

\begin{proof}
By (\ref{eq:bernstein}), we have that
\begin{equation}
tB^n_j(t) = \frac{j+1}{n+1}B^{n+1}_{j+1}(t)
\end{equation}
and
\begin{equation}
B^n_j(t) = \frac{n-j+1}{n+1}B^{n+1}_j(t)+\frac{j+1}{n+1}B^{n+1}_{j+1}(t).
\end{equation}
Therefore,
\begin{align*}
s\sum_{i,j=0}^n b_{ij}B^n_i(s)B^n_j(t) &= \sum_{i,j=0}^n \frac{i+1}{n+1}  b_{ij}B^{n+1}_{i+1}(s)B^n_j(t) \\
&= \sum_{i,j=0}^n \frac{(i+1)(n-j+1)}{(n+1)^2}b_{ij} B^{n+1}_{i+1}(s)B^{n+1}_j(t) \\
&\qquad +\sum_{i,j=0}^n \frac{(i+1)(j+1)}{(n+1)^2}b_{ij} B^{n+1}_{i+1}(s)B^{n+1}_{j+1}(t).
\end{align*}
Similarly, we have that
\begin{align*}
t\sum_{i,j=0}^n b_{ij}B^n_i(s)B^n_j(t) &= \sum_{i,j=0}^n \frac{j+1}{n+1}b_{ij}B^n_i(s)B^{n+1}_{j+1}(t) \\
&= \sum_{i,j=0}^n \frac{(n-i+1)(j+1)}{(n+1)^2} b_{ij} B^{n+1}_i(s)B^{n+1}_{j+1}(t) \\
&\qquad + \sum_{i,j=0}^n \frac{(i+1)(j+1)}{(n+1)^2} b_{ij} B^{n+1}_{i+1}(s)B^{n+1}_{j+1}(t).
\end{align*}
This implies that
\begin{align*}
(s-t)\sum_{i,j=0}^n b_{ij} B^n_i(s)B^n_j(t) &= \sum_{i,j=0}^n \frac{(i+1)(n-j+1)}{(n+1)^2}b_{ij} B^{n+1}_{i+1}(s)B^{n+1}_j(t) \\
&\quad -\sum_{i,j=0}^n \frac{(n-i+1)(j+1)}{(n+1)^2}b_{ij}B^{n+1}_i(s)B^{n+1}_{j+1}(t),
\end{align*}
and so
\begin{align*}
(s-t)\sum_{i,j=0}^n b_{ij}B^n_i(s)B^n_j(t) &= \sum_{i,j=0}^{n+1}\frac{i(n-j+1)}{(n+1)^2}b_{i-1,j}B^{n+1}_i(s)B^{n+1}_j(t) \\
&\quad -\sum_{i,j=0}^{n+1}\frac{j(n-i+1)}{(n+1)^2}b_{i,j-1}B^{n+1}_i(s)B^{n+1}_j(t).
\end{align*}
On the other hand,
\begin{equation}
v(s)w(t)-v(t)w(s) = \sum_{i,j=0}^{n+1} (\mathbf{v}_i\mathbf{w}_j-\mathbf{v}_j\mathbf{w}_i)B^{n+1}_i(s)B^{n+1}_j(t),
\end{equation}
and so the result follows by comparing coefficients in (\ref{eq:bezout}).
\end{proof}

We can form the first column and last row of $\Bez(v,w)$ by using (\ref{eq:recursion}) with $j=0$ and $i=n$. The rest of the columns can then be built by applying the recurrence relation. Therefore,

\begin{corollary}
\label{cor:bezoutcost}
$\Bez(v,w)$ can be constructed in $\mathcal{O}(n^2)$ operations.
\end{corollary}

By repeatedly applying (\ref{eq:recursion}), we obtain a closed form for the entries of the Bernstein--B\'{e}zout matrix.

\begin{corollary}
\label{cor:bernbezout}
If $v(t)=\sum_{i=0}^{n+1} \mathbf{v}_iB^{n+1}_i(t)$ and $w(t)=\sum_{i=0}^{n+1} \mathbf{w}_iB^{n+1}_i(t)$ are polynomials of degree at most $n+1$, then the entries $b_{ij}$ of the Bernstein--B\'{e}zout matrix $\Bez(v,w)$ generated by $v$ and $w$ are given by
\begin{equation}
\label{eq:bernbezout}
b_{ij} = \frac{1}{\binom{n}{i}\binom{n}{j}}\sum_{k=0}^{m_{ij}} \binom{n+1}{i+k+1}\binom{n+1}{j-k}\left( \mathbf{v}_{i+k+1}\mathbf{w}_{j-k}-\mathbf{v}_{j-k}\mathbf{w}_{i+k+1}\right),
\end{equation}
where $m_{ij} = \min\{j,n-i\}$.
\end{corollary}

Constructing $\Bez(v,w)$ by using (\ref{eq:bernbezout}) would require $\mathcal{O}(n^3)$ operations, and so Corollary~\ref{cor:bezoutcost} implies that (\ref{eq:bernbezout}) is not optimal for building $\Bez(v,w)$; however, since $\binom{n}{i}=0$ whenever $i>n$, the upper limit of the sum in (\ref{eq:bernbezout}) can be replaced with $k=n$, and so we recognize that $\Bez(v,w)$ is given by a difference of matrix products.

\begin{corollary}
\label{cor:factorization}
Let $v(t) = \sum_{i=0}^{n+1} \mathbf{v}_iB^{n+1}_i(t)$ and $w(t)=\sum_{i=0}^{n+1}\mathbf{w}_iB^{n+1}_i(t)$ be polynomials of degree at most $n+1$. Define the Toeplitz matrices $T^{v,n}$ and $T^{w,n}$ by
\begin{equation*}
T^{v,n}_{ij} = \binom{n+1}{j-i}\mathbf{v}_{j-i}\qquad \text{and}\qquad T^{w,n}_{ij} = \binom{n+1}{j-i}\mathbf{w}_{j-i},
\end{equation*}
define the Hankel matrices $H^{v,n}$ and $H^{w,n}$ by
\begin{equation*}
H^{v,n}_{ij} = \binom{n+1}{i+j+1}\mathbf{v}_{i+j+1}\qquad \text{and}\qquad H^{w,n}_{ij} = \binom{n+1}{i+j+1}\mathbf{w}_{i+j+1},
\end{equation*}
and let $\Delta^n=\diag\left(\binom{n}{j}\right)_{j=0}^n$. Then the Bernstein--B\'{e}zout matrix $\Bez(v,w)$ generated by $v$ and $w$ is given by
\begin{equation}
\label{eq:bezoutinverse}
\Bez(v,w) = \left(\Delta^n\right)^{-1}\left[ H^{v,n}T^{w,n}-H^{w,n}T^{v,n}\right]\left(\Delta^n\right)^{-1}.
\end{equation}
\end{corollary}

\begin{remark}
\label{rmk:connection}
In \cite{allenkirby2020mass}, it was shown that the inverse of the degree $n$ Bernstein mass matrix $M^n$ has the decomposition
\begin{equation}
\label{eq:massinverse}
\left(M^n\right)^{-1} = \left(\Delta^n\right)^{-1}\left[ \widetilde{T}^nH^n-T^n\widetilde{H}^n\right] \left(\Delta^n\right)^{-1},
\end{equation}
where $\widetilde{H}^n$ and $H^n$ are Hankel matrices, $\widetilde{T}^n$ and $T^n$ are Toeplitz matrices, and $\Delta^n = \diag\left(\binom{n}{j}\right)_{j=0}^n$. This decomposition was obtained from the monomial B\'{e}zout matrix corresponding to polynomials whose coefficients are contained in the last column of $\left(M^n\right)^{-1}$. It is interesting to note the similarities between (\ref{eq:bezoutinverse}) and (\ref{eq:massinverse}) despite the different matrices and bases being considered.
\end{remark}

Given a polynomial $v(t)=\sum_{k=0}^{n+1} \mathbf{v}_k B^{n+1}_k(t)$ of degree at most $n+1$, we can express $v$ in the monomial basis as $v(t)=\sum_{k=0}^{n+1} \widetilde{\mathbf{v}}_kt^k$. It was shown in \cite{farouki2000legendre} that
\begin{equation}
\label{eq:montobern}
\mathbf{v}_k = \frac{1}{\binom{n+1}{k}} \sum_{\ell=0}^k \binom{n-\ell+1}{k-\ell} \widetilde{\mathbf{v}}_{\ell}.
\end{equation}
This implies that
\begin{align*}
H^{v,n}_{ij} = \binom{n+1}{i+j+1}\mathbf{v}_{i+j+1} &= \sum_{k=0}^{i+j+1} \binom{n-k+1}{i+j-k+1}\widetilde{\mathbf{v}}_k \\
&= \sum_{k=0}^{i+j+1} \binom{n-k+1}{n-i-j}\widetilde{\mathbf{v}}_k.
\end{align*}
Similarly,
\begin{equation*}
T^{v,n}_{ij} = \sum_{k=0}^{j-i}\binom{n-k+1}{j-i-k}\widetilde{\mathbf{v}}_ k.
\end{equation*}
By Vieta's formula (see, for example, \cite{rotman2015algebra}),
\begin{equation}
\label{eq:vieta}
\prod_{i=0}^n (x-\mathbf{x}_i) = \sum_{k=0}^n (-1)^{n-k+1}\sigma_{n-k+1}(\mathbf{x})x^k,
\end{equation}
where
\begin{equation}
\sigma_k(\mathbf{x}) = \begin{cases} 1, & \text{if}\ k=0; \\ \sum_{0\leq i_0<\dots<i_{k-1}\leq n} \mathbf{x}_{i_0}\cdots \mathbf{x}_{i_{k-1}}, & \text{otherwise}; \end{cases}
\end{equation}
is the $k^{\text{th}}$ elementary symmetric function in the $n+1$ variables $\mathbf{x}_0,\dots,\mathbf{x}_n$. In addition, since
\begin{equation}
1 = \sum_{i=0}^{n+1} B^{n+1}_i(x)
\end{equation}
and
\begin{equation}
\frac{d}{dx}\left[ \prod_{i=0}^n (x-\mathbf{x}_i)\right]_{x=x_j} = \prod_{i\in\{0,\dots,n\}\setminus\{j\}} (\mathbf{x}_j-\mathbf{x}_i),
\end{equation}
we can combine the previous discussion with Corollary~\ref{cor:bezout} and Corollary~\ref{cor:factorization} to obtain a decomposition of the inverse of the Bernstein--Vandermonde matrix.

\begin{theorem}
\label{thm:factoredinverse}
Define the Hankel matrices $H^n$ and $\widetilde{H}^n(\mathbf{x})$ by
\begin{equation*}
H^n_{ij} = \binom{n+1}{i+j+1}\qquad \text{and}\qquad \widetilde{H}^n_{ij}(\mathbf{x}) = \sum_{k=0}^{i+j+1} (-1)^{n-k+1}\binom{n-k+1}{n-i-j}\sigma_{n-k+1}(\mathbf{x}),
\end{equation*}
define the Toeplitz matrices $T^n$ and $\widetilde{T}^n(\mathbf{x})$ by
\begin{equation*}
T^n_{ij} = \binom{n+1}{j-i}\qquad \text{and}\qquad \widetilde{T}^n_{ij}(\mathbf{x}) = \sum_{k=0}^{j-i} (-1)^{n-k+1} \binom{n-k+1}{j-i-k}\sigma_{n-k+1}(\mathbf{x}),
\end{equation*}
and define the diagonal matrices $D^n(\mathbf{x})$ and $\Delta^n$ by
\begin{equation*}
D^n(\mathbf{x}) = \diag\left( \prod_{i\in\{0,\dots,n\}\setminus\{j\}}(\mathbf{x}_j-\mathbf{x}_i)\right)_{j=0}^n\qquad \text{and}\qquad \Delta^n = \diag\left( \binom{n}{j}\right)_{j=0}^n.
\end{equation*}
In addition, let $\widetilde{V}^n(\mathbf{x})$ be the scaled Bernstein--Vandermonde matrix
\begin{equation*}
\widetilde{V}^n_{ij}(\mathbf{x}) = \mathbf{x}_i^j(1-\mathbf{x}_i)^{n-j}.
\end{equation*}
Then
\begin{equation}
\left(V^n(\mathbf{x})\right)^{-1} = \left(\Delta^n\right)^{-1}\left[ \widetilde{H}^n(\mathbf{x})T^n-H^n\widetilde{T}^n(\mathbf{x})\right]\left(\widetilde{V}^n(\mathbf{x})\right)^T\left(D^n(\mathbf{x})\right)^{-1}.
\end{equation}
\end{theorem}

\subsection{Equispaced Nodes}
\label{ssec:equispaced}

In Section~\ref{sec:dimensions}, we will derive a block $LU$ decomposition for the Bernstein--Vandermonde matrix associated to equispaced nodes on the $d$-simplex. This decomposition leads to a recursive, block-structured algorithm for the corresponding interpolation problem. The base for this algorithm is the one-dimensional Bernstein--Vandermonde matrix associated to equispaced nodes, and so we now briefly focus on the case where $\mathbf{x}_i=i/n$ for each $0\leq i\leq n$. For these nodes, we observe that (\ref{eq:vieta}) becomes
\begin{align*}
\prod_{i=0}^n (x-i/n) = \frac{1}{n^{n+1}}\prod_{i=0}^{n+1} (nx-i) &= \frac{1}{n^{n+1}}\sum_{k=0}^{n+1} s(n+1,k)(nx)^k \\
&= \sum_{k=0}^{n+1} \frac{s(n+1,k)}{n^{n-k+1}}x^k,
\end{align*}
where $s(n,k)$ are the (signed) Stirling numbers of the first kind \cite{stegun1965handbook}. In addition, we have that
\begin{equation}
\prod_{i\in\{0,\dots,n\}\setminus\{j\}}(j/n-i/n) = (-1)^{n-j} \frac{j!(n-j)!}{n^n}.
\end{equation}

\begin{corollary}
\label{cor:equispacedinverse}
Define the Hankel matrices $H^n$ and $\widetilde{H}^n$ by
\begin{equation*}
H^n_{ij} = \binom{n+1}{i+j+1}\qquad \text{and}\qquad \widetilde{H}^n_{ij} = \sum_{k=0}^{i+j+1}\binom{n-k+1}{n-i-j}\frac{s(n+1,k)}{n^{n-k+1}},
\end{equation*}
define the Toeplitz matrices $T^n$ and $\widetilde{T}^n$ by
\begin{equation*}
T^n_{ij} = \binom{n+1}{j-i}\qquad \text{and}\qquad \widetilde{T}^n_{ij} = \sum_{k=0}^{j-i} \binom{n-k+1}{j-i-k}\frac{s(n+1,k)}{n^{n-k+1}},
\end{equation*}
and define the diagonal matrices $D^n$ and $\Delta^n$ by
\begin{equation*}
D^n = \diag\left( (-1)^{n-j}\left[j!(n-j)!\right]\right)_{j=0}^n\qquad \text{and}\qquad \Delta^n = \diag\left( \binom{n}{j}\right)_{j=0}^n.
\end{equation*}
In addition, let $\widetilde{V}^n$ be the scaled Bernstein--Vandermonde matrix
\begin{equation*}
\widetilde{V}^n_{ij} = i^j(n-i)^{n-j}.
\end{equation*}
Then the inverse of $V^n=V^n(0,1/n,\dots,1)$ is given by
\begin{equation}
\left(V^n\right)^{-1} = \left(\Delta^n\right)^{-1}\left[ \widetilde{H}^nT^n-H^n\widetilde{T}^n\right]\left(\widetilde{V}^n\right)^T\left(D^n\right)^{-1}.
\end{equation}
\end{corollary}

\section{Applying the inverse}
\label{sec:apply}

Now, we describe several approaches to applying $\left(V^n(\mathbf{x})\right)^{-1}$ to a vector.

\paragraph{$LU$ factorization} We can decompose $V^n(\mathbf{x})$ as
\begin{equation}
V^n(\mathbf{x}) = L^n(\mathbf{x})U^n(\mathbf{x}),
\end{equation}
where $L^n(\mathbf{x})$ and $U^n(\mathbf{x})$ are lower and upper triangular matrices, respectively. Widely available in libraries, computing $L^n(\mathbf{x})$ and $U^n(\mathbf{x})$ requires $\mathcal{O}(n^3)$ operations, and each of the subsequent triangular solves require $\mathcal{O}(n^2)$ operations to perform.

\paragraph{Exact inverse} In light of Corollary~\ref{cor:bezout}, we can directly form $\left(V^n(\mathbf{x})\right)^{-1}$. By Theorem~\ref{thm:recursion}, we can form the inverse in $\mathcal{O}(n^2)$ operations. The inverse can then be applied to any vector in $\mathcal{O}(n^2)$ operations using the standard algorithm.

\paragraph{DFT-based application} By Theorem~\ref{thm:factoredinverse}, we can express the inverse of $V^n(\mathbf{x})$ in terms of its (scaled) transpose and Hankel, Toeplitz, and diagonal matrices. The matrices $H^n(\mathbf{x})$ and $T^n(\mathbf{x})$ can be formed in $\mathcal{O}(n)$ operations, but the matrices $\widetilde{H}^n(\mathbf{x})$ and $\widetilde{T}^n(\mathbf{x})$ require $\mathcal{O}(n^2)$ operations to form, since each entry involves a sum. Even though the Hankel and Toeplitz matrices can be applied to a vector in $\mathcal{O}(n\log n)$ operations via circulant embedding and a couple of FFT/iFFT \cite{vogel2002fft}, the scaled transpose still requires $\mathcal{O}(n^2)$ operations to apply to a vector.

\paragraph{Newton algorithm} In \cite{ainsworth2016newton}, Ainsworth and Sanchez gave a fast algorithm for solving $V^n(\mathbf{x})\mathbf{c}=\mathbf{b}$ via a recursion relation for the B\'{e}zier control points of the Newton form of the Lagrangian interpolant. The numerical results given suggest that the algorithm is stable even for high polynomial degree, and so we include it for comparison.

\section{Conditioning and accuracy}
\label{sec:cond}

In \cite{allenkirby2020mass}, we gave a partial explanation of the relatively high accuracy, compared to its large 2-norm condition number, of working with the Bernstein mass matrix. We will use similar reasoning to investigate the performance of the Bernstein--Vandermonde matrix.

Recall that for an integer $n\geq 0$, the Bernstein mass matrix $M^n$ is the $(n+1)\times(n+1)$ matrix given by
\begin{equation}
\label{eq:mass}
M^n_{ij} = \int_0^1 B^n_i(x)B^n_j(x)dx.
\end{equation}
The mass matrix (for Bernstein or any other family of polynomials) plays an important role in connecting the $L^2$ topology on the finite-dimensional space to linear algebra.  To see this, we first define mappings connecting polynomials of degree $n$ to $\mathbb{R}^{n+1}$.  Given any $\mathbf{c} \in \mathbb{R}^{n+1}$, we let $\pi(\mathbf{c})$ be the polynomial expressed in the Bernstein basis with coefficients contained in $\mathbf{c}$:
\begin{equation}
\pi(\mathbf{c})(x) = \sum_{i=0}^n \mathbf{c}_i B^n_i(x).
\end{equation}
We let $\mathbf{\Pi}$ be the inverse of this mapping, sending any polynomial of degree at most $n$ to the vector of $n+1$ coefficients with respect to the Bernstein basis.

Now, let $p(x)$ and $q(x)$ be polynomials of degree $n$ with expansion coefficients $\mathbf{\Pi}(p) = \mathbf{p}$ and $\mathbf{\Pi}(q) = \mathbf{q}$.  Then the $L^2$ inner product of $p$ and $q$ is given by the $M^n$-weighted inner product of $\mathbf{p}$ and $\mathbf{q}$, for
\begin{equation}
\label{eq:innprod}
\int_0^1 p(x) q(x) dx
=
\sum_{i,j=0}^n \mathbf{p}_i \mathbf{q}_j \int_0^1 B^n_i(x) B^n_j(x) dx
= \mathbf{p}^T M^n \mathbf{q}.
\end{equation}
Similarly, if
\begin{equation}
\| \mathbf{p} \|_{M^n} = \sqrt{ \mathbf{p}^T M^n \mathbf{p} }
\end{equation}
is the $M^n$-weighted vector norm, then we have for $p = \pi(\mathbf{p})$,
\begin{equation}
\label{eq:topology}
\| p \|_{L^2} = \| \mathbf{p} \|_{M^n}.
\end{equation}

We can interpret the Bernstein--Vandermonde matrix $V^n(\mathbf{x})$ as an operator mapping polynomials of degree at most $n$ to $\mathbb{R}^{n+1}$ via $p\mapsto V^n(\mathbf{x})\mathbf{\Pi}(p)$. Therefore, it makes sense to measure $p$ in the $L^2$ norm and $V^n(\mathbf{x})\mathbf{\Pi}(p)$ in the 2-norm. By (\ref{eq:topology}), we can also consider $\mathbf{\Pi}(p)$ in the $M^n$-norm, which leads us to define the operator norm
\begin{equation}
\label{eq:Mto2}
\| A \|_{M^n\rightarrow 2} = \max_{\mathbf{y}\neq 0} \frac{ \| A\mathbf{y} \|_2 }{\| \mathbf{y} \|_{M^n}},
\end{equation}
and going in the opposite direction,
\begin{equation}
\label{eq:2toM}
\| A \|_{2\rightarrow M^n} = \max_{\mathbf{y}\neq 0} \frac{ \| A\mathbf{y} \|_{M^n}}{\| \mathbf{y} \|_2}.
\end{equation}
These two norms naturally combine to define a new condition number
\begin{equation}
\label{eq:kappaMto2}
\kappa_{M^n\rightarrow 2}(A) = \| A \|_{M^n\rightarrow 2} \left\| A^{-1} \right\|_{2\rightarrow M^n}.
\end{equation}

Since $M^n$ is symmetric and positive definite, it has a well-defined positive square root via the spectral decomposition.

\begin{lemma}
\label{lem:kapparelation}
\begin{equation}
\kappa_{M^n\rightarrow 2}\left(V^n(\mathbf{x})\right) = \kappa_2 \left( V^n(\mathbf{x})\left(M^n\right)^{-1/2}\right).
\end{equation}
\end{lemma}

\begin{proof}
Observe that if $\mathbf{y}\neq 0$, then
\begin{align*}
\frac{\left\| \left(V^n(\mathbf{x})\right)^{-1} \mathbf{y} \right\|^2_{M^n}}{ \| \mathbf{y} \|^2_2} &= \frac{ \mathbf{y}^T\left(V^n(\mathbf{x})\right)^{-T} M^n \left(V^n(\mathbf{x})\right)^{-1}\mathbf{y}}{\| \mathbf{y} \|^2_2} \\
&= \frac{ \left( \left(M^n\right)^{1/2} \left(V^n(\mathbf{x})\right)^{-1} \mathbf{y}\right)^T\left( \left(M^n\right)^{1/2} \left(V^n(\mathbf{x})\right)^{-1} \mathbf{y}\right)}{\| \mathbf{y} \|^2_2} \\
&= \frac{ \left\| \left(M^n\right)^{1/2}\left( V^n(\mathbf{x})\right)^{-1} \mathbf{y}\right\|^2_2}{ \| \mathbf{y} \|^2_2 },
\end{align*}
and so
\begin{equation}
\left\| \left(V^n(\mathbf{x})\right)^{-1}\right\|_{2\rightarrow M^n} = \left\| \left(M^n\right)^{1/2}\left(V^n(\mathbf{x})\right)^{-1} \right\|_2.
\end{equation}
Let $\mathbf{z} = \left(M^n\right)^{1/2} \mathbf{y}$. Then $\mathbf{z}\neq 0$ and
\begin{align*}
\frac{ \left\| V^n(\mathbf{x})\mathbf{y}\right\|^2_2}{\left\| \mathbf{y}\right\|^2_{M^n}} &= \frac{ \mathbf{y}^T\left(V^n(\mathbf{x})\right)^TV^n(\mathbf{x})\mathbf{y}}{ \mathbf{y}^T M^n \mathbf{y} } \\
&= \frac{ \left( V^n(\mathbf{x})\left(M^n\right)^{-1/2}\mathbf{z}\right)^T\left( V^n(\mathbf{x})\left(M^n\right)^{-1/2}\mathbf{z}\right)}{\mathbf{z}^T\mathbf{z}} \\
&= \frac{ \left\| V^n(\mathbf{x})\left(M^n\right)^{-1/2}\mathbf{z}\right\|^2_2}{ \| \mathbf{z} \|^2_2},
\end{align*}
and so
\begin{equation}
\left\| V^n(\mathbf{x})\right\|_{M^n\rightarrow 2} = \left\| V^n(\mathbf{x})\left(M^n\right)^{-1/2}\right\|_2.
\end{equation}
It follows that
\begin{align*}
\kappa_{M^n\rightarrow 2}\left(V^n(\mathbf{x})\right) &= \left\| V^n(\mathbf{x})\right\|_{M^n\rightarrow 2}\left\| \left(V^n(\mathbf{x})\right)^{-1}\right\|_{2\rightarrow M^n} \\
&= \left\| V^n(\mathbf{x})\left(M^n\right)^{-1/2}\right\|_2 \left\| \left(M^n\right)^{1/2}\left(V^n(\mathbf{x})\right)^{-1} \right\|_2 \\
&=  \left\| V^n(\mathbf{x})\left(M^n\right)^{-1/2}\right\|_2  \left\| \left(V^n(\mathbf{x})\left(M^n\right)^{-1/2}\right)^{-1}\right\|_2 \\
&= \kappa_2\left( V^n(\mathbf{x})\left(M^n\right)^{-1/2}\right).
\end{align*}
\end{proof}

To analyze $\kappa_{2}\left(V^n(\mathbf{x})\left(M^n\right)^{-1/2}\right)$, we will need reference to the Legendre polynomials, mapped from their typical home on $[-1,1]$ to $[0,1]$. For $0\leq j\leq n$, let $L^j(x)$ denote the Legendre polynomial of degree $n$ over $[0,1]$, scaled so that $L^n(1)=1$ and
\begin{equation}
\label{eq:LegendreL2norm}
\| L^j \|^2_{L^2} = \frac{1}{2j+1}.
\end{equation}
The Legendre--Vandermonde matrix is the $(n+1)\times (n+1)$ matrix $\widehat{V}^n(\mathbf{x})$ given by
\begin{equation}
\label{eq:LegendreVandermonde}
\widehat{V}^n_{ij}(\mathbf{x}) = L^j(\mathbf{x}_i).
\end{equation}
Given any polynomial $p$ of degree at most $n$, let $\mathbf{\Theta}(p)$ denote the vector of $n+1$ coefficients with respect to the Legendre basis. Define $T^n$ to be the $(n+1)\times (n+1)$ matrix satisfying $T^n\mathbf{\Theta}(p)= \mathbf{\Pi}(p)$ for all polynomials $p$ of degree at most $n$. In particular, we have the relationship
\begin{equation}
\label{eq:Vandermondeconversion}
\widehat{V}^n(\mathbf{x}) = V^n(\mathbf{x})T^n.
\end{equation}

Let $\{\lambda^n_j\}_{j=0}^n$ be the set of eigenvalues of $M^n$. In \cite{allenkirby2020mass}, it was shown that $M^n$ admits the spectral decomposition
\begin{equation}
\label{eq:spectral}
M^n = Q^n\Lambda^n\left(Q^n\right)^T,
\end{equation}
where
\begin{equation}
Q^n = T^n\diag\left( \sqrt{(2j+1)\lambda^n_j}\right)_{j=0}^n
\end{equation}
is an orthogonal matrix and $\Lambda^n = \diag\left(\lambda^n_j\right)_{j=0}^n$ contains the eigenvalues. This decomposition combined with Lemma~\ref{lem:kapparelation} and the transformation given in (\ref{eq:Vandermondeconversion}) suggest a relationship between the $M^n\rightarrow 2$ condition number of $V^n(x)$ and the $2$-norm condition number of $\widehat{V}^n(\mathbf{x})$ scaled by a diagonal matrix.

\begin{lemma}
\label{lem:kappa2Legendre}
\begin{equation}
\label{eq:kappa2Legendre}
\kappa_{M^n\rightarrow 2}\left(V^n(\mathbf{x})\right) = \kappa_2\left( \widehat{V}^n(\mathbf{x})\diag\left( \sqrt{2j+1}\right)_{j=0}^n\right).
\end{equation}
\end{lemma}

\begin{proof}
We have that
\begin{align*}
V^n(\mathbf{x})\left(M^n\right)^{-1/2} &= V^n(\mathbf{x})Q^n\left(\Lambda^n\right)^{-1/2}\left(Q^n\right)^T \\
&= V^n(\mathbf{x})T^n\diag\left(\sqrt{(2j+1)\lambda^n_j}\right)_{j=0}^n\diag\left(\frac{1}{\sqrt{\lambda^n_j}}\right)_{j=0}^n\left(Q^n\right)^T \\
&= \widehat{V}^n(\mathbf{x})\diag\left(\sqrt{2j+1}\right)_{j=0}^n\left(Q^n\right)^T.
\end{align*}
Since $Q^n$ is orthogonal, the result follows from Lemma~\ref{lem:kapparelation}.
\end{proof}

In \cite{gautschi1983condition}, Gautschi computed the condition number in the Frobenius norm $\| \cdot \|_F$ for Vandermonde matrices associated with families of orthonormal polynomials. Since the diagonal matrix in Lemma~\ref{lem:kappa2Legendre} contains the reciprocal of the $L^2$ norms of the Legendre polynomials, we can adapt his arguments to bound $\left(\widehat{V}^n(\mathbf{x})\diag\left( \sqrt{2j+1}\right)_{j=0}^n\right)^{-1}$ in the Frobenius norm. In addition, the Legendre polynomials have the special property that $|L^j(x)|\leq 1$ for all $0\leq x\leq 1$, and so we can bound the Frobenius norm of $\widehat{V}^n(\mathbf{x})\diag\left( \sqrt{2j+1}\right)_{j=0}^n$.

For each integer $0\leq j\leq n$, let $\ell^{j,n}$ be the $j^{\text{th}}$ Lagrange polynomial with respect to $\mathbf{x}$; that is,
\begin{equation}
\ell^{j,n}(x) = \prod_{i\in\{0,\dots,n\}\setminus\{j\}} \frac{x-\mathbf{x}_i}{\mathbf{x}_j-\mathbf{x}_i}.
\end{equation}
Define $\mathbf{w}^n\in\mathbb{R}^{n+1}$ by
\begin{equation}
\mathbf{w}^n_j = \| \ell^{j,n} \|_{L^2}.
\end{equation}

\begin{theorem}
\label{thm:kappaMto2}
\begin{equation}
\kappa_{M^n\rightarrow 2}(V^n(\mathbf{x})) \leq (n+1)^{3/2}\| \mathbf{w}^n \|_2.
\end{equation}
\end{theorem}

\begin{proof}
Since $|L^j(x)|\leq 1$ for all $0\leq x\leq 1$, we have that
\begin{align*}
\left\| \widehat{V}^n(\mathbf{x})\diag\left(\sqrt{2j+1}\right)_{j=0}^n\right\|_F &= \left( \sum_{i,j=0}^n (2j+1)\left(L^j(\mathbf{x}_i)\right)^2\right)^{1/2} \\
&\leq \left( \sum_{i,j=0}^n (2j+1) \right)^{1/2} \\
&= (n+1)^{3/2}.
\end{align*}
Since
\begin{equation}
\ell^{j,n}(\mathbf{x}_i) = \begin{cases} 1, & \text{if}\ i=j; \\ 0, & \text{if}\ i\neq j; \end{cases}
\end{equation}
we have that $\left(\widehat{V}^n(\mathbf{x})\right)^{-1}_{ij} = \mathbf{\Theta}\left(\ell^{j,n}\right)_i$. Therefore, by (\ref{eq:LegendreL2norm}), we have that
\begin{align*}
\| \mathbf{w}^n \|_2 &= \left( \sum_{j=0}^n \left(\mathbf{w}^n_j\right)^2\right)^{1/2} \\
&= \left(\sum_{j=0}^n \int_0^1 \left(\ell^{j,n}(x)\right)^2dx\right)^{1/2} \\
&= \left(\sum_{j=0}^n \int_0^1 \sum_{i,k=0}^n \mathbf{\Theta}(\ell^{j,n})_i\mathbf{\Theta}(\ell^{j,n})_kL^i(x)L^k(x)dx\right)^{1/2} \\
&= \left(\sum_{j=0}^n \sum_{i,k=0}^n \mathbf{\Theta}(\ell^{j,n})_i\mathbf{\Theta}(\ell^{j,n})_k \int_0^1 L^i(x)L^k(x)dx\right)^{1/2} \\
&= \left(\sum_{i,j=0}^n \mathbf{\Theta}(\ell^{j,n})^2_i \int_0^1 (L^i(x))^2dx\right)^{1/2} \\
&= \left(\sum_{i,j=0}^n \frac{\mathbf{\Theta}(\ell^{j,n})^2_i}{2i+1}\right)^{1/2} \\
&= \left\| \diag\left( \frac{1}{\sqrt{2i+1}}\right)_{i=0}^n \left(\widehat{V}^n(\mathbf{x})\right)^{-1}\right\|_F \\
&= \left\| \left(\widehat{V}^n(\mathbf{x})\diag\left(\sqrt{2j+1}\right)_{j=0}^n\right)^{-1}\right\|_F.
\end{align*}
Since the condition number in the 2 norm is bounded above by the condition number in the Frobenius norm, the result follows from Lemma~\ref{lem:kappa2Legendre}.
\end{proof}

\begin{figure}
\begin{center}
\begin{tikzpicture}
\begin{semilogyaxis}[legend style={at={(0.05,0.95)}, anchor=north west}, xlabel=$n$, ylabel=Condition Numbers, xmin=1, xmax=20, ymin=0]
\addplot [mark=square] table [x=n, y=k2LD, col sep=comma] {conditioning.dat};
\addplot [mark=triangle] table [x=n, y=ub, col sep=comma] {conditioning.dat};
\addplot [mark=o] table [x=n, y=k2V, col sep=comma] {conditioning.dat};
\legend{$\kappa_{M^n\rightarrow 2}(V^n(\mathbf{x}))$, Estimate, $\kappa_2(V^n(\mathbf{x}))$}
\end{semilogyaxis}
\end{tikzpicture}
\end{center}
\caption{Comparsion of the condition number of $V^n$ in the $M^n\rightarrow 2$ norm and the $2$ norm for $1\leq n\leq 20$, where $V^n$ is the matrix described in Subsection~\ref{ssec:equispaced}. We also include the estimate given in Theorem~\ref{thm:kappaMto2}. We use Lemma~\ref{lem:kappa2Legendre} to compute $\kappa_{M^n\rightarrow 2}(V^n(\mathbf{x}))$.}
\label{fig:kappa}
\end{figure}

\section{Higher dimensions}
\label{sec:dimensions}

Now, in the case of equispaced nodes on the $d$-simplex, we can use block-recursive structure as discussed in \cite{kirby2012fast} to give low-complexity algorithms for the inversion of simplicial Bernstein--Vandermonde matrices. This gives, at least for the equispaced lattice, an alternate algorithm for solving simplicial Bernstein interpolation problems to the one worked out in \cite{ainsworth2011bernstein}.

\subsection{Notation and Preliminaries}
\label{ssec:notation}

For an integer $d\geq 1$, let $S_d$ be a nondegenerate simplex in $\mathbb{R}^d$. Let $\{\mathbf{v}_i\}_{i=0}^d\subset \mathbb{R}^d$ be the vertices of $S_d$, and let $\{\mathbf{b}_i\}_{i=0}^d$ denote the barycentric coordinates of $S_d$. Each $\mathbf{b}_i$ is an affine map from $\mathbb{R}^d$ to $\mathbb{R}$ such that
\begin{equation}
\mathbf{b}_i(\mathbf{v}_j) = \begin{cases} 1, & \text{if}\ i=j; \\ 0, & \text{if}\ i\neq j; \end{cases}
\end{equation}
for each vertex $\mathbf{v}_j$. Each $\mathbf{b}_i$ is nonnegative on $S_d$, and
\begin{equation}
\sum_{i=0}^d \mathbf{b}_i = 1.
\end{equation}
A multiindex $\mathbf{\beta}$ of length $d+1$ is a $(d+1)$-tuple of nonnegative integers, written
\begin{equation}
\mathbf{\beta} = (\mathbf{\beta}_0,\dots,\mathbf{\beta}_d).
\end{equation}
The order of $\mathbf{\beta}$, denoted $|\mathbf{\beta}|$, is given by
\begin{equation}
|\mathbf{\beta}| = \sum_{i=0}^d \mathbf{\beta}_i.
\end{equation}
The factorial $\mathbf{\beta}!$ of a multiindex $\mathbf{\beta}$ is defined by
\begin{equation}
\mathbf{\beta}! = \prod_{i=0}^d \mathbf{\beta}_i.
\end{equation}
For a multiindex $\mathbf{\beta}$ of length $d+1$, denote by $\mathbf{\beta}'$ the multiindex of length $d$ given by
\begin{equation}
\mathbf{\beta}' = (\mathbf{\beta}_1, \dots, \mathbf{\beta}_d).
\end{equation}
Given a nonnegative integer $b$ and a multiindex $\mathbf{\beta}' = (\mathbf{\beta}_1,\dots, \mathbf{\beta}_d)$ of length $d$, define a new multiindex $b\vdash\mathbf{\beta}$ of length $d+1$ by
\begin{equation}
b\vdash\mathbf{\beta}' = (b,\mathbf{\beta}_1,\dots,\mathbf{\beta}_d).
\end{equation}
In particular,
\begin{equation}
\mathbf{\beta} = \mathbf{\beta}_0\vdash\mathbf{\beta}'.
\end{equation}
Multiindices have a natural partial ordering given by
\begin{equation}
\label{eq:ordering}
\mathbf{\beta} \leq \widetilde{\mathbf{\beta}}\qquad \text{if and only if}\qquad \mathbf{\beta}_i\leq \widetilde{\mathbf{\beta}}_i\quad \text{for all}\quad 0\leq i\leq d.
\end{equation}
The Bernstein polynomials of degree $n$ on the $d$-simplex $S_d$ are defined by
\begin{equation}
B^n_{\mathbf{\beta}} = \frac{n!}{\mathbf{\beta}!} \prod_{i=0}^d \mathbf{b}_i^{\mathbf{\beta}_i}.
\end{equation}
The complete set of Bernstein polynomials $\{B^n_{\mathbf{\beta}}\}_{|\mathbf{\beta}|=n}$ form a basis for polynomials in $d$ variables of complete degree at most $n$.

If $n_0\leq n$, then any polynomial expressed in the basis $\{B^{n_0}_{\mathbf{\beta}}\}_{|\mathbf{\beta}|=n_0}$ can also be expressed in the basis $\{B^n_{\mathbf{\beta}}\}_{|\mathbf{\beta}|=n}$. We denote by $E^{d,n_0,n}$ the $\binom{n+d}{d}\times\binom{n_0+d}{d}$ matrix that maps the coefficients of the degree $n_0$ representation to the coefficients of the degree $n$ representation. The matrix $E^{d,n_0,n}$ is sparse and can be applied matrix-free \cite{kirby2017fast}, if desired.

For a nonnegative integer $m$ and a set of distinct nodes $\{\mathbf{x}^m_{\mathbf{\alpha}}\}_{|\mathbf{\alpha}|=m}\subset S_d$, define the Bernstein--Vandermonde matrix $V^{d,m,n}$ to be the $\binom{m+d}{d}\times\binom{n+d}{d}$ matrix given by
\begin{equation}
V^{d,m,n}_{\mathbf{\alpha}\mathbf{\beta}} = B^n_{\mathbf{\beta}}(\mathbf{x}^m_{\mathbf{\alpha}})
\end{equation}

\begin{theorem}
\label{thm:elev}
Let $d,m,n_0$, and $n$ be nonnegative integers with $d\geq 1$ and $n_0\leq n$, and let $\{\mathbf{x}^m_{\mathbf{\alpha}}\}_{|\mathbf{\alpha}|=m}$ be distinct nodes contained in the $d$-simplex $S_d$. Then
\begin{equation}
\label{eq:elev}
V^{d,m,n}E^{d,n_0,n}=V^{d,m,n_0}.
\end{equation}
\end{theorem}

\begin{proof}
Given a polynomial $p\in\Span\{B^{n_0}_{\mathbf{\beta}}\}_{|\mathbf{\beta}|=n_0}$, the matrix $V^{d,m,n_0}$ evaluates $p$ at each of the nodes $\mathbf{x}^m_{\mathbf{\alpha}}$. On the other hand, the matrix $V^{d,m,n}E^{d,n_0,n}$ evaluates the degree $n$ representation of $p$ at each of the nodes $\mathbf{x}^m_{\mathbf{\alpha}}$. Since we are evaluating at the same nodes and the polynomial has not been modified, both evaluations must give the same result.
\end{proof}

The partial ordering (\ref{eq:ordering}) implies a natural way to order the entries of $V^{d,m,n}$ and also imposes a block structure on $V^{d,m,n}$ by dividing the matrix into sections where $\mathbf{\alpha}_0$ and $\mathbf{\beta}_0$ are constant. For integers $0\leq \mathbf{\alpha}_0\leq m$ and $0\leq \mathbf{\beta}_0\leq n$, let $V^{d,m,n}_{\mathbf{\alpha}_0\mathbf{\beta}_0}$ denote the $\binom{m-\mathbf{\alpha}_0+d}{d}\times\binom{n-\mathbf{\beta}_0+d}{d}$ submatrix of $V^{d,m,n}$ whose entries satisfy
\begin{equation}
\left(V^{d,m,n}_{\mathbf{\alpha}_0\mathbf{\beta}_0}\right)_{\mathbf{\alpha}'\mathbf{\beta}'} = V^{d,m,n}_{\left(\mathbf{\alpha}_0\vdash\mathbf{\alpha}'\right)\left(\mathbf{\beta}_0\vdash\mathbf{\beta}'\right)}.
\end{equation}

\subsection{Degree Reduction}
\label{ssec:reduction}

We now consider the case of equispaced nodes on the $d$-simplex $S^d$. For this case, the entries of the Bernstein--Vandermonde matrix are given by
\begin{equation}
\label{eq:entries}
V^{d,m,n}_{\mathbf{\alpha}\mathbf{\beta}} = \frac{n!}{\mathbf{\beta}!}\frac{1}{m^n} \prod_{i=0}^d \mathbf{\alpha}_i^{\mathbf{\beta}_i}.
\end{equation}
This representation of the entries allows us to represent the submatrix $V^{d,m,n}_{\mathbf{\alpha}_0\mathbf{\beta}_0}$ in terms of a lower-dimensional Bernstein--Vandermonde matrix.

\begin{theorem}
\label{thm:reduction}
Let $d>1$ and $m,n\geq 0$ be integers, and let $\mathbf{\alpha}_0,\mathbf{\beta}_0$ be integers with $0\leq\mathbf{\alpha}_0\leq m$ and $0\leq\mathbf{\beta}_0\leq n$. Then
\begin{equation}
\label{eq:reduction}
V^{d,m,n}_{\mathbf{\alpha}_0\mathbf{\beta}_0} = m_{\mathbf{\alpha}_0\mathbf{\beta}_0}V^{d-1,m-\mathbf{\alpha}_0,n-\beta_0},
\end{equation}
where
\begin{equation}
m_{\mathbf{\alpha}_0\mathbf{\beta}_0} = \binom{n}{\mathbf{\beta}_0}(\mathbf{\alpha}_0/m)^{\mathbf{\beta}_0}(1-\mathbf{\alpha}_0/m)^{n-\mathbf{\beta}_0}.
\end{equation}
\end{theorem}

\begin{proof}
If $\mathbf{\alpha}_0=m$ and $\mathbf{\beta}_0\neq n$, then both sides equal zero and we are done. Otherwise, by (\ref{eq:entries}), we have that
\begin{equation}
V^{d-1,m-\mathbf{\alpha}_0,n-\mathbf{\beta}_0}_{\mathbf{\alpha}'\mathbf{\beta}'} = \frac{(n-\mathbf{\beta}_0)!}{\mathbf{\beta}'!}\frac{1}{(m-\mathbf{\alpha}_0)^{n-\mathbf{\beta}_0}}\prod_{i=1}^d \mathbf{\alpha}_i^{\mathbf{\beta}_i}.
\end{equation}
On the other hand,
\begin{align*}
\left(V^{d,m,n}_{\mathbf{\alpha}_0\mathbf{\beta}_0}\right)_{\mathbf{\alpha}'\mathbf{\beta}'} &= V^{d,m,n}_{\mathbf{\alpha}\mathbf{\beta}} \\
&= \frac{n!}{\mathbf{\beta}!}\frac{1}{m^n} \prod_{i=0}^d \mathbf{\alpha}_i^{\mathbf{\beta}_i} \\
&= \frac{n!}{\mathbf{\beta}_0!(n-\mathbf{\beta}_0)!}\frac{(n-\mathbf{\beta}_0)!}{\mathbf{\beta}'!} \frac{\mathbf{\alpha}_0^{\mathbf{\beta}_0}}{m^{\mathbf{\beta}_0}} \frac{(m-\mathbf{\alpha}_0)^{n-\mathbf{\beta}_0}}{m^{n-\mathbf{\beta}_0}} \frac{1}{(m-\mathbf{\alpha}_0)^{n-\mathbf{\beta}_0}}\prod_{i=1}^d \mathbf{\alpha}_i^{\mathbf{\beta}_i},
\end{align*}
where we have separated terms containing $\mathbf{\alpha}_0$ and $\mathbf{\beta}_0$ and multiplied and divided  by $(n-\mathbf{\beta}_0)!$ and $(m-\mathbf{\alpha}_0)^{n-\mathbf{\beta}_0}$. Since
\begin{equation}
\frac{n!}{\mathbf{\beta}_0!(n-\mathbf{\beta}_0)!}\frac{\mathbf{\alpha}_0^{\mathbf{\beta}_0}}{m^{\mathbf{\beta}_0}}\frac{(m-\alpha_0)^{n-\mathbf{\beta}_0}}{m^{n-\mathbf{\beta}_0}} = \binom{n}{\mathbf{\beta}_0}(\mathbf{\alpha}_0/m)^{\mathbf{\beta}_0}(1-\mathbf{\alpha}_0/m)^{n-\mathbf{\beta}_0},
\end{equation}
we have the desired result.
\end{proof}

The fact that the blocks are multiples of lower-dimensional matrices is analogous to what has been observed for other matrices related to Bernstein polynomials. For example, the Bernstein--Vandermonde matrix associated with the zeroes of Legendre polynomials has a similar structure \cite{kirby2012fast}, and so does the Bernstein mass matrix \cite{kirby2017fast}.

We recognize that $m_{\mathbf{\alpha}_0\mathbf{\beta}_0}$ describes a univariate Bernstein polynomial of degree $n$ being evaluated at equispaced points. Therefore, if $V^n$ is the one-dimensional Bernstein--Vandermonde matrix associated with equispaced nodes as described in Subsection~\ref{ssec:equispaced}, then we have that $V^{d,n,n}$ admits the block structure
\begin{equation}
\label{eq:block}
V^{d,n,n} =
\begin{pmatrix}
V^n_{00}V^{d-1,n,n} & V^n_{01}V^{d-1,n,n-1} & \cdots & V^n_{0n}V^{d-1,n,0} \\
V^n_{10}V^{d-1,n-1,n} & V^n_{11}V^{d-1,n-1,n-1} & \cdots & V^n_{1n}V^{d-1,n-1,0} \\
\vdots & \vdots & \ddots & \vdots \\
V^n_{n0}V^{d-1,0,n} & V^n_{n1}V^{d-1,0,n-1} & \cdots & V^n_{nn}V^{d-1,0,0}
\end{pmatrix}.
\end{equation}
We can use this block structure and Theorem~\ref{thm:elev} to perform block Gaussian elimination on $V^{d,n,n}$. This method was used in \cite{kirby2017fast} to obtain the block $LU$ decomposition of the Bernstein mass matrix. In the same way, we have the following:

\begin{theorem}
\label{thm:LU}
Let $V^n$ be the one-dimensional Bernstein--Vandermonde matrix associated with equispaced nodes as described in Subsection~\ref{ssec:equispaced}. Suppose $V^n=L^nU^n$ is the $LU$ decomposition of $V^n$. Then
\begin{equation}
V^{d,n,n}=L^{d,n}U^{d,n},
\end{equation}
where $L^{d,n}$ is the block lower triangular matrix with blocks given by
\begin{equation}
L^{d,n}_{\mathbf{\alpha}_0\mathbf{\beta}_0} = L^n_{\mathbf{\alpha}_0\mathbf{\beta}_0}V^{d-1,n-\mathbf{\alpha}_0,n-\mathbf{\beta}_0}
\end{equation}
and $U^{d,n}$ is the block upper triangular matrix with blocks given by
\begin{equation}
U^{d,n}_{\mathbf{\alpha}_0\mathbf{\beta}_0} = U^n_{\mathbf{\alpha}_0\mathbf{\beta}_0}E^{d-1,n-\mathbf{\beta}_0,n-\mathbf{\alpha}_0}.
\end{equation}
\end{theorem}

\begin{proof}
By Theorem~\ref{thm:elev}, if $0\leq \mathbf{\alpha}_0,\mathbf{\beta}_0\leq n$ and $\bm{\gamma}_0\leq \mathbf{\beta}_0$, then
\begin{align*}
L^{d,n}_{\mathbf{\alpha}_0\bm{\gamma}_0}U^{d,n}_{\bm{\gamma}_0\mathbf{\beta}_0} &=  L^n_{\mathbf{\alpha}_0\bm{\gamma}_0}U^n_{\bm{\gamma}_0\mathbf{\beta}_0}V^{d-1,n-\mathbf{\alpha}_0,n-\bm{\gamma}_0}E^{d-1,n-\mathbf{\beta}_0,n-\bm{\gamma}_0} \\
&= L^n_{\mathbf{\alpha}_0\bm{\gamma}_0}U^n_{\bm{\gamma}_0\mathbf{\beta}_0} V^{d-1,n-\mathbf{\alpha}_0,n-\mathbf{\beta}_0}.
\end{align*}
Therefore,
\begin{align*}
\left(L^{d,n}U^{d,n}\right)_{\mathbf{\alpha}_0\mathbf{\beta}_0} &= \sum_{\bm{\gamma}_0=0}^{\mathbf{\beta}_0} L^{d,n}_{\mathbf{\alpha}_0\bm{\gamma}_0}U^{d,n}_{\bm{\gamma}_0\mathbf{\beta}_0} \\
&= \left( \sum_{\bm{\gamma}_0=0}^{\mathbf{\beta}_0} L^n_{\mathbf{\alpha}_0\bm{\gamma}_0}U^n_{\bm{\gamma}_0\mathbf{\beta}_0}\right) V^{d-1,n-\mathbf{\alpha}_0,n-\mathbf{\beta}_0} \\
&= V^n_{\mathbf{\alpha}_0\mathbf{\beta}_0}V^{d-1,n-\mathbf{\alpha}_0,n-\mathbf{\beta}_0} \\
&= V^{d,n,n}_{\mathbf{\alpha}_0\mathbf{\beta}_0}.
\end{align*}
\end{proof}

\section{Numerical results}

Now, we consider the accuracy of the methods described above on several problems. For all of the problems, we chose random solution vectors, computed the right-hand side by matrix multiplication, and then attempted to recover the solution. The numerical results are run in double precision arithmetic on a 2014 Macbook Air running macOS 10.13 and using Python 2.7.13. Cholesky factorization and FFTs are performed using the \texttt{numpy} (v1.12.0) function calls.  Also, because our code is a mix of pure Python and low-level compiled libraries, timings are not terribly informative.  Consequently, we focus on assessing the stability and accuracy of our methods.  If future work leads to more stable fast algorithms, greater care will be afforded to tuning our implementations for performance.

In Figure~\ref{fig:equispacedinfo}, we consider the case of equispaced nodes. When considering the Euclidean norm of the error (Figure~\ref{subfig:equispaced2err}), the $LU$ factorization and the Newton algorithm have the best perfomance, followed by the DFT-based application and multiplying by the inverse. We observe that the $LU$ and Newton methods have comparable performance, as do the DFT-based algorithm and multiplying by the inverse. This is in contrast to what was observed in \cite{allenkirby2020mass}, where a similar DFT-based algorithm quickly became unstable. The same behavior can be observed when comparing the Euclidean norm of the residual (Figure~\ref{subfig:equispaced2res}), although some separation does occur between the Newton method and the $LU$ factorization. Since the solution vector can be viewed as the Bernstein coefficients of the interpolation polynomial, we also measure the $L^2$ difference between the exact and computed solutions (Figure~\ref{subfig:equispacedMerr}); equivalently, we measure the relative $M^n$ error, where $M^n$ is the Bernstein mass matrix given in (\ref{eq:mass}). All four solution methods have very similar behavior in the $M^n$ norm as they do when considering the Euclidean norm of the residual.

To ensure that the behavior of the solution methods does not depend on the choice of nodes, we also considered the case where the nodes $\mathbf{x}_j$ are randomly selected from $[j/(n+1),(j+1)/(n+1))$ for each $0\leq j\leq n$ (Figure~\ref{fig:randominfo}); however, there was no significant difference in the quantities measured between this case and the equispaced case.

We also used the block $LU$ decomposition given in Theorem~\ref{thm:LU} combined with the one-dimensional $LU$ algorithm to solve the interpolation problem for equispaced nodes on the $d$-simplex for $d=2$ and $d=3$ (Figure~\ref{fig:blockinfo}). Due to the recursive nature of the algorithm, the quantities measured are very similar to the ones observed for equispaced nodes.

\begin{figure}
\centering
\begin{subfigure}{0.475\linewidth}
\centering
\begin{tikzpicture}[scale=0.7]
\begin{semilogyaxis}[legend style={at={(0.05,0.95)}, anchor=north west}, xlabel=$n$, ylabel=$\| \mathbf{c}-\widehat{\mathbf{c}} \|_2 / \| \mathbf{c} \|_2$, xmin=1, xmax=20, ymin=1e-18]
\addplot [mark=square] table [x=n, y=BezoutL2err, col sep=comma] {equispaced_info.dat};
\addplot [mark=triangle] table [x=n, y=DFTL2err, col sep=comma] {equispaced_info.dat};
\addplot [mark=diamond] table [x=n, y=LUL2err, col sep=comma] {equispaced_info.dat};
\addplot[mark=o] table [x=n, y=NewtonL2err, col sep=comma]{equispaced_info.dat};
\legend{B\'{e}zout, DFT, $LU$, Newton}
\end{semilogyaxis}
\end{tikzpicture}
\caption{Error in the 2-norm.}
\label{subfig:equispaced2err}
\end{subfigure}
\hfill
\begin{subfigure}{0.475\linewidth}
\centering
\begin{tikzpicture}[scale=0.7]
\begin{semilogyaxis}[legend style={at={(0.05,0.95)}, anchor=north west}, xlabel=$n$, ylabel=$\| \mathbf{c}-\widehat{\mathbf{c}} \|_{M^n} / \| \mathbf{c} \|_{M^n}$, xmin=1, xmax=20, ymin=1e-18]
\addplot [mark=square] table [x=n, y=BezoutMerr, col sep=comma] {equispaced_info.dat};
\addplot [mark=triangle] table [x=n, y=DFTMerr, col sep=comma] {equispaced_info.dat};
\addplot [mark=diamond] table [x=n, y=LUMerr, col sep=comma] {equispaced_info.dat};
\addplot[mark=o] table [x=n, y=NewtonMerr, col sep=comma]{equispaced_info.dat};
\legend{B\'{e}zout, DFT, $LU$, Newton}
\end{semilogyaxis}
\end{tikzpicture}
\caption{Error in the $M^n$ norm.}
\label{subfig:equispacedMerr}
\end{subfigure}
\vskip\baselineskip
\begin{subfigure}{\linewidth}
\centering
\begin{tikzpicture}[scale=0.7]
\begin{semilogyaxis}[legend style={at={(0.05,0.95)}, anchor=north west}, xlabel=$n$, ylabel=$\| V^n\widehat{\mathbf{c}}-\mathbf{b} \|_2$, xmin=1, xmax=20, ymin=1e-18]
\addplot [mark=square] table [x=n, y=Bezoutres, col sep=comma] {equispaced_info.dat};
\addplot [mark=triangle] table [x=n, y=DFTres, col sep=comma] {equispaced_info.dat};
\addplot [mark=diamond] table [x=n, y=LUres, col sep=comma] {equispaced_info.dat};
\addplot[mark=o] table [x=n, y=Newtonres, col sep=comma]{equispaced_info.dat};
\legend{B\'{e}zout, DFT, $LU$, Newton}
\end{semilogyaxis}
\end{tikzpicture}
\caption{Residual in the 2-norm.}
\label{subfig:equispaced2res}
\end{subfigure}
\caption{Error/residual in using the methods described in Section~\ref{sec:apply} to solve $V^n\mathbf{c}=\mathbf{b}$ for $1\leq n\leq 20$, where $V^n$ is the matrix described in Subsection~\ref{ssec:equispaced} and $\mathbf{b}$ is a random vector in $[-1,1]^{n+1}$. B\'{e}zout refers to Corollary~\ref{cor:bezout}, DFT refers to Corollary~\ref{cor:equispacedinverse}, $LU$ refers to $LU$ decomposition of $V^n$, and Newton refers to the Ainsworth--Sanchez algorithm. We use $\widehat{\mathbf{c}}$ to denote the computed solution.}
\label{fig:equispacedinfo}
\end{figure}

\begin{figure}
\centering
\begin{subfigure}{0.475\linewidth}
\centering
\begin{tikzpicture}[scale=0.7]
\begin{semilogyaxis}[legend style={at={(0.05,0.95)}, anchor=north west}, xlabel=$n$, ylabel=$\| \mathbf{c}-\widehat{\mathbf{c}} \|_2 / \| \mathbf{c} \|_2$, xmin=1, xmax=20, ymin=1e-18]
\addplot [mark=square] table [x=n, y=BezoutL2err, col sep=comma] {random_info.dat};
\addplot [mark=triangle] table [x=n, y=DFTL2err, col sep=comma] {random_info.dat};
\addplot [mark=diamond] table [x=n, y=LUL2err, col sep=comma] {random_info.dat};
\addplot[mark=o] table [x=n, y=NewtonL2err, col sep=comma]{random_info.dat};
\legend{B\'{e}zout, DFT, $LU$, Newton}
\end{semilogyaxis}
\end{tikzpicture}
\caption{Error in the 2-norm.}
\label{subfig:random2err}
\end{subfigure}
\hfill
\begin{subfigure}{0.475\linewidth}
\centering
\begin{tikzpicture}[scale=0.7]
\begin{semilogyaxis}[legend style={at={(0.05,0.95)}, anchor=north west}, xlabel=$n$, ylabel=$\| \mathbf{c}-\widehat{\mathbf{c}} \|_{M^n} / \| \mathbf{c} \|_{M^n}$, xmin=1, xmax=20, ymin=1e-18]
\addplot [mark=square] table [x=n, y=BezoutMerr, col sep=comma] {random_info.dat};
\addplot [mark=triangle] table [x=n, y=DFTMerr, col sep=comma] {random_info.dat};
\addplot [mark=diamond] table [x=n, y=LUMerr, col sep=comma] {random_info.dat};
\addplot[mark=o] table [x=n, y=NewtonMerr, col sep=comma]{random_info.dat};
\legend{B\'{e}zout, DFT, $LU$, Newton}
\end{semilogyaxis}
\end{tikzpicture}
\caption{Error in the $M^n$ norm.}
\label{subfig:randomMerr}
\end{subfigure}
\vskip\baselineskip
\begin{subfigure}{\linewidth}
\centering
\begin{tikzpicture}[scale=0.7]
\begin{semilogyaxis}[legend style={at={(0.05,0.95)}, anchor=north west}, xlabel=$n$, ylabel=$\| V^n(\mathbf{x})\widehat{\mathbf{c}}-\mathbf{b} \|_2$, xmin=1, xmax=20, ymin=1e-18]
\addplot [mark=square] table [x=n, y=Bezoutres, col sep=comma] {random_info.dat};
\addplot [mark=triangle] table [x=n, y=DFTres, col sep=comma] {random_info.dat};
\addplot [mark=diamond] table [x=n, y=LUres, col sep=comma] {random_info.dat};
\addplot[mark=o] table [x=n, y=Newtonres, col sep=comma]{random_info.dat};
\legend{B\'{e}zout, DFT, $LU$, Newton}
\end{semilogyaxis}
\end{tikzpicture}
\caption{Residual in the 2-norm.}
\label{subfig:random2res}
\end{subfigure}
\caption{Error/residual in using the methods described in Section~\ref{sec:apply} to solve $V^n(\mathbf{x})\mathbf{c}=\mathbf{b}$ for $1\leq n\leq 20$, where $V^n(\mathbf{x})$ is the Bernstein--Vandermonde matrix associated to $\mathbf{x}$, the nodes $\mathbf{x}_j$ are randomly selected from $[j/(n+1),(j+1)/(n+1))$ for $0\leq j\leq n$, and $\mathbf{b}$ is a random vector in $[-1,1]^{n+1}$. B\'{e}zout refers to Corollary~\ref{cor:bezout}, DFT refers to Theorem~\ref{thm:factoredinverse}, $LU$ refers to $LU$ decomposition of $V^n$, and Newton refers to the Ainsworth--Sanchez algorithm. We use $\widehat{\mathbf{c}}$ to denote the computed solution.}
\label{fig:randominfo}
\end{figure}

\begin{figure}
\centering
\begin{subfigure}{0.475\linewidth}
\centering
\begin{tikzpicture}[scale=0.7]
\begin{semilogyaxis}[legend style={at={(0.05,0.95)}, anchor=north west}, xlabel=$n$, ylabel=$\| \mathbf{c}-\widehat{\mathbf{c}} \|_2 / \| \mathbf{c} \|_2$, xmin=1, xmax=20, ymin=1e-18]
\addplot [mark=square] table [x=n, y=2dL2err, col sep=comma] {blockLU_info.dat};
\addplot [mark=triangle] table [x=n, y=3dL2err, col sep=comma] {blockLU_info.dat};
\legend{$d=2$, $d=3$}
\end{semilogyaxis}
\end{tikzpicture}
\caption{Error in the 2-norm.}
\label{subfig:block2err}
\end{subfigure}
\hfill
\begin{subfigure}{0.475\linewidth}
\centering
\begin{tikzpicture}[scale=0.7]
\begin{semilogyaxis}[legend style={at={(0.05,0.95)}, anchor=north west}, xlabel=$n$, ylabel=$\| \mathbf{c}-\widehat{\mathbf{c}} \|_{M^{d,n}} / \| \mathbf{c} \|_{M^{d,n}}$, xmin=1, xmax=20, ymin=1e-18]
\addplot [mark=square] table [x=n, y=2dMerr, col sep=comma] {blockLU_info.dat};
\addplot [mark=triangle] table [x=n, y=3dMerr, col sep=comma] {blockLU_info.dat};
\legend{$d=2$, $d=3$}
\end{semilogyaxis}
\end{tikzpicture}
\caption{Error in the $M^{d,n}$ norm.}
\label{subfig:blockMerr}
\end{subfigure}
\vskip\baselineskip
\begin{subfigure}{\linewidth}
\centering
\begin{tikzpicture}[scale=0.7]
\begin{semilogyaxis}[legend style={at={(0.05,0.95)}, anchor=north west}, xlabel=$n$, ylabel=$\| V^{d,n,n}\widehat{\mathbf{c}}-\mathbf{b} \|_2$, xmin=1, xmax=20, ymin=1e-18]
\addplot [mark=square] table [x=n, y=2dres, col sep=comma] {blockLU_info.dat};
\addplot [mark=triangle] table [x=n, y=3dres, col sep=comma] {blockLU_info.dat};
\legend{$d=2$, $d=3$}
\end{semilogyaxis}
\end{tikzpicture}
\caption{Residual in the 2-norm.}
\label{subfig:block2res}
\end{subfigure}
\caption{Error/residual in using the block $LU$ decomposition given in Theorem~\ref{thm:LU} to solve $V^{d,n,n}\mathbf{c}=\mathbf{b}$ for $1\leq n\leq 20$ and $d=2,3$, where $V^{d,n,n}$ is the Bernstein--Vandermonde matrix given in (\ref{eq:entries}) and $\mathbf{b}$ is a random vector in $[-1,1]^{\binom{n+d}{d}}$. We use $\widehat{\mathbf{c}}$ to denote the computed solution.}
\label{fig:blockinfo}
\end{figure}

\section{Conclusion}

We have studied several algorithms for the inversion of the univariate Bernstein--Vandermonde matrix. These algorithms, while less stable than algorithms discovered previously, provide insight into the structure of the Bernstein--Vandermonde matrix and are remarkably similar to algorithms derived for the Bernstein mass matrix. In addition, we have used a block $LU$ decomposition of the Bernstein--Vandermonde matrix corresponding to equispaced nodes on the $d$-simplex to give a recursive, block-structured algorithm with comparable accuracy to the one-dimensional algorithm. Moreover, we have given a new perspective on the conditioning of the Bernstein--Vandermonde matrix, indicating that the interpolation problem is better-conditioned with respect to the $L^2$ norm than the Euclidean norm. In the future, we hope to expand this perspective to other polynomial problems and continue the development of fast and accurate methods for problems involving Bernstein polynomials.

\clearpage

\bibliographystyle{plain}

redpajama_set_name: RedPajamaArXiv


text:
THE JUNGLE BOOK's Justin Marks May Pen TOP GUN 2 Script
by Perri Nemiroff September 9, 2014
We've been talking about a Top Gun sequel for years and years now, but it looks as though we might actually have some legitimate movement on the project courtesy of Jerry Bruckheimer's move from Disney to Paramount. It's being reporting that Top Gun 2 may have just found a screenwriter, The Jungle Book scribe, Justin Marks.
Apparently Tom Cruise will return as Maverick and take part in a storyline that highlights the relevance of old fashioned pilots in a war environment loaded with high-tech drones. Hit the jump for more on the plans for Top Gun 2.
According to THR, Marks isn't locked yet, but he is currently in negotiations to pen the script. We've also got Skydance's David Ellison on board to produce with Dana Goldberg stepping in as an executive producer. Peter Craig wrote a draft of the script a couple of years ago, but after Tony Scott's passing, the project stalled.
After all these years, it's impossible not to be wary of Top Gun 2's potential, but, then again, if they haven't stopped pushing for it nearly 30 years after the original's release, it seems unlikely that they ever will. Plus, the whole drone warfare vs. traditional fighter pilot scenario seems like an appropriate way to go. It's topical and would give our main man something fresh and interesting to contend with.
But even though Bruckheimer does seem determined to push this thing forward, it's best to assume that Top Gun 2 is still a long way off. The Jungle Book is in production so presumably Marks could jump into this one pretty fast (unless one of the other projects he's attached to is the real deal), but even if he churns out a script sooner rather than later, there's always a chance that Bruckheimer and co. won't go for it. Hopefully that's not the case, but after all this waiting, I'm holding out for a green light until I take this one seriously.
Top Gun 2 Justin Marks
THE FINEST HOURS Starring Chris Pine, Casey Affleck and Holliday Grainger Nearing…
THE FORGER Clip: John Travolta and Tye Sheridan Share a Deep Father/Son…
• Dana Goldberg • David Ellison • Jerry Bruckheimer • Justin Marks • Movie News • Tom Cruise • Top Gun • Top Gun 2

redpajama_set_name: RedPajamaCommonCrawl


text:
Brunt väder betyder gråväder eller storm med dimma. Uttrycket nämns av en del skalder i samband med de lovsjungna kungarnas resor över Östersjön. "För brun storm till Sigtuna" styr Magnus den gode sitt skepp från Ryssland.

Källor
 Åke Ohlmarks, Fornnordiskt lexikon (Tiden, 1995).

Meteorologi

redpajama_set_name: RedPajamaWikipedia


Part of a huge ongoing charity drive, participants from across the games industry are invited to get involved providing a constant stream of support for GamesAid as well as adding some extra excitement to the start of the Premier League Season.
Sign up to the Last Man Standing Challenge footie challenge now!
It's just £10 to take part, with half the generated money going directly to GamesAid and the rest going straight into a prize fund. More people means more money for charity and also more money for the winners, so everyone can get involved.
To sign up simply follow this link to the Last Man Standing Challenge.
You will have to pay by card but your details will NOT be known to us or other members. There will be updates on Aardvark Swift's Facebook page, so look out for those, too.
Each week you pick a Premiership team to win, if your team wins you progress to the next week, if you draw or lose you're out! The competition will start when the 2014/15 Premier League season gets underway on the 16th August, with a new round starting every few weeks once everyone's been knocked out and a winner declared.
Entries HAVE to be in by 11:45 on the 16th August so make sure you don't miss out on earning GamesAid (and yourself) a few quid! Strictly over 18's only.
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One theme of 2017 was the internationalization of ID. Meanwhile we are colonizing the mainstream media, which, revile us as they may, can't seem to leave us alone.
Rather than get all caught up in esoteric musing, I suggest an entry level test for determining whether an entity has any moral value.


It's important to heat your QuadFusion print head to 235C and allow it to sit for several minuets BEFORE your first print. This will allow you to not only determine if the print head has been connected properly, but it will also allow it to prime for it's first use.
To do this simply set the temperature to 235C for Tool 0 in the upper left area of your Duet Web Control console.
Since this is the VERY first time the QuadFusion head has been heated, it may emit an odor. This odor should be slight and it should dissipate within a few minutes. Please make sure your work area is well ventilated during the first heating of your QuadFusion print head.
First, we will have to upload the G-code file we produced in the previous guide. Connect to the Duet Web Console on the printer. Then, press the G-code Files button and then press the Upload G-code File(s) button as shown in the image above. This will open a window which will allow you to select the G-code file you sliced. Open the file and wait for it to upload.
Once the file is uploaded, you can click the file in order to print it. A window will pop-up confirming that you are about to print something. Click Yes.
Now, the print has started. First, the printer will reach it's designated printing temperature that you set in Cura. This might take a while and the printer will not start printing until all components have reached temperature. When the print starts the Duet Web Console will shift over to the Print Status tab on the Duet Web Console. This tab allows you to monitor certain settings of the print. Two of the functions on this tab are very useful: Z Baby Stepping and Speed Factor , these allow you to change the height of the Z and speed while printing. If you notice that a layer is not sticking, try to reduce the speed.
Your print should now be able to run to completion.


As the LPGA touches down on U.S. soil for the first time in 2018, U.S.-born players already have won three of the season's first four stops.
That's the best start by the American faction since 2007, when Paula Creamer, Stacy Prammanasudh and Meaghan Francella swept the first three.
You have to go back 2 ½ years, in fact, to find the last time U.S. players had three wins in any stretch of four tournaments. It was summer 2015 when Kris Tamulis (Alabama), Jessica Korda (Malaysia) and Lexi Thompson (South Korea) teamed up, sandwiched around Lydia Ko's victory at the Evian Championship.
A win by an American this week would mark the first USA hat trick since 2014 when Wie (U.S. Women's Open), Stacy Lewis (Arkansas Championship), and Mo Martin (Women's British Open) posed with a trophy three weeks in a row.


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text:
From Oct 15 - 22 VOTE
Regional Councillor - Ward 2
Why Am I Running?
Tracy MacCharles. Former Ontario MPP and Cabinet Minister
I have known Gary Strange since our high school days. When I was first elected MPP for Pickering - Scarborough East it was very evident that Gary was active in our community and cared about the people in our community. Gary has volunteered his time and expertise to many initiatives including as an active member and President with the Ajax-Pickering Board of Trade. He has been a tireless community leader with groups such as Rotary and with the Durham West Arts Centre team to help make the Durham Performing and Visual Arts Center a reality. Gary is always ready to give of his time to Pickering organizations that help make our community stronger. Gary has also been a consistent advocate for Pickering's fair share of infrastructure and transit Investments. He has a good working relationship with local businesses, elected officials at all levels of government and community groups. Gary believes in strong public service and will be strong representative for Ward 2 and Pickering!
The definition of a volunteer who steps up, time and time again, to make his community, and country, a better place. In addition to all he does for Durham and, specifically, the City of Pickering, he is a tireless advocate for increasing support available for our ill and injured Veterans, First Responders and their families — our nations finest citizens. If ever a time, we could use a lot more people in politics like Gary Strange.
Peter Vanderyagt, Past Chair of the Transportation and Infrastructure Discussion Group
Gary was an integral part of our transportation and infrastructure group. This committee initiated by the Hon. Tracy MacCharles MPP, consisted of representatives from municipal and provincial government, business leaders, members of Metrolinx, TTC and Durham Transit. Raising his family in Pickering, operating a successful insurance business, and serving as the president of the Ajax Pickering Board of Trade, all provided Gary with significant insight into the needs of the Durham Region. He was very articulate in outlining some of the challenges of moving people, goods and services throughout the region. He was also one of those members who always had suggestions to offer when discussing solutions to issues being raised at the table. In particular, he often provided our committee with written documentation to highlight these issues and possible solutions. Having operated his own business, he was aware that there is no such thing as an 8 hour work day. Gary could be reached by cell phone throughout the day and evening to respond to enquiries about his proposals. Frequently our followup conversations took place while he was driving about the city as he made the most of his time balancing his commitments and responsibilities. I know that once Gary makes a commitment to complete a task it will be done. The people of Durham can depend on Gary to do his job with integrity,intensity and accountability.
IN THIS FOR YOU!
Gary Strange © 2020 All rights reserved.


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NEWSLETTER No 8 JULY 1994
THE EVACUEES' SENTIMENTAL JOURNEY
The eyes of the media will be on Dunning next September 2 and 3 when the Society sponsors a first ever Evacuees' Reunion. It will be 55 years since World War II was declared and 300 city children were suddenly evacuated here and billeted in local homes. Probably only 20 to 30 of the surviving evacuees, plus spouses, will be attending and we're asking members and other villagers to consider opening their homes and their hearts to these visitors for a night or two, just as Dunning took in these evacuees 55 years ago.
The event will start on Friday when, for the cameras and the schoolchildren, the evacuees will be lined up in the school playground to meet their hosts. The visitors will then go to local schools to tell pupils of their wartime experiences and in the evening will meet at the School with Historical Society members. Saturday will be taken up with nostalgic tours of the village and an evening Big Band dance at the Hall to which all the village is invited. If you'd be prepared to provide a bed for an ex-evacuee, would you please contact Nancy Hurry at 684 385 or Rita Laing 684 484.
MUSEUM OF MEMORIES: YOU'RE INVITED TO LEND ITEMS AND TO ATTEND A JULY 1-12 MUSEUM OF WORKPLACE MEMORIES AT THE SCOUT HUT. DETAILS ON BACK PAGE. CONTACTS KEN LAING 684 598 DAVID DOIG 684 321, SHEENA PROFF 684 550.
LETTER FROM DUNOON
Dear Editor
I read Newsletter No. 7 with great interest, particularly about my late grandfather James Hepburn's cows and the reminiscences of a former evacuee, coming under that category myself. I vaguely recollect my grandfather, an old gentleman with a white beard, a widower by that time who used to play me tunes on his violin. The byre in which he kept his cows after their repast in the cow park down the Perth Road, is still there in Upper Granco Street, although latterly my own father used it as a garage for his car when we were on holiday in the village. My Father used to keep hens and grow vegetables on ground at the side of the Burn at the bottom of Lower Granco Street opposite Angus Isdale's house. The hen house was built on stilts, with round metal discs to keep the rats from getting at the hens at night. My brother and I also had a spell as evacuees to the village; we had been born there but had moved away. My brother Donald (now resident in Toronto) was domiciled with our Uncle David Hepburn up the'Dragon' and I went to Boghall Farm on the road to Path of Condie and a considerable walk from the village school.
My brother complained that I had all the animals on the farm and he had only "an old blind dog" (my Aunt Rate's terrier was by this time blind) and my cousin Albert 'Paddy' Whyte rectified this matter somewhat by re-locating my pet hedgehog from the Farm to Burnside. It was many years later before I found out what had happened to this hedgehog.....
As Ron Freeland said, school lessons were somewhat irregular due to the influx of evacuees, which gave us more time of course to 'guddle' for trout at the quarry green, bathe in 'the Polly' or just play in the Den often as far up as the Tory Brig' passing 'Jenny's Gush' on the way. Of course the 'sandy hole' in the public park on the Dunnock was very popular with us and visitors too...
My uncle David Hepburn was a man of many talents, an accomplished 'booler', Sunday School Superintendent and local correspondent for the PA and SRWS. I will be eternally grateful to him for his regular letters of encouragement to me many years later during a year I spent in Switzerland with Tuberculosis from which I eventually recovered.
James R Hepburn, 48 Royal Crescent, Dunoon
REMINISCENCES OF A DUNNING EVACUEE: PART THREE
(Last of 3 articles by Ron Freeland, Culloden, recalling when he, his mother and sisters were evacuees in Dunning)
There was great excitement in the village one Sunday as a troop of gunners of the Royal Artillery, complete with a battery of six 26-pounder anti-tank guns, limbers of ammunition and other vehicles rolled in and took up residence in the Glebe of the Rev. McKinnon's manse. The remaining evacuees (many had returned home to Glasgow by this time) and village children soon made friends with the soldiers. Over the next year or two these lads became part of the village scene and were invited into many homes, where friendships blossomed.
A regular feature of village life was the Friday or Saturday evening concert often held in the town hall. These were very well attended by soldiers and villagers and there was rarely a lack of volunteers willing to 'do a turn' of singing, dancing, sketches or monologues. One performer always in great demand was my sister Margaret who, although only 11 or 12 years of age, was a 'seasoned trouper', having appeared with a concert party back home in Glasgow. At the end of these concerts the hall was cleared of benches and chairs and the grown-ups would then enjoy themselves dancing on into the wee hours, music being supplied by a small local band led by accordionist Bob Malcolm.
Our soldier friends' contribution to village life during the 18 or so months of their stay in Dunning was greatly appreciated and it was indeed a sad day when orders came for them to move out.
I believe the company went to join the Eighth Army under Montgomery in the desert campaigns of North Africa. These lads were however, by no means, the only military presence to be found in the village during the war. A large hutted camp up at Cask housed hundreds of Polish soldiers, many of whom found their way over to attend dances and concerts in our village hall. I should imagine that a number of the local girls were captivated by the many handsome Polish lads whose manners always seemed impeccable. Indeed my pal's sister, Maidie Scott, was one of those who took the plunge, marrying Lee, a handsome officer and a true gentleman. 1 trust they had a long happy life together. One incident I recall was the day the great Polish hero General Sikorski visited Gask on a tour of inspection. There was great excitement among the troops as his plane touched down (at that time there was an airstrip near the camp) and they were very proud to tell us all about it next day.
Like all small communities Dunning had her share of characters, without whom life in war-time would have been much bleaker. One of these was "Dam' dicht", the village's first air raid warden. This was old Charlie Laing who, on the approach of enemy raiders, was on his bicycle rattling around in the dark before the wail of the warning siren had died away. It was a case of "pity help anyone whose blackout was not 100%". Anyone showing a chink of light had Charlie knocking loudly on the door shouting (through an unfortunate speech impediment) "Pit oot dat dam' dicht!" He was certainly a most conscientious warden, as well as being a much respected figure in the village.
Another character, greatly loved, especially among the evacuees, was "Auntie Belie". Miss Flockhart owned the confectioner's shop and tea-room next to Lewis Angus the butcher. She was one of the hardy souls who had taken in six evacuees and also invited many of us to visit her as often as we liked. On reflection, I've no doubt we ah proved to be a welcome boost to her sales of sweets and lemonade but to us she was a larger than life figure who generated love and affection. She swept many a "puir wee mite" (her words) up to her ample bosom for a cuddle. Auntie Belie also took care of her bed-ridden sister, Maggie, and we often wondered how she managed to cope with such a houseful.
From time to time her two sons, twins John and Waiter, who were valets to the 'gentry' in London would return on holiday. On many occasions they would gather together as many evacuees as they could muster and take us all up to the Park for sports and games. The two of them were tireless in their efforts to keep us amused and we children always looked forward to their appearance in Dunning.
Wattie Crowe was also a well-kent figure as he cycled around the district delivering letters or telegrams, the contents of which I'm sure must often have dispensed great joy but sometimes abject sorrow. I'll always remember his reply to a pretty young miss who called as he cycled past "Any letters for me, Wattie?" "Sorry dear, nothing today, but had I known you were expecting one I would've written it myself!"
Many a fine day during the summer holidays found Wilfred James and me crawling through the undergrowth hunting rabbits up in the extensive grounds of Keltie Castle. Wilf and his mother came from England to live with his grandfather after the war broke out. The old man was the ghillie (or gamekeeper) on the estate, and had been for many years. He was a venerable old gentleman about six feet tall and sported a fairly long white beard and moustache. Complete with deerstalker hat, Norfolk tweeds and plus-fours, he could have passed as the brother of George Bernard Shaw! I remember well the day he showed me through the Armoury and seeing the rows of rifles, fowling-pieces, shotguns etc. He owned even a large elephant-gun! The old man regaled us with tales of bygone days when he would take parties of the 'gentry' out on the hills after deer, pheasants, grouse etc. He told us of a certain little girl who often came up from the South during the summer holidays with members of her family and who stayed in Keltie Castle. These were the Bowes-Lyons and the little girl of course was one day to become our Queen (now the Queen Mother).
With regards to the aristocracy, I wonder how many of the older villagers remember the amusing sight of the Honourable Cecily Polio on some occasions pulling along behind her through the village a little home-made, four-wheeled "bogey" on her way to the grocer or butcher to uplift the weekly rations.
On Monday evening, the 2nd March, 1942, 1744 (Auchterarder) Flight of the Air Training Corps gained two more recruits in the persons of Gordon (Bud) Sharpies and myself. Although the age of admission was 15 years, we were both accepted a few months short of that. One of the CO's greatest ambitions was to form a pipe-band and Gordon and I enrolled as drummers. Gordon chose the tenor drum while my choice was the side or snare drum. In time my other pals Jimmy Scott and Wilf James became proficient pipers in the ATC band. Many an evening I spent accompanying Jimmy with my drum down by the burnside (far enough away from habitation so no-one could complain!) We had built up quite a repertoire of tunes by the time Jimmy and I proudly played through the streets of Dunning a few years hence on VE-Day, 8th May, 1945 when Germany capitulated. My many happy memories of the ATC culminated in the honour of being chosen to represent 1744 Flight at the Victory March Past in Hyde Park, London, in June 1945. The salute was taken by HM King George VI.
My pipe-band tuition stood me in good stead when 1 was asked by Bob Malcolm to join his dance-band as its drummer. We played for dances in Dunning hall, also at Auchterarder, Gask, Forteviot and Forgandenny. For an evening's work, usually from 7.30pm till 2am we were paid the sum of fl each. This was really good money considering that as a 16-year-old, I was at that time earning 15/- for a full week's work as a grocer. This dance-band fee, however, was well-earned when one considers we had no motor transport and had to rely on cycling to reach the various venues. This with our instruments strapped to our backs!
Early in 1943 Gordon Sharpies and I were approached by Tom Reid, the Head Observer of the Royal Observer Corps unit based in Dunning, asking us if we would like to enrol. He had been finding some difficulty with recruitment as most of the eligible males in the village were already members of the Home Guard. Bud and I were soon full members of the ROC. We were still under 15, possibly the youngest members of the Corps in Scotland, if not the UK. Our'watch' involved the plotting of all aircraft both enemy and friendly which entered our sector tan area covering 10 miles radius from our post). This we did from our little listening-post situated on a hillock in a field up at Bob Mailer's Findony Farm. Enclosed in a small wooden stockade which housed a hut for resting when off-duty and an outside platform on which the sighting instrument was manned throughout 14 hours, each observer did 6-hour shifts working in pairs and sharing the duties equally among them. Gordon and I and the dozen or so other observers spent many a bitterly cold hour exposed to all weathers up at Post H2 ("How Two") during the next two years but of course we felt we were "doing our bit" in the war effort and besides we were keen on aeroplanes. Although we plotted a fair number of night raids I only once saw a "bandit" in daylight. This was on a clear summer evening when a twin-engined Junker Ju 188 flew right over us and across the village at a height of less than 100 feet. Obviously on a reconnaissance mission, he knew by flying so low he would be unable to be picked up by radar. I wonder if he made it home?
I should like to add a few glimpses of various 'happenings' which may trigger off memories for older Dunning friends. For instance, that Sunday which saw our attempts to repel The Invasion of Dunning'. This was around the summer of '42 when our local Home Guard were charged with preventing 'the enemy' taking over the village. The latter turned out to be a company of troops of the Highland Light Infantry representing the invaders. Gordon and I, being Observers, were naturally detailed to act as look-outs and were situated up in the St. Serfs belfry tower. After a long wait the 'enemy' duly appeared from the direction of Duncrub and we gave the warning. All I remember now are members of our own 'Dads' Army' rushing about throwing thunderflashes (a type of large firework simulating grenades which exploded with a very loud report) at the attackers. I'm sorry to say I don't know who eventually won that day!
Who remembers now the summer afternoon in 1943 when two Tiger Moth aircraft 'buzzed' the village? We had certainly never seen anything fly so low and in fact they actually flew around St. Serfs steeple fully ten feet below the clock faces! After about twenty minutes jinking about they disappeared, chasing one another up the Dragon. Many amazed villagers, having witnessed the performance, had congregated in the square when we heard that one had crashed up the Kippen Road. I took off in that direction and soon discovered that one of the pilots had apparently noticed our Observer Post at Findony and was in the process of showing off his skills when a wing-tip hit the hedge and the aircraft, after catapulting a few times, finished upside down in the field about ten yards from the Post. Amazingly he had escaped serious injury and had been able to climb out of the wreckage. What old Bob Mailer had to say about these proceedings was not recorded and I imagine the telephone lines from our Post to the Centre would be buzzing. An RAF vehicle arrived soon after and whisked off the culprit, whilst a Queen Mary (a 60 foot long articulated vehicle) came next day and removed one rather bent Tiger Moth. We never did hear what became of that pilot.
A couple of months earlier, on the night of Friday, 13th March, 1941, occurred an event which will remain forever etched in my memory. That evening seemed that the whole of the Luftwaffe was flying over our village. For over an hour and a half no fewer than ninety-six waves, each consisting of four or five planes, passed over, their ominous drone (which always seemed peculiar to German aircraft engines and whose sound was always instantly recognised by those of us who lived through the war) brought a cold chill of fear to the hearts of us far below. This huge armada flew in over the North Sea, crossing the coast below Aberdeen then turning south right down Strathearn on to their main targets of Glasgow and Clydebank. Damage and casualties caused there were considerable and the BBC announced next day that extensive fires were still burning. My older sister Jean travelled up to Dunning the next weekend and was able to recount to us the horrors of that terrible night. The blitz continued the very next night as the raiders returned to the same areas. I'm dad to report however that the 'Jerries' failed to prevent the locals and the soldiers from enjoying their usual Saturday night dance in the Dunning hall.
My mother, sisters and I had moved from the Hurrys' where we were first billeted to rented rooms in a terraced house on Perth Road. Then after 2 or 3 years we moved to the lower apartments in the Commercial Building on Auchterarder Road. After leaving school on turning fifteen I worked full-time with Mrs. Hurry in their family shop, doling out the weekly rations. If I remember correctly the coupons in one's ration-book allowed usually one egg per week and 1lb jam per month. These amounts, with 1/4lb meat in the butcher's (augmented by sausage, haggis etc.) varied only slightly throughout the war years.
In the autumn of 1943 I 'transferred' to James Milne, butcher, across the Square. I worked in that shop for only about six months durirlg that extremely cold winter of'43-'44 and soon began to realise that perhaps I wasn't quite cut out for the butchering trade. In fact, I was more than happy to accept an offer to join the office staff of a firm in Perth managed by our ATC commander officer. There I remained for eighteen months until my call-up to the Forces in 1945.
By then, of course, there were very few of the original evacuees left in Dunning and on my reporting for duty with the Air Force my mother and younger sister Lily left for home leaving my other sister Margaret in Dunning where she was in employment as housemaid at St Serfs manse.
I read in a magazine recently a rather sad account of a woman who as a girl had been evacuated to Arran and had been treated rather harshly by the billeting officer there. It made me think how fortunate we had been in having Mr Benzies to sort us out and find us homes with the good people of Dunning. I will always have a warm spot in my heart for the old village and its inhabitants and this recounting of my personal memories has brought to my mind again very clearly many personalities and events of half a century ago.
Ron Freeland, Culloden
OUR MUSEUM OF MEMORIES IS IMMINENT
A while ago 94 year old Dunning native, Henry Campbell wrote from Toronto:
There are very few people in Dunning who remember the days when there was no such thing as Radio or Television. So with no wireless I, with a great number of citizens read a lot. I devoured countless books, magazines etc. I read numerous articles about the Wireless and it dawned on me that it was going to have an enormous effect on the field of communication. I couldn't wait to get in on this so I got a Permit to operate an Experimental Station then bought a Crystal Set. My old friend Jim Crow put up an aerial for me. Sad to say all I ever was able to receive was in Morse Code. However, it was the first receiver of wireless messages in the village of Dunning.
Dunning's first wireless receiver and Mr Campbell's Permit will be among many nostalgic items on display when the DPHS holds a ten day Museum of Village Memories. Admission will be free, and there'll be mystery objects and a special contest and draw open to all visitors. Most of the exhibits will be on loan from generous members and villagers. The Museum of Village Memories will be held in the Scout Hut on the Kirk Wynd, Dunning, and will be open from 2.00 to 4.00 each week day, 11.00 to 4.00 Saturdays and Sundays and 6.30 to 8.00 Tuesday and Thursday evenings from Friday, July 1, 1994 to Sunday, July 10.
The following was sent by a Canadian member, Mrs Helen (Robertson) Laidlaw. It was written by her sister, the late Mrs Betty Grant.
In a wee village, doon in the glen,
Nestled a hoose, oor wee but and ben.
Five lassies, three laddies, oor Mum and Dad
It was hame tae us a', and whit fun we had.
We wirna rich and wirna poor
The wolf was aye kept awa' frae the door.
Oor Mither looked efter the coos that she lo'ed
Oor Faither felled trees oot in the wood.
The family as weel had tae lend a hand
Deliverin' milk in spotless tin cans,
Muckin' the byre or workin' oot doors,
Thae wir a few of our daily chores.
It wasna a' work. We played lots of.games.
Some of them had awfu' queer names
Like 'Catty and Duggy','Coont ten and awa',
'Burnin' the bible' or 'Catchin' a craw'.
Playin' doon at the burn, or up at the swings
Thae wir oor pleasures, a' common enough things
But tae us it meant mair, we were happy, content
Playing' at hooses, wi an auld rug for a tent.
But guid things don't last, and all too soon
We left oor wee village for Edinburgh toon
Tae seek fame and fortune and gie us mair scope
Mum was ambitious and for us she had hopes.
It worked oot the way she hoped that it would.
We all found work, aye the whole eight of the brood.
But for years we a' hankered for the wee misty glen
And the guid times we shared in that wee but and ben.
Oft times we still gang for a drive doon the glen,
Visit the Dragon, the Granco and gae through the Pen
But the people and landmarks are just not the same
The village of Dunning is noo only a name.
(Betty and Helen Robertson lived on Circus Sheet and at Durnbrac, and in a coming newsletter the now Helen Robertson Laidlaw, who resides in New Lowell, Ontario, will relate her personal recollections of a happy childhood spent in Dunning Mayl924-April 1937.)
NEW COMMITTEE:
Elected at the May 27 AGM as new Secretary of the DPHS 1994-95 executive committee was Shona Sinclair. Continuing in office are chair Lorne Wallace, vice-chair Colin Young, treasurer Jane Young and members Finella Wilson, Albie Sinclair, Judi Slater, Sheena Proff, Kirsty Macnab and Louise and Catherine Crowe. New committee members are Kirsty and David Doig and Grace McFarlane. Heartfelt thanks for their fine efforts in establishing the Society were voted to outgoing founding DPHS committee members Jill Tanner, Ken Laing, Janet Crowe and Patricia Wallace.
PLEASANT MEMORIES
Special guest at the AGM was Scottish graveyard authority Betty Willshcr, who led us on a brief tour of St Serfs and then at nearby but seldom visited Aberdalgie and Dupplin kirkyard divided us into five teams to record a quick survey of the gravestones. She writes: " I must say I had a moment of real happiness when I saw everyone working away in that beautiful graveyard. Their findings showed how much can be accomplished in a short time by people working together". The notes of the survey have been handed over to the Rev. Colin Williamson of Aberdalgie.
Other pleasant recent happenings were a well-attended trip to the Dunning Den with David Doig, and a trip to Alloa as guests of the Clackmannan Field Society where the Earl of Mar was one of several excellent guides to this fascinating town. And then in early June, Colin Young led a sunlit trip to several of the historic crofts located in the Ochils above Dunning.
THE SUMMER-AUTUMN PROGRAMME
1-12 July A ten day Museum of Village Memories starting Friday July 1 at the Scout Hut on Kirk Wynd, with historic items loaned by villager, Open on weekday afternoons 2 to 4pm, on Saturdays and Sundays 11am to 4pm, and Tuesday and Thursday evenings 6.30-8pm. Admission free with special contests and draw.
2-3 Sept Evacuees' Reunion. Friday, an assembly and media event followed by talks to local schools. Friday at 7.30pm a meeting with DPHS members at the Primary School. Saturday the evacuees tour the village and starting at 7.30pm the whole village is invited to a 'Sentimental Journey' Big Band dance at the Village Hall.
Sept TBA An Archaeological Fieldwalk, a follow-up to last year's highly successful fieldwalk which unearthed evidence of a 4,000 year old settlement here. To be led by Perth Museum's Mike King on a weekend date to be announced. Likely location: lain Philip's farm, Leadketty. If you want to participate, please contact Shona Sinclair 684 566.
Sept 29 Thursday, 7.30pm. A visit to the new Perth Library's archives and local history department. If you're interested, please call Grace McFarland 684 376.
Oct 20 Thursday, Village Hall, 7.30pm. Writer/broadcaster Rennie McOwan will discuss witches among other matters in a slide talk titled The Ochils - Are They Worth Protecting?
Nov 5 Saturday, Village Hall, a Coffee Morning, 10am to 12 noon. Featured: a special photographic exhibit loaned by John Crow.
Dec 1 Thursday, Dunning School, 7.30 pm. The Rev. Colin Williamson of Forteviot and Aberdalgie will update us on his community's attempt to keep the famous Dupplin Cross within the Parish.
(Other events like a new presentation by local historian Ken Laing, a Farm Film Night, and another video-making session are coming up in 1995. Details of the winter-spring programme in our new newsletter.)


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Editor Uncategorized 0
Huawei launches three Nova 6 smartphones, including a 5G model
Huawei just announced three new smartphones to join its already saturated Nova lineup. This time called the Nova 6, the phones are aimed at the mid-range market.
The cheapest device is definitely an indication of this. Coming in at around $315 in China, the Huawei Nova 6 SE looks a lot more like the Nova 5 series that preceded it.
A 6.4-inch IPS LCD also features a punch-hole selfie 16MP camera in the top left corner. Around the back, there's a quad camera array with a 48MP primary camera, 8MP ultrawide and two 2MP snappers for depth and macro shots.
A fingerprint sensor is mounted on the phone's right side too.
Internally, Huawei's using a Kirin 810 chipset, with a maximum of 8GB of RAM and 128GB of internal storage. It's powered by a 4200mAh battery — the same size as the Huawei P30 Pro.
The Nova 6 occupies a slightly higher step on the hierarchy, and will cost users around $450.
For that premium though, they get an oval selfie camera cutout up front with 32MP and 8MP sensors. The Nova 6's screen falls just short of 6.6-inches but does use AMOLED technology.
Around the back there are three snappers — a 40MP primary camera, 8MP telephoto, and an 8MP wide angle. These are arranged similarly to the Huawei P30 Pro.
Internally, the Nova 6 is all business.
It features a Kirin 990 chipset — the same silicon used in the Mate 30 — alongside up to 8GB and 128GB of internal storage. There's also a 4100mAh battery powering the lot.
Finally, the 5G version includes all the hardware you'd find on the Nova 6, but with a Balong 5000 5G chipset, up to 256GB of internal storage, and a 4200mAh battery.
It'll set users back at least $550.
There's no indication that Huawei will launch its Nova 6 range in South Africa, but considering the Nova 5T made its debut here, we wouldn't be surprised to see the Nova 6 5G lead the company's mid-tier 5G charge in 2020.
Remember, there's also a chance that the Nova 5T Pro could make its debut soon too.
Nevertheless, the Nova 6 family will go on sale in China next week.
Feature image: Huawei
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