# On the Total Positivity and Accurate Computations of \(r\)-Bell Polynomial Bases
## 2 Notations and Auxiliary Results
Theorem 4.1: _A nonsingular matrix \(A\) is totally positive (respectively, strictly totally positive) if and only if the Neville elimination of \(A\) and \(A^{T}\) can be performed without row exchanges, all the multipliers of the Neville elimination of \(A\) and \(A^{T}\) are nonnegative (respectively, positive) and the diagonal pivots of the Neville elimination of \(A\) are all positive._  
In [19], the bidiagonal factorization (5) of a nonsingular and totally positive \(A\in\mathbb{R}^{(n+1)\times(n+1)}\) is represented by defining a matrix \(BD(A)=(BD(A)_{i,j})_{1\leq i,j\leq n+1}\) such that  
\[BD(A)_{i,j}:=\left\{ \begin{aligned} & m_{i,j},&\text{if $i>j$,}\\ & p_{i,i},&\text{if $i=j$,}\\ &\widetilde{m}_{j,i},&\text{if $i<j$.}\end{aligned} \right. \tag{11}\]  
This representation will allow us to define algorithms adapted to the totally positive structure and provide accurate computations with the matrix. Let us observe that, for the matrix \(A\) in (4), the bidiagonal decomposition (5) can be written as follows:  
\[\left( \begin{array}{ccccc}2&4&12\\ 8&26&138\\ 56&262&1704\end{array} \right)=\left( \begin{array}{ccccc}1&0&0\\ 0&1&0\\ 0&7&1\end{array} \right)\left( \begin{array}{ccccc}1&0&0\\ 4&1&0\\ 0&8&1\end{array} \right)\left( \begin{array}{ccccc}2&0&0\\ 0&10&0\\ 0&0&18\end{array} \right)\left( \begin{array}{ccccc}1&2&0\\ 0&1&6\\ 0&0&1\end{array} \right)\left( \begin{array}{ccccc}1&0&0\\ 0&1&3\\ 0&0&1\end{array} \right).\]  
Moreover, since the diagonal pivots and multipliers of the Neville elimination of \(A\) and \(A^{T}\) are all positive, we conclude that \(A\) is strictly totally positive. The above factorization can be represented in the following matrix form:\[BD(A)=\left( \begin{array}{ccc}2&2&3\\ 4&10&6\\ 7&8&18\end{array} \right).\]  
Let us also recall that a real value \(x\) is computed to high relative accuracy whenever the computed value \(\tilde{x}\) satisfies  
\[\left|\frac{x-\tilde{x}}{x}\right|<Ku,\]  
where \(u\) is the unit round-off and \(K>0\) is a constant, which is independent of the arithmetic precision. High relative accuracy implies a great accuracy in the computations, since the relative errors have the same order of the machine precision and the accuracy is not affected by the dimension or the conditioning of the problem we are solving. A sufficient condition to assure that an algorithm can be computed to high relative accuracy is the non-inaccurate cancellation condition, sometimes denoted as the NIC condition, which is satisfied if the algorithm does not require inaccurate subtractions and only evaluates products, quotients, sums of numbers of the same sign, subtractions of numbers of the opposite sign, or subtraction of initial data (cf. [18; 19]). If the bidiagonal factorization (5) of a nonsingular and totally positive matrix \(A\) can be computed to high relative accuracy, the computation of its eigenvalues and singular values, the computation of \(A^{-1}\), and even the resolution of systems of linear equations \(Ax=b\), for vectors \(b\) with alternating signs, can be also computed to high relative accuracy, using the algorithms provided in [26]. Let \((u_{0},\ldots,u_{n})\) be a basis of a space \(U\) of functions defined on \(I\subseteq R\). Given a sequence of parameters \(x_{1}<\ldots<x_{n+1}\) on \(I\), the corresponding collocation matrix is defined by  
\[M(x_{1},\ldots,x_{n+1}):=\big{(}u_{j-1}(x_{i})\big{)}_{1\leq i,j\leq n+1}. \tag{12}\]  
The system \((u_{0},\ldots,u_{n})\) of functions defined on \(I\subseteq\mathbb{R}\) is totally positive if all of its collocation matrices \(M(x_{1},\ldots,x_{n+1})\) are totally positive. If the space \(U\) is formed by \(n\)-times continuously differentiable functions and \(x\in I\), the Wronskian matrix at \(x\) is defined by:  
\[W(u_{0},\ldots,u_{n})(x):=(u_{j-1}^{(i-1)}(x))_{1\leq i,j\leq n+1}, \tag{13}\]  
where \(u^{(i)}(x)\), \(i\leq n\), denotes the \(i\)-th derivative of \(u\) at \(x\).