# On the Total Positivity and Accurate Computations of \(r\)-Bell Polynomial Bases
## 2 Notations and Auxiliary Results  
Let us recall that a matrix is totally positive (respectively, strictly totally positive) if all its minors are nonnegative (respectively, positive). The Neville elimination is an alternative procedure to Gaussian elimination (see [16; 17; 25]). Given a nonsingular matrix \(A=(a_{i,j})_{1\leq i,j\leq n+1}\), its Neville elimination calculates a sequence of \((n+1)\times(n+1)\) matrices \(A^{(k)}\), \(k=1,\ldots,n+1\), so that their elements below the main diagonal of the \(k-1\) first columns are zeros and then \(A^{(n+1)}\) is upper triangular. From \(A^{(k)}=(a^{(k)}_{i,j})_{1\leq i,j\leq n+1}\), the Neville elimination computes \(A^{(k+1)}=(a^{(k+1)}_{i,j})_{1\leq i,j\leq n+1}\) as follows,  
\[a^{(k+1)}_{i,j}:= \begin{cases}a^{(k)}_{i,j},&\text{if}\ \ 1\leq i\leq k,\\ a^{(k)}_{i,j}-\frac{a^{(k)}_{i,k}}{a^{(k)}_{i-1,k}}a^{(k)}_{i-1,j^{\prime}}& \text{if}\ \ k+1\leq i,j\leq n+1,\ \text{and}\ a^{(k)}_{i-1,k}\
eq 0,\\ a^{(k)}_{i,j},&\text{if}\ \ k+1\leq i\leq n+1,\ \text{and}\ a^{(k)}_{i-1,k}=0, \end{cases}  \tag{1}\]  
with \(A^{(1)}:=A\). The element  
\[p_{i,j}:=a^{(j)}_{i,j},\quad 1\leq j\leq i\leq n+1, \tag{2}\]  
is called the \((i,j)\) pivot and we say that \(p_{i,i}\) is the \(i\)-th diagonal pivot of the Neville elimination of \(A\). Whenever all pivots are nonzero, no row exchanges are needed in the Neville elimination procedure. The value  
\[m_{i,j}:= \begin{cases}a^{(j)}_{i,j}/a^{(j)}_{i-1,j}=p_{i,j}/p_{i-1,j},&\text{ if}\ \ a^{(j)}_{i-1,j}\
eq 0,\\ 0,&\text{if}\ \ a^{(j)}_{i-1,j}=0,\end{cases}  \tag{3}\]  
for \(1\leq j<i\leq n+1\), is the \((i,j)\) multiplier of the Neville elimination of \(A\). The Neville elimination procedure of a nonsingular matrix is illustrated with the following example:  
\[A=A^{(1)}=\left( \begin{array}{ccc}2&4&12\\ 8&26&138\\ 56&262&1704\end{array} \right)\to A^{(2)}=\left( \begin{array}{ccc}2&4&12\\ 0&10&90\\ 0&80&738\end{array} \right)\to A^{(3)}=\left( \begin{array}{ccc}2&4&12\\ 0&10&90\\ 0&0&18\end{array} \right). \tag{4}\]  
The multipliers and the diagonal pivots are \(m_{2,1}=4\), \(m_{3,1}=7\), \(m_{3,2}=8\), \(p_{1,1}=2\), \(p_{2,2}=10\), and \(p_{3,3}=18\). By Theorem 4.2 and the arguments of p. 116 of [17], a nonsingular totally positive \(A\in\mathbb{R}^{(n+1)\times(n+1)}\) can be written as follows,  
\[A=F_{n}F_{n-1}\ldots F_{1}DG_{1}G_{2}\ldots G_{n}, \tag{5}\]  
where \(F_{i}\in\mathbb{R}^{(n+1)\times(n+1)}\) and \(G_{i}\in\mathbb{R}^{(n+1)  
\[\widehat{F}_{i}=\left( \begin{array}{ccccccccc}1&&&&&&&\\ 0&1&&&&\\ &\ddots&\ddots&&\\ &&0&1&&\\ &&&-m_{i+1,j}&1&&\\ &&&&-m_{i+2,j}&1&&\\ &&&&\ddots&\ddots&\\ &&&&m_{n+1,i}&1\end{array} \right), \tag{8}\]  
\[\widehat{G}_{i}^{T}=\left( \begin{array}{ccccccccc}1&&&&&&&\\ 0&1&&&&\\ &\ddots&\ddots&&\\ &&0&1&&\\ &&&-\widetilde{m}_{i+1,i}&1&&\\ &&&-\widetilde{m}_{i+2,i}&1&&\\ &&&&\ddots&\ddots&\\ &&&&\widetilde{m}_{n+1,i}&1\end{array} \right). \tag{9}\]  
Let us also observe that, if a matrix \(A\in\mathbb{R}^{(n+1)\times(n+1)}\) is nonsingular and totally positive, its transpose \(A^{T}\) is also nonsingular and totally positive, and  
\[A^{T}=\bar{F}_{n}\bar{F}_{n-1}\ldots\bar{F}_{1}D\bar{G}_{1}\bar{G}_{2}\ldots \bar{G}_{n}, \tag{10}\]  
where \(\bar{F}_{i}=G_{i}^{T}\), \(\bar{G}_{i}=F_{i}^{T}\) and \(F_{i}\), \(G_{i}\)\(i=1,\ldots,n\), are the lower and upper triangular bidiagonal matrices in (5). The total positivity property of a given matrix can be characterized by analyzing the sign of the diagonal pivots and multipliers of its Neville elimination, as shown in Theorem 4.1, Corollary 5.5 of [25] and the arguments of p. 116 of [17].