# On the Total Positivity and Accurate Computations of \(r\)-Bell Polynomial Bases
## 4 Accurate Computations with \(r\)-Bell Bases
: Given \(0<x_{1}<\ldots<x_{n+1}\), by formulae (39) and (41), the collocation matrices in (44) satisfy  
\[B=V(L^{r}_{n})^{T},\quad\bar{B}=V(L^{r}_{n})^{T}, \tag{45}\]  
where \(V\) is the Vandermonde matrix (42), and \(L^{r}_{n}\) and \(L^{r}_{n}\) are \(r\)-Stirling matrices of the first and second kind, respectively. It is well known that \(V\) is strictly totally positive at \(0<x_{1}<\ldots<x_{n+1}\) and its decomposition (5) can be computed to high relative accuracy (see [19]). By Theorem 2 (respectively, Theorem 3), \(L^{r}_{n}\) (respectively, \(\bar{L}^{r}_{n}\)) is a nonsingular totally positive matrix and its decomposition (5) can be computed to high relative accuracy. Taking into account these facts, \((L^{r}_{n})^{T}\) (respectively, \((\bar{L}^{r}_{n})^{T}\)) is a nonsingular totally positive matrix and its bidiagonal decomposition can be also computed to high relative accuracy (see (10)). Therefore, we can deduce that \(B\) (respectively, \(\bar{B}\)) is a totally positive matrix since it is the product of totally positive matrices (see Theorem 3.1 of [13]). Moreover, using Algorithm 5.1 of [26], if the decomposition (5) of two nonsingular totally positive matrices is provided to high relative accuracy, then the decomposition of the product can be obtained to high relative accuracy. Consequently, \(B\) (respectively, \(\bar{B}\)) and its decomposition (5) can be obtained to high relative accuracy. Corollary 1 of [20] provides the factorization (5) of \(W:=W(m_{0},\ldots,m_{n})(x)\), the Wronskian matrix of the monomial basis \((m_{0},\ldots,m_{n})\) at \(x\in\mathbb{R}\). For the matrix representation \(BD(W)\), we have  
\[BD(W)_{i,j}:= \begin{cases}x,&\text{if }i<j,\\ (i-1)!,&\text{if }i=j,\\ 0,&\text{if }i>j.\end{cases}  \tag{46}\]  
Taking into account the sign of the entries of \(BD(W)\) and Theorem 1, one can derive that the Wronskian matrix of the monomial basis is totally positive for any \(x>0\). Moreover, the computation of (46) satisfies the NIC condition and \(W\) can be computed to high relative accuracy. Using (46), computations to high relative accuracy when solving algebraic problems related to \(W\) have been achieved in [20] for \(x>0\). Using formula (39), it can be checked that  
\[W(B^{r}_{0},\ldots,B^{r}_{n})(x)=W(m_{0},\ldots,m_{n})(x)(L^{r}_{n})^{T},\,W (\bar{B}^{r}_{0},\ldots,\bar{B}^{r}_{n})(x)=W(m_{0},\ldots,m_{n})(x)(L^{r}_{n} )^{T}.\]  
So, with the reasoning of the proof of Theorem 4, the next result follows and we can also guarantee computations to high relative accuracy when solving algebraic problems related to Wronkian matrices of \(r\)-Bell polynomial bases at positive values. **Theorem 5**. : _Let \((B^{r}_{0},\ldots,B^{r}_{n})\) and \((\bar{B}^{r}_{0},\ldots,\bar{B}^{r}_{n})\) be the \(r\)-Bell bases of the first and second kind, respectively. For any \(x>0\), the Wronskian matrices \(W^{r}_{n}:=W(B^{r}_{0},\ldots,B^{r}_{n})(x)\) and \(\bar{W}^{r}_{n}:=W(B^{r}_{0},\ldots,B^{r}_{n})(x)\) are \(r\)-Stirling matrices of the first and second kind._  
Proof. : By Theorem 4, the first and second kind are \(r\)-Stirling matrices of the second kind, respectively. By Lemma 4, the second kind \(W(\tilde{B}_{0}^{r},\ldots,\tilde{B}_{n}^{r})(x)\) are nonsingular totally positive and their bidiagonal decomposition (5) can be computed to high relative accuracy._  
It is well known that the polynomial space \(\mathbf{P}^{n}([0,1])\) is a Hilbert space under the inner product  
\[\langle p,q\rangle:=\int_{0}^{1}p(x)q(x)\,dx, \tag{47}\]  
and the Gramian matrix of the monomial basis \((m_{0},\ldots,m_{n})\) with respect to (47) is:  
\[H_{n}:=\left(\int_{0}^{1}x^{i+j-2}\,dx\right)_{1\leq i,j\leq n+1}=\left(\frac{ 1}{i+j-1}\right)_{1\leq i,j\leq n+1}. \tag{48}\]  
The matrix \(H_{n}\) is the \((n+1)\times(n+1)\) Hilbert matrix. In Numerical Linear Algebra, Hilbert matrices are well-known Hankel matrices. Their inverses and determinants have explicit formulas; however, they are very ill-conditioned for moderate values of their dimension.