# On the Total Positivity and Accurate Computations of \(r\)-Bell Polynomial Bases
## 4 Accurate Computations with \(r\)-Bell Bases
For all considered matrices, we have compared the smallest eigenvalue and singular value obtained using the proposed bidiagonal decompositions provided by Algorithms 3-5 with the functions TNEigenvalues and TNSingularvalues, and the smallest eigenvalue and singular value computed with the Matlab commands eig and svd, respectively. Note that the eigenvalues of the Wronskian matrices are exact. Furthermore, the singular values of the Gramian matrices considered coincide with their eigenvalues (since Gramian matrices are symmetric). The relative errors are shown in Figure 2. Note that our approach accurately computes the smallest eigenvalue and singular value regardless of the 2-norm condition number of the considered matrices. In contrast, the Matlab commands eig and svd return results that are not accurate at all. **Computation of inverses**. Further to this, for all considered matrices, we have compared the inverse obtained using the proposed bidiagonal decompositions provided by Algorithms 3-5 with the function TNInverseExpand and the inverse computed with the Matlab command inv. As shown in Figure 3, our procedure provides very accurate results. On the contrary, the results obtained with Matlab reflect poor accuracy. **Resolution of linear systems**. Finally, for all considered matrices, we have compared the solution of the linear systems \(B^{r}_{n}c=d\), \(\tilde{B}^{r}_{n}c=d\), \(W^{r}_{n}c=d\), \(\tilde{W}^{r}_{n}c=d\), \(G^{r}_{n}c=d\) and \(\tilde{G}^{r}_{n}c=d\) where \(d=((-1)^{i+1}d_{i})_{1\leq i\leq n+1}\) and \(d_{i}\), \(i=1,\ldots,n+1\), are random nonnegative integer values, obtained using the proposals for bidiagonal decompositions provided by Algorithms 3-5 with the function TNSolve and the solutions obtained within the Matlab command \(\backslash\). As opposed to the results obtained with the command \(\backslash\), the proposed

Figure 1: The 2-norm condition number of collocation, Wronskian, and Gramian matrices of r-Bell bases of the first and second kind. totally positive. They also play a significant role in determining the linear transformation between monomial and \(r\)-Bell polynomial bases. The paper further discusses an efficient algorithm for computing the bidiagonal factorization of \(r\)-Stirling matrices to high relative accuracy. This factorization is used for solving relevant algebraic problems involving collocation, Wronskian, and Gramian matrices associated with \(r\)-Bell polynomial bases. To validate the proposed procedure and its usefulness, the paper presents numerical experiments that confirm its accuracy. Conceptualization, E.M., J.M.P. and B.R. ; methodology, E.M., J.M.P. and B.R. ; investigation, E.M., J.M.P. and B.R. ; writing--original draft, E.M., J.M.P. and B.R. ; writing--review and editing, E.M., J.M.P. and B.R. All authors have read and agreed to the published version of the manuscript. This research was partially supported by Spanish research grants PGC2018-096321-B-I00 (MCIU/AEI) and RED2022-134176-T (MCI/AEI) and by Gobierno de Aragon (E41_23R). The data and codes used in this work are available under request. The authors declare no conflict of interest. ## References

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