Conditioning and accurate computations with Pascal matrices

Algorithm 960

“…, 1/( n n )), and P L = (( i j )) 0≤i, j≤n is the lower triangular Pascal matrix andP L = (( i j )/2 i ) 0≤i, j≤n . In Section 2 of Alonso et al [2013], a bidiagonal decomposition of the lower triangular Pascal matrix P L was presented. ).…”

Almost strictly totally negative matrices: An algorithmic characterization

“…If j t = j 1 = 2, by (13) we have j − 2 ≤ i − n, that is i ≥ n − 2 + j(≥ n − 1). Therefore the minor is det B[n − 1, n|1, 2] = b n−1,1 b n2 > 0.…”

“…This is the case of matrices that admit a bidiagonal decomposition (see [12][13][14]). HRA means that the relative errors of the computations are of the order of machine precision, independently of the size of the condition number of matrix.…”

“…t ε i h −  i h t +1 = ε i h −1 ε i h , where indices i h t , j h t ,  i h t ,  j ht are given by conditions corresponding to(13) and(14).…”

An effective Lie–Kolchin Theorem for quasi-unipotent matrices

“…An immediate consequence of the Cauchy-Binet formula is that the product of two totally nonnegative matrices is again totally nonnegative. We define the lower triangular Pascal matrix L n (see [2]) to be the nˆn matrix whose entries are given by pL n q i, j "ˆi´1 j´1˙. With our convention, it becomes clear that if i ă j then the corresponding entry of L n is zero, so that L n is indeed lower triangular.…”