Customizable triangular factorizations of matrices (Q1826818)

The author obtains a number of matrix factorizations, where some of the factors are customized. Here `customized' means having special entries which are particularly well suited for some applications, or allowing fast numerical calculations. A typical result in the paper is the following: Given the customized diagonal entries of an upper triangular matrix \(U\) as \(d_1,d_2,\ldots,d_N\), an \(N\times N\) matrix \(A\) has a PLUS factorization \(A= \text{PLUS}\) if and only if \(\det A=\pm d_1d_2\cdots d_N\neq0\), where \(P\) is a permutation or an upper pseudo-permutation matrix, \(L\) is a unit lower triangular matrix, \(S\) is a unit single-row matrix with \(N-1\) elements in the customized positions of \(S(N,k)\) for \(k=1,2,3,\ldots,N-1\), or \(S=I+e_Ns_N^T\).