A proximity matrix, morse digits.dat, for the ten Morse code symbols representing the first ten digits (data from Rothkopf, 1957). 33 9.1 Order-constrained least-squares approximations to the digit proximity data of Shepard et al. (1975); the upper-triangular portion is anti-Robinson and the lower-triangular portion is strongly-anti-The 45 subsets listed according to increasing diameter values ...

Unlike the method without pivoting, Gaussian elimination with partial pivoting consecutively applies row permutation to matrix A in order to avoid possible akk diagonal entries of matrix A being equal to zero. Gaussian elimination with partial pivoting solves the matrix equation Ax = b decomposing matrix A into a lower L and upper U triangular matrices such that PA = LU, where P is a row permut...

This paper considers the problem of transforming a triangular integer input matrix to canonical Hermite and Smith normal form. We provide algorithms and prove deterministic running times for both transformation problems that are optimal in the matrix dimension. The algorithms are easily implemented, assume standard integer arithmetic, and admit excellent performance in practice. The results pre...

In this paper, we present an algorithm for computing the characteristic polynomial of the pencil (A ? sE). It is shown that after a preliminary reduction of the matrices A and E to, respectively, an upper Hessenberg and an upper triangular matrix, the problem of computing the characteristic polynomial is transformed to the solution of certain triangular systems of linear algebraic equations. We...

We show that for piecewise hereditary algebras, the periodicity of the Coxeter transformation implies the non-negativity of the Euler form. Contrary to previous assumptions, the condition of piecewise heredity cannot be omitted, even for triangular algebras, as demonstrated by incidence algebras of posets. We also give a simple, direct proof, that certain products of reflections, defined for an...