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New Contributions to Semipositive and Minimally Semipositive Matrices
No full textSemipositive matrices (matrices that map at least one nonnegative vector to a positive vector) and minimally semipositive matrices (semipositive matrices whose no column-deleted submatrix is semipositive) are well studied in matrix theory. In this article, this notion is revisited and new results are presented. It is shown that the set of all minimally semipositive matrices contains a basis for the linear space of all matrices. Apart from considerations involving principal pivot transforms and the Schur complement, results on semipositivity and/or minimal semipositivity for the following classes of matrices are presented: intervals of rectangular matrices, skew-symmetric and almost skew-symmetric matrices, copositive matrices, -matrices, almost -matrices and almost -matrices
The Best-Fitting Uniform: Balancing Legislative Standards and Judicial Processes in Veterans Treatment Courts
No full textFast verified computation for the solvent of the quadratic matrix equation
No full textTwo fast algorithms for numerically computing an interval matrix containing the solvent of the quadratic matrix equation with square matrices , , and are proposed. These algorithms require only cubic complexity, verify the uniqueness of the contained solvent, and do not involve iterative process. Let \ap{X} be a numerical approximation to the solvent. The first and second algorithms are applicable when and A\ap{X}+B are nonsingular and numerically computed eigenvector matrices of \ap{X}^T and \ap{X} + \inv{A}B, and \ap{X}^T and \inv{(A\ap{X}+B)}A are not ill-conditioned, respectively. The first algorithm moreover verifies the dominance and minimality of the contained solvent. Numerical results show efficiency of the algorithms
Convergence of a modified Newton method for a matrix polynomial equation arising in stochastic problem
No full textThe Newton iteration is considered for a matrix polynomial equation which arises in stochastic problem. In this paper, it is shown that the elementwise minimal nonnegative solution of the matrix polynomial equation can be obtained using Newton\u27s method if the equation satisfies the sufficient condition, and the convergence rate of the iteration is quadratic if the solution is simple. Moreover, it is shown that the convergence rate is at least linear if the solution is non-simple, but a modified Newton method whose iteration number is less than the pure Newton iteration number can be applied. Finally, numerical experiments are given to compare the effectiveness of the modified Newton method and the standard Newton method
Determinantal representations of elliptic curves via Weierstrass elliptic functions
No full textHelton and Vinnikov proved that every hyperbolic ternary form admits a symmetric derminantal representation via Riemann theta functions. In the case the algebraic curve of the hyperbolic ternary form is elliptic, the determinantal representation of the ternary form is formulated by using Weierstrass -functions in place of Riemann theta functions. An example of this approach is given