β Scribed by Jovan Stefanovski
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We show how Van Loan's method for annulling the (2,1) block of skew-Hamiltonian matrices by symplecticorthogonal similarity transformation generalizes to general matrices and provides a numerical algorithm for solving the general quadratic matrix equation:
For skew-Hamiltonian matrices we find their canonical form under a similarity transformation and find the class of all symplectic-orthogonal similarity transformations for annulling the (2,1) block and simultaneously bringing the (1,1) block to Hessenberg form.
We present a structure-preserving algorithm for the solution of continuous-time algebraic Riccati equation. Unlike other methods in the literature, the final transformed Hamiltonian matrix is not in Hamiltonian-Schur form.
Three applications are presented: (a) for a special system of partial differential equations of second order for a single unknown function, we obtain the matrix of partial derivatives of second order of the unknown function by only algebraic operations and differentiation of functions; (b) for a similar transformation of a complex matrix into a symmetric (and three-diagonal) one by applying only finite algebraic transformations; and (c) for finite-step reduction of the eigenvalues-eigenvectors problem of a Hermitian matrix to the eigenvalueseigenvectors problem of a real symmetric matrix of the same dimension.
π SIMILAR VOLUMES
A generalization of the Broadwell models for the discrete Boltzmann equation with linear and quadratic terms is investigated. We prove that there exists a time-global solution to this model in one space-dimension for locally bounded initial data, using a maximum principle of solutions. The boundedne