Licio Hernanes Bezerra

arXiv2023

An iterative method to solve Lyapunov equations

We present here a new splitting method to solve Lyapunov equations of the type $AP + PA^T=-BB^T$ in a Kronecker product form. Although that resulting matrix is of order $n^2$, each iteration of the method demands only two operations with the matrix $A$: a multiplication of the form $(A-σI) \hat{B}$ and a inversion of the form $(A-σI)^{-1}\hat{B}$. We see that for some choice of a parameter the iteration matrix is such that all their eigenvalues are in absolute value less than 1, which means that it should converge without depending on the starting vector. Nevertheless we present a theorem that enables us how to get a good starting vector for the method.

math.NA↗

arXiv2010

Vandermonde factorizations of a regular Hankel matrix and their application on the computation of Bézier curves

In this paper, a new method to compute a Bézier curve of degree n = 2m-1 is introduced, here formulated as a set of points whose coordinates are calculated from two Hankel forms in $\C^m$. From Vandermonde factorizations of the two associated Hankel matrices $H_x$ and $H_y$, the Hankel forms can be easily calculated, thus yielding points on the Bézier curve. Here, a new proof of the existence of a Vandermonde factorization of regular Hankel matrix is given from Pascal matrices techniques. But, even when the Hankel matrix associated to the form is singular, the method can still be used by shifting its skew-diagonal and counteracting it after, which is pratically done without costs.. By comparing this new method with a Pascal matrix method and Casteljau's, we see that the results suggest that this new method is very effective with regard to accuracy and time of computation for various values of n.

math.NA↗