Complex Chebyshev polynomials on circular sectors

AbstractAll those complex Chebyshev polynomials on circular sectors are given explicitly for which the set of extremal points is a certain set E or a subset of E. The case where zn is the Chebyshev polynomial on a circular sector is completely characterized. Some graphs representing the absolute value and the argument of certain selected Chebyshev polynomials on circular sectors are presented, demonstrating in particular the heterogeneous behavior of best complex Chebyshev approximations

Zero points of quadratic matrix polynomials

summary:Our aim is to classify and compute zeros of the quadratic two sided matrix polynomials, i.e. quadratic polynomials whose matrix coefficients are located at both sides of the powers of the matrix variable. We suppose that there are no multiple terms of the same degree in the polynomial $\mathbf{p}$, i.e., the terms have the form ${\mathbf{A}}_j{\mathbf{X}}^j{\mathbf{B}}_j$, where all quantities ${\mathbf{X}},{\mathbf{A}}_j,{\mathbf{B}}_j,j=0,1,\ldots,N,$ are square matrices of the same size. Both for classification and computation, the essential tool is the description of the polynomial $\mathbf{p}$ by a matrix equation $\mathbf{P}(\mathbf{X}) := \mathbf{A}(\mathbf{X})\mathbf{X}+\mathbf{B}(\mathbf{X})$, where $\mathbf{A}(\mathbf{X})$ is determined by the coefficients of the given polynomial $\mathbf{p}$ and $\mathbf{P}, \mathbf{X},\mathbf{B}$ are real column vectors. This representation allows us to classify five types of zero points of the polynomial $\mathbf{p}$ in dependence on the rank of the matrix $\mathbf{A}$. This information can be for example used for finding all zeros in the same class of equivalence if only one zero in that class is known. For computation of zeros, we apply Newtons method to $\mathbf{P}(\mathbf{X}) = \mathbf{0}.

Complex Chebyshev polynomials on circular sectors

AbstractAll those complex Chebyshev polynomials on circular sectors are given explicitly for which the set of extremal points is a certain set E or a subset of E. The case where zn is the Chebyshev polynomial on a circular sector is completely characterized. Some graphs representing the absolute value and the argument of certain selected Chebyshev polynomials on circular sectors are presented, demonstrating in particular the heterogeneous behavior of best complex Chebyshev approximations

Complex Chebyshev polynomials and generalizations with an application to the optimal choice of interpolating knots in complex planar splines

AbstractThis paper contains a brief account on complex planar splines which are complex valued functions defined piecewise on a grid. For noncontinuous (so called nonconforming) splines the problem of the placement of knots at which these splines are required to be continuous is investigated. It is shown that this problem reduces to finding complex Chebyshev polynomials under the additional requirement that the zeros of the polynomials are on the boundary of the corresponding domains. It is proved that the zeros of a generalized Chebyshev polynomial are in the convex hull of the domain on which the Chebyshev polynomials are defined. Some open problems are stated. A numerical and graphical display for the optimal location of three and six points on certain triangles is provided

The Conjugate Gradient Algorithm Applied to Quaternion-Valued Matrices

The well known conjugate gradient algorithm (cg-algorithm), introduced by Hestenes & Stiefel, [1952] intended for real, symmetric, positive definite matrices works as well for complex matrices and has the same typical convergence behavior. It will also work, not generally, but in many cases for hermitean, but not necessarily positive definite matrices. We shall show, that the same behavior is still valid if we apply the cg-algorithm to matrices with quaternion entries. We particularly investigate the early stop of the cg-algorithm in this case and we develop error estimates. We have to present some basic facts about quaternions and about matrices with quaternion entries, in particular, about eigenvalues of such matrices. We also present some numerical examples of quaternion systems solved by the cg-algorithm

Zero points of quadratic matrix polynomials

summary:Our aim is to classify and compute zeros of the quadratic two sided matrix polynomials, i.e. quadratic polynomials whose matrix coefficients are located at both sides of the powers of the matrix variable. We suppose that there are no multiple terms of the same degree in the polynomial $\mathbf{p}$, i.e., the terms have the form ${\mathbf{A}}_j{\mathbf{X}}^j{\mathbf{B}}_j$, where all quantities ${\mathbf{X}},{\mathbf{A}}_j,{\mathbf{B}}_j,j=0,1,\ldots,N,$ are square matrices of the same size. Both for classification and computation, the essential tool is the description of the polynomial $\mathbf{p}$ by a matrix equation $\mathbf{P}(\mathbf{X}) := \mathbf{A}(\mathbf{X})\mathbf{X}+\mathbf{B}(\mathbf{X})$, where $\mathbf{A}(\mathbf{X})$ is determined by the coefficients of the given polynomial $\mathbf{p}$ and $\mathbf{P}, \mathbf{X},\mathbf{B}$ are real column vectors. This representation allows us to classify five types of zero points of the polynomial $\mathbf{p}$ in dependence on the rank of the matrix $\mathbf{A}$. This information can be for example used for finding all zeros in the same class of equivalence if only one zero in that class is known. For computation of zeros, we apply Newtons method to $\mathbf{P}(\mathbf{X}) = \mathbf{0}.