Hermitian matrices of three parameters: Perturbing coalescing eigenvalues and a numerical method

In this work we consider Hermitian matrix-valued functions of 3 (real) parameters, and are interested in generic coalescing points of eigenvalues, conical intersections. Unlike our previous works [L. Dieci, A. Papini and A. Pugliese, Approximating coalescing points for eigenvalues of Hermitian matrices of three parameters, SIAM J. Matrix Anal. Appl., 2013] and [L. Dieci and A. Pugliese, Hermitian matrices depending on three parameters: Coalescing eigenvalues, Linear Algebra Appl., 2012], where we worked directly with the Hermitian problem and monitored variation of the geometric phase to detect conical intersections inside a sphere-like region, here we consider the following construction: (i) Associate to the given problem a real symmetric problem, twice the size, all of whose eigenvalues are now (at least) double, (ii) perturb this enlarged problem so that, generically, each pair of consecutive eigenvalues coalesce along curves, and only there, (iii) analyze the structure of these curves, and show that there is a small curve, nearly planar, enclosing the original conical intersection point. We will rigorously justify all of the above steps. Furthermore, we propose and implement an algorithm following the above approach, and illustrate its performance in locating conical intersections

On the Injectivity of Mean Value Mapping between Convex Quadrilaterals

We prove that Mean Value mapping between convex quadrilaterals is injective, affirmatively proving a conjecture stated in M. S. Floater and J. Kosinka, On the injectivity of Wachspress and mean value mappings between convex polygons, Adv. in Comp. Math. 32 (2010), 163-174

On the Error in QR Integration

This is the published version, also available here: http://dx.doi.org/10.1137/06067818X.An important change of variables for a linear time varying system $\dot x=A(t)x, t\ge 0$, is that induced by the QR-factorization of the underlying fundamental matrix solution: $X=QR$, with Q orthogonal and R upper triangular (with positive diagonal). To find this change of variable, one needs to solve a nonlinear matrix differential equation for Q. Practically, this means finding a numerical approximation to Q by using some appropriate discretization scheme, whereby one attempts to control the local error during the integration. Our contribution in this work is to obtain global error bounds for the numerically computed Q. These bounds depend on the local error tolerance used to integrate for Q, and on structural properties of the problem itself, but not on the length of the interval over which we integrate. This is particularly important, since—in principle—Q may need to be found on the half-line $t\ge 0$

Decompositions and coalescing eigenvalues of symmetric definite pencils depending on parameters

In this work, we consider symmetric positive definite pencils depending on two parameters. That is, we are concerned with the generalized eigenvalue problem $A(x)-\lambda B(x)$, where $A$ and $B$ are symmetric matrix valued functions in ${\mathbb R}^{n\times n}$, smoothly depending on parameters $x\in \Omega\subset {\mathbb R}^2$; further, $B$ is also positive definite. In general, the eigenvalues of this multiparameter problem will not be smooth, the lack of smoothness resulting from eigenvalues being equal at some parameter values (conical intersections). We first give general theoretical results on the smoothness of eigenvalues and eigenvectors for the present generalized eigenvalue problem, and hence for the corresponding projections, and then perform a numerical study of the statistical properties of coalescing eigenvalues for pencils where $A$ and $B$ are either full or banded, for several bandwidths. Our numerical study will be performed with respect to a random matrix ensemble which respects the underlying engineering problems motivating our study.Comment: 34 pages, 4 figure

Lyapunov Spectral Intervals: Theory and Computation

This is the published version, also available here: http://dx.doi.org/10.1137/S0036142901392304.Different definitions of spectra have been proposed over the years to characterize the asymptotic behavior of nonautonomous linear systems. Here, we consider the spectrum based on exponential dichotomy of Sacker and Sell [J. Differential Equations, 7 (1978), pp. 320--358] and the spectrum defined in terms of upper and lower Lyapunov exponents. A main goal of ours is to understand to what extent these spectra are computable. By using an orthogonal change of variables transforming the system to upper triangular form, and the assumption of integral separation for the diagonal of the new triangular system, we justify how popular numerical methods, the so-called continuous QR and SVD approaches, can be used to approximate these spectra. We further discuss how to verify the property of integral separation, and hence how to a posteriori infer stability of the attained spectral information. Finally, we discuss the algorithms we have used to approximate the Lyapunov and Sacker--Sell spectra and present some numerical results

Nonnegative moment coordinates on finite element geometries

In this paper, we introduce new generalized barycentric coordinates (coined as {\em moment coordinates}) on nonconvex quadrilaterals and convex hexahedra with planar faces. This work draws on recent advances in constructing interpolants to describe the motion of the Filippov sliding vector field in nonsmooth dynamical systems, in which nonnegative solutions of signed matrices based on (partial) distances are studied. For a finite element with $n$ vertices (nodes) in $\mathbb{R}^2$, the constant and linear reproducing conditions are supplemented with additional linear moment equations to set up a linear system of equations of full rank $n$, whose solution results in the nonnegative shape functions. On a simple (convex or nonconvex) quadrilateral, moment coordinates using signed distances are identical to mean value coordinates. For signed weights that are based on the product of distances to edges that are incident to a vertex and their edge lengths, we recover Wachspress coordinates on a convex quadrilateral. Moment coordinates are also constructed on a convex hexahedra with planar faces. We present proofs in support of the construction and plots of the shape functions that affirm its properties

Unitary Integrators and Applications to Continuous Orthonormalization Techniques

This is the published version, also available here: http://dx.doi.org/10.1137/0731014.In this paper the issue of integrating matrix differential systems whose solutions are unitary matrices is addressed. Such systems have skew-Hermitian coefficient matrices in the linear case and a related structure in the nonlinear case. These skew systems arise in a number of applications, and interest originates from application to continuous orthogonal decoupling techniques. In this case, the matrix system has a cubic nonlinearity. Numerical integration schemes that compute a unitary approximate solution for all stepsizes are studied. These schemes can be characterized as being of two classes: automatic and projected unitary schemes. In the former class, there belong those standard finite difference schemes which give a unitary solution; the only ones are in fact the Gauss–Legendre point Runge–Kutta (Gauss RK) schemes. The second class of schemes is created by projecting approximations computed by an arbitrary scheme into the set of unitary matrices. In the analysis of these unitary schemes, the stability considerations are guided by the skew-Hermitian character of the problem. Various error and implementation issues are considered, and the methods are tested on a number of examples