Rémi Vaillancourt

Behind and beyond the Matlab ODE suite

Abstract The paper explains the concepts of order and absolute stability of numerical methods for solving systems of first-order ordinary differential equations (ODE) of the form describes the phenomenon of problem stiffness , and reviews explicit Runge-Kutta methods, and explicit and implicit linear multistep methods. It surveys the five numerical methods contained in the Matlab ODE suite (three for nonstiff problems and two for stiff problems) to solve the above system, lists the available options, and uses the odedemo command to demonstrate the methods. One stiff ode code in Matlab can solve more general equations of the form M ( t ) y ′ = f ( t , y ) provided the Mass option is on.

Complex random matrices and Rician channel capacity

Eigenvalue densities of complex noncentral Wishart matrices are investigated to study an open problem in information theory. Specifically, the largest, smallest, and joint eigenvalue densities of complex noncentral Wishart matrices are derived. These densities are expressed in terms of complex zonal polynomials and invariant polynomials. A connection between the complex Wishart matrix theory and information theory is given. This facilitates evaluation of the most important information-theoretic measure, the so-called ergodic channel capacity. In particular, the capacity of multiple-input multiple-output (MIMO) Rician distributed channels is investigated. We consider both spatially correlated and uncorrelated MIMO Rician channels and derive exact and easily computable tight upper bound formulas for ergodic capacities. Numerical results are also given, which show how the channel correlation degrades the capacity of the communication system.

Windowed Fourier transform of two-dimensional quaternionic signals

In this paper, we generalize the classical windowed Fourier transform (WFT) to quaternion-valued signals, called the quaternionic windowed Fourier transform (QWFT). Using the spectral representation of the quaternionic Fourier transform (QFT), we derive several important properties such as reconstruction formula, reproducing kernel, isometry, and orthogonality relation. Taking the Gaussian function as window function we obtain quaternionic Gabor filters which play the role of coefficient functions when decomposing the signal in the quaternionic Gabor basis. We apply the QWFT properties and the (right-sided) QFT to establish a Heisenberg type uncertainty principle for the QWFT. Finally, we briefly introduce an application of the QWFT to a linear time-varying system.

Two-dimensional quaternion wavelet transform

Abstract In this paper we introduce the continuous quaternion wavelet transform (CQWT). We express the admissibility condition in terms of the (right-sided) quaternion Fourier transform. We show that its fundamental properties, such as inner product, norm relation, and inversion formula, can be established whenever the quaternion wavelets satisfy a particular admissibility condition. We present several examples of the CQWT. As an application we derive a Heisenberg type uncertainty principle for these extended wavelets.

Convective instability boundary of Couette flow between rotating porous cylinders with axial and radial flows

The convective instability boundary of a circular Couette flow in the annular region bounded by two co- or counter-rotating coaxial cylinders with angular velocities ω1 and ω2, respectively, is studied in the presence of an axial flow due to a constant axial pressure gradient and a radial flow through the permeable walls of the cylinders. A linear stability analysis is carried out for positive and negative radial Reynolds numbers corresponding to outward and inward radial flows, respectively. Axisymmetric and non-axisymmetric disturbances are considered. In the particular case of no axial flow, the Couette flow is stabilized by an inward, or a strong outward, radial flow, but destabilized by a weak outward radial flow. Non-axisymmetric disturbances lead to instability for some negative values of μ=ω2/ω1. Bifurcation diagrams for combined radial and axial flows are more complicated. For particular values of the parameters of the problem, the Couette flow has regions of stabilization and destabilization in ...

Polynomial zerofinding iterative matrix algorithms

Abstract Newberys method is completed to a method for the construction of a (complex) symmetric or nonsymmetric matrix with a given characteristic polynomial. The methods of Fiedler, Schmeisser, and Dorfler and Schmeisser for similar constructions of symmetric matrices are reviewed. Polynomials found in the literature are solved iteratively by one of Fiedlers methods with initial values supplied either by Schmeissers method, or taken on a large circle or randomly in a region of the complex plane. The determinental equations are solved by the QR algorithm. Fiedlers method used iteratively exhibits fast convergence to simple roots, even in the presence of multiple roots. If, at some iteration step, the values of the iterates, which are converging to a multiple root, are averaged according to the Hull-Mathon procedure, then fast convergence is also attained for multiple roots. This combination appears to have nice features for polynomials of small to moderate degree.

A composite polynomial zerofinding matrix algorithm

Abstract A globally convergent matrix algorithm is presented for finding the real and complex zeros of a (complex) polynomial p(x). A combination of Schmeissers and Fiedlers matrices is used in the algorithm. First, the greatest common divisor, g(x), of p(x) and its derivative, p′(x), is obtained and used to reduce p(x) to a polynomial q(x) = p(x) q(x) with simple zeros. Second, the zeros of q(x) are computed either by finding the eigenvalues of the first block of Schmeissers matrix or by applying Fiedlers algorithm recursively. In either case, the eigenvalues are obtained by the QR algorithm. Third, the multiplicity of each zero of p(x) is calculated by means of Lagouanelles modified limiting formula. Sample numerical results are given to demonstrate the effectiveness of the algorithm.

Attractive cycles in the iteration of meromorphic functions

SummaryThe existence of attractive cycles constitutes a serious impediment to the solution of nonlinear equations by iterative methods. This problem is illustrated in the case of the solution of the equationz tanz=c, for complex values ofc, by Newtons method. Relevant results from the theory of the iteration of rational functions are cited and extended to the analysis of this case, in which a meromorphic function is iterated. Extensive numerical results, including many attractive cycles, are summarized.

Convolution Theorems for Quaternion Fourier Transform: Properties and Applications

General convolution theorems for two-dimensional quaternion Fourier transforms (QFTs) are presented. It is shown that these theorems are valid not only for real-valued functions but also for quaternion-valued functions. We describe some useful properties of generalized convolutions and compare them with the convolution theorems of the classical Fourier transform. We finally apply the obtained results to study hypoellipticity and to solve the heat equation in quaternion algebra framework.

Microlocal filtering with multiwavelets

Abstract Hyperfunctions in R n are intuitively considered as sums of boundary values of holomorphic functions defined in infinitesimal wedges in C n. Orthonormal multiwavelets, which are a generalization of orthonormal single wavelets, generate a multiresolution analysis by means of several scaling functions. Microlocal analysis is briefly reviewed and a multiwavelet system adapted to microlocal filtering is proposed. A rough estimate of the microlocal content of functions or signals is obtained from their multiwavelet expansions. A fast algorithm for multiwavelet microlocal filtering is presented and several numerical examples are considered.