Jacobi-like Algorithm Research Articles

Simultaneous Diagonalization via Congruence of Hermitian Matrices: Some Equivalent Conditions and a Numerical Solution

This paper aims at solving the Hermitian SDC problem, i.e., that of \textit{simultaneously diagonalizing via $*$-congruence} a collection of finitely many (not need pairwise commute) Hermitian matrices. Theoretically, we provide some equivalent conditions for that such a matrix collection can be simultaneously diagonalized via $^*$-congruence.% by a nonsingular matrix. Interestingly, one of such conditions leads to the existence of a positive definite solution to a semidefinite program (SDP). From practical point of view, we propose an algorithm for numerically solving such problem. The proposed algorithm is a combination of (1) a positive semidefinite program detecting whether the initial Hermitian matrices are simultaneously diagonalizable via $*$-congruence, and (2) a Jacobi-like algorithm for simultaneously diagonalizing via $*$-congruence the commuting normal matrices derived from the previous stage. Illustrating examples by hand/coding in \textsc{Matlab} are also presented.

The Use of Solvable Directed Graphs in a Jacobi-like Algorithm

In this paper, we introduce a Jacobi-like algorithm (we call D-NJLA) to reduce a real nonsymmetric n × n matrix to a real upper triangular form by the help of solvable directed graphs. This method uses only real arithmetic and a sequence of orthogonal similarity transformations and achieves ultimate quadratic convergence. A theoretical analysis is constructed and some experimental results are given.

A block Jacobi method for complex skew-symmetric matrices with applications in the time-dependent variational principle

An iterative Jacobi-like algorithm is described for transforming a skew-symmetric complex matrix A of even dimension into Murnaghan’s normal form. The decomposition allows to determine the singular values of A and to solve the system of linear equations Ax=b in a least square sense without accidentally destroying the skew-symmetry. Complex skew-symmetric matrices arise in the context of the time-dependent variational principle (TDVP), that maps a quantum mechanical system to a Hamiltonian system in a high-dimensional curved phase space. When the skew-symmetric phase space metric becomes singular because of parameter redundancies, the equations of motion have to be solved in a least square sense. The presented algorithm ensures that the symplectic structure is retained also in the least square solution. As a test case it is applied to studying the deflection of a photoelectron by a hydrogen atom using the TDVP. A Fortran implementation of the skew-Jacobi algorithm is provided as supplementary material. Program summaryProgram title: skew-JacobiProgram Files doi:http://dx.doi.org/10.17632/sgsyn672nf.1Licensing provisions: MIT licenseProgramming language: python and Fortran 90Nature of problem: Computing the singular values and associated vectors for a complex skew-symmetric matrix of even dimensionSolution method: A sequence of block Jacobi rotations is constructed that iteratively reduces the matrix to normal form. The skew-symmetry is exploited so that singular values are produced in pairs.

Non-orthogonal Simultaneous Diagonalization of K-Order Complex Tensors for Source Separation

Source separation in the statistical framework is usually managed using tensor decompositions or matrix joint diagonalization. In this letter, we propose one of the first coordinate algorithms for non-orthogonal simultaneous diagonalization of any order complex tensors. It relies on the optimization of an inverse criterion and on a particular decomposition allowing to derive each parameter in an independent way. In the framework of digital telecommunication source separation, computer simulations show the interest of using sets of high-order tensors. They also illustrate the overall interesting performances of the proposed algorithm in comparison to a Jacobi-like algorithm of matrix joint diagonalization and to a canonical polyadic decomposition algorithm.

Interference‐leakage based non‐cooperative beamforming with low‐dimensional approximation

In this study, the authors have studied downlink multiuser non-cooperative beamforming in a time-division duplexing-based multicell system with a non-linear Jacobi-like iterative algorithm, where each base station is to minimise the interference leakage to adjacent cells in forming beams with signal-to-interference-plus-noise ratio constraints for users in its cell. The resulting approach is referred to as altruistic non-cooperative beamforming (ANB). To reduce the computational complexity for ANB per cell, a low-dimensional beamforming approach, which can be easily solved as it has a standard downlink multiuser beamforming formulation, has also been considered at the expense of degraded performance.

A Decoupled Jacobi-Like Algorithm for Non-Unitary Joint Diagonalization of Complex-Valued Matrices

We consider the problem of non-orthogonal joint diagonalization of a set of complex matrices. This appears in many signal processing problems and is instrumental in source separation. We propose a new Jacobi-like algorithm based both on a special parameterization of the diagonalizing matrix and on an adapted local criterion. The optimization scheme is based on an alternate estimation of the useful parameters. Numerical simulations illustrate the overall very good performances of the proposed algorithm in comparison to two other Jacobi-like algorithms and to a global algorithm existing in the literature.

Optimal combination of fourth-order cumulant based contrasts for blind separation of noncircular signals

In this paper, we have considered the problem of blind source separation of noncircular signals. We have proposed a new Jacobi-like algorithm that achieves optimization of the optimal combination of complex fourth-order cumulant based contrasts. We have investigated the application to separation of non-Gaussian sources using fourth order cumulants. And computer simulations applied to noncircular signals illustrate the proposed approach.

Nonorthogonal Joint Diagonalization by Combining Givens and Hyperbolic Rotations

A new algorithm for computing the nonorthogonal joint diagonalization of a set of matrices is proposed for independent component analysis and blind source separation applications. This algorithm is an extension of the Jacobi-like algorithm first proposed in the joint approximate diagonalization of eigenmatrices (JADE) method for orthogonal joint diagonalization. The improvement consists mainly in computing a mixing matrix of determinant one and columns of equal norm instead of an orthogonal mixing matrix. This target matrix is constructed iteratively by successive multiplications of not only Givens rotations but also hyperbolic rotations and diagonal matrices. The algorithm performance, evaluated on synthetic data, compares favorably with existing methods in terms of speed of convergence and complexity.

Jitter-Robust Orthogonal Hermite Pulses for Ultra-Wideband Impulse Radio Communications

The design of a class of jitter-robust, Hermite polynomial-based, orthogonal pulses for ultra-wideband impulse radio (UWB-IR) communications systems is presented. A unified and exact closed-form expression of the auto- and cross-correlation functions of Hermite pulses is provided. Under the assumption that jitter values are sufficiently smaller than pulse widths, this formula is used to decompose jitter-shifted pulses over an orthonormal basis of the Hermite space. For any given jitter probability density function (pdf), the decomposition yields an equivalent distribution of N-by-N matrices which simplifies the convolutional jitter channel model onto a multiplicative matrix model. The design of jitter-robust orthogonal pulses is then transformed into a generalized eigendecomposition problem whose solution is obtained with a Jacobi-like simultaneous diagonalization algorithm applied over a subset of samples of the channel matrix distribution. Examples of the waveforms obtained with the proposed design and their improved auto- and cross-correlation functions are given. Simulation results are presented, which demonstrate the superior performance of a pulse-shape modulated (PSM-) UWB-IR system using the proposed pulses, over the same system using conventional orthogonal Hermite pulses, in jitter channels with additive white Gaussian noise (AWGN).

Rounding error and perturbation bounds for the symplectic QR factorization

To compute the eigenvalues of a skew-symmetric matrix A, we can use a one-sided Jacobi-like algorithm to enhance accuracy. This algorithm begins by a suitable Cholesky-like factorization of A, A=G T JG . In some applications, A is given implicitly in that form and its natural Cholesky-like factor G is immediately available, but “tall”, i.e., not of full row rank. This factor G is unsuitable for the Jacobi-like process. To avoid explicit computation of A, and possible loss of accuracy, the factor has to be preprocessed by a QR-like factorization. In this paper we present the symplectic QR algorithm to achieve such a factorization, together with the corresponding rounding error and perturbation bounds. These bounds fit well into the relative perturbation theory for skew-symmetric matrices given in factorized form.

Jacobi-like algorithm for blind signal separation of convolutive mixtures

A new solution to the blind recovery of independent source signals from their convolutive mixtures is presented. In the case of instantaneous mixtures, a robust solution referred to as second order blind identification (SOBI) has been proposed previously; here, this technique is extended to the convolutive mixture case.

A generalized ICA algorithm

In this paper, we consider the source separation problem through a block algorithm based on the maximization of contrast functions. We propose a new contrast with parameterized cross-cumulants. It allows us to put three classical contrasts in a common framework. Following the same spirit of the independent component analysis (ICA) algorithm, we derive the analytical solution for the case of two sources. Finally, a computer simulation is performed to illustrate the behavior of a Jacobi-like algorithm for the maximization of the new contrast.

Numerical Methods for Simultaneous Diagonalization

A Jacobi-like algorithm for simultaneous diagonalization of commuting pairs of complex normal matrices by unitary similarity transformations is presented. The algorithm uses a sequence of similarity transformations by elementary complex rotations to drive the off-diagonal entries to zero. Its asymptotic convergence rate is shown to be quadratic and numerically stable. It preserves the special structure of real matrices, quaternion matrices, and real symmetric matrices.

Rational interpolation via orthogonal polynomials

Relations between rational interpolants and Hankel matrices are investigated. A modification of a Jacobi-like algorithm for rational interpolation proposed in [1] is developed. The proposed algorithm eliminates all the auxiliary assumptions required to implement the algorithm of [1]. In this way, it can be used to construct all existing rational interpolants with the cost of O( N 2) arithmetic operations where N + 1 is the number of data points.

A symplectic acceleration method for the solution of the algebraic riccati equation on a parallel computer

We give a cubic acceleration method for improving the current symplectic Jacobi-like algorithm for computing the Hamiltonian-Schur decomposition of a Hamiltonian matrix and finding the positive semidefinite solution of the Riccati equation. The acceleration method can speed up the rate of convergence at the end of the symplectic Jacobi-like process when the norm of the current strictly J-lower triangle has become sufficiently small; it has high parallelism and takes O( n) computational time when implemented on a mesh-connected n × n array processor system. A quantitative analysis of convergence and numerical comparisons of one Jacobi sweep versus one correction step are presented.

An efficient Jacobi-like algorithm for parallel eigenvalue computation

A very fast Jacobi-like algorithm for the parallel solution of symmetric eigenvalue problems is proposed. It becomes possible by not focusing on the realization of the Jacobi rotation with a CORDIC processor, but by applying approximate rotations and adjusting them to single steps of the CORDIC algorithm, i.e., only one angle of the CORDIC angle sequence defines the Jacobi rotation in each step. This angle can be determined by some shift, add and compare operations. Although only linear convergence is obtained for the most simple version of the new algorithm, the overall operation count (shifts and adds) decreases dramatically. A slow increase of the number of involved CORDIC angles during the runtime retains quadratic convergence. >

A parallel algorithm for the eigenvalues and eigenvectors of a general complex matrix

A new parallel Jacobi-like algorithm is developed for computing the eigenvalues of a general complex matrix. Most parallel methods for this problem typically display only linear convergence, Sequential `norm-reducing' algorithms also exist and they display quadratic convergence in most cases. The new algorithm is a parallel form of the `norm-reducing' algorithm due to Eberlein. It is proven that the asymptotic convergence rate of this algorithm is quadratic. Numerical experiments are presented which demonstrate the quadratic convergence of the algorithm and certain situations where the convergence is slow are also identified. The algorithm promises to be very competitive on a variety of parallel architectures. In particular, the algorithm can be implemented usingn 2/4 processors, takingO(n log2 n) time for random matrices.

A Jacobi-like algorithm for computing the generalized Schur form of a regular pencil

We develop a Jacobi-like scheme for computing the generalized Schur form ofa regular pencil of matrices σB − A. The method starts with a preliminary triangularization of the matrix B and iteratively reduces A to triangular form, while maintaining B triangular. The scheme heavily relies on the technique of Stewart for computing the Schur form of an arbitrary matrix A. Just as Stewart's algorithm, this one can efficiently be implemented in parallel on a square array of processors. This explains some of its peculiarities, and at the same time yields further insight in Stewart's algorithm.

A Jacobi-Like Algorithm for Computing the Schur Decomposition of a Nonhermitian Matrix

This paper describes an iterative method for reducing a general matrix to upper triangular form by unitary similarity transformations. The method is similar to Jacobi's method for the symmetric eigenvalue problem in that it uses plane rotations to annihilate off-diagonal elements, and when the matrix is Hermitian it reduces to a variant of Jacobi's method. Although the method cannot compete with the QR algorithm in serial implementation, it admits of a parallel implementation in which a double sweep of the matrix can be done in time proportional to the order of the matrix.

A norm-reducing Jacobi-like algorithm for the eigenvalues of non-normal matrices

A new norm decreasing Jacobi-like method for reducing a non-normal matrix to a normal one is described. The method is an improved version of the Huang-Gregory's procedure [6], in which certain norm reducing non-optimal steps are replaced by the optimal ones, which correspond no more to regular matrix transformations. The method renders itself particularly effective in dealing with defective matrices of special forms. Theory and experiments alike indicate that this algorithm is in all computational aspects — accuracy, convergence rate, computing time, complexity of the computer program — better or at least as good as the original one. Both procedures, the original and the improved one, end in all of the authors computed examples with the diagonal matrix containing the eigenvalues of the non-normal initial matrix.