Matrix Diagonalization Algorithm Research Articles

A Relation between Krylov and Nielsen Complexity.

Krylov complexity and Nielsen complexity are successful approaches to quantifying quantum evolution complexity that have been actively pursued without much contact between the two lines of research. The two quantities are motivated by quantum chaos and quantum computation, respectively, while the relevant mathematics is as different as matrix diagonalization algorithms and geodesic flows on curved manifolds. We demonstrate that, despite these differences, there is a relation between the two quantities. Namely, the time average of Krylov complexity of state evolution can be expressed as a trace of a certain matrix, which also controls an upper bound on Nielsen complexity with a specific custom-tailored penalty schedule adapted to the Krylov basis.

Application of the sequential matrix diagonalization algorithm to high-dimensional functional MRI data

This paper introduces an adaptation of the sequential matrix diagonalization (SMD) method to high-dimensional functional magnetic resonance imaging (fMRI) data. SMD is currently the most efficient statistical method to perform polynomial eigenvalue decomposition. Unfortunately, with current implementations based on dense polynomial matrices, the algorithmic complexity of SMD is intractable and it cannot be applied as such to high-dimensional problems. However, for certain applications, these polynomial matrices are mostly filled with null values and we have consequently developed an efficient implementation of SMD based on a GPU-parallel representation of sparse polynomial matrices. We then apply our “sparse SMD” to fMRI data, i.e. the temporal evolution of a large grid of voxels (as many as 29,328 voxels). Because of the energy compaction property of SMD, practically all the information is concentrated by SMD on the first polynomial principal component. Brain regions that are activated over time are thus reconstructed with great fidelity through analysis-synthesis based on the first principal component only, itself in the form of a sequence of polynomial parameters.

Large scale shell model calculations for the yrast line of 138Xe

We have adopted an importance sampling iterative matrix diagonalization algorithm to compute a large scale shell model calculation of the yrast spectrum of 138Xe up to high spin thereby extending a previous calculation confined to low-lying angular momenta. An effective nucleon–nucleon interaction derived from the CD-Bonn nucleon-nucleon potential is used to compute energies, E2 transition probabilities, and occupation numbers. A satisfactory agreement with the experimental data is reached.

Large–scale shell model calculations for 140Xe

This paper presents the results of a large-scale shell model calculations of the yrast spectrum of 140Xe. We extend the previous calculations confined to low-lying angular momenta to high-spin states apply- ing the same importance sampling iterative matrix diagonalization algorithm. Excitation energies and transi- tion probabilities are obtained by using an effective nucleon-nucleon interaction derived from the CD-Bonn nucleon-nucleon potential. A satisfactory agreement with the experimental data and the previous results for low lying states is achieved.

Diagonalization of complex symmetric matrices: Generalized Householder reflections, iterative deflation and implicit shifts

We describe a matrix diagonalization algorithm for complex symmetric (not Hermitian) matrices, A̲=A̲T, which is based on a two-step algorithm involving generalized Householder reflections based on the indefinite inner product 〈u̲,v̲〉∗=∑iuivi. This inner product is linear in both arguments and avoids complex conjugation. The complex symmetric input matrix is transformed to tridiagonal form using generalized Householder transformations (first step). An iterative, generalized QL decomposition of the tridiagonal matrix employing an implicit shift converges toward diagonal form (second step). The QL algorithm employs iterative deflation techniques when a machine-precision zero is encountered “prematurely” on the super-/sub-diagonal. The algorithm allows for a reliable and computationally efficient computation of resonance and antiresonance energies which emerge from complex-scaled Hamiltonians, and for the numerical determination of the real energy eigenvalues of pseudo-Hermitian and PT-symmetric Hamilton matrices. Numerical reference values are provided. Program summaryProgram Title: HTDQLSProgram Files doi:http://dx.doi.org/10.17632/x24wjxtrsg.1Licensing provisions: GPLv3Programming language: Fortran 90 using fixed form notationNature of problem: Calculating the eigenvalues and optionally the eigenvectors of complex symmetric (non-Hermitian), densely populated matrices.Solution method: The complex symmetric (not Hermitian) input matrix is diagonalized in two steps. First step: The matrix is tridiagonalized via a series of (n−2) generalized Householder reflections, where n is the rank of the input matrix. Second step: The tridiagonal matrix is diagonalized via a generalization of the “chasing the bulge” technique, which is an iterative process utilizing an implicitly shifted initial rotation followed by (n−2) Givens rotations. This technique is an implementation of QL factorization, and converges roughly as [(λi−σi)∕(λi+1−σi)]j where λi is the eigenvalue located in the (i,i) position of the final diagonal matrix and the eigenvalues are ordered (|λ1|<|λ2|<…<|λn|), and j is the iteration. The “educated guess” σi for the eigenvalue λi is obtained from the analytic determination of the eigenvalues of (k×k)-submatrices of A̲, in the vicinity of the ith element, where k=0,1,2,3 (here, k=0 means that the implicit shift vanishes, σi=0). The routine optionally calculates the rotation matrix Z̲, such that Z̲−1A̲Z̲=D̲ where A̲ is the input matrix and D̲ is the diagonal matrix containing the eigenvalues. The ith column of Z̲ then is the eigenvector of A̲ corresponding to the eigenvalue found at the element D̲(i,i), in the ith position on the diagonal of the matrix D̲.Unusual features:For simplicity, the “wrapper” program which contains an example application and the HTDQLS routine are distributed in the same file.

GPU parallelization of the sequential matrix diagonalization algorithm and its application to high-dimensional data

This paper presents the parallelization on a GPU of the sequential matrix diagonalization (SMD) algorithm, a method for diagonalizing polynomial covariance matrices, which is the most recent technique for polynomial eigenvalue decomposition. We first parallelize with CUDA the calculation of the polynomial covariance matrix. Then, following a formal transformation of the polynomial matrix multiplication code—extensively used by SMD—we insert in this code the cublasDgemm function of CUBLAS library. Furthermore, a specialized cache memory system is implemented within the GPU to greatly limit the PC-to-GPU transfers of slices of polynomial matrices. The resulting SMD code can be applied efficiently over high-dimensional data. The proposed method is verified using sequences of images of airplanes with varying spatial orientation. The performance of the parallel codes for polynomial covariance matrix generation and SMD is evaluated and reveals speedups of up to 161 and 67, respectively, relative to sequential execution on a PC.

Sequential Matrix Diagonalization Algorithms for Polynomial EVD of Parahermitian Matrices

For parahermitian polynomial matrices, which can be used, for example, to characterize space-time covariance in broadband array processing, the conventional eigenvalue decomposition (EVD) can be generalized to a polynomial matrix EVD (PEVD). In this paper, a new iterative PEVD algorithm based on sequential matrix diagonalization (SMD) is introduced. At every step the SMD algorithm shifts the dominant column or row of the polynomial matrix to the zero lag position and eliminates the resulting instantaneous correlation. A proof of convergence is provided, and it is demonstrated that SMD establishes diagonalization faster and with lower order operations than existing PEVD algorithms.

Spectroscopy of neutron-rich Te and Xe isotopes within a new shell model context

Te and Xe isotopes above the $N=82$ shell closure are investigated within a large-scale shell model approach based on an iterative matrix diagonalization algorithm. The spectra and transition strengths, computed using a realistic Hamiltonian, are in overall agreement with the available experimental data. The calculation predicts an increasing neutron weight in the lowest collective ${2}_{1}^{+}$ state of the isotopes as they depart from the doubly magic ${}^{132}$Sn and move toward the neutron drip line. Such a neutron dominance is predicted to cause a breaking of the neutron-proton exchange symmetry and a dramatic drop of the strengths of the $E2$ and $M1$ transitions among the excited ${2}^{+}$ states. This drop establishes a strong asymmetry between Te and Xe isotopes above and below the $N=82$ shell closure.

Complex nuclear spectra in a large scale shell model approach

We report on a shell model implementation of an iterative matrix diagonalization algorithm in the spin uncoupled scheme. A new importance sampling is adopted which brings the eigenvalues to convergence with about 10% of the basis states. The method is shown to be able to provide an exhaustive description of the low-energy spectroscopic properties of 132–134Xe isotopes and of the spectrum of 130Xe.

Importance-sampling diagonalization algorithm for large-scale shell model calculations onN=80isotones

An iterative matrix diagonalization algorithm endowed with an importance sampling, recently reformulated and tested through large-scale shell model calculations on some heavy Xe isotopes, is here applied to $N=80$ isotones. For these nuclei, the iterative sampling procedure is shown to converge more rapidly than in the case of Xe isotopes. The improved convergence rate enables us to produce accurate eigenvalues and eigenfunctions even in spaces of dimensions $N\ensuremath{\sim}{10}^{9}$. It is therefore possible to offer an exhaustive description of low-energy spectroscopic properties for all $N=80$ isotones from ${}^{132}$Te up to ${}^{140}$Nd through the calculation of spectra and transition strengths. The results are fairly consistent with the available experimental data and with other shell model calculations based on the Lanczos algorithm.

Nuclear spectra in a large scale shell model approach

An iterative matrix diagonalization algorithm is briefly outlined. Its implementation in the spin uncoupled scheme for large scale shell model calculations is discussed with special attention at the convergence properties of the iterative process which establish the domain of applicability of the method. Within this domain, the method is able to generate a large number of eigenstates and, therefore, to provide an exhaustive description of low-lying nuclear spectra. The performance of the method is illustrated by its application to 130–134Xe isotopes.

Matrix diagonalization algorithm and its applicability to the nuclear shell model

An importance-sampling iterative algorithm for diagonalizing shell model Hamiltonian matrices is reviewed and implemented in a spin uncoupled basis. Shell model spaces of dimensions up to $N\ensuremath{\lesssim}{10}^{9}$ are considered. The analysis shows that about $10%$ of the basis states are enough to bring the eigenvalues to convergence. This fraction of states, however, is insufficient to lead to convergence of the transition strengths, thereby limiting the applicability of the method to not too large spaces. In its domain of validity, the method yields a large number of eigensolutions and can be usefully adopted for rather complete studies of low-energy spectroscopy. This is done here for ${}^{132,134}$Xe isotopes. The calculation yields spectra and electromagnetic responses in fairly good agreement with the available experimental data and unveils the properties of the low-energy states of these isotopes, including their proton-neutron symmetry.

An upgraded algorithm for shell model calculations and its implementation in medium-heavy nuclei

A matrix diagonalization algorithm we developed in the past years has been revised and implemented for large scale shell model calculations using a spin uncoupled basis so as to obtain a sparse Hamiltonian matrix. The dimensions of such a matrix are reduced drastically by a procedure that combines an effective projection of the good angular momentum with an importance sampling specific of the method. Its numerical implementation on 116Sn illustrates the simplicity of the method and the fast convergence of the iterative procedure.

Spectral Chebyshev Collocation for the Poisson and Biharmonic Equations

This paper is concerned with the spectral Chebyshev collocation solution of the Dirichlet problems for the Poisson and biharmonic equations in a square. The collocation schemes are solved at a cost of $2N^3+O(N^2\log N)$ operations using an appropriate set of basis functions, a matrix diagonalization algorithm, and fast Fourier transforms. For the biharmonic problem, the resulting Schur complement system is solved by a preconditioned biconjugate gradient method. An application of the Poisson spectral preconditioner is discussed for the solution of a variable coefficient spectral problem. Numerical results confirm the efficiency of the proposed algorithms and the spectral and polynomial accuracy of the collocation schemes for smooth and singular solutions, respectively.

Theoretical studies and efficient algorithm of semi-blind ICI equalization for OFDM

The intercarrier interference (ICI) due to the Doppler frequency shift, sampling clock offset, time-varying multipath fading and local oscillator frequency offset becomes the major difficulty for the data transmission via the wireless orthogonal frequency division multiplexing (OFDM) systems. The existing ICI mitigation schemes involve the frequency-domain channel estimation/equalization or the additional coding and therefore require the pilot symbols which reduce the throughput. The frequency-domain channel estimation/equalization relies on the huge matrix inversion with high computational complexity especially for the OFDM technologies possessing many subcarriers such as digital video broadcasting (DVB) systems and wireless metropolitan-area networks (WMAN). In our previous work, we proposed a semi-blind ICI equalization scheme using the joint multiple matrix diagonalization (JMMD) algorithm and empirically showed that the proposed method significantly improved the symbol error rates for QPSK- and 16QAM-OFDM systems. In this paper, we discuss the sufficient condition for the theoretical ICI equalizability and also propose an alternative semi-blind ICI equalization method based on the joint approximate diagonalization of eigen-matrices (JADE) algorithm, which is much more computationally efficient than our previous method.

Robust Blind Beamforming Algorithm Using Joint Multiple Matrix Diagonalization

The objective of the blind beamforming is to restore the unknown source signals simply based on the observations, without a priori knowledge of the source signals and the mixing matrix. In this paper, we propose a new joint multiple matrix diagonalization (JMMD) algorithm for the robust blind beamforming. This new JMMD algorithm is based on the iterative eigen decomposition of the fourth-order cumulant matrices. Therefore, it can avoid the problems of the stability and the misadjustment, which arise from the conventional steepest-descent approaches for the constant-modulus or cumulant optimization. Our Monte Carlo simulations show that our proposed algorithm significantly outperforms the ubiquitous joint approximate diagonalization of eigen-matrices algorithm, relying on the Givens rotations for the phase-shift keying source signals in terms of signal-to-interference-and-noise ratio for a wide variety of signal-to-noise ratios

Novel Semi-Blind ICI Equalization Algorithm for Wireless OFDM Systems

Intercarrier interference is deemed as one of the crucial problems in the wireless orthogonal frequency division multiplexing (OFDM) systems. The conventional ICI mitigation schemes involve the frequency-domain channel estimation or the additional coding, both of which require the spectral overhead and hence lead to the significant throughput reduction. Besides, the OFDM receivers using the ICI estimation rely on a large-dimensional matrix inverter with high computational complexity especially for many subcarriers such as digital video broadcasting (DVB) systems and wireless metropolitan-area networks (WMAN). To the best of our knowledge, no semi-blind ICI equalization has been addressed in the existing literature. Thus, in this paper, we propose a novel semi-blind ICI equalization scheme using the joint multiple matrix diagonalization (JMMD) algorithm to greatly reduce the intercarrier interference in OFDM. However, the well-known phase and permutation indeterminacies emerge in all blind equalization schemes. Hence we also design a few OFDM pilot blocks and propose an iterative identification method to determine the corresponding phase and permutation variants in our semi-blind scheme. Our semi-blind ICI equalization algorithm integrating the JMMD with the additional pilot-based iterative identification is very promising for the future high-throughput OFDM systems. Through Monte Carlo simulations, the QPSK-OFDM system with our proposed semi-blind ICI equalizer can achieve significantly better performance with symbol error rate reduction in several orders-of-magnitude. For the 16QAM-OFDM system, our scheme can also improve the performance over the plain OFDM system to some extent.

QUANTUM GENETIC ALGORITHM METHOD IN SELF-CONSISTENT ELECTRONIC STRUCTURE CALCULATIONS OF A QUANTUM DOT WITH MANY ELECTRONS

In this study, we have calculated energy levels of an N-electron quantum dot. For this purpose, we have used two different techniques, matrix diagonalization and quantum genetic algorithm, to obtain simultaneous solutions of the coupled Schrödinger and Poisson equation in the Hartree approximation. We have determined single particle energy levels, total energy, chemical potential and capacitive energy. We have also compared the results, demonstrated the applicability of QGA to many-electron quantum systems and evaluated its effectiveness.

On the interplay between inner and outer Krylov spaces applied to large matrix diagonalization algorithms

Large-scale matrix diagonalization techniques are reviewed in the context of Lagrangian function optimization. Based in the effective Hamiltonian theory, the idea of separation of a general matrix–vector problem in inner and outer space components relative to a given model (or reference) space is introduced to improve the Lagrange–Newton–Raphson targeted diagonalization scheme ( LNRd). A new effective Krylov space Lagrange–Newton–Raphson diagonalization ( EKS-LNRd) algorithm is proposed and compared with the previous LNRd and Davidson methods and some test cases are shown and discussed.

Canonical Van Vleck Quasidegenerate Perturbation Theory with Trigonometric Variables

A canonical Van Vleck quasidegenerate perturbation theory that uses trigonometric transformed generators of the wave operator is presented. The equations for the first-order corrections to the wave operator generator are similar to those used in the stable Jacobi matrix diagonalization algorithm. Use of the trigonometric transformed variables can be seen to eliminate numerical instabilities due to zero-order quasidegeneracy. The separate question of orbital energies for use with the Møller−Plesset partitioned form of this quasidegenerate perturbation theory is also addressed. Results of calculations, through second order, for electron correlation energy in benchmark molecules are presented (i.e., DZ H2O, 6-31G NH2, DZP CH2, (DZ+)P F2, DZP O2). Remarkably high accuracy is observed already in the second-order implementation.