Symmetric Band Matrices Research Articles

ITVOLT: An iterative solver for the time-dependent Schrödinger equation

We present a novel approach for solving the time-dependent Schrödinger equation (TDSE). The method we propose converts the TDSE to an equivalent Volterra integral equation; introducing a global Lagrange interpolation of the integrand transforms the equation to a linear system, which is then solved iteratively. In this paper, we derive the method, explore its performance on several examples, and discuss the corresponding numerical details. Program summaryProgram Title: Iterative Volterra Propagator (ITVOLT)CPC Library link to program files:https://doi.org/10.17632/jzd5xpv6t7.1Developer's repository link:https://github.com/ry-schneider/Iterative_Volterra_Propagator.gitLicensing provisions: MITProgramming language: Modern FortranNature of problem: ITVOLT is a solver for the time-dependent Schrödinger equation (TDSE). More broadly, it can be applied to any problem that can be written as a Volterra integral equation.Solution method: ITVOLT solves a Volterra integral equation representation of the TDSE by reducing it to a linear system via Lagrange interpolation of the integrand. It then solves the system iteratively with one of several iteration schemes.Additional comments including restrictions and unusual features: General subroutines assume that the Hamiltonian for the TDSE being solved can be represented by a symmetric banded matrix whose time-dependent component is some scalar function of time (for example a pulse) multiplied by a fixed matrix.

SUSY transfer matrix approach for the real symmetric 1d random band matrices

This paper adapts the recently developed rigorous application of the supersymmetric transfer matrix approach for the Hermitian 1d band matrices to the case of the orthogonal symmetry. We consider N×N block band matrices consisting of W×W random Gaussian blocks (parametrized by j,k∈Λ=[1,n]∩Z, N=nW) with a fixed entry’s variance Jjk=W−1(δj,k+βΔj,k) in each block. Considering the limit W,n→∞, we prove that the behaviour of the second correlation function of characteristic polynomials of such matrices in the bulk of the spectrum exhibit a crossover near the threshold W∼ N.

Symmetric Polynomials in the Symplectic Alphabet and the Change of Variables $z_j = x_j + x_j^{-1}$

Given a symmetric polynomial $P$ in $2n$ variables, there exists a unique symmetric polynomial $Q$ in $n$ variables such that\[P(x_1,\ldots,x_n,x_1^{-1},\ldots,x_n^{-1})=Q(x_1+x_1^{-1},\ldots,x_n+x_n^{-1}).\] We denote this polynomial $Q$ by $\Phi_n(P)$ and show that $\Phi_n$ is an epimorphism of algebras. We compute $\Phi_n(P)$ for several families of symmetric polynomials $P$: symplectic and orthogonal Schur polynomials, elementary symmetric polynomials, complete homogeneous polynomials, and power sums. Some of these formulas were already found by Elouafi (2014) and Lachaud (2016).
 The polynomials of the form $\Phi_n(\operatorname{s}_{\lambda/\mu}^{(2n)})$, where $\operatorname{s}_{\lambda/\mu}^{(2n)}$ is a skew Schur polynomial in $2n$ variables, arise naturally in the study of the minors of symmetric banded Toeplitz matrices, when the generating symbol is a palindromic Laurent polynomial, and its roots can be written as $x_1,\ldots,x_n,x^{-1}_1,\ldots,x^{-1}_n$. Trench (1987) and Elouafi (2014) found efficient formulas for the determinants of symmetric banded Toeplitz matrices. We show that these formulas are equivalent to the result of Ciucu and Krattenthaler (2009) about the factorization of the characters of classical groups.

A fast spectral divide‐and‐conquer method for banded matrices

AbstractBased on the spectral divide‐and‐conquer algorithm by Nakatsukasa and Higham [SIAM J. Sci. Comput., 35(3):A1325–A1349, 2013], we propose a new algorithm for computing all the eigenvalues and eigenvectors of a symmetric banded matrix with small bandwidth, with the eigenvectors given implicitly as a product of orthonormal matrices stored in the so‐called hierarchically off‐diagonal low‐rank (HODLR) format. For this purpose, we combine our previous work on the fast computation of spectral projectors in the HODLR format, with a novel technique for extracting a basis for the range of such a HODLR matrix. Preliminary numerical experiments demonstrate that our algorithm exhibits quasi‐linear complexity for matrices that can be efficiently represented in the HODLR format throughout the divide‐and‐conquer algorithm, and allows for conveniently dealing with such large‐scale matrices.

Nullities for a class of 0-1 symmetric Toeplitz band matrices

Let $S(n,k)$ denote the $n \times n$ symmetric Toeplitz band matrix whose first $k$ superdiagonals and first $k$ subdiagonals have all entries $1$, and whose remaining entries are all $0$. For all $n > k >0$ with $k$ even, we give formulas for the nullity of $S(n,k)$. As an application, it is shown that over half of these matrices $S(n,k)$ are nonsingular. For the purpose of rapid computation, we devise an algorithm that quickly computes the nullity of $S(n,k)$ even for extremely large values of $n$ and $k$, when $k$ is even. The algorithm is based on a connection between the nullspace vectors of $S(n,k)$ and the cycles in a certain directed graph.

Solution of the symmetric band partial inverse eigenvalue problem for the damped mass spring system

ABSTRACT The structured partial quadratic inverse eigenvalue problem (SPQIEP) is to construct the structured quadratic matrix polynomial using the partial eigendata. The structures arising in physical applications include symmetry, band (tridiagonal, diagonal, pentagonal) etc. The construction of the structured matrix polynomial is the most difficult aspect of this problem and the research on structured inverse eigenvalue problem is rare. In this paper, the symmetric band partial quadratic inverse eigenvalue problem (SBPQIEP) for the damped mass spring system is considered. This problem concerns in finding the symmetric band matrices , and C with bandwidth p from m ( ) prescribed eigenpairs so that the corresponding quadratic matrix polynomial has the given eigenpairs as its eigenvalues and eigenvectors. In general, SBPQIEP is very hard to solve due to the additional band structure constraint. We propose a novel method, based on the matrix-vectorization and Kronecker product of matrices for solving this problem. Furthermore, explicit expressions for general solutions are presented. Numerical experiments on a spring mass problem are presented to illustrate the applicability and the practical usefulness of the proposed method.

Parallel gaussian elimination of symmetric positive definite band matrices for shared-memory multicore architectures

This study presents a new parallel Gaussian elimination approach for symmetric positive definite band systems. For each task, the appropriate start time and adequate processor are determined. Unnecessary dependencies between tasks are eliminated. Simultaneously, all processors perform their associated tasks with precedence constraints under consideration. Our main goal is to obtain a high degree of parallelism by balancing the load of processors and reducing the total idle and parallel execution times. The theoretical lower bounds for parallel execution time and number of processors required to execute the precedence graph at an optimal time are also computed. The validity of our investigation is confirmed by carrying out several experiments on a shared-memory multicore architecture using OpenMP. Practical results prove the efficiency of the proposed method.

Random band matrices in the delocalized phase, III: averaging fluctuations

We consider a general class of symmetric or Hermitian random band matrices $$H=(h_{xy})_{x,y \in \llbracket 1,N\rrbracket ^d}$$ in any dimension $$d\ge 1$$ , where the entries are independent, centered random variables with variances $$s_{xy}=\mathbb {E}|h_{xy}|^2$$ . We assume that $$s_{xy}$$ vanishes if $$|x-y|$$ exceeds the band width W, and we are interested in the mesoscopic scale with $$1\ll W\ll N$$ . Define the generalized resolvent of H as $$G(H,Z):=(H - Z)^{-1}$$ , where Z is a deterministic diagonal matrix with entries $$Z_{xx}\in \mathbb {C}_+$$ for all x. Then we establish a precise high-probability bound on certain averages of polynomials of the resolvent entries. As an application of this fluctuation averaging result, we give a self-contained proof for the delocalization of random band matrices in dimensions $$d\ge 2$$ . More precisely, for any fixed $$d\ge 2$$ , we prove that the bulk eigenvectors of H are delocalized in certain averaged sense if $$N\le W^{1+\frac{d}{2}}$$ . This improves the corresponding results in He and Marcozzi (Diffusion profile for random band matrices: a short proof, 2018. arXiv:1804.09446 ) that imposed the assumption $$N\ll W^{1+\frac{d}{d+1}}$$ , and the results in Erdős and Knowles (Ann Henri Poincare12(7):1227–1319, 2011; Commun Math Phys 303(2): 509–554, 2011) that imposed the assumption $$N\ll W^{1+\frac{d}{6}}$$ . For 1D random band matrices, our fluctuation averaging result was used in Bourgade et al. (J Stat Phys 174:1189–1221, 2019; Random band matrices in the delocalized phase, I: quantum unique ergodicity and universality, 2018. arXiv:1807.01559 ) to prove the delocalization conjecture and bulk universality for random band matrices with $$N\ll W^{4/3}$$ .

Random Band Matrices in the Delocalized Phase I: Quantum Unique Ergodicity and Universality

Consider $N\times N$ symmetric one-dimensional random band matrices with general distribution of the entries and band width $W \geq N^{3/4+\varepsilon}$ for any $\varepsilon>0$. In the bulk of the spectrum and in the large $N$ limit, we obtain the following results. (i) The semicircle law holds up to the scale $N^{-1+\varepsilon}$ for any $\varepsilon>0$. (ii) The eigenvalues locally converge to the point process given by the Gaussian orthogonal ensemble at any fixed energy. (iii) All eigenvectors are delocalized, meaning their ${\rm L}^\infty$ norms are all simultaneously bounded by $N^{-\frac{1}{2}+\varepsilon}$ (after normalization in ${\rm L}^2$) with overwhelming probability, for any $\varepsilon>0$. (iv )Quantum unique ergodicity holds, in the sense that the local ${\rm L}^2$ mass of eigenvectors becomes equidistributed with overwhelming probability. We extend the mean-field reduction method \cite{BouErdYauYin2017}, which required $W=\Omega(N)$, to the current setting $W \ge N^{3/4+\varepsilon}$. Two new ideas are: (1) A new estimate on the generalized resolvent of band matrices when $W \geq N^{3/4+\varepsilon}$. Its proof, along with an improved fluctuation average estimate, will be presented in parts 2 and 3 of this series \cite {BouYanYauYin2018,YanYin2018}. (2) A strong (high probability) version of the quantum unique ergodicity property of random matrices. For its proof, we construct perfect matching observables of eigenvector overlaps and show they satisfying the eigenvector moment flow equation \cite{BouYau2017} under the matrix Brownian motions.

Random matrices with exchangeable entries

We consider ensembles of real symmetric band matrices with entries drawn from an infinite sequence of exchangeable random variables, as far as the symmetry of the matrices permits. In general, the entries of the upper triangular parts of these matrices are correlated and no smallness or sparseness of these correlations is assumed. It is shown that the eigenvalue distribution measures still converge to a semicircle but with random scaling. We also investigate the asymptotic behavior of the corresponding [Formula: see text]-operator norms. The key to our analysis is a generalization of a classic result by de Finetti that allows to represent the underlying probability spaces as averages of Wigner band ensembles with entries that are not necessarily centered. Some of our results appear to be new even for such Wigner band matrices.

Evidence of the Poisson/Gaudin–Mehta phase transition for band matrices on global scales

We prove that the Poisson/Gaudin–Mehta phase transition conjectured to occur when the bandwidth of an [Formula: see text] symmetric band matrix grows like [Formula: see text] is naturally observable in the rate of convergence of the level density to the Wigner semi-circle law. Specifically, we show for periodic and non-periodic band matrices the rate of convergence of the fourth moment of the level density is independent of the boundary conditions in the localized regime [Formula: see text] with a rate of [Formula: see text] for both cases, whereas in the delocalized regime [Formula: see text] where boundary effects become important, the rate of convergence for the two ensembles differs significantly, slowing to [Formula: see text] for non-periodic band matrices. Additionally, we examine the case of thick non-periodic band matrices [Formula: see text], showing that the fourth moment is maximally deviated from the Wigner semi-circle law when [Formula: see text], and provide numerical evidence that the eigenvector statistics also exhibit critical behavior at this point.

A symmetrical formulation of coupled BEM/FEM for submerged structures based on acoustical reciprocity

The paper presents a structural acoustics formulation for elastic structures submerged in water. The structural equation is described by a finite element method where the linear displacement variables on the wetted surface are chosen locally as one normal displacement and the other two displacements tangent to the wetted surface, whereas the rest of rotational displacements or degrees of freedom not contacting with water are defined globally. A boundary element formulation describing the acoustic loading on the structure is expressed as a function of normal velocity or displacement on the wetted surface. The coupling of the FEM and BEM is through the normal velocity, or equivalently, the normal displacement on the wetted surface. An acoustical reciprocity is used to prove that the associated acoustic loading expressed as the normal displacement of the wetted structure is a complex symmetric matrix. This matrix can be viewed an acoustic element whose degrees of freedom are the normal displacements of the wetted surface. Thus, the coupled FEM and BEM becomes a symmetric banded matrix formulation, leading to an efficient numerical way to solve the equation. A capped cylindrical shell with periodical ring stiffeners and bulkheads submerged in water is used to demonstrate the present numerical method.

A class of inverse eigenvalue problems for real symmetric banded matrices with odd bandwidth

In this paper, we propose and discuss a class of inverse eigenvalue problems for real symmetric banded matrices with odd bandwidth. For an odd 2p+1 with a positive integer p, the problem is to construct an n×n real symmetric banded matrix with bandwidth 2p+1 whose m×m leading principal submatrix is a given m×m real symmetric banded matrix with bandwidth 2p+1 and spectrum is a given set of real numbers {λi}i=1n, where the number of distinct real numbers of {λi}i=1n is 2k when m=pk, or 2k+1 when pk<m<p(k+1), where m,n and k are positive integers and m<n. We point out that the well-known double dimensional (DD) problem is a special case of our proposed inverse eigenvalue problems. The necessary and sufficient condition for the solvability of the above inverse eigenvalue problem is derived, and the target real symmetric banded matrix can be constructed by the block Lanczos algorithms when the inverse eigenvalue problem is solvable. Several numerical examples show that our algorithms are feasible. Some concluding remarks are introduced.

A class of non-symmetric band determinants with the Gaussian q-binomial coefficients

A class of symmetric band matrices of bandwidth 2r+1 with the binomial coefficients entries was studied earlier. We consider a class of non-symmetric band matrices with the Gaussian q-binomial coefficients whose upper bandwith is s and lower bandwith is r. We give explicit formulæ for the determinant, the inverse (along with its infinity-norm when q → 1) and the LU -decomposition of the class. We use the celebrated q-Zeilberger algorithm and unimodality property to prove claimed results.

A fixed-point method for approximate projection onto the positive semidefinite cone

The projection of a symmetric matrix onto the positive semidefinite cone is an important problem with application in many different areas such as economy, physics and, directly, semidefinite programming. This problem has analytical solution, but it relies on the eigendecomposition of a given symmetric matrix which clearly becomes prohibitive for larger dimension and dense matrices. We present a fixed-point iterative method for computing an approximation of such projection. Each iteration requires matrix–matrix products whose costs may be much less than O(n3) for certain structured matrices. Numerical experiments showcase the attractiveness of the proposed approach for sparse symmetric banded matrices.

Time domain equalization method for TS-OFDM signal under higher mobile environments

In higher time-varying fading channels, the signal quality of orthogonal frequency division multiplexing (OFDM) technique would be degraded relatively due to the occurrence of inter-carrier interference (ICI). To solve this problem, this paper firstly proposes a new design of time domain training sequence (TS) in the estimation of channel impulse response (CIR) for the TS-OFDM signal which can reduce the leakage of power spectrum density (PSD) at the outside of OFDM allocated frequency bandwidth with keeping higher CIR estimation accuracy. Secondly, this paper proposes a time domain equalization (TDE) method which can achieve better bit error rate (BER) performance with keeping lower computation complexity even in higher time-varying fading channels. The salient feature of the proposed TDE method is to employ a partial differentiation for the time domain CIR matrix for solving the maximum likelihood (ML) equation in which the time domain CIR matrix becomes a symmetric banded matrix. From this feature, a low-complexity parallel block inverse matrix algorithm can be employed in the calculation of inverse matrix in keeping the same accuracy as that of the direct inverse matrix calculation. This paper presents various computer simulation results to demonstrate the effectiveness of the proposed TDE method for the TS-OFDM signal as compared with conventional frequency domain equalization (FDE) methods under higher mobile environments.

Fast Computation of Spectral Projectors of Banded Matrices

We consider the approximate computation of spectral projectors for symmetric banded matrices. While this problem has received considerable attention, especially in the context of linear scaling electronic structure methods, the presence of small relative spectral gaps challenges existing methods based on approximate sparsity. In this work, we show how a data-sparse approximation based on hierarchical matrices can be used to overcome this problem. We prove a priori bounds on the approximation error and propose a fast algorithm based on the QR-based dynamically weighted Halley (QDWH) algorithm, along the lines of works by Nakatsukasa and colleagues. Numerical experiments demonstrate that the performance of our algorithm is robust with respect to the spectral gap. A preliminary MATLAB implementation becomes faster than eig already for matrix sizes of a few thousand.

Sparse solution of the Lyapunov equation for large-scale interconnected systems

We consider the problem of computing an approximate banded solution of the continuous-time Lyapunov equation A¯X¯+X¯A¯T=P¯, where the coefficient matrices A¯ and P¯ are large, symmetric banded matrices. The (sparsity) pattern of A¯ describes the interconnection structure of a large-scale interconnected system. Recently, it has been shown that the entries of the solution X¯ are spatially localized or decaying away from a banded pattern. We show that the decay of the entries of X¯ is faster if the condition number of A¯ is smaller. By exploiting the decay of entries of X¯, we develop two computationally efficient methods for approximating X¯ by a banded matrix. For a well-conditioned and sparse banded A¯, the computational and memory complexities of the methods scale linearly with the state dimension. We perform extensive numerical experiments that confirm this, and that demonstrate the effectiveness of the developed methods. The methods proposed in this paper can be generalized to (sparsity) patterns of A¯ and P¯ that are more general than banded matrices. The results of this paper open the possibility for developing computationally efficient methods for approximating the solution of the large-scale Riccati equation by a sparse matrix.

Universality of the second mixed moment of the characteristic polynomials of the 1D band matrices: Real symmetric case

We prove that the asymptotic behavior of the second mixed moment of the characteristic polynomials of the N × N 1D Gaussian real symmetric band matrices with the width of the band W ≫ N1/2 coincides with those for the Gaussian orthogonal ensemble. Here, we adapt the approach of Shcherbina [Commun. Math. Phys. 328, 45–82 (2014)], where the Hermitian case was considered.

Avoiding Communication in Successive Band Reduction

The running time of an algorithm depends on both arithmetic and communication (i.e., data movement) costs, and the relative costs of communication are growing over time. In this work, we present sequential and distributed-memory parallel algorithms for tridiagonalizing full symmetric and symmetric band matrices that asymptotically reduce communication compared to previous approaches. The tridiagonalization of a symmetric band matrix is a key kernel in solving the symmetric eigenvalue problem for both full and band matrices. In order to preserve structure, tridiagonalization routines use annihilate-and-chase procedures that previously have suffered from poor data locality and high parallel latency cost. We improve both by reorganizing the computation and obtain asymptotic improvements. We also propose new algorithms for reducing a full symmetric matrix to band form in a communication-efficient manner. In this article, we consider the cases of computing eigenvalues only and of computing eigenvalues and all eigenvectors.