Block Tridiagonal Matrix Research Articles

A Semianalytical Spectral Element Method for the Analysis of 3-D Layered Structures

A semianalytical spectral element method (SEM) is proposed for electromagnetic simulations of 3-D layered structures. 2-D spectral elements are employed to discretize the cross section of a layered structure, and the Legendre transformation is then used to cast the semidiscretized problem from the Lagrangian system into the Hamiltonian system. A Riccati equation-based high precision integration method is utilized to perform integration along the longitudinal direction, which is the undiscretized direction, to generate the stiffness matrix of the whole layered structure. The final system of equations by the semianalytical SEM will take the form of a set of linear equations with a block tri-diagonal matrix, which can be solved efficiently by the block Thomas algorithm. Numerical examples demonstrate the high efficiency and accuracy of the proposed method.

Development of a High-speed Eigenvalue-solver for Constant Plasma Monitoring on a Cell Cluster System

Abstract We developed a high speed eigenvalue solver that is an essential part of a plasma stability analysis system for fusion reactors on a Cell cluster system. In order to achieve continuous operation of fusion reactors, we must evaluate the state of plasma within the characteristic confinement time of the plasma density and temperature in fusion reactors. This is because we can prevent plasma from being disrupted by controlling the confining magnetic field, if we can determine the state of the plasma within the characteristic confinement time. Therefore, we introduced a Cell processor that has high computational power and high performance/cost, in order to achieve constant monitoring of fusion reactors. Furthermore, we developed a novel eigenvalue solver, which usually consumes most of the plasma evaluation time, to achieve high performance of our Cell cluster system. The eigensolver is based on the conjugate gradient (CG) method and was designed by considering three levels of parallelism, which we refer to as Intra-processor, Inner-processor, and SIMD parallel. In addition, we developed a new CG acceleration method, called locally complete LU. This method has the same acceleration performance as complete LU, which is one of the best acceleration methods, without any reduction in parallel performance. Finally, we succeeded in obtaining our target performance: we were able to solve a block tri-diagonal Hermitian matrix containing 1024 diagonal blocks, where the size of each block was 128 × 128, within a second. Therefore, we have found a suitable candidate for achieving a satisfactory monitoring system.

Precise integration method for solving singular perturbation problems

This paper presents a precise method for solving singularly perturbed boundary-value problems with the boundary layer at one end. The method divides the interval evenly and gives a set of algebraic equations in a matrix form by the precise integration relationship of each segment. Substituting the boundary conditions into the algebraic equations, the coefficient matrix can be transformed to the block tridiagonal matrix. Considering the nature of the problem, an efficient reduction method is given for solving singular perturbation problems. Since the precise integration relationship introduces no discrete error in the discrete process, the present method has high precision. Numerical examples show the validity of the present method.

BCYCLIC: A parallel block tridiagonal matrix cyclic solver

A block tridiagonal matrix is factored with minimal fill-in using a cyclic reduction algorithm that is easily parallelized. Storage of the factored blocks allows the application of the inverse to multiple right-hand sides which may not be known at factorization time. Scalability with the number of block rows is achieved with cyclic reduction, while scalability with the block size is achieved using multithreaded routines (OpenMP, GotoBLAS) for block matrix manipulation. This dual scalability is a noteworthy feature of this new solver, as well as its ability to efficiently handle arbitrary (non-powers-of-2) block row and processor numbers. Comparison with a state-of-the art parallel sparse solver is presented. It is expected that this new solver will allow many physical applications to optimally use the parallel resources on current supercomputers. Example usage of the solver in magneto-hydrodynamic (MHD), three-dimensional equilibrium solvers for high-temperature fusion plasmas is cited.

Qd block algorithm

The block qd algorithm is studied in order to obtain some properties about the asymptotic behavior of some eigenvalues of a block tridiagonal positive definite symmetric matrix. We prove that the eigenvalues of the first block on the block diagonal of the decomposition given by the block qd algorithm at the different stages of this algorithm constitute strictly increasing sequences and those of the last block constitute strictly decreasing sequences. Moreover the convergence of this qd algorithm is proved under certain assumptions.

On twisted factorizations of block tridiagonal matrices

Abstract Non-symmetric and symmetric twisted block factorizations of block tridiagonal matrices are discussed. In contrast to non-blocked factorizations of this type, localized pivoting strategies can be integrated which improves numerical stability without causing any extra fill-in. Moreover, the application of such factorizations for approximating an eigenvector of a block tridiagonal matrix, given an approximation of the corresponding eigenvalue, is outlined. A heuristic strategy for determining a suitable starting vector for the underlying inverse iteration process is proposed.

Extremal algebraic connectivities of certain caterpillar classes and symmetric caterpillars

A caterpillar is a tree in which the removal of all pendant vertices makes it a path. Let d ≥ 3 and n ≥ 6 be given. Let Pd−1 be the path of d − 1 vertices and Sp be the star of p + 1 vertices. Let p = [p1,p2,...,pd−1] such that p1 ≥ 1, p2 ≥ 1, ...,pd−1 ≥ 1. Let C (p) be the caterpillar obtained from the stars Sp1,Sp2, ...,Spd−1 and the path Pd−1 by identifying the root of Spi with the i−vertex of Pd−1. Let n > 2(d − 1) be given. LetC = {C (p) : p1 + p2 + ... + pd−1 = n − d + 1}andS = {C(p) ∈ C : pj = pd−j, j = 1,2,··· ,⌊d − 1 2⌋}.In this paper, the caterpillars in C and in S having the maximum and the minimum algebraic connectivity are found. Moreover, the algebraic connectivity of a caterpillar in S as the smallest eigenvalue of a 2 × 2 - block tridiagonal matrix of order 2s × 2s if d = 2s + 1 or d = 2s + 2 is characterized.

Geometry of Weyl theory for Jacobi matrices with matrix entries

A Jacobi matrix with matrix entries is a self-adjoint block tridiagonal matrix with invertible blocks on the off-diagonals. The Weyl surface describing the dependence of Green’s matrix on the boundary conditions is interpreted as the set of maximally isotropic subspaces of a quadratic form given by the Wronskian. Analysis of the possibly degenerate limit quadratic form leads to the limit point/limit surface theory of maximal symmetric extensions for semi-infinite Jacobi matrices with matrix entries with arbitrary deficiency indices. The resolvent of the extensions is calculated explicitly.

Optimal Portfolio Execution Strategies and Sensitivity to Price Impact Parameters

When liquidating a portfolio of large blocks of risky assets, an institutional investor wants to minimize the cost as well as the risk of execution. An optimal execution strategy minimizes a weighted combination of the expected value and the variance of the execution cost, where the weight is given by a nonnegative risk aversion parameter. The execution cost is determined from price impact functions. In particular, a linear price impact model is defined by the temporary impact matrix $\Omega$ and the permanent impact matrix $\Gamma$, which represent the expected price depression caused by trading assets at a unit rate. In this paper, we analyze the sensitivity of the optimal execution strategy to estimation errors in the impact matrices under a linear price impact model. We show that, instead of depending on $\Omega$ and $\Gamma$ individually, the optimal execution strategy is determined by the combined impact matrix $\Theta=\frac{1}{\tau}\,\bigl(\Omega+\Omega^{T}\bigr)-\Gamma$, where $\tau$ is the time length between consecutive trades. We prove that the minimum expected execution cost strategy is the naive execution strategy, independent of perturbations, when the permanent impact matrix $\Gamma$ is symmetric and the combined impact matrix $\Theta$ is positive definite. We provide upper bounds on the size of change in the optimal execution strategy in a general setting. These upper bounds are in terms of the changes in the impact matrices, the eigenvalues of a block tridiagonal matrix defined by $\Theta$, the risk aversion parameter, and the covariance matrix. These upper bounds indicate that, when the covariance matrix is positive definite, a large risk aversion parameter reduces the sensitivity of the optimal execution strategy. Moreover, when the permanent impact matrix $\Gamma$ and its perturbation are symmetric, the optimal execution strategy is asymptotically not sensitive to estimation errors when either the minimum eigenvalue of the covariance matrix or the minimum eigenvalue of $\Theta$ is large. In addition, our computational results confirm that the sensitivity of the optimal execution strategy to the perturbations decreases when $\Gamma$ and the perturbed permanent impact matrix are symmetric. Moreover, the change in the efficient frontier increases as the risk aversion parameter decreases for asymmetric perturbations. We consistently observe that imposing short selling constraints reduces the sensitivity of the optimal execution strategy and the efficient frontier to the perturbations.

On Minimal Eigenvalues of a Class of Tridiagonal Matrices

It is known that the worst case near-far resistance of optimum multiuser detectors for asynchronous Gaussian multiple-access channels can be expressed in terms of a class of block-tridiagonal matrices, and the minimal eigenvalues of such a class of block-tridiagonal matrices serve as a good measure of the worst case near-far resistance. In this paper, we focus on the two-user scenario where each block-tridiagonal matrix under consideration is a tridiagonal matrix. We derive closed-form expressions for the minimal eigenvalues of such a class of tridiagonal matrices in terms of the largest real solution of a trigonometric equation in <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[0,pi]</i> . We also obtain lower bounds and upper bounds on the minimal eigenvalues which improve on previously known results in the literature.

Optimal block-tridiagonalization of matrices for coherent charge transport

Numerical quantum transport calculations are commonly based on a tight-binding formulation. A wide class of quantum transport algorithms require the tight-binding Hamiltonian to be in the form of a block-tridiagonal matrix. Here, we develop a matrix reordering algorithm based on graph partitioning techniques that yields the optimal block-tridiagonal form for quantum transport. The reordered Hamiltonian can lead to significant performance gains in transport calculations, and allows to apply conventional two-terminal algorithms to arbitrarily complex geometries, including multi-terminal structures. The block-tridiagonalization algorithm can thus be the foundation for a generic quantum transport code, applicable to arbitrary tight-binding systems. We demonstrate the power of this approach by applying the block-tridiagonalization algorithm together with the recursive Green’s function algorithm to various examples of mesoscopic transport in two-dimensional electron gases in semiconductors and graphene.

Spectral averaging techniques for Jacobi matrices with matrix entries

A Jacobi matrix with matrix entries is a self-adjoint block tridiagonal matrix with invertible blocks on the off-diagonals. Averaging over boundary condi- tions leads to explicit formulae for the averaged spectral measure which can potentially be useful for spectral analysis. Furthermore, another variant of spectral averaging over coupling constants for these operators is presented.

OS0126 異物を含む一様断面弾性体に対する有限要素解析の効率化(弾性数理解析とその応用,オーガナイズドセッション)

This research proposes an efficient dynamic finite element method (FEM) for an elastic object with uniform cross-section including a different substance. In this method, the object to be analyzed is at first sliced into layers with the same thickness and then each layer is discretized into finite elements of the same pattern. This way of discretization makes the matrix into repetitive block tridiagonal matrices. The present method can obtain the solution by the repetition method. The way can save the computational time by utilising this characteristic of the repetitive block tridiagonal matrix. An example problem was solved with this method to demonstrate its effectiveness.

Stability of block LU factorization for block tridiagonal matrices

It is showed that if A is I-block diagonally dominant (II-block diagonally dominant), then the reduced matrix S preserves the same property. We also give a sufficient condition for the reduced matrix S also to be a block H -matrix when A is a block H -matrix, and some properties on the comparison matrices μ I ( A ( k ) ) , μ I I ( A ( k ) ) , μ I ( L ) , and μ I ( U ) are obtained. Finally, error analysis of block LU factorization for block tridiagonal matrix is presented.

Fast algorithm and numerical simulation for ray-tracing in 3D structure

Beginning with the method of whole path iterative ray-tracing and according to the positive definiteness of the coefficient matrix of the systems of linear equations, a symmetry block tridiagonal matrix was decomposed into the product of block bidiagonal triangular matrix and its transpose by means of Cholesky decomposition. Then an algorithm for solving systems of block bidiagonal triangular linear equations was given, which is not necessary to treat with the zero elements out of banded systems. A fast algorithm for solving the systems of symmetry block tridiagonal linear equations was deduced, which can quicken the speed of ray-tracing. Finally, the simulation based on this algorithm for ray-tracing in three dimensional media was carried out. Meanwhile, the segmentally-iterative ray-tracing method and banded method for solving the systems of block tridiagonal linear equations were compared in the same model mentioned above. The convergence condition was assumed that the L-2 norm summation for mk, 1 and mk, 2 in the whole ray path was limited in 10−6. And the calculating speeds of these methods were compared. The results show that the calculating speed of this algorithm is faster than that of conventional method and the calculated results are accurate enough. In addition, its precision can be controlled according to the requirement of ray-tracing

Determinants of block tridiagonal matrices

An identity is proven that evaluates the determinant of a block tridiagonal matrix with (or without) corners as the determinant of the associated transfer matrix (or a submatrix of it).

Properties of generalized matrix sequence

In case of block-tridiagonal matrix, the problem of calculation of a generalized double-point matrix sequence is examined. The general form of inverse matrix of the bordered matrix is obtained when the initial matrix is singular. The criterion of existence of the generalized matrix sequence is found, and the algorithm of calculation of the sequence and the structure elements of the block-tridiagonal matrices are given.

Fast approximate solution of Bloch equation for simulation of RF artifacts in Magnetic Resonance Imaging

The technique used to spot information in Magnetic Resonance Imaging (MRI) uses electromagnetic fields. Even minor perturbations of these magnetic fields can disturb the imaging process and may render clinical images inaccurate or useless. Modelling and numerical simulation of the effects of static field inhomogeneities are now well established. Less attention has been paid to mathematical modelling of the effects of radio-frequency (RF) field inhomogeneities in the imaging process. When considering RF field inhomogeneities, the major difficulty is that the mathematical expression of the magnetisation vector is not anymore explicitly known in contrast with the unperturbed case. Indeed, the Bloch equation becomes an ordinary differential equation with nonconstant coefficients that cannot be solved analytically. The use of standard numerical schemes for ordinary differential equations to compute the magnetisation vector appears to be costly and not well suited for MRI image simulation. In this paper, we present an original method for solving the Bloch equation based on a truncated series expansion of the solution. The computational cost of the method reduces to the computation of the eigenelements of a block tridiagonal matrix of a very small size.

Block tridiagonal matrix inversion and fast transmission calculations

A method for the inversion of block tridiagonal matrices encountered in electronic structure calculations is developed, with the goal of efficiently determining the matrices involved in the Fisher–Lee relation for the calculation of electron transmission coefficients. The new method leads to faster transmission calculations compared to traditional methods, as well as freedom in choosing alternate Green’s function matrix blocks for transmission calculations. The new method also lends itself to calculation of the tridiagonal part of the Green’s function matrix. The effect of inaccuracies in the electrode self-energies on the transmission coefficient is analyzed and reveals that the new algorithm is potentially more stable towards such inaccuracies.

Parallel block tridiagonalization of real symmetric matrices

Two parallel block tridiagonalization algorithms and implementations for dense real symmetric matrices are presented. Block tridiagonalization is a critical pre-processing step for the block tridiagonal divide-and-conquer algorithm for computing eigensystems and is useful for many algorithms desiring the efficiencies of block structure in matrices. For an “effectively” sparse matrix, which frequently results from applications with strong locality properties, a heuristic parallel algorithm is used to transform it into a block tridiagonal matrix such that the eigenvalue errors remain bounded by some prescribed accuracy tolerance. For a dense matrix without any usable structure, orthogonal transformations are used to reduce it to block tridiagonal form using mostly level 3 BLAS operations. Numerical experiments show that block tridiagonal structure obtained from this algorithm directly affects the computational complexity of the parallel block tridiagonal divide-and-conquer eigensolver. Reduction to block tridiagonal form provides significantly lower execution times, as well as memory traffic and communication cost, over the traditional reduction to tridiagonal form for eigensystem computations.