Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “matrices”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 1,027 records · Page 57
All sourcesNASA NTRSDOE OSTI
DOE OSTI2018

An iterative method for computing a symplectic SVD-like decomposition

In this paper, we present an effective iterative method for computing symplectic SVD-like decomposition for a 2n-by-m rectangular real matrix A. The main purpose here is a block-power iterative method with the ortho-symplectic SR decomposition in the normalization step. We compute an k-block symplectic SVD-like decomposition, namely A{sub k}=S{sub k}Σ{sub k}V{sub k}{sup T} where S{sub k}∈ℝ{sup 2n×2k} is symplectic and V{sub k}∈ℝ{sup m×2k} is orthogonal. For large matrices, we usually have to compute only part of eigenvalues. The main interest of the proposed method is to efficiently compute the desired number of ordered in magnitude eigenvalues of structured matrices.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

An evaluation of low-cost heuristics for matrix bandwidth and profile reductions

Hundreds of heuristics have been proposed to resolve the problems of bandwidth and profile reductions since the 1960s. We found 132 heuristics that have been applied to these problems in reviews of the literature. Among them, 14 were selected for which no other simulation or comparison revealed that the heuristic could be superseded by any other algorithm in the analyzed articles with respect to bandwidth or profile reduction. We also considered the computational costs of the heuristics during this process. Therefore, these 14 heuristics were selected as potentially being the best low-cost methods to solve the bandwidth and/or profile reduction problems. Results of the 14 selected heuristics are evaluated in this work. For evaluation on the set of test problems, a metric based on the relative percentage distance to the best possible bandwidth or profile is proposed. The most promising heuristics for several application areas are identified. Moreover, it was found that the FNCHC and GPS heuristics showed the best overall results in reducing the bandwidth of symmetric and asymmetric matrices among the evaluated heuristics, respectively. In addition, the NSloan and MPG heuristics showed the best overall results in reducing the profile of symmetric and asymmetric matrices among the heuristics among the evaluated heuristics, respectively.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Numerical approach based on fractional-order Lagrange polynomials for solving a class of fractional differential equations

In this study, we propose a new set of fractional functions based on the Lagrange polynomials to solve a class of fractional differential equations. Fractional differential equations are the best tools for modelling natural phenomenon that are elaborated by fractional calculus. Therefore, we need an accurate and efficient technique for solving them. The main purpose of this article is to generalize new functions based on Lagrange polynomials to the fractional calculus. At first, we present a new representation of Lagrange polynomials and in continue, we propose a new set of fractional-order functions which are called fractional-order Lagrange polynomials (FLPs). Besides, a general formulation for operational matrices of fractional integration and derivative of FLPs on arbitrary nodal points are extracted. These matrices are obtained using Laplace transform. The initial value problems is reduced to the system of algebraic equations using the operational matrix of fractional integration and collocation method. Also, we find the upper bound of error vector for the fractional integration operational matrix and we indicate convergence of approximations of FLPs. Illustrative examples are included to demonstrate the validity and applicability of the proposed technique.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Nekrasov tensors and nonsingular H-tensors

Nekrasov matrices and nonsingular H-matrices are closely related. In this paper, Nekrasov tensors and S-Nekrasov tensors are proved to be nonsingular H-tensors. And tensors are generalized Nekrasov tensors if and only if they are nonsingular H-tensors. Furthermore, an iterative criterion for identifying nonsingular H-tensors is provided.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Application of hat basis functions for solving two-dimensional stochastic fractional integral equations

This article concerns with a computational scheme to solve two-dimensional stochastic fractional integral equations (2DSFIEs), numerically. In these equations, the fractional integral is considered in the Riemann–Liouville sense. The proposed method is essentially based on two-dimensional hat basis functions and its fractional operational matrices. The fractional-order operational matrices of integration are applied to reduce the solution of 2DSFIEs to the solution of a system of linear equations which can be solved using a direct method or iterative method. Some results concerning the convergence analysis associated with the proposed technique are discussed. In addition, we establish the rate of convergence of this approach for solving 2DSFIEs is O(h{sup 2}). Finally, some examples are solved using present method to indicate the pertinent features of the method.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2013

The Kronecker product in terms of Hubbard operators and the Clebsch–Gordan decomposition of SU(2)×SU(2)

We review the properties of the Kronecker (direct, or tensor) product of square matrices A⊗B⊗C⋯ in terms of Hubbard operators. In its simplest form, a Hubbard operator X{sub n}{sup i,j} can be expressed as the n-square matrix which has entry 1 in position (i,j) and zero in all other entries. The algebra and group properties of the observables that define a multipartite quantum system are notably straightforward in such a framework. In particular, we use the Kronecker product in Hubbard notation to get the Clebsch–Gordan decomposition of the product group SU(2)×SU(2). Finally, the n-dimensional irreducible representations so obtained are used to derive closed forms of the Clebsch–Gordan coefficients that rule the addition of angular momenta. Our results can be further developed in many different directions. -- Highlights: •The Kronecker product is studied in terms of Hubbard operators. •Complicated calculations involving large matrices are reduced to simple relations of subscripts. •The algebraic properties of the quantum observables of multipartite systems are studied. •The Clebsch–Gordan coefficients are given in terms of hypergeometric {sub 3}F{sub 2} functions. •The results can be further developed in many different directions.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2018

Construction of distinct discrete time scattering quantum walk formulations on the honeycomb lattice

Here we address the scattering construction of discrete time quantum walks on the honeycomb lattice. We write the system general (unitary) one step time evolution operator in terms of: (i) complete arbitrary scattering matrices Γ{sup (j,k)}, defined on the sites (j,k) of the lattice and; and (ii) topological directional functions Φ, which represent the distinct ways we can keep track of the propagation directions along the three bonds attached to each (j,k). By imposing the Φ’s to comply with all the honeycomb translational and point group symmetries (in the case of Γ{sup (j,k)}=Γ), we obtain in total ten independent model versions, presenting different dynamical features. To study some of their traits, we consider the idea of ‘characteristic paths’ CPs (closely related to classical random walks), determining the CPs for each one of the ten formulations. We then discuss many numerical examples of time evolution dynamics for our scattering quantum walks. For the calculations we use few Γ’s, as the Discrete Fourier Transform (DFT) and Grover matrices.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2013

A fast algorithm for sparse matrix computations related to inversion

We have developed a fast algorithm for computing certain entries of the inverse of a sparse matrix. Such computations are critical to many applications, such as the calculation of non-equilibrium Green’s functions G{sup r} and G{sup <} for nano-devices. The FIND (Fast Inverse using Nested Dissection) algorithm is optimal in the big-O sense. However, in practice, FIND suffers from two problems due to the width-2 separators used by its partitioning scheme. One problem is the presence of a large constant factor in the computational cost of FIND. The other problem is that the partitioning scheme used by FIND is incompatible with most existing partitioning methods and libraries for nested dissection, which all use width-1 separators. Our new algorithm resolves these problems by thoroughly decomposing the computation process such that width-1 separators can be used, resulting in a significant speedup over FIND for realistic devices — up to twelve-fold in simulation. The new algorithm also has the added advantage that desired off-diagonal entries can be computed for free. Consequently, our algorithm is faster than the current state-of-the-art recursive methods for meshes of any size. Furthermore, the framework used in the analysis of our algorithm is the first attempt to explicitly apply the widely-used relationship between mesh nodes and matrix computations to the problem of multiple eliminations with reuse of intermediate results. This framework makes our algorithm easier to generalize, and also easier to compare against other methods related to elimination trees. Finally, our accuracy analysis shows that the algorithms that require back-substitution are subject to significant extra round-off errors, which become extremely large even for some well-conditioned matrices or matrices with only moderately large condition numbers. When compared to these back-substitution algorithms, our algorithm is generally a few orders of magnitude more accurate, and our produced round-off errors stay at a reasonable level.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2011

Ultraviolet absorption and luminescence of matrix-isolated adenine

We have investigated the absorption, the fluorescence and phosphorescence emission and the fluorescence lifetimes of adenine in low-temperature argon and nitrogen matrices at 15 K. Compared to other environments the absorption spectrum shows higher intensity at the shortest wavelengths, and a weak apparent absorption peak is observed at 280 nm. The resolved fluorescence excitation spectrum has five peaks at positions corresponding to those observed in the absorption spectrum. The position of the fluorescence maximum depends on the excitation wavelength. Excitation below 220 nm displays a fluorescence maximum at 305 nm, while for excitations at higher wavelengths the maximum occurs at 335 nm. The results suggest that multiple-emission excited electronic states are populated in low-temperature gas matrices. Excitation at 265 nm produces a phosphorescence spectrum with a well-resolved vibrational structure and a maximum at 415 nm. The fluorescence decays corresponding to excitation at increasing energy of each resolved band could be fit with a double exponential, with the shorter and longer lifetimes ranging from 1.7 to 3.3 ns and from 12 to 23 ns, respectively. Only for the excitation at 180 nm one exponential is required, with the calculated lifetimes of 3.3 ns. The presented results provide an experimental evidence of the existence of multiple site-selected excited electronic states, and may help elucidate the possible deexcitation pathways of adenine. The additional application of synchrotron radiation proved to result in a significant enhancement of the resolution and spectral range of the phenomena under investigation.

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2010

The symmetries of the fine gradings of sl(n{sup k},C) associated with direct product of Pauli groups

A grading of a Lie algebra is called fine if it could not be further refined. For a fine grading of a simple Lie algebra, we define its Weyl group to describe the symmetry of this grading. It is already known that the Weyl group of the fine grading of sl(n,C) induced by the action of the group {Pi}{sub n} of the generalized Pauli matrices of rank n is SL(2,Z{sub n}), where Z{sub n} is the cyclic group of order n. In this paper, we consider the fine grading of sl(n{sup k},C) induced by the action of the group of k-fold tensor product of the generalized Pauli matrices of rank n. We prove that its Weyl group is Sp(2k,Z{sub n}) and is generated by transvections; therefore, this generalizes the previous result.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2010

Quantum wavepacket ab initio molecular dynamics: Generalizations using an extended Lagrangian treatment of diabatic states coupled through multireference electronic structure

We present a generalization to our previously developed quantum wavepacket ab initio molecular dynamics (QWAIMD) method by using multiple diabatic electronic reduced single particle density matrices, propagated within an extended Lagrangian paradigm. The Slater determinantal wavefunctions associated with the density matrices utilized may be orthogonal or nonorthogonal with respect to each other. This generalization directly results from an analysis of the variance in electronic structure with quantum nuclear degrees of freedom. The diabatic electronic states are treated here as classical parametric variables and propagated simultaneously along with the quantum wavepacket and classical nuclei. Each electronic density matrix is constrained to be N-representable. Consequently two sets of new methods are derived: extended Lagrangian-QWAIMD (xLag-QWAIMD) and diabatic extended Lagrangian-QWAIMD (DxLag-QWAIMD). In both cases, the instantaneous potential energy surface for the quantum nuclear degrees of freedom is constructed from the diabatic states using an on-the-fly nonorthogonal multireference formalism. By introducing generalized grid-based electronic basis functions, we eliminate the basis set dependence on the quantum nucleus. Subsequent reuse of the two-electron integrals during the on-the-fly potential energy surface computation stage yields a substantial reduction in computational costs. Specifically, both xLag-QWAIMD and DxLag-QWAIMD turn out to be about two orders of magnitude faster than our previously developed time-dependent deterministic sampling implementation of QWAIMD. Energy conservation properties, accuracy of the associated potential surfaces, and vibrational properties are analyzed for a family of hydrogen bonded systems.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2011

Laplacians in polar matrix coordinates and radial fermionization in higher dimensions

We consider the quantum mechanical Hamiltonian of two, space indexed, Hermitian matrices. By introducing matrix valued polar coordinates, we obtain the form of the Laplacian acting on invariant states. For potentials depending only on the eigenvalues of the radial matrix, we establish that the radially invariant sector is equivalent to a system of noninteracting 2 + 1 dimensional fermions and obtain its density description. For a larger number of matrices, the presence of a repulsive radial intereigenvalue potential is identified.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2013

Partial transpose of random quantum states: Exact formulas and meanders

We investigate the asymptotic behavior of the empirical eigenvalues distribution of the partial transpose of a random quantum state. The limiting distribution was previously investigated via Wishart random matrices indirectly (by approximating the matrix of trace 1 by the Wishart matrix of random trace) and shown to be the semicircular distribution or the free difference of two free Poisson distributions, depending on how dimensions of the concerned spaces grow. Our use of Wishart matrices gives exact combinatorial formulas for the moments of the partial transpose of the random state. We find three natural asymptotic regimes in terms of geodesics on the permutation groups. Two of them correspond to the above two cases; the third one turns out to be a new matrix model for the meander polynomials. Moreover, we prove the convergence to the semicircular distribution together with its extreme eigenvalues under weaker assumptions, and show large deviation bound for the latter.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2014

A realistic renormalizable supersymmetric E₆ model

A complete realistic model based on the supersymmetric version of E₆ is presented. It consists of three copies of matter 27, and a Higgs sector made of 2×(27+27⁻)+351´+351´⁻ representations. An analytic solution to the equations of motion is found which spontaneously breaks the gauge group into the Standard Model. The light fermion mass matrices are written down explicitly as non-linear functions of three Yukawa matrices. This contribution is based on Ref. [1].

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2015

Exponential of a matrix, a nonlinear problem, and quantum gates

We describe solutions of the matrix equation exp(z(A − I{sub n})) = A, where z ∈ ℂ. Applications in quantum computing are given. Both normal and non-normal matrices are studied. For normal matrices, the Lambert W-function plays a central role.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2015

An exact formulation of the time-ordered exponential using path-sums

We present the path-sum formulation for the time-ordered exponential of a time-dependent matrix. The path-sum formulation gives the time-ordered exponential as a branched continued fraction of finite depth and breadth. The terms of the path-sum have an elementary interpretation as self-avoiding walks and self-avoiding polygons on a graph. Our result is based on a representation of the time-ordered exponential as the inverse of an operator, the mapping of this inverse to sums of walks on a graphs, and the algebraic structure of sets of walks. We give examples demonstrating our approach. We establish a super-exponential decay bound for the magnitude of the entries of the time-ordered exponential of sparse matrices. We give explicit results for matrices with commonly encountered sparse structures.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2015

Optimal ancilla-free Pauli+V circuits for axial rotations

We address the problem of optimal representation of single-qubit rotations in a certain unitary basis consisting of the so-called V gates and Pauli matrices. The V matrices were proposed by Lubotsky, Philips, and Sarnak [Commun. Pure Appl. Math. 40, 401–420 (1987)] as a purely geometric construct in 1987 and recently found applications in quantum computation. They allow for exceptionally simple quantum circuit synthesis algorithms based on quaternionic factorization. We adapt the deterministic-search technique initially proposed by Ross and Selinger to synthesize approximating Pauli+V circuits of optimal depth for single-qubit axial rotations. Our synthesis procedure based on simple SL{sub 2}(ℤ) geometry is almost elementary.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2016

Efficient magnetic fields for supporting toroidal plasmas

The magnetic field that supports tokamak and stellarator plasmas must be produced by coils well separated from the plasma. However, the larger the separation, the more difficult it is to produce a given magnetic field in the plasma region, so plasma configurations should be chosen that can be supported as efficiently as possible by distant coils. The efficiency of an externally generated magnetic field is a measure of the field's shaping component magnitude at the plasma compared to the magnitude near the coils; the efficiency of a plasma equilibrium can be measured using the efficiency of the required external shaping field. Counterintuitively, plasma shapes with low curvature and spectral width may have low efficiency, whereas plasma shapes with sharp edges may have high efficiency. Two precise measures of magnetic field efficiency, which correctly identify such differences in difficulty, will be examined. These measures, which can be expressed as matrices, relate the externally produced normal magnetic field on the plasma surface to the either the normal field or current on a distant control surface. A singular value decomposition (SVD) of either matrix yields an efficiency ordered basis for the magnetic field distributions. Calculations are carried out for both tokamak and stellarator cases. For axisymmetric surfaces with circular cross-section, the SVD is calculated analytically, and the range of poloidal and toroidal mode numbers that can be controlled to a given desired level is determined. If formulated properly, these efficiency measures are independent of the coordinates used to parameterize the surfaces.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗