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NASA NTRS1988

On conjugate gradient type methods and polynomial preconditioners for a class of complex non-Hermitian matrices

Conjugate gradient type methods are considered for the solution of large linear systems Ax = b with complex coefficient matrices of the type A = T + i(sigma)I where T is Hermitian and sigma, a real scalar. Three different conjugate gradient type approaches with iterates defined by a minimal residual property, a Galerkin type condition, and an Euclidian error minimization, respectively, are investigated. In particular, numerically stable implementations based on the ideas behind Paige and Saunder's SYMMLQ and MINRES for real symmetric matrices are proposed. Error bounds for all three methods are derived. It is shown how the special shift structure of A can be preserved by using polynomial preconditioning. Results on the optimal choice of the polynomial preconditioner are given. Also, some numerical experiments for matrices arising from finite difference approximations to the complex Helmholtz equation are reported.

Freund, Roland↗

NASA NTRS1989

Bunch-Kaufman factorization for real symmetric indefinite banded matrices

The Bunch-Kaufman algorithm for factoring symmetric indefinite matrices was rejected for banded matrices because it destroys the banded structure of the matrix. Herein, it is shown that for a subclass of real symmetric matrices which arise in solving the generalized eigenvalue problem using Lanczos's method, the Bunch-Kaufman algorithm does not result in major destruction of the bandwidth. Space time complexities of the algorithm are given and used to show that the Bunch-Kaufman algorithm is a significant improvement over LU factorization.

Jones, Mark T.↗

NASA NTRS1990

Characterization of two MMIC GaAs switch matrices at microwave frequencies

Monolithic GaAs microwave switch matrices for use in satellite switched, time division multiple access communication systems were developed. Two monolithic GaAs MESFET switch matrices were fabricated; one for switching operation at intermediate frequencies, 3.5 to 6.0 GHz, and another for switching at radio frequencies, 17.7 to 20.2 GHz. Key switch parameters were measured for both switch matrices.

Fujikawa, Gene↗

NASA NTRS1989

Preconditioning matrices for the pseudospectral approximation of first-order operators

The behavior of the eigenvalues of preconditioning matrices for the pseudospectral approximation to the derivative operator has been analyzed in one and two dimensions. The one-dimensional analysis resulted in real and positive eigenvalues for the selected tridiagonal matrices. In the two-dimensional analysis, the eigenvalues of the selected block-diagonal matrices behaved well, but the preconditioner is full and therefore not suitable for applications. The Richardson scheme has been applied in the unpreconditioned as well as the preconditioned version to find the solution of the model problem.

Funaro, D.↗

NASA NTRS1990

Characterization of two MMIC GaAs switch matrices at microwave frequencies

Fujikawa, Gene↗

NASA NTRS1993

Damage detection using experimentally measured mass and stiffness matrices

A method is presented for locating physical damage or change in a structure using experimentally measured mass and stiffness matrices. The approach uses a recently developed algorithm for transforming a state-space realization into a second order structural model with physical displacements as the generalized coordinates. This is accomplished by first rotating a state-space model of the identified structural dynamics into modal coordinates and approximating the mass normalized modal vectors for the output measurement set. Next, the physical mass, damping and stiffness matrices are synthesized directly from the measured modal parameters. This yields experimental mass and stiffness matrices for the structure without the use of a finite element model or a numerical search. The computed mass and stiffness are asymptotically equivalent to a static condensation of the global physical coordinate model. Techniques for solving the inverse connectivity problem are then developed whereby it is possible to assess the stiffness in a region of the structure bounded by several sensors. Applications to both simulated data and experimental data are used to discuss the effectiveness of the approach.

Peterson, L. D.↗

NASA NTRS1993

Visual optimization of DCT quantization matrices for individual images

Many image compression standards (JPEG, MPEG, H.261) are based on the Discrete Cosine Transform (DCT). However, these standards do not specify the actual DCT quantization matrix. We have previously provided mathematical formulae to compute a perceptually lossless quantization matrix. Here I show how to compute a matrix that is optimized for a particular image. The method treats each DCT coefficient as an approximation to the local response of a visual 'channel'. For a given quantization matrix, the DCT quantization errors are adjusted by contrast sensitivity, light adaptation, and contrast masking, and are pooled non-linearly over the blocks of the image. This yields an 8x8 'perceptual error matrix'. A second non-linear pooling over the perceptual error matrix yields total perceptual error. With this model we may estimate the quantization matrix for a particular image that yields minimum bit rate for a given total perceptual error, or minimum perceptual error for a given bit rate. Custom matrices for a number of images show clear improvement over image-independent matrices. Custom matrices are compatible with the JPEG standard, which requires transmission of the quantization matrix.

Watson, Andrew B.↗

NASA NTRS1995

Visualization of newt aragonitic otoconial matrices using transmission electron microscopy

Otoconia are calcified protein matrices within the gravity-sensing organs of the vertebrate vestibular system. These protein matrices are thought to originate from the supporting or hair cells in the macula during development. Previous studies of mammalian calcitic, barrel-shaped otoconia revealed an organized protein matrix consisting of a thin peripheral layer, a well-defined organic core and a flocculent matrix inbetween. No studies have reported the microscopic organization of the aragonitic otoconial matrix, despite its protein characterization. Pote et al. (1993b) used densitometric methods and inferred that prismatic (aragonitic) otoconia have a peripheral protein distribution, compared to that described for the barrel-shaped, calcitic otoconia of birds, mammals, and the amphibian utricle. By using tannic acid as a negative stain, we observed three kinds of organic matrices in preparations of fixed, decalcified saccular otoconia from the adult newt: (1) fusiform shapes with a homogenous electron-dense matrix; (2) singular and multiple strands of matrix; and (3) more significantly, prismatic shapes outlined by a peripheral organic matrix. These prismatic shapes remain following removal of the gelatinous matrix, revealing an internal array of organic matter. We conclude that prismatic otoconia have a largely peripheral otoconial matrix, as inferred by densitometry.

NASA Discipline Neuroscience↗

NASA NTRS2009

Nano-Fiber Reinforced Enhancements in Composite Polymer Matrices

Nano-fibers are used to reinforce polymer matrices to enhance the matrix dependent properties that are subsequently used in conventional structural composites. A quasi isotropic configuration is used in arranging like nano-fibers through the thickness to ascertain equiaxial enhanced matrix behavior. The nano-fiber volume ratios are used to obtain the enhanced matrix strength properties for 0.01,0.03, and 0.05 nano-fiber volume rates. These enhanced nano-fiber matrices are used with conventional fiber volume ratios of 0.3 and 0.5 to obtain the composite properties. Results show that nano-fiber enhanced matrices of higher than 0.3 nano-fiber volume ratio are degrading the composite properties.

Chamis, Christos C.↗

DOE OSTI2014

STRUMPACK -- STRUctured Matrices PACKage

STRUMPACK - STRUctured Matrices PACKage - is a package for computations with sparse and dense structured matrix, i.e., matrices that exhibit some kind of low-rank property, in particular Hierarchically Semi Separable structure (HSS). Such matrices appear in many applications, e.g., FEM, BEM, Integral equations. etc. Exploiting this structure using certain compression algorithms allow for fast solution of linear systems and/or fast computation of matrix-vector products, which are the two main building blocks of matrix computations. STRUMPACK has presently two main components: a distributed-memory dense matrix computations package and a shared-memory sparse direct solver.

Source record↗

DOE OSTI2026

Quantum block encoding for one-pair semiseparable matrices

Quantum block encoding (QBE) is a crucial step in the development of most quantum algorithms, as it provides an embedding of a given matrix into a suitable larger unitary matrix. Historically, the development of efficient techniques for QBE has mostly focused on sparse matrices; less effort has been devoted to data-sparse (e.g., rank-structured) matrices. In this work we examine a particular case of rank structure, namely, one-pair semiseparable matrices. We present a new block encoding approach that relies on a suitable factorization of the given matrix as the product of triangular and diagonal factors. To encode the matrix, the algorithm needs $2\log(N)+7$ ancillary qubits. Assuming that the data input oracles can be implemented with polylogarithmic depth, or that a QRAM input model is available, our proposed method requires $\mathcal{O}({\rm polylog} (N))$ time and has an error of $\mathcal{O}(N^2)$, where $N$ is the matrix size.

Antonioli, Giacomo [Pisa U.; CERN] (ORCID:00090000↗

DOE OSTI2018

Algorithmic Patterns for $$\mathcal {H}$$-Matrices on Many-Core Processors

In this work, we consider the reformulation of hierarchical ( $\mathcal {H}$ ) matrix algorithms for many-core processors with a model implementation on graphics processing units (GPUs). $\mathcal {H}$ matrices approximate specific dense matrices, e.g., from discretized integral equations or kernel ridge regression, leading to log-linear time complexity in dense matrix–vector products. The parallelization of $\mathcal {H}$ matrix operations on many-core processors is difficult due to the complex nature of the underlying algorithms. While previous algorithmic advances for many-core hardware focused on accelerating existing $\mathcal {H}$ matrix CPU implementations by many-core processors, we here aim at totally relying on that processor type. As main contribution, we introduce the necessary parallel algorithmic patterns allowing to map the full $\mathcal {H}$ matrix construction and the fast matrix–vector product to many-core hardware. In this work, crucial ingredients are space filling curves, parallel tree traversal and batching of linear algebra operations. The resulting model GPU implementation hmglib is the, to the best of the authors knowledge, first entirely GPU-based Open Source $\mathcal {H}$ matrix library of this kind. We investigate application examples as present in kernel ridge regression, Gaussian Process Regression and kernel-based interpolation. In this context, an in-depth performance analysis and a comparative performance study against a standard multi-core CPU $\mathcal {H}$ matrix library highlights profound speedups of our many-core parallel approach.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2018

Binormal Matrices

A square complex matrix A is said to be binormal if the associated matrices A{sup *}A and AA{sup *} commute. This matrix class yields a meaningful finite-dimensional extension of the concept of normality. The paper can be regarded as a survey of properties of binormal matrices.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2018

Localized Pisot Matrices and Joint Approximations of Algebraic Numbers

A development of the simplex-module algorithm for expansion of algebraic numbers in multidimensional continued fractions is proposed. To this end, localized Pisot matrices are constructed, whose eigenvalues with moduli less than one are contained in an interval of small length. Such Pisot matrices generate continued fractions whose convergents are arbitrarily close to the best approximations.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2014

Universality for a global property of the eigenvectors of Wigner matrices

Let M{sub n} be an n×n real (resp. complex) Wigner matrix and U{sub n}Λ{sub n}U{sub n}{sup *} be its spectral decomposition. Set (y{sub 1},y{sub 2}⋯,y{sub n}){sup T}=U{sub n}{sup *}x, where x = (x{sub 1}, x{sub 2}, ⋅⋅⋅, x{sub n}){sup T} is a real (resp. complex) unit vector. Under the assumption that the elements of M{sub n} have 4 matching moments with those of GOE (resp. GUE), we show that the process X{sub n}(t)=√((βn)/2 )∑{sub i=1}{sup ⌊nt⌋}(|y{sub i}|{sup 2}−1/n ) converges weakly to the Brownian bridge for any x satisfying ‖x‖{sub ∞} → 0 as n → ∞, where β = 1 for the real case and β = 2 for the complex case. Such a result indicates that the orthogonal (resp. unitary) matrices with columns being the eigenvectors of Wigner matrices are asymptotically Haar distributed on the orthogonal (resp. unitary) group from a certain perspective.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2014

Optimization of the determinant of the Vandermonde matrix and related matrices

Various techniques for interpolation of data, moment matching in stochastic applications and various methods in numerical analysis can be described using Vandermonde matrices. For this reason the properties of the determinant of the Vandermonde matrix and related matrices are interesting. Here the extreme points of the Vandermonde determinant, and related determinants, on some simple surfaces such as the unit sphere are analyzed, both numerically and analytically. Some results are also visualized in various dimensions. The extreme points of the Vandermonde determinant are also related to the roots of certain orthogonal polynomials such as the Hermite polynomials.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2010

Measurement-based quantum phase estimation algorithm for finding eigenvalues of non-unitary matrices

We propose a quantum algorithm for finding eigenvalues of non-unitary matrices. We show how to construct, through interactions in a quantum system and projective measurements, a non-Hermitian or non-unitary matrix and obtain its eigenvalues and eigenvectors. This proposal combines ideas of frequent measurement, measured quantum Fourier transform, and quantum state tomography. It provides a generalization of the conventional phase estimation algorithm, which is limited to Hermitian or unitary matrices.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2010

Nonsupersymmetric strong coupling background from the large N quantum mechanics of two matrices coupled via a Yang-Mills interaction

We derive a planar sector of the large N nonsupersymmetric background of the quantum mechanical Hamiltonian of two Hermitian matrices coupled via a Yang-Mills interaction, in terms of the density of eigenvalues of one of the matrices. This background satisfies an implicit nonlinear integral equation, with a perturbative small coupling expansion and a solvable large coupling solution, which is obtained. The energy of system and the expectation value of several correlators are obtained in this strong coupling limit. They are free of infrared divergences.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗