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NASA NTRS1988

Parallel computer methods for eigenvalue extraction

Akl, Fred↗

NASA NTRS1989

Parallel Eigenvalue extraction

A new numerical algorithm for the solution of large-order eigenproblems typically encountered in linear elastic finite element systems is presented. The architecture of parallel processing is utilized in the algorithm to achieve increased speed and efficiency of calculations. The algorithm is based on the frontal technique for the solution of linear simultaneous equations and the modified subspace eigenanalysis method for the solution of the eigenproblem. Assembly, elimination and back-substitution of degrees of freedom are performed concurrently, using a number of fronts. All fronts converge to and diverge from a predefined global front during elimination and back-substitution, respectively. In the meantime, reduction of the stiffness and mass matrices required by the modified subspace method can be completed during the convergence/divergence cycle and an estimate of the required eigenpairs obtained. Successive cycles of convergence and divergence are repeated until the desired accuracy of calculations is achieved. The advantages of this new algorithm in parallel computer architecture are discussed.

Akl, Fred A.↗

NASA NTRS1993

Computation of selected eigenvalues of generalized eigenvalue problems

The Arnoldi (1951) and Lanczos (1950) iterative algorithms are used to obtain a few selected eigenvalues of the generalized eigenvalue problem A(x) = lambdaB(x), where A and B are n x n nonsymmetric banded complex matrices. A shift-and-invert strategy is used to increase the rate of convergence toward the desired eigenvalues, and the two approaches are evaluated in view of their performance in a model problem concerning the linear stability of compressible boundary layers. A general scheme for computing the eigenvalues that lie within a 'box' in the complex plane is presented.

Nayar, Narinder↗

NASA NTRS1982

A general algorithm for solving the algebraic Riccati equation

The generalized eigenvalue problem provides a suitable framework for reliable solutions of many system theoretic, control, and estimation problems. A general algorithm for solving the matrix algebraic Riccati equation (ARE) which utilizes a pencil structure is described here. This algorithm avoids unnecessary inversion of cost or transition matrices, making it a numerically sound way to solve for the gains and/or ARE with singular quadratic costs, for cases satisfying detectability and stabilizability conditions. Examples are solution with discrete dead-beat control, noiseless measurements in Kalman filters and time-delays in discrete-time systems, which cause difficulties in the Hamiltonian standard eigenvalue problem formulation. The ARE algorithm implementatiton and numerical examples are shown.

Walker, R. A.↗

DOE OSTI2025

Fermion determinants on a quantum computer

We present a quantum algorithm to compute the logarithm of the determinant of the fermion matrix, assuming access to a classical lattice gauge field configuration. The algorithm uses the quantum eigenvalue transform, and quantum mean estimation, giving a query complexity that scales like O ( V log ( V ) ) in the matrix dimension V . Published by the American Physical Society 2025

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

NASA NTRS1981

Multigrid solvers on parallel computers

Massively parallel computers, as considered in this investigation, are not yet available. However, a large-scale parallel computer cannot usefully be designed before the hypothetical algorithms which will employ it are studied. Most of the studies of parallel partial differential equations (PDE) solvers are based on solution techniques much slower (on sequential machines) than multigrid methods. Multigrid methods are highly parallelizable. Each of their processes can simultaneously be performed at all grid points. The present investigation is concerned with a preliminary exploration of the potential of multigrid, or, more generally, Multi-Level Adaptive Techniques (MLAT) on computers with many processors. Basic processes are considered, taking into account coarse-grid approximation, relaxation, coarse-grid corrections, full multigrid algorithms, nonlinear problems and eigenvalue problems, fine-to-coarse correction, and chains of problems. Details of parallel multigrid processing are also examined.

Brandt, A.↗

NASA NTRS2004

Open-Source Software for Modeling of Nanoelectronic Devices

The Nanoelectronic Modeling 3-D (NEMO 3-D) computer program has been upgraded to open-source status through elimination of license-restricted components. The present version functions equivalently to the version reported in "Software for Numerical Modeling of Nanoelectronic Devices" (NPO-30520), NASA Tech Briefs, Vol. 27, No. 11 (November 2003), page 37. To recapitulate: NEMO 3-D performs numerical modeling of the electronic transport and structural properties of a semiconductor device that has overall dimensions of the order of tens of nanometers. The underlying mathematical model represents the quantum-mechanical behavior of the device resolved to the atomistic level of granularity. NEMO 3-D solves the applicable quantum matrix equation on a Beowulf-class cluster computer by use of a parallel-processing matrix vector multiplication algorithm coupled to a Lanczos and/or Rayleigh-Ritz algorithm that solves for eigenvalues. A prior upgrade of NEMO 3-D incorporated a capability for a strain treatment, parameterized for bulk material properties of GaAs and InAs, for two tight-binding submodels. NEMO 3-D has been demonstrated in atomistic analyses of effects of disorder in alloys and, in particular, in bulk In(x)Ga(1-x)As and in In(0.6)Ga(0.4)As quantum dots.

Oyafuso, Fabiano↗

NASA NTRS2003

Numerical Modeling of Nanoelectronic Devices

Nanoelectronic Modeling 3-D (NEMO 3-D) is a computer program for numerical modeling of the electronic structure properties of a semiconductor device that is embodied in a crystal containing as many as 16 million atoms in an arbitrary configuration and that has overall dimensions of the order of tens of nanometers. The underlying mathematical model represents the quantummechanical behavior of the device resolved to the atomistic level of granularity. The system of electrons in the device is represented by a sparse Hamiltonian matrix that contains hundreds of millions of terms. NEMO 3-D solves the matrix equation on a Beowulf-class cluster computer, by use of a parallel-processing matrix vector multiplication algorithm coupled to a Lanczos and/or Rayleigh-Ritz algorithm that solves for eigenvalues. In a recent update of NEMO 3-D, a new strain treatment, parameterized for bulk material properties of GaAs and InAs, was developed for two tight-binding submodels. The utility of the NEMO 3-D was demonstrated in an atomistic analysis of the effects of disorder in alloys and, in particular, in bulk In(x)Ga(l-x)As and in In0.6Ga0.4As quantum dots.

Klimeck, Gerhard↗

NASA NTRSDate not supplied

TPSAS-NF1676L-33433-DND

Reducing aircraft noise is a major objective in the field of computational aeroacoustics. When designing next generation quiet aircraft, it is important to be able to accurately and efficiently predict the acoustic scattering by an aircraft body from a given noise source. Acoustic liners are an effective tool for aircraft noise reduction, and are characterized by a complex valued frequency-dependent impedance. Converted into the time-domain using Fourier transforms, an impedance boundary condition can be used to simulate the acoustic scattering of geometric bodies treated with acoustic liners. This work considers using either an impedance boundary condition or admittance boundary condition to allow acoustic scattering problems to be modeled with geometries consisting of both un-lined and lined surfaces; admittance is defined to be the inverse of impedance. Three different acoustic liner models will be discussed: the Extended Helmholtz Resonator Model, the Three-Parameter Impedance Model, and a Broadband Model. In the Extended Helmholtz Resonator Model and Three-Parameter Impedance Model, the liner impedance is specified at a single frequency; the Broadband Model allows for the investigation of multiple frequencies simultaneously. The impedance and admittance boundary conditions for each model will be derived and coupled with a time-domain boundary integral equation. The scattering solution will be obtained iteratively using spatial and temporal basis functions. Stability will be demonstrated through eigenvalue analysis. Moreover, fast algorithms and high performance computing are considered to both assess and reduce the computational cost of the simulation.

Michelle E Rodio↗

NASA NTRSDate not supplied

Investigating the Feasibility and Stability for Modeling Broadband Acoustic Wave Scattering using a Time-Domain Boundary Integral Equation

Reducing aircraft noise is a major objective in the field of computational aeroacoustics. Acoustic liners are an effective tool for reducing aircraft noise and are characterized by a frequency-dependent impedance value. Converted into the time-domain using Fourier transforms, an impedance boundary condition can be used to simulate the acoustic wave scattering by geometric bodies treated with acoustic liners. A Broadband Impedance Model is discussed in which the liner impedance is specified along a wide range of frequencies. An impedance boundary condition is derived and coupled with a time-domain boundary integral equation to model acoustic scattering by a flat plate. It is assumed the flat plate has surfaces treated with acoustic liners. The stability of the numerical algorithm is assessed using eigenvalue analysis.

mathmatics↗

NASA NTRS1991

A look-ahead variant of the Lanczos algorithm and its application to the quasi-minimal residual method for non-Hermitian linear systems

The Lanczos algorithm can be used both for eigenvalue problems and to solve linear systems. However, when applied to non-Hermitian matrices, the classical Lanczos algorithm is susceptible to breakdowns and potential instabilities. In addition, the biconjugate gradient (BCG) algorithm, which is the natural generalization of the conjugate gradient algorithm to non-Hermitian linear systems, has a second source of breakdowns, independent of the Lanczos breakdowns. Here, we present two new results. We propose an implementation of a look-ahead variant of the Lanczos algorithm which overcomes the breakdowns by skipping over those steps where a breakdown or a near-breakdown would occur. The new algorithm can handle look-ahead steps of any length and requires the same number of matrix-vector products and inner products per step as the classical Lanczos algorithm without look-ahead. Based on the proposed look-ahead Lanczos algorithm, we then present a novel BCG-like approach, the quasi-minimal residual (QMR) method, which avoids the second source of breakdowns in the BCG algorithm. We present details of the new method and discuss some of its properties. In particular, we discuss the relationship between QMR and BCG, showing how one can recover the BCG iterates, when they exist, from the QMR iterates. We also present convergence results for QMR, showing the connection between QMR and the generalized minimal residual (GMRES) algorithm, the optimal method in this class of methods. Finally, we give some numerical examples, both for eigenvalue computations and for non-Hermitian linear systems.

Nachtigal, Noel M.↗

NASA NTRS1981

Eigenvector determination by iterative optical methods

Three power algorithms are considered by which an iterative optical processor can be used to compute the eigenvalues and eigenvectors of a matrix. The algorithms are appropriate for three applications: (1) calculating the largest eigenvalue; (2) calculating the eigenvalues in order of decreasingly dominant eigenvalues; and (3) calculating the eigenvector whose eigenvalue is closest to a given value.

Vijaya Kumar, B. V. K.↗

DOE OSTI2026

Laplace Transform–Based Quantum Eigenvalue Transformation via Linear Combination of Hamiltonian Simulation

Eigenvalue transformations, which include solving time-dependent differential equations as a special case, have a wide range of applications in scientific and engineering computation. While quantum algorithms for singular value transformations are well studied, eigenvalue transformations are distinct, especially for nonnormal matrices. Here, we propose an efficient quantum algorithm for performing a class of eigenvalue transformations that can be expressed as a certain type of matrix Laplace transformation. This allows us to significantly extend the recently developed linear combination of Hamiltonian simulation method [D. An, J.-P. Liu, and L. Lin, Phys. Rev. Lett., 131 (2023), 150603; D. An, A. M. Childs, and L. Lin, Commun. Math. Phys. 407, 19 (2026)] to represent a wider class of eigenvalue transformations, such as powers of the matrix inverse, 𝐴 −𝑘 , and the exponential of the matrix inverse, 𝑒 −𝐴 −1 . The latter can be interpreted as the solution of a mass-matrix differential equation of the form form 𝐴⁢𝑢′⁡⁡(𝑡) =−𝑢⁡(𝑡). We demonstrate that our eigenvalue transformation approach can solve this problem without explicitly inverting 𝐴, thereby reducing the computational complexity.

Laplace transform↗

NASA NTRS2004

Computational Aerothermodynamic Simulation Issues on Unstructured Grids

The synthesis of physical models for gas chemistry and turbulence from the structured grid codes LAURA and VULCAN into the unstructured grid code FUN3D is described. A directionally Symmetric, Total Variation Diminishing (STVD) algorithm and an entropy fix (eigenvalue limiter) keyed to local cell Reynolds number are introduced to improve solution quality for hypersonic aeroheating applications. A simple grid-adaptation procedure is incorporated within the flow solver. Simulations of flow over an ellipsoid (perfect gas, inviscid), Shuttle Orbiter (viscous, chemical nonequilibrium) and comparisons to the structured grid solvers LAURA (cylinder, Shuttle Orbiter) and VULCAN (flat plate) are presented to show current capabilities. The quality of heating in 3D stagnation regions is very sensitive to algorithm options in general, high aspect ratio tetrahedral elements complicate the simulation of high Reynolds number, viscous flow as compared to locally structured meshes aligned with the flow.

Gnoffo, Peter A.↗

NASA NTRS1993

Eigenvalue routines in NASTRAN: A comparison with the Block Lanczos method

The NASA STRuctural ANalysis (NASTRAN) program is one of the most extensively used engineering applications software in the world. It contains a wealth of matrix operations and numerical solution techniques, and they were used to construct efficient eigenvalue routines. The purpose of this paper is to examine the current eigenvalue routines in NASTRAN and to make efficiency comparisons with a more recent implementation of the Block Lanczos algorithm by Boeing Computer Services (BCS). This eigenvalue routine is now available in the BCS mathematics library as well as in several commercial versions of NASTRAN. In addition, CRAY maintains a modified version of this routine on their network. Several example problems, with a varying number of degrees of freedom, were selected primarily for efficiency bench-marking. Accuracy is not an issue, because they all gave comparable results. The Block Lanczos algorithm was found to be extremely efficient, in particular, for very large size problems.

Tischler, V. A.↗

NASA NTRS1976

A study of sound transmission in curved duct bends by the Galerkin method

A computational procedure based on the Galerkin method is developed to study the sound transmission and reflection characteristics of circularly curved bends of rectangular ducts. This procedure is adaptable to routine computer calculation. An important component of the computer program consists of a QR iterative algorithm which solves the matrix eigenvalue problem generated by the Galerkin method. The present procedure produced very satisfactory convergent results when applied even to cases where the incident wave has high frequency and high wave mode numbers. Some properties of the Galerkin solution are investigated. Its relationship to the classical solution by the method of separation of variables is discussed.

Tam, C. K. W.↗

NASA NTRS1984

The ERODYN and QRPIG computer programs

The role of the ERODYN computer program in providing error analyses involving orbital, geodetic, and geophysical parameters is discussed. It was designed to operate as a companion program to the GEODYN orbit determination and parameter estimating program. The Q R Partitioned Eigenvalue/Eigenvector analysis program (QRPIG) is designed to process symmetric matrices with an out-of-core partitioning algorithm for the eigenvectors and eigenvalues of the matrices.

Felsentreger, T. L.↗

NASA NTRS1991

Block Lanczos Algorithm For Gyroscopic Systems

Report describes details of implementation of procedure for accurate and economical computation of natural frequencies and associated vibrational modes of elastic structure rotating along arbitrary axis. Block version of Lanczos algorithm derived for solution of eigenvalue and eigenvector problems. Fully exploits sparsity of associated matrices and employs only real numbers in all relevant computations. Capable of determining multiple roots and proves to be most efficient when compared to other similar existing techniques.

Gupta, Kajal K.↗