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NASA NTRS2019

Observations on the Computation of Eigenvalue and Eigenvector Jacobians

Many scientific and engineering problems benefit from analytic expressions for eigenvalue and eigenvector derivatives with respect to the elements of the parent matrix. While there exists extensive literature on the calculation of these derivatives, which take the form of Jacobian matrices, there are a variety of deficiencies that have yet to be addressed — including the need for both left and right eigenvectors, limitations on the matrix structure, and issues with complex eigenvalues and eigenvectors. This work addresses these deficiencies by proposing a new analytic solution for the eigenvalue and eigenvector derivatives. The resulting analytic Jacobian matrices are numerically efficient to compute and are valid for the general complex case. It is further shown that this new general result collapses to previously known relations for the special cases of real symmetric matrices and real diagonal matrices. Finally, the new Jacobian expressions are validated using forward finite differencing and performance is compared with another technique.

Jacobian↗

NASA NTRS1977

Robustness of linear quadratic state feedback designs in the presence of system uncertainty

The paper deals with the problem of expressing the robustness (stability) property of a linear quadratic state feedback (LQSF) design quantitatively in terms of bounds on the perturbations (modeling errors or parameter variations) in the system matrices so that the closed-loop system remains stable. Nonlinear time-varying and linear time-invariant perturbations are considered. The only computation required in obtaining a measure of the robustness of an LQSF design is to determine the eigenvalues of two symmetric matrices determined when solving the algebraic Riccati equation corresponding to the LQSF design problem. Results are applied to a complex dynamic system consisting of the flare control of a STOL aircraft. The design of the flare control is formulated as an LQSF tracking problem.

Patel, R. V.↗

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 NTRS1975

Matrix bandwidth and profile reduction

This program, REDUCE, reduces the bandwidth and profile of sparse symmetric matrices, using row and corresponding column permutations. It is a realization of the algorithm described by the authors elsewhere. It was extensively tested and compared with several other programs and was found to be considerably faster than the others, superior for bandwidth reduction and as satisfactory as any other for profile reduction.

Crane, H. L., Jr.↗

NASA NTRS1975

ALARM: A highly efficient eigenvalue extraction routine for very large matrixes

A highly efficient computer program, called ALARM, for the determination of eigenvalues and eigenvectors of large symmetric matrices (over 10,000 degrees of freedom), was described. As such, it is highly useful for analyzing complex finite element and finite difference idealizations encountered in structural dynamics and acoustic probelms. The program is based upon a scheme which reduces a large matrix to an equivalent tridiagonal one of much smaller size. The main strength of the algorithm lies in its ability to retain all the information required to obtain the eigenvalues at either end of the original matrix spectrum.

Ojalvo, I. U.↗

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 NTRS1988

Solving large sparse eigenvalue problems on supercomputers

An important problem in scientific computing consists in finding a few eigenvalues and corresponding eigenvectors of a very large and sparse matrix. The most popular methods to solve these problems are based on projection techniques on appropriate subspaces. The main attraction of these methods is that they only require the use of the matrix in the form of matrix by vector multiplications. The implementations on supercomputers of two such methods for symmetric matrices, namely Lanczos' method and Davidson's method are compared. Since one of the most important operations in these two methods is the multiplication of vectors by the sparse matrix, methods of performing this operation efficiently are discussed. The advantages and the disadvantages of each method are compared and implementation aspects are discussed. Numerical experiments on a one processor CRAY 2 and CRAY X-MP are reported. Possible parallel implementations are also discussed.

Philippe, Bernard↗

NASA NTRS1993

Massively parallel computation of RCS with finite elements

One of the promising combinations of finite element approaches for scattering problems uses Whitney edge elements, spherical vector wave-absorbing boundary conditions, and bi-conjugate gradient solution for the frequency-domain near field. Each of these approaches may be criticized. Low-order elements require high mesh density, but also result in fast, reliable iterative convergence. Spherical wave-absorbing boundary conditions require additional space to be meshed beyond the most minimal near-space region, but result in fully sparse, symmetric matrices which keep storage and solution times low. Iterative solution is somewhat unpredictable and unfriendly to multiple right-hand sides, yet we find it to be uniformly fast on large problems to date, given the other two approaches. Implementation of these approaches on a distributed memory, message passing machine yields huge dividends, as full scalability to the largest machines appears assured and iterative solution times are well-behaved for large problems. We present times and solutions for computed RCS for a conducting cube and composite permeability/conducting sphere on the Intel ipsc860 with up to 16 processors solving over 200,000 unknowns. We estimate problems of approximately 10 million unknowns, encompassing 1000 cubic wavelengths, may be attempted on a currently available 512 processor machine, but would be exceedingly tedious to prepare. The most severe bottlenecks are due to the slow rate of mesh generation on non-parallel machines and the large transfer time from such a machine to the parallel processor. One solution, in progress, is to create and then distribute a coarse mesh among the processors, followed by systematic refinement within each processor. Elimination of redundant node definitions at the mesh-partition surfaces, snap-to-surface post processing of the resulting mesh for good modelling of curved surfaces, and load-balancing redistribution of new elements after the refinement are auxiliary steps expected to result in a robust low i/o system for very large finite element problems.

Parker, Jay↗

NASA NTRS1992

Towards a fast implementation of spectral nested dissection

We describe the spectral nested dissection (SND) algorithm, a new algorithm for computing orderings appropriate for parallel factorization of sparse, symmetric matrices. The algorithm makes use of spectral properties of the Laplacian matrix associated with the given matrix to compute separators. We evaluate the quality of the spectral orderings with respect to several measures: fill, elimination tree height, height and weight balances of elimination trees, and clique tree heights. We use some very large structural analysis problems as test cases and demonstrate on these real applications (such as the Space Shuttle Solid Rocket Booster) that spectral orderings compare quite favorably with commonly used orderings, outperforming them by a wide margin for some of these measures. The only disadvantage of SND is its relatively long execution time. We will present some recent efforts to improve the execution time using both a multilevel and a hybrid approach. We use SND in computing a multifrontal numerical factorization with the different orderings on an eight processor Cray Y-MP and show its effectiveness. We believe that spectral nested dissection is a major breakthrough in terms of generating efficient sparse orderings for parallel machines.

Pothen, Alex↗

NASA NTRS1997

Two Improved Algorithms for Envelope and Wavefront Reduction

Two algorithms for reordering sparse, symmetric matrices or undirected graphs to reduce envelope and wavefront are considered. The first is a combinatorial algorithm introduced by Sloan and further developed by Duff, Reid, and Scott; we describe enhancements to the Sloan algorithm that improve its quality and reduce its run time. Our test problems fall into two classes with differing asymptotic behavior of their envelope parameters as a function of the weights in the Sloan algorithm. We describe an efficient 0(nlogn + m) time implementation of the Sloan algorithm, where n is the number of rows (vertices), and m is the number of nonzeros (edges). On a collection of test problems, the improved Sloan algorithm required, on the average, only twice the time required by the simpler Reverse Cuthill-Mckee algorithm while improving the mean square wavefront by a factor of three. The second algorithm is a hybrid that combines a spectral algorithm for envelope and wavefront reduction with a refinement step that uses a modified Sloan algorithm. The hybrid algorithm reduces the envelope size and mean square wavefront obtained from the Sloan algorithm at the cost of greater running times. We illustrate how these reductions translate into tangible benefits for frontal Cholesky factorization and incomplete factorization preconditioning.

Kumfert, Gary↗

NASA NTRS1992

Algorithms For Compression Of Polarimetric-Radar Image Data

Two sets of algorithms provide moderate improvements in compression and decompression of scattering-matrix data from polarimetric imaging radar system. Algorithms operate on original single-look scattering matrices, do not symmetrize scattering matrices and, preserve asymmetrical data on background-noise and equipment effects.

Van Zyl, Jakob J.↗

NASA NTRS1991

Diffraction image formation in optical systems with polarization aberrations. II - Amplitude response matrices for rotationally symmetric systems

In the previous paper in this series (McGuire and Chipman, 1990), a formulation was established for the calculation and analysis of diffraction image quality in polarizing optical systems illuminated with partially polarized, partially coherent light. In the present paper, the effect of second- and fourth-order polarization aberrations on the image plane diffraction patterns are examined. The amplitude response matrix is calculated for optical systems with small numerical apertures. Numerical results are presented for optical systems with circular apertures for three of the aberration types.

Mcguire, James P., Jr.↗

NASA NTRS1975

Diagonalization and simultaneous symmetrization of the gas-dynamic matrices

The hyperbolic nature of the unsteady, inviscid, gas-dynamic equations implies the existence of a similarity transformation for diagonalizing an arbitrary linear combination of coefficient matrices. It is shown that the individual matrices are simultaneously symmetrized by the similarity transformation. The transformations and their norms can be applied to the well-posedness of the Cauchy problem, linear stability theory for finite-difference approximations, and simplification of block-tridiagonal systems that arise in implicit time-split algorithms.

Warming, R. F.↗

NASA NTRS1993

Efficient, massively parallel eigenvalue computation

In numerical simulations of disordered electronic systems, one of the most common approaches is to diagonalize random Hamiltonian matrices and to study the eigenvalues and eigenfunctions of a single electron in the presence of a random potential. An effort to implement a matrix diagonalization routine for real symmetric dense matrices on massively parallel SIMD computers, the Maspar MP-1 and MP-2 systems, is described. Results of numerical tests and timings are also presented.

Huo, Yan↗

NASA NTRS1976

Symmetric stiffness matrix for incompressible hyperelastic materials

Symmetric structure matrices are derived for solving plane strain and axisymmetric problems involving incompressible hyperelastic materials. An infinite hollow cylinder subjected to internal pressure is considered as an example. Displacement and hydrostatic pressure profiles are calculated using the Newton-Raphson iteration technique. The results are in good agreement with the exact curves.

Takamatsu, T.↗

NASA NTRS1984

Finite Element Prediction of Acoustic Scattering and Radiation from Submerged Elastic Structures

A finite element formulation is derived for the scattering and radiation of acoustic waves from submerged elastic structures. The formulation uses as fundamental unknowns the displacement in the structure and a velocity potential in the field. Symmetric coefficient matrices result. The outer boundary of the fluid region is terminated with an approximate local wave-absorbing boundary condition which assumes that outgoing waves are locally planar. The finite element model is capable of predicting only the near-field acoustic pressures. Far-field sound pressure levels may be determined by integrating the surface pressures and velocities over the wet boundary of the structure using the Helmholtz integral. Comparison of finite element results with analytic results show excellent agreement. The coupled fluid-structure problem may be solved with general purpose finite element codes by using an analogy between the equations of elasticity and the wave equation of linear acoustics.

Everstine, G. C.↗

NASA NTRS1992

Design of dissipative low-authority controllers using an eigensystem assignment technique

A novel method for the design of dissipative, low-authority controllers has been developed. The method uses a sequential approach along with eigensystem assignment to compute rate and position gain matrices that assign a number of closed-loop poles of the system to desired locations. Because the feedback gain matrices are symmetric and nonnegative definite, the closed-loop stability is always guaranteed regardless of the model order or parameter inaccuracies. The resulting (nominal) closed-loop system can have specified damping ratios for m modes, which makes the plant amenable to high-authority controller design, using methods such as LQG/LTR or H-infinity. A numerical example is worked out for a flexible structure in order to demonstrate the proposed technique.

Maghami, P. G.↗

NASA NTRS1992

Applications of multiple-constraint matrix updates to the optimal control of large structures

Low-authority control or vibration suppression in large, flexible space structures can be formulated as a linear feedback control problem requiring computation of displacement and velocity feedback gain matrices. To ensure stability in the uncontrolled modes, these gain matrices must be symmetric and positive definite. In this paper, efficient computation of symmetric, positive-definite feedback gain matrices is accomplished through the use of multiple-constraint matrix update techniques originally developed for structural identification applications. Two systems were used to illustrate the application: a simple spring-mass system and a planar truss. From these demonstrations, use of this multiple-constraint technique is seen to provide a straightforward approach for computing the low-authority gains.

Smith, S. W.↗