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At least 451 records · Page 25

Estimating Microwave Delay by Atmospheric Water

Tropospheric path delays for microwave very-long-baseline interferometry (VLBI) estimated with algorithm that determines and explicitly integrates simple water-vapor distribution based on temperature data from water-vapor radiometer (WVR) and emission model. Although computationally complex, method readily accommodates even dramatic changes in observation conditions, emission model, and WVR equipment. Algorithm accommodates changes in observation conditions, emission model, and radiometer hardware.

Robinson, S. E.↗

Experiments in global navigation and control of a free-flying space robot

This paper reviews initial work at the Stanford University Aerospace Robotics Laboratory (ARL) in developing and controlling a free-flying space robot. The objective of this project is to develop a laboratory version of a space robot that is capable of performing target tracking, acquisition, and manipulation. In particular, this paper focuses on the problems associated with capturing a free-floating object that is initially out of reach of the robot. A set of rules is presented for generating an appropriate intercept trajectory. A controller architecture suitable for carrying out these motions is also described. Finally the results of computer simulations illustrating the behavior of these algorithms are shown along with a description of the physical hardware on which they will be tested.

Ullman, M.↗

Discreteness noise versus force errors in N-body simulations

A low accuracy in the force calculation per time step of a few percent for each particle pair is sufficient for collisionless N-body simulations. Higher accuracy is made meaningless by the dominant discreteness noise in the form of two-body relaxation, which can be reduced only by increasing the number of particles. Since an N-body simulation is a Monte Carlo procedure in which each particle-particle force is essentially random, i.e., carries an error of about 1000 percent, the only requirement is a systematic averaging-out of these intrinsic errors. We illustrate these assertions with two specific examples in which individual pairwise forces are deliberately allowed to carry significant errors: tree-codes on supercomputers and algorithms on special-purpose machines with low-precision hardware.

Hernquist, Lars↗

Optical position measurement for a large gap magnetic suspension system: Design and performance analysis

An optical measurement system (OMS) has been designed and tested for a large gap magnetic suspension system (LGMSS). The LGMSS will be used to study control laws for magnetic suspension systems for vibration isolation and pointing applications. The LGMSS features six degrees of freedom and consists of a planar array of electromagnets that levitate and position a cylindrical element containing a permanent magnet core. The OMS provides information on the location and orientation of the element to the LGMSS control system to stabilize suspension. The hardware design of this optical sensing system and the tracking algorithms are presented. The results of analyses and experiments are presented that define the accuracy limits of the optical sensing system and that quantify the errors in position estimation.

Welch, Sharon S.↗

Evolutionary Computation Applied to the Tuning of MEMS Gyroscopes

We propose a tuning method for MEMS gyroscopes based on evolutionary computation to efficiently increase the sensitivity of MEMS gyroscopes through tuning and, furthermore, to find the optimally tuned configuration for this state of increased sensitivity. The tuning method was tested for the second generation JPL/Boeing Post-resonator MEMS gyroscope using the measurement of the frequency response of the MEMS device in open-loop operation.

evolvable hardware↗

Reactive, Safe Navigation for Lunar and Planetary Robots

When humans return to the moon, Astronauts will be accompanied by robotic helpers. Enabling robots to safely operate near astronauts on the lunar surface has the potential to significantly improve the efficiency of crew surface operations. Safely operating robots in close proximity to astronauts on the lunar surface requires reactive obstacle avoidance capabilities not available on existing planetary robots. In this paper we present work on safe, reactive navigation using a stereo based high-speed terrain analysis and obstacle avoidance system. Advances in the design of the algorithms allow it to run terrain analysis and obstacle avoidance algorithms at full frame rate (30Hz) on off the shelf hardware. The results of this analysis are fed into a fast, reactive path selection module, enforcing the safety of the chosen actions. The key components of the system are discussed and test results are presented.

Utz, Hans↗

Enigma Version 12

Enigma Version 12 software combines model building, animation, and engineering visualization into one concise software package. Enigma employs a versatile user interface to allow average users access to even the most complex pieces of the application. Using Enigma eliminates the need to buy and learn several software packages to create an engineering visualization. Models can be created and/or modified within Enigma down to the polygon level. Textures and materials can be applied for additional realism. Within Enigma, these models can be combined to create systems of models that have a hierarchical relationship to one another, such as a robotic arm. Then these systems can be animated within the program or controlled by an external application programming interface (API). In addition, Enigma provides the ability to use plug-ins. Plugins allow the user to create custom code for a specific application and access the Enigma model and system data, but still use the Enigma drawing functionality. CAD files can be imported into Enigma and combined to create systems of computer graphics models that can be manipulated with constraints. An API is available so that an engineer can write a simulation and drive the computer graphics models with no knowledge of computer graphics. An animation editor allows an engineer to set up sequences of animations generated by simulations or by conceptual trajectories in order to record these to highquality media for presentation. Enigma Version 12 Lyndon B. Johnson Space Center, Houston, Texas 28 NASA Tech Briefs, September 2013 Planetary Protection Bioburden Analysis Program NASA's Jet Propulsion Laboratory, Pasadena, California This program is a Microsoft Access program that performed statistical analysis of the colony counts from assays performed on the Mars Science Laboratory (MSL) spacecraft to determine the bioburden density, 3-sigma biodensity, and the total bioburdens required for the MSL prelaunch reports. It also contains numerous tools that report the data in various ways to simplify the reports required. The program performs all the calculations directly in the MS Access program. Prior to this development, the data was exported to large Excel files that had to be cut and pasted to provide the desired results. The program contains a main menu and a number of submenus. Analyses can be performed by using either all the assays, or only the accountable assays that will be used in the final analysis. There are three options on the first menu: either calculate using (1) the old MER (Mars Exploration Rover) statistics, (2) the MSL statistics for all the assays, or This software implements penetration limit equations for common micrometeoroid and orbital debris (MMOD) shield configurations, windows, and thermal protection systems. Allowable MMOD risk is formulated in terms of the probability of penetration (PNP) of the spacecraft pressure hull. For calculating the risk, spacecraft geometry models, mission profiles, debris environment models, and penetration limit equations for installed shielding configurations are required. Risk assessment software such as NASA's BUMPERII is used to calculate mission PNP; however, they are unsuitable for use in shield design and preliminary analysis studies. The software defines a single equation for the design and performance evaluation of common MMOD shielding configurations, windows, and thermal protection systems, along with a description of their validity range and guidelines for their application. Recommendations are based on preliminary reviews of fundamental assumptions, and accuracy in predicting experimental impact test results. The software is programmed in Visual Basic for Applications for installation as a simple add-in for Microsoft Excel. The user is directed to a graphical user interface (GUI) that requires user inputs and provides solutions directly in Microsoft Excel workbooks. This work was done by Shannon Ryan of the USRA Lunar and Planetary Institute for Johnson Space Center. Further information is contained in a TSP (see page 1). MSC- 24582-1 Micrometeoroid and Orbital Debris (MMOD) Shield Ballistic Limit Analysis Program Lyndon B. Johnson Space Center, Houston, Texas Commercially, because it is so generic, Enigma can be used for almost any project that requires engineering visualization, model building, or animation. Models in Enigma can be exported to many other formats for use in other applications as well. Educationally, Enigma is being used to allow university students to visualize robotic algorithms in a simulation mode before using them with actual hardware.

Shores, David↗

Planetary Protection Bioburden Analysis Program

This program is a Microsoft Access program that performed statistical analysis of the colony counts from assays performed on the Mars Science Laboratory (MSL) spacecraft to determine the bioburden density, 3-sigma biodensity, and the total bioburdens required for the MSL prelaunch reports. It also contains numerous tools that report the data in various ways to simplify the reports required. The program performs all the calculations directly in the MS Access program. Prior to this development, the data was exported to large Excel files that had to be cut and pasted to provide the desired results. The program contains a main menu and a number of submenus. Analyses can be performed by using either all the assays, or only the accountable assays that will be used in the final analysis. There are three options on the first menu: either calculate using (1) the old MER (Mars Exploration Rover) statistics, (2) the MSL statistics for all the assays, or This software implements penetration limit equations for common micrometeoroid and orbital debris (MMOD) shield configurations, windows, and thermal protection systems. Allowable MMOD risk is formulated in terms of the probability of penetration (PNP) of the spacecraft pressure hull. For calculating the risk, spacecraft geometry models, mission profiles, debris environment models, and penetration limit equations for installed shielding configurations are required. Risk assessment software such as NASA's BUMPERII is used to calculate mission PNP; however, they are unsuitable for use in shield design and preliminary analysis studies. The software defines a single equation for the design and performance evaluation of common MMOD shielding configurations, windows, and thermal protection systems, along with a description of their validity range and guidelines for their application. Recommendations are based on preliminary reviews of fundamental assumptions, and accuracy in predicting experimental impact test results. The software is programmed in Visual Basic for Applications for installation as a simple add-in for Microsoft Excel. The user is directed to a graphical user interface (GUI) that requires user inputs and provides solutions directly in Microsoft Excel workbooks. This work was done by Shannon Ryan of the USRA Lunar and Planetary Institute for Johnson Space Center. Further information is contained in a TSP (see page 1). MSC- 24582-1 Micrometeoroid and Orbital Debris (MMOD) Shield Ballistic Limit Analysis Program Lyndon B. Johnson Space Center, Houston, Texas Commercially, because it is so generic, Enigma can be used for almost any project that requires engineering visualization, model building, or animation. Models in Enigma can be exported to many other formats for use in other applications as well. Educationally, Enigma is being used to allow university students to visualize robotic algorithms in a simulation mode before using them with actual hardware. This work was done by David Shores and Sharon P. Goza of Johnson Space Center; Cheyenne McKeegan, Rick Easley, Janet Way, and Shonn Everett of MEI Technologies; Mark Manning of PTI; and Mark Guerra, Ray Kraesig, and William Leu of Tietronix Software, Inc. For further information, contact the JSC Innovation Partnerships Office at (281) 483-3809. MSC-24211-1 Spitzer Telemetry Processing System NASA's Jet Propulsion Laboratory, Pasadena, California The Spitzer Telemetry Processing System (SirtfTlmProc) was designed to address objectives of JPL's Multi-mission Image Processing Lab (MIPL) in processing spacecraft telemetry and distributing the resulting data to the science community. To minimize costs and maximize operability, the software design focused on automated error recovery, performance, and information management. The system processes telemetry from the Spitzer spacecraft and delivers Level 0 products to the Spitzer Science Center. SirtfTlmProc is a unique system with automated error notification and recovery, with a real-time continuous service that can go quiescent after periods of inactivity. The software can process 2 GB of telemetry and deliver Level 0 science products to the end user in four hours. It provides analysis tools so the operator can manage the system and troubleshoot problems. It automates telemetry processing in order to reduce staffing costs. This work was done by Alice Stanboli, Elmain M. Martinez, and James M. McAuley of Caltech for NASA's Jet Propulsion Laboratory. For more information, contact iaoffice @jpl.nasa.gov. This software is available for commercial licensing. Please contact Dan Broderick at Daniel.F. Broderick@jpl.nasa.gov. Refer to NPO-47803. NASA Tech Briefs, September 2013 29 This rapid response computer program predicts Orbiter Wing Leading Edge (WLE) damage caused by ice or foam impact during a Space Shuttle launch (Program "IMPACT2"). The program was developed after the Columbia accident in order to assess quickly WLE damage due to ice, foam, or metal impact (if any) during a Shuttle launch. IMPACT2 simulates an impact event in a few minutes for foam impactors, and in seconds for ice and metal impactors. The damage criterion is derived from results obtained from one sophisticated commercial program, which requires hours to carry out simulations of the same impact events. The program was designed to run much faster than the commercial program with prediction of projectile threshold velocities within 10 to 15% of commercial-program values. The mathematical model involves coupling of Orbiter wing normal modes of vibration to nonlinear or linear springmass models. IMPACT2 solves nonlinear or linear impact problems using classical normal modes of vibration of a target, and nonlinear/ linear time-domain equations for the projectile. Impact loads and stresses developed in the target are computed as functions of time. This model is novel because of its speed of execution. A typical model of foam, or other projectile characterized by material nonlinearities, impacting an RCC panel is executed in minutes instead of hours needed by the commercial programs. Target damage due to impact can be assessed quickly, provided that target vibration modes and allowable stress are known. This work was done by Robert Clark, Jr., Paul Cotter, and Constantine Michalopoulos of The Boeing Company for Johnson Space Center. For further information, contact the JSC Innovation Partnerships Office at (281) 483-3809. MSC-24988-1 Wing Leading Edge RCC Rapid Response Damage Prediction Tool (IMPACT2) Lyndon B. Johnson Space Center, Houston, Texas (3) the MSL statistics for only the accountable assays. Other options on the main menu include a data editing form and utility programs that produce various reports requested by the microbiologists and the project, and tools to generate the groupings for the final analyses. The analyses can be carried out in three ways: Each assay can be treated separately, the assays can be collectively treated for the whole zone as a group, or the assays can be collected in groups designated by the JPL Planetary Protection Manager. The latter approach was used to generate the final report because assays on the same equipment or similar equipment can be assumed to have been exposed to the same environment and cleaning. Thus, the statistics are improved by having a larger population, thereby reducing the standard deviation by the square root of N. For each method mentioned above, three reports are available. The first is a detailed report including all the data. This version was very useful in verifying the calculations. The second is a brief report that is similar to the full detailed report, but does not print out the data. The third is a grand total and summary report in which each assay requires only one line. For the first and second reports, most of the calculations are performed in the report section itself. For the third, all the calculations are performed directly in the query bound to the report. All the numerical results were verified by comparing them with Excel templates, then exporting the data from the Planetary Protection Analysis program to Excel.

Beaudet, Robert A.↗

SMAP Mission: Changes in the RFI Environment

The Soil Moisture Active/Passive satellite microwave radiometer has been providing measurements of L-band thermal emission from Earth for more than 2 years. SMAP retrieves surface soil moisture from its brightness temperature measurements, and continues to provide science products to the user community. Even though the SMAP radiometer operates in a protected band, its measurements are still corrupted by Radio Frequency Interference (RFI) caused by illegal in-band transmissions or out-of-band emissions. The SMAP radiometer was designed to include special hardware to enable RFI detection and filtering using multiple detection algorithms. Given the good overall performance of SMAP algorithms to detect RFI sources, an automatic tool to report source properties automatically was developed and is now operational. This paper provides a preliminary analysis of the outputs of this reporting tool with a particular focus on the evolution of the RFI environment observed by SMAP during its period of operations.

L-band↗

Enabling Intelligent Data Downlink Prioritization of In-Situ Observations through Generalizable and Computationally Inexpensive Anomaly Detection

High-fidelity measurements of magnetic fields and other observed properties, such as energetic particle fluxes, are a necessary component to our understanding of the highly dynamic near-Earth space environment. As our desire to study smaller-scale phenomena such as shocks and dipolorizations has increased, we have been driven to take and telemeter measurements at higher cadences. Unfortunately, many missions are unable to downlink all their captured data due to the well-known data transmission bottleneck at the DSN. These missions must then prioritize their high-cadence data such that the most scientifically useful intervals are transmitted. One simple prioritization technique uses the spacecraft position to telemeter data from only the region of interest. Although easy to implement, this method does not leverage the available scientific data and can omit intervals of useful scientific data when they lie outside the region of interest. The Magnetospheric Multiscale Mission (MMS) uses mission-specific parameterization of several data products to automatically prioritize scientifically useful intervals. Then, MMS verifies the automatically selected intervals by having a domain expert manually select intervals for downlink. The overall complexity required by this technique make it prohibitive for deployment on low-cost platforms (i.e., CubeSats) or on future missions featuring large constellations of satellites such as the Geospace Dynamics Constellation (GDC). We present preliminary results for a simple, generic, and data-driven method of downlink prioritization for magnetic field (and other) measurements. Specifically, Principal Components Analysis (PCA) and One-Class Support Vector Machines (OC-SVMs) are used to detect intervals containing anomalous activity, which can then be prioritized for subsequent downlink. The computational simplicity of this algorithm makes it an excellent candidate for implementation on spaceflight hardware, as well as provide generalizability to a broad range of missions and data products. Initial analysis of this technique has been performed using magnetic field measurements from the Magnetospheric Multiscale Mission and CASSIOP, where it automatically identified scientifically interesting intervals containing Alfvén waves and EMIC activity.

Matthew G. Finley↗

Enabling Intelligent Data Downlink Prioritization of In-Situ Observations through Generalizable and Computationally Inexpensive Anomaly Detection

High-fidelity measurements of magnetic fields and other observed properties, such as energetic particle fluxes, are a necessary component to our understanding of the highly dynamic near-Earth space environment. As our desire to study smaller-scale phenomena such as shocks and dipolorizations has increased, we have been driven to take and telemeter measurements at higher cadences. Unfortunately, many missions are unable to downlink all their captured data due to the well-known data transmission bottleneck at the DSN. These missions must then prioritize their high-cadence data such that the most scientifically useful intervals are transmitted. One simple prioritization technique uses the spacecraft position to telemeter data from only the region of interest. Although easy to implement, this method does not leverage the available scientific data and can omit intervals of useful scientific data when they lie outside the region of interest. The Magnetospheric Multiscale Mission (MMS) uses mission-specific parameterization of several data products to automatically prioritize scientifically useful intervals. Then, MMS verifies the automatically selected intervals by having a domain expert manually select intervals for downlink. The overall complexity required by this technique make it prohibitive for deployment on low-cost platforms (i.e., CubeSats) or on future missions featuring large constellations of satellites such as the Geospace Dynamics Constellation (GDC). We present preliminary results for a simple, generic, and data-driven method of downlink prioritization for magnetic field (and other) measurements. Specifically, Principal Components Analysis (PCA) and One-Class Support Vector Machines (OC-SVMs) are used to detect intervals containing anomalous activity, which can then be prioritized for subsequent downlink. The computational simplicity of this algorithm makes it an excellent candidate for implementation on spaceflight hardware, as well as provide generalizability to a broad range of missions and data products. Initial analysis of this technique has been performed using magnetic field measurements from the Magnetospheric Multiscale Mission and CASSIOP, where it automatically identified scientifically interesting intervals containing Alfvén waves and EMIC activity.

Matthew G. Finley↗

FPGA Coprocessor for Accelerated Classification of Images

An effort related to that described in the preceding article focuses on developing a spaceborne processing platform for fast and accurate onboard classification of image data, a critical part of modern satellite image processing. The approach again has been to exploit the versatility of recently developed hybrid Virtex-4FX field-programmable gate array (FPGA) to run diverse science applications on embedded processors while taking advantage of the reconfigurable hardware resources of the FPGAs. In this case, the FPGA serves as a coprocessor that implements legacy C-language support-vector-machine (SVM) image-classification algorithms to detect and identify natural phenomena such as flooding, volcanic eruptions, and sea-ice break-up. The FPGA provides hardware acceleration for increased onboard processing capability than previously demonstrated in software. The original C-language program demonstrated on an imaging instrument aboard the Earth Observing-1 (EO-1) satellite implements a linear-kernel SVM algorithm for classifying parts of the images as snow, water, ice, land, or cloud or unclassified. Current onboard processors, such as on EO-1, have limited computing power, extremely limited active storage capability and are no longer considered state-of-the-art. Using commercially available software that translates C-language programs into hardware description language (HDL) files, the legacy C-language program, and two newly formulated programs for a more capable expanded-linear-kernel and a more accurate polynomial-kernel SVM algorithm, have been implemented in the Virtex-4FX FPGA. In tests, the FPGA implementations have exhibited significant speedups over conventional software implementations running on general-purpose hardware.

Pingree, Paula J.↗

Preliminary Structural Design Using Topology Optimization with a Comparison of Results from Gradient and Genetic Algorithm Methods

In this paper, genetic algorithm based and gradient-based topology optimization is presented in application to a real hardware design problem. Preliminary design of a planetary lander mockup structure is accomplished using these methods that prove to provide major weight savings by addressing the structural efficiency during the design cycle. This paper presents two alternative formulations of the topology optimization problem. The first is the widely-used gradient-based implementation using commercially available algorithms. The second is formulated using genetic algorithms and internally developed capabilities. These two approaches are applied to a practical design problem for hardware that has been built, tested and proven to be functional. Both formulations converged on similar solutions and therefore were proven to be equally valid implementations of the process. This paper discusses both of these formulations at a high level.

Burt, Adam O.↗

Transistor Level Circuit Experiments using Evolvable Hardware

The Jet Propulsion Laboratory (JPL) performs research in fault tolerant, long life, and space survivable electronics for the National Aeronautics and Space Administration (NASA). With that focus, JPL has been involved in Evolvable Hardware (EHW) technology research for the past several years. We have advanced the technology not only by simulation and evolution experiments, but also by designing, fabricating, and evolving a variety of transistor-based analog and digital circuits at the chip level. EHW refers to self-configuration of electronic hardware by evolutionary/genetic search mechanisms, thereby maintaining existing functionality in the presence of degradations due to aging, temperature, and radiation. In addition, EHW has the capability to reconfigure itself for new functionality when required for mission changes or encountered opportunities. Evolution experiments are performed using a genetic algorithm running on a DSP as the reconfiguration mechanism and controlling the evolvable hardware mounted on a self-contained circuit board. Rapid reconfiguration allows convergence to circuit solutions in the order of seconds. The paper illustrates hardware evolution results of electronic circuits and their ability to perform under 230 C temperature as well as radiations of up to 250 kRad.

self configuration↗

NASA Tech Briefs, March 2013

Topics covered include: Remote Data Access with IDL Data Compression Algorithm Architecture for Large Depth-of-Field Particle Image Velocimeters Vectorized Rebinning Algorithm for Fast Data Down-Sampling Display Provides Pilots with Real-Time Sonic-Boom Information Onboard Algorithms for Data Prioritization and Summarization of Aerial Imagery Monitoring and Acquisition Real-time System (MARS) Analog Signal Correlating Using an Analog-Based Signal Conditioning Front End Micro-Textured Black Silicon Wick for Silicon Heat Pipe Array Robust Multivariable Optimization and Performance Simulation for ASIC Design; Castable Amorphous Metal Mirrors and Mirror Assemblies; Sandwich Core Heat-Pipe Radiator for Power and Propulsion Systems; Apparatus for Pumping a Fluid; Cobra Fiber-Optic Positioner Upgrade; Improved Wide Operating Temperature Range of Li-Ion Cells; Non-Toxic, Non-Flammable, -80 C Phase Change Materials; Soft-Bake Purification of SWCNTs Produced by Pulsed Laser Vaporization; Improved Cell Culture Method for Growing Contracting Skeletal Muscle Models; Hand-Based Biometric Analysis; The Next Generation of Cold Immersion Dry Suit Design Evolution for Hypothermia Prevention; Integrated Lunar Information Architecture for Decision Support Version 3.0 (ILIADS 3.0); Relay Forward-Link File Management Services (MaROS Phase 2); Two Mechanisms to Avoid Control Conflicts Resulting from Uncoordinated Intent; XTCE GOVSAT Tool Suite 1.0; Determining Temperature Differential to Prevent Hardware Cross-Contamination in a Vacuum Chamber; SequenceL: Automated Parallel Algorithms Derived from CSP-NT Computational Laws; Remote Data Exploration with the Interactive Data Language (IDL); Mixture-Tuned, Clutter Matched Filter for Remote Detection of Subpixel Spectral Signals; Partitioned-Interval Quantum Optical Communications Receiver; and Practical UAV Optical Sensor Bench with Minimal Adjustability.

Source record↗

Automatic holographic droplet analysis for liquid fuel sprays

The basic scheme for automated holographic analysis involves an optical system for reconstruction of the three dimensional real image of the droplet field, a spatial scanning system to transport a digitizing X-y image sensor through the real image, and processing algorithms for droplet recognition which establish the droplet sizes and positions. The hardware for system demonstrated includes the expanded and collimated beam from a 5 mW helium-neon laser for hologram reconstruction, an imaging lens for magnification of the real image field, and a video camera and digitizer providing 512-by-512 pixel resolution with 8-bit digitization. A mechanical stage is used to scan the hologram in three dimensional space, maintaining constant image magnification. A test droplet hologram is used for development and testing of the image processing algorithms.

Stanton, A. C.↗

Binary tree eigen solver in finite element analysis

This paper presents a transputer-based binary tree eigensolver for the solution of the generalized eigenproblem in linear elastic finite element analysis. The algorithm is based on the method of recursive doubling, which parallel implementation of a number of associative operations on an arbitrary set having N elements is of the order of o(log2N), compared to (N-1) steps if implemented sequentially. The hardware used in the implementation of the binary tree consists of 32 transputers. The algorithm is written in OCCAM which is a high-level language developed with the transputers to address parallel programming constructs and to provide the communications between processors. The algorithm can be replicated to match the size of the binary tree transputer network. Parallel and sequential finite element analysis programs have been developed to solve for the set of the least-order eigenpairs using the modified subspace method. The speed-up obtained for a typical analysis problem indicates close agreement with the theoretical prediction given by the method of recursive doubling.

Akl, F. A.↗