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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.

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At least 19 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2022

Effects of short-interval reburns in the boreal forest on soil bacterial communities compared to long-interval reburns

Abstract Increasing fire frequency in some biomes is leading to fires burning in close succession, triggering rapid vegetation change and altering soil properties. We studied the effects of short-interval (SI) reburns on soil bacterial communities of the boreal forest of northwestern Canada using paired sites (n = 44). Both sites in each pair had burned in a recent fire; one site had burned within the previous 20 years before the recent fire (SI reburn) and the other had not. Paired sites were closely matched in prefire ecosite characteristics, prefire tree species composition, and stand structure. We hypothesized that there would be a significant effect of short vs. long fire-free intervals on community composition and that richness would not be consistently different between paired sites. We found that Blastococcus sp. was consistently enriched in SI reburns, indicating its role as a strongly ‘pyrophilous’ bacterium. Caballeronia sordidicola was consistently depleted in SI reburns. The depletion of this endophytic diazotroph raises questions about whether this is contributing to—or merely reflects—poor conifer seedling recolonization post-fire at SI reburns. While SI reburns had no significant effect on richness, dissimilarity between short- and long-interval pairs was significantly correlated with difference in soil pH, and there were small significant changes in overall community composition.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2021

Modeling interval trendlines: Symbolic singular spectrum analysis for interval time series

Abstract In this article we propose an extension of singular spectrum analysis for interval‐valued time series. The proposed methods can be used to decompose and forecast the dynamics governing a set‐valued stochastic process. The resulting components on which the interval time series is decomposed can be understood as interval trendlines, cycles, or noise. Forecasting can be conducted through a linear recurrent method, and we devised generalizations of the decomposition method for the multivariate setting. The performance of the proposed methods is showcased in a simulation study. We apply the proposed methods so to track the dynamics governing the Argentina Stock Market (MERVAL) in real time, in a case study over a period of turbulence that led to discussions of the government of Argentina with the International Monetary Fund.

de Carvalho, Miguel↗
DOE OSTI2024

Short-Term Load Forecasting Considering EV Charging Loads with Prediction Interval Evaluation

Short-term load forecasting plays a critical role in power system planning and operation. Along with the electrification of various loads, electricity demands are becoming increasingly hard to predict. Notably, the recent rise in electric vehicles (EVs) has further contributed to this unpredictability. To address this issue, this paper proposes a probabilistic load forecasting strategy utilizing Gaussian process regression, structured in a day-ahead manner. While many works focus on deterministic prediction, probabilistic forecasting offers additional insights into variability and uncertainty, enabling more flexible and reliable operation for power systems. To enhance the accuracy of the load forecasting model, the inputs include features related to EV charging habits as well as commonly used weather information. The load forecasting results are evaluated using various metrics, including conventional ones that assess the accuracy of point forecasts, as well as additional metrics that test the reliability of prediction intervals. The proposed load forecasting method is finally tested on real residential power consumption data and EV charging data sampled from real-world sources. The results prove that the new features can greatly improve the performance of the load forecasting method.

electrical vehicle↗
DOE OSTI2018

The influence of interval arithmetic on the shape of uncertainly defined domains modelled by closed curves

The paper presents an unambiguous modelling of imprecisely defined shapes of closed curves using classical and directed interval arithmetic. The authors focus on the development of an effective strategy of modelling interval smooth closed curves (which enforce C{sup 2} continuity in points at which adjacent interval segments join) using interval cubic Bézier segments. For this purpose, algebraic relationships between Bézier and de Boor control points, formerly known for precisely defined curves, are generalized. We obtain interval control points that define interval closed curves. Additionally, the reliability of such way of modelling of closed curves is examined. We directly apply classical and directed interval arithmetic to mentioned relationships and try to solve obtained interval systems of algebraic equations. However, we obtain ambiguous solutions. Therefore, we propose our new strategy of modification of directed interval arithmetic to obtain reliable and unambiguous shapes of interval closed curves.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1992

Optimal parallel algorithms for problems modeled by a family of intervals

A family of intervals on the real line provides a natural model for a vast number of scheduling and VLSI problems. Recently, a number of parallel algorithms to solve a variety of practical problems on such a family of intervals have been proposed in the literature. Computational tools are developed, and it is shown how they can be used for the purpose of devising cost-optimal parallel algorithms for a number of interval-related problems including finding a largest subset of pairwise nonoverlapping intervals, a minimum dominating subset of intervals, along with algorithms to compute the shortest path between a pair of intervals and, based on the shortest path, a parallel algorithm to find the center of the family of intervals. More precisely, with an arbitrary family of n intervals as input, all algorithms run in O(log n) time using O(n) processors in the EREW-PRAM model of computation.

Olariu, Stephan↗
NASA NTRS1997

Optimal Approximation of Quadratic Interval Functions

Measurements are never absolutely accurate, as a result, after each measurement, we do not get the exact value of the measured quantity; at best, we get an interval of its possible values, For dynamically changing quantities x, the additional problem is that we cannot measure them continuously; we can only measure them at certain discrete moments of time t(sub 1), t(sub 2), ... If we know that the value x(t(sub j)) at a moment t(sub j) of the last measurement was in the interval [x-(t(sub j)), x + (t(sub j))], and if we know the upper bound D on the rate with which x changes, then, for any given moment of time t, we can conclude that x(t) belongs to the interval [x-(t(sub j)) - D (t - t(sub j)), x + (t(sub j)) + D (t - t(sub j))]. This interval changes linearly with time, an is, therefore, called a linear interval function. When we process these intervals, we get an expression that is quadratic and higher order w.r.t. time t, Such "quadratic" intervals are difficult to process and therefore, it is necessary to approximate them by linear ones. In this paper, we describe an algorithm that gives the optimal approximation of quadratic interval functions by linear ones.

Koshelev, Misha↗
NASA NTRS1998

Mechanisms underlying very-low-frequency RR-interval oscillations in humans

BACKGROUND: Survival of post-myocardial infarction patients is related inversely to their levels of very-low-frequency (0.003 to 0.03 Hz) RR-interval variability. The physiological basis for such oscillations is unclear. In our study, we used blocking drugs to evaluate potential contributions of sympathetic and vagal mechanisms and the renin-angiotensin-aldosterone system to very-low-frequency RR-interval variability in 10 young healthy subjects. METHODS AND RESULTS: We recorded RR intervals and arterial pressures during three separate sessions, with the patient in supine and 40 degree upright tilt positions, during 20-minute frequency (0.25 Hz) and tidal volume-controlled breathing after intravenous injections: saline (control), atenolol (0.2 mg/kg, beta-adrenergic blockade), atropine sulfate (0.04 mg/kg, parasympathetic blockade), atenolol and atropine (complete autonomic blockade), and enalaprilat (0.02 mg/kg, ACE blockade). We integrated fast Fourier transform RR-interval spectral power at very low (0.003 to 0.03 Hz), low (0.05 to 0. 15 Hz), and respiratory (0.2 to 0.3 Hz) frequencies. Beta-adrenergic blockade had no significant effect on very-low- or low-frequency RR-interval power but increased respiratory frequency power 2-fold. ACE blockade had no significant effect on low or respiratory frequency RR-interval power but modestly (approximately 21%) increased very-low-frequency power in the supine (but not upright tilt) position (P<0.05). The most profound effects were exerted by parasympathetic blockade: Atropine, given alone or with atenolol, abolished nearly all RR-interval variability and decreased very-low-frequency variability by 92%. CONCLUSIONS: Although very-low-frequency heart period rhythms are influenced by the renin-angiotensin-aldosterone system, as low and respiratory frequency RR-interval rhythms, they depend primarily on the presence of parasympathetic outflow. Therefore the prognostic value of very-low-frequency heart period oscillations may derive from the fundamental importance of parasympathetic mechanisms in cardiovascular health.

Non-NASA Center↗
NASA NTRS2012

Bootstrap Prediction Intervals in Non-Parametric Regression with Applications to Anomaly Detection

Prediction intervals provide a measure of the probable interval in which the outputs of a regression model can be expected to occur. Subsequently, these prediction intervals can be used to determine if the observed output is anomalous or not, conditioned on the input. In this paper, a procedure for determining prediction intervals for outputs of nonparametric regression models using bootstrap methods is proposed. Bootstrap methods allow for a non-parametric approach to computing prediction intervals with no specific assumptions about the sampling distribution of the noise or the data. The asymptotic fidelity of the proposed prediction intervals is theoretically proved. Subsequently, the validity of the bootstrap based prediction intervals is illustrated via simulations. Finally, the bootstrap prediction intervals are applied to the problem of anomaly detection on aviation data.

Kumar, Sricharan↗
DOE OSTI2023

Technical Specification Surveillance Interval Extension Using Self-Diagnostics

As part of the Light Water Reactor Sustainability program, an ongoing research effort is being conducted on technical specifications surveillance interval extension of digital equipment in nuclear power plants. The research team is led by Idaho National Laboratory and includes Pacific Northwest National Laboratory, Technology Resources, and Oak Ridge National Laboratory. This research focuses on developing methods for applying the U.S. Nuclear Regulatory Commission (NRC)–approved guidance to implement a licensee-controlled, risk-informed surveillance frequency change program in digital instrumentation and control (I&C) systems that include self-diagnostics and online monitoring (OLM) capabilities. Although approved methods exist for extending technical specifications (TS) surveillance test intervals (STIs) for general equipment, including analog I&C equipment, gaps remain in technology and guidance on crediting newer digital equipment’s internal self-diagnostics and OLM characteristics. Previous research described a general methodology for crediting internal self-diagnostics for extending surveillance test intervals. The methodology used self-diagnostics to detect—and credited recovery from—failure. Self-diagnostics were also applied for performance monitoring during the extended surveillance interval. This report discusses the status of recent activities to evaluate the previously developed methodology using a pilot study. Although both a utility partner for a pilot study and a specific digital asset were identified in FY2020, delays in obtaining proprietary information resulted in a limited ability to fully evaluate the methodology, and further interactions were complicated by the COVID pandemic. Therefore, at that time, the use of public-domain information—along with current processes for surveillance interval extension through a surveillance frequency control program—identified the need to fully assess diagnostic coverage as part of the pilot study. Furthermore, self-diagnostics were also identified as a potential option to replace the drift analyses conducted as part of current STI extension procedures. In FY2022, the project was reconstituted with the industry partner, and information and data were made available by the industry partner to the research team for review. The shared information included failure event descriptions and data for a digital I&C system since its implementation, as well as recent STI extension interval reports developed by the utility partner on that digital I&C system. This report presents an evaluation of this information and data and describes an application of the proposed methodology cited above. The methodology seeks to take advantage of the self-diagnostics and OLM capabilities to reduce risk or reduce the level of qualitative monitoring assessment needed to perform a risk-informed STI extension using existing NRC approved guidance or both. Addressing these issues of STI extension by crediting self-diagnostics is likely to result in benefits for current and future nuclear power plant (NPP) operations, including lowering the barriers to adoption of digital I&C systems and increasing cost savings by deferring or eliminating unneeded preventive maintenance (tasks or checks or activities). Specifically, self-diagnostic and OLM capabilities of newer digital equipment being installed in non-safety and safety applications are designed to detect failures, provide early warning of potential failures, and notify plant operators to take appropriate action to reduce out of service (OOS) time thus protecting safety margins. Moreover, the equipment is expected to provide information that time-related operational degradation is identified early to ensure timely and planned corrective actions instead of a reactive and unplanned approach ahead of an extended-surveillance interval.

42 ENGINEERING↗
NASA NTRS1987

A new technique to determine the correlation between the QT interval and heart-rate for control and SIDS babies

The ability of the autonomic nervous system to alter the QT interval in response to heart rate changes is essential to cardiovascular control. An accurate way to determine the relation between QT intervals and their corresponding RR intervals is described. A computer algorithm measures the RR intervals using digital filtering and cross-correlating the QRS sections of consecutive waveforms. The QT intervals is calculated by choosing a section of, the ECG that includes the T wave and cross-correlating it with all the consecutive T waves. At least 4000 pairs of QT-RR intervals are computed for each subject and a best fit correlation function determines the relations between the QT and RR intervals. This technique enables to establish a precise correlation between RR and QT in order to distinguish between control and SIDS babies.

Non-NASA Center↗
NASA NTRS2000

Fractal correlation properties of R-R interval dynamics and mortality in patients with depressed left ventricular function after an acute myocardial infarction

BACKGROUND: Preliminary data suggest that the analysis of R-R interval variability by fractal analysis methods may provide clinically useful information on patients with heart failure. The purpose of this study was to compare the prognostic power of new fractal and traditional measures of R-R interval variability as predictors of death after acute myocardial infarction. METHODS AND RESULTS: Time and frequency domain heart rate (HR) variability measures, along with short- and long-term correlation (fractal) properties of R-R intervals (exponents alpha(1) and alpha(2)) and power-law scaling of the power spectra (exponent beta), were assessed from 24-hour Holter recordings in 446 survivors of acute myocardial infarction with a depressed left ventricular function (ejection fraction </=35%). During a mean+/-SD follow-up period of 685+/-360 days, 114 patients died (25.6%), with 75 deaths classified as arrhythmic (17.0%) and 28 as nonarrhythmic (6.3%) cardiac deaths. Several traditional and fractal measures of R-R interval variability were significant univariate predictors of all-cause mortality. Reduced short-term scaling exponent alpha(1) was the most powerful R-R interval variability measure as a predictor of all-cause mortality (alpha(1) <0.75, relative risk 3.0, 95% confidence interval 2.5 to 4.2, P<0.001). It remained an independent predictor of death (P<0.001) after adjustment for other postinfarction risk markers, such as age, ejection fraction, NYHA class, and medication. Reduced alpha(1) predicted both arrhythmic death (P<0.001) and nonarrhythmic cardiac death (P<0.001). CONCLUSIONS: Analysis of the fractal characteristics of short-term R-R interval dynamics yields more powerful prognostic information than the traditional measures of HR variability among patients with depressed left ventricular function after an acute myocardial infarction.

Non-NASA Center↗
NASA NTRS1999

Cardiac interbeat interval dynamics from childhood to senescence : comparison of conventional and new measures based on fractals and chaos theory

BACKGROUND: New methods of R-R interval variability based on fractal scaling and nonlinear dynamics ("chaos theory") may give new insights into heart rate dynamics. The aims of this study were to (1) systematically characterize and quantify the effects of aging from early childhood to advanced age on 24-hour heart rate dynamics in healthy subjects; (2) compare age-related changes in conventional time- and frequency-domain measures with changes in newly derived measures based on fractal scaling and complexity (chaos) theory; and (3) further test the hypothesis that there is loss of complexity and altered fractal scaling of heart rate dynamics with advanced age. METHODS AND RESULTS: The relationship between age and cardiac interbeat (R-R) interval dynamics from childhood to senescence was studied in 114 healthy subjects (age range, 1 to 82 years) by measurement of the slope, beta, of the power-law regression line (log power-log frequency) of R-R interval variability (10(-4) to 10(-2) Hz), approximate entropy (ApEn), short-term (alpha(1)) and intermediate-term (alpha(2)) fractal scaling exponents obtained by detrended fluctuation analysis, and traditional time- and frequency-domain measures from 24-hour ECG recordings. Compared with young adults (<40 years old, n=29), children (<15 years old, n=27) showed similar complexity (ApEn) and fractal correlation properties (alpha(1), alpha(2), beta) of R-R interval dynamics despite lower spectral and time-domain measures. Progressive loss of complexity (decreased ApEn, r=-0.69, P<0.001) and alterations of long-term fractal-like heart rate behavior (increased alpha(2), r=0.63, decreased beta, r=-0.60, P<0.001 for both) were observed thereafter from middle age (40 to 60 years, n=29) to old age (>60 years, n=29). CONCLUSIONS: Cardiac interbeat interval dynamics change markedly from childhood to old age in healthy subjects. Children show complexity and fractal correlation properties of R-R interval time series comparable to those of young adults, despite lower overall heart rate variability. Healthy aging is associated with R-R interval dynamics showing higher regularity and altered fractal scaling consistent with a loss of complex variability.

NASA Discipline Cardiopulmonary↗
NASA NTRS1998

Abnormalities of the QT interval in primary disorders of autonomic failure

BACKGROUND: Experimental evidence shows that activation of the autonomic nervous system influences ventricular repolarization and, therefore, the QT interval on the ECG. To test the hypothesis that the QT interval is abnormal in autonomic dysfunction, we examined ECGs in patients with severe primary autonomic failure and in patients with congenital dopamine beta-hydroxylase (DbetaH) deficiency who are unable to synthesize norepinephrine and epinephrine. SUBJECTS AND METHODS: Maximal QT and rate-corrected QT (QTc) intervals and adjusted QTc dispersion [(maximal QTc - minimum QTc on 12 lead ECG)/square root of the number of leads measured] were determined in blinded fashion from ECGs of 67 patients with primary autonomic failure (36 patients with multiple system atrophy [MSA], and 31 patients with pure autonomic failure [PAF]) and 17 age- and sex-matched healthy controls. ECGs of 5 patients with congenital DbetaH deficiency and 6 age- and sex-matched controls were also analyzed. RESULTS: Patients with MSA and PAF had significantly prolonged maximum QTc intervals (492+/-58 ms(1/2) and 502+/-61 ms(1/2) [mean +/- SD]), respectively, compared with controls (450+/-18 ms(1/2), P < .05 and P < .01, respectively). A similar but not significant trend was observed for QT. QTc dispersion was also increased in MSA (40+/-20 ms(1/2), P < .05 vs controls) and PAF patients (32+/-19 ms(1/2), NS) compared with controls (21+/-5 ms(1/2)). In contrast, patients with congenital DbetaH deficiency did not have significantly different RR, QT, QTc intervals, or QTc dispersion when compared with controls. CONCLUSIONS: Patients with primary autonomic failure who have combined parasympathetic and sympathetic failure have abnormally prolonged QT interval and increased QT dispersion. However, QT interval in patients with congenital DbetaH deficiency was not significantly different from controls. It is possible, therefore, that QT abnormalities in patients with primary autonomic failure are not solely caused by lesions of the sympathetic nervous system, and that the parasympathetic nervous system is likely to have a modulatory role in ventricular repolarization.

Non-NASA Center↗
NASA NTRS1996

Fundamental relations between short-term RR interval and arterial pressure oscillations in humans

BACKGROUND: One of the principal explanations for respiratory sinus arrhythmia is that it reflects arterial baroreflex buffering of respiration-induced arterial pressure fluctuations. If this explanation is correct, then elimination of RR interval fluctuations should increase respiratory arterial pressure fluctuations. METHODS AND RESULTS: We measured RR interval and arterial pressure fluctuations during normal sinus rhythm and fixed-rate atrial pacing at 17.2+/-1.8 (SEM) beats per minute greater than the sinus rate in 16 healthy men and 4 healthy women, 20 to 34 years of age. Measurements were made during controlled-frequency breathing (15 breaths per minute or 0.25 Hz) with subjects in the supine and 40 degree head-up tilt positions. We characterized RR interval and arterial pressure variabilities in low-frequency (0.05 to 0.15 Hz) and respiratory-frequency (0.20 to 0.30 Hz) ranges with fast Fourier transform power spectra and used cross-spectral analysis to determine the phase relation between the two signals. As expected, cardiac pacing eliminated beat-to-beat RR interval variability. Against expectations, however, cardiac pacing in the supine position significantly reduced arterial pressure oscillations in the respiratory frequency (systolic, 6.8+/-1.8 to 2.9 +/-0.6 mm Hg2/Hz, P=.017). In contrast, cardiac pacing in the 40 degree tilt position increased arterial pressure variability (systolic, 8.0+/-1.8 to 10.8 +/-2.6, P=.027). Cross-spectral analysis showed that 40 degree tilt shifted the phase relation between systolic pressure and RR interval at the respiratory frequency from positive to negative (9 +/-7 degrees versus -17+/-11 degrees, P=.04); that is, in the supine position, RR interval changes appeared to lead arterial pressure changes, and in the upright position, RR interval changes appeared to follow arterial pressure changes. CONCLUSIONS: These results demonstrate that respiratory sinus arrhythmia can actually contribute to respiratory arterial pressure fluctuations. Therefore, respiratory sinus arrhythmia does not represent simple baroreflex buffering of arterial pressure.

NASA Discipline Cardiopulmonary↗
NASA NTRS1993

Important influence of respiration on human R-R interval power spectra is largely ignored

Frequency-domain analyses of R-R intervals are used widely to estimate levels of autonomic neural traffic to the human heart. Because respiration modulates autonomic activity, we determined for nine healthy subjects the influence of breathing frequency and tidal volume on R-R interval power spectra (fast-Fourier transform method). We also surveyed published literature to determine current practices in this burgeoning field of scientific inquiry. Supine subjects breathed at rates of 6, 7.5, 10, 15, 17.1, 20, and 24 breaths/min and with nominal tidal volumes of 1,000 and 1,500 ml. R-R interval power at respiratory and low (0.06-0.14 Hz) frequencies declined significantly as breathing frequency increased. R-R interval power at respiratory frequencies was significantly greater at a tidal volume of 1,500 than 1,000 ml. Neither breathing frequency nor tidal volume influenced average R-R intervals significantly. Our review of studies reporting human R-R interval power spectra showed that 51% of the studies controlled respiratory rate, 11% controlled tidal volume, and 11% controlled both respiratory rate and tidal volume. The major implications of our analyses are that breathing parameters strongly influence low-frequency as well as respiratory frequency R-R interval power spectra and that this influence is largely ignored in published research.

NASA Discipline Cardiopulmonary↗
DOE OSTI2022

Wabash CarbonSAFE. Geologic Analysis of the Potosi Dolomite Reservoir Interval and Potential Confining Units, at the Wabash #1 Well, Vigo Co., Indiana, Subtask 7.2, Technical Report

The Wabash CarbonSAFE project drilled the Wabash #1 stratigraphic test well (ID# 168045) at the Wabash Valley Resources (WVR) IGCC facility in Vigo County, Indiana, to evaluate the feasibility of commercial-scale CO 2 storage near the site. The Mt. Simon Sandstone was the initial target for storage evaluation and was found to have generally poor reservoir qualities in the Wabash #1 well. Focus was then placed on the secondary target, the Potosi Dolomite. For storage and containment evaluation an extensive suite of geophysical logs was collected throughout the borehole, and whole cores were obtained from the Mt. Simon Sandstone, Eau Claire Formation, and Maquoketa Group. The Maquoketa Group is identified as a regionally extensive secondary sealing interval above the Mt. Simon Sandstone and as a primary seal for the Potosi Dolomite storage complex. Due to concerns over potential lost circulation zones in the Potosi Dolomite during drilling, no core samples or Formation Micro Imager (FMI) logs were acquired in the Potosi Dolomite. However, cased hole well testing was performed over a 20 feet (6 m) interval within the Potosi Dolomite during which a fluid swab sample was obtained. Approximately 35 miles of 2D seismic information was acquired in the project area near Wabash #1 to aid in reservoir and caprock characterization. The highly porous and permeable zones in the Potosi Dolomite and lower Oneota Dolomite observed within the Wabash #1 well are, for the purpose of this report, considered the same reservoir package and referred to as the Potosi reservoir interval. This report presents a regional depositional context for the Potosi reservoir interval, the overlying confining zones in the Knox Group, and Maquoketa Group primary seal. This context is developed through detailed information obtained from the Wabash #1 well such as lithologic data collected from cuttings and core, geophysical logging, geomechanical analysis of core samples, and well testing and fluid sampling within the Potosi reservoir interval. Information discussed in this report was compiled as a basis for the development of static and dynamic geologic models of the Potosi Dolomite storage complex. The static and dynamic modeling of CO 2 injection in the Potosi reservoir interval are discussed in a separate report (Dessenberger et al., 2022) under the Wabash CarbonSAFE project.

29 ENERGY PLANNING, POLICY, AND ECONOMY↗
NASA NTRS1971

A contracting-interval program for the Danilewski method

The concept of contracting-interval programs is applied to finding the eigenvalues of a matrix. The development is a three-step process in which (1) a program is developed for the reduction of a matrix to Hessenberg form, (2) a program is developed for the reduction of a Hessenberg matrix to colleague form, and (3) the characteristic polynomial with interval coefficients is readily obtained from the interval of colleague matrices. This interval polynomial is then factored into quadratic factors so that the eigenvalues may be obtained. To develop a contracting-interval program for factoring this polynomial with interval coefficients it is necessary to have an iteration method which converges even in the presence of controlled rounding errors. A theorem is stated giving sufficient conditions for the convergence of Newton's method when both the function and its Jacobian cannot be evaluated exactly but errors can be made proportional to the square of the norm of the difference between the previous two iterates. This theorem is applied to prove the convergence of the generalization of the Newton-Bairstow method that is used to obtain quadratic factors of the characteristic polynomial.

Harris, J. D.↗