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2018 Contributo in Atti di convegno metadata only access

3D Wet Refractivity Monitoring Using Gnss Tomography Technique Constrained with Airs Data

Pedro Benevides ; Joao Catalao ; Giovanni Nico ; Pedro Miranda

A Global Navigational Satellite System (GNSS) tomography experiment has been performed for 1 week, introducing Atmospheric Infrared Sounder (AIRS) remote sensing data to initiate and update a 3D wet refractivity hourly solution series of the troposphere. Some qualitative and quantitate studies have been performed, taking advantage of a local radiosonde campaign with a 4-hour sampling data. 3D wet refractivity maps with an accuracy close to 2 g/m 3 are obtained.

GNSS atmosphere
2018 Contributo in Atti di convegno metadata only access

Assimilation of Insar Propagation Delay Maps in High-Resolution Numerical Weather Model: Imaging of Water Vapor Structures in Atmosphere

Pedro Mateus ; Giovanni Nico ; João Catalão

In this work we present a methodology to estimate the 3D distribution of water vapor in atmosphere based on the use of SAR interferometry (InSAR) and Sentinel-l data. Maps of propagation delay in atmosphere are assimilated in a high resolution Numerical Weather Model to enhance the forecast of atmosphere parameters. These are used to compute the atmosphere refractivity. Furthermore, 3D maps of hydrometers in atmosphere are derived after the assimilation of InSAR data. Both refractivity and hydrometeors maps are used to map 3D Water vapor patterns in atmosphere. Examples of InSAR signatures of water vapor in atmosphere are shown. We show how the 3D maps liquid refractivity and hydrometeors can be a useful tool to map moisture in atmosphere in case of convective phenomena in atmosphere.

extreme weather event NWP sar
2018 Articolo in rivista metadata only access

Measurement of Pier Deformation Patterns by Ground-Based SAR Interferometry: Application to a Bollard Pull Trial

Nico Giovanni ; Cifarelli Giuseppe ; Miccoli Gianluca ; Soccodato Filippo ; Feng Weike ; Sato Motoyuki ; Miliziano Salvatore ; Marini Maurizio

In this paper, we describe a new methodology for the nondestructive measurement of absolute displacements of a pier during a bollard pull trial by ground-based synthetic aperture radar (GBSAR) interferometry. This technique measures displacement patterns with a submillimeter precision in any weather conditions, operating at a distance up to 4 km from the target area. Bollard pull trials are performed to study the deformation response of a pier when a static pull is applied by a tug to a bollard on the pier edge. The precise measurement of the pulling force and the corresponding displacement pattern of the pier around the bollard is a useful piece of information for the back-analysis studies during the assessment phases of recently built piers. An experiment is carried out to measure pier's displacements at 12 co-located corner reflectors (CRs) and surveying prisms, by SAR interferometry and topographic techniques during a bollard pull trial. The GBSAR results have been validated at the CR locations using the displacement measurements provided by topographic survey. The pulling force applied to the bollard is measured by a load cell specifically customized to precisely measure the pulling force during the trial. Results demonstrate that GBSAR systems can provide a useful tool for the assessment of harbor infrastructures, such as piers, measuring absolute displacements with near-real time capabilities.

Bollard pull trial ground-based synthetic aperture radar (GBSAR) SAR interferometry
2018 Articolo in rivista metadata only access

NOVEL METEOR SIMULATION AND OBSERVATION TECHNIQUES THAT EMERGED FROM BIG-SKY-EARTH COST ACTION

Butka P ; Gritsevich M ; Vinkovic D ; Cellino A ; Bertaina M ; Monkola S ; MorenoIbanez M ; Nico G ; Nina A ; Sreckovic V ; Mitrovic S T

The cooperation of scientists in Big-Sky-Earth COST Action creates an emergent group of researchers with relation to meteor science. Selected cases of development of novel approaches and techniques for meteor simulation and observation are presented.

meteors NWP
2018 Articolo in rivista metadata only access

Assimilating InSAR Maps of Water Vapor to Improve Heavy Rainfall Forecasts: A Case Study With Two Successive Storms

Mateus Pedro ; Miranda Pedro M A ; Nico Giovanni ; Catalao Joao ; Pinto Paulo ; Tome Ricardo

Very high resolution precipitable water vapor maps obtained by the Sentinel-1 A synthetic aperture radar (SAR), using the SAR interferometry (InSAR) technique, are here shown to have a positive impact on the performance of severe weather forecasts. A case study of deep convection which affected the city of Adra, Spain, on 6-7 September 2015, is successfully forecasted by the Weather Research and Forecasting model initialized with InSAR data assimilated by the three-dimensional variational technique, with improved space and time distributions of precipitation, as observed by the local weather radar and rain gauge. This case study is exceptional because it consisted of two severe events 12hr apart, with a timing that allows for the assimilation of both the ascending and descending satellite images, each for the initialization of each event. The same methodology applied to the network of Global Navigation Satellite System observations in Iberia, at the same times, failed to reproduce observed precipitation, although it also improved, in a more modest way, the forecast skill. The impact of precipitable water vapor data is shown to result from a direct increment of convective available potential energy, associated with important adjustments in the low-level wind field, favoring its release in deep convection. It is suggested that InSAR images, complemented by dense Global Navigation Satellite System data, may provide a new source of water vapor data for weather forecasting, since their sampling frequency could reach the subdaily scale by merging different SAR platforms, or when future geosynchronous radar missions become operational.

data assimilation SAR interferometry severe weather events atmospheric moisture Weather Research and Forecasting (WRF) precipitation
2018 Abstract in Atti di convegno metadata only access

The Big Data Era in Sky and Earth Observation Cost Action (BIG-SKY-EARTH)

Giovanni Nico ; Dejan Vinkovic ; Marco Quartulli ; Amaia Gil ; Pedro Benevides ; Vasco Conde ; Joao Catalao ; Anna Kontu ; Maria Gritsevich

Big Data Era in Sky and Earth Observation (BIG-SKY-EARTH, http://www.bigskyearth.eu) is COST Action that aims at setting the ground for a long-term networking between astronomy and remote sensing research communities in the area of Big Data utilization. The purpose of BIG-SKY-EARTH is to emphasize similarities between these disciplines and boost the communication within and between the emerging field of astroinformatics and its older Earth Observation counterpart geoinformatics, in close collaboration with computer scientists. The Action is now entering its final year and the results are visible on several scales. There are many examples of concrete "industrial cross-pollination" stories where BIG-SKY-EARTH facilitated exchange of methods and knowledge between network participants. For example, remote sensing and astronomy big data repositories for meteorological nowcasting, thermosolar energy production forecasting, astronomy big data analytics libraries for wind farm predictive maintenance visualization, astronomy and remote sensing C-based stack for scalable numerical analysis used in advanced manufacturing analytics, GPU analytics for remote sensing and industrial analytics, or developing astronomy platform on the top of commercial remote sensing airship to enable transfer the same technology to a high-resolution remote sensing platform. Some of those collaborations expanded into research papers or even project proposals for H2020 based on partnerships between academia and industry, including developing new types of astronomy and remote sensing research based on innovative airship technologies. The Action has also organized three training schools so far: "Big Data Processing" (Oberpfaffenhofen, Germany), "Big Data Visualization" (Preston, UK), "Big Data GPU Analytics" (San Sebastián, Spain). On the level of the entire networking, the Action is also working on the book "Big Data in AstroGeoInformatics" and accompanying code and algorithm repository. Altogether, the established level of activity and interests for further collaboration suggest that this networking will actively continue also after the official end of COST funding. This presentation will also show two examples of research activities that the presenter started thanks to BIG-SKY-EARTH. The first example focuses on the Precipitable Water Vapor (PWV) estimated from Sentinel-1 images using the SAR interferometry technique. Large databases of high resolution Sentinel-1 PWV maps will need to be analyzed before their assimilation in Numerical Weather Models and use for the estimation of geophysical parameters. This research started during an STSM visit at the Finnish Geospatial Research Institute led to the first tools for the analysis of PWV time series in terms of terrain topography and landcover and the visualization of atmosphere thermodynamic quantities [1]. The second example is on the mapping of the Snow Water Equivalent (SWE) using Sentinel-1 SAR images[2-4]. References: [1]G.Nico,A.Gil,M.Quartulli,P.Mateus,J.Catalao,Merging InSAR and GNSS meteorology:how can we mine InSAR and GNSS databases to extract and visualize information on atmosphere processes?,Proc.of Big Data from Space(BIDS),375-378,2017 [2]V.Conde,G.Nico,P.Mateus,J.Catalao,A.Kontu,M.Gritsevich,Snow Water Equivalent Retrieval Using Synthetic Aperture Radar(SAR) Interferometry,Proc. 8th EARSeL workshop on Land Ice and Snow,2017 [3]V.Conde,G.Nico,J.Catalao,A.Kontu,M.Gritsevich,Wide-area mapping of snow water equivalent by Sentinel-1&2 data,Geophysical Research Abstracts Vol.19, EGU2017-9580-1,2017 [4]V.Conde,G.Nico,P.Mateus,J.Catalão,A.Kontu,M.Gritsevich,On the estimation of temporal changes of snow water equivalent by spaceborne SAR interferometry: a new application for the Sentinel-1 mission,Journal of Hydrology and Hydromechanics,DOI:10.2478/johh-2018-0003,2017

Big data
2018 Abstract in Atti di convegno metadata only access

First example of Sentinel-1 InSAR PWV maps assimilation into a high resolution NWP to improve the forecast of convective system in atmosphere

Giovanni Nico ; Pedro Mateus ; Joao Catalao ; Pedro Miranda

We study the impact of assimilating very high-resolution Precipitable Water Vapor (PWV) maps into a non-hydrostatic Numerical Weather Prediction (NWP) model by the three-dimensional variational (3D-var) technique. PWV maps are obtained by processing the Sentinel-1 Synthetic Aperture Radar (SAR), using the SAR interferometry (InSAR) technique. Changes in the 3D distribution of water vapor, temperature and wind are studied to explain the onset of a deep convection phenomenon. Sentinel-1 images are used to build a time series of PWV maps having a spatial resolution up to 25 m and a time sampling of 6 days. We show that a sub-daily time sampling can be attained if data from different SAR platforms and/or orbits are used, or when future geosynchronous SAR satellites will become operational. The Weather Research Forecasting Data Assimilation (WRFDA) model is used to implement the 3D-Var technique. The finer 3-km domain is centered over the area of interest. A two-way nesting procedure was used. The initial and boundary conditions are set using ECMWF forecasting over Europe are available at very high resolution (0.1°). The InSAR PWV map are assimilated only on the fine domain (3-km). A model spin-up for 6h. For the assimilation the model is initiated at the time of SAR acquisitions and run for 12 hours. The background error covariance matrix B was computed by the National Meteorological Centre (NMC) method, for the finer-resolution domain, where the model perturbations were given by the differences between forecasts (e.g., T + 24 minus T + 12) valid at the same time over a period of one month. We discuss the improvement of the InSAR PWV assimilation in terms of model thermodynamics. Changes in the Convective Available Potential Energy (CAPE), Convective Inhibition (CIN) and Severe Weather Threat Index (SWEAT) are evaluated and used to improve the detection of deep convection onset. A thorough statistical analysis is performed comparing the WRF output with the results obtained by assimilating InSAR and GNSS-based PWV measurements. We show that the assimilation of InSAR data provides an improvement in terms of precipitation and forecast skill score. We analyze also the changes in the 3D distribution of hydrometeors that in the case of storms can significantly contribute to the measured PWV. A case study of deep convection which affected the city of Adra, Spain, on 6-7 September 2015 is presented. The advantage and limitations of assimilating InSAR data into the mesoscale model are discussed. Reference: P. Mateus, J. Catalão, and G. Nico, "Sentinel-1 Interferometric SAR Mapping of Precipitable Water Vapor Over a Country-Spanning Area", IEEE Transactions on Geoscience and Remote Sensing, 55(5), 2993-2999, 2017.

NWP SAR Assimilation
2018 Articolo in rivista metadata only access

Monitoring Strategies of Earth Dams by Ground-Based Radar Interferometry: How to Extract Useful Information for Seismic Risk Assessment

Di Pasquale Andrea ; Nico Giovanni ; Pitullo Alfredo ; Prezioso Giuseppina

The aim of this paper is to describe how ground-based radar interferometry can provide displacement measurements of earth dam surfaces and of vibration frequencies of its main concrete infrastructures. In many cases, dams were built many decades ago and, at that time, were not equipped with in situ sensors embedded in the structure when they were built. Earth dams have scattering properties similar to landslides for which the Ground-Based Synthetic Aperture Radar (GBSAR) technique has been so far extensively applied to study ground displacements. In this work, SAR and Real Aperture Radar (RAR) configurations are used for the measurement of earth dam surface displacements and vibration frequencies of concrete structures, respectively. A methodology for the acquisition of SAR data and the rendering of results is described. The geometrical correction factor, needed to transform the Line-of-Sight (LoS) displacement measurements of GBSAR into an estimate of the horizontal displacement vector of the dam surface, is derived. Furthermore, a methodology for the acquisition of RAR data and the representation of displacement temporal profiles and vibration frequency spectra of dam concrete structures is presented. For this study a Ku-band ground-based radar, equipped with horn antennas having different radiation patterns, has been used. Four case studies, using different radar acquisition strategies specifically developed for the monitoring of earth dams, are examined. The results of this work show the information that a Ku-band ground-based radar can provide to structural engineers for a non-destructive seismic assessment of earth dams.

ground-based radar Synthetic Aperture Radar (SAR) Real Aperture Radar (RAR) SAR interferometry earth dam
2018 Contributo in Atti di convegno metadata only access

NON-DESTRUCTIVE MONITORING STRATEGIES OF HISTORICAL CONSTRUCTIONS AND TANGIBLE CULTURALE HERITAGE BASED ON GROUND-BASED SAR INTERFEROMETRY

Giovanni Nico ; Olimpia Masci ; Evgeny Panidi

Tangible cultural heritage, historical buildings and bridges have an important cultural significance and economic value within the tourism industry and the identity of local communities. The preservation and the assessment of their structural health are important issues which call for multidisciplinary teams and non-invasive monitoring techniques due the uniqueness and historical values of these man-made structures. Numerical models used to study the structural behavior of these historical buildings and bridges under different adverse conditions (eg intense traffic flow, natural hazard events, chemical pollution or simply aging) can benefit from accurate measurements of mechanical properties such as displacements and vibration frequencies, both bringing information about the static and dynamical behavior of such historical constructions. This work presents some results of structural monitoring of man-made structures by Ground-based Synthetic Aperture Radar (GBSAR) interferometry techniques. A ku-band GBSAR interferometer is used to derive displacement maps of the monitored target, with a sub-millimeter precisions. Furthermore, GBSAR interferometry is used to measure vibration frequencies of vertical and horizontal structures, such bell towers, towers, bridges and historical walls. The main advantage of this technique is its capability to operate in any weather and sun-illumination condition, in a truly Non-Destructive Monitoring (NDM) approach, ie without installing any reflector on the observed target.

Synthetic Aperture Radar (SAR) Ground-based SAR (GBSAR) SAR interferometry Non-Destructive Monitoring (NDM) Structural Health Monitoring (SHM) metrology displacement vibration frequency monuments cultural heritage
2018 Contributo in Atti di convegno metadata only access

DIFFERENCES IN THE SOLAR X-RAY FLARE INDUCED TECD INCREASE WITH REGARDS TO GEOGRAPHICAL LOCATION

Aleksandra Nina ; Vladimir M ade ; Giovanni Nico ; Luka Popovi

In this paper we analyze the influence of the geographical position on the increase of the total electron content in the ionospheric D-region during solar X-ray flares. We modeled the total electron content using data related to signals whose propagation paths lie in the mid and both mid and low latitude ionosphere. The obtained results indicate a larger increase of the total electron content in the perturbed equatorial D-region where the solar radiation is more pronounced and causes a larger electron density gradient with altitude.

ionosphere modelling SAR GNSS VHF
2018 Contributo in Atti di convegno metadata only access

Tools for the real time visualization and analysis of Ground-based SAR data: application to the monitoring of landslides

Giovanni Nico ; Uro Kosti ; Andrea Di Pasquale

This paper is focused on visualization of the information extracted by GBSAR data acquired in landslide areas. It describes the way Graphical Processing Units (GPUs) can be used to generate and visualize accurate GBSAR images and displacement maps in near real time. Examples of GBSAR images, as radar coordinates and rendered on Digital Surface Models (DSMs), coherence and displacement maps are shown.

GPU; SAR;
2018 Contributo in Atti di convegno metadata only access

Evaluation of rainfall forecasts combining GNSS precipitable water vapor with ground and remote sensing meteorological variables in a neural network approach

P Benevides ; João Catalão ; Giovanni Nico ; Pedro MA Miranda

In this study, an experiment aimed to integrate Global Navigation Satellite System (GNSS) atmospheric data with meteorological data into a neural network system is performed. Precipitable Water Vapor (PWV) estimates derived from GNSS are combined with surface pressure, surface temperature and relative humidity obtained continuously from ground-based meteorological stations. The work aims to develop a methodology to forecast short-term intense rainfall. Hence, all the data is sampled at one hour interval. A continuous time series of 3 years of GNSS data from one station in Lisbon, Portugal, is processed. Meteorological data from a nearby meteorological station are collected. Remote sensing data of cloud top from SEVIRI is used, providing collocated data also on an hourly basis. A 3 year time series of hourly accumulated precipitation data are also available for evaluation of the neural network results. In previous studies, it was found that time varying PWV is correlated with rainfall, with a strong increase of PWV peaking just before intense rainfall, and with a strong decrease afterwards. However, a significant amount of false positives was found, meaning that the evolution of PWV does not contain enough information to infer future rain. In this work a multilayer fitting network is used to process the GNSS and meteorological data inputs in order to estimate the target outputs, given by the hourly precipitation. It is found that the combination of GNSS data and meteorological variables processed by neural network improves the detection of heavy rainfall events and reduces the number of false positives.

Neural networks classification GNSS
2018 Rapporto di ricerca / Relazione scientifica metadata only access

Diffusion driven X-ray two-dimensional pattern denoising

We propose the use of a mathematical model in order to denoise X-ray twodimensional patterns. The model, which makes use of a generalized diffusion equation whose diffusion constant depends on the image gradients, enables to obtain an efficient reduction of pattern noise as witnessed by the computed peak of signal to noise ratio. The corresponding MATLAB code is made available.

image processing
2018 Contributo in Atti di convegno metadata only access

Learning Gaussian Graphical Models by symmetric parallel regression technique

De Canditiis ; Daniela ; Guardasole Armando

In this contribution we deal with the problem of learning an undi- rected graph which encodes the conditional dependence relationship be- tween variables of a complex system, given a set of observations of this system. This is a very central problem of modern data analysis and it comes out every time we want to investigate a deeper relationship be- tween random variables, which is different from the classical dependence usually measured by the covariance. In particular, in this contribution we deal with the case of Gaussian Graphical Models (GGMs) for which the system of variables has a mul- tivariate gaussian distribution. We revise some of the existing techniques for such a problem and propose a smart implementation of the symmetric parallel regression technique which turns out to be very competitive for learning sparse GGMs under high dimensional data regime.

Gaussian Graphical Models (GGM) Grouped-Lasso penalty
2018 Articolo in rivista metadata only access

The sparse method of simulated quantiles: An application to portfolio optimization

Stolfi P ; Bernardi M ; Petrella L

The sparse multivariate method of simulated quantiles (S-MMSQ) is applied to solve a portfolio optimization problem under value-at-risk constraints where the joint returns follow a multivariate skew-elliptical stable distribution. The S-MMSQ is a simulation-based method that is particularly useful for making parametric inference in some pathological situations where the maximum likelihood estimator is difficult to compute. The method estimates parameters by minimizing the distance between quantile-based statistics evaluated on true and synthetic data, simulated from the postulated model, penalized by adding the smoothly clipped absolute deviation l-penalty in order to achieve sparsity. The S-MMSQ aims to efficiently handle the problem of estimating large-dimensional distributions with intractable likelihood, such as the stable distributions that have been widely applied in finance to model financial returns.

portfolio optimisation; sparse method of simulated qualntiles scad
2018 Contributo in volume (Capitolo o Saggio) metadata only access

Robust Time-Varying Undirected Graphs

Mauro Bernardi ; Paola Stolfi

Undirected graphs are useful tools for the analysis of sparse and high-dimensional data sets. In this setting the sparsity helps in reducing the complexity of the model. However, sparse graphs are usually estimated under the Gaussian paradigm thereby leading to estimates that are very sensitive to the presence of outlying observations. In this paper we deal with sparse time-varying undirected graphs, namely sparse graphs whose structure evolves over time. Our contribution is to provide a robustification of these models, in particular we propose a robust estimator which minimises the ?-divergence. We provide an algorithm for the parameter estimation and we investigate the rate of convergence of the proposed estimator.

Divergence Kernel Methods Robust Methods Dynamic models
2018 Contributo in volume (Capitolo o Saggio) metadata only access

Approximate EM algorithm for sparse estimation of multivariate location-scale mixture of normal

Mauro Bernardi ; Paola Stolfi

Parameter estimation of distributions with intractable density, such as the Elliptical Stable, often involves high-dimensional integrals requiring numerical integration or approximation. This paper introduces a novel Expectation-Maximisation algorithm for fitting such models that exploits the fast Fourier integration for computing the expectation step. As a further contribution we show that by slightly modifying the objective function, the proposed algorithm also handle sparse estimation of non-Gaussian models. The method is subsequently applied to the problem of selecting the asset within a sparse non-Gaussian portfolio optimisation framework.

Sparse estimation Multivariate heavy--tailed distributions Expectation-Maximisation
2018 Presentazione / Comunicazione non pubblicata (convegno, evento, webinar...) metadata only access

Distribution and trend estimation of MIPAS ESA V7 carbon tetrachloride data and preliminary results of variability of new species derived with MIPAS ESA V8 processor

MIPAS on ENVISAT performed almost continuous measurements of atmospheric composition for approximately 10 years, from June 2002 to April 2012. ESA processor, based on the algorithm ORM (Optimized Retrieval Model), originally designed for the Near Real Time analysis, is currently used for the reanalysis of the full MIPAS mission. Version 7 of the full mission data was released in 2016, but further improvements have been recently performed in ORM V8 to be used in next full mission reanalysis. For these latest releases (V7 and V8) L1 data corrected for reducing the instrumental drift are used.TheinstrumentaldriftisduetoMIPASphotometricdetectorsnonlinearitiesthatchangewithtimeduetothe ageing of the instrument. Numerous species are retrieved from MIPAS measurements. Among them, CCl4 has been recently studied. This species has received increasing interest due to the so called "mystery of CCl4", since it was found that its atmospheric concentration at the surface declines with a rate significantly smaller than its lifetime-limited rate. Indeed there is a discrepancy between the atmospheric observations and the estimated distribution based on the reported production and consumption. MIPAS products generated with Version 7 of the L2 ESA algorithm were used to estimate CCl4 distributions, its trend, and atmospheric lifetime in the upper troposphere / lower stratosphere (UTLS) region. The trends derived by these observations between 2002 and 2012 as a function of both latitude and altitude confirm the decline of atmospheric mixing ratios, in agreement with ground based observations. Stratospheric trend derived from the MIPAS data are non-uniform, with some positive trends even being found in the middle stratosphere, mainly at high altitudes in the Southern Hemisphere. The variability in stratospheric trends reflects the impact of variability in stratospheric transport on trace gases and their temporal evolution.In addition to CCl4, some preliminary results obtained with the latest version of the processor (V8), that performs the analysis of a larger number of species and takes into account horizontal inhomogeneities, will be shown.

MIPAS trend carbontetrachloride Envisat
2018 Articolo in rivista metadata only access

Entropic lattice Boltzmann model for charged leaky dielectric multiphase fluids in electrified jets

We present a lattice Boltzmann model for charged leaky dielectric multiphase fluids in the context of electrified jet simulations, which are of interest for a number of production technologies including electrospinning. The role of nonlinear rheology on the dynamics of electrified jets is considered by exploiting the Carreau model for pseudoplastic fluids. We report exploratory simulations of charged droplets at rest and under a constant electric field, and we provide results for charged jet formation under electrospinning conditions.

lattice Boltzmann model Electrospinning pseudoplastic fluids
2018 Articolo in rivista metadata only access

The QR Steps with Perfect Shifts

Nicola Mastronardi ; Paul Van Dooren

In this paper we revisit the problem of performing a QR-step on an unreduced Hessenberg matrix H when we know an "exact" eigenvalue ?0 of H. Under exact arithmetic, this eigenvalue will appear on diagonal of the transformed Hessenberg matrix H~ and will be decoupled from the remaining part of the Hessenberg matrix, thus resulting in a deflation. But it is well known that in finite precision arithmetic the so-called perfect shift can get blurred and that the eigenvalue ?0 can then not be deflated and/or is perturbed significantly. In this paper, we develop a new strategy for computing such a QR step so that the deflation is almost always successful. We also show how to extend this technique to double QR-steps with complex conjugate shifts.

Hessenberg form QR step eigenvalue problem perfect shift