Solving secular and polynomial equations: A multiprecision algorithm

We present an algorithm for the solution of polynomial equations and secular equations of the form $S(x)=0$ for $S(x)=sum_{i=1}^nrac{a_i}{x-b_i}-1=0$, which provides guaranteed approximation of the roots with any desired number of digits. It relies on the combination of two different strategies for dealing with the precision of the floating point computation: the strategy used in the package MPSolve of D.~Bini and G.~Fiorentino [Numer.~Algo.~23, 2000] and the strategy used in the package Eigensolve of S.~Fortune [J. Symb. Comput.~33, 2002]. The algorithm is based on the Ehrlich-Aberth (EA) iteration, and on several results introduced in the paper. In particular, we extend the concept and the properties of root-neighborhoods from polynomials to secular functions, provide perturbation results of the roots, obtain an effective stop condition for the EA iteration and guaranteed a posteriori error ounds. We provide an implementation, released in the package MPSolve 3.0, based on the GMP library. From the many numerical experiments it turns out that our code is generally much faster than MPSolve 2.0 and of the package Eigensolve. For certain polynomials, like the Mandelbrot or the partition polynomials the acceleration is dramatic. The algorithm exploits the parallel architecture of the computing platform

On a class of matrix pencils and ℓ-ifications equivalent to a given matrix polynomial

A new class of linearizations and $ell$-ifications for m×m matrix polynomials $P(x)$ of degree $n$ is proposed. The $ell$-ifications in this class have the form $A(x)=D(x)+(eotimes I_m)W(x)$ where $D$ is a block diagonal matrix polynomial with blocks $B_i(x)$ of size $m$, $W$ is an $m imes qm$ matrix polynomial and $e=(1,…,1)^t in C^q$, for a suitable integer $q$. The blocks $B_i(x)$ can be chosen a priori, subjected to some restrictions. Under additional assumptions on the blocks $B_i(x)$ the matrix polynomial $A(x)$ is a strong $ell$-ification, i.e., the reversed polynomial of $A(x)$ defined by A^# (x):=x^{deg ⁡A(x)}A(x^{-1}) is an $ell$-ification of P^# (x). The eigenvectors of the matrix polynomials $P(x)$ and $A(x)$ are related by means of explicit formulas. Some practical examples of $ell$-ifications are provided. A strategy for choosing $B_i(x)$ in such a way that $A(x)$ is a well conditioned linearization of $P(x)$ is proposed. Some numerical experiments that validate the theoretical results are reported

On the decay of the off-diagonal singular values in cyclic reduction

It was recently observed in [10] that the singular values of the off-diagonal blocks of the matrix sequences generated by the Cyclic Reduction algorithm decay exponentially. This property was used to solve, with a higher efficiency, certain quadratic matrix equations encountered in the analysis of queuing models. In this paper, we provide a theoretical bound to the basis of this exponential decay together with a tool for its estimation based on a rational interpolation problem. Numerical experiments show that the bound is often accurate in practice. Applications to solving n × n block tridiagonal block Toeplitz systems with n × n quasiseparable blocks and certain generalized Sylvester equations in O(n 2 log n) arithmetic operations are shown

A family of modified regularizing circulant preconditioners for image reconstruction problems

Regularizing iterative methods for image restoration problems with 2D band Toeplitz matrices are examined. They often require the use of suitable preconditioners. Circulant preconditioners, which can be applied with a low cost and can be easily adapted to cope with the noise, are widely used. In this paper three of them are taken into consideration and modified in order to obtain a better convergence rate, still retaining regularizing properties. A large numerical experimentation validates the theoretical results

How future surgery will benefit from SARS-COV-2-related measures: a SPIGC survey conveying the perspective of Italian surgeons

COVID-19 negatively affected surgical activity, but the potential benefits resulting from adopted measures remain unclear. The aim of this study was to evaluate the change in surgical activity and potential benefit from COVID-19 measures in perspective of Italian surgeons on behalf of SPIGC. A nationwide online survey on surgical practice before, during, and after COVID-19 pandemic was conducted in March-April 2022 (NCT:05323851). Effects of COVID-19 hospital-related measures on surgical patients' management and personal professional development across surgical specialties were explored. Data on demographics, pre-operative/peri-operative/post-operative management, and professional development were collected. Outcomes were matched with the corresponding volume. Four hundred and seventy-three respondents were included in final analysis across 14 surgical specialties. Since SARS-CoV-2 pandemic, application of telematic consultations (4.1% vs. 21.6%; p < 0.0001) and diagnostic evaluations (16.4% vs. 42.2%; p < 0.0001) increased. Elective surgical activities significantly reduced and surgeons opted more frequently for conservative management with a possible indication for elective (26.3% vs. 35.7%; p < 0.0001) or urgent (20.4% vs. 38.5%; p < 0.0001) surgery. All new COVID-related measures are perceived to be maintained in the future. Surgeons' personal education online increased from 12.6% (pre-COVID) to 86.6% (post-COVID; p < 0.0001). Online educational activities are considered a beneficial effect from COVID pandemic (56.4%). COVID-19 had a great impact on surgical specialties, with significant reduction of operation volume. However, some forced changes turned out to be benefits. Isolation measures pushed the use of telemedicine and telemetric devices for outpatient practice and favored communication for educational purposes and surgeon-patient/family communication. From the Italian surgeons' perspective, COVID-related measures will continue to influence future surgical clinical practice

Infected pancreatic necrosis: outcomes and clinical predictors of mortality. A post hoc analysis of the MANCTRA-1 international study

: The identification of high-risk patients in the early stages of infected pancreatic necrosis (IPN) is critical, because it could help the clinicians to adopt more effective management strategies. We conducted a post hoc analysis of the MANCTRA-1 international study to assess the association between clinical risk factors and mortality among adult patients with IPN. Univariable and multivariable logistic regression models were used to identify prognostic factors of mortality. We identified 247 consecutive patients with IPN hospitalised between January 2019 and December 2020. History of uncontrolled arterial hypertension (p = 0.032; 95% CI 1.135-15.882; aOR 4.245), qSOFA (p = 0.005; 95% CI 1.359-5.879; aOR 2.828), renal failure (p = 0.022; 95% CI 1.138-5.442; aOR 2.489), and haemodynamic failure (p = 0.018; 95% CI 1.184-5.978; aOR 2.661), were identified as independent predictors of mortality in IPN patients. Cholangitis (p = 0.003; 95% CI 1.598-9.930; aOR 3.983), abdominal compartment syndrome (p = 0.032; 95% CI 1.090-6.967; aOR 2.735), and gastrointestinal/intra-abdominal bleeding (p = 0.009; 95% CI 1.286-5.712; aOR 2.710) were independently associated with the risk of mortality. Upfront open surgical necrosectomy was strongly associated with the risk of mortality (p < 0.001; 95% CI 1.912-7.442; aOR 3.772), whereas endoscopic drainage of pancreatic necrosis (p = 0.018; 95% CI 0.138-0.834; aOR 0.339) and enteral nutrition (p = 0.003; 95% CI 0.143-0.716; aOR 0.320) were found as protective factors. Organ failure, acute cholangitis, and upfront open surgical necrosectomy were the most significant predictors of mortality. Our study confirmed that, even in a subgroup of particularly ill patients such as those with IPN, upfront open surgery should be avoided as much as possible. Study protocol registered in ClinicalTrials.Gov (I.D. Number NCT04747990)

Diversity and ethics in trauma and acute care surgery teams: results from an international survey

Background Investigating the context of trauma and acute care surgery, the article aims at understanding the factors that can enhance some ethical aspects, namely the importance of patient consent, the perceptiveness of the ethical role of the trauma leader, and the perceived importance of ethics as an educational subject. Methods The article employs an international questionnaire promoted by the World Society of Emergency Surgery. Results Through the analysis of 402 fully filled questionnaires by surgeons from 72 different countries, the three main ethical topics are investigated through the lens of gender, membership of an academic or non-academic institution, an official trauma team, and a diverse group. In general terms, results highlight greater attention paid by surgeons belonging to academic institutions, official trauma teams, and diverse groups. Conclusions Our results underline that some organizational factors (e.g., the fact that the team belongs to a university context or is more diverse) might lead to the development of a higher sensibility on ethical matters. Embracing cultural diversity forces trauma teams to deal with different mindsets. Organizations should, therefore, consider those elements in defining their organizational procedures. Level of evidence Trauma and acute care teams work under tremendous pressure and complex circumstances, with their members needing to make ethical decisions quickly. The international survey allowed to shed light on how team assembly decisions might represent an opportunity to coordinate team member actions and increase performance

Search for dark matter produced in association with bottom or top quarks in √s = 13 TeV pp collisions with the ATLAS detector

A search for weakly interacting massive particle dark matter produced in association with bottom or top quarks is presented. Final states containing third-generation quarks and miss- ing transverse momentum are considered. The analysis uses 36.1 fb−1 of proton–proton collision data recorded by the ATLAS experiment at √s = 13 TeV in 2015 and 2016. No significant excess of events above the estimated backgrounds is observed. The results are in- terpreted in the framework of simplified models of spin-0 dark-matter mediators. For colour- neutral spin-0 mediators produced in association with top quarks and decaying into a pair of dark-matter particles, mediator masses below 50 GeV are excluded assuming a dark-matter candidate mass of 1 GeV and unitary couplings. For scalar and pseudoscalar mediators produced in association with bottom quarks, the search sets limits on the production cross- section of 300 times the predicted rate for mediators with masses between 10 and 50 GeV and assuming a dark-matter mass of 1 GeV and unitary coupling. Constraints on colour- charged scalar simplified models are also presented. Assuming a dark-matter particle mass of 35 GeV, mediator particles with mass below 1.1 TeV are excluded for couplings yielding a dark-matter relic density consistent with measurements