Relative Condition Number Research Articles

Fast evaluation and root finding for polynomials with floating-point coefficients

Evaluating or finding the roots of a polynomial f(z)=f0+⋯+fdzd with floating-point number coefficients is a ubiquitous problem. By using a piecewise approximation of f obtained with a careful use of the Newton polygon of f, we improve state-of-the-art upper bounds on the number of operations to evaluate and find the roots of a polynomial. In particular, if the coefficients of f are given with m significant bits, we provide for the first time an algorithm that finds all the roots of f with a relative condition number lower than 2m, using a number of bit operations quasi-linear in the bit-size of the floating-point representation of f. Notably, our new approach handles efficiently polynomials with coefficients ranging from 2−d to 2d, both in theory and in practice.

Ultrasparse Ultrasparsifiers and Faster Laplacian System Solvers

In this paper we provide an O ( m loglog O (1) n log (1/ϵ))-expected time algorithm for solving Laplacian systems on n -node m -edge graphs, improving upon the previous best expected runtime of \(O(m \sqrt {\log n} \mathrm{log log}^{O(1)} n \log (1/\epsilon)) \) achieved by (Cohen, Kyng, Miller, Pachocki, Peng, Rao, Xu 2014). To obtain this result we provide efficient constructions of low spectral stretch graph approximations with improved stretch and sparsity bounds. As motivation for this work, we show that for every set of vectors in \(\mathbb {R}^d \) (not just those induced by graphs) and all integer k > 1 there exist an ultra-sparsifier with d − 1 + O ( d / k ) re-weighted vectors of relative condition number at most k 2 . For small k , this improves upon the previous best known multiplicative factor of \(k \cdot \tilde{O}(\log d) \) , which is only known for the graph case. Additionally, in the graph case we employ our low-stretch subgraph construction to obtain n − 1 + O ( n / k )-edge ultrasparsifiers of relative condition number k 1 + o (1) for k = ω (log δ n ) for any δ > 0: this improves upon the previous work for k = o (exp (log 1/2 − δ n )).

On the conditioning of some structured generalized eigenvalue problems

This work continues the analysis of the conditioning of a Hankel structured generalized eigenvalue problem (GEP) started in [1]. The considered generalized eigenvalue problem appears in exponential analysis and sparse interpolation.
 We generalize the proof in [1] and add expressions for the relative condition numbers of two reformulations of the GEP, a reformulation as a Loewner GEP valid for general complex data, and a compression to a Hankel+Toeplitz GEP in the case of real data. Both reformulations are compared to the original Hankel GEP. The analysis is concluded with ample numerical illustrations.

PGRASS-Solver: A Graph Spectral Sparsification-Based Parallel Iterative Solver for Large-Scale Power Grid Analysis

With the increase in the complexity of VLSI chips, power grid analysis has become a challenging task, because linear equations of extremely large size need to be solved. Recent graph sparsification based solvers have shown promising performance for power grid analysis. However, existing graph sparsification algorithms are implemented in serial computing, while factorization and backward/forward substitution of the sparsifier’s Laplacian matrix are hard to parallelize. On the other hand, partition based iterative methods which are inherently parallel lack a direct control of the relative condition number of the preconditioner and consume more memory. In this work, we propose a novel parallel iterative solver called pGRASS-Solver. We first propose a practically-efficient parallel graph sparsification algorithm. Then domain decomposition method (DDM) is utilized to solve the sparsifier’s Laplacian matrix. To further improve the efficiency, a variant of DDM which employs partial Cholesky factorization and Schur complement matrix sparsification is proposed. Thus, we obtain an efficient parallel preconditioner, which not only leads to fast convergence but also enjoys ease of parallelization. Numerous experiments are conducted to illustrate the superior efficiency of the proposed pGRASS-Solver for large-scale power grid analysis, showing an average 6.8X speedup over a recent parallel iterative solver [1]. Moreover, it solves a real-world power grid matrix with 0.36 billion nodes and 8.7 billion nonzeros within 20 minutes on a 16-core machine, which is 10.9X faster than the best result of sequential graph sparsification based solver [2].

Elucidation of critical pH-dependent structural changes in Botulinum Neurotoxin E

Botulinum Neurotoxins (BoNT) are the most potent toxins currently known. However, they also have therapeutic applications for an increasing number of motor related conditions due to their specificity, and low diffusion into the system. Although the start- and end- points for the BoNT mechanism of action are well-studied, a critical step remains poorly understood. It is theorised that BoNTs undergo a pH-triggered conformational shift, activating the neurotoxin by priming it to form a transmembrane (TM) channel. To test this hypothesis, we combined molecular dynamics (MD) simulations and small-angle x-ray scattering (SAXS), revealing a new conformation of serotype E (BoNT/E). This conformation was exclusively observed in simulations below pH 5.5, as determined by principal component analysis (PCA), and its theoretical SAXS profile matched an experimental SAXS profile obtained at pH 4. Additionally, a localised secondary structural change was observed in MD simulations below pH 5.5, in a region previously identified as instrumental for membrane insertion for serotype A (BoNT/A). These changes were found at a critical pH value for BoNTs in vivo, and may be relevant for their therapeutic use.

Rational Approximations in Robust Preconditioning of Multiphysics Problems

Multiphysics or multiscale problems naturally involve coupling at interfaces which are manifolds of lower dimensions. The block-diagonal preconditioning of the related saddle-point systems is among the most efficient approaches for numerically solving large-scale problems in this class. At the operator level, the interface blocks of the preconditioners are fractional Laplacians. At the discrete level, we propose to replace the inverse of the fractional Laplacian with its best uniform rational approximation (BURA). The goal of the paper is to develop a unified framework for analysis of the new class of preconditioned iterative methods. As a final result, we prove that the proposed preconditioners have optimal computational complexity O(N), where N is the number of unknowns (degrees of freedom) of the coupled discrete problem. The main theoretical contribution is the condition number estimates of the BURA-based preconditioners. It is important to note that the obtained estimates are completely analogous for both positive and negative fractional powers. At the end, the analysis of the behavior of the relative condition numbers is aimed at characterizing the practical requirements for minimal BURA orders for the considered Darcy–Stokes and 3D–1D examples of coupled problems.

Clinical and pathophysiological aspects of disseminated intravascular coagulation

Disseminated intravascular coagulation (DIC) is a complex process, the course and outcome of which is determined by triggering mechanisms, the reactivity of the body and a number of related conditions. As a result of these interactions, the clinical and laboratory manifestations of DIC vary greatly, making it difficult to diagnose and correct it. With DIC, there are no standard, especially pathognomonic, clinical symptoms, there are no generally accepted laboratory diagnostic criteria, as well as specific therapy. The development of methods for treating DIC and assessing their effectiveness is hampered by the fact that, firstly, the outcome of DIC depends more on its immediate cause than on treatment; secondly, the usually recommended therapeutic effects (heparin therapy, infusion of antithrombin III concentrates, protein C, etc.) have never been the subject of objective randomized follow-up studies, with the exception, perhaps, of the use of antithrombin III drugs [4]. Undoubtedly, advances in the prevention, diagnosis and treatment of DIC directly depend on the depth of understanding of its pathogenetic mechanisms that unfold at the molecular and cellular levels. Modern ideas about the pathogenesis of DIC syndrome are, in one way or another, a development of the idea of ​​continuous intravascular coagulation [5], which under some conditions from latent physiological becomes massive pathological with distinct clinical and laboratory manifestations [1, 2, 7].

A real triple dqds algorithm for the nonsymmetric tridiagonal eigenvalue problem

The paper discusses the following topics: attractions of the real tridiagonal case, relative eigenvalue condition number for matrices in factored form, dqds, triple dqds, error analysis, new criteria for splitting and deflation, eigenvectors of the balanced form, twisted factorizations and generalized Rayleigh quotient iteration. We present our fast real arithmetic algorithm and compare it with alternative published approaches.

Страны Западного полушария и глобальная продовольственная проблема: состояние и перспективы

The modern civilization has faced a number of problems of a planetary scale, and not only its condition, but, perhaps, its very existence will depend on solving these problems. One of the most urgent ones is the global food problem (GFP) – the ability of humanity to provide itself with sufficient food while fulfilling a number of related conditions within the framework of the concept of sustainable development. The article examines the state and possibility of solving the GFP in the Western Hemisphere, both in the region as a whole and in the context of individual countries. Following the structure of the GFP factors, the availability of natural resources for agricultural purposes, as well as the volume and sectoral structure of production of main agricultural products are analyzed. The author assesses the near-term prospects of the possibility of providing the population of individual countries with sufficient food by using model calculations that depend on two factors: availability of arable land and economic well-being (per capita GDP). Special attention is also paid to the analysis of the environmental aspect of the GFP, in assessing both the current state of the problem in the region and the prospects for its solution. This factor is seen as one of the keys to development of food systems in several Latin American countries. Certain attention is also paid to some "new" GFP factors, such as the problem of food losses (quite relevant for developed countries, especially for the United States), as well as logistics. Disruptions of logistical chains during market shocks (for example, during the COVID-19 pandemic) has posed a major problem for countries with highly developed food markets. In conclusion, the region's overall position in the global food system is seen as quite strong. At the same time, the differences between some countries on a number of indicators that determine the level of food security is very significant. Thus, the region’s prospects in terms of solving the GPP will largely depend on the ability/determination of individual countries to offset the negative impact of a number of GFP factors.

Transient Induction of Fever in the Imiquimod C57BL/6 Mouse Model of Psoriasis-Like Disease Involves IL-1 and IL-6 but Not IL-36

Psoriasis embodies a number of related chronic inflammatory skin conditions. Plaque psoriasis is the most common form and affects 1–3% of the population in many Western countries, whereas generalized pustular psoriasis (GPP) is a much rarer severe form. In addition to skin manifestations, patients with GPP also exhibit several signs of systemic disease, including fever, elevated serum CRP levels, and general malaise (Marrakchi et al., 2011; Onoufriadis et al., 2011). GPP is associated with mutations in the gene encoding the IL-36Ra (Marrakchi et al., 2011; Onoufriadis et al., 2011).

Protective effect of garlic juice on renal function and lipid profile in rats fed with high-fat diet

Background: Hyperlipidemia is one of the most challenging clinical disorders and is known to be a causative factor in a number of related conditions. Garlic has traditionally been used to lower serum lipids in hyperlipidemia patients. The present study evaluates the renoprotective role of garlic against induced changes in rat kidneys as a result of a high-fat diet (HFD). Materials and Methods: Twenty adult male Wistar rats were arranged into 4 groups: Group 1 (control) were fed a normal rat diet; Group 2 were fed an HFD (butter in a dose of 20g/100g food); Group 3 were fed fresh garlic juice (GJ) in their diet (6g/100g food); and Group 4 were fed with butter and GJ in their diet (HFD + GJ). The experimental period was 8 weeks. Serum lipid profiles and renal function tests were carried out and evaluated. Results: The HFD significantly increased body weight, total serum cholesterol, triglycerides, low-density lipoprotein cholesterol (LDL-C), uric acid, creatinine, and blood urea nitrogen (BUN), and decreased total protein and albumin, as compared to the control. In the HFD + GJ group, normal body weight was restored; serum levels of total cholesterol, creatinine, BUN, and albumin were similar to the control, and serum levels of triglyceride, LDL-C, uric acid, and total protein were partially restored to the levels of the control. Conclusions: The incorporation of GJ into an HFD resulted in improved lipid profile and kidney function. Hence, the consumption of GJ may be a useful supplement for renal protection in hyperlipidemic patients.

The condition number of a function relative to a set

The condition number of a differentiable convex function, namely the ratio of its smoothness to strong convexity constants, is closely tied to fundamental properties of the function. In particular, the condition number of a quadratic convex function is the square of the aspect ratio of a canonical ellipsoid associated to the function. Furthermore, the condition number of a function bounds the linear rate of convergence of the gradient descent algorithm for unconstrained convex minimization. We propose a condition number of a differentiable convex function relative to a reference convex set and distance function pair. This relative condition number is defined as the ratio of relative smoothness to relative strong convexity constants. We show that the relative condition number extends the main properties of the traditional condition number both in terms of its geometric insight and in terms of its role in characterizing the linear convergence of first-order methods for constrained convex minimization. When the reference set X is a convex cone or a polyhedron and the function f is of the form $$f = g\circ A$$ , we provide characterizations of and bounds on the condition number of f relative to X in terms of the usual condition number of g and a suitable condition number of the pair (A, X).

GRASS: Graph Spectral Sparsification Leveraging Scalable Spectral Perturbation Analysis

Spectral graph sparsification aims to find ultra-sparse subgraphs whose Laplacian matrix can well approximate the original Laplacian eigenvalues and eigenvectors. In recent years, spectral sparsification techniques have been extensively studied for accelerating various numerical and graph-related applications. Prior nearly-linear-time spectral sparsification methods first extract low-stretch spanning tree from the original graph to form the backbone of the sparsifier, and then recover small portions of spectrally-critical off-tree edges to the spanning tree to significantly improve the approximation quality. However, it is not clear how many off-tree edges should be recovered for achieving a desired spectral similarity level within the sparsifier. Motivated by recent graph signal processing techniques, this work proposes a similarity-aware spectral graph sparsification framework that leverages efficient spectral off-tree edge embedding and filtering schemes to construct spectral sparsifiers with guaranteed spectral similarity (relative condition number) level. An iterative graph densification scheme is also introduced to facilitate efficient and effective filtering of off-tree edges for highly ill-conditioned problems. The proposed method has been validated using various kinds of graphs obtained from public domain sparse matrix collections relevant to very large-scale integration computer-aided design, finite element analysis, as well as social and data networks frequently studied in many machine learning and data mining applications. For instance, a sparse SDD matrix with 40 million unknowns and 180 million nonzeros can be solved (1E-3 accuracy level) within 2 min using a single CPU core and about 6-GB memory.

Structured generalized eigenvalue condition numbers for parameterized quasiseparable matrices

In this paper, when A and B are {1;1}-quasiseparable matrices, we consider the structured generalized relative eigenvalue condition numbers of the pair $$(A, \, B)$$ with respect to relative perturbations of the parameters defining A and B in the quasiseparable and the Givens-vector representations of these matrices. A general expression is derived for the condition number of the generalized eigenvalue problems of the pair $$(A,\, B)$$ , where A and B are any differentiable function of a vector of parameters with respect to perturbations of such parameters. Moreover, the explicit expressions of the corresponding structured condition numbers with respect to the quasiseparable and Givens-vector representation via tangents for $$\{1; 1\}$$ -quasiseparable matrices are derived. Our proposed condition numbers can be computed efficiently by utilizing the recursive structure of quasiseparable matrices. We investigate relationships between various condition numbers of structured generalized eigenvalue problem when A and B are {1;1}-quasiseparable matrices. Numerical results show that there are situations in which the unstructured condition number can be much larger than the structured ones.

Survival of the fittest

The case for exercise in preventive health is now comprehensively made: For all nine of the National Health Priority Areas (NHPAs) exercise therapy is shown to lower risk. However exercise can also be used effectively as both secondary prevention and treatment for these conditions, as well as a number of related conditions.

Error Analysis and Model Adaptivity for Flows in Gas Networks

Abstract In the simulation and optimization of natural gas flow in a pipeline network, a hierarchy of models is used that employs different formulations of the Euler equations. While the optimization is performed on piecewise linear models, the flow simulation is based on the one to three dimensional Euler equations including the temperature distributions. To decide which model class in the hierarchy is adequate to achieve a desired accuracy, this paper presents an error and perturbation analysis for a two level model hierarchy including the isothermal Euler equations in semilinear form and the stationary Euler equations in purely algebraic form. The focus of the work is on the effect of data uncertainty, discretization, and rounding errors in the numerical simulation of these models and their interaction. Two simple discretization schemes for the semilinear model are compared with respect to their conditioning and temporal stepsizes are determined for which a well-conditioned problem is obtained. The results are based on new componentwise relative condition numbers for the solution of nonlinear systems of equations. More- over, the model error between the semilinear and the algebraic model is computed, the maximum pipeline length is determined for which the algebraic model can be used safely, and a condition is derived for which the isothermal model is adequate.

Complexity of Sparse Polynomial Solving: Homotopy on Toric Varieties and the Condition Metric

This paper investigates the cost of solving systems of sparse polynomial equations by homotopy continuation. First, a space of systems of n-variate polynomial equations is specified through n monomial bases. The natural locus for the roots of those systems is known to be a certain toric variety. This variety is a compactification of $$(\mathbb {C}\setminus \{0\})^n$$ , dependent on the monomial bases. A toric Newton operator is defined on that toric variety. Smale’s alpha theory is generalized to provide criteria of quadratic convergence. Two condition numbers are defined, and a higher derivative estimate is obtained in this setting. The Newton operator and related condition numbers turn out to be invariant through a group action related to the momentum map. A homotopy algorithm is given and is proved to terminate after a number of Newton steps which is linear on the condition length of the lifted homotopy path. This generalizes a result from Shub (Found Comput Math 9(2):171–178, 2009. https://doi.org/10.1007/s10208-007-9017-6 ).

On structured componentwise condition numbers for Hamiltonian eigenvalue problems

In this paper, we introduce the structured componentwise perturbation analysis to Hamiltonian eigenvalue problems. The explicit expressions for the relative structured componentwise condition number for Hamiltonian eigenvalue problems are derived. Also, we will compare the relative structured componentwise condition number with the relative unstructured componentwise condition number and the relative structured normwise condition number, respectively. We show the superiority of the relative structured componentwise condition number through numerical examples.

Conditioning of the matrix-matrix exponentiation

If A has no eigenvalues on the closed negative real axis, and B is arbitrary square complex, the matrix-matrix exponentiation is defined as A B := e log(A)B . It arises, for instance, in Von Newmann’s quantum-mechanical entropy, which in turn finds applications in other areas of science and engineering. In this paper, we revisit this function and derive new related results. Particular emphasis is devoted to its Frechet derivative and conditioning. We propose a new definition of bivariate matrix function and derive some general results on their Frechet derivatives, which hold, not only to the matrix-matrix exponentiation but also to other known functions, such as means of two matrices, second order Frechet derivatives and some iteration functions arising in matrix iterative methods. The numerical computation of the Frechet derivative is discussed and an algorithm for computing the relative condition number of A B is proposed. Some numerical experiments are included.

Incomplete factorization by local exact factorization (ILUE)

This study proposes a new preconditioning strategy for symmetric positive (semi-)definite SP(S)D matrices referred to as incomplete factorization by local exact factorization (ILUE). The investigated technique is based on exact LU decomposition of small-sized local matrices associated with a splitting of the domain into overlapping or non-overlapping subdomains. The ILUE preconditioner is defined and its relative condition number estimated. Numerical tests on linear systems arising from the finite element (FE) discretization of a second order elliptic boundary value problem in mixed form demonstrate the advantage of the new algorithm, even for problems with highly oscillatory permeability coefficients, against the classical ILU(p) and ILUT(τ) incomplete factorization preconditioners.