Deflation Technique Research Articles

The Exact Root Algorithm for Computing the Real Roots of an Nth Degree Polynomial

Problem statement: The need to find an efficient and reliable algorithm for computing the exact real roots of the steady-state polynomial encountered in the investigation of temperature profiles in biological tissues during Microwave heating and other similar cases as found in the literature gave rise to this study. Approach: The algorithm (simply called ERA-Exact Root Algorithm) adopted polynomial deflation technique and uses Newton-Raphson iterative procedure though with a modified termination rule. A general formula was specified for finding the initial approximation so as to overcome the limitation of local convergence which is inherent in Newton’s method. Results: A new algorithm for finding the real roots of an nth degree polynomial at a practically low computational cost was obtained. Conclusion/Recommendations: ERA is simple, flexible, easy to use and has clear benefits and preferences to a number of existing methods.

Cerebrovascular Regulation During Transient Hypotension and Hypertension in Humans

The cerebrovasculature dilates or constricts in response to acute blood pressure changes to stabilize cerebral blood flow across a range of blood pressures. It is unclear, however, whether such dynamic cerebral autoregulation (dCA) is equally effective in responding to falling versus rising blood pressure. In this study we applied a pharmacological approach to evaluate dCA gain to transient hypotension and hypertension and compared this method with 2 established indices of dCA that do not explicitly differentiate between dCA efficacy and falling versus rising blood pressure. Middle cerebral arterial velocity and blood pressure recordings were made in 26 healthy volunteers randomized to 2 protocols. In 10 subjects, dCA gain to transient hypotension induced with intravenous nitroprusside was compared with dCA gain to transient hypertension induced with intravenous phenylephrine. In 16 subjects, dCA gain to transient hypotension induced with intravenous nitroprusside was compared with the rate of regulation and autoregulatory index derived from transient hypotension induced with the thigh cuff deflation technique. dCA gain to transient hypotension induced with intravenous nitroprusside was unrelated to dCA gain to transient hypertension induced with intravenous phenylephrine (r=0.06; P=0.87) and was consistently greater than dCA gain to transient hypertension induced with intravenous phenylephrine (0.57+/-0.16 versus 0.31+/-0.20 cm/s per millimeter of mercury; P<0.01). However, dCA gain to transient hypotension induced with intravenous nitroprusside was inversely related to the rate of regulation (r=-0.52; P=0.037) and autoregulatory index (r=-0.66; P=0.005). These data indicate that, under our laboratory conditions, dCA appears to be inherently nonlinear with disparate efficacy against rising and falling blood pressure, and dCA gain derived from pharmacologically induced transient hypotension correlates with established nonpharmacological indices of dCA.

Reliable root detection with the qd-algorithm: When Bernoulli, Hadamard and Rutishauser cooperate

When using Rutishauser's qd-algorithm for the determination of the roots of a polynomial (originally the poles of a meromorphic function), or for related problems, conditions have been formulated for the interpretation of the computed q- and e-values. For a correct interpretation, the so-called critical indices play a crucial role. They index a column of e-values that tends to zero because of a jump in modulus among the poles. For more than 50 years the qd-algorithm in exact arithmetic was considered to be fully understood. In this presentation we push the detailed theoretical investigation of the qd-algorithm even further and we present a new aspect that seems to have been overlooked. We indicate a new element that makes a column of e-values tend to zero, namely a jump in multiplicity among equidistant poles. This result is obtained by combining the qd-algorithm with a deflation technique, and hence mainly relying on Bernoulli's method and Hadamard's formally orthogonal polynomials. Our results round up the theoretical analysis of the qd-algorithm as formulated in its original form, and are of importance in a variety of practical applications as outlined in the introduction.

A NEW UNDERSTANDING OF THE QR METHOD

The QR method is one of the most common methods for calculating the eigenvalues of a square matrix, however its understanding would require complicated and sophisticated mathematical logics. In this article, we present a simple way to understand QR method only with a minimal mathematical knowledge. A deflation technique is introduced, and its combination with the power iteration leads to extracting all the eigenvectors. The orthogonal iteration is then shown to be compatible with the combination of deflation and power iteration. The connection of QR method to orthogonal iteration is then briefly reviewed. Our presentation is unique and easy to understand among many accounts for the QR method by introducing the orthogonal iteration in terms of deflation and power iteration.

An Implicit Multishift $QR$-Algorithm for Hermitian Plus Low Rank Matrices

Hermitian plus possibly non-Hermitian low rank matrices can be efficiently reduced into Hessenberg form. The resulting Hessenberg matrix can still be written as the sum of a Hermitian plus low rank matrix. In this paper we develop a new implicit multishift $QR$-algorithm for Hessenberg matrices, which are the sum of a Hermitian plus a possibly non-Hermitian low rank correction. The proposed algorithm exploits both the symmetry and low rank structure to obtain a $QR$-step involving only $\mathcal{O}(n)$ floating point operations instead of the standard $\mathcal{O}(n^2)$ operations needed for performing a $QR$-step on a Hessenberg matrix. The algorithm is based on a suitable $\mathcal{O}(n)$ representation of the Hessenberg matrix. The low rank parts present in both the Hermitian and low rank part of the sum are compactly stored by a sequence of Givens transformations and a few vectors. Due to the new representation, we cannot apply classical deflation techniques for Hessenberg matrices. A new, efficient technique is developed to overcome this problem. Some numerical experiments based on matrices arising in applications are performed. The experiments illustrate effectiveness and accuracy of both the $QR$-algorithm and the newly developed deflation technique.

Distal balloon deflation technique: A new method to facilitate entry of balloon catheter, stent, and guiding catheter to distal lesion

We occasionally encountered difficult cases of balloon and stent delivery to the distal lesion because of narrow stent strut, severe calcification, or tortuosity of the proximal section. We describe a new technique to deliver balloon, stent, and guiding catheter to the distal lesion using a distal balloon deflation. This technique offers a potential alternative for cases in which the delivery of balloon and stent is difficult.

Fast numerical solutions of patient‐specific blood flows in 3D arterial systems

The study of hemodynamics in arterial models constructed from patient-specific medical images requires the solution of the incompressible flow equations in geometries characterized by complex branching tubular structures. The main challenge with this kind of geometries is that the convergence rate of the pressure Poisson solver is dominated by the graph depth of the computational grid. This paper presents a deflated preconditioned conjugate gradients (DPCG) algorithm for accelerating the pressure Poisson solver. A subspace deflation technique is used to approximate the lowest eigenvalues along tubular domains. This methodology was tested with an idealized cylindrical model and three patient-specific models of cerebral arteries and aneurysms constructed from medical images. For these cases, the number of iterations decreased by up to a factor of 16, while the total CPU time was reduced by up to 4 times when compared with the standard PCG solver.

Swedish Historical National Accounts, 1800–2000: Principles and Implications of a New Generation

A new set of data and estimates of historical national accounts for Sweden is concluded with the project Structural Change in the Swedish Economy, 1800–1980: Construction and Analysis of National Product Series. This article provides a short overview of earlier efforts, beginning in the 1930s, to construct historical national accounts and an account of the present project. The new estimates resulted in partly new representations of economic growth and change. They are also compared with earlier data. Furthermore, effects of the deflation techniques ( i.e., double deflation) are analyzed by comparing the series with those resulting from single deflation. Finally, structural changes are analyzed using modern time series analysis.

Deflation acceleration of lattice QCD simulations

Close to the chiral limit, many calculations in numerical lattice QCD can potentially be accelerated using low-mode deflation techniques. In this paper it is shown that the recently introduced domain-decomposed deflation subspaces can be propagated along the field trajectories generated by the Hybrid Monte Carlo (HMC) algorithm with a modest effort. The quark forces that drive the simulation may then be computed using a deflation-accelerated solver for the lattice Dirac equation. As a consequence, the computer time required for the simulations is significantly reduced and an improved scaling behaviour of the simulation algorithm with respect to the quark mass is achieved.

An automated method for measuring static pressure–volume curves of the respiratory system and its application in healthy lungs and after lung damage by oleic acid infusion

Elastic pressure/volume (PV) curves of the respiratory system have attracted increasing interest, because they may be helpful to optimize ventilator settings in patients undergoing mechanical ventilation. Clinically applicable methods need to be fast, use routinely available equipment, draw the inspiratory and expiratory PV curve limbs, separate the resistive and viscoelastic properties of the respiratory system from the elastic properties, and provide reproducible measurements. This paper presents a computer-controlled method for rapid measurements of static PV curves using a long inflation–deflation with pauses, and its evaluation in six pigs before and after lung damage caused by oleic acid. The method is fast, i.e. 20.5 ± 1.9 s (mean ± SD) in healthy lungs and 17.7 ± 4.1 s in diseased lungs, this including inspiratory and expiratory pauses of 1.1 s duration. In addition the only equipment used was a clinical ventilator and a PC. For healthy and damaged lungs expiratory PV curve limbs were very reproducible and were at higher volume than the inspiratory limbs, indicating hysteresis. For damaged lungs inspiratory PV limbs were reproducible. For healthy lungs the inspiratory limbs were reproducible but only after the first inflation–deflation. It is possible that during the first inflation alveoli are recruited which are not derecruited on deflation, shifting the inspiratory limb of the PV curve. The paused long inflation–deflation technique provides a quick, automated measurement of static PV curves on both inspiratory and expiratory limbs using routinely available equipment in the intensive care unit.

Perturbation-Based Eigenvector Updates for On-Line Principal Components Analysis and Canonical Correlation Analysis

Principal components analysis is an important and well-studied subject in statistics and signal processing. Several algorithms for solving this problem exist, and could be mostly grouped into one of the following three approaches: adaptation based on Hebbian updates and deflation, optimization of a second order statistical criterion (like reconstruction error or output variance), and fixed point update rules with deflation. In this study, we propose an alternate approach that avoids deflation and gradient-search techniques. The proposed method is an on-line procedure based on recursively updating the eigenvector and eigenvalue matrices with every new sample such that the estimates approximately track their true values as would be calculated analytically from the current sample estimate of the data covariance matrix. The perturbation technique is theoretically shown to be applicable for recursive canonical correlation analysis, as well. The performance of this algorithm is compared with that of a structurally similar matrix perturbation-based method and also with a few other traditional methods like Sanger's rule and APEX.

On deflation and singular symmetric positive semi-definite matrices

For various applications, it is well-known that the deflated ICCG is an efficient method for solving linear systems with invertible coefficient matrix. We propose two equivalent variants of this deflated ICCG which can also solve linear systems with singular coefficient matrix, arising from discretization of the discontinuous Poisson equation with Neumann boundary conditions. It is demonstrated both theoretically and numerically that the resulting methods accelerate the convergence of the iterative process. Moreover, in practice the singular coefficient matrix has often been made invertible by modifying the last element, since this can be advantageous for the solver. However, the drawback is that the condition number becomes worse-conditioned. We show that this problem can completely be remedied by applying the deflation technique with just one deflation vector.

RLS-based adaptive algorithms for generalized eigen-decomposition

The aim of this paper is to develop efficient online adaptive algorithms for the generalized eigen-decomposition problem which arises in a variety of modern signal processing applications. First, we reinterpret the generalized eigen-decomposition problem as an unconstrained minimization problem by constructing a novel cost function. Second, by applying projection approximation method and recursive least-square (RLS) technique to the cost function, a parallel adaptive algorithm for a basis for the r-dimensional (r>0) dominant generalized eigen-subspace and a sequential algorithm based on deflation technique for the first r-dominant generalized eigenvectors are derived. These algorithms can be viewed as counterparts of the extended projection approximation subspace tracking (PAST) and PASTd algorithms, respectively. Furthermore, we modify the parallel algorithm to explicitly estimate the first r-generalized eigenvectors in parallel, not the generalized eigen-subspace. More important, the modified parallel algorithm can be used to extract multiple generalized eigenvectors of two nonstationary sequences, while the proposed sequential algorithm lacks this ability because of slow convergence of minor generalized eigenvectors due to error propagation of the deflation technique. Third, following convergence analysis methods for PAST and PASTd, we prove the asymptotic convergence properties of the proposed algorithms. Finally, computer simulations are performed to investigate the accuracy and the speed advantages of the proposed algorithms.

Lung Function Tests in Neonates and Infants with Chronic Lung Disease: Forced Expiratory Maneuvers

This fourth paper in a review series on the role of lung function testing in infants and young children with acute neonatal disorders and chronic lung disease of infancy (CLDI) addresses measurements of forced expiration using rapid thoraco-abdominal compression (RTC) techniques and the forced deflation technique. Following orientation of the reader to the subject area, we focus our comments on the areas of inquiry proposed in the introductory paper to this series. The quality of the published literature is reviewed critically, and recommendations are provided to guide future investigation in this field. All studies on infants and young children with CLDI using forced expiratory or deflation maneuvers demonstrated that forced flows at low lung volume remain persistently low through the first 3 years of life. Measurement of maximal flow at functional residual capacity (V'maxFRC) is the most commonly used method for assessing airway function in infants, but is highly dependent on lung volume and airway tone. Recent studies suggested that the raised volume RTC technique, which assesses lung function over an extended volume range as in older children, may be a more sensitive means of discriminating changes in airway function in infants with respiratory disease. The forced deflation technique allows investigation of pulmonary function during the early development of CLDI in intubated subjects, but its invasive nature precludes its use in the routine setting. For all techniques, there is an urgent need to establish suitable reference data and evaluate within- and between-occasion repeatability, prior to establishing the clinical usefulness of these techniques in assessing baseline airway function and/or response to interventions in subjects with CLDI.

Intracranial Pressure Response to Induced Hypertension: Role of Dynamic Pressure Autoregulation

Induced hypertension is commonly used to improve cerebral perfusion, but this treatment may have the deleterious side effect of raising intracranial pressure (ICP). We tested the hypothesis that dynamic pressure autoregulation testing could identify patients who might develop increased ICP during induced hypertension. Twenty-two studies were performed in 21 patients. Baseline dynamic testing of autoregulation by cuff deflation and carotid compression techniques was performed. After phenylephrine was infused to increase mean arterial pressure by 20 to 30 mm Hg, cuff deflation tests were repeated. The average increase in mean arterial pressure was 32.2 +/- 16.1 mm Hg. This increase was accompanied by increased flow velocity (P < 0.001), brain tissue PO2 (P = 0.011), and regional cerebral blood flow (P = 0.008). Also, dynamic pressure autoregulation consistently improved (P = 0.015). Induced hypertension caused increased ICP (iICP) in 12 patients and a decrease in ICP (dICP) in 9. Baseline jugular venous oxygen saturation in the iICP group was 82 +/- 10% compared with 70 +/- 10% in dICP patients (P = 0.02). Baseline dynamic autoregulatory index for the cuff deflation tests (1.8 +/- 1.4) and baseline transient hyperemic response ratio for the carotid compression tests (1.11 +/- 0.07) were significantly lower in iICP patients (dICP group: autoregulatory index 3.2 +/- 1.7, P = 0.06; transient hyperemic response ratio 1.26 +/- 0.11, P = 0.009). Flow velocity increased more with the increase in blood pressure in the iICP group than in the dICP group: 19.0 +/- 6.8 cm/s versus 10.2 +/- 6.3 cm/s (P = 0.007). The patients who had an increase in ICP with induced hypertension had a greater degree of impairment of autoregulation and induced hypertension resulted in a greater increase in flow velocity.

The generalized eigendecomposition approach to the blind source separation problem

This paper proposes a novel formulation of the generalized eigendecomposition (GED) approach to blind source separation (BSS) problems. The generalized eigendecomposition algorithms consider the estimation of a pair of correlation matrices (a matrix pencil) using observed sensor signals. Each of various algorithms proposed in the literature uses a different approach to form the pencil. This study proposes a linear algebra formulation which exploits the definition of congruent matrix pencils and shows that the solution and its constraints are independent of the way the matrix pencil is computed. Also an iterative eigendecomposition algorithm, that updates separation parameters on a sample-by-sample basis, is developed. It comprises of: (1) performing standard eigendecompositions based on power and deflation techniques; (2) computing a transformation matrix using spectral factorization. Another issue discussed in this work is the influence of the length of the data segment used to estimate the pencil. The algorithm is applied to artificially mixed audio data and it is shown that the separation performance depends on the eigenvalue spread. The latter varies with the number of samples used to estimate the eigenvalues.

A quadratically convergent QR-like method without shifts for the Hermitian eigenvalue problem

We propose a new QR-like algorithm, symmetric squared QR (SSQR) method, that can be readily parallelized using commonly available parallel computational primitives such as matrix–matrix multiplication and QR decomposition. The algorithm converges quadratically and the quadratic convergence is achieved through a squaring technique without utilizing any kind of shifts. We provide a rigorous convergence analysis of SSQR and derive structures for several of the important quantities generated by the algorithm. We also discuss various practical implementation issues such as stopping criteria and deflation techniques. We demonstrate the convergence behavior of SSQR using several numerical examples.

A Rank-Revealing Method with Updating, Downdating, and Applications

A new rank-revealing method is proposed. For a given matrix and a threshold for near-zero singular values, by employing a globally convergent iterative scheme as well as a deflation technique the method calculates approximate singular values below the threshold one by one and returns the approximate rank of the matrix along with an orthonormal basis for the approximate null space. When a row or column is inserted or deleted, algorithms for updating/downdating the approximate rank and null space are straightforward, stable, and efficient. Numerical results exhibiting the advantages of our code over existing packages based on two-sided orthogonal rank-revealing decompositions are presented. Also presented are applications of the new algorithm in numerical computation of the polynomial GCD as well as identification of nonisolated zeros of polynomial systems.

Auxiliary field quantum Monte-Carlo simulation of interacting electrons in quantum dots

Accurate auxiliary field quantum Monte-Carlo (AFQMC) simulations of interacting electrons in quantum dots are reported. Two different formulations of this approach are presented both of which have been designed specifically for application to quantum dots. A deflation technique for calculation of anti-symmetrized traces is introduced. The auxiliary field is sampled with a hybrid algorithm and the artificial dynamics needed for use with the present formulation of AFQMC is described. The constrained path approximation is used to control the sign problem. Results for the ground state energy of two spin-polarised, interacting electrons are presented and are found to agree well with exact diagonalization results for a wide range of screening lengths. The sign problem does not appear in the regime of small screening length.

Measurement of Forced Expiratory Flows and Lung Volumes

After completing this article, readers should be able to: 1. Describe the information that measurements of forced expiratory flow can provide in spontaneously breathing and mechanically ventilated infants. 2. Delineate how lung volume can be measured. The measurements of forced expiratory flows and lung volumes are important to the evaluation of patients who have respiratory symptoms. Until recently, measurements have been restricted to adults and children old enough to cooperate with testing maneuvers. Newer techniques have enabled such measurements to be made in infants and led to improved understanding of normal lung and airway growth during infancy and early childhood, when dramatic changes occur. The measurements have had important clinical uses in the evaluation and treatment of infants who have respiratory disease, but their use has been limited because they require specialized equipment and expertise as well as sedation of the infant. In this article, we discuss the techniques required for measurement of forced expiratory flows and lung volumes, their limitations, and applications in clinical practice. Detection of airflow obstruction, by physical examination or by quantitative measurement techniques, is facilitated by increasing expiratory flow rates. Measurement of forced expiratory flows is useful because flow rates are inversely proportional to the fourth power of the radius of the airway and, therefore, are indicative of airway size when flow is limited. Disease states affecting the airways, including asthma, bronchopulmonary dysplasia (BPD), or congenital lesions such as pulmonary hypoplasia or vascular airway compression, are expected to decrease expiratory flow. Infants typically are sedated for testing to maintain relaxed breathing with a face mask placed over the mouth and nose. This sedation often is perceived by parents and physicians as one of the barriers to such testing. (1) However, sedation can be achieved safely using oral chloral hydrate at a dose of 80 to 100 mg/kg. …