High Relative Accuracy Research Articles

Accurate Eigenvalues of Some Generalized Sign Regular Matrices via Relatively Robust Representations

In this paper, we consider how to accurately solve the nonsymmetric eigenvalue problem for a class of generalized sign regular matrices including extremely ill-conditioned quasi-Cauchy and quasi-Vandermonde matrices. The problem of performing accurate computations with structured matrices is very much a representation problem. We first develop a relatively robust representation (RRR) for this class of matrices by introducing a free parameter, which exceeds an essential threshold, into an indefinite factorization. We then design a new $$O(n^{3})$$ algorithm to compute all the eigenvalues of such matrices with high relative accuracy, as warranted by the RRR. Error analysis and numerical experiments are performed to illustrate the high relative accuracy.

Immunoglobulin G N-Glycan Analysis by Ultra-Performance Liquid Chromatography.

Glycomics is a new subspecialty in omics system research that offers significant potential in discovering next-generation biomarkers for disease susceptibility, drug target discovery, and precision medicine. Alternative IgG N-glycans have been reported in several common chronic diseases and suggested to have great potential in clinical applications (i.e., biomarkers for diagnosis and prediction of diseases). IgG N-glycans are widely characterized using the method of hydrophilic interaction chromatography (HILIC) ultra-performance liquid chromatography (UPLC). UPLC is a stable detection technology with good reproducibility and high relative quantitative accuracy. In addition, the structure of IgG N-glycan is clearly separated, and glycan composition and relative abundance in plasma are characterized.

Accurate computations for eigenvalues of products of Cauchy-polynomial-Vandermonde matrices

In this paper, we consider the product eigenvalue problem for the class of Cauchy-polynomial-Vandermonde (CPV) matrices arising in a rational interpolation problem. We present the explicit expressions of minors of CPV matrices. An algorithm is designed to accurately compute the bidiagonal decomposition for strictly totally positive CPV matrices and their additive inverses. We then illustrate the sign regularity of the bidiagonal decomposition to show that all the eigenvalues of a product involving such matrices are computed to high relative accuracy. Numerical experiments are given to confirm the claimed high relative accuracy.

Welding defect detection based on local image enhancement

Welding defect detection in a radiographic image is an important topic in the field of industrial non-destructive testing. To improve the accuracy of welding defect segmentation, a local image enhancement approach is proposed. In this algorithm, the requirement of contrast enhancement is considered when extracting the weld seam and segmenting the weld defect. The whole defect detection is conducted by three procedures: image enhancement, welding seam extraction, and defect segmentation. Firstly, a method for determining the Localised Pixel Inhomogeneity Factor (LPIF) is proposed. Then, based on the results of LPIF, the Otsu method is applied to segment the welding seam and defects are, identified by region growing algorithm. The authors compared LPIF with histogram equalisation, adaptive histogram equalisation, and contrast-limited adaptive histogram equalisation algorithms and assessed its performance by using indicators such as image contrast, image definition, and edge intensity. Moreover, the authors compared the segmentation results of the enhanced defect images with the original image to further study the method's effect on weld defect segmentation. More than 70 images containing various types of defects are tested. The experimental results demonstrate that the quality of enhanced defect images is improved significantly, and has a high relative segmentation accuracy of more than 92%.

Trajectory Definition with High Relative Accuracy (HRA) by Parametric Representation of Curves in Nano-Positioning Systems.

Nanotechnology applications demand high accuracy positioning systems. Therefore, in order to achieve sub-micrometer accuracy, positioning uncertainty contributions must be minimized by implementing precision positioning control strategies. The positioning control system accuracy must be analyzed and optimized, especially when the system is required to follow a predefined trajectory. In this line of research, this work studies the contribution of the trajectory definition errors to the final positioning uncertainty of a large-range 2D nanopositioning stage. The curve trajectory is defined by curve fitting using two methods: traditional CAD/CAM systems and novel algorithms for accurate curve fitting. This novel method has an interest in computer-aided geometric design and approximation theory, and allows high relative accuracy (HRA) in the computation of the representations of parametric curves while minimizing the numerical errors. It is verified that the HRA method offers better positioning accuracy than commonly used CAD/CAM methods when defining a trajectory by curve fitting: When fitting a curve by interpolation with the HRA method, fewer data points are required to achieve the precision requirements. Similarly, when fitting a curve by a least-squares approximation, for the same set of given data points, the HRA method is capable of obtaining an accurate approximation curve with fewer control points.

Highly accurate doubling algorithm for quadratic matrix equation from quasi-birth-and-death process

A highly accurate doubling algorithm to solve the most fundamental quadratic matrix equation in the quasi-birth-and-death (qbd) process is developed. It follows from the general framework of the doubling algorithm for the first standard form (SF1) but can be implemented to compute the minimal nonnegative solution with high entrywise relative accuracy for all entries, large or tiny. The algorithm is globally and quadratically convergent, except for qbd equations in the critical case where convergence is linear with the linear rate 1/2. Numerical examples are presented to demonstrate and confirm our claims. The development here parallels the recent work of Xue and Li (2017) [3] on the M-matrix algebraic Riccati equation.

Accurate solutions of weighted least squares problems associated with rank-structured matrices

In this paper, we consider how to accurately solve the weighted least squares (WLS) problem associated with a class of rank-structured matrices admitting bidiagonal representations (BRs) from weighted polynomial regression. We develop a new algorithm to solve the structured WLS problem, provided that BRs are available. A mechanism is exploited to guarantee that all the solution components are computed with a desirable high accuracy. Forward error analysis and numerical experiments are performed to confirm the high accuracy. In particular, our random examples demonstrate the high relative accuracy of each computed solution component, independently of the ill-conditioning of weighted matrices.

Accurate bidiagonal decomposition of totally positive h-Bernstein-Vandermonde matrices and applications

An algorithm for the computation of the bidiagonal decomposition of strictly totally positive h-Bernstein-Vandermonde matrices is presented. These matrices are collocation matrices of h-Bernstein bases (a generalization of the Bernstein basis) of the space of polynomials of degree less than or equal to n. The algorithm is fast (its computational complexity is O(n2)) and has high relative accuracy. The computation of this bidiagonal decomposition is used as the initial stage for the accurate and efficient solution of several linear algebra problems with these matrices: linear system solving, eigenvalue computation and inverse computation. Numerical experiments are also included.

First measurements with the K-monochromator at the European XFEL.

Hard X-ray free-electron lasers (XFELs) generate intense coherent X-ray beams by passing electrons through undulators, i.e. very long periodic magnet structures, which extend over hundreds of meters. The SASE1 and SASE2 undulator systems of the European XFEL consist of 35 segments with variable-gap planar undulators which are initially tuned to precise on-axis magnetic field strengths in a magnetic measurement laboratory to keep an important quality parameter - the K-value variation from segment to segment - below a certain limit (3 × 10-4 for 12 keV photon energy). After tunnel installation only photon-based methods can determine the K-values of undulator segments with a similar accuracy. The synchrotron radiation from a single or a few segments can be spectrally filtered by a dedicated crystal monochromator (K-monochromator) and recorded with a photodiode or with an imager that provides 2D information, tuned for high sensitivity to detect low photon densities from distant single undulator segments. This instrumentation is applied for electron orbit analysis and optimization, and adjustment of individual undulators in terms of their central magnetic axis with respect to the electron beam. Single undulator segments were analysed by scanning the monochromator crystal angle and detecting the steepest slope of a photodiode signal. Alternatively, in the imaging method, an imager recorded the radiation cone of electrons passing through the undulator segment. From the spatial distribution of the radiation, the K-parameter was determined with a sufficiently high relative accuracy.

Collecting and processing of barometric data from smartphones for potential use in numerical weather prediction data assimilation

AbstractThe potential for use of crowd‐sourced data in the atmospheric sciences is vastly expanding, including observations from smartphones with barometric sensors. Smartphone pressure observations can potentially help improve numerical weather prediction and aid forecasters. In this contribution a method of collecting data from smartphones is presented, other methods are discussed and guidelines are derived from the experience. Quality control is vital when using crowd‐sourced data. Screening methods aimed at smartphone pressure observations are presented. Results from previous studies, showing a substantial but long‐term stable bias in combination with high relative accuracy, are confirmed. The collection of Danish smartphone pressure observations has been very successful, with over 6 million observations during a 7 week period. Case studies show that distinct weather patterns can be seen in unprocessed data. The screening method developed reduces the observational noise but filters out the majority of observations. Assimilating smartphone pressure observations in a single case study, using the 3D variational data assimilation system of the HARMONIE numerical weather prediction system, proved to decrease the bias of surface pressure in the model without increasing the root mean square error and the skill of accumulated precipitation increased. It is found that the altitude assignment of smartphones needs improvement.

Accurate Algorithms for Bessel Matrices

In this paper, we prove that any collocation matrix of Bessel polynomials at positive points is strictly totally positive, that is, all its minors are positive. Moreover, an accurate method to construct the bidiagonal factorization of these matrices is obtained and used to compute with high relative accuracy the eigenvalues, singular values and inverses. Similar results for the collocation matrices for the reverse Bessel polynomials are also obtained. Numerical examples illustrating the theoretical results are included.

A qd-type method for computing generalized singular values of BF matrix pairs with sign regularity to high relative accuracy

A qd-type method for computing generalized singular values of BF matrix pairs with sign regularity to high relative accuracy

On the complex Falk–Langemeyer method

A new algorithm for the simultaneous diagonalization of two complex Hermitian matrices is derived. It is a proper generalization of the known Falk–Langemeyer algorithm which was originally derived in 1960 for a pair of positive definite matrices. It is proved that the complex Falk–Langemeyer algorithm is defined for a pair of Hermitian matrices which make a definite pair. Special attention is paid to the stability of the formulas for the transformation parameters in the case when the pivot submatrices are almost proportional. The numerical tests show the high relative accuracy of the method if both matrices are definite and if the condition numbers of DAADA and DBBDB are small for some diagonal matrices DA and DB.

Accurate inverses of Nekrasov Z-matrices

We present a parametrization of a Nekrasov Z-matrix that allows us to compute its inverse with high relative accuracy. Numerical examples illustrating the accuracy of the method are included.

Advancing Short‐Term Forecasts of Ice Conditions in the Beaufort Sea

AbstractNumerical experiments with a regional configuration of the CICE model (Los‐Alamos Community sea Ice CodE) in the Beaufort Sea assimilating Special Sensor Microwave Imager/Sounder ice concentration (IC) and CryoSat ice thickness (IT) data acquired for September–December of 2015 are presented. We explore sensitivity of the 24‐hr IT/IC forecast skill to the system updates, which include introduction of the IT assimilation capability, flow‐dependent correlations, and Gaussianization of IC innovations. Experiments with IC data assimilation have shown that the flow‐dependent correlations provide 5–7% improvement of the forecast skill during the freezing period (10 September to 10 November) while Gaussianization contributes an additional improvement of 3–4% in most of the cases. In winter (11 November to 31 December) IC assimilation did not produce any statistically significant improvement of the skill due to the loss of dynamical information in the IC fields associated saturation of the ice cover. In contrast, IT assimilation provides larger improvement in December compared to October–November due to the better coverage of the Beaufort Sea by observations and their higher relative accuracy in winter. Comparison of the IT forecast fields with independent in situ observations by two upward looking sonars demonstrates statistically insignificant improvements. Much better improvement (15–25%) is observed when comparing monthly mean IT assimilation runs against independent Advanced Microwave Scanning Radiometer for Earth observing system (AMSR‐E) data. Introduction of the heuristic in situ IC/IT correlations into the background covariance model did not produce any improvements of the forecast skill.

Accurate eigenvalues and exact zero Jordan blocks of totally nonnegative matrices

We present a new algorithm which, given a bidiagonal decomposition of a totally nonnegative matrix, computes all its eigenvalues to high relative accuracy in floating point arithmetic in \(O(n^3)\) time. It also computes exactly the Jordan blocks corresponding to zero eigenvalues in up to \(O(n^4)\) time.

On the Global Convergence of the Complex HZ Method

The paper considers a Jacobi method for solving the generalized eigenvalue problem $Ax=\lambda Bx$, where $A$ and $B$ are complex Hermitian matrices and $B$ is positive definite. The method is a proper generalization of the standard Jacobi method for the Hermitian matrix $A$ to the matrix pair $(A,B)$. The paper derives the method and proves its global convergence under the large class of generalized serial pivot strategies. If both matrices are positive definite, it can be implemented as a one-sided method. It then solves the initial problem as the generalized singular value problem. Its main application is to serve as a kernel algorithm in a block Jacobi method for the same problem with large matrices $A$ and $B$. The block Jacobi methods are methods of choice on contemporary CPU and GPU computing architectures. The proposed algorithm is very efficient on pairs of almost diagonal matrices, and diagonalization of such matrices is the main task of the kernel algorithm. The numerical tests indicate the high relative accuracy of the method on certain pairs of positive definite matrices.

On the Accuracy of the Element‐wise Jacobi Methods for PGEP

AbstractWe analyze the relative accuracy of two new element‐wise Jacobi‐type methods for the positive definite generalized eigenvalue problem Ax = λBx, where A and B are symmetric matrices and B is positive definite. A detailed error analysis is used, and the appropriate numerical tests are performed. If A and B are well‐behaved positive definite matrices then the transformation parameters will have small relative errors and numerical tests indicate the high relative accuracy of the methods.

Complex Falk‐Langemeyer Method

AbstractThe known Falk‐Langemeyer method for the simultaneous diagonalization of two positive definite symmetric matrices is generalized to work with complex matrices. It is shown that the derived method is well defined for the Hermitian matrices which make a definite pair. Special attention is paid to the stability of the formulas for the computational parameters when the pivot submatrices are close to being proportional. Numerical tests indicate the high relative accuracy of the method provided that both matrices are definite and well‐behaved, i.e. if they can be well‐scaled symmetrically.

A New Method for Calibration of Gain Variation in a Detector System

Abstract Transit spectroscopy of habitable planets orbiting late-type stars requires high relative spectrophotometric accuracy between wavelengths during transit/eclipse observation. The spectrophotometric signal is affected not only by image movement and deformation due to wavefront error but also by electrical variation in the detector system. These time-variation components, coupled to the transit signal, distort the measurements of atmospheric composition in transit spectroscopy. Here we propose a new concept for improvement of spectrophotometric accuracy through the calibration of the time-variation components in the detector system by developing densified pupil spectroscopy that provides multiple spectra of the star–planet system. Owing to a group of pixels exposed by the object light (i.e., science pixels), pixel-to-pixel variations can be smoothed out through an averaging operation; thus, only common time-variation components over the science pixels remain. In addition, considering that the detector plane is optically conjugated to the pupil plane, a pupil mask can completely block astronomical light coming into residual pixels. The common time-variation components are reconstructed with the residual pixels and reduced into a random term. Applying the densified pupil spectrograph with a mid-infrared detector system to a large space cryogenic telescope such as the Origins Space Telescope, we show that the system nearly achieves photon noise–limited performance and detects absorption features through transmission spectroscopy and secondary eclipse of terrestrial planets orbiting M-type stars at 10 pc with 60 transit observations. Thus, the proposed method contributes to the measurement of planetary habitability and biosignatures of the nearby transiting habitable candidates.