Rice condition numbers of QR and Cholesky factorizations

A condition number is an amplification coefficient due to errors in computing. Thus the theory of condition numbers plays an important role in error analysis. In this paper, following the approach of Rice, condition numbers are defined for factors of some matrix factorizations such as the Cholesky factorization of a symmetric positive definite matrix and QR factorization of a general matrix. The condition numbers are derived by a technique of analytic expansion of the factor dependent on one parameter and matrix-vector equation. Condition numbers of the Cholesky and QR factors are different from the ones previously introduced by other authors, but similar to Chang's results. In Cholesky factorization, corresponding with the condition number of the factor matrix L , K _L is a low bound of Stewart's condition number K .

Structured condition numbers and statistical condition estimation for the LDU factorization

In this article, we consider the structured condition numbers for LDU, factorization by using the modified matrix-vector approach and the differential calculus, which can be represented by sets of parameters. By setting the specific norms and weight parameters, we present the expressions of the structured normwise, mixed, componentwise condition numbers and the corresponding results for unstructured ones. In addition, we investigate the statistical estimation of condition numbers of LDU factorization using the probabilistic spectral norm estimator and the small-sample statistical condition estimation method, and devise three algorithms. Finally, we compare the structured condition numbers with the corresponding unstructured ones in numerical experiments.

Effective condition numbers and small sample statistical condition estimation for the generalized Sylvester equation

Abstract In this paper, we investigate the effective condition numbers for the generalized Sylvester equation （AX - YB, DX - YE） = （C,F）, where A,D ∈ Rm×m B,E ∈ Rn×n and C,F ∈ Rm×n. We apply the small sample statistical method for the fast condition estimation of the generalized Sylvester equation, which requires （9（m2n ＋ mn2） flops, comparing with （-O（m3 ＋ n3） flops for the generalized Schur and generalized Hessenberg- Schur methods for solving the generalized Sylvester equation. Numerical examples illustrate the sharpness of our perturbation bounds.

STRUCTURED CONDITION NUMBERS FOR THE TIKHONOV REGULARIZATION OF DISCRETE ILL-POSED PROBLEMS

The possibly most popular regularization method for solving the least squares problem rain ‖Ax - b‖2 with a highly ill-conditioned or rank deficient coefficient matrix A is the x Tikhonov regularization method. In this paper we present the explicit expressions of the normwise, mixed and componentwise condition numbers for the Tikhonov regularization when A has linear structures. The structured condition numbers in the special cases of nonlinear structure i.e. Vandermonde and Cauchy matrices are also considered. Some comparisons between structured condition numbers and unstructured condition numbers are made by numerical experiments. In addition, we also derive the normwise, mixed and componentwise condition numbers for the Tikhonov regularization when the coefficient matrix, regularization matrix and right-hand side vector are all perturbed, which generalize the results obtained by Chu et al. [Numer. Linear Algebra Appl., 18 （2011）, 87-103].

Location method of ill-conditioned microseismic source and its engineering application

Microseismic event location is one of the core parameters in microseismic monitoring,and the accuracy of localization will directly affect the effectiveness of engineering applications.However,limited by spatial factors,the geometry of the sensor installation will be close to linear,which makes the localization equation suffer from the pathological problem,and the localization accuracy is greatly reduced.To address this problem,the reasons for the pathological problem are analyzed from the perspective of the objective function residuals and coefficient matrix.The pathological problem is caused by the combined effect of the poorer sensor array and data errors,and its residual isosurface shows a conical distribution,and as the residual value decreases,the apex of the isosurface gradually extends to the far side,and the localization results do not converge.For this reason,an improved regularized Newton downhill localization algorithm is proposed.In this method,firstly,the Newtonian downhill method is improved so that the magnitudes of the seismic source parameters are the same,and the condition number of the coefficient matrix is reduced;then,the L-curve method is used to calculate the regularization factor for the pathological equations,and the coefficient matrix is improved;finally,the pathological equations are regularized,and the seismic source coordinates are obtained by the improved Newtonian downhill method.The results of engineering applications show that compared with the traditional algorithm based on automatic of P-arrival picking,the number of effective microseismic events calculated by the proposed localization algorithm is increased by 194.7%,and the localization accuracy is substantially improved.The proposed algorithm reduces the problem of low accuracy of S-arrival picking and allows localization using only P-wave arrival.The method reduces the quality requirements of the data and significantly improves the utilization of microseismic events and positioning accuracy.

Geometric Condition of 3UPS-S Parallel Mechanism in Singular Configuration

The existing researches on singularity of parallel mechanism are mostly limited to the property and regularity of singularity locus and there is no further research into the geometric relationship between uncontrolled kinematic screw and parallel mechanism in singularity. A 3UPS-S parallel mechanism is presented which fulfils 3-DOF in rotation. The regularity of nutation angle singularity is analyzed based on the Jacobian matrix, and the singularity surface of 3UPS-S parallel mechanisms is obtained. By applying the concept of reciprocal product in screw theory, the singular kinematic screw is derived when 3UPS-S parallel mechanism is in singularity. The geometric relationship between singular kinematic screw and singular configuration of 3UPS-S parallel mechanism is investigated by using programs in MATLAB. It is revealed that there are two kinds of situation. Firstly, the three limbs of 3UPS-S parallel mechanism intersect the singular kinematic screw in space simultaneously; Secondly, two limbs cross the singular kinematic screw while the third limb parallels with that screw. It is concluded that the nutation angle singularity of 3UPS-S parallel mechanism belongs to the singular linear complexes. This paper sheds light into and clarifies the geometric relationship between singular kinematic screw and singular configuration of 3UPS-S parallel mechanism.

An Improved Semi-Orthogonal User Selection Algorithm Based on Condition Number for Multiuser MIMO Systems

In Multiple-Input Multiple-Out （MIMO） systems, the user selection algorithm plays an important role in the realization of multiplexing gain. In this paper, an improved Semi-orthogonal User Selection algorithm based on condition number is proposed. Besides, a new MIMO pre- coding scheme is designed. The proposed SUS- CN （SUS with condition number） algorithm outperforms the SUS algorithm for the selection of users with better matrix inversion property, thus a higher information rate for selected user pair is achieved. The designed MIMO precoding matrix brings benefits of the power equality at transmitted terminals, the limited dynamic range of the power over time, and a better power efficiency. The simulation results give the key insights into the im- pact of the different condition number value and users on the sum-rate capacity.

Novel method based on ant colony opti mization for solving ill-conditioned linear systems of equations

A novel method based on ant colony optimization （ACO）, algorithm for solving the ill-conditioned linear systems of equations is proposed. ACO is a parallelized bionic optimization algorithm which is inspired from the behavior of real ants. ACO algorithm is first introduced, a kind of positive feedback mechanism is adopted in ACO. Then, the solu- tion problem of linear systems of equations was reformulated as an unconstrained optimization problem for solution by an ACID algorithm. Finally, the ACID with other traditional methods is applied to solve a kind of multi-dimensional Hilbert ill-conditioned linear equations. The numerical results demonstrate that ACO is effective, robust and recommendable in solving ill-conditioned linear systems of equations.

ON THE APPROXIMATE COMPUTATION OF EXTREME EIGENVALUES AND THE CONDITION NUMBER OF NONSINGULAR MATRICES

From the formulas of the conjugate gradient, a similarity between a symmetric positive definite (SPD) matrix A and a tridiagonal matrix B is obtained. The elements of the matrix B are determined by the parameters of the conjugate gradient. The computation of eigenvalues of A is then reduced to the case of the tridiagonal matrix B. The approximation of extreme eigenvalues of A can be obtained as a 'by-product' in the computation of the conjugate gradient if a computational cost of O(s) arithmetic operations is added, where s is the number of iterations This computational cost is negligible compared with the conjugate gradient. If the matrix A is not SPD, the approximation of the condition number of A can be obtained from the computation of the conjugate gradient on AT A. Numerical results show that this is a convenient and highly efficient method for computing extreme eigenvalues and the condition number of nonsingular matrices.

On the Preconditioning Properties of RHSS Preconditioner for Saddle-Point Linear Systems

In this paper,for the regularized Hermitian and skew-Hermitian splitting(RHSS)preconditioner introduced by Bai and Benzi(BIT Numer Math 57:287–311,2017)for the solution of saddle-point linear systems,we analyze the spectral properties of the preconditioned matrix when the regularization matrix is a special Hermitian positive semidefinite matrix which depends on certain parameters.We accurately describe the numbers of eigenvalues clustered at(0,0)and(2,0),if the iteration parameter is close to 0.An estimate about the condition number of the corresponding eigenvector matrix,which partly determines the convergence rate of the RHSS-preconditioned Krylov subspace method,is also studied in this work.

Generation of Complex Matrices Belonging to Matrix Groups with Controlled Condition Number

The authors study the generation of matrices with complex entries belonging to some matrix groups, mainly those that are defined by a scalar product space. These matrices have useful applications in quantum mechanical problems and complex control problems. In this work, the authors try to generate matrices such that： （1） the condition number of these types of matrices is controlled and （2） The algorithm used to generate these matrices preserves their structure.

A New Class of Biased Linear Estimators in Deficient-rank Linear Models

In this paper, we define a new class of biased linear estimators of the vector of unknown parameters in the deficient_rank linear model based on the spectral decomposition expression of the best linear minimun bias estimator. Some important properties are discussed. By appropriate choices of bias parameters, we construct many interested and useful biased linear estimators, which are the extension of ordinary biased linear estimators in the full_rank linear model to the deficient_rank linear model. At last, we give a numerical example in geodetic adjustment.

Re ned rigorous perturbation bounds for the SR decomposition

In this article,some new rigorous perturbation bounds for the SR decomposition un-der normwise or componentwise perturbations for a given matrix are derived.Also,the explicit expressions for the mixed and componentwise condition numbers are presented by utilizing the block matrix-vector equation approach.Hypothetical and trial results demonstrate that these new bounds are constantly more tightly than the comparing ones in the literature.

Novel method of improving the alignment accuracy of SINS on revolving mounting base

In the process of initial alignment for a strapdown inertial navigation system （SINS） on a stationary base, the east gyro drift rate is an important factor affecting the alignment accuracy of the azimuth misalignment angle. When the Kalman filtering algorithm is adopted in initial alignment, it yields a constant error in the estimation of the azimuth misalignment angle because the east gyro drift rate cannot be estimated. To improve the alignment accuracy, a novel alignment method on revolving mounting base is proposed. The Kalman filtering algorithm of extending the measured values is studied. The theory of spectral condition number is utilized to analyze the degrees of observability of states. Simulation results show that the estimation accuracy of the azimuth misalignment angle is greatly improved through revolving mounting base, and the proposed method is efficient in initial alignment for a medium accurate SINS.

Diffusion tensor imaging with multiple diffusion-weighted gradient directions

Diffusion tensor MRI （DT-MRI or DTI） is emerging as an important non-invasive technology for elucidating intemal brain structures. It has recently been utilized to diagnose a series of diseases that affect the integrity of neural systems to provide a basis for neuroregenerative studies. Results from the present study suggested that neural tissue is reconstructed with multiple diffusion-weighted gradient directions DTI, which varies from traditional imaging methods that utilize 6 gradient directions. Simultaneously, the diffusion tensor matrix is obtained by multiple linear regressions from an equation of echo signal intensity. The condition number value and standard deviation of fractional anisotropy for each scheme can be used to evaluate image quality. Results demonstrated that increasing gradient direction to some extent resulted in improved effects. Therefore, the traditional 6 and 15 directions should not be considered optimal scan protocols for clinical DTI application. In a scheme with 20 directions, the condition number and standard deviation of fractional anisotropy of the encoding gradients matrix were significantly reduced, and resulted in more clearly and accurately displayed neural tissue. Results demonstrated that the scheme with 20 diffusion gradient directions provided better accuracy of structural renderings and could be an optimal scan protocol for clinical DTI application.

A complete solution of an improved universal 3D coordinate similarity transformation model

When linearizing three-dimensional(3 D)coordinate similarity transformation model with large rotations,we usually encounter the ill-posed normal matrix which may aggravate the instability of solutions.To alleviate the problem,a series of conversions are contributed to the 3 D coordinate similarity transformation model in this paper.We deduced a complete solution for the 3 D coordinate similarity transformation at any rotation with the nonlinear adjustment methodology,which involves the errors of the common and the non-common points.Furthermore,as the large condition number of the normal matrix resulted in an intractable form,we introduced the bary-centralization technique and a surrogate process for deterministic element of the normal matrix,and proved its benefit for alleviating the condition number.The experimental results show that our approach can obtain the smaller condition number to stabilize the convergence of the interested parameters.Especially,our approach can be implemented for considering the errors of the common and the non-common points,thus the accuracy of the transformed coordinates improves.

Domain Decomposition for Wavelet Single Layer on Geometries with Patches

We focus on the single layer formulation which provides an integral equation of the first kind that is very badly conditioned. The condition number of the unpreconditioned system increases exponentially with the multiscale levels. A remedy utilizing overlapping domain decompositions applied to the Boundary Element Method by means of wavelets is examined. The width of the overlapping of the subdomains plays an important role in the estimation of the eigenvalues as well as the condition number of the additive domain decomposition operator. We examine the convergence analysis of the domain decomposition method which depends on the wavelet levels and on the size of the subdomain overlaps. Our theoretical results related to the additive Schwarz method are corroborated by numerical outputs.

Stability of GM(1,1) power model on vector transformation

The morbidity problem of the GM（1,1） power model in parameter identification is discussed by using multiple and rotation transformation of vectors. Firstly we consider the morbidity problem of the special matrix and prove that the condition number of the coefficient matrix is determined by the ratio of lengths and the included angle of the column vector, which could be adjusted by multiple and rotation transformation to turn the matrix to a well-conditioned one. Then partition the corresponding matrix of the GM（1,1） power model in accordance with the column vector and regulate the matrix to a well-conditioned one by multiple and rotation transformation of vectors, which completely solve the instability problem of the GM（1,1） power model. Numerical results show that vector transformation is a new method in studying the stability problem of the GM（1,1） power model.

2D Minimum Compliance Topology Optimization Based on a Region Partitioning Strategy

This paper presents an extended sequential element rejection and admission(SERA)topology optimizationmethod with a region partitioning strategy.Based on the partitioning of a design domain into solid regions and weak regions,the proposed optimizationmethod sequentially implements finite element analysis(FEA)in these regions.After standard FEA in the solid regions,the boundary displacement of the weak regions is constrained using the numerical solution of the solid regions as Dirichlet boundary conditions.This treatment can alleviate the negative effect of the material interpolation model of the topology optimization method in the weak regions,such as the condition number of the structural global stiffness matrix.For optimization,in which the forward problem requires nonlinear structural analysis,a linear solver can be applied in weak regions to avoid numerical singularities caused by the over-deformedmesh.To enhance the robustness of the proposedmethod,the nonmanifold point and island are identified and handled separately.The performance of the proposed method is verified by three 2D minimum compliance examples.

Novel high-performance element in the electromagnetic finite-element method——node-edge element

It is known in the computational electromagnetics （CEM） that the node element has a relative wellconditioned matrix, but suffers from the spurious solution problem; whereas the edge element has no spurious solutions, but usually produces an ill-conditioned matrix. Particularly, when the mesh is over dense, the iterative solution of the matrix equation from edge element converges very slowly. Based on the node element and edge element, a node-edge element is presented, which has no spurious solutions and better-conditioned matrix. Numerical experiments demonstrate that the proposed node-edge element is more efficient than now-widely used edge element.