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 setti...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.展开更多
In this paper,we consider the indefinite least squares problem with quadratic constraint and its condition numbers.The conditions under which the problem has the unique solution are first presented.Then,the normwise,m...In this paper,we consider the indefinite least squares problem with quadratic constraint and its condition numbers.The conditions under which the problem has the unique solution are first presented.Then,the normwise,mixed,and componentwise condition numbers for solution and residual of this problem are derived.Numerical example is also provided to illustrate these results.展开更多
In this paper, we investigate the condition numbers for indefinite least squares problem with multiple right-hand sides. The normwise, mixed and componentwise condition numbers and the corresponding structured conditi...In this paper, we investigate the condition numbers for indefinite least squares problem with multiple right-hand sides. The normwise, mixed and componentwise condition numbers and the corresponding structured condition numbers are presented. The structured matrices under consideration include the linear structured matrices, such as the Toeplitz, Hankel, symmetric, and tridiagonal matrices, and the nonlinear structured matrices, such as the Vandermonde and Cauchy matrices. Numerical examples show that the structured condition numbers are tighter than the unstructured ones.展开更多
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 n...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 .展开更多
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 ...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.展开更多
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 t...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.展开更多
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...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.展开更多
In this paper,we propose evaluation approaches for the spatial target localization precision based on the observation matrix conditional number.Three evaluation approaches for the spatial target localization precision...In this paper,we propose evaluation approaches for the spatial target localization precision based on the observation matrix conditional number.Three evaluation approaches for the spatial target localization precision are derived including relative condition numbers,absolute condition numbers,and volume condition numbers by properties of vectors and matrix norms.The theoretical analysis shows that the proposed methods are the upper certainty bound of the magnification of measurement error.Meanwhile,the proposed methods are able to account for variations of localization accuracy by exploiting geometric variations in the composition between the target and measurement stations.Finally,the proposed methods perform better,compared with the traditional evaluation methods.This is of great significance for the accuracy evaluation of high-precision measuring equipment,optimization of workstation layout and geometric configuration.Simulation experiments corroborate the effectiveness of the proposed methods.展开更多
We present results of numerical simulations of the tensor-valued elliptic-parabolic PDE model for biological network formation.The numerical method is based on a nonlinear finite difference scheme on a uniform Cartesi...We present results of numerical simulations of the tensor-valued elliptic-parabolic PDE model for biological network formation.The numerical method is based on a nonlinear finite difference scheme on a uniform Cartesian grid in a two-dimensional(2D)domain.The focus is on the impact of different discretization methods and choices of regularization parameters on the symmetry of the numerical solution.In particular,we show that using the symmetric alternating direction implicit(ADI)method for time discretization helps preserve the symmetry of the solution,compared to the(non-symmetric)ADI method.Moreover,we study the effect of the regularization by the isotropic background permeability(r>0),showing that the increased condition number of the elliptic problem due to decreasing value of r leads to loss of symmetry.We show that in this case,neither the use of the symmetric ADI method preserves the symmetry of the solution.Finally,we perform the numerical error analysis of our method making use of the Wasserstein distance.展开更多
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 ...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.展开更多
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 ...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].展开更多
In this paper, the unitarily invariant norm \\.\\ on C-mxn is used. We first discuss the problem under what case, a rectangular matrix A has minimum condition number K(A) = \\A\\ \\A(+)\\, where A(+) designates the Mo...In this paper, the unitarily invariant norm \\.\\ on C-mxn is used. We first discuss the problem under what case, a rectangular matrix A has minimum condition number K(A) = \\A\\ \\A(+)\\, where A(+) designates the Moore-Penrose inverse of A; and under what condition, a square matrix A has minimum condition number for its eigenproblem? Then we consider the second problem, i.e., optimum of K(A) = \\A\\ \\A(-1)\\(2) in error estimation.展开更多
In this paper, we investigate the condition numbers for the generalized matrix inversion and the rank deficient linear least squares problem： minx ｜｜Ax- b｜｜2, where A is an m-by-n （m ≥ n） rank deficient matrix...In this paper, we investigate the condition numbers for the generalized matrix inversion and the rank deficient linear least squares problem： minx ｜｜Ax- b｜｜2, where A is an m-by-n （m ≥ n） rank deficient matrix. We first derive an explicit expression for the condition number in the weighted Frobenius norm ｜｜ [AT,βb] ｜｜F of the data A and b, where T is a positive diagonal matrix and β is a positive scalar. We then discuss the sensitivity of the standard 2-norm condition numbers for the generalized matrix inversion and rank deficient least squares and establish relations between the condition numbers and their condition numbers called level-2 condition numbers.展开更多
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...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.展开更多
To mitigate the ill effects and obtain more accurate and stable solution of the observation equations,the so-called truncated singular-value decomposition(TSVD)method is introduced by modifying the small(nonzero)singu...To mitigate the ill effects and obtain more accurate and stable solution of the observation equations,the so-called truncated singular-value decomposition(TSVD)method is introduced by modifying the small(nonzero)singular values of the coefficients matrix.The proposed method directly disposed the ill-conditioning observation equations,which differs considerably from the traditional normal equation method.An application explains that TSVD can mitigate the ill conditioning.展开更多
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 th...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.展开更多
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 s...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.展开更多
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 es...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.展开更多
In this study,an iterative algorithm is proposed to solve the nonlinear matrix equation X+A∗eXA=In.Explicit expressions for mixed and componentwise condition numbers with their upper bounds are derived to measure the ...In this study,an iterative algorithm is proposed to solve the nonlinear matrix equation X+A∗eXA=In.Explicit expressions for mixed and componentwise condition numbers with their upper bounds are derived to measure the sensitivity of the considered nonlinear matrix equation.Comparative analysis for the derived condition numbers and the proposed algorithm are presented.The proposed iterative algorithm reduces the number of iterations significantly when incorporated with exact line searches.Componentwise condition number seems more reliable to detect the sensitivity of the considered equation than mixed condition number as validated by numerical examples.展开更多
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 ang...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.展开更多
基金Supported by the National Natural Science Foundation of China(11671060).
文摘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.
基金Supported by the National Natural Science Foundation of China(Grant No.11671060)the Fundamental Research Funds for the Central Universities(Grant No.106112015CDJXY100003)
文摘In this paper,we consider the indefinite least squares problem with quadratic constraint and its condition numbers.The conditions under which the problem has the unique solution are first presented.Then,the normwise,mixed,and componentwise condition numbers for solution and residual of this problem are derived.Numerical example is also provided to illustrate these results.
基金Supported by the National Natural Science Foundation of China(Grant No.11671060)the Fundamental Research Funds for the Central Universities(Grant No.106112015CDJXY100003)
文摘In this paper, we investigate the condition numbers for indefinite least squares problem with multiple right-hand sides. The normwise, mixed and componentwise condition numbers and the corresponding structured condition numbers are presented. The structured matrices under consideration include the linear structured matrices, such as the Toeplitz, Hankel, symmetric, and tridiagonal matrices, and the nonlinear structured matrices, such as the Vandermonde and Cauchy matrices. Numerical examples show that the structured condition numbers are tighter than the unstructured ones.
文摘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 .
基金This paper was supported by the National Natural Science Foundation of China under Grant No.61390513 and 61201225,and National Science and Technology Major Project of China under Grant No.2013ZX03003004,the Natural Science Foundation of Shanghai under Grant No.12ZR1450800,and sponsored by Shanghai Pujiang Program under Grant No.13PJD030.It was also supported by the Fundamental Research Funds for the Central Universities under Grant No.20140767,the Program for Young Excellent Talents in Tongji University under Grant No.2013KJ007,and 'Chen Guang' project supported by Shanghai Municipal Education Commission and Shanghai Education Development Foundation under Grant No.13CG18
文摘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.
文摘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.
文摘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.
基金supported by the National Natural Science Foundation of China(62203458)the Postgraduate Scientific Research Innovation Project of Hunan Province(CX20220024,CX20230014).
文摘In this paper,we propose evaluation approaches for the spatial target localization precision based on the observation matrix conditional number.Three evaluation approaches for the spatial target localization precision are derived including relative condition numbers,absolute condition numbers,and volume condition numbers by properties of vectors and matrix norms.The theoretical analysis shows that the proposed methods are the upper certainty bound of the magnification of measurement error.Meanwhile,the proposed methods are able to account for variations of localization accuracy by exploiting geometric variations in the composition between the target and measurement stations.Finally,the proposed methods perform better,compared with the traditional evaluation methods.This is of great significance for the accuracy evaluation of high-precision measuring equipment,optimization of workstation layout and geometric configuration.Simulation experiments corroborate the effectiveness of the proposed methods.
文摘We present results of numerical simulations of the tensor-valued elliptic-parabolic PDE model for biological network formation.The numerical method is based on a nonlinear finite difference scheme on a uniform Cartesian grid in a two-dimensional(2D)domain.The focus is on the impact of different discretization methods and choices of regularization parameters on the symmetry of the numerical solution.In particular,we show that using the symmetric alternating direction implicit(ADI)method for time discretization helps preserve the symmetry of the solution,compared to the(non-symmetric)ADI method.Moreover,we study the effect of the regularization by the isotropic background permeability(r>0),showing that the increased condition number of the elliptic problem due to decreasing value of r leads to loss of symmetry.We show that in this case,neither the use of the symmetric ADI method preserves the symmetry of the solution.Finally,we perform the numerical error analysis of our method making use of the Wasserstein distance.
基金supported by National Natural Science Foundation of China(Grant Nos.11001045,10926107 and 11271084)Specialized Research Fund for the Doctoral Program of Higher Education of MOE(Grant No. 20090043120008)+4 种基金Training Fund of NENU’S Scientific Innovation Project of Northeast Normal University(Grant No. NENU-STC08009)Program for Changjiang Scholars and Innovative Research Team in Universitythe Programme for Cultivating Innovative Students in Key Disciplines of Fudan University(973 Program Project)(Grant No. 2010CB327900)Doctoral Program of the Ministry of Education(Grant No.20090071110003)Shanghai Science & Technology Committee and Shanghai Education Committee(Dawn Project)
文摘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.
基金The authors would like to thank the anonymous referees for their valu- able suggestions and comments. This work was supported by the National Natural Science Foundation of China （No. 11571004 and No. 11171371）.
文摘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].
文摘In this paper, the unitarily invariant norm \\.\\ on C-mxn is used. We first discuss the problem under what case, a rectangular matrix A has minimum condition number K(A) = \\A\\ \\A(+)\\, where A(+) designates the Moore-Penrose inverse of A; and under what condition, a square matrix A has minimum condition number for its eigenproblem? Then we consider the second problem, i.e., optimum of K(A) = \\A\\ \\A(-1)\\(2) in error estimation.
基金The first author is supported by the National Natural Science Foundation of China Under grant 10471027 Shanghai Education＇Committee. The third author is partially supported by Natural ScienceEngineering Research Council of Canada and supported by Shanghai Key Laboratory of Contemporary Applied Mathematics of Fudan University during Sanzheng Qiao＇s visit.
文摘In this paper, we investigate the condition numbers for the generalized matrix inversion and the rank deficient linear least squares problem： minx ｜｜Ax- b｜｜2, where A is an m-by-n （m ≥ n） rank deficient matrix. We first derive an explicit expression for the condition number in the weighted Frobenius norm ｜｜ [AT,βb] ｜｜F of the data A and b, where T is a positive diagonal matrix and β is a positive scalar. We then discuss the sensitivity of the standard 2-norm condition numbers for the generalized matrix inversion and rank deficient least squares and establish relations between the condition numbers and their condition numbers called level-2 condition numbers.
基金supported by Aeronautical Science Foundation of China(Grant No.20081651025)
文摘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.
基金Project Supported by the National Natural Science Foundation of China for Distinguished Yound Scholars of China(40125013)NationalNatural Science Foundation of China(40074006,40274001)Natural Science Foundation of Henan Province,China(004051300,0211051100)
文摘To mitigate the ill effects and obtain more accurate and stable solution of the observation equations,the so-called truncated singular-value decomposition(TSVD)method is introduced by modifying the small(nonzero)singular values of the coefficients matrix.The proposed method directly disposed the ill-conditioning observation equations,which differs considerably from the traditional normal equation method.An application explains that TSVD can mitigate the ill conditioning.
文摘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.
基金The work is partially supported by the National Natural Science Foundation of China (No. 11801362).
文摘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.
文摘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.
文摘In this study,an iterative algorithm is proposed to solve the nonlinear matrix equation X+A∗eXA=In.Explicit expressions for mixed and componentwise condition numbers with their upper bounds are derived to measure the sensitivity of the considered nonlinear matrix equation.Comparative analysis for the derived condition numbers and the proposed algorithm are presented.The proposed iterative algorithm reduces the number of iterations significantly when incorporated with exact line searches.Componentwise condition number seems more reliable to detect the sensitivity of the considered equation than mixed condition number as validated by numerical examples.
文摘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.
