The problem of joint eigenvalue estimation for the non-defective commuting set of matrices A is addressed. A procedure revealing the joint eigenstructure by simultaneous diagonalization of. A with simultaneous Schur d...The problem of joint eigenvalue estimation for the non-defective commuting set of matrices A is addressed. A procedure revealing the joint eigenstructure by simultaneous diagonalization of. A with simultaneous Schur decomposition （SSD） and balance procedure alternately is proposed for performance considerations and also for overcoming the convergence difficulties of previous methods based only on simultaneous Schur form and unitary transformations, it is shown that the SSD procedure can be well incorporated with the balancing algorithm in a pingpong manner, i. e., each optimizes a cost function and at the same time serves as an acceleration procedure for the other. Under mild assumptions, the convergence of the two cost functions alternately optimized, i. e., the norm of A and the norm of the left-lower part of A is proved. Numerical experiments are conducted in a multi-dimensional harmonic retrieval application and suggest that the presented method converges considerably faster than the methods based on only unitary transformation for matrices which are not near to normality.展开更多
Direct numerical simulations are carried out with different disturbance forms introduced into the inlet of a flat plate boundary layer with the Mach number 4.5. According to the biorthogonal eigenfunction system of th...Direct numerical simulations are carried out with different disturbance forms introduced into the inlet of a flat plate boundary layer with the Mach number 4.5. According to the biorthogonal eigenfunction system of the linearized Navier-Stokes equations and the adjoint equations, the decomposition of the direct numerical simulation results into the discrete normal mode is easily realized. The decomposition coefficients can be solved by doing the inner product between the numerical results and the eigenfunctions of the adjoint equations. For the quadratic polynomial eigenvalue problem, the inner product operator is given in a simple form, and it is extended to an Nth-degree polynomial eigenvalue problem. The examples illustrate that the simplified mode decomposition is available to analyze direct numerical simulation results.展开更多
We demonstrate that, when computing the LDU decomposition (a typical example of a direct solution method), it is possible to obtain the derivative of a determinant with respect to an eigenvalue of a non-symmetric matr...We demonstrate that, when computing the LDU decomposition (a typical example of a direct solution method), it is possible to obtain the derivative of a determinant with respect to an eigenvalue of a non-symmetric matrix. Our proposed method augments an LDU decomposition program with an additional routine to obtain a program for easily evaluating the derivative of a determinant with respect to an eigenvalue. The proposed method follows simply from the process of solving simultaneous linear equations and is particularly effective for band matrices, for which memory requirements are significantly reduced compared to those for dense matrices. We discuss the theory underlying our proposed method and present detailed algorithms for implementing it.展开更多
In this paper, we obtain a formula for the derivative of a determinant with respect to an eigenvalue in the modified Cholesky decomposition of a symmetric matrix, a characteristic example of a direct solution method i...In this paper, we obtain a formula for the derivative of a determinant with respect to an eigenvalue in the modified Cholesky decomposition of a symmetric matrix, a characteristic example of a direct solution method in computational linear algebra. We apply our proposed formula to a technique used in nonlinear finite-element methods and discuss methods for determining singular points, such as bifurcation points and limit points. In our proposed method, the increment in arc length (or other relevant quantities) may be determined automatically, allowing a reduction in the number of basic parameters. The method is particularly effective for banded matrices, which allow a significant reduction in memory requirements as compared to dense matrices. We discuss the theoretical foundations of our proposed method, present algorithms and programs that implement it, and conduct numerical experiments to investigate its effectiveness.展开更多
A modified multiple-component scattering power decomposition for analyzing polarimetric synthetic aperture radar(PolSAR)data is proposed.The modified decomposition involves two distinct steps.Firstly,ei⁃genvectors of ...A modified multiple-component scattering power decomposition for analyzing polarimetric synthetic aperture radar(PolSAR)data is proposed.The modified decomposition involves two distinct steps.Firstly,ei⁃genvectors of the coherency matrix are used to modify the scattering models.Secondly,the entropy and anisotro⁃py of targets are used to improve the volume scattering power.With the guarantee of high double-bounce scatter⁃ing power in the urban areas,the proposed algorithm effectively improves the volume scattering power of vegeta⁃tion areas.The efficacy of the modified multiple-component scattering power decomposition is validated using ac⁃tual AIRSAR PolSAR data.The scattering power obtained through decomposing the original coherency matrix and the coherency matrix after orientation angle compensation is compared with three algorithms.Results from the experiment demonstrate that the proposed decomposition yields more effective scattering power for different PolSAR data sets.展开更多
In order to extract the fault feature of the bearing effectively and prevent the impact components caused by bearing damage being interfered with by discrete frequency components and background noise,a method of fault...In order to extract the fault feature of the bearing effectively and prevent the impact components caused by bearing damage being interfered with by discrete frequency components and background noise,a method of fault feature extraction based on cepstrum pre-whitening(CPW)and a quantitative law of symplectic geometry mode decomposition(SGMD)is proposed.First,CPW is performed on the original signal to enhance the impact feature of bearing fault and remove the periodic frequency components from complex vibration signals.The pre-whitening signal contains only background noise and non-stationary shock caused by damage.Secondly,a quantitative law that the number of effective eigenvalues of the Hamilton matrix is twice the number of frequency components in the signal during SGMD is found,and the quantitative law is verified by simulation and theoretical derivation.Finally,the trajectory matrix of the pre-whitening signal is constructed and SGMD is performed.According to the quantitative law,the corresponding feature vector is selected to reconstruct the signal.The Hilbert envelope spectrum analysis is performed to extract fault features.Simulation analysis and application examples prove that the proposed method can clearly extract the fault feature of bearings.展开更多
A new hybrid Freeman/eigenvalue decomposition based on the orientation angle compensation and the various extended volume models for polarimetric synthetic aperture radar(PolSAR) data are presented. There are three st...A new hybrid Freeman/eigenvalue decomposition based on the orientation angle compensation and the various extended volume models for polarimetric synthetic aperture radar(PolSAR) data are presented. There are three steps in the novel version of the three-component model-based decomposition.Firstly, two special unitary transform matrices are applied on the coherency matrix for deorientation to decrease the correlation between the co-polarized term and the cross-polarized term.Secondly, two new conditions are proposed to distinguish the manmade structures and the nature media after the orientation angle compensation. Finally, in order to adapt to the scattering properties of different media, five different volume scattering models are used to decompose the coherency matrix. These new conditions pre-resolves man-made structures, which is beneficial to the subsequent selection of a more suitable volume scattering model.Fully PolSAR data on San Francisco are used in the experiments to prove the efficiency of the proposed hybrid Freeman/eigenvalue decomposition.展开更多
A new method for estimating the bounds of eigenvalues ispresented. In order to show that the method proposed is as effectiveas Qiu's an undamping spring-mass system with 5 nodes and 5 degrees ofreedom is given. To...A new method for estimating the bounds of eigenvalues ispresented. In order to show that the method proposed is as effectiveas Qiu's an undamping spring-mass system with 5 nodes and 5 degrees ofreedom is given. To illustrate that the present method can beapplied to structures which cannot be treated by non-negativedecomposition, a plane frame with 202 nodes and 357 beam elements isgiven. The results show that the present method is effective forestimating the bounds of eigenvalues and is more common than Qiu's.展开更多
The correlation matrix, which is widely used in eigenvalue decomposition （EVD） or singular value decomposition （SVD）, usually can be denoted by R = E[yiy＇i]. A novel method for constructing the correlation matrix...The correlation matrix, which is widely used in eigenvalue decomposition （EVD） or singular value decomposition （SVD）, usually can be denoted by R = E[yiy＇i]. A novel method for constructing the correlation matrix R is proposed. The proposed algorithm can improve the resolving power of the signal eigenvalues and overcomes the shortcomings of the traditional subspace methods, which cannot be applied to low SNR. Then the proposed method is applied to the direct sequence spread spectrum （DSSS） signal＇s signature sequence estimation. The performance of the proposed algorithm is analyzed, and some illustrative simulation results are presented.展开更多
In this paper, we present a unified approach to decomposing a special class of block tridiagonal matrices <i>K</i> (<i>α</i> ,<i>β</i> ) into block diagonal matrices using similar...In this paper, we present a unified approach to decomposing a special class of block tridiagonal matrices <i>K</i> (<i>α</i> ,<i>β</i> ) into block diagonal matrices using similarity transformations. The matrices <i>K</i> (<i>α</i> ,<i>β</i> )∈ <i>R</i><sup><i>pq</i>× <i>pq</i></sup> are of the form <i>K</i> (<i>α</i> ,<i>β</i> = block-tridiag[<i>β B</i>,<i>A</i>,<i>α B</i>] for three special pairs of (<i>α</i> ,<i>β</i> ): <i>K</i> (1,1), <i>K</i> (1,2) and <i>K</i> (2,2) , where the matrices <i>A</i> and <i>B</i>, <i>A</i>, <i>B</i>∈ <i>R</i><sup><i>p</i>× <i>q</i></sup> , are general square matrices. The decomposed block diagonal matrices <img src="Edit_00717830-3b3b-4856-8ecd-a9db983fef19.png" width="15" height="15" alt="" />(<i>α</i> ,<i>β</i> ) for the three cases are all of the form: <img src="Edit_71ffcd27-6acc-4922-b5e2-f4be15b9b8dc.png" width="15" height="15" alt="" />(<i>α</i> ,<i>β</i> ) = <i>D</i><sub>1</sub> (<i>α</i> ,<i>β</i> ) ⊕ <i>D</i><sub>2</sub> (<i>α</i> ,<i>β</i> ) ⊕---⊕ <i>D</i><sub>q</sub> (<i>α</i> ,<i>β</i> ) , where <i>D<sub>k</sub></i> (<i>α</i> ,<i>β</i> ) = <i>A</i>+ 2cos ( <i>θ<sub>k</sub></i> (<i>α</i> ,<i>β</i> )) <i>B</i>, in which <i>θ<sub>k</sub></i> (<i>α</i> ,<i>β</i> ) , k = 1,2, --- q , depend on the values of <i>α</i> and <i>β</i>. Our decomposition method is closely related to the classical fast Poisson solver using Fourier analysis. Unlike the fast Poisson solver, our approach decomposes <i>K</i> (<i>α</i> ,<i>β</i> ) into <i>q</i> diagonal blocks, instead of <i>p</i> blocks. Furthermore, our proposed approach does not require matrices <i>A</i> and <i>B</i> to be symmetric and commute, and employs only the eigenvectors of the tridiagonal matrix <i>T</i> (<i>α</i> ,<i>β</i> ) = tridiag[<i>β b</i>, <i>a</i>,<i>αb</i>] in a block form, where <i>a</i> and <i>b</i> are scalars. The transformation matrices, their inverses, and the explicit form of the decomposed block diagonal matrices are derived in this paper. Numerical examples and experiments are also presented to demonstrate the validity and usefulness of the approach. Due to the decoupled nature of the decomposed matrices, this approach lends itself to parallel and distributed computations for solving both linear systems and eigenvalue problems using multiprocessors.展开更多
We present in this paper a new method for solving polynomial eigenvalue problem. We give methods that decompose a skew-Hamiltonian matrix using Cholesky like-decomposition. We transform first the polynomial eigenvalue...We present in this paper a new method for solving polynomial eigenvalue problem. We give methods that decompose a skew-Hamiltonian matrix using Cholesky like-decomposition. We transform first the polynomial eigenvalue problem to an equivalent skew-Hamiltonian/Hamiltonian pencil. This process is known as linearization. Decomposition of the skew-Hamiltonian matrix is the fundamental step to convert a structured polynomial eigenvalue problem into a standard Hamiltonian eigenproblem. Numerical examples are given.展开更多
基金The National Natural Science Foundation of China(No.60572072,60496311),the National High Technology Researchand Development Program of China (863Program ) ( No.2003AA123310),the International Cooperation Project on Beyond 3G Mobile of China (No.2005DFA10360).
文摘The problem of joint eigenvalue estimation for the non-defective commuting set of matrices A is addressed. A procedure revealing the joint eigenstructure by simultaneous diagonalization of. A with simultaneous Schur decomposition （SSD） and balance procedure alternately is proposed for performance considerations and also for overcoming the convergence difficulties of previous methods based only on simultaneous Schur form and unitary transformations, it is shown that the SSD procedure can be well incorporated with the balancing algorithm in a pingpong manner, i. e., each optimizes a cost function and at the same time serves as an acceleration procedure for the other. Under mild assumptions, the convergence of the two cost functions alternately optimized, i. e., the norm of A and the norm of the left-lower part of A is proved. Numerical experiments are conducted in a multi-dimensional harmonic retrieval application and suggest that the presented method converges considerably faster than the methods based on only unitary transformation for matrices which are not near to normality.
基金supported by the National Natural Science Foundation of China(Nos.1133200711202147+2 种基金and 9216111)the Specialized Research Fund for the Doctoral Program of Higher Education(No.20120032120007)the Open Fund from State Key Laboratory of Aerodynamics(Nos.SKLA201201 and SKLA201301)
文摘Direct numerical simulations are carried out with different disturbance forms introduced into the inlet of a flat plate boundary layer with the Mach number 4.5. According to the biorthogonal eigenfunction system of the linearized Navier-Stokes equations and the adjoint equations, the decomposition of the direct numerical simulation results into the discrete normal mode is easily realized. The decomposition coefficients can be solved by doing the inner product between the numerical results and the eigenfunctions of the adjoint equations. For the quadratic polynomial eigenvalue problem, the inner product operator is given in a simple form, and it is extended to an Nth-degree polynomial eigenvalue problem. The examples illustrate that the simplified mode decomposition is available to analyze direct numerical simulation results.
文摘We demonstrate that, when computing the LDU decomposition (a typical example of a direct solution method), it is possible to obtain the derivative of a determinant with respect to an eigenvalue of a non-symmetric matrix. Our proposed method augments an LDU decomposition program with an additional routine to obtain a program for easily evaluating the derivative of a determinant with respect to an eigenvalue. The proposed method follows simply from the process of solving simultaneous linear equations and is particularly effective for band matrices, for which memory requirements are significantly reduced compared to those for dense matrices. We discuss the theory underlying our proposed method and present detailed algorithms for implementing it.
文摘In this paper, we obtain a formula for the derivative of a determinant with respect to an eigenvalue in the modified Cholesky decomposition of a symmetric matrix, a characteristic example of a direct solution method in computational linear algebra. We apply our proposed formula to a technique used in nonlinear finite-element methods and discuss methods for determining singular points, such as bifurcation points and limit points. In our proposed method, the increment in arc length (or other relevant quantities) may be determined automatically, allowing a reduction in the number of basic parameters. The method is particularly effective for banded matrices, which allow a significant reduction in memory requirements as compared to dense matrices. We discuss the theoretical foundations of our proposed method, present algorithms and programs that implement it, and conduct numerical experiments to investigate its effectiveness.
基金Supported by the National Natural Science Foundation of China(62376214)the Natural Science Basic Research Program of Shaanxi(2023-JC-YB-533)Foundation of Ministry of Education Key Lab.of Cognitive Radio and Information Processing(Guilin University of Electronic Technology)(CRKL200203)。
文摘A modified multiple-component scattering power decomposition for analyzing polarimetric synthetic aperture radar(PolSAR)data is proposed.The modified decomposition involves two distinct steps.Firstly,ei⁃genvectors of the coherency matrix are used to modify the scattering models.Secondly,the entropy and anisotro⁃py of targets are used to improve the volume scattering power.With the guarantee of high double-bounce scatter⁃ing power in the urban areas,the proposed algorithm effectively improves the volume scattering power of vegeta⁃tion areas.The efficacy of the modified multiple-component scattering power decomposition is validated using ac⁃tual AIRSAR PolSAR data.The scattering power obtained through decomposing the original coherency matrix and the coherency matrix after orientation angle compensation is compared with three algorithms.Results from the experiment demonstrate that the proposed decomposition yields more effective scattering power for different PolSAR data sets.
基金Supported by the National Basic Research Program(973 Program)of China(2013CB329402)the National Natural Science Foundation of China(61473215,61472306,61271302,61272282,61272176)
基金The National Natural Science Foundation of China(No.52075095).
文摘In order to extract the fault feature of the bearing effectively and prevent the impact components caused by bearing damage being interfered with by discrete frequency components and background noise,a method of fault feature extraction based on cepstrum pre-whitening(CPW)and a quantitative law of symplectic geometry mode decomposition(SGMD)is proposed.First,CPW is performed on the original signal to enhance the impact feature of bearing fault and remove the periodic frequency components from complex vibration signals.The pre-whitening signal contains only background noise and non-stationary shock caused by damage.Secondly,a quantitative law that the number of effective eigenvalues of the Hamilton matrix is twice the number of frequency components in the signal during SGMD is found,and the quantitative law is verified by simulation and theoretical derivation.Finally,the trajectory matrix of the pre-whitening signal is constructed and SGMD is performed.According to the quantitative law,the corresponding feature vector is selected to reconstruct the signal.The Hilbert envelope spectrum analysis is performed to extract fault features.Simulation analysis and application examples prove that the proposed method can clearly extract the fault feature of bearings.
基金supported by the National Natural Science Foundation of China(41704118 11747032)+2 种基金the Natural Science Basic Research Plan in Shaanxi Province of China(2017JQ6065 2017JQ4017)the Special Scientific Research Project of Shaanxi Provincial Education Department(18JK0549)
文摘A new hybrid Freeman/eigenvalue decomposition based on the orientation angle compensation and the various extended volume models for polarimetric synthetic aperture radar(PolSAR) data are presented. There are three steps in the novel version of the three-component model-based decomposition.Firstly, two special unitary transform matrices are applied on the coherency matrix for deorientation to decrease the correlation between the co-polarized term and the cross-polarized term.Secondly, two new conditions are proposed to distinguish the manmade structures and the nature media after the orientation angle compensation. Finally, in order to adapt to the scattering properties of different media, five different volume scattering models are used to decompose the coherency matrix. These new conditions pre-resolves man-made structures, which is beneficial to the subsequent selection of a more suitable volume scattering model.Fully PolSAR data on San Francisco are used in the experiments to prove the efficiency of the proposed hybrid Freeman/eigenvalue decomposition.
基金the National Natural Science Foundation (No.19872028)the Mechanical Technology Development Foundation of China
文摘A new method for estimating the bounds of eigenvalues ispresented. In order to show that the method proposed is as effectiveas Qiu's an undamping spring-mass system with 5 nodes and 5 degrees ofreedom is given. To illustrate that the present method can beapplied to structures which cannot be treated by non-negativedecomposition, a plane frame with 202 nodes and 357 beam elements isgiven. The results show that the present method is effective forestimating the bounds of eigenvalues and is more common than Qiu's.
文摘The correlation matrix, which is widely used in eigenvalue decomposition （EVD） or singular value decomposition （SVD）, usually can be denoted by R = E[yiy＇i]. A novel method for constructing the correlation matrix R is proposed. The proposed algorithm can improve the resolving power of the signal eigenvalues and overcomes the shortcomings of the traditional subspace methods, which cannot be applied to low SNR. Then the proposed method is applied to the direct sequence spread spectrum （DSSS） signal＇s signature sequence estimation. The performance of the proposed algorithm is analyzed, and some illustrative simulation results are presented.
文摘In this paper, we present a unified approach to decomposing a special class of block tridiagonal matrices <i>K</i> (<i>α</i> ,<i>β</i> ) into block diagonal matrices using similarity transformations. The matrices <i>K</i> (<i>α</i> ,<i>β</i> )∈ <i>R</i><sup><i>pq</i>× <i>pq</i></sup> are of the form <i>K</i> (<i>α</i> ,<i>β</i> = block-tridiag[<i>β B</i>,<i>A</i>,<i>α B</i>] for three special pairs of (<i>α</i> ,<i>β</i> ): <i>K</i> (1,1), <i>K</i> (1,2) and <i>K</i> (2,2) , where the matrices <i>A</i> and <i>B</i>, <i>A</i>, <i>B</i>∈ <i>R</i><sup><i>p</i>× <i>q</i></sup> , are general square matrices. The decomposed block diagonal matrices <img src="Edit_00717830-3b3b-4856-8ecd-a9db983fef19.png" width="15" height="15" alt="" />(<i>α</i> ,<i>β</i> ) for the three cases are all of the form: <img src="Edit_71ffcd27-6acc-4922-b5e2-f4be15b9b8dc.png" width="15" height="15" alt="" />(<i>α</i> ,<i>β</i> ) = <i>D</i><sub>1</sub> (<i>α</i> ,<i>β</i> ) ⊕ <i>D</i><sub>2</sub> (<i>α</i> ,<i>β</i> ) ⊕---⊕ <i>D</i><sub>q</sub> (<i>α</i> ,<i>β</i> ) , where <i>D<sub>k</sub></i> (<i>α</i> ,<i>β</i> ) = <i>A</i>+ 2cos ( <i>θ<sub>k</sub></i> (<i>α</i> ,<i>β</i> )) <i>B</i>, in which <i>θ<sub>k</sub></i> (<i>α</i> ,<i>β</i> ) , k = 1,2, --- q , depend on the values of <i>α</i> and <i>β</i>. Our decomposition method is closely related to the classical fast Poisson solver using Fourier analysis. Unlike the fast Poisson solver, our approach decomposes <i>K</i> (<i>α</i> ,<i>β</i> ) into <i>q</i> diagonal blocks, instead of <i>p</i> blocks. Furthermore, our proposed approach does not require matrices <i>A</i> and <i>B</i> to be symmetric and commute, and employs only the eigenvectors of the tridiagonal matrix <i>T</i> (<i>α</i> ,<i>β</i> ) = tridiag[<i>β b</i>, <i>a</i>,<i>αb</i>] in a block form, where <i>a</i> and <i>b</i> are scalars. The transformation matrices, their inverses, and the explicit form of the decomposed block diagonal matrices are derived in this paper. Numerical examples and experiments are also presented to demonstrate the validity and usefulness of the approach. Due to the decoupled nature of the decomposed matrices, this approach lends itself to parallel and distributed computations for solving both linear systems and eigenvalue problems using multiprocessors.
文摘We present in this paper a new method for solving polynomial eigenvalue problem. We give methods that decompose a skew-Hamiltonian matrix using Cholesky like-decomposition. We transform first the polynomial eigenvalue problem to an equivalent skew-Hamiltonian/Hamiltonian pencil. This process is known as linearization. Decomposition of the skew-Hamiltonian matrix is the fundamental step to convert a structured polynomial eigenvalue problem into a standard Hamiltonian eigenproblem. Numerical examples are given.
