The generalized inverse A T ,S of a matrix A is a {2}-inverse of A with the prescribed range T an... more The generalized inverse A T ,S of a matrix A is a {2}-inverse of A with the prescribed range T and null space S. A representation for the generalized inverse A T ,S has been recently developed with the condition σ(GA|T ) ⊂ (0,∞), where G is a matrix with R(G) = T and N(G) = S. In this note, we remove the above condition. Three types of iterative methods for A T ,S are presented if σ(GA|T ) is a subset of the open right half-plane and they are extensions of existing computational procedures of A T ,S , including special cases such as the weighted Moore-Penrose inverse AM,N and the Drazin inverse AD . Numerical examples are given to illustrate our results.
This paper is contributed to characterizations of positive semi-definite Hankel tensors. We first... more This paper is contributed to characterizations of positive semi-definite Hankel tensors. We first show that if a lower-order Hankel tensor is positive semi-definite, then its associated higher-order Hankel tensor with the same generating vector, where the higher order is a multiple of the lower order, is also positive semi-definite. This is proved via a convolution formula, which is an analytic interpretation of Hankel tensor-vector products. Then we investigate those Hankel tensors whose associated Hankel matrices are positive semi-definite, called strong Hankel tensors. It is verified that each strong Hankel tensor admits an augmented Vandermonde decomposition with all positive coefficients.
The main purpose of this note is to investigate some kinds of nonlinear complementarity problems ... more The main purpose of this note is to investigate some kinds of nonlinear complementarity problems (NCP). For the structured tensors, such as, symmetric positive definite tensors and copositive tensors, we derive the existence theorems on a solution of these kinds of nonlinear complementarity problems. We prove that a unique solution of the NCP exists under the condition of diagonalizable tensors.
This paper is contributed to a fast algorithm for Hankel tensor-vector products. For this purpose... more This paper is contributed to a fast algorithm for Hankel tensor-vector products. For this purpose, we first discuss a special class of Hankel tensors that can be diagonalized by the Fourier matrix, which is called \emph{anti-circulant} tensors. Then we obtain a fast algorithm for Hankel tensor-vector products by embedding a Hankel tensor into a larger anti-circulant tensor. The computational complexity is about $\mathcal{O}(m^2 n \log mn)$ for a square Hankel tensor of order $m$ and dimension $n$, and the numerical examples also show the efficiency of this scheme. Moreover, the block version for multi-level block Hankel tensors is discussed as well. Finally, we apply the fast algorithm to exponential data fitting and the block version to 2D exponential data fitting for higher performance.
Combining the modified matrix-vector equation approach with the technique of Lyapunov majorant fu... more Combining the modified matrix-vector equation approach with the technique of Lyapunov majorant function and the Banach fixed point principle, we obtain new rigorous perturbation bounds for the LU and QR factorizations with normwise or componentwise perturbations in the given matrix, where the componentwise perturbations have the form of backward error resulting from the standard factorization algorithms. Each of the new rigorous perturbation bounds is a rigorous version of the first-order perturbation bound derived by the matrix-vector equation approach in the literature, and we present their explicit expressions. These bounds improve the results given by Chang & Stehl\'{e} (2010). Moreover, we derive new sharper first-order perturbation bounds including two optimal ones for the LU factorization, and provide the explicit expressions of the optimal first-order perturbation bounds for the LU and QR factorizations.
We present a stability analysis of Gohberg-Semencul-Trench type formulae for the Moore-Penrose an... more We present a stability analysis of Gohberg-Semencul-Trench type formulae for the Moore-Penrose and group inverses of singular Toeplitz matrices. We develop a fast algorithm for the computation of the Moore-Penrose inverse based on a Gohberg-Semencul-Trench type formula and the LSQR method. For the group inverse, the DGMRES method is used to perform the fast computation. Numerical tests show that the fast algorithms designed here are at least as good as the known Newton iteration.
In this paper, we present new perturbation analysis and randomized algorithms for the total least... more In this paper, we present new perturbation analysis and randomized algorithms for the total least squares (TLS) problem. Our new perturbation results are sharper than the earlier ones. We prove that the condition numbers obtained by Zhou et al. [Numer. Algorithm, 51 (2009), pp. 381-399], Barboulin and Gratton [SIAM J. Matrix Anal. Appl., 32 (2011), pp. 685-699], Li and Jia [Linear Algebra Appl., 435 (2011), pp. 674-686] are mathematically equivalent. Statistical condition estimates (SCE) are applied to the TLS problem. Motivated by the recently popular probabilistic algorithms for low-rank approximations, we develop randomized algorithms for the TLS and the truncated total least squares (TTLS) solutions of large-scale discrete ill-posed problems, which can greatly reduce the computational time and still keep good accuracy. Its efficiency is well demonstrated by numerical examples.
ABSTRACT We consider the numerical solution of the rational algebraic Riccati equations in $\math... more ABSTRACT We consider the numerical solution of the rational algebraic Riccati equations in $\mathbb{R}^n$, arising from stochastic optimal control in continuous- and discrete-time. Applying the homotopy method, we continue from the stabilizing solutions of the deterministic algebraic Riccati equations, which are readily available. The associated differential equations require the solutions of some generalized Lyapunov or Stein equations, which can be solved by the generalized Smith methods, of $O(n^3)$ computational complexity and $O(n^2)$ memory requirement. For large-scale problems, the sparsity and structures in the relevant matrices further improve the efficiency of our algorithms. In comparison, the alternative (modified) Newton's methods require a difficult initial stabilization step. Some illustrative numerical examples are provided.
We present perturbation bounds for the relative error in the eigenval-ues of diagonalizable and s... more We present perturbation bounds for the relative error in the eigenval-ues of diagonalizable and singular matrices. These are extensions of the related results on non-singular matrices.
Chemical composition, cholesterol levels, fatty acid profile, meat taste, and quality parameters ... more Chemical composition, cholesterol levels, fatty acid profile, meat taste, and quality parameters were evaluated in 48 buck kids from goats of the Guanzhong Dairy breed (Group G) and their crosses (Group F1: 1/2 Boermale symbolx1/2 Guanzhong Dairyfemale symbol; Group F2: 3/4 Boermale symbolx1/4 Guanzhong Dairyfemale symbol; Group F3: 7/8 Boermale symbolx1/8 Guanzhong Dairyfemale symbol) at different ages of slaughter (6, 8 and 10 months). Results indicated that grading hybridization (P<0.05) affected meat nutritive value. The muscle of hybrid goats had lower crude fat and cholesterol, higher crude protein, and greater proportion of C18:2 and C18:3 than that of Group G at each age. Group F1 goats had better (P<0.05) desirable fatty acid (DFA) and polyunsaturated fatty acid (PUFA) to saturated fatty acid (SFA) ratios and greater (C18:0+C18:1/C16:0) ratios (P<0.01) than those of the other genotypes. Furthermore, the muscles of hybrid goats were tenderer and juicier compared to ...
ABSTRACT Some new rigorous perturbation bounds for the Cholesky-like factorization of skew-symmet... more ABSTRACT Some new rigorous perturbation bounds for the Cholesky-like factorization of skew-symmetric matrix are obtained when the original matrix has the normwise or componentwise perturbations, where the componentwise perturbation has the form of backward rounding error for the Cholesky-like factorization algorithm. These bounds can be much tighter than the existing one. Two numerical examples are provided to illustrate these results.
The generalized inverse A T ,S of a matrix A is a {2}-inverse of A with the prescribed range T an... more The generalized inverse A T ,S of a matrix A is a {2}-inverse of A with the prescribed range T and null space S. A representation for the generalized inverse A T ,S has been recently developed with the condition σ(GA|T ) ⊂ (0,∞), where G is a matrix with R(G) = T and N(G) = S. In this note, we remove the above condition. Three types of iterative methods for A T ,S are presented if σ(GA|T ) is a subset of the open right half-plane and they are extensions of existing computational procedures of A T ,S , including special cases such as the weighted Moore-Penrose inverse AM,N and the Drazin inverse AD . Numerical examples are given to illustrate our results.
This paper is contributed to characterizations of positive semi-definite Hankel tensors. We first... more This paper is contributed to characterizations of positive semi-definite Hankel tensors. We first show that if a lower-order Hankel tensor is positive semi-definite, then its associated higher-order Hankel tensor with the same generating vector, where the higher order is a multiple of the lower order, is also positive semi-definite. This is proved via a convolution formula, which is an analytic interpretation of Hankel tensor-vector products. Then we investigate those Hankel tensors whose associated Hankel matrices are positive semi-definite, called strong Hankel tensors. It is verified that each strong Hankel tensor admits an augmented Vandermonde decomposition with all positive coefficients.
The main purpose of this note is to investigate some kinds of nonlinear complementarity problems ... more The main purpose of this note is to investigate some kinds of nonlinear complementarity problems (NCP). For the structured tensors, such as, symmetric positive definite tensors and copositive tensors, we derive the existence theorems on a solution of these kinds of nonlinear complementarity problems. We prove that a unique solution of the NCP exists under the condition of diagonalizable tensors.
This paper is contributed to a fast algorithm for Hankel tensor-vector products. For this purpose... more This paper is contributed to a fast algorithm for Hankel tensor-vector products. For this purpose, we first discuss a special class of Hankel tensors that can be diagonalized by the Fourier matrix, which is called \emph{anti-circulant} tensors. Then we obtain a fast algorithm for Hankel tensor-vector products by embedding a Hankel tensor into a larger anti-circulant tensor. The computational complexity is about $\mathcal{O}(m^2 n \log mn)$ for a square Hankel tensor of order $m$ and dimension $n$, and the numerical examples also show the efficiency of this scheme. Moreover, the block version for multi-level block Hankel tensors is discussed as well. Finally, we apply the fast algorithm to exponential data fitting and the block version to 2D exponential data fitting for higher performance.
Combining the modified matrix-vector equation approach with the technique of Lyapunov majorant fu... more Combining the modified matrix-vector equation approach with the technique of Lyapunov majorant function and the Banach fixed point principle, we obtain new rigorous perturbation bounds for the LU and QR factorizations with normwise or componentwise perturbations in the given matrix, where the componentwise perturbations have the form of backward error resulting from the standard factorization algorithms. Each of the new rigorous perturbation bounds is a rigorous version of the first-order perturbation bound derived by the matrix-vector equation approach in the literature, and we present their explicit expressions. These bounds improve the results given by Chang & Stehl\'{e} (2010). Moreover, we derive new sharper first-order perturbation bounds including two optimal ones for the LU factorization, and provide the explicit expressions of the optimal first-order perturbation bounds for the LU and QR factorizations.
We present a stability analysis of Gohberg-Semencul-Trench type formulae for the Moore-Penrose an... more We present a stability analysis of Gohberg-Semencul-Trench type formulae for the Moore-Penrose and group inverses of singular Toeplitz matrices. We develop a fast algorithm for the computation of the Moore-Penrose inverse based on a Gohberg-Semencul-Trench type formula and the LSQR method. For the group inverse, the DGMRES method is used to perform the fast computation. Numerical tests show that the fast algorithms designed here are at least as good as the known Newton iteration.
In this paper, we present new perturbation analysis and randomized algorithms for the total least... more In this paper, we present new perturbation analysis and randomized algorithms for the total least squares (TLS) problem. Our new perturbation results are sharper than the earlier ones. We prove that the condition numbers obtained by Zhou et al. [Numer. Algorithm, 51 (2009), pp. 381-399], Barboulin and Gratton [SIAM J. Matrix Anal. Appl., 32 (2011), pp. 685-699], Li and Jia [Linear Algebra Appl., 435 (2011), pp. 674-686] are mathematically equivalent. Statistical condition estimates (SCE) are applied to the TLS problem. Motivated by the recently popular probabilistic algorithms for low-rank approximations, we develop randomized algorithms for the TLS and the truncated total least squares (TTLS) solutions of large-scale discrete ill-posed problems, which can greatly reduce the computational time and still keep good accuracy. Its efficiency is well demonstrated by numerical examples.
ABSTRACT We consider the numerical solution of the rational algebraic Riccati equations in $\math... more ABSTRACT We consider the numerical solution of the rational algebraic Riccati equations in $\mathbb{R}^n$, arising from stochastic optimal control in continuous- and discrete-time. Applying the homotopy method, we continue from the stabilizing solutions of the deterministic algebraic Riccati equations, which are readily available. The associated differential equations require the solutions of some generalized Lyapunov or Stein equations, which can be solved by the generalized Smith methods, of $O(n^3)$ computational complexity and $O(n^2)$ memory requirement. For large-scale problems, the sparsity and structures in the relevant matrices further improve the efficiency of our algorithms. In comparison, the alternative (modified) Newton&#39;s methods require a difficult initial stabilization step. Some illustrative numerical examples are provided.
We present perturbation bounds for the relative error in the eigenval-ues of diagonalizable and s... more We present perturbation bounds for the relative error in the eigenval-ues of diagonalizable and singular matrices. These are extensions of the related results on non-singular matrices.
Chemical composition, cholesterol levels, fatty acid profile, meat taste, and quality parameters ... more Chemical composition, cholesterol levels, fatty acid profile, meat taste, and quality parameters were evaluated in 48 buck kids from goats of the Guanzhong Dairy breed (Group G) and their crosses (Group F1: 1/2 Boermale symbolx1/2 Guanzhong Dairyfemale symbol; Group F2: 3/4 Boermale symbolx1/4 Guanzhong Dairyfemale symbol; Group F3: 7/8 Boermale symbolx1/8 Guanzhong Dairyfemale symbol) at different ages of slaughter (6, 8 and 10 months). Results indicated that grading hybridization (P<0.05) affected meat nutritive value. The muscle of hybrid goats had lower crude fat and cholesterol, higher crude protein, and greater proportion of C18:2 and C18:3 than that of Group G at each age. Group F1 goats had better (P<0.05) desirable fatty acid (DFA) and polyunsaturated fatty acid (PUFA) to saturated fatty acid (SFA) ratios and greater (C18:0+C18:1/C16:0) ratios (P<0.01) than those of the other genotypes. Furthermore, the muscles of hybrid goats were tenderer and juicier compared to ...
ABSTRACT Some new rigorous perturbation bounds for the Cholesky-like factorization of skew-symmet... more ABSTRACT Some new rigorous perturbation bounds for the Cholesky-like factorization of skew-symmetric matrix are obtained when the original matrix has the normwise or componentwise perturbations, where the componentwise perturbation has the form of backward rounding error for the Cholesky-like factorization algorithm. These bounds can be much tighter than the existing one. Two numerical examples are provided to illustrate these results.
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