Structured generalized eigenvalue condition numbers for parameterized quasiseparable matrices

Diao, Huaian

doi:10.1007/s10543-019-00748-5

Cited by 6 publications

(4 citation statements)

References 32 publications

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“…In the following theorem, we will study the relationship between K |Ω GV |,|Ω B | (A(Ω GV ), B(Ω B )) and K |Ω QS |,|Ω B | (A(Ω QS ), B(Ω B )). The similar conclusions had been obtained for eigenvalue, generalized eigenvalue computations and linear system solving for {1;1}-quasiseparable matrices in [11,16,17], respectively. Before that, we need to review Lemma 5.1, which describes the perturbation magnitude relationship between the Givens-vector representation via tangents given in Definition 2.3 and the quasiseparable representation given in Definition 2.1.…”

Section: Proofsupporting

confidence: 80%

“…Recently, structured componentwise condition numbers for low-rank structured matrices have been introduced for eigenvalue problems [17], linear systems [16], and generalized eigenvalue problems [11] with parameterized quasiseparable matrices [43]. In this paper, we consider the structured componentwise condition numbers of the multiple right-hand side linear system (1.2) when the coefficient matrix A is a quasiseparable matrix.…”

Section: Introductionmentioning

confidence: 99%

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Structured condition number for multiple right-hand side linear systems with parameterized quasiseparable coefficient matrix

Diao

2020

Journal of Computational and Applied Mathematics

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In this paper, we consider the structured perturbation analysis for multiple right-hand side linear systems with parameterized coefficient matrix. Especially, we present the explicit expressions for structured condition numbers for multiple right-hand sides linear systems with {1;1}-quasiseparable coefficient matrix in the quasiseparable and the Givens-vector representations. In addition, the comparisons of these two condition numbers between themselves, and with respect to unstructured condition number are investigated. Moreover, the effective structured condition number for multiple righthand sides linear systems with {1;1}-quasiseparable coefficient matrix is proposed. The relationships between the effective structured condition number and structured condition numbers with respect to the quasiseparable and the Givens-vector representations are also studied. Numerical experiments show that there are situations in which the effective structured condition number can be much smaller than the unstructured ones.

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Section: Proofsupporting

confidence: 80%

Section: Introductionmentioning

confidence: 99%

Structured condition number for multiple right-hand side linear systems with parameterized quasiseparable coefficient matrix

Diao

2020

Journal of Computational and Applied Mathematics

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“…Dopico and Pomés [19,20] have introduced structured condition numbers for linear systems and eigenvalue problems based on the general quasiseparable and Givens-vector representation and obtained the explicit expressions of the structured condition number using those representations. The structured condition numbers of generalized eigenvalues and multiple right-hand side linear systems with low-rank structured matrices can be found in [16,40].…”

Section: Huaian Diao Lei LI and Qingle Mengmentioning

confidence: 99%

Structured condition numbers for Sylvester matrix equation with parameterized quasiseparable matrices

Diao,

Li,

Meng

2023

CAC

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This paper considers the structured perturbation analysis of Sylvester matrix equation with low-rank structures. When the coefficient matrix and the right-hand side of Sylvester matrix equation are {1; 1}-quasiseparable matrices, we propose the structured condition numbers and obtain explicit expressions for these structured condition numbers using the general parameter representation and the tangent-based Givens-vector representation. By comparing different condition numbers of Sylvester matrix equation, we analyze their mathematical relationship. Numerical experiments demonstrate that the structured condition number is significantly smaller than the unstructured condition number when the elements in the general representation of {1; 1}quasiseparable matrices have different scales. This suggests that structured algorithms for low-rank structured matrix equations can effectively reduce the loss of numerical solution accuracy.

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“…A numerical example in Section 8 shows that our bound can be more accurate than (5.9). Now we consider the condition number of the generalized eigenvalue λ (e.g., [13,14,18]). A definition similar to the one in [14] is first given as follows cond(λ) := lim…”

Section: Linear Perturbation Bounds For Diagonal Elements Of T and Rmentioning

confidence: 99%

Componentwise perturbation analysis for the generalized Schur decomposition

Li¹,

Wei²

2021

Preprint

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By defining two important terms called basic perturbation vectors and obtaining their linear bounds, we obtain the linear componentwise perturbation bounds for unitary factors and upper triangular factors of the generalized schur decomposition. The perturbation bounds for the diagonal elements of the upper triangular factors and the generalized invariant subspace are also derived. From the former, we present an upper bound and a condition number of the generalized eigenvalue. Furthermore, with numerical iterative method, the nonlinear componentwise perturbation bounds of the generalized Schur decomposition are also provided. Numerical examples are given to test the obtained bounds. Among them, we compare our upper bound and condition number of the generalized eigenvalue with their counterparts given in the literature. Numerical results show that they are very close to each other but our results don't need to calculate the left and right eigenvectors.

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Structured generalized eigenvalue condition numbers for parameterized quasiseparable matrices

Cited by 6 publications

References 32 publications

Structured condition number for multiple right-hand side linear systems with parameterized quasiseparable coefficient matrix

Structured condition number for multiple right-hand side linear systems with parameterized quasiseparable coefficient matrix

Structured condition numbers for Sylvester matrix equation with parameterized quasiseparable matrices

Componentwise perturbation analysis for the generalized Schur decomposition

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