Bounds for norms of the matrix inverse and the smallest singular value

Morača, Nenad

doi:10.1016/j.laa.2007.12.026

Cited by 35 publications

(22 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Rg i j m θ Rg i j and η * i m η i , then by using triangular inequality we have (e) Using (a)-(d), considering ε as a design parameter and [51,52], it can be easily deduced that u and u ε defined respectively in (26) and ( Proof. Consider the following Lyapunov function candidate:…”

Section: The Proposed Observer-based Methodsmentioning

confidence: 99%

Observer-based adaptive interval type-2 fuzzy control of uncertain MIMO nonlinear systems with unknown asymmetric saturation actuators

Shahnazi

2016

Neurocomputing

View full text Add to dashboard Cite

Section: The Proposed Observer-based Methodsmentioning

confidence: 99%

Observer-based adaptive interval type-2 fuzzy control of uncertain MIMO nonlinear systems with unknown asymmetric saturation actuators

Shahnazi

2016

Neurocomputing

View full text Add to dashboard Cite

“…Specifically, row j + ( k − 1) J includes the diagonal term

1 MathClass-bin+ (||, μ_{j MathClass-punc, k}^{n MathClass-bin+ 1}) MathClass-bin+ (||, λ_{j MathClass-punc, k}^{n MathClass-bin+ 1}) MathClass-bin+ γ_{k}

at column j + ( k − 1) J , the off‐diagonal term

MathClass-bin- (||, μ_{j MathClass-punc, k}^{n MathClass-bin+ 1})

at column ( j + 1) + ( k − 1) J , and the off‐diagonal term

MathClass-bin- (||, λ_{j MathClass-punc, k}^{n MathClass-bin+ 1})

at column j + kJ . Therefore, matrix A is a strictly diagonally dominant matrix, that is, a matrix whose diagonal term absolute value in each row is greater than the sum of the absolute values of all the off‐diagonal terms of that row . A theorem on strictly diagonally dominant matrices states that the maximum norm of the inverse matrix, ∥ A −1 ∥ ∞ , has the upper bound

MathClass-rel∥ bold A^{- 1} {MathClass-rel∥}_{MathClass-rel\infty} MathClass-rel\leq \frac{1}{\underset{normalmin}{msub} [MathClass-rel| a_{ii} MathClass-rel| MathClass-bin- r_{i} (bold-italicA)]} MathClass-punc,

where N = JK is the number of rows, a ii is the i th diagonal term, and

r_{i} (bold-italicA) MathClass-rel= \underset{MathClass-op\sum}{msub} MathClass-rel| a_{ij} MathClass-rel|

.…”

Section: Time Backward Differencing Stability Analysismentioning

confidence: 99%

“…Therefore, matrix A is a strictly diagonally dominant matrix, that is, a matrix whose diagonal term absolute value in each row is greater than the sum of the absolute values of all the off‐diagonal terms of that row . A theorem on strictly diagonally dominant matrices states that the maximum norm of the inverse matrix, ∥ A −1 ∥ ∞ , has the upper bound

MathClass-rel∥ bold A^{- 1} {MathClass-rel∥}_{MathClass-rel\infty} MathClass-rel\leq \frac{1}{\underset{normalmin}{msub} [MathClass-rel| a_{ii} MathClass-rel| MathClass-bin- r_{i} (bold-italicA)]} MathClass-punc,

where N = JK is the number of rows, a ii is the i th diagonal term, and

r_{i} (bold-italicA) MathClass-rel= \underset{MathClass-op\sum}{msub} MathClass-rel| a_{ij} MathClass-rel|

.…”

Section: Time Backward Differencing Stability Analysismentioning

confidence: 99%

“…Specifically, row j C .k 1/J includes the diagonal term 1 Cˇ nC1 j;kˇCˇ nC1 j;kˇC k at column j C .k 1/J , the off-diagonal term ˇ nC1 j;kˇa t column .j C 1/ C .k 1/J , and the off-diagonal term ˇ nC1 j;kˇa t column j C kJ . Therefore, matrix A is a strictly diagonally dominant matrix, that is, a matrix whose diagonal term absolute value in each row is greater than the sum of the absolute values of all the off-diagonal terms of that row [17]. A theorem on strictly diagonally dominant matrices states that the maximum norm of the inverse matrix, kA 1 k 1 , has the upper bound…”

Section: Variable Coefficients: Truncation Error Analysismentioning

confidence: 99%

See 1 more Smart Citation

Exact stability conditions in upwinding‐scheme FDTD for the Boltzman transport equation

Chamanara

Caloz

2013

Int J Numerical Modelling

View full text Add to dashboard Cite

SUMMARYAn in-depth stability analysis of the FDTD method under the upwinding scheme for the Boltzmann transport equation (BTE) under the relaxation time approximation is provided. Both time forward and time backward difference equations are considered. In the time forward differencing case, a sufficient stability condition is derived for the BTE with variable coefficients, and a necessary and sufficient condition is derived for the BTE with constant coefficients. In the time backward differencing case, it is shown that the differencing equations are unconditionally stable. It is shown numerically that the previously reported stability conditions in the literature are not accurate.

show abstract

“…We notice that the results of Theorem 3 (and the corollary) were recently rediscovered [5]. At the same time, the M -matrices are included into the broader class of matrices of monotone type.…”

mentioning

confidence: 97%

Norm estimates for the inverses of matrices of monotone type and totally positive matrices

Volkov

Miroshnichenko

2009

Sib Math J

View full text Add to dashboard Cite

We give estimates of the infinity norm of the inverses of matrices of monotone type and totally positive matrices.Keywords: matrix of monotone type, M -matrix, totally positive matrix, diagonal dominance, inverse matrix, norm of a matrix In many problems of numerical analysis there is a need to estimate some norm of the matrix A −1 , the inverse to a given nonsingular matrix A = (a ij ) ∈ R n×n . It is usually not difficult to compute (or estimate) the norm of A. The estimation of the norm of A −1 = (a ij ) is a far more difficult task for any norm when A −1 is not known explicitly.A sufficiently simple method for estimation of the ∞-norm of the inverse is known for the matrices with diagonal dominance. This method was proposed in [1] in connection with the necessity to derive some error bounds of interpolation by cubic splines. Many papers have appeared, since then the estimate from [1] was improved and refined in various particular cases. Especially many articles in this area were devoted to the so-called matrices of monotone type, M -matrices.In this paper we consider a broad class of matrices and extend some results on the norm estimation for the inverses of M -matrices to the class. Furthermore, similar estimates are derived for totally positive matrices.Denote bythe diagonal dominance of each row, and putAs usual, we call a matrix A diagonally dominant provided that R * (A) 0.In spite of the availability of many papers refining (2), the estimate (2) is, however, attainable. Namely, if the main diagonal entries of A are positive, all off-diagonal entries are nonpositive, and the diagonal dominance of all rows is the same; then the norm A −1 ∞ is found exactly.

show abstract

Bounds for norms of the matrix inverse and the smallest singular value

Cited by 35 publications

References 8 publications

Observer-based adaptive interval type-2 fuzzy control of uncertain MIMO nonlinear systems with unknown asymmetric saturation actuators

Observer-based adaptive interval type-2 fuzzy control of uncertain MIMO nonlinear systems with unknown asymmetric saturation actuators

Exact stability conditions in upwinding‐scheme FDTD for the Boltzman transport equation

Norm estimates for the inverses of matrices of monotone type and totally positive matrices

Contact Info

Product

Resources

About