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

Volkov, Yu. S.; Miroshnichenko, V. L.

doi:10.1007/s11202-009-0108-2

Cited by 23 publications

(11 citation statements)

References 6 publications

(7 reference statements)

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“…with A −1 ∞ = 1/β if all β i = β and Ã −1 ∞ = 1/β if allβ i =β; see, e.g., [35]. Sinceβ i ≥ β i with some strict inequalities holding, the bound for Ã −1 ∞ can be smaller than that for A −1 ∞ .…”

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confidence: 99%

An Approach to Making SPAI and PSAI Preconditioning Effective for Large Irregular Sparse Linear Systems

Jia¹,

Zhang²

2013

SIAM J. Sci. Comput.

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We investigate the SPAI and PSAI preconditioning procedures and shed light on two important features of them: (i) For the large linear system Ax = b with A irregular sparse, i.e., with A having s relatively dense columns, SPAI may be very costly to implement, and the resulting sparse approximate inverses may be ineffective for preconditioning. PSAI can be effective for preconditioning but may require excessive storage and be unacceptably time consuming; (ii) the situation is improved drastically when A is regular sparse, that is, all of its columns are sparse. In this case, both SPAI and PSAI are efficient. Moreover, SPAI and, especially, PSAI are more likely to construct effective preconditioners. Motivated by these features, we propose an approach to making SPAI and PSAI more practical for Ax = b with A irregular sparse. We first split A into a regular sparseÃ and a matrix of low rank s. Then exploiting the Sherman-Morrison-Woodbury formula, we transform Ax = b into s + 1 new linear systems with the same coefficient matrixÃ, use SPAI and PSAI to compute sparse approximate inverses ofÃ efficiently and apply Krylov iterative methods to solve the preconditioned linear systems. Theoretically, we consider the non-singularity and conditioning ofÃ obtained from some important classes of matrices. We show how to recover an approximate solution of Ax = b from those of the s + 1 new systems and how to design reliable stopping criteria for the s + 1 systems to guarantee that the approximate solution of Ax = b satisfies a desired accuracy. Given the fact that irregular sparse linear systems are common in applications, this approach widely extends the practicability of SPAI and PSAI. Numerical results demonstrate the considerable superiority of our approach to the direct application of SPAI and PSAI to Ax = b.

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confidence: 99%

An Approach to Making SPAI and PSAI Preconditioning Effective for Large Irregular Sparse Linear Systems

Jia¹,

Zhang²

2013

SIAM J. Sci. Comput.

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“…It follows from (16) and (27) that E → 0 as h i → 0. Thus we conclude that method (11) converges and the order of the convergence of method (11) is at least quadratic.…”

Section: Convergence Of the Methodsmentioning

confidence: 95%

Variable Mesh Size Exponential Finite Difference Method for the Numerical Solutions of Two Point Boundary Value Problems

Pandey

2015

bspm

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In this article, we presented a non-uniform mesh size high order exponential finite difference scheme for the numerical solutions of two point boundary value problems with Dirichlet's boundary conditions. Under appropriate conditions, we have discussed the local truncation error and the convergence of the proposed method. Numerical experiments have been carried out to demonstrate the use and high order computational efficiency of the present method in several model problems. Numerical results showed that the proposed method is accurate and convergent. The order of accuracy is at least cubic which is in good agreement with the theoretically established order of the method.

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“…Also by row sum criterion matrix J is for sufficiently small h monotone [21]. For the bound of J , we define [22]- [24], …”

Section: Convergence Of the Non-standard Difference Methodsmentioning

confidence: 99%

Non-Standard Difference Method for Numerical Solution of Linear Fredholm Integro-Differential Type Two-Point Boundary Value Problems

Pandey¹

2015

OALib

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In this article we have considered Fredholm integro-differential equation type second-order boundary value problems and proposed a rational difference method for numerical solution of the problems. The composite trapezoidal quadrature and non-standard difference method are used to convert Fredholm integro-differential equation into a system of equations. The numerical results in experiment on some model problems show the simplicity and efficiency of the method. Numerical results showed that the proposed method is convergent and at least second-order of accurate.

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

Cited by 23 publications

References 6 publications

An Approach to Making SPAI and PSAI Preconditioning Effective for Large Irregular Sparse Linear Systems

An Approach to Making SPAI and PSAI Preconditioning Effective for Large Irregular Sparse Linear Systems

Variable Mesh Size Exponential Finite Difference Method for the Numerical Solutions of Two Point Boundary Value Problems

Non-Standard Difference Method for Numerical Solution of Linear Fredholm Integro-Differential Type Two-Point Boundary Value Problems

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