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Cox, A.J.; Higham, Nicholas J.

doi:10.1023/a:1022385611904

Cited by 17 publications

(2 citation statements)

References 24 publications

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“…Let us remark that if c = 0 we recover the known result for least squares problems given in [27,Theorem 1.1]. We also note that the parametrization of the set of perturbations E is similar to that obtained for equality constrained least squares problems in [9], where the constraint is however not present.…”

Section: Backward Error Analysissupporting

confidence: 80%

On the iterative solution of systems of the form $A^T A x=A^Tb+c$

Calandra,

Gratton,

Riccietti

et al. 2019

Preprint

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Given a full column rank matrix A ∈ R m×n (m ≥ n), we consider a special class of linear systems of the form A ⊤ Ax = A ⊤ b + c with x, c ∈ R n and b ∈ R m . The occurrence of c in the right-hand side of the equation prevents the direct application of standard methods for least squares problems. Hence, we investigate alternative solution methods that, as in the case of normal equations, take advantage of the peculiar structure of the system to avoid unstable computations, such as forming A ⊤ A explicitly. We propose two iterative methods that are based on specific reformulations of the problem and we provide explicit closed formulas for the structured condition number related to each problem. These formula allow us to compute a more accurate estimate of the forward error than the standard one used for generic linear systems, that does not take into account the structure of the perturbations. The relevance of our estimates is shown on a set of synthetic test problems. Numerical experiments highlight both the increased robustness and accuracy of the proposed methods compared to the standard conjugate gradient method. It is also found that the new methods can compare to standard direct methods in terms of solution accuracy.

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Section: Backward Error Analysissupporting

confidence: 80%

On the iterative solution of systems of the form $A^T A x=A^Tb+c$

Calandra,

Gratton,

Riccietti

et al. 2019

Preprint

View full text Add to dashboard Cite

show abstract

“…Let us remark that if c = 0 we recover the known result for least squares problems given in [29,Theorem 1.1]. Note also that the parameterization of the set of perturbations \scrE is similar to that obtained for equality constrained least squares problems in [10], even if there is no constraint in [10].…”

Section: Proof Let Us Define M = [((R T \Otimes (supporting

confidence: 77%

On Iterative Solution of the Extended Normal Equations

Calandra¹,

Gratton²,

Riccietti³

et al. 2020

SIAM J. Matrix Anal. Appl.

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Given a full-rank matrix A \in \BbbR m\times n (m \geq n), we consider a special class of linear systems A T Ax = A T b + c with x, c \in \BbbR n and b \in \BbbR m , which we refer to as the extended normal equations. The occurrence of c gives rise to a problem with a different conditioning from the standard normal equations and prevents direct application of standard methods for least squares. Hence, we seek more insights on theoretical and practical aspects of the solution of such problems. We propose an explicit formula for the structured condition number, which allows us to compute a more accurate estimate of the forward error than the standard one used for generic linear systems, which does not take into account the structure of the perturbations. The relevance of our estimate is shown on a set of synthetic test problems. Then, we propose a new iterative solution method that, as in the case of normal equations, takes advantage of the structure of the system to avoid unstable computations such as forming A T A explicitly. Numerical experiments highlight the increased robustness and accuracy of the proposed method compared to standard iterative methods. It is also found that the new method can compare to standard direct methods in terms of solution accuracy.

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