A three-stage variable-shift iteration for polynomial zeros and its relation to generalized rayleigh iteration

Jenkins, M. A.; Traub, Joseph F.

doi:10.1007/bf02163334

Cited by 154 publications

(78 citation statements)

References 4 publications

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“…Our implementation uses the Jenkins and Traub root finder (Jenkins and Traub, 1970), and we found that this algorithm is numerically very stable.…”

Section: Problem Statementmentioning

confidence: 99%

W-PnP Method: Optimal Solution for the Weak-Perspective n-Point Problem and Its Application to Structure from Motion

Hajder

2017

Proceedings of the 12th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Application

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Abstract:Camera calibration is a key problem in 3D computer vision since the late 80's. Most of the calibration methods deal with the (perspective) pinhole camera model. This is not a simple goal: the problem is nonlinear due to the perspectivity. The strategy of these methods is to estimate the intrinsic camera parameters first; then the extrinsic ones are computed by the so-called PnP method. Finally, the accurate camera parameters are obtained by slow numerical optimization. In this paper, we show that the weak-perspective camera model can be optimally calibrated without numerical optimization if the L 2 norm is used. The solution is given by a closed-form formula, thus the estimation is very fast. We call this method as the Weak-Perspective n-Point (W-PnP) algorithm. Its advantage is that it simultaneously estimates the two intrinsic weak-perspective camera parameters and the extrinsic ones. We show that the proposed calibration method can be utilized as the solution for a subproblem of 3D reconstruction with missing data. An alternating least squares method is also defined that optimizes the camera motion using the proposed optimal calibration method.

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“…Our implementation uses the Jenkins and Traub root finder (Jenkins and Traub, 1970), and we found that this algorithm is numerically very stable.…”

Section: Problem Statementmentioning

confidence: 99%

W-PnP Method: Optimal Solution for the Weak-Perspective n-Point Problem and Its Application to Structure from Motion

Hajder

2017

Proceedings of the 12th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Application

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“…, a(i) being the step-size parameter (174), Laguerre's method (110; 132), and the JenkinsTraub algorithm (136). One can deflate the input polynomial via its numerical division by x − z to extend these algorithms to approximating the other roots.…”

Section: Root-finding Via Functional Iterationsmentioning

confidence: 99%

Algebraic and Numerical Algorithms

Emiris¹,

Pan²,

Tsigaridas³

2009

Chapman &Amp; Hall/CRC Applied Algorithms and Data Structures Series

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“…The data were computed by using the Jenkins and Traub root-finding method (see [25]), appropriate in this case since the polynomials has real coefficients, and they are similar to those given in [34,Ch. 8] where the distribution of zeros of Szegö polynomials for this measure is also considered: From Figures 1 and 2 one can observe that the zeros of φ 10 (z) are located on the circle z : |z| = q 1/2 whereas the zeros of φ 11 (z) are located on the circle z : |z| = q −1/2 , in accordance with Proposition 2.2 and the Mazel-GeronimoHayes theorem (see [32]).…”

Section: ✷ 6 Numerical Examplesmentioning

confidence: 99%

A matrix approach to the computation of quadrature formulas on the unit circle

Cantero¹,

Cruz-Barroso²,

González-Vera³

2008

Applied Numerical Mathematics

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In this paper we consider a general sequence of orthogonal Laurent polynomials on the unit circle and we first study the equivalences between recurrences for such families and Szegő's recursion and the structure of the matrix representation for the multiplication operator in Λ when a general sequence of orthogonal Laurent polynomials on the unit circle is considered. Secondly, we analyze the computation of the nodes of the Szegő quadrature formulas by using Hessenberg and five-diagonal matrices. Numerical examples concerning the family of Rogers-Szegő q-polynomials are also analyzed.

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A three-stage variable-shift iteration for polynomial zeros and its relation to generalized rayleigh iteration

Cited by 154 publications

References 4 publications

W-PnP Method: Optimal Solution for the Weak-Perspective n-Point Problem and Its Application to Structure from Motion

W-PnP Method: Optimal Solution for the Weak-Perspective n-Point Problem and Its Application to Structure from Motion

Algebraic and Numerical Algorithms

A matrix approach to the computation of quadrature formulas on the unit circle

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