Using Sturm Sequences to Bracket Real Roots of Polynomial Equations

Hook, D. G.; McĂree, P. R.

doi:10.1016/b978-0-08-050753-8.50089-9

Cited by 37 publications

(41 citation statements)

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“…We create a polynomial in a single variable and find its solutions by calculating the roots only in a selected interval using Sturm-sequences [13]. This significantly speeds up Gröbner basis solvers.…”

Section: Gröbner Basis Solversmentioning

confidence: 99%

“…Note that in all our solutions we only replace the eigenvalue action matrix computation, performed in the existing Gröbner basis solver, with the proposed "matrix FGLM" method or with the Danilevsky method for computing the coefficients of the characteristic polynomial, followed by the computation of the roots of the obtained single-variable polynomial using the Sturm-sequences [13].…”

Section: Minimal Solversmentioning

confidence: 99%

“…After finding the coefficients of the characteristic polynomial p Mx i (x i ) of the action matrix M xi , as described in Section 5.2, we can use Sturm-sequences [13] to find its roots on the interval, which is feasible for the variable x i .…”

Section: Characteristic Polynomial Of the Action Matrixmentioning

confidence: 99%

“…After finding the coefficients of the single-variable polynomial we can use Sturm-sequences [13] to find its roots on the interval, which is feasible for the variable x i , e.g. only positive values for the focal length.…”

Section: Modified Matrix Fglm Algorithmmentioning

confidence: 99%

“…This is done by first transforming the problem to a polynomial in a single variable and then by calculating the roots of this polynomial only on a selected interval using efficient Sturm-sequences [13]. This way we save huge amount of work spent in calculating solutions which are usually latter dropped.…”

Section: Introductionmentioning

confidence: 99%

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Making minimal solvers fast

Bujnak¹,

Kúkelová

Pajdla

2012

2012 IEEE Conference on Computer Vision and Pattern Recognition

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show abstract

Section: Gröbner Basis Solversmentioning

confidence: 99%

Section: Minimal Solversmentioning

confidence: 99%

Section: Characteristic Polynomial Of the Action Matrixmentioning

confidence: 99%

Section: Modified Matrix Fglm Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Making minimal solvers fast

Bujnak¹,

Kúkelová

Pajdla

2012

2012 IEEE Conference on Computer Vision and Pattern Recognition

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A fast, robust solution to the Stewart platform forward kinematics

McĂree

Daniel

1996

J. Robotic Syst.

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A kinematic view of loop closure

et al. 2004

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We consider the problem of loop closure, i.e., of finding the ensemble of possible backbone structures of a chain segment of a protein molecule that is geometrically consistent with preceding and following parts of the chain whose structures are given. We reduce this problem of determining the loop conformations of six torsions to finding the real roots of a 16th degree polynomial in one variable, based on the robotics literature on the kinematics of the equivalent rotator linkage in the most general case of oblique rotators. We provide a simple intuitive view and derivation of the polynomial for the case in which each of the three pair of torsional axes has a common point. Our method generalizes previous work on analytical loop closure in that the torsion angles need not be consecutive, and any rigid intervening segments are allowed between the free torsions. Our approach also allows for a small degree of flexibility in the bond angles and the peptide torsion angles; this substantially enlarges the space of solvable configurations as is demonstrated by an application of the method to the modeling of cyclic pentapeptides. We give further applications to two important problems. First, we show that this analytical loop closure algorithm can be efficiently combined with an existing loop-construction algorithm to sample loops longer than three residues. Second, we show that Monte Carlo minimization is made severalfold more efficient by employing the local moves generated by the loop closure algorithm, when applied to the global minimization of an eight-residue loop. Our loop closure algorithm is freely available at http://dillgroup. ucsf.edu/loop_closure/.

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Using Sturm Sequences to Bracket Real Roots of Polynomial Equations

Cited by 37 publications

References 0 publications

Making minimal solvers fast

Making minimal solvers fast

A fast, robust solution to the Stewart platform forward kinematics

A kinematic view of loop closure

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