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## LAGUERRE-LIKE METHODS FOR THE SIMULTANEOUS APPROXIMATION OF POLYNOMIAL MULTIPLE ZEROS (2004)

### Citations

592 |
Introduction to Interval Computations,
- Alefeld, Herzberger
- 1983
(Show Context)
Citation Context ...) ( )1/ 2 1/ 21/ 2 / 2 1/ 2 / 2 1/ 2: {| | ;| | | | } { | | ;| | | | }.i iZ c e c c r c e c c rθ θ= − − − − −Us(4.3)sFor more details about the properties of complex circular arithmetic see the books =-=[3]-=- ands[21].sLet us assume that the denominator of (4.2) does not contain the origin, and thensthe set on the right hand side of (4.2) defines a closed disk. This suggests the followingsiterative method... |

53 | Computing multiple roots of inexact polynomials
- Zeng
- 2005
(Show Context)
Citation Context ... respectively. We will not consider here the problem ofsdetermination of the order of multiplicity; the reader interested in this topic may findsseveral efficient procedures in [10]-[12], [14], [26], =-=[27]-=-. However, we have used somesof these procedures in practical realization of numerical examples, three of them arespresented in this paper.sSubstituting the exact zeros appearing in the sums 1,i∑sands... |

33 |
A family of root finding methods
- Hansen, Patrick
- 1977
(Show Context)
Citation Context ...ot only of applied mathematicssincluding the theory of optimizations but also in many branches of engineering sciences,sphysics, computer science, astronomy, finance, and so on. As noted in [5], [6], =-=[8]-=-, [9],s[15], [16], [18], [23], Laguerre's method belongs to the most powerful methods forssolving polynomial equations. The convergence characteristics of this method weresextensively investigated in ... |

30 |
Iteration methods for finding all zeros of a polynomial simultaneously
- Aberth
- 1973
(Show Context)
Citation Context ...1 . These multiple zerossmake a cluster [swhich is an additional difficulty. We started from 4 initialsapproximations equidistantly spaced on the circle | | 1− 2 1.9, 2, 2.1] 10z =s(Aberth's approach =-=[1]-=-). It issevident that these approximations are rather far from the sought zeros. Despite thissinconvenient situation, after 8 iterations the method (3.3) produced reasonably goodsapproximations which ... |

28 |
Iterative Methods For Simultaneous Inclusion of Polynomial Zeros
- Petkovic
- 1989
(Show Context)
Citation Context ...s (expressed by the quantity (0)ρ ) and their size. This fact issof great practical importance.sMore details about iterative methods for the simultaneous inclusion ofspolynomial zeros may be found in =-=[17]-=- and references cited there.s42 M. Petković, L. Rančić, D. Milošević / Laguerre-Like Methods for Polynomial Multiple ZerossImproved methods (II)sIt is worth noting thatsthe convergence ofsthe interval... |

18 |
Laguerre's Method Applied to the Matrix Eigenvalue Problem
- Parlett
- 1964
(Show Context)
Citation Context ...ed mathematicssincluding the theory of optimizations but also in many branches of engineering sciences,sphysics, computer science, astronomy, finance, and so on. As noted in [5], [6], [8], [9],s[15], =-=[16]-=-, [18], [23], Laguerre's method belongs to the most powerful methods forssolving polynomial equations. The convergence characteristics of this method weresextensively investigated in literature; refer... |

16 | Ten methods to bound multiple roots of polynomials
- Rump
(Show Context)
Citation Context ... ,νµ µ respectively. We will not consider here the problem ofsdetermination of the order of multiplicity; the reader interested in this topic may findsseveral efficient procedures in [10]-[12], [14], =-=[26]-=-, [27]. However, we have used somesof these procedures in practical realization of numerical examples, three of them arespresented in this paper.sSubstituting the exact zeros appearing in the sums 1,i... |

10 |
Some modifications of Laguerre’s method
- Hansen, Patrick, et al.
- 1977
(Show Context)
Citation Context ...ly of applied mathematicssincluding the theory of optimizations but also in many branches of engineering sciences,sphysics, computer science, astronomy, finance, and so on. As noted in [5], [6], [8], =-=[9]-=-,s[15], [16], [18], [23], Laguerre's method belongs to the most powerful methods forssolving polynomial equations. The convergence characteristics of this method weresextensively investigated in liter... |

9 |
Solution of Equations in Euclidean and Banach
- Ostrowski
- 1973
(Show Context)
Citation Context ... applied mathematicssincluding the theory of optimizations but also in many branches of engineering sciences,sphysics, computer science, astronomy, finance, and so on. As noted in [5], [6], [8], [9],s=-=[15]-=-, [16], [18], [23], Laguerre's method belongs to the most powerful methods forssolving polynomial equations. The convergence characteristics of this method weresextensively investigated in literature;... |

8 |
la méthode Laguerre pour l’approximation des racines de certaines équations algébriques et sur la critique d’Hermite
- Bodewig, Sur
- 1946
(Show Context)
Citation Context ...ird order method forsfinding a multiple zero,s( ) 1/ 221, 2, 1, ˆi i i i i i nz z n n µδ δ δµ = − ⎡ ⎤⎛ ⎞−± −⎢ ⎥⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦i ,sactually the counterpart of Laguerre's method which was known to Bodewig =-=[4]-=- (see,salso, [8]).sFrom this reason, all methods derived from the fixed point relation (2.5) willsbe called Laguerre-like methods, shorter (L).sThe fixed point relation (2.5) is suitable for the const... |

8 |
A modification of newton’s method for analytic mappings having multiple zeros
- Kravanja, Haegemans
- 1999
(Show Context)
Citation Context ...ities 1,..., ,νµ µ respectively. We will not consider here the problem ofsdetermination of the order of multiplicity; the reader interested in this topic may findsseveral efficient procedures in [10]-=-=[12]-=-, [14], [26], [27]. However, we have used somesof these procedures in practical realization of numerical examples, three of them arespresented in this paper.sSubstituting the exact zeros appearing in ... |

8 |
A method for finding the zeros of polynomials using a companion matrix
- Niu, Sakurai
- 2003
(Show Context)
Citation Context ...1,..., ,νµ µ respectively. We will not consider here the problem ofsdetermination of the order of multiplicity; the reader interested in this topic may findsseveral efficient procedures in [10]-[12], =-=[14]-=-, [26], [27]. However, we have used somesof these procedures in practical realization of numerical examples, three of them arespresented in this paper.sSubstituting the exact zeros appearing in the su... |

7 |
On the convergence speed of some algorithms for the simultaneous approximation of polynomial zeros
- Alefeld, Herzberger
- 1974
(Show Context)
Citation Context ...-step method (3.3) is convergent with thesconvergence order equal to four.sThe proof of this theorem is extensive but elementary and can be found in [25].sUsing the approach of Alefeld and Herzberger =-=[2]-=-, the following theorem can besproved for the single-step method (3.4):sTheorem 3.3. If the inequalities (3.9) are valid, then the lower bound of the R -order ofsconvergence of the iterative method (3... |

7 | Z.: The quasi-Laguerre iteration
- Du, Jin, et al.
- 1997
(Show Context)
Citation Context ...ractice, not only of applied mathematicssincluding the theory of optimizations but also in many branches of engineering sciences,sphysics, computer science, astronomy, finance, and so on. As noted in =-=[5]-=-, [6], [8], [9],s[15], [16], [18], [23], Laguerre's method belongs to the most powerful methods forssolving polynomial equations. The convergence characteristics of this method weresextensively invest... |

7 |
Parallel Laguerre iterations: The complex case
- Gargantini
- 1976
(Show Context)
Citation Context ...− −⎢ ⎥⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦ = − ⎡ ⎤⎛ ⎞⎛ ⎞−+ − − +⎢ ⎥⎜ ⎟⎜ ⎟−⎢ ⎥⎝ ⎠⎝ ⎠⎣ ⎦is(4.2)sWe recall that the square root of a disk { ; }Z c r=s( | | )ic c e θ=snot containing 0s(that is,s) is the union of two disks (see =-=[7]-=-), | |c r> ( ) ( )1/ 2 1/ 21/ 2 / 2 1/ 2 / 2 1/ 2: {| | ;| | | | } { | | ;| | | | }.i iZ c e c c r c e c c rθ θ= − − − − −Us(4.3)sFor more details about the properties of complex circular arithmetic s... |

6 |
Generalizations of Laguerre's method: higher order methods
- Foster
- 1981
(Show Context)
Citation Context ...ce, not only of applied mathematicssincluding the theory of optimizations but also in many branches of engineering sciences,sphysics, computer science, astronomy, finance, and so on. As noted in [5], =-=[6]-=-, [8], [9],s[15], [16], [18], [23], Laguerre's method belongs to the most powerful methods forssolving polynomial equations. The convergence characteristics of this method weresextensively investigate... |

4 |
Improving the Van de Vel root-finding method
- King
- 1983
(Show Context)
Citation Context ...iplicities 1,..., ,νµ µ respectively. We will not consider here the problem ofsdetermination of the order of multiplicity; the reader interested in this topic may findsseveral efficient procedures in =-=[10]-=--[12], [14], [26], [27]. However, we have used somesof these procedures in practical realization of numerical examples, three of them arespresented in this paper.sSubstituting the exact zeros appearin... |

3 | On computing zeros of analytic functions and related problems in structured numerical linear algebra - Kravanja - 1999 |

3 |
Laguerre-like inclusion method for polynomial zeros
- Petković
(Show Context)
Citation Context ...hematicssincluding the theory of optimizations but also in many branches of engineering sciences,sphysics, computer science, astronomy, finance, and so on. As noted in [5], [6], [8], [9],s[15], [16], =-=[18]-=-, [23], Laguerre's method belongs to the most powerful methods forssolving polynomial equations. The convergence characteristics of this method weresextensively investigated in literature; references ... |

3 |
A higher order family for the simultaneous inclusion of multiple zeros of polynomials", Numerical Algorithms
- Petković, Milošević
(Show Context)
Citation Context ... to thesconvergence order of the considered methods, mainly well match the convergencesbehavior of these methods in practice. A more detailed comparative analysis for intervalsmethods may be found in =-=[20]-=-. Besides, a number of numerical examples (including thespresented examples) show that the proposed Laguerre-like methods belong to the mostspowerful iterative methods with the convergence order equal... |

3 |
On the guaranteed convergence of Laguerre-like method
- Petković, Petković, et al.
(Show Context)
Citation Context ...cssincluding the theory of optimizations but also in many branches of engineering sciences,sphysics, computer science, astronomy, finance, and so on. As noted in [5], [6], [8], [9],s[15], [16], [18], =-=[23]-=-, Laguerre's method belongs to the most powerful methods forssolving polynomial equations. The convergence characteristics of this method weresextensively investigated in literature; references mentio... |

1 |
Iterative methods for the simultaneous inclusion of polynomial zeros
- Milošević
- 2005
(Show Context)
Citation Context ...), Ostrowski-like method (O) and Halley-like methods(H) which have a similar structure and have the same order of convergence. The explicitsformulas that define the last three methods can be found in =-=[13]-=-. In addition, we alsostested the corresponding single-step variants of these four methods.sThe zeros of Psare 1 1,ζ = −s2 2 ,iζ =s3 1 ,iζ = +s4 1 ,iζ = −s5 3iζ = −sof thesmultiplicities 1 2,µ =s2 3,µ... |

1 |
Laguerre-like method for the simultaneous inclusion of multiple zeros of a polynomial
- Petković, Milošević
(Show Context)
Citation Context ...erion for the selection of a proper disk was statedsin [7] (see, also, [18]).sUnder suitable initial conditions which take into consideration the distributionsand size of initial inclusion disks (see =-=[19]-=-), in each iteration the simultaneous intervalsmethod (4.4) enables the inclusion ( )ki iZζ ∈sfor all νIi ∈ . In this way an automaticscomputation of rigorous error bound (given by the radii of result... |

1 | Square-root families for the simultaneous approximations of polynomial multiple zeros", Novi Sad
- Rančić, Petković
(Show Context)
Citation Context ...+ − + + +sTo compare the obtained results, we also tested several methods of the fourth order:sEuler-like method (E), Ostrowski-like method (O) and Halley-like method (H). Thesesmethods, presented in =-=[24]-=-, belong to the same class since they have the similar structuresand the same order of convergence. Besides, we also tested single-step variants of thesesfour methods.sThe exact zeros of the above pol... |

1 |
Simultaneous methods for solving algebraic equations
- Rančić
- 2005
(Show Context)
Citation Context ... −s(3.9)shold for every 1,...,i ν= , then the total-step method (3.3) is convergent with thesconvergence order equal to four.sThe proof of this theorem is extensive but elementary and can be found in =-=[25]-=-.sUsing the approach of Alefeld and Herzberger [2], the following theorem can besproved for the single-step method (3.4):sTheorem 3.3. If the inequalities (3.9) are valid, then the lower bound of the ... |