## Some tests of generalized bisection (1987)

Venue: | ACM Trans. Math. Software |

Citations: | 52 - 2 self |

### BibTeX

@ARTICLE{Kearfott87sometests,

author = {I?. Baker Kearfott},

title = {Some tests of generalized bisection},

journal = {ACM Trans. Math. Software},

year = {1987},

volume = {13},

pages = {197--220}

}

### OpenURL

### Abstract

This paper addresses the task of reliably finding approximations to all solutions to a system of nonlinear equations within a region defined by bounds on each of the individual coordinates. Various forms of generalized bisection were proposed some time ago for this task. This paper systematically compares such generalized bisection algorithms to themselves, to continuation methods, and to hybrid steepest descent/quasi-Newton methods. A specific algorithm containing novel “expansion ” and “exclusion ” steps is fully described, and the effectiveness of these steps is evaluated. A test problem consisting of a small, high-degree polynomial system that is appropriate for generalized bisection, but very difticult for continuation methods, is presented. This problem forms part of a set of 17 test problems from published literature on the methods being compared; this test set is fully described here.

### Citations

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Citation Context ...mpare with continuation methods, we use the published results in [19], and tabulate results with problems 1, 2, 3, and 4. We can only obtain an idea of the relative behavior of these techniques since =-=(1)-=- numerous problem types have been omitted. (2) improvements have been made to the continuation method in [19] (cf., [20]), and (3) since improvements can be made in our method of bisection. Because bi... |

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Citation Context ...independent of the point) are contained in B. (The chord method is an iteration of the form Xk+i t Xk - YF(Xk), where Y is held fixed relative to X,, and is only an approximate inverse to F’(Xk); see =-=[23]-=-, p. 181.) Thus, if B C B, the Schauder fixed point theorem and the fact that stationary points of the chord method correspond to solutions of (1.1) show that there is a solution of (1.1) in B. Furthe... |

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Citation Context ...h code compiled with the Fortran “H” compiler. To get a very rough idea of the total number of instructions executed, we used the LINPACK benchmark timings of these machines and compilers reported in =-=[4]-=-; based on this benchmark, the IBM PC executes floating-point operations 150 times slower than the IBM 3033. Thus, in Table 5.1, the actual CPU times in hours:min.:sec. for the PC appear in the column... |

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Citation Context ... L in Algorithm 3.1 and in slower execution times. 4. THE TEST SET The 17 test problems were chosen from diverse published reports of computational tests. These included tests of quasi-Newton methods =-=[18]-=-; a GarciaZangwill continuation method [ 191, a simplicial bisection method [5]; a “simple” box bisection method [21]; a problem arising from robot kinematics [12]; a problem arising from a technique ... |

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Citation Context ...a sophisticated code (embodying improvements over that in [19]) for solving polynomial systems of equations; we used the standard tolerances supplied with the code. This method is fully documented in =-=[20]-=-. The code successfully found the 12 real solutions while it traversed the 126 paths. However, due to excessive numbers of steps, it terminated prematurely on many of the paths leading to the singular... |

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Citation Context ...ethod starting in any of the subregions will converge to a unique root within that subregion. In that sense, bisection is optimal. The method in this paper employs an interval form of Newton’s method =-=[13]-=- and is thus similar to algorithms proposed by Moore et al. [15], Hansen (see [6, 7,8,9], etc.), Alefeld et al. [2], etc., although we do not emphasize the interval arithmetic aspects of the method. H... |

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Citation Context ... subregion. In that sense, bisection is optimal. The method in this paper employs an interval form of Newton’s method [13] and is thus similar to algorithms proposed by Moore et al. [15], Hansen (see =-=[6, 7,8,9]-=-, etc.), Alefeld et al. [2], etc., although we do not emphasize the interval arithmetic aspects of the method. Here, we test the method on 17 problems that include artificial and real-world examples f... |

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Citation Context ...ion within the nonnegative unit box, but has other, nonphysical solutions in larger domains. ACM Transactions on Mathematical Software, Vol. 13, No. 3, September 1987.s11. A robot kinematics problem (=-=[24]-=-): where Some Tests of Generalized Bisection l 207 + Ly3Xl + LyqX2 + (Y5X4 + 0&3X7 + Ly7 = 0 f2 = %3xl3c3 + aSx2x3 + (YlOxl + (Yllx2 + al2x4 + (Y]3 = 0 f3 = ~14&$3 + (Yl5xl + (Y161t2 = 0 f4 = q7x1 + a... |

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Citation Context ...e root within that subregion. In that sense, bisection is optimal. The method in this paper employs an interval form of Newton’s method [13] and is thus similar to algorithms proposed by Moore et al. =-=[15]-=-, Hansen (see [6, 7,8,9], etc.), Alefeld et al. [2], etc., although we do not emphasize the interval arithmetic aspects of the method. Here, we test the method on 17 problems that include artificial a... |

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Citation Context ... the other comparisons here.) Careful comparison on more problems will prove useful. 5.5 Comparison with Quasi-Newton Software Here, we compare Algorithm 3.1 to the routine HYBFLJl from MINPACK (cf., =-=[17]-=-) on problems 3, 4, 15, and 17, which are problems 2, 8, 1, and 14, respectively, in the standard test set distributed with MINPACK (cf., [18]). The standard test driver for MINPACK supplies starting ... |

16 |
Box-bisection for solving second-degree systems and the problem of clustering
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Citation Context ...ished reports of computational tests. These included tests of quasi-Newton methods [18]; a GarciaZangwill continuation method [ 191, a simplicial bisection method [5]; a “simple” box bisection method =-=[21]-=-; a problem arising from robot kinematics [12]; a problem arising from a technique for analyzing bifurcation points [lo]; and a problem used to test an interval Newton method that is somewhat more sop... |

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Citation Context ... subregion. In that sense, bisection is optimal. The method in this paper employs an interval form of Newton’s method [13] and is thus similar to algorithms proposed by Moore et al. [15], Hansen (see =-=[6, 7,8,9]-=-, etc.), Alefeld et al. [2], etc., although we do not emphasize the interval arithmetic aspects of the method. Here, we test the method on 17 problems that include artificial and real-world examples f... |

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Citation Context ...cluded tests of quasi-Newton methods [18]; a GarciaZangwill continuation method [ 191, a simplicial bisection method [5]; a “simple” box bisection method [21]; a problem arising from robot kinematics =-=[12]-=-; a problem arising from a technique for analyzing bifurcation points [lo]; and a problem used to test an interval Newton method that is somewhat more sophisticated than (3.7), 191. The methods in the... |

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Citation Context ...rious times in published experiments, but we cite references where appropriate. ACM Transactions on Mathematical Software, Vol. 13, No. 3, September 1987.s4.1 Problems Used to Test a Homotopy Method (=-=[19]-=-) 1. Cubic-Parabola: Some Tests of Generalized Bisection l 205 4x: - 3X* - x2 = 0 XT -x2=0 Initial Box: [-2, 21 X [-2, 21 Remark: This problem has three solutions in the box. It poses a problem for th... |

5 |
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Citation Context ...ures convergence), for a savings in execution time. We mention here that more sophisticated versions of the right-hand side of (3.7) have been proven to also lead to root inclusion tests (see [9] and =-=[16]-=-). Also, ACM Transactions on Mathematical Software, Vol. 13, No. 3, September 1987.s204 - Ft. Baker Kearfott Neumaier develops a general theory of when interval Newton methods based on interval invers... |

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Citation Context ...on is optimal. The method in this paper employs an interval form of Newton’s method [13] and is thus similar to algorithms proposed by Moore et al. [15], Hansen (see [6, 7,8,9], etc.), Alefeld et al. =-=[2]-=-, etc., although we do not emphasize the interval arithmetic aspects of the method. Here, we test the method on 17 problems that include artificial and real-world examples from the literature on quasi... |

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Citation Context ...roblems were chosen from diverse published reports of computational tests. These included tests of quasi-Newton methods [18]; a GarciaZangwill continuation method [ 191, a simplicial bisection method =-=[5]-=-; a “simple” box bisection method [21]; a problem arising from robot kinematics [12]; a problem arising from a technique for analyzing bifurcation points [lo]; and a problem used to test an interval N... |

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Citation Context ...e note that alternate approaches to interval floating-point arithmetic include: (i) use of a precompiler that supports the interval data type (cf., [3]) in conjunction with a subroutine package (cf., =-=[25]-=-); and (ii) use of a compiler, such as CDC’s M77 compiler or Pascal-SC, that supports an interval data type. We believe the first alternative would result in execution times similar to those reported ... |

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Citation Context ...cted rounding execute more slowly than our scheme. We note that alternate approaches to interval floating-point arithmetic include: (i) use of a precompiler that supports the interval data type (cf., =-=[3]-=-) in conjunction with a subroutine package (cf., [25]); and (ii) use of a compiler, such as CDC’s M77 compiler or Pascal-SC, that supports an interval data type. We believe the first alternative would... |

1 | On a general technique for finding directions proceeding from bifurcation points - KEARFOTT - 1984 |