## An exploration of homotopy solving in Maple (2003)

Venue: | Proc. of the Sixth Asian Symp. on Comp. Math. (ASCM 2003). Lect. Note Series on Comput. by World Sci. Publ. 10 |

Citations: | 2 - 2 self |

### BibTeX

@INPROCEEDINGS{Hazaveh03anexploration,

author = {K. Hazaveh and D. J. Jeffrey and G. J. Reid and S. M. Watt and A. D. Wittkopf},

title = {An exploration of homotopy solving in Maple},

booktitle = {Proc. of the Sixth Asian Symp. on Comp. Math. (ASCM 2003). Lect. Note Series on Comput. by World Sci. Publ. 10},

year = {2003}

}

### OpenURL

### Abstract

Homotopy continuation methods find approximate solutions of a given system by a continuous deformation of the solutions of a related exactly solvable system. There has been much recent progress in the theory and implementation of such path following methods for polynomial systems. In particular, exactly solvable related systems can be given which enable the computation of all isolated roots of a given polynomial system. Extension of such methods to determine manifolds of solutions has also been recently achieved. This progress, and our own research on extending continuation methods to identifying missing constraints for systems of differential equations, motivated us to implement higher order continuation methods in the computer algebra language Maple. By higher order, we refer to the iterative scheme used to solve for the roots of the homotopy equation at each step. We provide examples for which the higher order iterative scheme achieves a speed up when compared with the standard second order scheme. We also demonstrate how existing Maple numerical ODE solvers can be used to give a predictor only continuation method for solving polynomial systems. We apply homotopy continuation to determine the missing constraints in a system of nonlinear PDE, which is to our knowledge, the first published instance of such a calculation. 1.

### Citations

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Citation Context ...ferential equations in [14], and initiated a study of Numerical Jet Geometry, using homotopy methods. Despite the availability of very well developed implementations for homotopy continuation methods =-=[26,12] -=-surprisingly little has been implemented in the context of computer algebra systems for numerical solutions for polynomial systems. We note that, for example, Gröbner bases in Maple are limited to po... |

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Citation Context ... of such a calculation. 1. Introduction Newton’s local method for square polynomial systems is a classical method for finding a root of systems with finitely many roots. Homotopy continuation method=-=s [1]-=- deform the known roots of a related system into the roots of the system of interest, and can calculate all the (isolated complex) roots of such systems. Recent developments by Sommese, Verschelde and... |

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Citation Context ...he known roots of a related system into the roots of the system of interest, and can calculate all the (isolated complex) roots of such systems. Recent developments by Sommese, Verschelde and Wampler =-=[20,19]-=- include the extension of such homotopy methods to non-square (over- and 145sNovember 27, 2003 18:9 WSPC/Trim Size: 9in x 6in for Proceedings ascm2003 146 under-determined systems), and characterize t... |

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Citation Context ...he known roots of a related system into the roots of the system of interest, and can calculate all the (isolated complex) roots of such systems. Recent developments by Sommese, Verschelde and Wampler =-=[20,19]-=- include the extension of such homotopy methods to non-square (over- and 145sNovember 27, 2003 18:9 WSPC/Trim Size: 9in x 6in for Proceedings ascm2003 146 under-determined systems), and characterize t... |

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Citation Context ...s defined by the geometric theory of PDE (see [16] and the references therein). This system first appeared in an article to illustrate a new exact elimination algorithm for simplifying systems of PDE =-=[15].-=- We present, for the first time, a PDE example of the interpolation-free homotopy method described in [14, §6.4], made possible by the works [21,22]. We confine ourselves to the case p = 2. In terms ... |

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Citation Context ... elimination algorithm for simplifying systems of PDE [15]. We present, for the first time, a PDE example of the interpolation-free homotopy method described in [14, §6.4], made possible by the works=-= [21,22]-=-. We confine ourselves to the case p = 2. In terms of jet coordinates (which are formal indeterminates corresponding to derivatives of the dependent variables, etc.) this is a differential polynomial ... |

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Citation Context ... elimination algorithm for simplifying systems of PDE [15]. We present, for the first time, a PDE example of the interpolation-free homotopy method described in [14, §6.4], made possible by the works=-= [21,22]-=-. We confine ourselves to the case p = 2. In terms of jet coordinates (which are formal indeterminates corresponding to derivatives of the dependent variables, etc.) this is a differential polynomial ... |

37 | Numerical algebraic geometry
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Citation Context ...2003 146 under-determined systems), and characterize the components or manifolds of solutions of such systems. This is part of the rapidly developing area of Numerical Algebraic Geometry initiated in =-=[23].-=- This yields new methods for problems which have been traditionally approached with symbolic methods from Computer Algebra, such as factorization [3], Gröbner bases and the completion of systems of p... |

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Citation Context ...inear system of PDE for u = u(x, y): ∂2u ∂y2 − ∂2u = 0, ∂x∂y � ∂u ∂x � p 157 + ∂u − u = 0. (6.1) ∂x The aim is to complete (6.1) to an involutive system as defined by the geo=-=metric theory of PDE (see [16]-=- and the references therein). This system first appeared in an article to illustrate a new exact elimination algorithm for simplifying systems of PDE [15]. We present, for the first time, a PDE exampl... |

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Citation Context ...m DR = 1 dim π 2 (D 2 R) = 1 dim π(D 2 R) = 1 dim D 2 R = 1 For a projection of the output system π ℓ (D k R) to be involutive, it should satisfy a projected dimension test and have involutive sy=-=mbol [30]. Verifyin-=-g the projected dimension test involves checking for the maximum ℓ ≤ k, if it exists, such that its dimension satisfies dim π ℓ (D k R) = dim π ℓ+1 (D k+1 R). This is first satisfied when k ... |

12 |
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Citation Context ...rea of Numerical Algebraic Geometry initiated in [23]. This yields new methods for problems which have been traditionally approached with symbolic methods from Computer Algebra, such as factorization =-=[3],-=- Gröbner bases and the completion of systems of partial differential equations. We have extended this work to systems of differential equations in [14], and initiated a study of Numerical Jet Geometr... |

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Citation Context ...trol in ODE integrators. Another view of homotopy solving is that of solving a differential equation on a manifold (that is, solving a differential-algebraic equation). Both Visconti [28] and Arponen =-=[24]-=- have implemented DAE solving methods in Maple, and it would be of interest to use these in homotopy solving. Differentiating the ODE (3.4) yields expressions for the higher order derivatives: Hx wher... |

10 |
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Citation Context ... = 0, and Halley’s method cannot be used for a function satisfying 2(f ′ ) 2 − ff ′′ = 0, which means any function of the form f(x) = 1/(Ax + B). For the vector case, we use Cartesian-tensor=-= notation [9]. -=-When applying these results to homotopy methods, we shall give equivalent results in vector-matrix notation. Let f : R m → R m be a vector function, with component functions fi. Let f depend upon th... |

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Citation Context ...ods from Computer Algebra, such as factorization [3], Gröbner bases and the completion of systems of partial differential equations. We have extended this work to systems of differential equations in=-= [14]-=-, and initiated a study of Numerical Jet Geometry, using homotopy methods. Despite the availability of very well developed implementations for homotopy continuation methods [26,12] surprisingly little... |

8 |
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Citation Context ...onal residual control in ODE integrators. Another view of homotopy solving is that of solving a differential equation on a manifold (that is, solving a differential-algebraic equation). Both Visconti =-=[28]-=- and Arponen [24] have implemented DAE solving methods in Maple, and it would be of interest to use these in homotopy solving. Differentiating the ODE (3.4) yields expressions for the higher order der... |

6 | CMPSm: A continuation method for polynomial systems (Matlab version
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Citation Context ...e by T.Y. Li and his team [12] which enables highly efficient computation of mixed volumes. An efficient and parallel implementation of polyhedral continuation in Matlab is the work of Kim and Kojima =-=[10]-=-. We have made use of the LinearAlgebra Package in Maple, and in particular its interface to NAG Library routines. Using the predictor and corrector methods given above, we now wish to step from t = 0... |

5 |
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Citation Context ...h in turn has components xj. We wish to solve fi(x) = 0, starting from an initial estimate x (0) . We direct the reader to the literature where a multivariate Halley method of the type below is given =-=[5]. The Taylor s-=-eries for f about x (0) can be written using ∆j = xj −x (0) j : fi(x (0) ) = fi(x (0) ) + fi,k(x (0) )∆k + 1 2 fi,kh(x (0) )∆k∆h + · · · . (2.5) Let ˇ fki be the inverse of fi,k, defined... |

5 |
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(Show Context)
Citation Context ... ≤ t ≤ 1 (3.4) where Hx is the Jacobian of H with respect to x1, . . . , xn. Naturally this has led to the use of ODE software for higher order predictors in continuation methods [1, Section 6.3] =-=and [13]-=-. One extreme is not to use any corrector steps. For convergence the path should stay in the basin of attraction of the sought after isolated root at t = 1. To ensure this, most approaches use a combi... |

5 | Graphic and numerical comparison between iterative methods
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(Show Context)
Citation Context ... schemes Newton’s method to find solutions of a single nonlinear equation f(x) = 0 is well known; it is also well known that the method is second order and that higher-order methods have been derive=-=d [25,7]-=-. Here we start by giving a uniform treatment of the higher-order scalar schemes, as a preparation for the vector case. Consider solving the scalar equation f(x) = 0, given an initial estimate x0 for ... |

2 |
Naumann (eds.) Automatic Differentiation 2000: From Simulation to Optimization
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Citation Context ...se, an iteration requires a matrix inverse, making the iteration more expensive. In addition higher order derivatives for polynomial systems can be cheaply obtained (e.g. by automatic differentiation =-=[4]-=-) and this opens the possibility that the thirdorder method will be more efficient. If an initial estimate x (0) is further away from the root, and the convergence theorems do not apply, then we must ... |

2 |
Methodus nova, accurata & facilis inveniendi radices aequationum quarumcunque generaliter, sine praevia reductione
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Citation Context ...] as a starting point. We compare the methods, and apply them to a variety of problems arising in polynomial system solving. For scalar functions, higher-order schemes are often called Halley methods =-=[7], be-=-cause of Halley’s discovery in Newton’s era. Higher-order schemes allow more rapid convergence and larger step sizes in processes such as homotopy solution techniques. In this paper, we first pres... |

2 | Homotopies and polynomial system solving I. Basic Principles
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(Show Context)
Citation Context ...tions to find roots of this truncated function in a given domain. Root counts are verified by using Cauchy’s integral formula, using numerical quadrature, around the boundary of the domain. Kotsirea=-=s [11]-=- has developed a multivariate fixed step homotopy method in Maple. We have implemented a variable step homotopy continuation method in Maple, both for second and third orders, using the code of Smith ... |

1 |
Computing Roots of Truncated Zeta Functions
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- 2002
(Show Context)
Citation Context ... of Numerical Algebraic Geometry (such as the computation of mixed volumes) still are not implemented in that context. Existing Homotopy implementations in Maple include the univariate program of Fee =-=[6]-=-. In that work Fee truncates the Riemann zeta function, and uses a very efficient homotopy method he has developed for analytic functions to find roots of this truncated function in a given domain. Ro... |

1 | Integer roots for integerpower-content calculations
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(Show Context)
Citation Context ...an isolated non-singular root x (e) , then with each iteration a second-order method will approximately double the number of digits that are correct, while a thirdorder method will triple that number =-=[8]-=-. Thus, for example, an estimate that is correct to 1 digit can be improved to 8 digits in 3 second-order steps or 2 third-order steps. Since the computation of the second derivative term is often exp... |

1 |
Further Development in HomotopySolve for Maple 7
- Smith
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(Show Context)
Citation Context ... has developed a multivariate fixed step homotopy method in Maple. We have implemented a variable step homotopy continuation method in Maple, both for second and third orders, using the code of Smith =-=[18]-=- as a starting point. We compare the methods, and apply them to a variety of problems arising in polynomial system solving. For scalar functions, higher-order schemes are often called Halley methods [... |