Maple Computer Algebra System Research Papers (original) (raw)

A fundamental task in geodesy is solving systems 1 of equations. Many geodetic problems are represented as sys-2 tems of multivariate polynomials. A common problem in 3 solving such systems is improper initial starting values for 4 iterative methods, leading to convergence to solutions with 5 no physical meaning, or to convergence that requires global 6 methods. Though symbolic methods such as Groebner bases 7 or resultants have been shown to be very efficient, i.e., provid-8 ing solutions for determined systems such as 3-point prob-9 lem of 3D affine transformation, the symbolic algebra can be 10 very time consuming, even with special Computer Algebra 11 Systems (CAS). This study proposes the Linear Homotopy 12 method that can be implemented easily in high-level com-13 puter languages like C++ and Fortran that are faster than CAS 14 B. Paláncz (B) by at least two orders of magnitude. Using Mathematica, the 15 power of Homotopy is demonstrated in solving three nonlin-16 ear geodetic problems: resection, GPS positioning, and affine 17 transformation. The method enlarging the domain of conver-18 gence is found to be efficient, less sensitive to rounding of 19 numbers, and has lower complexity compared to other local 20 methods like Newton-Raphson. 21 Keywords Homotopy · Nonlinear systems of equations · 22 GPS positioning · Resection · Affine transformation 23 Journal: 190 MS: 0346 TYPESET DISK LE CP Disp.:2009/9/15 Pages: 17 Layout: Large Author Proof u n c o r r e c t e d p r o o f B. Paláncz et al. (Bates et al. 2008), HOM4PS (Lee et al. 2008), HOMPACK 47 (Watson et al. 1997), PHCpack (Verschelde 1999), and PHoM 48 (Gunji et al. 2004). Other codes are written in Maple (see, 49 e.g., Leykin and Verschelde 2004) and Matlab (Decarolis 50 et al. 2002). Recently, codes for fixed point and Newton 51 homotopy were developed in Mathematica, too, see Binous 52 (2007a,b). 53 In geodesy, several algebraic procedures have been put 54 forward for solving nonlinear systems of equations (see, e.g., 55 Awange and Grafarend 2005; Paláncz et al. 2008a). The pro-56 cedures suggested in these studies, such as Groebner bases 57 and resultant approaches, are, however, normally restricted 58 by the size of the nonlinear systems of equations involved. 59 In most cases the symbolic computations are time consum-60 ing. These symbolic computations were necessitated by the 61 failure to obtain suitable starting values for numerical itera-62 tive procedures. In situations where large systems of equa-63 tions are to be solved (e.g., affine transformation), and where 64 symbolic methods are insufficient, there exists the need to 65 investigate the suitability of other alternatives. One such alter-66 native is the linear homotopy. 67 In this study linear homotopy continuation methods are 68 introduced (see Kotsireas 2001). The definitions and basic 69 ideas are considered and illustrated. The general algorithms 70 of linear homotopy method are considered: iterated solution 71 of homotopy equations via Newton-Raphson method; and as 72 an initial value problem of a system of ordinary differential 73 equations. The efficiency of these methods are illustrated here 74 by solving polynomial equation systems in geodesy, namely 75 solution of 3D resection, GPS navigation, and 3D affine trans-76 formation problems. Our computations were carried out with 77 Mathematica and Fermat computer algebra systems on HP 78 workstation xw 4100 with XP operation system, 3 GHz P4 79 Intel processor and 1 GB RAM. The details can be found 80 in MathSource (Wolfram Research Inc.) in Paláncz (2008). 81 Although the mathematical background of the algorithms is 82 discussed in this paper, it can be also found in the literature, 83 e.g., Sommese et al. (2005). The application of the imple-84 mented Mathematica functions is illustrated in the Appendix. 85 The study is organized as follows: In Sect. 2, we pres-86 ent the definition and basic concepts of homotopy required 87 to understand the solution of nonlinear equations presented 88 in Sect. 3. For simplicity purposes, a quadratic and a third-89 degree polynomial equation are used to illustrate the 90 approach. Once the example has been made clear, Sect. 4 91 demonstrates how the technique is applied to solve three non-92 linear geodetic problems. The results are discussed in Sect. 5 93 and concluded in Appendix.