Linear homotopy solution of nonlinear systems of equations in geodesy     

Béla Paláncz  Joseph L. Awange  Piroska Zaletnyik  Robert H. Lewis 《Journal of Geodesy》2010,84(1):79-95

A fundamental task in geodesy is solving systems of equations. Many geodetic problems are represented as systems of multivariate polynomials. A common problem in solving such systems is improper initial starting values for iterative methods, leading to convergence to solutions with no physical meaning, or to convergence that requires global methods. Though symbolic methods such as Groebner bases or resultants have been shown to be very efficient, i.e., providing solutions for determined systems such as 3-point problem of 3D affine transformation, the symbolic algebra can be very time consuming, even with special Computer Algebra Systems (CAS). This study proposes the Linear Homotopy method that can be implemented easily in high-level computer languages like C++ and Fortran that are faster than CAS by at least two orders of magnitude. Using Mathematica, the power of Homotopy is demonstrated in solving three nonlinear geodetic problems: resection, GPS positioning, and affine transformation. The method enlarging the domain of convergence is found to be efficient, less sensitive to rounding of numbers, and has lower complexity compared to other local methods like Newton–Raphson.  相似文献   

Biogeochemistry of trace metals (Mn, Sr, Rb, Ba, Cu, Zn, Pb and Cd) in a river-wetland-lake system (Balaton Region, Hungary)     

Françoise Elbaz-Poulichet  Attila Nagy  Tibor Cserny  Piroska Pomogyi 《Aquatic Geochemistry》1996,2(4):379-402

Mn, Sr, Ba, Rb, Cu, Zn, Pb and Cd concentrations have been measured seasonally in the water and deposited sediments of the system comprising: Zala river (main input) — Lakes Kis-Balaton 1 and 2 (small artificial lakes created in a former bay of Lake Balaton) — Keszthely bay (hypertrophic part of Lake Balaton). The concentrations of the trace elements together with pH, alkalinity, dissolved cations (Ca²⁺, Mg²⁺, Na⁺, and K⁺), dissolved inorganic ligands (Cl^–, SO₄ ^2–), particulate Al, Ca, inorganic and organic carbon are used to assess the contamination of the study area and biogeochemical processes controlling trace element concentrations. Thermodynamic speciation calculations have also been utilized to enhance our understanding of the system. In the sediments Rb, Ba, Cu and Zn concentrations were mainly controlled by the abundance of the aluminosilicate fraction. Strontium was mainly associated with the calcium carbonate fraction. The aluminosilicate fraction constitutes a major sink for Mn and Cd but the concentration of these elements are also strongly related to calcite precipitation. The main processes that control the dissolved distribution of trace elements in the Balaton system were: solid phase formation (carbonate) for Mn; coprecipitation with calcite for Sr, Ba, Rb and possibly Mn and Cd; adsorption/desorption processes (pH dependent) for Zn and Pb; solubilization of Mn and precipitation of Cd and Cu in reed covered wetland areas where anoxic conditions were probably existing during the warm season. A preliminary budget of atmospheric and river input to Lake Balaton has also been outlined. Although Lake Balaton, is subjected to anthropogenic inputs mainly from agricultural and domestic activities, their impact on trace element concentrations in the Balaton system is very limited due to the efficiency of removal processes (i.e. adsorption and co-precipitation) and to high sedimentation rates and strong sediment re-suspension. Anthropogenic inputs are only detected for Pb.  相似文献   

Dixon resultant’s solution of systems of geodetic polynomial equations     

Béla Paláncz  Piroska Zaletnyik  Joseph L. Awange  Erik W. Grafarend 《Journal of Geodesy》2008,82(8):505-511

The Dixon resultant is proposed as an alternative to Gröbner basis or multipolynomial resultant approaches for solving systems of polynomial equations inherent in geodesy. Its smallness in size, high density (ratio on the number of nonzero elements to the number of all elements), speed, and robustness (insensitive to combinatorial sequence and monomial order, e.g., Gröbner basis) makes it extremely attractive compared to its competitors. Using 3D-intersection and conformal C ₇ datum transformation problems, we compare its performance to those of the Sturmfels’s resultant and Gröbner basis. For the 3D-intersection problem, Sturmfels’s resultant needed 0.578 s to solve a 6  ×  6 resultant matrix whose density was 0.639, the Dixon resultant on the other hand took 0.266 s to solve a 4  ×  4 resultant matrix whose density was 0.870. For the conformal C ₇ datum transformation problem, the Dixon resultant took 2.25 s to compute a quartic polynomial in scale parameter whereas the computaton of the Gröbner basis fails. Using relative coordinates to compute the quartic polynomial in scale parameter, the Gröbner basis needed 0.484 s, while the Dixon resultant took 0.016 s. This highlights the robustness of the Dixon resultant (i.e., the capability to use both absolute and relative coordinates with any order of variables) as opposed to Gröbner basis, which only worked well with relative coordinates, and was sensitive to the combinatorial sequence and order of variables. Geodetic users uncomfortable with lengthy expressions of Gröbner basis or multipolynomial resultants, and who aspire to optimize on the attractive features of Dixon resultant, may find it useful.  相似文献