Abstract
A two-step fifth and a multi-step order iterative method are derived, for finding the solution of system of nonlinear equations. The new two-step fifth order method requires two functions, two first order derivatives, and the multi-step methods needs a additional function per step. The performance of this method has been tested with finding solutions to several test problems then applied to solving pseudorange nonlinear equations on Global Navigation Satellite Signal (GNSS). To solve the problem, at least four satellite’s measurements are needed to locate the user position and receiver time offset. In this work, a number of satellites from 4 to 8 are considered such that the number of equations is more than the number of unknown variables to calculate the user position. Moreover, the Geometrical Dilution of Precision (GDOP) values are computed based on the satellite selection algorithm (fuzzy logic method) which could be able to bring the best suitable combination of satellites. We have restricted the number of satellites to 4 to 6 for solving the pseudorange equations to get better GDOP value even after increasing the number of satellites beyond six also yields a 0.4075 GDOP value. Actually, the conventional methods utilized in the position calculation module of the GNSS receiver typically converge with six iterations for finding the user position whereas the proposed method takes only three iterations which really decreases the computation time which provide quicker position calculation. A practical study was done to evaluate the computation efficiency index (CE) and efficiency index (IE) of the new model. From the simulation outcomes, it has been noted that the new method is more efficient and converges 33% faster than the conventional iterative methods with good accuracy of 92%.
Highlights
Numerical analysis is a comprehensive subject which is interconnected with applied mathematics, various fields of science and engineering, medical, etc
We have considered four or more than four visible satellites scenario for linearizing the pseudorange equations for obtaining better Geometrical Dilution of Precision (GDOP) value and one can notice that the computational order calculation is not matching with the theoretical order when we increased the number of satellites more than four, most of the iterative methods does not match with the theoretical order but the new multistep iterative method described here approximately matches with the order and still preserves the accuracy
The conventional way of solving Global Positioning System (GPS) pseudorange equations involve Taylor series and iterative methods which typically take more than 6 iterations if the number of visible satellites are considered to be 4 but in the proposed work, the user position is computed within 3 iterations that significantly reduce the computation time in the position calculation module of the GPS receiver
Summary
Numerical analysis is a comprehensive subject which is interconnected with applied mathematics, various fields of science and engineering, medical, etc. F (x) = 0, where F : D ⊂ Rn → Rn , and this type of problem will be solved by the famous Newton’s method (2nd NR) which has second order convergence [1] by y(x(n) ) = x(n) − [ F 0 (x(n) )]−1 F (x(n) ), n = 0, 1, 2,. Traub [2] gave a double-step Newton’s type method (3rd TM) having convergence order 3 by two F, one F 0 evaluations z(x(n) ) = y(x(n) ) − [ F 0 (x(n) ))]−1 F (y(x(n) )),. Abad et al [3], a different combination was used to get a three-step fifth order method, where three functions, two Jacobian matrices and their inverses were evaluated, and it is given below x(n+1) = z(x(n) ) − [ F 0 (y(x(n) ))]−1 F (z(x(n) )).
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