Appendix

surjectivity defect (ds) of the CPT functions not including the datum defects (d-defects) was

For the sake of simplicity for readers, a simple CPT function has been chosen at first. After the initial estimation of the function, the estimated errors vector has been found. We have seen that the errors have had a periodic characteristic in time. So, a new wave defining the error characteristic and been able to carry by the first wave has been planned for expanding the CPTs. It is shown that we can expand a CPT function until ensuring statically equivalency between a priory and a posteriori variances. For instance, one may secure the equivalency of the variances if one

The convergence rates (upon the iteration numbers) of the LS estimation have been inspected according to the threshold (thres = 5e-12) which is a good value for the estimation of the nonlinear CPT function. An algorithm compiled by different compilers and run in different architectures (with 32 Bit or 64 Bit) changes the convergence rate of the estimations in such as the inconsistent scientific problems. It is also observed that the iteration numbers change when the 64-bit Python software is run on Linux platform which has a different framework than Windows. But, the numbers have not been given in the example part of the chapter. Contrary to inconsistency model, namely in a consistent one, the iteration numbers can take equivalence values in all circumstances. Another way to determine the inconsistency of a model is to obtain its condition number which is computed from a rate of maximum and minimum eigenvalues or of singular values under LS. If the condition number is close to one, the projected model is accepted as a consistent model.

We can use the Soft Computing Methods (SCM) if not an exact mathematical relationship between the data and unknowns. The mathematical model is established by the trial-and-error method in training part of SCM by means of arbitrary weights and activation functions depending on SCM expert forecasts. For the solution of the SCM model during the training, we can use least absolute residuals (LAR) and minmax absolute residuals (MAR) objective functions by the linear programming or the LS estimation as in hard computing method (HCM). In the state, the inconsistency problem can erase whatever the solution method (LAR, MAR or LS) is. The inconstancy can be removed by means of experiences gained from HCMs. Prior information is very important to select a suitable mathematical model for a scientific problem. For example, comparing a priori variance with a posteriori variance at the end of the estimation is a useful warning to the user to determine the correct mathematical model as seen

In numerical computation, there are two main phenomena which are the mathematical model (as a combination of functional and stochastic models) and objective function. The solution strategy is of no importance if the same mathematical model and objective function are preferred in the same problem of hard computing. All solution strategies always give same

results, only their solution time spans can be distinct from each other (Table 3).

I wish to thank TÜRKSAT A.S (https://www.turksat.com.tr/en) supporting this study.

from the expanded model in the example section of the chapter.

Acknowledgements

would expand more by a new wave in the last estimated model in the same manner.

eliminated by the LS objective function.

70 Optimization Algorithms - Examples



Epoch(j) tjt0 [h] Xj [km] Yj [km] Zj [km] tj [μsec] 17.50 40069.319938 10107.494927 9362.125250 712.976412 17.75 39115.191840 11567.526548 11492.356997 713.287987 18.00 37995.179006 12978.503915 13573.844046 713.599417 18.25 36714.092016 14334.463621 15597.779572 713.910828 18.50 35277.410608 15629.676058 17555.603122 714.221952 18.75 33691.260665 16858.668299 19439.034930 714.533584 19.00 31962.388796 18016.245942 21240.109087 714.845281 19.25 30098.134631 19097.513875 22951.205467 715.156764 19.50 28106.400907 20097.895874 24565.080300 715.468390 19.75 25995.621474 21013.152993 26074.895308 715.779824 20.00 23774.727299 21839.400671 27474.245306 716.091334 20.25 21453.110591 22573.124521 28757.184187 716.402951 20.50 19040.587166 23211.194732 29918.249189 716.714457 20.75 16547.357150 23750.879046 30952.483389 717.026029 21.00 13983.964160 24189.854253 31855.456321 717.337474 21.25 11361.253075 24526.216175 32623.282652 717.649172 21.50 8690.326544 24758.488074 33252.638847 717.960633 21.75 5982.500334 24885.627475 33740.777748 718.272113 22.00 3249.257698 24907.031342 34085.541002 718.583548 22.25 502.202871 24822.539587 34285.369282 718.895203 22.50 2246.986126 24632.436884 34339.310236 719.206528 22.75 4986.605239 24337.452758 34247.024111 719.518326 23.00 7704.972522 23938.759921 34008.787012 719.830016 23.25 10390.476491 23437.970866 33625.491749 720.141620 23.50 13031.624539 22837.132665 33098.646226 720.452998 23.75 15617.091194 22138.719998 32430.369356 720.764247

On Non-Linearity and Convergence in Non-Linear Least Squares

http://dx.doi.org/10.5772/intechopen.76313

Table 3. Space Fixed Coordinates of C06 inclined geostationary earth orbit in COMPASS (which is Chinese Global Positioning Satellite System) are transformed with respect to t0 from earth fixed coordinates downloaded from ftp:// ftp.glonass-iac.ru/MCC/PRODUCTS/17091/final/Sta19426.sp3 [26] {t0 = 2017.04.01–00:00:00 (Civil Calendar) = 1942–

518,400 (GPS week—week seconds)}.

Address all correspondence to: orhnkrt@gmail.com

Geomatics Engineering Department, Kocaeli University, Kocaeli, Turkey

Author details

Orhan Kurt

On Non-Linearity and Convergence in Non-Linear Least Squares http://dx.doi.org/10.5772/intechopen.76313 


Table 3. Space Fixed Coordinates of C06 inclined geostationary earth orbit in COMPASS (which is Chinese Global Positioning Satellite System) are transformed with respect to t0 from earth fixed coordinates downloaded from ftp:// ftp.glonass-iac.ru/MCC/PRODUCTS/17091/final/Sta19426.sp3 [26] {t0 = 2017.04.01–00:00:00 (Civil Calendar) = 1942– 518,400 (GPS week—week seconds)}.
