I'm doing a report on the mathematics of GPS, and I have the following equations to solve for $x$, $y$, $z$ and $t_b$:
I'm using Newton's method, with the Jacobian matrix of partial derivatives being as follows (by the power rule):
I'm using an initial guess of:
I've substituted in the initial guess to give what should be the first iteration using Newton's method:
Which simplifies to give:
However, this evaluates to:
The correct first iteration however, as calculated by Maxima, is:
Where have I gone wrong in this? I've self-taught myself most of this mathematics, and the fact that my iteration is so different from the correct one means I've probably done something completely wrong. Any help is appreciated.
This is not an answer but it is too long for a comment.
I repeated the calculations but set all numbers as rational. The results I obtained are $$x=-\frac{5092356599265080239}{142135566553125000}\approx-35.82746193$$ $$y=-\frac{51362394942211922323}{142135566553125000}\approx-361.3620165$$ $$z=-\frac{315241754157890072131}{7896420364062500}\approx-39922.10896$$ $$t=\frac{28067559633057906988253}{51100601628870148500000}\approx0.5492608450$$ which are identical to those Maxima gave.
Setting all numbers as decimals, I ended with the same results.
As copper.hat commented, the system is very badly conditioned ($J\approx 6\times 10^7$) and I wonder what could happen working with low precision. Could you precise this point ?
Just for your curiosity, the problem has two solutions $$\{x\to -41.7663,y\to -16.7878,z\to 6370.2,t\to -0.00319939\}\}$$ $$\{x\to -39.7413,y\to -134.276,z\to -9413.92,t\to 0.185171\}$$
Edit
If I may suggest, I should not use Newton for this kind of problem. Expand $f_1-f_2$, $f_1-f_3$, $f_1-f_3$; all squares disappear and you have $$f_1-f_2=-\frac{1312248584 }{5}t+6320 x-9580 y-3060 z+\frac{1172267578731}{62500}=0$$ $$f_1-f_3=-\frac{5536610464 }{5}t+4020 x+14180 y-13320 z+\frac{1276786201414}{15625}=0$$ $$f_1-f_4=-\frac{1509984672 }{5}t+7140 x-13860 y-3500 z+\frac{334297508818}{15625}=0$$ from which you can eliminate $y,z,t$ as linear functions of $x$; report these expressions in $f_1$ and you get a quadratic equation in $x$ $$x^2+\frac{1172085587221693444650100593 }{14380080939595292010312500}x+\frac{3729488115129024118430005986528113}{224688764 6811764376611328125000}=0 $$ Solve it and back to $y,z,t$.
This is a general method for this kind of equations.