I would like to know, whether any improved Newton Raphson method is available for non-linear overdetermined equations (So we use Jacobian matrix and pseudo inverse). Data used as measurements are noisy. The final outputs will be very close by, for eg: if x1,x2,x3,x4 are outputs it will be 200.5,200.51, 200.61,200.53. When I initialize Newton raphson with this values it diverges. I have to give far values to initialize. How to give a good guess for Newton Raphson.
Number of equations are 6 (non-linear, quadratic), number of outputs are 4.
Please guide
So, you have $n > 4$ equations $$f_i(x_1,x_2,x_3,x_4)=0 \qquad \text{for} \qquad i=1,2,\cdots,n$$ I should not try to solve the problem using Newton-Raphson but, instead, I should define $$\Phi(x_1,x_2,x_3,x_4)=\sum_{i=1}^n \big[f_i(x_1,x_2,x_3,x_4)\big]^2$$and I shall try to minimize $\Phi(x_1,x_2,x_3,x_4)$ providing the analytical Jacobian and, if possible, the analytical Hessian.