Nonlinear fitting of the function $h(t)=\frac{\alpha\beta}{d-vt-\frac{1}{2}at^2}$

Question

I have the following function: $$h(t)=\dfrac{\alpha\beta}{d-vt-\dfrac{1}{2}at^2}$$ with $\alpha$,$\beta$ constants. I have a set of experimental data: $$h_1(t_1),h_2(t_2),...h_N(t_N)$$ I need to find the set $(d,v,a)$ so that the error between the given $h(t)$ and the data is minimum. I tried the Newton Raphson method, but it is strongly dependent from the initial value vector. The same problem with the Levenberg Marquardt algorithm. Can someone suggest me a good numerical method to find the three parameters of $h(t)$? Thanks in advance.

Answer 1

You have $n$ data points$(t_i,h_i)$ and the model is $$h=\dfrac{\alpha\beta}{d-vt-\dfrac{1}{2}at^2}$$ Since $\alpha,\beta$ are constants, in a first step, consider the model to be $$z=\dfrac{\alpha\beta} h={d-vt-\dfrac{1}{2}at^2}={d-vt-bt^2}$$ So, defining for each data point $z_i=\dfrac{\alpha\beta} {h_i}$, a linear regression will provide reasonable estimates of $d$ and $v$ and $a=2b$.

Now you should be ready for solving the problem using almost any method.