Solve for rational coefficients

Question

Let $n \geq 0$ be and even integer. I have $2n + 5$ data points $(x_i,y_i)$ for $i = 1,\ldots,2n+5$.

I wish to find parameters $r, a_{n+1},\ldots, a_0, b_{n-1}, \ldots, b_0, c, d$ (I will call these old params) such that

$$y_i = r\sqrt{\frac{x_i^{n + 2} + a_{n + 1}x_i^{n+1} + a_nx_i^n + \ldots + a_0}{x_i^{n} + b_{n - 1}x_i^{n-1} + b_{n - 2}x_i^{n - 2} + \ldots + b_0}} + cx_i + d $$

I have tried to convert this system to be linear. i.e $$y_i^2x_i^n + \alpha_{n - 1} y_i^2x_i^{n - 1} + \ldots, \alpha_0y_i^2 + \beta_{n+1}y_ix_i^{n+1} + \ldots + \beta_0y_i + \gamma_{n+2}x_i^{n+2} + \ldots + \gamma_0 = 0$$

There are $3n + 5$ of these new params $\alpha_j, \beta_j, \gamma_j$ that is written in terms of previous params. When $n = 0$, I am able to solve for these new params and back out the old params. However, I don't know how to solve this problem when $n$ is greater than 0.

Answer 1

$y_i - cx_i - d= r\sqrt{\frac{x_i^{n + 2} + a_{n + 1}x_i^{n+1} + a_nx_i^n + \ldots + a_0}{x_i^{n} + b_{n - 1}x_i^{n-1} + b_{n - 2}x_i^{n - 2} + \ldots + b_0}} $

$y_i^2 - 2cx_i y_i -2dy_i +c^2x_i^2 + 2cdx_i + d^2= r^2\frac{x_i^{n + 2} + a_{n + 1}x_i^{n+1} + a_nx_i^n + \ldots + a_0}{x_i^{n} + b_{n - 1}x_i^{n-1} + b_{n - 2}x_i^{n - 2} + \ldots + b_0} $

$(y_i^2 - 2cx_i y_i -2dy_i +c^2x_i^2 + 2cdx_i + d^2)({x_i^{n} + b_{n - 1}x_i^{n-1} + b_{n - 2}x_i^{n - 2} + \ldots + b_0})= r^2(x_i^{n + 2} + a_{n + 1}x_i^{n+1} + a_nx_i^n + \ldots + a_0) $

$y_i^2({x_i^{n} + b_{n - 1}x_i^{n-1} + b_{n - 2}x_i^{n - 2} + \ldots + b_0}) - 2cx_i y_i({x_i^{n} + b_{n - 1}x_i^{n-1} + b_{n - 2}x_i^{n - 2} + \ldots + b_0}) -2dy_i({x_i^{n} + b_{n - 1}x_i^{n-1} + b_{n - 2}x_i^{n - 2} + \ldots + b_0}) +c^2x_i^2({x_i^{n} + b_{n - 1}x_i^{n-1} + b_{n - 2}x_i^{n - 2} + \ldots + b_0}) + 2cdx_i({x_i^{n} + b_{n - 1}x_i^{n-1} + b_{n - 2}x_i^{n - 2} + \ldots + b_0}) + d^2({x_i^{n} + b_{n - 1}x_i^{n-1} + b_{n - 2}x_i^{n - 2} + \ldots + b_0}) - r^2(x_i^{n + 2} + a_{n + 1}x_i^{n+1} + a_nx_i^n + \ldots + a_0) \\ =y_i^2({x_i^{n} + b_{n - 1}x_i^{n-1} + b_{n - 2}x_i^{n - 2} + \ldots + b_0}) - 2y_i({cx_i^{n+1} + cb_{n - 1}x_i^{n} + cb_{n - 2}x_i^{n - 2} + \ldots + cb_0x_i}) -2y_i({dx_i^{n} + db_{n - 1}x_i^{n-1} + db_{n - 2}x_i^{n - 2} + \ldots + db_0}) +(c^2{x_i^{n+2} + c^2b_{n - 1}x_i^{n+1} + c^2b_{n - 2}x_i^{n } + \ldots + c^2b_0x^2}) + 2cd({x_i^{n+1} + b_{n - 1}x_i^{n} + b_{n - 2}x_i^{n - 1} + \ldots + b_0x_i}) + d^2({x_i^{n} + b_{n - 1}x_i^{n-1} + b_{n - 2}x_i^{n - 2} + \ldots + b_0}) - r^2(x_i^{n + 2} + a_{n + 1}x_i^{n+1} + a_nx_i^n + \ldots + a_0)\\ =y_i^2({x_i^{n} + b_{n - 1}x_i^{n-1} + b_{n - 2}x_i^{n - 2} + \ldots + b_0}) +y_i(-2{cx_i^{n+1}-2 (cb_{n - 1}+d)x_i^{n} -2(cb_{n - 2}+db_{n - 1})x_i^{n - 1} + \ldots -2 (cb_0+db_1)x_i}-2 db_0) +(c^2{x_i^{n+2} + c^2b_{n - 1}x_i^{n+1} + c^2b_{n - 2}x_i^{n } + \ldots + c^2b_0x^2}) + 2cd({x_i^{n+1} + b_{n - 1}x_i^{n} + b_{n - 2}x_i^{n - 1} + \ldots + b_0x_i}) + d^2({x_i^{n} + b_{n - 1}x_i^{n-1} + b_{n - 2}x_i^{n - 2} + \ldots + b_0}) - r^2(x_i^{n + 2} + a_{n + 1}x_i^{n+1} + a_nx_i^n + \ldots + a_0)=0$

With all new param $\alpha, \beta, \gamma\space$ known, compare coefficients of $y_i^2$ terms, you get all b's. Then compare coefficients of $y_i$ terms, you get c and d, or no solution if any one equation is not equal. Then compare coefficients of terms without y, you get r and all a's.