Inverse interpolation

Given $(x_k,f_k)$ from a function $f(x)$, supose we want to find an approximation of the value of $x$ such that $f(x)=c$, where $c$ is a given value.

We will solve the equation $x=g(c)$ where $g$ is the inverse function of $f$. Then we will interpolate this function $g(y)$ and will evaluate it in $y=c$, or, in other words, if we use Newton's method we will put in the first column the values $f_j$ and in the second one the values $x_j$ and proceed the same way.

For example, let's suppose that we want to calculate a zero of the function $f(x)=x^3-15x+4$ knowing that this is close to $x=0.3$. Then we will do quadratic interpolation, for example, of the inverse of $f(x)$. We then first evaluate the function in three points close to $x=0.3$:

Now we fill in the table to calculate the divided differences of Newton, but exchanging the columns, obtaining the coefficients of the interpolating polynomial:

Thus the interpolating polynomial is:

$$\begin{array}{rl} P_3(y)=&0.2+0.0675\cdot(y-1.008)+0.00028963\cdot(y-1.008)(y-0.473) \\ =&0.2679019090-0.067654y+0.00028963y^2 \end{array}$$

So an approximation of the zero of the function is:

$$P_3(0)=0.2679019090$$