# 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 $$x=0.3$$:

 $$x$$ $$0.2$$ $$0.3$$ $$0.4$$ $$f(x)$$ $$1.008$$ $$-0.473$$ $$-1.936$$

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

 $$1.008$$ $$0.2$$ $$-0.0675$$ $$-0.473$$ $$0.3$$ $$0.00028963$$ $$-0.0684$$ $$-1.936$$ $$0.4$$

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: $$P_3(0)=0.2679019090$$$