# Numerically inverting an exponentially growing function (defined by Chebyshev polynomials)

Assume a function $M(t)$ strictly increasing, essentially growing exponentially, and asymptoptically growing at a known rate $\bar{g}$, i.e. $\lim_{t\to\infty}M'(t)/M(t) = \bar{g}$

In a set of awful integro-differential equations, I need to use the inverse of $M(t)$, denoted by $q(\cdot)$ for simplicity, i.e. $t = M^{-1}(M) \equiv q(M)$.

The function $M(t)$ will be solved numerically with some sort of spectral collocation method with a polynomial basis (see below). My general problem is: with a guess on $M(t)$, how can I find $q(M)$?

Since this equation is growing exponentially, I don't think I can use a polynomial basis for $M(t)$ directly. Instead, I suspect that I should guess a Chebyshev basis for $g(t)$ constraining for some large $T$ that $g(T) = \bar{g}$. If so, then $$M(t) \equiv \exp\left(\int_0^t g(\tau) d \tau \right)$$

Which works fine for my computations. In particular, it is easy with a Chebyshev basis to find the partial integrals $\int_0^t g(\tau) d \tau$ for $t$ nodes of the Chebyshev basis (through Calculating integrals for a function approximated by Chebyshev polynomials)

But if so, how can I find the $q(M)$ function, or a basis of it in $M$ space? Obviously, I could just evaluate $M(t)$ at the Chebyshev nodes of $t$ to find the appropriate $M$, and then use some sort of interpolation to find the approximating $q(M)$, but I would prefer an approach that has a chance of using auto-differentiation by throwing it into my collocation method.

For example, is there a way to translate this into a nonlinear ODE using the inverse-function theorem, and if so does this simplify substantially given that this is a Chebyshev basis?