We present a new algorithm to compute the $n$ middle coefficients of a $2n \times n$ product in $K(n)$ ring operations, where $K(n)$ is the number of operations needed by Karatsuba's algorithm for a full $n \times n$ product. Used in Newton iteration, and together with previous work of Mulders, Karp and Markstein, this algorithm enables one to compute an inverse in $\sim 0.904 \, K(n)$ operations, a quotient in $\sim 1.173 \, K(n)$ operations, and a square root in $\sim 0.891 \, K(n)$ operations. These results apply both to power series and polynomials.