We study the computation of the Petz-Augustin mean of order (\alpha \in (0,1) \cup (1,\infty)), defined as the minimizer of a weighted sum of (n) Petz-R'enyi divergences of order (\alpha) over the set of (d)-by-(d) quantum states, where the Petz-R'enyi divergence is a quantum generalization of the classical R'enyi divergence. We propose the first algorithm with a non-asymptotic convergence guarantee for solving this optimization problem. The iterates are guaranteed to converge to the Petz-Augustin mean at a linear rate of ( O\left( \lvert 1 - 1/\alpha \rvert^T \right) ) with respect to the Thompson metric for (\alpha\in(1/2,1)\cup(1,\infty)), where ( T ) denotes the number of iterations. The algorithm has an initialization time complexity of (O\left(nd^3\right)) and a per-iteration time complexity of (O\left(nd^2 + d^3\right)). Two applications follow. First, we propose the first iterative method with a non-asymptotic convergence guarantee for computing the Petz capacity of order (\alpha\in(1/2,1)), which generalizes the quantum channel capacity and characterizes the optimal error exponent in classical-quantum channel coding. Second, we establish that the Petz-Augustin mean of order (\alpha), when all quantum states commute, is equivalent to the equilibrium prices in Fisher markets with constant elasticity of substitution (CES) utilities of common elasticity (\rho=1-1/\alpha), and our proposed algorithm can be interpreted as a t\^{a}tonnement dynamic. We then extend the proposed algorithm to inhomogeneous Fisher markets, where buyers have different elasticities, and prove that it achieves a faster convergence rate compared to existing t\^{a}tonnement-type algorithms.