Matrix interpretations are widely used in automated complexity analysis. Certifying such analyses boils down to determining the growth rate of $A^n$ for a fixed non-negative rational matrix A. There exists a conceptually simple algorithm to determine the growth rate, but this algorithm has the disadvantage that it is based on algebraic number computations.

In this work we present an even simpler algorithm to compute the growth rate. Its soundness is based on a variant of a Perron–Frobenius theorem that has been conjectured in earlier work. So far it only has been proven for small matrices, and here we present a proof for the general case.

We further verify both the algorithm and the new Perron–Frobenius theorem in the proof assistant Isabelle/HOL, and integrate it into CeTA, a verified certifier for various properties, including complexity proofs. Because of the new results, CeTA no longer requires a verified implementation of algebraic numbers.

矩阵解释广泛用于自动复杂性分析。证明此类分析归结为确定增长率${\mathrm{一种}}^n$对于固定的非负有理矩阵A。存在一种概念上简单的算法来确定增长率，但该算法的缺点是它基于代数数计算。

在这项工作中，我们提出了一种更简单的算法来计算增长率。它的合理性基于 Perron-Frobenius 定理的一个变体，该定理在早期的工作中已经被推测出来。到目前为止，它只对小矩阵进行了证明，这里我们给出了一般情况的证明。

我们在证明助手 Isabelle/HOL 中进一步验证了算法和新的 Perron-Frobenius 定理，并将其集成到CeTA 中，CeTA是一个经过验证的各种属性的证明者，包括复杂性证明。由于新结果，CeTA不再需要经过验证的代数数实现。