We show that there exists no algorithm that decides for any bilinear system $(B,v)$ if the growth rate of $(B,v)$ is $1$. This answers a question of Bui who showed that if the coefficients are positive the growth rate is computable (i.e., there is an algorithm that outputs the sequence of digits of the growth rate of $(B,v)$). Our proof is based on a reduction of the computation of the joint spectral radius of a set of matrices to the computation of the growth rate of a bilinear system. We also use our reduction to deduce that there exists no algorithm that approximates the growth rate of a bilinear system with relative accuracy $\varepsilon$ in time polynomial in the size of the system and of $\varepsilon$. Our two results hold even if all the coefficients are nonnegative rationals.
翻译：我们显示,如果(B,v)美元增长率为1美元,那么没有任何算法决定任何双线系统$(B,v)$(B)$(B)$(B)$(B)$(B)$(B)$(B)$(B,v)$(B)$(B)$(B)$(B)$(B)$(B,v)$(B)$(B)$(B)$(Bi))的增长率。这回答了一个布伊的问题,他的问题显示,如果系数为正数,则增长率是可以计算(即,有算法输出(B,v)$(B,v)$(B)$(B,v)$(B)$(B,v)$(B)$(B)$(B,v)$(美元)的数值。我们的证据是基于减少计算一套基数组矩阵的光谱半径,以计算双线系统增长率。我们还利用我们的减法推论推论推论,没有一种算算出,在系统大小的双线系统上,在时间的多线系统的多线系统和美元($(z)的多线系统的计算中,即使所有系数都是非负数的。我们的两种结果维持着。即使所有系数都是非负。
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