We derive an analytical calculation formula for the channel capacity of a classical channel without any iteration while its existing algorithms require iterations and the number of iteration depends on the required precision level. Hence, our formula is its first analytical formula without any iteration. We apply the obtained formula to examples and see how the obtained formula works in these examples. Then, we extend it to the channel capacity of a classical-quantum (cq-) channel. Many existing studies proposed algorithms for a cq-channel and all of them require iterations. Our extended analytical algorithm have also no iteration and output the exactly optimum values.

翻译：我们推导出一种无需迭代即可计算经典信道容量的解析公式，而现有算法均需迭代，且迭代次数取决于所需精度。因此，该公式是首个无需任何迭代的解析公式。我们将所得公式应用于实例，考察其在实例中的运作方式，进而将其推广至经典-量子（cq）信道的容量计算。现有诸多研究提出的cq信道算法均需迭代，而我们的推广解析算法同样无需迭代，并能直接输出精确最优值。