Fast algorithms are proposed for calculating error bounds for a numerically computed Perron root and vector of an irreducible nonnegative matrix. Particular emphasis is put on the computational efficiency of these algorithms which has only square complexity under an assumption. The error bounds for the root and vector are based on the Collatz-Wielandt theorem and estimating a solution of a linear system whose coefficient matrix is an $M$-matrix, respectively. We introduce a technique for obtaining smaller error bounds. Numerical results show efficiency of the algorithms.