  The $p$-spectral radius of a uniform hypergraph covers many important
concepts, such as Lagrangian and spectral radius of the hypergraph, and is
crucial for solving spectral extremal problems of hypergraphs. In this paper,
we establish a spherically constrained maximization model and propose a
first-order conjugate gradient algorithm to compute the $p$-spectral radius of
a uniform hypergraph (CSRH). By the semialgebraic nature of the adjacency
tensor of a uniform hypergraph, CSRH is globally convergent and obtains the
global maximizer with a high probability. When computing the spectral radius of
the adjacency tensor of a uniform hypergraph, CSRH stands out among existing
approaches. Furthermore, CSRH is competent to calculate the $p$-spectral radius
of a hypergraph with millions of vertices and to approximate the Lagrangian of
a hypergraph. Finally, we show that the CSRH method is capable of ranking
real-world data set based on solutions generated by the $p$-spectral radius
model.
