This paper proposes two low-complexity iterative algorithms to compute the capacity of a single-user multiple-input multiple-output channel with per-antenna power constraint. The first method results from manipulating the optimality conditions of the considered problem and applying fixed-point iteration. In the second approach, we transform the considered problem into a minimax optimization program using the well-known MAC-BC duality, and then solve it by a novel alternating optimization method. In both proposed iterative methods, each iteration involves an optimization problem which can be efficiently solved by the water-filling algorithm. The proposed iterative methods are provably convergent. Complexity analysis and extensive numerical experiments are carried out to demonstrate the superior performance of the proposed algorithms over an existing approach known as the mode-dropping algorithm.