We present a new algorithm by which the Adomian polynomials can be determined for scalar-valued nonlinear polynomial functional in a Hilbert space. This algorithm calculates the Adomian polynomials without the complicated operations such as parametrization, expansion, regrouping, differentiation, etc. The algorithm involves only some matrix operations. Because of the simplicity in the mathematical operations, the new algorithm is faster and more efficient than the other algorithms previously reported in the literature. We also implement the algorithm in the MATHEMATICA code. The computing speed and efficiency of the new algorithm are compared with some other algorithms in the one-dimensional case.