Factorization of cp-rank-3 completely positive matrices

A symmetric positive semi-definite matrix A is called completely positive if there exists a matrix B with nonnegative entries such that A = BB⊤. If B is such a matrix with a minimal number p of columns, then p is called the cp-rank of A. In this paper we develop a finite and exact algorithm to factorize any matrix A of cp-rank 3. Failure of this algorithm implies that A does not have cp-rank 3. Our motivation stems from the question if there exist three nonnegative polynomials of degree at most four that vanish at the boundary of an interval and are orthonormal with respect to a certain inner product.