Optimal algorithms of Gram–Schmidt type

Three algorithms of Gram–Schmidt type are given that produce orthogonal decompositions of finite d-dimensional symmetric, alternating, or Hermitian forms over division rings. The first uses d3/3+O(d2) products in Δ. Next, that algorithm is adapted in two new directions. One is a nearly optimal sequential algorithm whose complexity matches the complexity of matrix multiplication. The other is a parallel algorithm.