Optimal algorithms of Gram-Schmidt type

TL;DR

This paper introduces three efficient Gram-Schmidt type algorithms for orthogonal decomposition of forms over division rings, achieving optimal or near-optimal complexity and including parallel implementations.

Contribution

It presents new algorithms that improve the efficiency of orthogonal decomposition, matching matrix multiplication complexity and enabling parallel computation.

Findings

First algorithm uses approximately d^3/3 operations with simple implementation

Second algorithm achieves optimal sequential complexity matching matrix multiplication

Third algorithm provides a parallel NC implementation with similar efficiency

Abstract

Three algorithms of Gram-Schmidt type are given that produce an orthogonal decomposition of finite $d$-dimensional symmetric, alternating, or Hermitian forms over division rings. The first uses $d^3/3+O(d^2)$ ring operations with very simple implementation. Next, that algorithm is adapted in two new directions. One is an optimal sequential algorithm whose complexity matches the complexity of matrix multiplication. The other is a parallel NC algorithm with similar complexity.