Accuracy and stability of quaternion Gaussian elimination

This paper is devoted to the accuracy and stability of quaternion Gaussian elimination (qGE). First, considering the noncommutativity of quaternion multiplications, we establish the rules of quaternion floating-point operations. After that, roundoff error analyses of quaternion LU and qGE with partial pivoting (qGEPP) are presented to show the backward stability of qGEPP is affected by the growth factor. Then, the upper bounds for growth factors of qGE with partial or complete pivoting, and the quaternion matrices with maximal growth factor are investigated. Finally, growth factors of some special matrices are also studied, and the theoretical results are confirmed through several numerical experiments. A conjecture about the growth factor of quaternion Hadamard matrix is given.