Symmetric Storage Gaxpy

Matrix Computation · 19 Jul 2021

Introduction

A matrix \(\color{black}{A\in\mathcal{R}^{n\times n}}\) is symmetric if \(\color{black}{A^{T}=A}\) and skew-symmetric if \(\color{black}{A^{T}=-A}\). Likewise, a matrix \(\color{black}{A\in\mathcal{C}^{n\times n}}\) is hermitian if \(\color{black}{A^{H}=A}\) and skew-hermitian if \(\color{black}{A^{H}=-A}\).

For such matrices, storage requirements can be halved by simply storing the lower triangle of elements.

For general \(\color{black}{n}\), we set

\[\color{black}{A.vec((n-j/2)(j-1)+i)=a_{ij} 1\le j\le i \le n}\]

for j = 1:n
    for i = 1:j-1
        y(i) = y(i) + A.vec((i-1)n - i(i-1)/2 + j)x(j)
    end
    for i = j:n
        y(i) = y(i) + A.vec((j-1)n - j(j-1)/2 + i)x(j)
    end
end

This algorithm requires the same \(\color{black}{2n^2}\) flops that an ordinary gaxpy requires.