A sparsity for decomposing a symmetric matrix (Q2365956)

A direct method based decomposition of a nonsingular symmetric matrix \(A\) is derived to reduce the complexity in inverting \(A\) and in solving large sparse sets of linear algebraic equations. \(A^{-1}\) is computed directly in the form of \(LDL^ T\), where \(L\) denotes a lower triangular matrix with unit diagonal coefficients and \(D\) is a diagonal matrix. Stair-shape sparsity of the matrix \(A\) employs a double bordered block diagonal form of sparse matrices in the form of element-based and block- based procedures. It can reduce the number of arithmetical operations and computer memory by preserving the structure of zeros of \(A\) when applying the proposed decomposition.