15th June 2022

# HSL_MP54: Parallel Cholesky solver

For a matrix that is full, symmetric and positive definite, this package performs parallel partial and complete factorisations and solutions of corresponding sets of equations, using OpenMP.

We consider the factorization $\mathbf{A} = \left(\begin{array}{cc} \mathbf{A}_{11} & \mathbf{A}_{21}^T \\ \mathbf{A}_{21} & \mathbf{A}_{22} \end{array}\right) = \left(\begin{array}{cc} \mathbf{L}_{11} & \\ \mathbf{ L}_{21} & \mathbf{I} \end{array}\right) \left(\begin{array}{cc} \mathbf{I} & \\ & \mathbf{S}_{22} \end{array}\right) \left(\begin{array}{cc} \mathbf{L}_{11}^T & \mathbf{L}_{21}^T \\ & \mathbf{I} \end{array}\right) = \mathbf{LSL}^T$ where $$\mathbf{A}$$ is order $$n$$, $$\mathbf{L}_{11}$$ is lower triangular and both $$\mathbf{A}_{11}$$ and $$\mathbf{L}_{11}$$ have order $$p\le n$$.

Subroutines are also provided for the complementary partial forward and backward substitutions, that is, solving $\mathbf{ LX = B} \quad\mathrm{and}\quad \mathbf{ L}^T\mathbf{X=B}.$

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