A New Algorithm for Computing the Exponential of a Block Triangular Matrix

Abstract: The exponential of block triangular matrices arises in a wide range of scientific computing applications, including exponential integrators for solving systems of ordinary differential equations, Hamiltonian systems in control theory, sensitivity analysis, and option pricing in finance. We propose a novel algorithm exploiting the block triangular structure for simultaneously computing the exponentials of the diagonal blocks and the off-diagonal block of the matrix exponential without direct involvement of the full block matrix in the computations. This approach generalizes the work of Al-Mohy and Higham on the Fr\'echet derivative of the matrix exponential. The generalization is established through a linear operator framework, facilitating efficient evaluation schemes and rigorous backward error analysis. The algorithm employs the scaling and squaring method using diagonal Pad\'e approximants with algorithmic parameters selected based on the backward error analysis. A key feature is that the selection of the scaling parameter relies solely on the maximal norm of the diagonal blocks with no dependence on the norm of the off-diagonal block. Numerical experiments confirm that the proposed algorithm consistently outperforms existing algorithms in both accuracy and efficiency, making it a preferred choice for computing the matrix exponential of block triangular matrices.