ETD

• aaa
• arnoldi
• automatic
• differentiation
• functions
• implementation
• julia
• krylov
• matrix
• parlett
• rational
• schur

We present MatFun, a Julia package for computing dense and sparse matrix functions fully automatically (no user input required, other than the code to compute the function $f$ and the matrix $A$ themselves). This is achieved by combining specifically chosen algorithms and some peculiar feature of Julia. For dense matrices, the Schur-Parlett algorithm has been implemented, leveraging Julia's automatic differentiation capabilities. The algorithm has also been improved from a performance standpoint, making the Parlett recurrence cache-oblivious and enabling the whole procedure to work mostly in real arithmetic, for real inputs. For sparse matrices, we implemented a Rational Krylov method, alongside the AAA Rational Approximation. Given a function's samples, AAA is often able to accurately identify its poles, which can then be used by the Rational Krylov method itself for the approximation of $f(A)b$. The accuracy and performance of the algorithms are evaluated, in comparison with existing specialized methods.