Worked on stabilizing the LU factorization workflow in the JuliaSparse/SparseArrays.jl repository, focusing on resolving a critical bug in the UMFPACK lu!() function when the reuse_symbolic option was set to false. Addressed undefined variable issues by refining the algorithm and introduced targeted regression tests to validate both reuse_symbolic settings, thereby reducing the risk of future regressions. Employed Julia and numerical methods to enhance the reliability of sparse linear algebra operations. Demonstrated a test-driven approach by expanding coverage and ensuring robust handling of edge cases, ultimately improving the stability and user experience for production deployments involving sparse matrices.