We introduce REPTILE, a compiler that performs tiling optimizations for programs expressed as mathematical recurrence equations. REPTILE recursively decomposes a recurrence program into a set of unique tiles and then simplifies each into a different set of recurrences. Given declarative user specifications of recurrence equations, optimizations, and optional mappings of recurrence subexpressions to external libraries calls, REPTILE generates C code that composes compiler-generated loops with calls to external hand-optimized libraries. We show that for direct linear solvers expressible as recurrence equations, the generated C code matches and often exceed the performance of standard hand-optimized libraries. We evaluate REPTILE’s generated C code against hand-optimized implementations of linear solvers in Intel MKL, as well as two non-solver recurrences from bioinformatics: Needleman-Wunsch and Smith-Waterman. When the user provides good tiling specifications, REPTILE achieves parity with MKL, achieving between 0.79–1.27x speedup for the LU decomposition, 0.97–1.21x speedup for the Cholesky decomposition, 1.61x–2.72x for lower triangular matrix inversion, 1.01–1.14x speedup for triangular solve with multiple right-hand sides, and 1.14–1.73x speedup over handwritten implementations of the bioinformatics recurrences.