RN-Functions and Related Concepts Adaptive Mixed-Precision Rational Krylov Eigensolvers with Residual- and Condition-Aware Precision Control for Large-Scale Nonlinear Eigenvalue Problems

Abstract

Large-scale nonlinear eigenvalue problems require repeated sparse shifted solves whose cost and memory traffic dominate rational Krylov eigensolvers. Uniform binary64 arithmetic is reliable but expensive, whereas uniform binary32 arithmetic can stagnate before stringent backward-error tolerances are reached.
This study develops RAMP-RK, an adaptive mixed-precision rational Krylov method. Sparse factorization and triangular solves begin in complex binary32, while basis storage, two-pass reorthogonalization, projected extraction, nonlinear residual evaluation, and certification remain in complex binary64. A precision-adequacy ratio combines the current normwise backward error with a projected generalized-eigenvalue condition surrogate. When binary32 becomes inadequate, the method selects either mixed-precision Newton polishing with iterative refinement or a guarded binary64 rational-Krylov restart. Three sparse benchmark families were tested for grid sizes 30 to 80, with linearized dimensions up to 19,200.
Uniform FP32 failed to meet the 10⁻¹⁰ tolerance on all 12 tested instances, whereas RAMP-RK converged on every case. For the largest problems, RAMP-RK achieved backward errors from 7.69×10⁻¹⁶ to 6.13×10⁻¹¹ and speedups of 1.15-1.45 over the FP64 baseline. On the strongly nonnormal family, condition-aware switching avoided seven unsuccessful Newton steps and reduced time from 1.534 s for a residual-only controller to 0.745 s.
Residual and sensitivity information can safely govern arithmetic precision inside rational Krylov NEP solvers. RAMP-RK preserves binary64-level certification while reducing high-precision shifted-solve work and avoiding unreliable low-precision corrections.