We consider the solution of the balancing-related frequency-weighted model and controller reduction problems using accuracy enhanced numerical algorithms. We propose first new stability-enforcing choices of the frequency-weighted grammians which can guarantee the stability of reduced models for two-sided frequency weights. Then we show that for the frequency-weighted controller reduction problems with standard stability and performance-enforcing frequency weights the computation of the frequency-weighted grammians can be done by solving reduced order Lyapunov equations. For both frequency-weighted model and controller reduction problems we indicate how to compute the grammians directly in terms of their Cholesky factors. This allows the extension of the square-root and balancing-free accuracy-enhancing techniques to the frequency-weighted case. (C) 2003 Elsevier Science Ltd. All rights reserved.