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We consider nonparametric adaptive spectral analysis of complex-valued data sequences with missing samples occurring in arbitrary patterns. We first present two high-resolution missing-data spectral estimation algorithms: the Iterative Adaptive Approach (IAA) and the Sparse Learning via Iterative Minimization (SLIM) method. Both algorithms can significantly improve the spectral estimation performance, including enhanced resolution and reduced sidelobe levels. Moreover, we consider fast implementations of these algorithms using the Conjugate Gradient (CG) technique and the Gohberg-Semencul-type (GS) formula. Our proposed implementations fully exploit the structure of the steering matrices and maximize the usage of the fast Fourier transform (FFT), resulting in much lower computational complexities as well as much reduced memory requirements. The effectiveness of the adaptive spectral estimation algorithms is demonstrated via several numerical examples including both 1-D spectral estimation and 2-D interrupted synthetic aperture radar (SAR) imaging examples.