Proportionate-type normalized least mean square (PtNLMS) algorithms are developed and analyzed. PtNLMS algorithms offer low computational complexity and fast convergence times, for sparse impulse responses in network and acoustic echo cancellation applications. New PtNLMS algorithms are developed by choosing gains that optimize user-defined criteria, such as mean square error, at all times. PtNLMS algorithms are extended from real-valued signals to complex-valued signals. The computational complexity of the presented algorithms is examined.