This document presents a method for updating the memory table of a distributed arithmetic (DA) adaptive FIR filter without compromising convergence speed or requiring additional memory resources. DA reduces computational workload by precomputing and storing filter coefficient sums in a lookup table (LUT). However, updating the table is challenging for adaptive filters. The proposed method exploits temporal locality and subexpression sharing to fully update the table with each new sample. It reduces computations and maintains fast convergence. The method enables efficient implementation of computationally-intensive discrete wavelet transforms using DA's parallel architecture and maximal LUT utilization. Simulation results show the discrete wavelet transform can be computed with high LUT utilization using this adaptive DA filter design.