# Low-Complexity Recursive Least-Squares Adaptive Algorithm Based on Tensorial Forms
## 3 Tensor-Based RLS Algorithms
### Tensor-Based Recursive Least-Squares Dichotomous Coordinate Descent Algorithm (RLS-DCD-T)  
We propose to use the combination between the dichotomous coordinate descent (DCD) iterations and the RLS method as an alternative solution for the systems of equations in (21). The RLS-DCD algorithm [20] was employed in the past due to its low complexity arithmetic workloads and improved numerical stability. The advantages obtained by using the generalized tensorial model applied with low-complexity RLS algorithms can lead to high convergence/tracking speeds and acceptable computational requirements, an overall design which is suitable for efficient hardware applications. As was similarly considered in [27; 20], we assume that each of the \(N\) filters corresponding to the tensor-based decomposition has generated a result at time index \(n-1\) for its associated systems in (21). A residual vector for each solution can be expressed as  
\[\mathbf{r}_{i}(n-1)=\mathbf{p}_{i}(n-1)-\mathbf{R}_{i}(n-1)\widehat{\mathbf{h }}_{i}(n-1) \tag{31}\]  
and several other notations can be employed to track the changes of the entities in (21) with respect to \(n\):\[\Delta\mathbf{R}_{i}(n) =\mathbf{R}_{i}(n)-\mathbf{R}_{i}(n-1), \tag{32}\] \[\Delta\mathbf{p}_{i}(n) =\mathbf{p}_{i}(n)-\mathbf{p}_{i}(n-1),\] (33) \[\Delta\mathbf{h}_{i}(n) =\mathbf{h}_{i}(n)-\widehat{\mathbf{h}}_{i}(n-1), \tag{34}\]  
where \(\Delta\mathbf{R}_{i}(n)\) is a \(L_{i}\times L_{i}\) matrix, and \(\Delta\mathbf{p}_{i}(n)\), respectively \(\Delta\mathbf{h}_{i}(n)\), are \(L_{i}\)-valued column vectors. The classical methods for directly solving the systems in (21) can be shifted towards determining the estimates \(\Delta\widehat{\mathbf{h}}_{i}(n)\) and adding the vectors to the estimates generated by the previous filter update process \(\widehat{\mathbf{h}}_{i}(n-1)\), in order to obtain new filter coefficients in \(\widehat{\mathbf{h}}_{i}(n)\). Consequently, for time index \(n\), (21) can be rewritten as:  
\[\mathbf{R}_{i}(n)\Big{[}\widehat{\mathbf{h}}_{i}(n-1)+\Delta\mathbf{h}_{i}(n) \Big{]}=\mathbf{p}_{i}(n). \tag{35}\]  
From (35), we can start formulating new systems of equations, similar to (21), with the purpose of determining the estimates \(\Delta\widehat{\mathbf{h}}_{i}(n)\). On the left side of (35), we extract the term  
\[\mathbf{R}_{i}(n)\Delta\mathbf{h}_{i}(n)=\mathbf{p}_{i}(n)-\mathbf{R}_{i}(n) \widehat{\mathbf{h}}_{i}(n-1)=\mathbf{p}_{0,i}(n). \tag{36}\]  
Furthermore, (36) can be developed by using (32) and (33), then (31), in order to eliminate \(\mathbf{p}_{i}(n)\), respectively \(\mathbf{R}_{i}(n)\), and we obtain:  
\[\mathbf{p}_{0,i}(n)=\mathbf{r}_{i}(n-1)+\Delta\mathbf{p}_{i}(n)-\Delta \mathbf{R}_{i}(n)\widehat{\mathbf{h}}_{i}(n-1). \tag{37}\]  
The reinterpretation of (21), which led to (36), generated \(N\) new systems of equations with the form  
\[\mathbf{R}_{i}(n)\Delta\mathbf{h}_{i}(n)=\mathbf{p}_{0,i}(n), \tag{38}\]  
where \(\Delta\widehat{\mathbf{h}}_{i}(n)\) are the new solutions that will be computed. The approach is equivalent to (21) because the new filter coefficients are determined with  
\[\widehat{\mathbf{h}}_{i}(n)=\widehat{\mathbf{h}}_{i}(n-1)+\Delta\widehat{ \mathbf{h}}_{i}(n). \tag{39}\]  
Moreover, the solutions obtained from (38) are based on \(\widehat{\mathbf{h}}_{i}(n-1)\) estimated at the previous time index, and the computation of the \(N\) increment vectors \(\Delta\widehat{\mathbf{h}}_{i}(n)\) is expected to require considerably less computational resources than the direct approaches. This makes the form in (38) suitable for iterative methods, such as the DCD iterations, which can also update the values of the residual vectors \(\mathbf{r}_{i}(n)\) using a minimum computational effort. They are required at the next time index, as can be noticed in (37).