This study refines a single-layer Feed-Forward Neural Network (FFNN) for the treatment of textile dye wastewater, concentrating on percentage decolorization (%DEC) and percentage chemical oxygen demand (%COD) reduction. The optimized neural network configuration comprises four input and one output neuron, fine-tuned based on the mean squared error (MSE). The training phase demonstrates a consistent MSE decline, reaching its lowest at epoch 209 for %DEC and epoch 34 for %COD, with corresponding MSEs of $1.799\times {10}^{-5}$ and $1.4\times {10}^{-3}$, respectively. The maximum absolute errors for %DEC and %COD were found to be $4.0787$ and $2.4486$, while the mean absolute errors were $0.4821$ and $0.7256$, respectively. In contrast to second-degree polynomial regression, the FFNN model exhibits enhanced predictive accuracy, as indicated by higher $R^2$ values of $0.99363$ for %DEC and $0.99716$ for %COD, and reduced error metrics.

Understanding the eigenvalue distribution of sequence Toeplitz matrices has advanced significantly in recent years. Notable contributors include Bogoya, Grudsky, Böttcher, and Maximenko, who have derived precise asymptotic expansions for these eigenvalues under certain conditions related to the generating function as the matrix size increases. Building on this foundation, the Stefano Serra-Capizzano conjectured that, under certain assumptions about $\mathrm{\Omega }$ and $\mathrm{\Phi }$, a similar expansion may hold for the eigenvalues of a sequence of preconditioned Toeplitz matrices $T_n^{-1}(\mathrm{\Phi })T_n(\mathrm{\Omega })$, given a monotonic ratio $r=\mathrm{\Omega }/\mathrm{\Phi }$. In contrast to current eigenvalue solvers, this work presents a novel method for efficiently calculating the eigenvalues of a sequence of large preconditioned banded symmetric Toeplitz matrices (PBST). Our algorithm uses a higher-order spline fitting extrapolation technique to gather spectral data from a smaller sequence of PBST matrices in order to forecast the spectrum of bigger matrices.