# Multithreading-Based Algorithm for High-Performance Tchebichef Polynomials with Higher Orders
## 3 The Proposed Multithreaded Algorithm
A high number of threads at small sizes (500 and 1000) leads to diminishing returns due to the threading overhead, and this applies to all of the numbers of cores considered. At larger sizes of 2000 and above, the results achieved by two or more cores show more improvement stability. The maximum improvement achieved by the two-core case is 4.4 at 32 threads at the 8000 size. On the other hand, the three-core case achieved an up to 5.79 improvement at 24 threads at the 8000 size. Clearly, the proposed algorithm, using multicore and multithreaded parallel TPs calculations, addresses the issue of long calculation times for large-scale dimensions. Not only does it solve this problem but it also improves the stability and accuracy of the solution, resulting in enhanced overall performance. Figure 11: Computation time and performance gain of the proposed algorithm for size 8000. ## 5 Conclusions

The computation of TPs represents a time-consuming operation, especially in high-order signals. This work proposed an approach to handle large-sized signals with performance enhancement while maintaining numerical stability. The proposed approach distributed the calculation workload among independent threads, thus taking advantage of the available parallel computing resources represented by the multicores. This approach divides the coefficients into four regions, where the first region represents an initialization. Parallelism within each of the three other regions is exploited and the workload is divided between a set of threads that run on the different cores. The proposed approach showed a clear improvement in performance, achieving an improvement of 7.2 times with respect to the unthreaded case. This maximum improvement was obtained at the highest size considered (8000) when the number of threads is set to 32 with 8 cores. It was noticed that increasing the number of threads does not necessarily result in performance improvement. A high number of threads introduces management overheads that, at some point, dimin

Figure 5:0 Improvement of the Multithreaded algorithm as a function of the number of threads for different numbers of cores and for sizes: (**a**) 500, (**b**) 1000, (**c**) 2000, (**d**) 4000, (**e**) 6000, and (**f**) 8000.

ish the speedup of parallel execution. This was clear in all sizes considered, but it was especially prominent at small sizes, namely 500 and 1000, where the improvement drops to 17.9% and 24.7% of the maximum improvement achieved. Multithreading-based TPs computation provides faster and more accurate calculations and brings new opportunities to algorithm designers. Moreover, implementing multithreaded-strategy-based orthogonal polynomial computations will be one of the important future directions in accelerating the computations in the field of signal processing. Methodology, S.H.A. and W.N.F. ; software, B.M.M., W.N.F. and A.H.A.-s.; validation, F.A.S., B.M.M., M.A. and S.H.A. ; formal analysis, W.N.F., F.A.S. and A.H.A.-s.; investigation, F.A.S., A.H.A.-s., B.M.M. and M.A. ; writing--original draft preparation, S.H.A., B.M.M., W.N.F. and A.H.A.-s; writing--review and editing, A.H.A.-s., F.A.S. and M.A. ; visualization, S.H.A., B.M.M. and M.A. ; supervision, S.H.A. ; project administration, W.N.F. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Not applicable. No new data were created or analyzed in this study. Data sharing is not applicable to this article. The authors declare no conflicts of interest. The following abbreviations are used in this manuscript:

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