We investigated a wider range of Winograd family convolution algorithms for Deep Neural Network. We presented the explicit Winograd convolution algorithm in general case (used the polynomials of the degrees higher than one). It allows us to construct more different versions in the aspect of performance than commonly used Winograd convolution algorithms and improve the accuracy and performance of convolution computations. We found that in @math this approach gives us better accuracy of image recognition while keeps the same number of general multiplications computed per single output point as the commonly used Winograd algorithm for a kernel of the size @math and output size equal to @math . We demonstrated that in @math it is possible to perform the convolution computation faster keeping the accuracy of image recognition the same as for direct convolution method. We tested our approach for a subset of @math images from Imaginet validation set. We present the results for three precision of computations @math , @math and @math .
 1. Introduction 2. Introduction to abstract algebra 3. Fast algorithms for the discrete Fourier transform 4. Fast algorithms based on doubling strategies 5. Fast algorithms for short convolutions 6. Architecture of filters and transforms 7. Fast algorithms for solving Toeplitz systems 8. Fast algorithms for trellis search 9. Numbers and fields 10. Computation in finite fields and rings 11. Fast algorithms and multidimensional convolutions 12. Fast algorithms and multidimensional transforms Appendices: A. A collection of cyclic convolution algorithms B. A collection of Winograd small FFT algorithms. For short data sequences, Winograd's convolution algorithms attaining the minimum number of multiplications also attain a low number of additions, making them very efficient. However, for longer lengths they require a larger number of additions. Winograd's approach is usually extended to longer lengths by using a nesting approach such as the Agarwal-Cooley (1977) or Split-Nesting algorithms. Although these nesting algorithms are organizationally quite simple, they do not make the greatest use of the factorability of the data sequence length. The algorithm we propose adheres to Winograd's original approach more closely than do the nesting algorithms. By evaluating polynomials over simple matrices we retain, in algorithms for longer lengths, the basic structure and strategy of Winograd's approach, thereby designing computationally refined algorithms. This tactic is arithmetically profitable because Winograd's approach is based on a theory of minimum multiplicative complexity. > Contents: Introduction to Abstract Algebra.- Tensor Product and Stride Permutation.- Cooley-Tukey FFF Algorithms.- Variants of FFT Algorithms and Their Implementations.- Good-Thomas PFA.- Linear and Cyclic Convolutions.- Agarwal-Cooley Convolution Algorithm.- Introduction to Multiplicative Fourier Transform Algorithms (MFTA).- MFTA: The Prime Case.- MFTA: Product of Two Distinct Primes.- MFTA: Transform Size N = Mr. M-Composite Integer and r-Prime.- MFTA: Transform Size N = p2.- Periodization and Decimation.- Multiplicative Character and the FFT.- Rationality.- Index.