Multiplicative complexity, convolution, and the DFT
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1. Introduction.- 1.1. An Overview of Multiplicative Complexity.- 1.2. Why Count Only Multiplications and Divisions?.- 1.3. Organization.- 2. Multiplicative Complexity of Linear and Bilinear Systems.- 2.1. Historical Perspective.- 2.2. Definitions and Basic Results.- 2.3. Semilinear Systems.- 2.4. Quadratic and Bilinear Systems.- 2.4.1. Properties of Quadratic Systems.- 2.4.2. Bilinear Systems and Noncommutative Algorithms.- 2.4.3. Direct Products and Direct Sums of Systems.- 2.5. Summary of Chapter 2.- 3. Convolution and Polynomial Multiplication.- 3.1. Aperiodic Convolution/Polynomial Multiplication.- 3.2. Polynomial Multiplication Modulo an Irreducible Polynomial.- 3.3. Polynomial Multiplication Modulo a General Polynomial.- 3.4. Products of a Fixed Polynomial with Several Polynomials.- 3.4.1. Equivalent Systems and Row Reduced Forms.- 3.4.2. Reduction and Inflation Mappings.- 3.4.3. Equivalence of Products with a Fixed Polynomial.- 3.4.4. Multiplicative Complexity Results.- 3.5. Products with Several Fixed Polynomials in the Same Ring.- 3.6. Products with Several Fixed Polynomials in Different Rings.- 3.7. Multivariate Polynomial Multiplication.- 3.7.1. Polynomial Products in Two Variables.- 3.7.2. Multidimensional Cyclic Convolution.- 3.8. Summary of Chapter 3.- 4. Constrained Polynomial Multiplication.- 4.1. General Input Constraints.- 4.2. Multiplication by a Symmetric Polynomial.- 4.3. Multiplication by an Antisymmetric Polynomial.- 4.4. Products of Two Symmetric Polynomials.- 4.5. Polynomial Multiplication with Restricted Outputs.- 4.5.1. Decimation of Outputs.- 4.5.2. Other Output Restrictions.- 4.6. Summary of Chapter 4.- 5. Multiplicative Complexity of Discrete Fourier Transform.- 5.1. The Discrete Fourier Transform.- 5.2. Prime Lengths.- 5.2.1. Rader's Permutation.- 5.2.2. Multiplicative Complexity.- 5.3. Powers of Prime Lengths.- 5.4. Power-of-Two Lengths.- 5.5. Arbitrary Lengths.- 5.6. DFTs with Complex-Valued Inputs.- 5.7. Multidimensional DFTs.- 5.8. Summary of Chapter 5.- 6. Restricted and Constrained DFTs.- 6.1. Restricting DFT Outputs to One Point.- 6.2. Constraining DFT Inputs to One Point.- 6.3. DFTs with Symmetric Inputs.- 6.4. Discrete Hartley Transform.- 6.5. Discrete Cosine Transform.- 6.6. Summary of Chapter 6.- Appendix A. Cyclotomic Polynomials and Their Properties.- Appendix B. Complexities of Multidimensional Cyclic Convolutions.- Appendix C. Programs for Computing Multiplicative Complexity.- Appendix D. Tabulated Complexities of the One-Dimensional DFT.- Problems.