Algorithms for implementing roots, inverse and inverse roots in hardware

In applications as in future MIMO communication systems a massive computation of complex matrix operations, such as QR decomposition, is performed. In these matrix operations, the functions roots, inverse and inverse roots are computed in large quantities. Therefore, to obtain high enough performance in such applications, efficient algorithms are highly important. Since these algorithms need to be realized in hardware it must also be ensured that they meet high requirements in terms of small chip area, low computation time and low power consumption. Power consumption is particularly important since many applications are battery powered. For most unary functions, directly applying an approximation methodology in a straightforward way will not lead to an efficient implementation. Instead, a dedicated algorithm often has to be developed. The functions roots, inverse and inverse roots are in this category. The developed approaches are founded on working in a floating-point format. For the roots functions also a change of number base is used. These procedures not only enable simpler solutions but also increased accuracy, since the approximation algorithm is performed on a mantissa of limited range. As a summarizing example the inverse square root is chosen. For comparison, the inverse square root is implemented using two methodologies: Harmonized Parabolic Synthesis and NewtonRaphson method. The novel methodology, Harmonized Parabolic Synthesis (HPS), is chosen since it has been demonstrated to provide very efficient approximations. The Newton-Raphson (NR) method is chosen since it is known for providing a very efficient implementation of the inverse square root. It is also commonly used in signal processing applications for computing approximations on fixed-point numbers of a limited range. Four implementations are made; HPS with 32 and 512 interpolation intervals and NR with 1 and 2 iterations. Summarizing the comparisons of the hardware performance, the implementations HPS 32, HPS 512 and NR 1 are comparable when it comes to hardware performance, while NR 2 is much worse. However, HPS 32 stands out in terms of better performance when it comes to the distribution of the error. Index Terms Approximation, unary functions, elementary functions, arithmetic computation, root, inverse, inverse roots, harmonized parabolic synthesis, Newton-Raphson method.