Composite Iterative Algorithm and Architecture for q-th Root Calculation

An algorithm for the q-th root extraction, q being any integer, is presented in this paper. The algorithm is based on an optimized implementation of X^{1/q} by a sequence of parallel and/or overlapped operations: (1) reciprocal, (2) digit-recurrence logarithm, (3) left-to-right carry-free multiplication and (4) on-line exponential. A detailed error analysis and two architectures are proposed, for low precision q and for higher precision q. The execution time and hardware requirements are estimated for single precision floating-point computations for several radices, this helps to determine which radices result in the most efficient implementations. The architectures proposed improve the features of other architectures for q-th root extraction.

[1]  Javier D. Bruguera,et al.  Algorithm and architecture for logarithm, exponential, and powering computation , 2004, IEEE Transactions on Computers.

[2]  José Alejandro Piñeiro Riobó.  Algorithms and architectures for elementary function computation , 2003 .

[3]  James Demmel,et al.  IEEE Standard for Floating-Point Arithmetic , 2008 .

[4]  Jie Cheng,et al.  High-performance architectures for elementary function generation , 2001, Proceedings 15th IEEE Symposium on Computer Arithmetic. ARITH-15 2001.

[5]  Naofumi Takagi.  Powering by a Table Look-Up and a Multiplication with Operand Modification , 1998, IEEE Trans. Computers.

[6]  Naofumi Takagi.  A Digit-Recurrence Algorithm for Cube Rooting , 2001 .

[7]  Kishor S. Trivedi,et al.  On-line algorithms for division and multiplication , 1975, 1975 IEEE 3rd Symposium on Computer Arithmetic (ARITH).

[8]  Tomás Lang,et al.  Fast Multiplication Without Carry-Propagate Addition , 1990, IEEE Trans. Computers.

[9]  Javier D. Bruguera,et al.  Faithful powering computation using table look-up and a fused accumulation tree , 2001, Proceedings 15th IEEE Symposium on Computer Arithmetic. ARITH-15 2001.

[10]  Jean-Michel Muller,et al.  Elementary Functions: Algorithms and Implementation , 1997 .

[11]  Javier D. Bruguera,et al.  A Digit-by-Digit Algorithm for mth Root Extraction , 2007, IEEE Transactions on Computers.

[12]  J. Muller.  Elementary Functions , 1997, Birkhäuser Boston.

[13]  Javier D. Bruguera,et al.  A Radix-2 Digit-by-Digit Architecture for Cube Root , 2008, IEEE Transactions on Computers.

[14]  Tomás Lang,et al.  On-the-Fly Rounding , 1992, IEEE Trans. Computers.

[15]  Jun Cao,et al.  High-performance hardware for function generation , 1997, Proceedings 13th IEEE Sympsoium on Computer Arithmetic.

[16]  I. S. Gradshteyn,et al.  1 – ELEMENTARY FUNCTIONS , 1980 .

[17]  M.D. Ercegovac.  On Digit-by-Digit Methods for Computing Certain Functions , 2007, 2007 Conference Record of the Forty-First Asilomar Conference on Signals, Systems and Computers.

[18]  Tomás Lang,et al.  Digit-Serial Arithmetic , 2004 .