linear algebra basic blocks

In embedded systems, ecient implementations of numerical algorithms typ- ically use the xed-point arithmetic rather than the standardized and costly oating-point arithmetic. But, xed-point programmers face two diculties: First, writing xed-point codes is tedious and error prone. Second, the low dynamic range of xed-point numbers leads to the persistent belief that xed- point computations are inherently inaccurate. In this article, we address these two limitations by introducing a methodology to design and implement tools that synthesize xed-point programs. To strengthen the user's condence in the synthesized code, analytic methods are presented to automatically assert its numerical quality. Furthermore, we use this framework to generate xed-point code for linear algebra basic blocks such as matrix multiplication and inversion. For example, the former task involves trade-os such as choosing to maximize the code's accuracy or minimize its size. For the two cases of matrix multi- plication and inversion, we describe, implement, and experiment with several algorithms to nd trade-os between the conicting goals.

[1]  Randy Yates,et al.  Fixed-Point Arithmetic: An Introduction , 2013 .

[2]  Matthias Mehlhose,et al.  Efficient fixed-point implementation of linear equalization for cooperative MIMO systems , 2009, 2009 17th European Signal Processing Conference.

[3]  Seehyun Kim,et al.  Fixed-point optimization utility for C and C++ based digital signal processing programs , 1998 .

[4]  Nicholas J. Higham,et al.  INVERSE PROBLEMS NEWSLETTER , 1991 .

[5]  Dong-U Lee,et al.  Optimized Custom Precision Function Evaluation for Embedded Processors , 2009, IEEE Transactions on Computers.

[6]  Milos D. Ercegovac,et al.  Digital Arithmetic , 2003, Wiley Encyclopedia of Computer Science and Engineering.

[7]  Thibault Hilaire,et al.  Sum-of-products evaluation schemes with fixed-point arithmetic, and their application to IIR filter implementation , 2012, Proceedings of the 2012 Conference on Design and Architectures for Signal and Image Processing.

[8]  Xin-She Yang,et al.  Introduction to Algorithms , 2021, Nature-Inspired Optimization Algorithms.

[9]  Ioannis Papaefstathiou,et al.  A fast parallel matrix multiplication reconfigurable unit utilized in face recognitions systems , 2009, 2009 International Conference on Field Programmable Logic and Applications.

[10]  R. Baker Kearfott,et al.  Introduction to Interval Analysis , 2009 .

[11]  Rob A. Rutenbar,et al.  Fast, Accurate Static Analysis for Fixed-Point Finite-Precision Effects in DSP Designs , 2003, ICCAD 2003.

[12]  Ryan Kastner,et al.  GUSTO: An automatic generation and optimization tool for matrix inversion architectures , 2010, TECS.

[13]  Zoran Nikolic,et al.  Design and Implementation of Numerical Linear Algebra Algorithms on Fixed Point DSPs , 2007, EURASIP J. Adv. Signal Process..

[14]  Daniel Ménard,et al.  Floating-to-Fixed-Point Conversion for Digital Signal Processors , 2006, EURASIP J. Adv. Signal Process..

[15]  Saudi Arabia,et al.  FPGA Design and Implementation of Matrix Multiplier Architectures for Image and Signal Processing Applications , 2010 .

[16]  Milos D. Ercegovac,et al.  Improving Goldschmidt Division, Square Root, and Square Root Reciprocal , 2000, IEEE Trans. Computers.

[17]  Sunil P. Khatri,et al.  Resource and delay efficient matrix multiplication using newer FPGA devices , 2006, GLSVLSI '06.

[18]  E. R. Hansen,et al.  A Generalized Interval Arithmetic , 1975, Interval Mathematics.

[19]  Gene H. Golub,et al.  Matrix computations , 1983 .

[20]  Joseph R. Guerci,et al.  Space-Time Adaptive Processing for Radar , 2003 .

[21]  Matthieu Martel,et al.  Code Size and Accuracy-aware Synthesis of Fixed-point Programs for Matrix Multiplication , 2014, PECCS.

[22]  Michael Ian Shamos,et al.  Closest-point problems , 1975, 16th Annual Symposium on Foundations of Computer Science (sfcs 1975).

[23]  Wonyong Sung,et al.  Simulation-based word-length optimization method for fixed-point digital signal processing systems , 1995, IEEE Trans. Signal Process..

[24]  Ling Qiu,et al.  A novel adaptive equalization algorithm for MIMO communication system , 2005, VTC-2005-Fall. 2005 IEEE 62nd Vehicular Technology Conference, 2005..

[25]  Wayne Luk,et al.  Ieee Transactions on Computer-aided Design of Integrated Circuits and Systems Accuracy Guaranteed Bit-width Optimization Abstract— We Present Minibit, an Automated Static Approach for Optimizing Bit-widths of Fixed-point Feedforward Designs with Guaranteed Accuracy. Methods to Minimize Both the In- , 2022 .