Affine arithmetic

Affine arithmetic (AA) is a computer arithmetic which was made to improve the performance of interval arithmetic.

Background
Today, the interval arithmetic technology which was made by SunagaT. Sunaga, Theory of interval algebra and its application to numerical analysis. (1958). RAAG memoirs, 29–46. and R. MooreMoore, R. E. (1979). Methods and applications of interval analysis. Society for Industrial and Applied Mathematics. is used in many areas including validated numerics.Jaulin, L. Kieffer, M., Didrit, O. Walter, E. (2001). Applied Interval Analysis. Berlin: Springer. Unfortunately, interval arithmetic is useless when numerical computation is repeated many times. Therefore, many experts have studied how to overcome this weakness. Affine arithmetic is one result of this movement.

Applications
Affine arithmetic is available in some interval arithmetic libraries like INTLAB.S.M. Rump: INTLAB - INTerval LABoratory. In Tibor Csendes, editor, Developments in Reliable Computing, pages 77-104. Kluwer Academic Publishers, Dordrecht, 1999.S.M. Rump, M. Kashiwagi: Implementation and improvements of affine arithmetic, Nonlinear Theory and Its Applications (NOLTA), IEICE, 2015. It is also used in the following fields:

 Electrical circuitsFemia, N., & Spagnuolo, G. (2000). True worst-case circuit tolerance analysis using genetic algorithms and affine arithmetic. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 47(9), 1285-1296.Lemke, A., Hedrich, L., Barke, E., & Barke, E. (2002, November). Analog circuit sizing based on formal methods using affine arithmetic. In Proceedings of the 2002 IEEE/ACM international conference on Computer-aided design (pp. 486-489). ACM.Ding, T., Trinchero, R., Manfredi, P., Stievano, I. S., & Canavero, F. G. (2015). How affine arithmetic helps beat uncertainties in electrical systems. IEEE Circuits and Systems Magazine, 15(4), 70-79.
 Mixed signal systemsGrimm, C., Heupke, W., & Waldschmidt, K. (2004, February). Refinement of mixed-signals systems with affine arithmetic. In Proceedings Design, Automation and Test in Europe Conference and Exhibition (Vol. 1, pp. 372-377). IEEE.Grimm, C., Heupke, W., & Waldschmidt, K. (2004). Analysis of mixed-signal systems with affine arithmetic. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 24(1), 118-123.Radojicic, C., Grimm, C., Schupfer, F., & Rathmair, M. (2013). Verification of mixed-signal systems with affine arithmetic assertions. VLSI Design, 2013, 5.Radojicic, C., & Grimm, C. (2016, June). Formal verification of mixed-signal designs using extended affine arithmetic. In 2016 12th Conference on Ph. D. Research in Microelectronics and Electronics (PRIME) (pp. 1-4). IEEE.
 Numerical methods for ordinary differential equationsY. Kanazawa and S. Oishi (2002), "A numerical method of proving the existence of solutions for nonlinear ODEs using affine arithmetic". Proc. SCAN'02 — 10th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics.
 OptimizationF. Messine and A. Mahfoudi (1998), "Use of affine arithmetic in interval optimization algorithms to solve multidimensional scaling problems". Proc. SCAN'98 — IMACS/GAMM International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics (Budapest, Hungary), 22–25.F. Messine (2002), "Extensions of affine arithmetic: Application to unconstrained global optimization". Journal of Universal Computer Science, 8 11, 992–1015.
 Power flowVaccaro, A., Canizares, C. A., & Villacci, D. (2009). An affine arithmetic-based methodology for reliable power flow analysis in the presence of data uncertainty. IEEE Transactions on Power Systems, 25(2), 624-632.Gu, W., Luo, L., Ding, T., Meng, X., & Sheng, W. (2014). An affine arithmetic-based algorithm for radial distribution system power flow with uncertainties. International Journal of Electrical Power & Energy Systems, 58, 242-245.Wang, S., Han, L., & Wu, L. (2014). Uncertainty tracing of distributed generations via complex affine arithmetic based unbalanced three-phase power flow. IEEE Transactions on Power Systems, 30(6), 3053-3062.Pirnia, M., Cañizares, C. A., Bhattacharya, K., & Vaccaro, A. (2014). A novel affine arithmetic method to solve optimal power flow problems with uncertainties. IEEE Transactions on Power Systems, 29(6), 2775-2783.Vaccaro, A., & Canizares, C. A. (2016). An affine arithmetic-based framework for uncertain power flow and optimal power flow studies. IEEE Transactions on Power Systems, 32(1), 274-288.Ding, T., Bo, R., Guo, Q., Sun, H., Wu, W., & Zhang, B. (2013). A non-iterative affine arithmetic methodology for interval power flow analysis of transmission network.Ding, T., Cui, H., Gu, W., & Wan, Q. (2012). An uncertainty power flow algorithm based on interval and affine arithmetic. Automation of Electric Power Systems, 13.Pirnia, M., Cañizares, C. A., Bhattacharya, K., & Vaccaro, A. (2012, July). An affine arithmetic method to solve the stochastic power flow problem based on a mixed complementarity formulation. In 2012 IEEE Power and Energy Society General Meeting (pp. 1-7). IEEE.
 Root-finding algorithmsT. Kikuchi and M. Kashiwagi (2001), "Elimination of non-existence regions of the solution of nonlinear equations using affine arithmetic". Proc. NOLTA'01 — 2001 International Symposium on Nonlinear Theory and its Applications.M. Kashiwagi (1998), "An all solution algorithm using affine arithmetic". NOLTA'98 — 1998 International Symposium on Nonlinear Theory and its Applications (Crans-Montana, Switzerland), 14–17.

Improvements
Some experts are trying to improve affine arithmetic. Their results are known as the extended affine arithmeticLiao, X., Liu, K., Le, J., Zhu, S., Huai, Q., Li, B., & Zhang, Y. (2020). Extended affine arithmetic-based global sensitivity analysis for power flow with uncertainties. International Journal of Electrical Power & Energy Systems, 115, 105440.Messine, F., & Touhami, A. (2006). A general reliable quadratic form: An extension of affine arithmetic. Reliable Computing, 12(3), 171-192.Goubault, E., & Putot, S. (2008). Perturbed affine arithmetic for invariant computation in numerical program analysis. arXiv preprint arXiv:0807.2961. or modified affine arithmetic.Shou, H., Lin, H., Martin, R., & Wang, G. (2003). Modified affine arithmetic is more accurate than centered interval arithmetic or affine arithmetic. In Mathematics of Surfaces (pp. 355-365). Springer, Berlin, Heidelberg.Shou, H., Lin, H., Martin, R. R., & Wang, G. (2006). Modified affine arithmetic in tensor form for trivariate polynomial evaluation and algebraic surface plotting. Journal of Computational and Applied Mathematics, 195(1-2), 155-171.

Libraries
This a list of libraries that supports affine arithmetic:
 kvOverview of kv – a C++ library for verified numerical computation, Masahide Kashiwagi, SCAN 2018.
 Libaffa
 YalAA 
 ASOL

References

Further Reading
Applications

 L. H. de Figueiredo and J. Stolfi (1996), "Adaptive enumeration of implicit surfaces with affine arithmetic". Computer Graphics Forum, 15  5,  287–296.
 W. Heidrich (1997), "A compilation of affine arithmetic versions of common math library functions". Technical Report 1997-3, Universität Erlangen-Nürnberg.
 L. Egiziano, N. Femia, and G.  Spagnuolo (1998), "New approaches to the true worst-case evaluation in circuit tolerance and sensitivity analysis — Part II: Calculation of the outer solution using affine arithmetic". Proc. COMPEL'98 — 6th Workshop on Computer in Power Electronics (Villa Erba, Italy),  19–22.
 W. Heidrich, Ph. Slusallek, and H.-P. Seidel (1998), "Sampling procedural shaders using affine arithmetic". ACM Transactions on Graphics, 17  3,  158–176.
 A. de Cusatis Jr., L. H. Figueiredo, and M. Gattass (1999), "Interval methods for ray casting surfaces with affine arithmetic". Proc. SIBGRAPI'99 — 12th Brazilian Symposium on Computer Graphics and Image Processing,  65–71. 
 I. Voiculescu, J. Berchtold, A. Bowyer, R. R. Martin, and Q. Zhang (2000), "Interval and affine arithmetic for surface location of power- and Bernstein-form polynomials". Proc. Mathematics of Surfaces IX,  410–423. Springer, .
 Q. Zhang and R. R. Martin (2000), "Polynomial evaluation using affine arithmetic for curve drawing". Proc. of Eurographics UK 2000 Conference,  49–56. .
 N. Femia and G. Spagnuolo (2000), "True worst-case circuit tolerance analysis using genetic algorithm and affine arithmetic — Part I". IEEE Transactions on Circuits and Systems, 47  9,  1285–1296.
 R. Martin, H. Shou, I. Voiculescu, and G. Wang (2001), "A comparison of Bernstein hull and affine arithmetic methods for algebraic curve drawing". Proc. Uncertainty in Geometric Computations,  143–154. Kluwer Academic Publishers, .
 A. Bowyer, R. Martin, H. Shou, and I. Voiculescu (2001), "Affine intervals in a CSG geometric modeller". Proc. Uncertainty in Geometric Computations,  1–14. Kluwer Academic Publishers, .
 L. H. de Figueiredo, J. Stolfi, and L. Velho (2003), "Approximating parametric curves with strip trees using affine arithmetic". Computer Graphics Forum, 22  2,  171–179.
 C. F. Fang, T. Chen, and R. Rutenbar (2003), "Floating-point error analysis based on affine arithmetic". Proc. 2003 International Conf. on Acoustic, Speech and Signal Processing.
 A. Paiva, L. H. de Figueiredo, and J. Stolfi (2006), "Robust visualization of strange attractors using affine arithmetic". Computers & Graphics, 30  6,  1020– 1026.

Surveys
 L. H. de Figueiredo and J. Stolfi (2004) "Affine arithmetic: concepts and applications." Numerical Algorithms 37 (1–4), 147–158.
 J. L. D. Comba and J. Stolfi (1993), "Affine arithmetic and its applications to computer graphics". Proc. SIBGRAPI'93 — VI Simpósio Brasileiro de Computação Gráfica e Processamento de Imagens (Recife, BR),  9–18.
 Nedialkov, N. S., Kreinovich, V., & Starks, S. A. (2004). Interval arithmetic, affine arithmetic, Taylor series methods: why, what next?. Numerical Algorithms, 37(1-4), 325-336.

Category:Numerical analysis
category:Computer science