KidzSearch Wiki

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 Sunaga[1] and R. Moore[2][3][4] is used in many areas including validated numerics.[5] But unfortunately, interval arithmetic is useless when numerical computation is repeated many times.[4] Therefore, many experts have studied how to overcome this weakness. Affine arithmetic is one result of this movement.

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, ISBN 1-85233-358-8.
  • Q. Zhang and R. R. Martin (2000), "Polynomial evaluation using affine arithmetic for curve drawing". Proc. of Eurographics UK 2000 Conference, 49–56. ISBN 0-9521097-9-4.
  • 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, ISBN 0-7923-7309-X.
  • 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, ISBN 0-7923-7309-X.
  • 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.

Improvements

Some experts are trying to improve affine arithmetic. Their results are known as the extended affine arithmetic[6][7][8] or modified affine arithmetic.[9][10]

Libraries

This a list of libraries that supports affine arithmetic:

References

  1. ↑ T. Sunaga, Theory of interval algebra and its application to numerical analysis. (1958). RAAG memoirs, 29–46.
  2. ↑ Stace, Clive A. (1991-10-03). Plant Taxonomy and Biosystematics. Cambridge University Press. ISBN 978-0-521-42785-2.
  3. ↑ Moore, R. E. (1979). Methods and applications of interval analysis. Society for Industrial and Applied Mathematics.
  4. ↑ 4.0 4.1 Moore, Ramon E. (2009). Introduction to interval analysis. R. Baker Kearfott, Michael J. Cloud. Philadelphia, PA: Society for Industrial and Applied Mathematics. ISBN 978-0-89871-669-6. OCLC 258333490.
  5. ↑ Jaulin, L. Kieffer, M., Didrit, O. Walter, E. (2001). Applied Interval Analysis. Berlin: Springer.
  6. ↑ Liao, 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.
  7. ↑ Messine, F., & Touhami, A. (2006). A general reliable quadratic form: An extension of affine arithmetic. Reliable Computing, 12(3), 171-192.
  8. ↑ Goubault, E., & Putot, S. (2008). Perturbed affine arithmetic for invariant computation in numerical program analysis. arXiv preprint arXiv:0807.2961.
  9. ↑ 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.
  10. ↑ 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.
  11. ↑ Overview of kv – a C++ library for verified numerical computation, Masahide Kashiwagi, SCAN 2018.

Further Reading

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.
Retrieved from "https://wiki.kidzsearch.com/w/index.php?title=Affine_arithmetic&oldid=5686191"