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 and R. Moore is used in many areas including validated numerics. 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. It is also used in the following fields:

- Electrical circuits
- Mixed signal systems
- Numerical methods for ordinary differential equations
- Optimization
- Power flow
- Root-finding algorithms

### Improvements

Some experts are trying to improve affine arithmetic. Their results are known as the extended affine arithmetic or modified affine arithmetic.

## Libraries

This a list of libraries that supports affine arithmetic:
- kv
- Libaffa
- YalAA
- ASOL

### 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.