INTLAB
INTLAB (INTerval LABoratory) is an interval arithmetic library[1][2][3][4] using MATLAB and GNU Octave, available in Windows and Linux, macOS. It was developed by S.M. Rump from Hamburg University of Technology. INTLAB was used to develop other MATLAB-based libraries such as VERSOFT[5] and INTSOLVER,[6] and it was used to solve some problems in the Hundred-dollar, Hundred-digit Challenge problems.[7]
Version history
[edit]- 12/30/1998 Version 1
- 03/06/1999 Version 2
- 11/16/1999 Version 3
- 03/07/2002 Version 3.1
- 12/08/2002 Version 4
- 12/27/2002 Version 4.1
- 01/22/2003 Version 4.1.1
- 11/18/2003 Version 4.1.2
- 04/04/2004 Version 5
- 06/04/2005 Version 5.1
- 12/20/2005 Version 5.2
- 05/26/2006 Version 5.3
- 05/31/2007 Version 5.4
- 11/05/2008 Version 5.5
- 05/08/2009 Version 6
- 12/12/2012 Version 7
- 06/24/2013 Version 7.1
- 05/10/2014 Version 8
- 01/22/2015 Version 9
- 12/07/2016 Version 9.1
- 05/29/2017 Version 10
- 07/24/2017 Version 10.1
- 12/15/2017 Version 10.2
- 01/07/2019 Version 11
- 03/06/2020 Version 12
Functionality
[edit]INTLAB can help users to solve the following mathematical/numerical problems with interval arithmetic.
- Numerical linear algebra[1][2][3][4] (Not only solving matrix systems or eigenvalue problems, INTLAB can handle the least squares, Hessian matrix,[1][3] and verify the positive definiteness of a given matrix[8])
- root-finding algorithm[1][3][4]
- Affine arithmetic[1][9]
- Solving ODEs rigorously (This feature includes external tools such as the AWA toolbox and the Taylor model toolbox)[1][3][10]
- Automatic differentiation[1][3][4][11]
- Numerical integration[1][3]
- Fast Fourier transform[1]
- Rigorously compute the gamma function[12]
Works cited by INTLAB
[edit]INTLAB is based on the previous studies of the main author, including his works with co-authors.
- S. M. Rump: Fast and Parallel Interval Arithmetic, BIT Numerical Mathematics 39(3), 539–560, 1999.
- S. Oishi, S. M. Rump: Fast verification of solutions of matrix equations, Numerische Mathematik 90, 755–773, 2002.
- T. Ogita, S. M. Rump, and S. Oishi. Accurate Sum and Dot Product, SIAM Journal on Scientific Computing (SISC), 26(6):1955–1988, 2005.
- S.M. Rump, T. Ogita, and S. Oishi. Fast High Precision Summation. Nonlinear Theory and Its Applications (NOLTA), IEICE, 1(1), 2010.
- S.M. Rump: Ultimately Fast Accurate Summation, SIAM Journal on Scientific Computing (SISC), 31(5):3466–3502, 2009.
- S.M. Rump, T. Ogita, and S. Oishi: Accurate Floating-point Summation I: Faithful Rounding. SIAM Journal on Scientific Computing (SISC), 31(1): 189–224, 2008.
- S. M. Rump, T. Ogita, and S. Oishi: Accurate Floating-point Summation II: Sign, K-fold Faithful and Rounding to Nearest. SIAM Journal on Scientific Computing (SISC), 31(2):1269–1302, 2008.
- S. M. Rump: Ultimately Fast Accurate Summation, SIAM Journal on Scientific Computing (SISC), 31(5):3466–3502, 2009.
- S. M. Rump. Accurate solution of dense linear systems, Part II: Algorithms using directed rounding. Journal of Computational and Applied Mathematics (JCAM), 242:185–212, 2013.
- S. M. Rump. Verified Bounds for Least Squares Problems and Underdetermined Linear Systems. SIAM Journal of Matrix Analysis and Applications (SIMAX), 33(1):130–148, 2012.
- S. M. Rump: Improved componentwise verified error bounds for least squares problems and underdetermined linear systems, Numerical Algorithms, 66:309–322, 2013.
- R. Krawzcyk, A. Neumaier: Interval slopes for rational functions and associated centered forms, SIAM Journal on Numerical Analysis 22, 604–616 (1985)
- S. M. Rump: Expansion and Estimation of the Range of Nonlinear Functions, Mathematics of Computation 65(216), pp. 1503–1512, 1996.
External links
[edit]See also
[edit]References
[edit]- ^ a b c d e f g h i S.M. Rump: INTLAB – INTerval LABoratory. In Tibor Csendes, editor, Developments in Reliable Computing, pages 77–104. Kluwer Academic Publishers, Dordrecht, 1999.
- ^ a b Moore, R. E., Kearfott, R. B., & Cloud, M. J. (2009). Introduction to Interval Analysis. Society for Industrial and Applied Mathematics.
- ^ a b c d e f g Rump, S. M. (2010). Verification methods: Rigorous results using floating-point arithmetic. Acta Numerica, 19, 287–449.
- ^ a b c d Hargreaves, G. I. (2002). Interval analysis in MATLAB. Numerical Algorithms, (2009.1).
- ^ Rohn, J. (2009). VERSOFT: verification software in MATLAB/INTLAB.
- ^ Montanher, T. M. (2009). Intsolver: An interval based toolbox for global optimization. Version 1.0.
- ^ Bornemann, F., Laurie, D., & Wagon, S. (2004). The SIAM 100-digit challenge: a study in high-accuracy numerical computing. Society for Industrial and Applied Mathematics.
- ^ S. M. Rump: Verffication of positive definiteness, BIT Numerical Mathematics, 46 (2006), 433–452.
- ^ S.M. Rump, M. Kashiwagi: Implementation and improvements of affine arithmetic, Nonlinear Theory and Its Applications (NOLTA), IEICE, 2015.
- ^ Lohner, R. J. (1987). Enclosing the solutions of ordinary initial and boundary value problems. Computer arithmetic, 225–286.
- ^ L.B. Rall: Automatic Differentiation: Techniques and Applications, Lecture Notes in Computer Science 120, Springer, 1981.
- ^ S.M. Rump. Verified sharp bounds for the real gamma function over the entire floating-point range. Nonlinear Theory and Its Applications (NOLTA), IEICE, Vol.E5-N, No. 3, July, 2014.