---
title: "Verification methods: rigorous results using floating-point arithmetic"
description: "Ordinary floating-point computations give approximate answers with no guarantee of accuracy. 225 references, cited by 9 papers here."
authors: ["Siegfried M. Rump"]
year: 2010
venue: "Acta Numerica 19:287–449"
source: "https://www.tuhh.de/ti3/publications.shtml?author=rump"
licence: "author's copy — read only"
summary_licence: "Elamur Cosmic License"
canonical: "https://research.elamur.ai/free/en/papers/rump-2010-verification-methods-rigorous-results-floating/"
---
[Elamur Research](https://md.research.elamur.ai/free/en/) · [this page in HTML](https://research.elamur.ai/free/en/papers/rump-2010-verification-methods-rigorous-results-floating/)

[home](https://md.research.elamur.ai/free/en/) / [papers](https://md.research.elamur.ai/free/en/papers/) / [2010](https://md.research.elamur.ai/free/en/papers/2010/) / Verification methods: rigorous results using floating-point arithmetic

# Verification methods: rigorous results using floating-point arithmetic

**Authors:** [Siegfried M. Rump](https://md.research.elamur.ai/free/en/authors/siegfried-m-rump/) · **Year:** 2010 · **Venue:** Acta Numerica 19:287–449 · **Publisher:** [Cambridge University Press](https://md.research.elamur.ai/free/en/publishers/cambridge-university-press/) · **Licence:** author's copy — read only

**Links:** [Publisher's page](https://doi.org/10.1017/s096249291000005x) · [PDF](https://www.tuhh.de/ti3/paper/rump/Ru10.pdf) · [OpenAlex](https://openalex.org/W2170345902)

**Citations:** 276 citations (OpenAlex, 2026-10-14) · 9 in our library, of the 149,439 papers whose bibliographies we read

## Summary
Ordinary floating-point computations give approximate answers with no guarantee of accuracy. Verification methods address this by producing mathematically rigorous bounds, for example proving that a true solution exists inside a computed enclosure. This survey by Rump explains how such methods are built and where they break down. It has three parts of similar size. Part 1 covers the basic tools: floating-point arithmetic, error-free transformations, directed rounding and interval arithmetic, along with historical remarks. Part 2 treats finite-dimensional problems such as dense and sparse linear systems, nonlinear equations, polynomial roots, eigenvalue bounds, quadrature, automatic differentiation and optimization. Part 3 handles infinite-dimensional problems, namely boundary value problems for differential equations. A recurring theme is that interval arithmetic earned a poor reputation through misuse, and that rigorous algorithms need different design principles from conventional numerical ones. Many algorithms are given as runnable MATLAB/INTLAB code, so readers can test them directly.

## Key points
- Published in Acta Numerica 19 (2010), pp. 287–449. The section on semilinear elliptic boundary value problems was written by Michael Plum (Karlsruhe).
- A probabilistic argument shows that interval Gaussian elimination (IGA) fails even for small, well-conditioned random matrices.
- Topics include M-matrices, simple and multiple roots and eigenvalues (including clusters), semidefinite programming, and enclosures of gradients, Hessians, Taylor coefficients and slopes.
- Global optimization is covered only briefly, with readers referred to Neumaier's 2004 Acta Numerica survey.
- In 2004, Bornemann, Laurie, Wagon and Waldvogel relied on INTLAB, the MATLAB toolbox for reliable computing, to solve half of Trefethen's SIAM 10 × 10-digit challenge problems.

*Our summary of the paper, in our own words — not its text. The work belongs to its authors; read it at the links above.*

## Institutions

- [Hamburg University of Technology](https://md.research.elamur.ai/free/en/universities/hamburg-university-of-technology/) — University, Hamburg, Germany · 3 works · 2,676 citations
- [Waseda University](https://md.research.elamur.ai/free/en/universities/waseda-university/) — University, Tokyo, Japan · 22 works · 869 citations

📚 Shelves (5): [🖥️ Computer Science](https://md.research.elamur.ai/free/en/shelves/computer-science/) · 19,503 works · [🤖 Artificial Intelligence](https://md.research.elamur.ai/free/en/shelves/computer-science/artificial-intelligence/) · 18,805 works · [➗ Mathematics](https://md.research.elamur.ai/free/en/shelves/mathematics/) · 233 works · [🧮 Computational Theory and Mathematics](https://md.research.elamur.ai/free/en/shelves/computer-science/computational-theory-and-mathematics/) · 496 works · [➗ Modeling and Simulation](https://md.research.elamur.ai/free/en/shelves/mathematics/modeling-and-simulation/) · 12 works

## Cite

Cite as: Siegfried M. Rump. 2010. Verification methods: rigorous results using floating-point arithmetic. Acta Numerica 19:287–449.

The paper itself, not this page. BibTeX from the library's record of the work.

```bibtex
@inproceedings{rump2010verification,
  title = {{Verification methods: rigorous results using floating-point arithmetic}},
  author = {Siegfried M. Rump},
  booktitle = {Acta Numerica 19:287–449},
  year = {2010},
  doi = {10.1017/s096249291000005x},
  url = {https://doi.org/10.1017/s096249291000005x}
}
```

- [← previous SemEval-2010 Task 8: Multi-Way Classification of Semantic Relations between Pairs of Nominals](https://md.research.elamur.ai/free/en/papers/hendrickx-2010-semeval-2010-task-8-multi/)
- [next → A Simple Domain-Independent Probabilistic Approach to Generation](https://md.research.elamur.ai/free/en/papers/angeli-2010-simple-domain-independent-probabilistic-approach/)
- [↑ back to papers of 2010](https://md.research.elamur.ai/free/en/papers/2010/)

## References

225 references, 3 of them with a card of ours. ↗ leads to the work, «find» searches for one we have no id for yet.

1. J. P. Abbott and R. P. Brent (1975), ‘Fast local convergence with single and multistep methods for nonlinear equations’, Austr. Math. Soc. B 19, 173–199. [find](https://scholar.google.com/scholar?q=J%2E+P%2E+Abbott+and+R%2E+P%2E+Brent+%281975%29%2C+%E2%80%98Fast+local+convergence+with+single+and+multistep+methods+for+nonlinear+equations%E2%80%99%2C+Austr%2E+Math%2E+Soc%2E+B+19%2C+173%E2%80%93199%2E)
2. ACRITH (1984), IBM High-Accuracy Arithmetic Subroutine Library (ACRITH), Release 1, IBM Deutschland GmbH, Böblingen. [find](https://scholar.google.com/scholar?q=ACRITH+%281984%29%2C+IBM+High%2DAccuracy+Arithmetic+Subroutine+Library+%28ACRITH%29%2C+Release+1%2C+IBM+Deutschland+GmbH%2C+Bo%CC%88blingen%2E)
3. R. A. Adams (1975), Sobolev Spaces, Academic Press, New York. [find](https://scholar.google.com/scholar?q=R%2E+A%2E+Adams+%281975%29%2C+Sobolev+Spaces%2C+Academic+Press%2C+New+York%2E)
4. G. Alefeld (1994), Inclusion methods for systems of nonlinear equations. In Topics in Validated Computations (J. Herzberger, ed.), Studies in Computational Mathematics, Elsevier, Amsterdam, pp. 7–26. [find](https://scholar.google.com/scholar?q=G%2E+Alefeld+%281994%29%2C+Inclusion+methods+for+systems+of+nonlinear+equations%2E+In+Topics+in+Validated+Computations+%28J%2E+Herzberger%2C+ed%2E%29%2C+Studies+in+Computational+Mathematics%2C+Elsevier%2C+Amsterdam%2C+pp%2E+7%E2%80%9326%2E)
5. G. Alefeld and J. Herzberger (1974), Einführung in die Intervallrechnung, BI Wis-senschaftsverlag. [find](https://scholar.google.com/scholar?q=G%2E+Alefeld+and+J%2E+Herzberger+%281974%29%2C+Einfu%CC%88hrung+in+die+Intervallrechnung%2C+BI+Wis%2Dsenschaftsverlag%2E)
6. G. Alefeld, V. Kreinovich and G. Mayer (1997), ‘On the shape of the symmetric, persymmetric, and skew-symmetric solution set’, SIAM J. Matrix Anal. Appl. 18, 693–705. [find](https://scholar.google.com/scholar?q=G%2E+Alefeld%2C+V%2E+Kreinovich+and+G%2E+Mayer+%281997%29%2C+%E2%80%98On+the+shape+of+the+symmetric%2C+persymmetric%2C+and+skew%2Dsymmetric+solution+set%E2%80%99%2C+SIAM+J%2E+Matrix+Anal%2E+Appl%2E+18%2C+693%E2%80%93705%2E)
7. G. Alefeld, V. Kreinovich and G. Mayer (2003), ‘On the solution sets of particular classes of linear interval systems’, J. Comput. Appl. Math. 152, 1–15. [find](https://scholar.google.com/scholar?q=G%2E+Alefeld%2C+V%2E+Kreinovich+and+G%2E+Mayer+%282003%29%2C+%E2%80%98On+the+solution+sets+of+particular+classes+of+linear+interval+systems%E2%80%99%2C+J%2E+Comput%2E+Appl%2E+Math%2E+152%2C+1%E2%80%9315%2E)
8. E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov and D. C. Sorensen (1995), LAPACK User’s Guide, Release 2.0, 2nd edn, SIAM, Philadelphia. [find](https://scholar.google.com/scholar?q=E%2E+Anderson%2C+Z%2E+Bai%2C+C%2E+Bischof%2C+J%2E+Demmel%2C+J%2E+Dongarra%2C+J%2E+Du+Croz%2C+A%2E+Greenbaum%2C+S%2E+Hammarling%2C+A%2E+McKenney%2C+S%2E+Ostrouchov+and+D%2E+C%2E+Sorensen+%281995%29%2C+LAPACK+User%E2%80%99s+Guide%2C+Release+2%2E0%2C+2nd+edn%2C+SIAM%2C)
9. M. V. A. Andrade, J. L. D. Comba and J. Stolfi (1994), Affine arithmetic. Extended abstract, presented at INTERVAL’94 , St. Petersburg. [find](https://scholar.google.com/scholar?q=M%2E+V%2E+A%2E+Andrade%2C+J%2E+L%2E+D%2E+Comba+and+J%2E+Stolfi+%281994%29%2C+Affine+arithmetic%2E+Extended+abstract%2C+presented+at+INTERVAL%E2%80%9994+%2C+St%2E+Petersburg%2E)
10. ARITHMOS (1986), ARITHMOS: Benutzerhandbuch, Siemens AG, Bibl.-Nr. U 2900-I-Z87-1 edition. [find](https://scholar.google.com/scholar?q=ARITHMOS+%281986%29%2C+ARITHMOS%3A+Benutzerhandbuch%2C+Siemens+AG%2C+Bibl%2E%2DNr%2E+U+2900%2DI%2DZ87%2D1+edition%2E)
11. M. Aschbacher (1994), Sporadic Groups, Cambridge University Press. [find](https://scholar.google.com/scholar?q=M%2E+Aschbacher+%281994%29%2C+Sporadic+Groups%2C+Cambridge+University+Press%2E)
12. A. Avizienis (1961), ‘Signed-digit number representations for fast parallel arithmetic’, Ire Trans. Electron. Comp. EC-10, 389–400. [find](https://scholar.google.com/scholar?q=A%2E+Avizienis+%281961%29%2C+%E2%80%98Signed%2Ddigit+number+representations+for+fast+parallel+arithmetic%E2%80%99%2C+Ire+Trans%2E+Electron%2E+Comp%2E+EC%2D10%2C+389%E2%80%93400%2E)
13. H. Bauer (1978), Wahrscheinlichkeitstheorie und Grundzüge der Maßtheorie, 3rd edn, de Gruyter, Berlin. [find](https://scholar.google.com/scholar?q=H%2E+Bauer+%281978%29%2C+Wahrscheinlichkeitstheorie+und+Grundzu%CC%88ge+der+Ma%C3%9Ftheorie%2C+3rd+edn%2C+de+Gruyter%2C+Berlin%2E)
14. O. Beaumont (2000), Solving interval linear systems with oblique boxes. Research report PI 1315, INRIA. [find](https://scholar.google.com/scholar?q=O%2E+Beaumont+%282000%29%2C+Solving+interval+linear+systems+with+oblique+boxes%2E+Research+report+PI+1315%2C+INRIA%2E)
15. H. Behnke (1989), Die Bestimmung von Eigenwertschranken mit Hilfe von Vari-ationsmethoden und Intervallarithmetik. Dissertation, Institut für Mathematik, TU Clausthal. [find](https://scholar.google.com/scholar?q=H%2E+Behnke+%281989%29%2C+Die+Bestimmung+von+Eigenwertschranken+mit+Hilfe+von+Vari%2Dationsmethoden+und+Intervallarithmetik%2E+Dissertation%2C+Institut+fu%CC%88r+Mathematik%2C+TU+Clausthal%2E)
16. H. Behnke and F. Goerisch (1994), Inclusions for eigenvalues of selfadjoint problems. In Topics in Validated Computations (J. Herzberger, ed.), Studies in Computational Mathematics, Elsevier, Amsterdam, pp. 277–322. [find](https://scholar.google.com/scholar?q=H%2E+Behnke+and+F%2E+Goerisch+%281994%29%2C+Inclusions+for+eigenvalues+of+selfadjoint+problems%2E+In+Topics+in+Validated+Computations+%28J%2E+Herzberger%2C+ed%2E%29%2C+Studies+in+Computational+Mathematics%2C+Elsevier%2C+Amsterda)
17. A. Ben-Tal and A. Nemirovskii (2001), Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications, SIAM, Philadelphia. [find](https://scholar.google.com/scholar?q=A%2E+Ben%2DTal+and+A%2E+Nemirovskii+%282001%29%2C+Lectures+on+Modern+Convex+Optimization%3A+Analysis%2C+Algorithms%2C+and+Engineering+Applications%2C+SIAM%2C+Philadelphia%2E)
18. F. Bernelli Zazzera, M. Vasile, M. Massari and P. Di Lizia (2004), Assessing the accuracy of interval arithmetic estimates in space flight mechanics. Final report, Ariadna id: 04/4105, Contract Number: 18851/05/NL/MV. [find](https://scholar.google.com/scholar?q=F%2E+Bernelli+Zazzera%2C+M%2E+Vasile%2C+M%2E+Massari+and+P%2E+Di+Lizia+%282004%29%2C+Assessing+the+accuracy+of+interval+arithmetic+estimates+in+space+flight+mechanics%2E+Final+report%2C+Ariadna+id%3A+04%2F4105%2C+Contract+Number)
19. Y. Bertot and P. Castéran (2004), Interactive Theorem Proving and Program Development, Coq’Art: The Calculus of Inductive Constructions, Texts in Theoretical Computer Science, Springer. [find](https://scholar.google.com/scholar?q=Y%2E+Bertot+and+P%2E+Caste%CC%81ran+%282004%29%2C+Interactive+Theorem+Proving+and+Program+Development%2C+Coq%E2%80%99Art%3A+The+Calculus+of+Inductive+Constructions%2C+Texts+in+Theoretical+Computer+Science%2C+Springer%2E)
20. M. Berz and K. Makino (1999), ‘New methods for high-dimensional verified quadrature’, Reliable Computing 5, 13–22. [find](https://scholar.google.com/scholar?q=M%2E+Berz+and+K%2E+Makino+%281999%29%2C+%E2%80%98New+methods+for+high%2Ddimensional+verified+quadrature%E2%80%99%2C+Reliable+Computing+5%2C+13%E2%80%9322%2E)
21. C. H. Bischof, A. Carle, G. Corliss and A. Griewank (1991), ADIFOR: Generating derivative codes from Fortran programs. Technical report, Mathematics and Computer Science Division, Argonne National Laboratory. [find](https://scholar.google.com/scholar?q=C%2E+H%2E+Bischof%2C+A%2E+Carle%2C+G%2E+Corliss+and+A%2E+Griewank+%281991%29%2C+ADIFOR%3A+Generating+derivative+codes+from+Fortran+programs%2E+Technical+report%2C+Mathematics+and+Computer+Science+Division%2C+Argonne+National+Lab)
22. B. Borchers (1999), ‘SDPLIB 1.2: A library of semidefinite programming test problems’, Optim. Methods Software 11, 683–690. [find](https://scholar.google.com/scholar?q=B%2E+Borchers+%281999%29%2C+%E2%80%98SDPLIB+1%2E2%3A+A+library+of+semidefinite+programming+test+problems%E2%80%99%2C+Optim%2E+Methods+Software+11%2C+683%E2%80%93690%2E)
23. F. Bornemann, D. Laurie, S. Wagon and J. Waldvogel (2004), The SIAM 100-Digit Challenge: A Study in High-Accuracy Numerical Computing, SIAM, Philadelphia. [find](https://scholar.google.com/scholar?q=F%2E+Bornemann%2C+D%2E+Laurie%2C+S%2E+Wagon+and+J%2E+Waldvogel+%282004%29%2C+The+SIAM+100%2DDigit+Challenge%3A+A+Study+in+High%2DAccuracy+Numerical+Computing%2C+SIAM%2C+Philadelphia%2E)
24. N. C. Börsken (1978), Komplexe Kreis-Standardfunktionen. Diplomarbeit, Frei-burger Intervall-Ber. 78/2, Institut für Angewandte Mathematik, Universität Freiburg. [find](https://scholar.google.com/scholar?q=N%2E+C%2E+Bo%CC%88rsken+%281978%29%2C+Komplexe+Kreis%2DStandardfunktionen%2E+Diplomarbeit%2C+Frei%2Dburger+Intervall%2DBer%2E+78%2F2%2C+Institut+fu%CC%88r+Angewandte+Mathematik%2C+Universita%CC%88t+Freiburg%2E)
25. K. D. Braune (1987), Hochgenaue Standardfunktionen für reelle und komplexe Punkte und Intervalle in beliebigen Gleitpunktrastern. Dissertation, Universität Karlsruhe. [find](https://scholar.google.com/scholar?q=K%2E+D%2E+Braune+%281987%29%2C+Hochgenaue+Standardfunktionen+fu%CC%88r+reelle+und+komplexe+Punkte+und+Intervalle+in+beliebigen+Gleitpunktrastern%2E+Dissertation%2C+Universita%CC%88t+Karlsruhe%2E)
26. B. Breuer, J. Horák, P. J. McKenna and M. Plum (2006), ‘A computer-assisted existence and multiplicity proof for travelling waves in a nonlinearly supported beam’, J. Diff. Equations 224, 60–97. [find](https://scholar.google.com/scholar?q=B%2E+Breuer%2C+J%2E+Hora%CC%81k%2C+P%2E+J%2E+McKenna+and+M%2E+Plum+%282006%29%2C+%E2%80%98A+computer%2Dassisted+existence+and+multiplicity+proof+for+travelling+waves+in+a+nonlinearly+supported+beam%E2%80%99%2C+J%2E+Diff%2E+Equations+224%2C+60%E2%80%9397%2E)
27. B. Breuer, P. J. McKenna and M. Plum (2003), ‘Multiple solutions for a semilinear boundary value problem: A computational multiplicity proof’, J. Diff. Equations 195, 243–269. [find](https://scholar.google.com/scholar?q=B%2E+Breuer%2C+P%2E+J%2E+McKenna+and+M%2E+Plum+%282003%29%2C+%E2%80%98Multiple+solutions+for+a+semilinear+boundary+value+problem%3A+A+computational+multiplicity+proof%E2%80%99%2C+J%2E+Diff%2E+Equations+195%2C+243%E2%80%93269%2E)
28. B. M. Brown, D. K. R. McCormack and A. Zettl (2000), ‘On a computer assisted proof of the existence of eigenvalues below the essential spectrum of the Sturm–Liouville problem’, J. Comput. Appl. Math. 125, 385–393. [find](https://scholar.google.com/scholar?q=B%2E+M%2E+Brown%2C+D%2E+K%2E+R%2E+McCormack+and+A%2E+Zettl+%282000%29%2C+%E2%80%98On+a+computer+assisted+proof+of+the+existence+of+eigenvalues+below+the+essential+spectrum+of+the+Sturm%E2%80%93Liouville+problem%E2%80%99%2C+J%2E+Comput%2E+Appl%2E+Math%2E+)
29. M. W. Browne (1988), Is a math proof a proof if no one can check it? The New York Times, December 1988, p. 1. [find](https://scholar.google.com/scholar?q=M%2E+W%2E+Browne+%281988%29%2C+Is+a+math+proof+a+proof+if+no+one+can+check+it%3F+The+New+York+Times%2C+December+1988%2C+p%2E+1%2E)
30. J. R. Bunch, J. W. Demmel and C. F. Van Loan (1989), ‘The strong stability of algorithms for solving symmetric linear systems’, SIAM J. Matrix Anal. Appl. 10, 494–499. [find](https://scholar.google.com/scholar?q=J%2E+R%2E+Bunch%2C+J%2E+W%2E+Demmel+and+C%2E+F%2E+Van+Loan+%281989%29%2C+%E2%80%98The+strong+stability+of+algorithms+for+solving+symmetric+linear+systems%E2%80%99%2C+SIAM+J%2E+Matrix+Anal%2E+Appl%2E+10%2C+494%E2%80%93499%2E)
31. F. Bünger (2008), private communication. [find](https://scholar.google.com/scholar?q=F%2E+Bu%CC%88nger+%282008%29%2C+private+communication%2E)
32. O. Caprani and K. Madsen (1978), ‘Iterative methods for interval inclusion of fixed points’, BIT Numer. Math. 18, 42–51. [find](https://scholar.google.com/scholar?q=O%2E+Caprani+and+K%2E+Madsen+%281978%29%2C+%E2%80%98Iterative+methods+for+interval+inclusion+of+fixed+points%E2%80%99%2C+BIT+Numer%2E+Math%2E+18%2C+42%E2%80%9351%2E)
33. F. Chatelin (1988), Analyse statistique de la qualité numérique et arithmétique de la résolution approchée d’équations par calcul sur ordinateur. Technical Report F.133, Centre Scientifique IBM–France. [find](https://scholar.google.com/scholar?q=F%2E+Chatelin+%281988%29%2C+Analyse+statistique+de+la+qualite%CC%81+nume%CC%81rique+et+arithme%CC%81tique+de+la+re%CC%81solution+approche%CC%81e+d%E2%80%99e%CC%81quations+par+calcul+sur+ordinateur%2E+Technical+Report+F%2E133%2C+Centre+Scientifique+IBM%E2%80%93)
34. Y. S. Choi and P. J. McKenna (1993), ‘A mountain pass method for the numerical solutions of semilinear elliptic problems’, Nonlinear Anal. Theory Methods Appl. 20, 417–437. [find](https://scholar.google.com/scholar?q=Y%2E+S%2E+Choi+and+P%2E+J%2E+McKenna+%281993%29%2C+%E2%80%98A+mountain+pass+method+for+the+numerical+solutions+of+semilinear+elliptic+problems%E2%80%99%2C+Nonlinear+Anal%2E+Theory+Methods+Appl%2E+20%2C+417%E2%80%93437%2E)
35. L. Collatz (1942), ‘Einschließungssatz für die charakteristischen Zahlen von Matrizen’, Math. Z. 48, 221–226. [find](https://scholar.google.com/scholar?q=L%2E+Collatz+%281942%29%2C+%E2%80%98Einschlie%C3%9Fungssatz+fu%CC%88r+die+charakteristischen+Zahlen+von+Matrizen%E2%80%99%2C+Math%2E+Z%2E+48%2C+221%E2%80%93226%2E)
36. G. F. Corliss and L. B. Rall (1987), ‘Adaptive, self-validating numerical quadrature’, SIAM J. Sci. Statist. Comput. 8, 831–847. [find](https://scholar.google.com/scholar?q=G%2E+F%2E+Corliss+and+L%2E+B%2E+Rall+%281987%29%2C+%E2%80%98Adaptive%2C+self%2Dvalidating+numerical+quadrature%E2%80%99%2C+SIAM+J%2E+Sci%2E+Statist%2E+Comput%2E+8%2C+831%E2%80%93847%2E)
37. G. Corliss, C. Faure, A. Griewank, L. Hascoët and U. Nauman (2002), Automatic Differentiation of Algorithms: From Simulation to Optimization, Springer. [find](https://scholar.google.com/scholar?q=G%2E+Corliss%2C+C%2E+Faure%2C+A%2E+Griewank%2C+L%2E+Hascoe%CC%88t+and+U%2E+Nauman+%282002%29%2C+Automatic+Differentiation+of+Algorithms%3A+From+Simulation+to+Optimization%2C+Springer%2E)
38. A. Cuyt, B. Verdonk, S. Becuwe and P. Kuterna (2001), ‘A remarkable example of catastrophic cancellation unraveled’, Computing 66, 309–320. [find](https://scholar.google.com/scholar?q=A%2E+Cuyt%2C+B%2E+Verdonk%2C+S%2E+Becuwe+and+P%2E+Kuterna+%282001%29%2C+%E2%80%98A+remarkable+example+of+catastrophic+cancellation+unraveled%E2%80%99%2C+Computing+66%2C+309%E2%80%93320%2E)
39. E. N. Dancer and S. S. Yan (2005), ‘On the superlinear Lazer–McKenna conjecture’, J. Diff. Equations 210, 317–351. [find](https://scholar.google.com/scholar?q=E%2E+N%2E+Dancer+and+S%2E+S%2E+Yan+%282005%29%2C+%E2%80%98On+the+superlinear+Lazer%E2%80%93McKenna+conjecture%E2%80%99%2C+J%2E+Diff%2E+Equations+210%2C+317%E2%80%93351%2E)
40. S. M. Rump Verification methods G. Darboux (1876), ‘Sur les développements en série des fonctions d’une seule variable’, J. des Mathématiques Pures et Appl. 3, 291–312. [find](https://scholar.google.com/scholar?q=S%2E+M%2E+Rump+Verification+methods+G%2E+Darboux+%281876%29%2C+%E2%80%98Sur+les+de%CC%81veloppements+en+se%CC%81rie+des+fonctions+d%E2%80%99une+seule+variable%E2%80%99%2C+J%2E+des+Mathe%CC%81matiques+Pures+et+Appl%2E+3%2C+291%E2%80%93312%2E)
41. M. Daumas, G. Melquiond and C. Muñoz (2005), Guaranteed proofs using interval arithmetic. In Proc. 17th IEEE Symposium on Computer Arithmetic (ARITH’05). [find](https://scholar.google.com/scholar?q=M%2E+Daumas%2C+G%2E+Melquiond+and+C%2E+Mun%CC%83oz+%282005%29%2C+Guaranteed+proofs+using+interval+arithmetic%2E+In+Proc%2E+17th+IEEE+Symposium+on+Computer+Arithmetic+%28ARITH%E2%80%9905%29%2E)
42. T. J. Dekker (1971), ‘A floating-point technique for extending the available precision’, Numer. Math. 18, 224–242. [find](https://scholar.google.com/scholar?q=T%2E+J%2E+Dekker+%281971%29%2C+%E2%80%98A+floating%2Dpoint+technique+for+extending+the+available+precision%E2%80%99%2C+Numer%2E+Math%2E+18%2C+224%E2%80%93242%2E)
43. J. B. Demmel (1989), On floating point errors in Cholesky. LAPACK Working Note 14 CS-89-87, Department of Computer Science, University of Tennessee, Knoxville, TN, USA. [find](https://scholar.google.com/scholar?q=J%2E+B%2E+Demmel+%281989%29%2C+On+floating+point+errors+in+Cholesky%2E+LAPACK+Working+Note+14+CS%2D89%2D87%2C+Department+of+Computer+Science%2C+University+of+Tennessee%2C+Knoxville%2C+TN%2C+USA%2E)
44. J. B. Demmel, B. Diament and G. Malajovich (2001), ‘On the complexity of computing error bounds’, Found. Comput. Math. 1, 101–125. [find](https://scholar.google.com/scholar?q=J%2E+B%2E+Demmel%2C+B%2E+Diament+and+G%2E+Malajovich+%282001%29%2C+%E2%80%98On+the+complexity+of+computing+error+bounds%E2%80%99%2C+Found%2E+Comput%2E+Math%2E+1%2C+101%E2%80%93125%2E)
45. J. B. Demmel, I. Dumitriu, O. Holtz and P. Koev (2008), Accurate and efficient expression evaluation and linear algebra. In Acta Numerica, Vol. 17, Cambridge University Press, pp. 87–145. [find](https://scholar.google.com/scholar?q=J%2E+B%2E+Demmel%2C+I%2E+Dumitriu%2C+O%2E+Holtz+and+P%2E+Koev+%282008%29%2C+Accurate+and+efficient+expression+evaluation+and+linear+algebra%2E+In+Acta+Numerica%2C+Vol%2E+17%2C+Cambridge+University+Press%2C+pp%2E+87%E2%80%93145%2E)
46. J. B. Demmel, Y. Hida, W. Kahan, X. S. Li, S. Mukherjee and E. J. Riedy (2004), Error bounds from extra precise iterative refinement. Report no. ucb/csd-04-1344, Computer Science Devision (EECS), University of California, Berkeley. [find](https://scholar.google.com/scholar?q=J%2E+B%2E+Demmel%2C+Y%2E+Hida%2C+W%2E+Kahan%2C+X%2E+S%2E+Li%2C+S%2E+Mukherjee+and+E%2E+J%2E+Riedy+%282004%29%2C+Error+bounds+from+extra+precise+iterative+refinement%2E+Report+no%2E+ucb%2Fcsd%2D04%2D1344%2C+Computer+Science+Devision+%28EECS%29%2C+Univ)
47. P. S. Dwyer (1951), Linear Computations, Wiley, New York/London. [find](https://scholar.google.com/scholar?q=P%2E+S%2E+Dwyer+%281951%29%2C+Linear+Computations%2C+Wiley%2C+New+York%2FLondon%2E)
48. C. Eckart and G. Young (1936), ‘The approximation of one matrix by another of lower rank’, Psychometrika 1, 211–218. [find](https://scholar.google.com/scholar?q=C%2E+Eckart+and+G%2E+Young+%281936%29%2C+%E2%80%98The+approximation+of+one+matrix+by+another+of+lower+rank%E2%80%99%2C+Psychometrika+1%2C+211%E2%80%93218%2E)
49. J.-P. Eckmann, H. Koch and P. Wittwer (1984), ‘A computer-assisted proof of universality for area-preserving maps’, Mem. Amer. Math. Soc. 47, 289. [find](https://scholar.google.com/scholar?q=J%2E%2DP%2E+Eckmann%2C+H%2E+Koch+and+P%2E+Wittwer+%281984%29%2C+%E2%80%98A+computer%2Dassisted+proof+of+universality+for+area%2Dpreserving+maps%E2%80%99%2C+Mem%2E+Amer%2E+Math%2E+Soc%2E+47%2C+289%2E)
50. P. Eijgenraam (1981), The solution of initial value problems using interval arithmetic. [find](https://scholar.google.com/scholar?q=P%2E+Eijgenraam+%281981%29%2C+The+solution+of+initial+value+problems+using+interval+arithmetic%2E)
51. B. Fazekas, M. Plum and C. Wieners (2005), Enclosure for biharmonic equation. In Dagstuhl Online Seminar Proceedings 05391. [http://drops.dagstuhl.de/portal/05391/](http://drops.dagstuhl.de/portal/05391/). [↗](http://drops.dagstuhl.de/portal/05391/)
52. L. H. de Figueiredo and J. Stolfi (2004), ‘Affine arithmetic: Concepts and applications’, Numer. Algorithms 37, 147–158. [find](https://scholar.google.com/scholar?q=L%2E+H%2E+de+Figueiredo+and+J%2E+Stolfi+%282004%29%2C+%E2%80%98Affine+arithmetic%3A+Concepts+and+applications%E2%80%99%2C+Numer%2E+Algorithms+37%2C+147%E2%80%93158%2E)
53. L. V. Foster (1994), ‘Gaussian elimination with partial pivoting can fail in practice’, SIAM J. Matrix Anal. Appl. 14, 1354–1362. [find](https://scholar.google.com/scholar?q=L%2E+V%2E+Foster+%281994%29%2C+%E2%80%98Gaussian+elimination+with+partial+pivoting+can+fail+in+practice%E2%80%99%2C+SIAM+J%2E+Matrix+Anal%2E+Appl%2E+14%2C+1354%E2%80%931362%2E)
54. L. Fousse, G. Hanrot, V. Lefèvre, P. Pélissier and P. Zimmermann (2005), MPFR: A multiple-precision binary floating-point library with correct rounding. Research Report RR-5753, INRIA. Code and documentation available at: [http://hal.inria.fr/inria-00000818](http://hal.inria.fr/inria-00000818). *) [↗](http://hal.inria.fr/inria-00000818)
55. A. Frommer (2001), Proving conjectures by use of interval arithmetic. In Perspectives on Enclosure Methods: SCAN 2000 (U. Kulisch et al., ed.), Springer. [find](https://scholar.google.com/scholar?q=A%2E+Frommer+%282001%29%2C+Proving+conjectures+by+use+of+interval+arithmetic%2E+In+Perspectives+on+Enclosure+Methods%3A+SCAN+2000+%28U%2E+Kulisch+et+al%2E%2C+ed%2E%29%2C+Springer%2E)
56. Z. Galias and P. Zgliczynski (1998), ‘Computer assisted proof of chaos in the Lorenz equations’, Physica D 115, 165–188. [find](https://scholar.google.com/scholar?q=Z%2E+Galias+and+P%2E+Zgliczynski+%281998%29%2C+%E2%80%98Computer+assisted+proof+of+chaos+in+the+Lorenz+equations%E2%80%99%2C+Physica+D+115%2C+165%E2%80%93188%2E)
57. I. Gargantini and P. Henrici (1972), ‘Circular arithmetic and the determination of polynomial zeros’, Numer. Math. 18, 305–320. [find](https://scholar.google.com/scholar?q=I%2E+Gargantini+and+P%2E+Henrici+%281972%29%2C+%E2%80%98Circular+arithmetic+and+the+determination+of+polynomial+zeros%E2%80%99%2C+Numer%2E+Math%2E+18%2C+305%E2%80%93320%2E)
58. B. Gidas, W. Ni and L. Nirenberg (1979), ‘Symmetry and related problems via the maximum principle’, Comm. Math. Phys. 68, 209–243. [find](https://scholar.google.com/scholar?q=B%2E+Gidas%2C+W%2E+Ni+and+L%2E+Nirenberg+%281979%29%2C+%E2%80%98Symmetry+and+related+problems+via+the+maximum+principle%E2%80%99%2C+Comm%2E+Math%2E+Phys%2E+68%2C+209%E2%80%93243%2E)
59. D. Gilbarg and N. S. Trudinger (1983), Elliptic Partial Differential Equations of Second Order, 2nd edn, Springer. *) published in ACM Transaction on Mathematical Software (TOMS) 33, article 13, 2007 [find](https://scholar.google.com/scholar?q=D%2E+Gilbarg+and+N%2E+S%2E+Trudinger+%281983%29%2C+Elliptic+Partial+Differential+Equations+of+Second+Order%2C+2nd+edn%2C+Springer%2E+%2A%29+published+in+ACM+Transaction+on+Mathematical+Software+%28TOMS%29+33%2C+article+13%2C+2007)
60. D. Goldberg (1991), ‘[What every computer scientist should know about floating-point arithmetic](https://md.research.elamur.ai/free/en/papers/goldberg-1991-what-every-computer-scientist-should/)’, ACM Comput. Surv. 23, 5–48.
61. M. J. C. Gordon (2000), [From LCF to HOL: A short history](https://md.research.elamur.ai/free/en/papers/gordon-2000-lcf-hol-short-history/). In Proof, Language, and Interaction: Essays in Honour of Robin Milner (G. Plotkin, C. Stirling and M. Tofte, eds), MIT Press. [http://www.cl.cam.ac.uk/˜mjcg/papers/HolHistory.html](http://www.cl.cam.ac.uk/˜mjcg/papers/HolHistory.html). [↗](http://www.cl.cam.ac.uk/˜mjcg/papers/HolHistory.html)
62. D. Gorenstein, R. Lyons and R. Solomon (1994), The Classification of the Finite Simple Groups, Vol. 40 of Math. Surveys Monographs, AMS, Providence, RI. [find](https://scholar.google.com/scholar?q=D%2E+Gorenstein%2C+R%2E+Lyons+and+R%2E+Solomon+%281994%29%2C+The+Classification+of+the+Finite+Simple+Groups%2C+Vol%2E+40+of+Math%2E+Surveys+Monographs%2C+AMS%2C+Providence%2C+RI%2E)
63. A. Griewank (2003), A mathematical view of automatic differentiation. In Acta Numerica, Vol. 12, Cambridge University Press, pp. 321–398. [find](https://scholar.google.com/scholar?q=A%2E+Griewank+%282003%29%2C+A+mathematical+view+of+automatic+differentiation%2E+In+Acta+Numerica%2C+Vol%2E+12%2C+Cambridge+University+Press%2C+pp%2E+321%E2%80%93398%2E)
64. P. Grisvard (1985), Elliptic Problems in Nonsmooth Domains, Pitman, Boston. [find](https://scholar.google.com/scholar?q=P%2E+Grisvard+%281985%29%2C+Elliptic+Problems+in+Nonsmooth+Domains%2C+Pitman%2C+Boston%2E)
65. E. Hansen (1969), The centered form. In Topics in Interval Analysis (E. Hansen, ed.), Oxford University Press, pp. 102–106. [find](https://scholar.google.com/scholar?q=E%2E+Hansen+%281969%29%2C+The+centered+form%2E+In+Topics+in+Interval+Analysis+%28E%2E+Hansen%2C+ed%2E%29%2C+Oxford+University+Press%2C+pp%2E+102%E2%80%93106%2E)
66. E. R. Hansen and R. Smith (1967), ‘Interval arithmetic in matrix computations II’, SIAM J. Numer. Anal. 4, 1–9. [find](https://scholar.google.com/scholar?q=E%2E+R%2E+Hansen+and+R%2E+Smith+%281967%29%2C+%E2%80%98Interval+arithmetic+in+matrix+computations+II%E2%80%99%2C+SIAM+J%2E+Numer%2E+Anal%2E+4%2C+1%E2%80%939%2E)
67. G. Hargreaves (2002), Interval analysis in MATLAB. Master’s thesis, University of Manchester. [http://www.manchester.ac.uk/mims/eprints](http://www.manchester.ac.uk/mims/eprints). [↗](http://www.manchester.ac.uk/mims/eprints)
68. J. Hass, M. Hutchings and R. Schlafly (1995), ‘The double bubble conjecture’, Electron. Res. Announc. Amer. Math. Soc. 1, 98–102. [find](https://scholar.google.com/scholar?q=J%2E+Hass%2C+M%2E+Hutchings+and+R%2E+Schlafly+%281995%29%2C+%E2%80%98The+double+bubble+conjecture%E2%80%99%2C+Electron%2E+Res%2E+Announc%2E+Amer%2E+Math%2E+Soc%2E+1%2C+98%E2%80%93102%2E)
69. D. J. Higham and N. J. Higham (1992a), ‘Componentwise perturbation theory for linear systems with multiple right-hand sides’, Linear Algebra Appl. 174, 111–129. [find](https://scholar.google.com/scholar?q=D%2E+J%2E+Higham+and+N%2E+J%2E+Higham+%281992a%29%2C+%E2%80%98Componentwise+perturbation+theory+for+linear+systems+with+multiple+right%2Dhand+sides%E2%80%99%2C+Linear+Algebra+Appl%2E+174%2C+111%E2%80%93129%2E)
70. D. J. Higham and N. J. Higham (1992b), ‘Backward error and condition of structured linear systems’, SIAM J. Matrix Anal. Appl. 13, 162–175. [find](https://scholar.google.com/scholar?q=D%2E+J%2E+Higham+and+N%2E+J%2E+Higham+%281992b%29%2C+%E2%80%98Backward+error+and+condition+of+structured+linear+systems%E2%80%99%2C+SIAM+J%2E+Matrix+Anal%2E+Appl%2E+13%2C+162%E2%80%93175%2E)
71. N. J. Higham (2002), Accuracy and Stability of Numerical Algorithms, 2nd edn, SIAM, Philadelphia. [find](https://scholar.google.com/scholar?q=N%2E+J%2E+Higham+%282002%29%2C+Accuracy+and+Stability+of+Numerical+Algorithms%2C+2nd+edn%2C+SIAM%2C+Philadelphia%2E)
72. J. Hölzl (2009), Proving real-valued inequalities by computation in Isabelle/HOL. Diplomarbeit, Fakultät für Informatik der Technischen Universität München. [find](https://scholar.google.com/scholar?q=J%2E+Ho%CC%88lzl+%282009%29%2C+Proving+real%2Dvalued+inequalities+by+computation+in+Isabelle%2FHOL%2E+Diplomarbeit%2C+Fakulta%CC%88t+fu%CC%88r+Informatik+der+Technischen+Universita%CC%88t+Mu%CC%88nchen%2E)
73. IEEE 754 (2008), ANSI/IEEE 754-2008: IEEE Standard for Floating-Point Arithmetic, New York. [find](https://scholar.google.com/scholar?q=IEEE+754+%282008%29%2C+ANSI%2FIEEE+754%2D2008%3A+IEEE+Standard+for+Floating%2DPoint+Arithmetic%2C+New+York%2E)
74. C. Jansson (1991), ‘Interval linear systems with symmetric matrices, skew-symmetric matrices, and dependencies in the right hand side’, Computing 46, 265–274. [find](https://scholar.google.com/scholar?q=C%2E+Jansson+%281991%29%2C+%E2%80%98Interval+linear+systems+with+symmetric+matrices%2C+skew%2Dsymmetric+matrices%2C+and+dependencies+in+the+right+hand+side%E2%80%99%2C+Computing+46%2C+265%E2%80%93274%2E)
75. C. Jansson (1994), On self-validating methods for optimization problems. In Topics in Validated Computations (J. Herzberger, ed.), Studies in Computational Mathematics, Elsevier, Amsterdam, pp. 381–438. [find](https://scholar.google.com/scholar?q=C%2E+Jansson+%281994%29%2C+On+self%2Dvalidating+methods+for+optimization+problems%2E+In+Topics+in+Validated+Computations+%28J%2E+Herzberger%2C+ed%2E%29%2C+Studies+in+Computational+Mathematics%2C+Elsevier%2C+Amsterdam%2C+pp%2E+381%E2%80%9343)
76. C. Jansson (1997), ‘Calculation of exact bounds for the solution set of linear interval systems’, Linear Algebra Appl. 251, 321–340. [find](https://scholar.google.com/scholar?q=C%2E+Jansson+%281997%29%2C+%E2%80%98Calculation+of+exact+bounds+for+the+solution+set+of+linear+interval+systems%E2%80%99%2C+Linear+Algebra+Appl%2E+251%2C+321%E2%80%93340%2E)
77. C. Jansson (2004a), ‘A rigorous lower bound for the optimal value of convex optimization problems’, J. Global Optim. 28, 121–137. [find](https://scholar.google.com/scholar?q=C%2E+Jansson+%282004a%29%2C+%E2%80%98A+rigorous+lower+bound+for+the+optimal+value+of+convex+optimization+problems%E2%80%99%2C+J%2E+Global+Optim%2E+28%2C+121%E2%80%93137%2E)
78. C. Jansson (2004b), ‘Rigorous lower and upper bounds in linear programming’, SIAM J. Optim. 14, 914–935. [find](https://scholar.google.com/scholar?q=C%2E+Jansson+%282004b%29%2C+%E2%80%98Rigorous+lower+and+upper+bounds+in+linear+programming%E2%80%99%2C+SIAM+J%2E+Optim%2E+14%2C+914%E2%80%93935%2E)
79. C. Jansson (2006), VSDP: A MATLAB software package for verified semidefinite programming. In NOLTA 2006 , pp. 327–330. [find](https://scholar.google.com/scholar?q=C%2E+Jansson+%282006%29%2C+VSDP%3A+A+MATLAB+software+package+for+verified+semidefinite+programming%2E+In+NOLTA+2006+%2C+pp%2E+327%E2%80%93330%2E)
80. C. Jansson (2009), ‘On verified numerical computations in convex programming’, Japan J. Indust. Appl. Math. 26, 337–363. [find](https://scholar.google.com/scholar?q=C%2E+Jansson+%282009%29%2C+%E2%80%98On+verified+numerical+computations+in+convex+programming%E2%80%99%2C+Japan+J%2E+Indust%2E+Appl%2E+Math%2E+26%2C+337%E2%80%93363%2E)
81. S. M. Rump Verification methods C. Jansson and J. Rohn (1999), ‘An algorithm for checking regularity of interval matrices’, SIAM J. Matrix Anal. Appl. 20, 756–776. [find](https://scholar.google.com/scholar?q=S%2E+M%2E+Rump+Verification+methods+C%2E+Jansson+and+J%2E+Rohn+%281999%29%2C+%E2%80%98An+algorithm+for+checking+regularity+of+interval+matrices%E2%80%99%2C+SIAM+J%2E+Matrix+Anal%2E+Appl%2E+20%2C+756%E2%80%93776%2E)
82. C. Jansson, D. Chaykin and C. Keil (2007), ‘Rigorous error bounds for the optimal value in semidefinite programming’, SIAM J. Numer. Anal. 46, 180–200. [http://link.aip.org/link/?SNA/46/180/1](http://link.aip.org/link/?SNA/46/180/1). [↗](http://link.aip.org/link/?SNA/46/180/1)
83. W. M. Kahan (1968), A more complete interval arithmetic. Lecture notes for a summer course at the University of Michigan. [find](https://scholar.google.com/scholar?q=W%2E+M%2E+Kahan+%281968%29%2C+A+more+complete+interval+arithmetic%2E+Lecture+notes+for+a+summer+course+at+the+University+of+Michigan%2E)
84. Y. Kanzawa and S. Oishi (1999a), ‘Imperfect singular solutions of nonlinear equations and a numerical method of proving their existence’, IEICE Trans. Fundamentals E82-A, 1062–1069. [find](https://scholar.google.com/scholar?q=Y%2E+Kanzawa+and+S%2E+Oishi+%281999a%29%2C+%E2%80%98Imperfect+singular+solutions+of+nonlinear+equations+and+a+numerical+method+of+proving+their+existence%E2%80%99%2C+IEICE+Trans%2E+Fundamentals+E82%2DA%2C+1062%E2%80%931069%2E)
85. Y. Kanzawa and S. Oishi (1999b), ‘Calculating bifurcation points with guaranteed accuracy’, IEICE Trans. Fundamentals E82-A, 1055–1061. [find](https://scholar.google.com/scholar?q=Y%2E+Kanzawa+and+S%2E+Oishi+%281999b%29%2C+%E2%80%98Calculating+bifurcation+points+with+guaranteed+accuracy%E2%80%99%2C+IEICE+Trans%2E+Fundamentals+E82%2DA%2C+1055%E2%80%931061%2E)
86. T. Kato (1966), Perturbation Theory for Linear Operators, Springer, New York. [find](https://scholar.google.com/scholar?q=T%2E+Kato+%281966%29%2C+Perturbation+Theory+for+Linear+Operators%2C+Springer%2C+New+York%2E)
87. R. B. Kearfott (1997), ‘Empirical evaluation of innovations in interval branch and bound algorithms for nonlinear systems’, SIAM J. Sci. Comput. 18, 574–594. [find](https://scholar.google.com/scholar?q=R%2E+B%2E+Kearfott+%281997%29%2C+%E2%80%98Empirical+evaluation+of+innovations+in+interval+branch+and+bound+algorithms+for+nonlinear+systems%E2%80%99%2C+SIAM+J%2E+Sci%2E+Comput%2E+18%2C+574%E2%80%93594%2E)
88. R. B. Kearfott, M. Dawande, K. Du and C. Hu (1992), ‘INTLIB: A portable Fortran-77 elementary function library’, Interval Comput. 3, 96–105. [find](https://scholar.google.com/scholar?q=R%2E+B%2E+Kearfott%2C+M%2E+Dawande%2C+K%2E+Du+and+C%2E+Hu+%281992%29%2C+%E2%80%98INTLIB%3A+A+portable+Fortran%2D77+elementary+function+library%E2%80%99%2C+Interval+Comput%2E+3%2C+96%E2%80%93105%2E)
89. R. B. Kearfott, J. Dian and A. Neumaier (2000), ‘Existence verification for singular zeros of complex nonlinear systems’, SIAM J. Numer. Anal. 38, 360–379. [find](https://scholar.google.com/scholar?q=R%2E+B%2E+Kearfott%2C+J%2E+Dian+and+A%2E+Neumaier+%282000%29%2C+%E2%80%98Existence+verification+for+singular+zeros+of+complex+nonlinear+systems%E2%80%99%2C+SIAM+J%2E+Numer%2E+Anal%2E+38%2C+360%E2%80%93379%2E)
90. R. B. Kearfott, M. T. Nakao, A. Neumaier, S. M. Rump, S. P. Shary and P. van Hentenfyck (2005) Standardized notation in interval analysis. In Proc. XIII Baikal International School–Seminar: Optimization Methods and their Applications, Vol. 4, Melentiev Energy Systems Institute SB RAS, Irkutsk. [find](https://scholar.google.com/scholar?q=R%2E+B%2E+Kearfott%2C+M%2E+T%2E+Nakao%2C+A%2E+Neumaier%2C+S%2E+M%2E+Rump%2C+S%2E+P%2E+Shary+and+P%2E+van+Hentenfyck+%282005%29+Standardized+notation+in+interval+analysis%2E+In+Proc%2E+XIII+Baikal+International+School%E2%80%93Seminar%3A+Optimizati)
91. C. Keil (2006), Lurupa: Rigorous error bounds in linear programming. In Algebraic and Numerical Algorithms and Computer-assisted Proofs (B. Buchberger, S. Oishi, M. Plum and S. M. Rump, eds), Vol. 05391 of Dagstuhl Seminar Proceedings, Internationales Begegnungs- und Forschungszentrum für Informati… [find](https://scholar.google.com/scholar?q=C%2E+Keil+%282006%29%2C+Lurupa%3A+Rigorous+error+bounds+in+linear+programming%2E+In+Algebraic+and+Numerical+Algorithms+and+Computer%2Dassisted+Proofs+%28B%2E+Buchberger%2C+S%2E+Oishi%2C+M%2E+Plum+and+S%2E+M%2E+Rump%2C+eds%29%2C+Vol%2E+053)
92. C. Keil and C. Jansson (2006), ‘Computational experience with rigorous error bounds for the Netlib linear programming library’, Reliable Computing 12, 303–321. [http://www.optimization-online.org/DB](http://www.optimization-online.org/DB) HTML/2004/12/1018.html. [↗](http://www.optimization-online.org/DB)
93. R. Klatte, U. Kulisch, A. Wiethoff, C. Lawo and M. Rauch (1993), C-XSC A C++ Class Library for Extended Scientific Computing, Springer, Berlin. [find](https://scholar.google.com/scholar?q=R%2E+Klatte%2C+U%2E+Kulisch%2C+A%2E+Wiethoff%2C+C%2E+Lawo+and+M%2E+Rauch+%281993%29%2C+C%2DXSC+A+C%2B%2B+Class+Library+for+Extended+Scientific+Computing%2C+Springer%2C+Berlin%2E)
94. O. Knüppel (1994), ‘PROFIL/BIAS: A fast interval library’, Computing 53, 277– 287. [find](https://scholar.google.com/scholar?q=O%2E+Knu%CC%88ppel+%281994%29%2C+%E2%80%98PROFIL%2FBIAS%3A+A+fast+interval+library%E2%80%99%2C+Computing+53%2C+277%E2%80%93+287%2E)
95. O. Knüppel (1998), PROFIL/BIAS and extensions, Version 2.0. Technical report, Institut für Informatik III, Technische Universität Hamburg–Harburg. [find](https://scholar.google.com/scholar?q=O%2E+Knu%CC%88ppel+%281998%29%2C+PROFIL%2FBIAS+and+extensions%2C+Version+2%2E0%2E+Technical+report%2C+Institut+fu%CC%88r+Informatik+III%2C+Technische+Universita%CC%88t+Hamburg%E2%80%93Harburg%2E)
96. D. E. Knuth (1969), The Art of Computer Programming: Seminumerical Algorithms, Vol. 2, Addison-Wesley, Reading, MA. [find](https://scholar.google.com/scholar?q=D%2E+E%2E+Knuth+%281969%29%2C+The+Art+of+Computer+Programming%3A+Seminumerical+Algorithms%2C+Vol%2E+2%2C+Addison%2DWesley%2C+Reading%2C+MA%2E)
97. L. V. Kolev and V. Mladenov (1997), Use of interval slopes in implementing an interval method for global nonlinear DC circuit analysis. Internat. J. Circuit Theory Appl. 12, 37–42. [find](https://scholar.google.com/scholar?q=L%2E+V%2E+Kolev+and+V%2E+Mladenov+%281997%29%2C+Use+of+interval+slopes+in+implementing+an+interval+method+for+global+nonlinear+DC+circuit+analysis%2E+Internat%2E+J%2E+Circuit+Theory+Appl%2E+12%2C+37%E2%80%9342%2E)
98. W. Krämer (1987), Inverse Standardfunktionen für reelle und komplexe Intervall-argumente mit a priori Fehlerabschätzung für beliebige Datenformate. Dissertation, Universität Karlsruhe. [find](https://scholar.google.com/scholar?q=W%2E+Kra%CC%88mer+%281987%29%2C+Inverse+Standardfunktionen+fu%CC%88r+reelle+und+komplexe+Intervall%2Dargumente+mit+a+priori+Fehlerabscha%CC%88tzung+fu%CC%88r+beliebige+Datenformate%2E+Dissertation%2C+Universita%CC%88t+Karlsruhe%2E)
99. W. Krämer (1991), Verified solution of eigenvalue problems with sparse matrices. In Proc. 13th World Congress on Computation and Applied Mathematics, pp. 32–33. [find](https://scholar.google.com/scholar?q=W%2E+Kra%CC%88mer+%281991%29%2C+Verified+solution+of+eigenvalue+problems+with+sparse+matrices%2E+In+Proc%2E+13th+World+Congress+on+Computation+and+Applied+Mathematics%2C+pp%2E+32%E2%80%9333%2E)
100. R. Krawczyk (1969a), ‘Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken’, Computing 4, 187–201. [find](https://scholar.google.com/scholar?q=R%2E+Krawczyk+%281969a%29%2C+%E2%80%98Newton%2DAlgorithmen+zur+Bestimmung+von+Nullstellen+mit+Fehlerschranken%E2%80%99%2C+Computing+4%2C+187%E2%80%93201%2E)
101. R. Krawczyk (1969b), ‘Fehlerabschätzung reeller Eigenwerte und Eigenvektoren von Matrizen’, Computing 4, 281–293. [find](https://scholar.google.com/scholar?q=R%2E+Krawczyk+%281969b%29%2C+%E2%80%98Fehlerabscha%CC%88tzung+reeller+Eigenwerte+und+Eigenvektoren+von+Matrizen%E2%80%99%2C+Computing+4%2C+281%E2%80%93293%2E)
102. R. Krawczyk and A. Neumaier (1985), ‘Interval slopes for rational functions and associated centered forms’, SIAM J. Numer. Anal. 22, 604–616. [find](https://scholar.google.com/scholar?q=R%2E+Krawczyk+and+A%2E+Neumaier+%281985%29%2C+%E2%80%98Interval+slopes+for+rational+functions+and+associated+centered+forms%E2%80%99%2C+SIAM+J%2E+Numer%2E+Anal%2E+22%2C+604%E2%80%93616%2E)
103. V. Kreinovich, A. V. Lakeyev and S. I. Noskov (1993), ‘Optimal solution of interval linear systems is intractable (NP-hard)’, Interval Comput. 1, 6–14. [find](https://scholar.google.com/scholar?q=V%2E+Kreinovich%2C+A%2E+V%2E+Lakeyev+and+S%2E+I%2E+Noskov+%281993%29%2C+%E2%80%98Optimal+solution+of+interval+linear+systems+is+intractable+%28NP%2Dhard%29%E2%80%99%2C+Interval+Comput%2E+1%2C+6%E2%80%9314%2E)
104. V. Kreinovich, A. Neumaier and G. Xiang (2008), ‘Towards a combination of interval and ellipsoid uncertainty’, Vych. Techn. (Computational Technologies) 13, 5–16. [find](https://scholar.google.com/scholar?q=V%2E+Kreinovich%2C+A%2E+Neumaier+and+G%2E+Xiang+%282008%29%2C+%E2%80%98Towards+a+combination+of+interval+and+ellipsoid+uncertainty%E2%80%99%2C+Vych%2E+Techn%2E+%28Computational+Technologies%29+13%2C+5%E2%80%9316%2E)
105. U. Kulisch (1981), Computer Arithmetic in Theory and Practice, Academic Press. [find](https://scholar.google.com/scholar?q=U%2E+Kulisch+%281981%29%2C+Computer+Arithmetic+in+Theory+and+Practice%2C+Academic+Press%2E)
106. M. La Porte and J. Vignes (1974), ‘Etude statistique des erreurs dans l’arith-métique des ordinateurs: Application au contrôle des résultats d’algorithmes númeriques’, Numer. Math. 23, 63–72. [find](https://scholar.google.com/scholar?q=M%2E+La+Porte+and+J%2E+Vignes+%281974%29%2C+%E2%80%98Etude+statistique+des+erreurs+dans+l%E2%80%99arith%2Dme%CC%81tique+des+ordinateurs%3A+Application+au+contro%CC%82le+des+re%CC%81sultats+d%E2%80%99algorithmes+nu%CC%81meriques%E2%80%99%2C+Numer%2E+Math%2E+23%2C+63%E2%80%9372%2E)
107. O. A. Ladyzhenskaya and N. N. Uralt́seva (1968), Linear and Quasilinear Elliptic Equations, Academic Press, New York. [find](https://scholar.google.com/scholar?q=O%2E+A%2E+Ladyzhenskaya+and+N%2E+N%2E+Uralt%CC%81seva+%281968%29%2C+Linear+and+Quasilinear+Elliptic+Equations%2C+Academic+Press%2C+New+York%2E)
108. J. Lahmann and M. Plum (2004), ‘A computer-assisted instability proof for the Orr–Sommerfeld equation with Blasius profile’, Z. Angew. Math. Mech. 84, 188–204. [find](https://scholar.google.com/scholar?q=J%2E+Lahmann+and+M%2E+Plum+%282004%29%2C+%E2%80%98A+computer%2Dassisted+instability+proof+for+the+Orr%E2%80%93Sommerfeld+equation+with+Blasius+profile%E2%80%99%2C+Z%2E+Angew%2E+Math%2E+Mech%2E+84%2C+188%E2%80%93204%2E)
109. C. W. H. Lam, L. Thiel and S. Swiercz (1989), ‘The nonexistence of finite projective planes of order 10’, Canad. J. Math. 41, 1117–1123. [find](https://scholar.google.com/scholar?q=C%2E+W%2E+H%2E+Lam%2C+L%2E+Thiel+and+S%2E+Swiercz+%281989%29%2C+%E2%80%98The+nonexistence+of+finite+projective+planes+of+order+10%E2%80%99%2C+Canad%2E+J%2E+Math%2E+41%2C+1117%E2%80%931123%2E)
110. N. J. Lehmann (1963), ‘Optimale Eigenwerteinschließung’, Numer. Math. 5, 246– 272. [find](https://scholar.google.com/scholar?q=N%2E+J%2E+Lehmann+%281963%29%2C+%E2%80%98Optimale+Eigenwerteinschlie%C3%9Fung%E2%80%99%2C+Numer%2E+Math%2E+5%2C+246%E2%80%93+272%2E)
111. X. S. Li, J. W. Demmel, D. H. Bailey, G. Henry, Y. Hida, J. Iskandar, W. Kahan, S. Y. Kang, A. Kapur, M. C. Martin, B. J. Thompson, T. Tung and D. J. Yoo (2002), ‘Design, implementation and testing of extended and mixed precision BLAS’, ACM Trans. Math. Software 28, 152–205. [find](https://scholar.google.com/scholar?q=X%2E+S%2E+Li%2C+J%2E+W%2E+Demmel%2C+D%2E+H%2E+Bailey%2C+G%2E+Henry%2C+Y%2E+Hida%2C+J%2E+Iskandar%2C+W%2E+Kahan%2C+S%2E+Y%2E+Kang%2C+A%2E+Kapur%2C+M%2E+C%2E+Martin%2C+B%2E+J%2E+Thompson%2C+T%2E+Tung+and+D%2E+J%2E+Yoo+%282002%29%2C+%E2%80%98Design%2C+implementation+and+testing+of)
112. E. Loh and W. Walster (2002), ‘Rump’s example revisited’, Reliable Computing 8, 245–248. [find](https://scholar.google.com/scholar?q=E%2E+Loh+and+W%2E+Walster+%282002%29%2C+%E2%80%98Rump%E2%80%99s+example+revisited%E2%80%99%2C+Reliable+Computing+8%2C+245%E2%80%93248%2E)
113. R. Lohner (1988), Einschließung der Lösung gewöhnlicher Anfangs- und Randwert-aufgaben und Anordnungen. PhD thesis, University of Karlsruhe. [find](https://scholar.google.com/scholar?q=R%2E+Lohner+%281988%29%2C+Einschlie%C3%9Fung+der+Lo%CC%88sung+gewo%CC%88hnlicher+Anfangs%2D+und+Randwert%2Daufgaben+und+Anordnungen%2E+PhD+thesis%2C+University+of+Karlsruhe%2E)
114. Maple (2009), Release 13, Reference Manual. [find](https://scholar.google.com/scholar?q=Maple+%282009%29%2C+Release+13%2C+Reference+Manual%2E)
115. S. Markov and K. Okumura (1999), The contribution of T. Sunaga to interval analysis and reliable computing. In Developments in Reliable Computing (T. Csendes, ed.), Kluwer, pp. 167–188. [find](https://scholar.google.com/scholar?q=S%2E+Markov+and+K%2E+Okumura+%281999%29%2C+The+contribution+of+T%2E+Sunaga+to+interval+analysis+and+reliable+computing%2E+In+Developments+in+Reliable+Computing+%28T%2E+Csendes%2C+ed%2E%29%2C+Kluwer%2C+pp%2E+167%E2%80%93188%2E)
116. Mathematica (2009), Release 7.0, Reference Manual. [find](https://scholar.google.com/scholar?q=Mathematica+%282009%29%2C+Release+7%2E0%2C+Reference+Manual%2E)
117. MATLAB (2004), User’s Guide, Version 7, The MathWorks. [find](https://scholar.google.com/scholar?q=MATLAB+%282004%29%2C+User%E2%80%99s+Guide%2C+Version+7%2C+The+MathWorks%2E)
118. R. E. Moore (1962), Interval arithmetic and automatic error analysis in digital computing. Dissertation, Stanford University. [find](https://scholar.google.com/scholar?q=R%2E+E%2E+Moore+%281962%29%2C+Interval+arithmetic+and+automatic+error+analysis+in+digital+computing%2E+Dissertation%2C+Stanford+University%2E)
119. R. E. Moore (1966), Interval Analysis, Prentice-Hall, Englewood Cliffs. [find](https://scholar.google.com/scholar?q=R%2E+E%2E+Moore+%281966%29%2C+Interval+Analysis%2C+Prentice%2DHall%2C+Englewood+Cliffs%2E)
120. R. E. Moore (1977), ‘A test for existence of solutions for non-linear systems’, SIAM J. Numer. Anal. 4, 611–615. [find](https://scholar.google.com/scholar?q=R%2E+E%2E+Moore+%281977%29%2C+%E2%80%98A+test+for+existence+of+solutions+for+non%2Dlinear+systems%E2%80%99%2C+SIAM+J%2E+Numer%2E+Anal%2E+4%2C+611%E2%80%93615%2E)
121. R. E. Moore (1999), ‘The dawning’, Reliable Computing 5, 423–424. [find](https://scholar.google.com/scholar?q=R%2E+E%2E+Moore+%281999%29%2C+%E2%80%98The+dawning%E2%80%99%2C+Reliable+Computing+5%2C+423%E2%80%93424%2E)
122. S. M. Rump Verification methods R. E. Moore, R. B. Kearfott and M. J. Cloud (2009), Introduction To Interval Analysis, Cambridge University Press. [find](https://scholar.google.com/scholar?q=S%2E+M%2E+Rump+Verification+methods+R%2E+E%2E+Moore%2C+R%2E+B%2E+Kearfott+and+M%2E+J%2E+Cloud+%282009%29%2C+Introduction+To+Interval+Analysis%2C+Cambridge+University+Press%2E)
123. J.-M. Muller, N. Brisebarre, F. de Dinechin, C.-P. Jeannerod, V. Lefèvre, G. Melquiond, R. Revol, D. Stehlé and S. Torres (2009), Handbook of Floating-Point Arithmetic, Birkhäuser, Boston. [find](https://scholar.google.com/scholar?q=J%2E%2DM%2E+Muller%2C+N%2E+Brisebarre%2C+F%2E+de+Dinechin%2C+C%2E%2DP%2E+Jeannerod%2C+V%2E+Lefe%CC%80vre%2C+G%2E+Melquiond%2C+R%2E+Revol%2C+D%2E+Stehle%CC%81+and+S%2E+Torres+%282009%29%2C+Handbook+of+Floating%2DPoint+Arithmetic%2C+Birkha%CC%88user%2C+Boston%2E)
124. K. Nagatou, M. T. Nakao and N. Yamamoto (1999), ‘An approach to the numerical verification of solutions for nonlinear elliptic problems with local uniqueness’, Numer. Funct. Anal. Optim. 20, 543–565. [find](https://scholar.google.com/scholar?q=K%2E+Nagatou%2C+M%2E+T%2E+Nakao+and+N%2E+Yamamoto+%281999%29%2C+%E2%80%98An+approach+to+the+numerical+verification+of+solutions+for+nonlinear+elliptic+problems+with+local+uniqueness%E2%80%99%2C+Numer%2E+Funct%2E+Anal%2E+Optim%2E+20%2C+543%E2%80%93565%2E)
125. M. T. Nakao (1988), ‘A numerical approach to the proof of existence of solutions for elliptic problems’, Japan J. Appl. Math. 5, 313–332. [find](https://scholar.google.com/scholar?q=M%2E+T%2E+Nakao+%281988%29%2C+%E2%80%98A+numerical+approach+to+the+proof+of+existence+of+solutions+for+elliptic+problems%E2%80%99%2C+Japan+J%2E+Appl%2E+Math%2E+5%2C+313%E2%80%93332%2E)
126. M. T. Nakao (1993), Solving nonlinear elliptic problems with result verification using an H −1 type residual iteration. Computing (Suppl.) 9, 161–173. [find](https://scholar.google.com/scholar?q=M%2E+T%2E+Nakao+%281993%29%2C+Solving+nonlinear+elliptic+problems+with+result+verification+using+an+H+%E2%88%921+type+residual+iteration%2E+Computing+%28Suppl%2E%29+9%2C+161%E2%80%93173%2E)
127. M. T. Nakao and N. Yamamoto (1995), ‘Numerical verifications for solutions to elliptic equations using residual iterations with higher order finite elements’, J. Comput. Appl. Math. 60, 271–279. [find](https://scholar.google.com/scholar?q=M%2E+T%2E+Nakao+and+N%2E+Yamamoto+%281995%29%2C+%E2%80%98Numerical+verifications+for+solutions+to+elliptic+equations+using+residual+iterations+with+higher+order+finite+elements%E2%80%99%2C+J%2E+Comput%2E+Appl%2E+Math%2E+60%2C+271%E2%80%93279%2E)
128. M. T. Nakao, K. Hashimoto and Y. Watanabe (2005), ‘A numerical method to verify the invertibility of linear elliptic operators with applications to nonlinear problems’, Computing 75, 1–14. [find](https://scholar.google.com/scholar?q=M%2E+T%2E+Nakao%2C+K%2E+Hashimoto+and+Y%2E+Watanabe+%282005%29%2C+%E2%80%98A+numerical+method+to+verify+the+invertibility+of+linear+elliptic+operators+with+applications+to+nonlinear+problems%E2%80%99%2C+Computing+75%2C+1%E2%80%9314%2E)
129. N. S. Nedialkov (1999), Computing rigorous bounds on the solution of an initial value problem for an ordinary differential equation. PhD dissertation, University of Toronto, Canada. [find](https://scholar.google.com/scholar?q=N%2E+S%2E+Nedialkov+%281999%29%2C+Computing+rigorous+bounds+on+the+solution+of+an+initial+value+problem+for+an+ordinary+differential+equation%2E+PhD+dissertation%2C+University+of+Toronto%2C+Canada%2E)
130. N. S. Nedialkov and K. R. Jackson (2000), ODE software that computes guaranteed bounds on the solution. In Advances in Software Tools for Scientic Computing (H. P. Langtangen, A. M. Bruaset and E. Quak, eds), Springer, pp. 197–224. [find](https://scholar.google.com/scholar?q=N%2E+S%2E+Nedialkov+and+K%2E+R%2E+Jackson+%282000%29%2C+ODE+software+that+computes+guaranteed+bounds+on+the+solution%2E+In+Advances+in+Software+Tools+for+Scientic+Computing+%28H%2E+P%2E+Langtangen%2C+A%2E+M%2E+Bruaset+and+E%2E+Qua)
131. NETLIB (2009), Linear Programming Library. [http://www.netlib.org/lp](http://www.netlib.org/lp). [↗](http://www.netlib.org/lp)
132. A. Neumaier (1974), ‘Rundungsfehleranalyse einiger Verfahren zur Summation endlicher Summen’, Z. Angew. Math. Mech. 54, 39–51. [find](https://scholar.google.com/scholar?q=A%2E+Neumaier+%281974%29%2C+%E2%80%98Rundungsfehleranalyse+einiger+Verfahren+zur+Summation+endlicher+Summen%E2%80%99%2C+Z%2E+Angew%2E+Math%2E+Mech%2E+54%2C+39%E2%80%9351%2E)
133. A. Neumaier (1984), ‘New techniques for the analysis of linear interval equations’, Linear Algebra Appl. 58, 273–325. [find](https://scholar.google.com/scholar?q=A%2E+Neumaier+%281984%29%2C+%E2%80%98New+techniques+for+the+analysis+of+linear+interval+equations%E2%80%99%2C+Linear+Algebra+Appl%2E+58%2C+273%E2%80%93325%2E)
134. A. Neumaier (1987), ‘Overestimation in linear interval equations’, SIAM J. Numer. Anal. 24, 207–214. [find](https://scholar.google.com/scholar?q=A%2E+Neumaier+%281987%29%2C+%E2%80%98Overestimation+in+linear+interval+equations%E2%80%99%2C+SIAM+J%2E+Numer%2E+Anal%2E+24%2C+207%E2%80%93214%2E)
135. A. Neumaier (1988), ‘An existence test for root clusters and multiple roots’, Z. Angew. Math. Mech. 68, 256–257. [find](https://scholar.google.com/scholar?q=A%2E+Neumaier+%281988%29%2C+%E2%80%98An+existence+test+for+root+clusters+and+multiple+roots%E2%80%99%2C+Z%2E+Angew%2E+Math%2E+Mech%2E+68%2C+256%E2%80%93257%2E)
136. A. Neumaier (1989), ‘Rigorous sensitivity analysis for parameter-dependent systems of equations’, J. Math. Anal. Appl. 144, 16–25. [find](https://scholar.google.com/scholar?q=A%2E+Neumaier+%281989%29%2C+%E2%80%98Rigorous+sensitivity+analysis+for+parameter%2Ddependent+systems+of+equations%E2%80%99%2C+J%2E+Math%2E+Anal%2E+Appl%2E+144%2C+16%E2%80%9325%2E)
137. A. Neumaier (1990), Interval Methods for Systems of Equations, Encyclopedia of Mathematics and its Applications, Cambridge University Press. [find](https://scholar.google.com/scholar?q=A%2E+Neumaier+%281990%29%2C+Interval+Methods+for+Systems+of+Equations%2C+Encyclopedia+of+Mathematics+and+its+Applications%2C+Cambridge+University+Press%2E)
138. A. Neumaier (1993), ‘The wrapping effect, ellipsoid arithmetic, stability and confidence regions’, Computing Supplementum 9, 175–190. [find](https://scholar.google.com/scholar?q=A%2E+Neumaier+%281993%29%2C+%E2%80%98The+wrapping+effect%2C+ellipsoid+arithmetic%2C+stability+and+confidence+regions%E2%80%99%2C+Computing+Supplementum+9%2C+175%E2%80%93190%2E)
139. A. Neumaier (2001), Introduction to Numerical Analysis, Cambridge University Press. [find](https://scholar.google.com/scholar?q=A%2E+Neumaier+%282001%29%2C+Introduction+to+Numerical+Analysis%2C+Cambridge+University+Press%2E)
140. A. Neumaier (2002), ‘Grand challenges and scientific standards in interval analysis’, Reliable Computing 8, 313–320. [find](https://scholar.google.com/scholar?q=A%2E+Neumaier+%282002%29%2C+%E2%80%98Grand+challenges+and+scientific+standards+in+interval+analysis%E2%80%99%2C+Reliable+Computing+8%2C+313%E2%80%93320%2E)
141. A. Neumaier (2003), ‘Enclosing clusters of zeros of polynomials’, J. Comput. Appl. Math. 156, 389–401. [find](https://scholar.google.com/scholar?q=A%2E+Neumaier+%282003%29%2C+%E2%80%98Enclosing+clusters+of+zeros+of+polynomials%E2%80%99%2C+J%2E+Comput%2E+Appl%2E+Math%2E+156%2C+389%E2%80%93401%2E)
142. A. Neumaier (2004), Complete search in continuous global optimization and constraint satisfaction. In Acta Numerica, Vol. 13, Cambridge University Press, pp. 271–369. [find](https://scholar.google.com/scholar?q=A%2E+Neumaier+%282004%29%2C+Complete+search+in+continuous+global+optimization+and+constraint+satisfaction%2E+In+Acta+Numerica%2C+Vol%2E+13%2C+Cambridge+University+Press%2C+pp%2E+271%E2%80%93369%2E)
143. A. Neumaier (2009), FMathL: Formal mathematical language. [http://www.mat.univie.ac.at/˜neum/FMathL.html](http://www.mat.univie.ac.at/˜neum/FMathL.html). [↗](http://www.mat.univie.ac.at/˜neum/FMathL.html)
144. A. Neumaier (2010), ‘Improving interval enclosures’, to appear in Reliable Computing. [find](https://scholar.google.com/scholar?q=A%2E+Neumaier+%282010%29%2C+%E2%80%98Improving+interval+enclosures%E2%80%99%2C+to+appear+in+Reliable+Computing%2E)
145. A. Neumaier and T. Rage (1993), ‘Rigorous chaos verification in discrete dynamical systems’, Physica D 67, 327–346. [find](https://scholar.google.com/scholar?q=A%2E+Neumaier+and+T%2E+Rage+%281993%29%2C+%E2%80%98Rigorous+chaos+verification+in+discrete+dynamical+systems%E2%80%99%2C+Physica+D+67%2C+327%E2%80%93346%2E)
146. A. Neumaier and O. Shcherbina (2004), ‘Safe bounds in linear and mixed-integer programming’, Math. Program. A 99, 283–296. [find](https://scholar.google.com/scholar?q=A%2E+Neumaier+and+O%2E+Shcherbina+%282004%29%2C+%E2%80%98Safe+bounds+in+linear+and+mixed%2Dinteger+programming%E2%80%99%2C+Math%2E+Program%2E+A+99%2C+283%E2%80%93296%2E)
147. W. Oettli and W. Prager (1964.), ‘Compatibility of approximate solution of linear equations with given error bounds for coefficients and right-hand sides’, Numer. Math. 6, 405–409 [find](https://scholar.google.com/scholar?q=W%2E+Oettli+and+W%2E+Prager+%281964%2E%29%2C+%E2%80%98Compatibility+of+approximate+solution+of+linear+equations+with+given+error+bounds+for+coefficients+and+right%2Dhand+sides%E2%80%99%2C+Numer%2E+Math%2E+6%2C+405%E2%80%93409)
148. T. Ogita, S. Oishi and Y. Ushiro (2001), ‘Fast verification of solutions for sparse monotone matrix equations’, Comput. Suppl. 15, 175–187. [find](https://scholar.google.com/scholar?q=T%2E+Ogita%2C+S%2E+Oishi+and+Y%2E+Ushiro+%282001%29%2C+%E2%80%98Fast+verification+of+solutions+for+sparse+monotone+matrix+equations%E2%80%99%2C+Comput%2E+Suppl%2E+15%2C+175%E2%80%93187%2E)
149. S. Oishi (1998), private communication. [find](https://scholar.google.com/scholar?q=S%2E+Oishi+%281998%29%2C+private+communication%2E)
150. S. Oishi (2000), Numerical Methods with Guaranteed Accuracy (in Japanese), Corona-sya. [find](https://scholar.google.com/scholar?q=S%2E+Oishi+%282000%29%2C+Numerical+Methods+with+Guaranteed+Accuracy+%28in+Japanese%29%2C+Corona%2Dsya%2E)
151. S. Oishi and S. M. Rump (2002), ‘Fast verification of solutions of matrix equations’, Numer. Math. 90, 755–773. [find](https://scholar.google.com/scholar?q=S%2E+Oishi+and+S%2E+M%2E+Rump+%282002%29%2C+%E2%80%98Fast+verification+of+solutions+of+matrix+equations%E2%80%99%2C+Numer%2E+Math%2E+90%2C+755%E2%80%93773%2E)
152. T. Okayama, T. Matsuo and M. Sugihara (2009), Error estimates with explicit constants for sinc approximation, sinc quadrature and sinc indefinite integration. Technical Report METR2009-01, The University of Tokyo. [find](https://scholar.google.com/scholar?q=T%2E+Okayama%2C+T%2E+Matsuo+and+M%2E+Sugihara+%282009%29%2C+Error+estimates+with+explicit+constants+for+sinc+approximation%2C+sinc+quadrature+and+sinc+indefinite+integration%2E+Technical+Report+METR2009%2D01%2C+The+Univers)
153. J. B. Oliveira and L. H. de Figueiredo (2002) ‘Interval computation of Viswanath’s constant’, Reliable Computing 8, 131–138. [find](https://scholar.google.com/scholar?q=J%2E+B%2E+Oliveira+and+L%2E+H%2E+de+Figueiredo+%282002%29+%E2%80%98Interval+computation+of+Viswanath%E2%80%99s+constant%E2%80%99%2C+Reliable+Computing+8%2C+131%E2%80%93138%2E)
154. F. Ordóñez and R. M. Freund (2003), ‘Computational experience and the explanatory value of condition measures for linear optimization’, SIAM J. Optim. 14, 307–333. [find](https://scholar.google.com/scholar?q=F%2E+Ordo%CC%81n%CC%83ez+and+R%2E+M%2E+Freund+%282003%29%2C+%E2%80%98Computational+experience+and+the+explanatory+value+of+condition+measures+for+linear+optimization%E2%80%99%2C+SIAM+J%2E+Optim%2E+14%2C+307%E2%80%93333%2E)
155. M. Overton (2001), Numerical Computing with IEEE Floating Point Arithmetic, SIAM, Philadelphia. [find](https://scholar.google.com/scholar?q=M%2E+Overton+%282001%29%2C+Numerical+Computing+with+IEEE+Floating+Point+Arithmetic%2C+SIAM%2C+Philadelphia%2E)
156. A. Ovseevich and F. Chernousko (1987), ‘On optimal ellipsoids approximating reachable sets’, Problems of Control and Information Theory 16, 125–134. [find](https://scholar.google.com/scholar?q=A%2E+Ovseevich+and+F%2E+Chernousko+%281987%29%2C+%E2%80%98On+optimal+ellipsoids+approximating+reachable+sets%E2%80%99%2C+Problems+of+Control+and+Information+Theory+16%2C+125%E2%80%93134%2E)
157. M. Payne and R. Hanek (1983), ‘Radian reduction for trigonometric functions’, SIGNUM Newsletter 18, 19–24. [find](https://scholar.google.com/scholar?q=M%2E+Payne+and+R%2E+Hanek+%281983%29%2C+%E2%80%98Radian+reduction+for+trigonometric+functions%E2%80%99%2C+SIGNUM+Newsletter+18%2C+19%E2%80%9324%2E)
158. K. Petras (2002), ‘Self-validating integration and approximation of piecewise analytic functions’, J. Comput. Appl. Math. 145, 345–359. [find](https://scholar.google.com/scholar?q=K%2E+Petras+%282002%29%2C+%E2%80%98Self%2Dvalidating+integration+and+approximation+of+piecewise+analytic+functions%E2%80%99%2C+J%2E+Comput%2E+Appl%2E+Math%2E+145%2C+345%E2%80%93359%2E)
159. M. Plum (1992), ‘Numerical existence proofs and explicit bounds for solutions of nonlinear elliptic boundary value problems’, Computing 49, 25–44. [find](https://scholar.google.com/scholar?q=M%2E+Plum+%281992%29%2C+%E2%80%98Numerical+existence+proofs+and+explicit+bounds+for+solutions+of+nonlinear+elliptic+boundary+value+problems%E2%80%99%2C+Computing+49%2C+25%E2%80%9344%2E)
160. M. Plum (1994), ‘Enclosures for solutions of parameter-dependent nonlinear elliptic boundary value problems: Theory and implementation on a parallel computer’, Interval Comput. 3, 106–121. [find](https://scholar.google.com/scholar?q=M%2E+Plum+%281994%29%2C+%E2%80%98Enclosures+for+solutions+of+parameter%2Ddependent+nonlinear+elliptic+boundary+value+problems%3A+Theory+and+implementation+on+a+parallel+computer%E2%80%99%2C+Interval+Comput%2E+3%2C+106%E2%80%93121%2E)
161. M. Plum (1995), ‘Existence and enclosure results for continua of solutions of parameter-dependent nonlinear boundary value problems’, J. Comput. Appl. Math. 60, 187–200. [find](https://scholar.google.com/scholar?q=M%2E+Plum+%281995%29%2C+%E2%80%98Existence+and+enclosure+results+for+continua+of+solutions+of+parameter%2Ddependent+nonlinear+boundary+value+problems%E2%80%99%2C+J%2E+Comput%2E+Appl%2E+Math%2E+60%2C+187%E2%80%93200%2E)
162. M. Plum (1996), Enclosures for two-point boundary value problems near bifurcation points. In Scientific Computing and Validated Numerics: Proc. International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics, SCAN-95 (G. Alefeld et al., eds), Vol. 90 of Math. Res., Akadem… [find](https://scholar.google.com/scholar?q=M%2E+Plum+%281996%29%2C+Enclosures+for+two%2Dpoint+boundary+value+problems+near+bifurcation+points%2E+In+Scientific+Computing+and+Validated+Numerics%3A+Proc%2E+International+Symposium+on+Scientific+Computing%2C+Compute)
163. S. M. Rump Verification methods M. Plum (1997), ‘Guaranteed numerical bounds for eigenvalues’, In Spectral Theory and Computational Methods of Sturm–Liouville Problems: Proc. 1996 Conference, Knoxville, TN, USA (D. Hinton et al., eds), Vol. 191 of Lect. Notes Pure Appl. Math., Marcel Dekker, New Yor… [find](https://scholar.google.com/scholar?q=S%2E+M%2E+Rump+Verification+methods+M%2E+Plum+%281997%29%2C+%E2%80%98Guaranteed+numerical+bounds+for+eigenvalues%E2%80%99%2C+In+Spectral+Theory+and+Computational+Methods+of+Sturm%E2%80%93Liouville+Problems%3A+Proc%2E+1996+Conference%2C+Knoxvill)
164. M. Plum (2008), ‘Existence and multiplicity proofs for semilinear elliptic boundary value problems by computer assistance’, DMV Jahresbericht 110, 19–54. [find](https://scholar.google.com/scholar?q=M%2E+Plum+%282008%29%2C+%E2%80%98Existence+and+multiplicity+proofs+for+semilinear+elliptic+boundary+value+problems+by+computer+assistance%E2%80%99%2C+DMV+Jahresbericht+110%2C+19%E2%80%9354%2E)
165. M. Plum and C. Wieners (2002), ‘New solutions of the Gelfand problem’, J. Math. Anal. Appl. 269, 588–606. [find](https://scholar.google.com/scholar?q=M%2E+Plum+and+C%2E+Wieners+%282002%29%2C+%E2%80%98New+solutions+of+the+Gelfand+problem%E2%80%99%2C+J%2E+Math%2E+Anal%2E+Appl%2E+269%2C+588%E2%80%93606%2E)
166. S. Poljak and J. Rohn (1993), ‘Checking robust nonsingularity is NP-hard’, Math. Control, Signals, and Systems 6, 1–9. [find](https://scholar.google.com/scholar?q=S%2E+Poljak+and+J%2E+Rohn+%281993%29%2C+%E2%80%98Checking+robust+nonsingularity+is+NP%2Dhard%E2%80%99%2C+Math%2E+Control%2C+Signals%2C+and+Systems+6%2C+1%E2%80%939%2E)
167. L. B. Rall (1981), Automatic Differentiation: Techniques and Applications, Vol. 120 of Lecture Notes in Computer Science, Springer. [find](https://scholar.google.com/scholar?q=L%2E+B%2E+Rall+%281981%29%2C+Automatic+Differentiation%3A+Techniques+and+Applications%2C+Vol%2E+120+of+Lecture+Notes+in+Computer+Science%2C+Springer%2E)
168. H. Ratschek and J. Rokne (1984), Computer Methods for the Range of Functions, Halsted Press, New York. [find](https://scholar.google.com/scholar?q=H%2E+Ratschek+and+J%2E+Rokne+%281984%29%2C+Computer+Methods+for+the+Range+of+Functions%2C+Halsted+Press%2C+New+York%2E)
169. A. Rauh, E. Auer and E. P. Hofer (2006), ValEncIA-IVP: A comparison with other initial value problem solvers. In Proc. 12th GAMM–IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics, SCAN, Duisburg. [find](https://scholar.google.com/scholar?q=A%2E+Rauh%2C+E%2E+Auer+and+E%2E+P%2E+Hofer+%282006%29%2C+ValEncIA%2DIVP%3A+A+comparison+with+other+initial+value+problem+solvers%2E+In+Proc%2E+12th+GAMM%E2%80%93IMACS+International+Symposium+on+Scientific+Computing%2C+Computer+Arithme)
170. K. Rektorys (1980), Variational methods in mathematics. In Science and Engineering, 2nd edn, Reidel, Dordrecht. [find](https://scholar.google.com/scholar?q=K%2E+Rektorys+%281980%29%2C+Variational+methods+in+mathematics%2E+In+Science+and+Engineering%2C+2nd+edn%2C+Reidel%2C+Dordrecht%2E)
171. F. N. Ris (1972), Interval analysis and applications to linear algebra. PhD dissertation, Oxford University. [find](https://scholar.google.com/scholar?q=F%2E+N%2E+Ris+%281972%29%2C+Interval+analysis+and+applications+to+linear+algebra%2E+PhD+dissertation%2C+Oxford+University%2E)
172. R. H. Risch (1969), ‘The problem of integration in finite terms’, Trans. Amer. Math. Soc. 139, 167–189. [find](https://scholar.google.com/scholar?q=R%2E+H%2E+Risch+%281969%29%2C+%E2%80%98The+problem+of+integration+in+finite+terms%E2%80%99%2C+Trans%2E+Amer%2E+Math%2E+Soc%2E+139%2C+167%E2%80%93189%2E)
173. J. Rohn (1994), NP-hardness results for linear algebraic problems with interval data. In Topics in Validated Computations (J. Herzberger, ed.), Studies in Computational Mathematics, Elsevier, Amsterdam, pp. 463–471. [find](https://scholar.google.com/scholar?q=J%2E+Rohn+%281994%29%2C+NP%2Dhardness+results+for+linear+algebraic+problems+with+interval+data%2E+In+Topics+in+Validated+Computations+%28J%2E+Herzberger%2C+ed%2E%29%2C+Studies+in+Computational+Mathematics%2C+Elsevier%2C+Amsterda)
174. J. Rohn (2005), A handbook of results on interval linear problems. [http://www.cs.cas.cz/rohn/handbook](http://www.cs.cas.cz/rohn/handbook). [↗](http://www.cs.cas.cz/rohn/handbook)
175. J. Rohn (2009a), ‘Forty necessary and sufficient conditions for regularity of interval matrices: A survey’, Electron. J. Linear Algebra 18, 500–512. [find](https://scholar.google.com/scholar?q=J%2E+Rohn+%282009a%29%2C+%E2%80%98Forty+necessary+and+sufficient+conditions+for+regularity+of+interval+matrices%3A+A+survey%E2%80%99%2C+Electron%2E+J%2E+Linear+Algebra+18%2C+500%E2%80%93512%2E)
176. J. Rohn (2009b), VERSOFT: Verification software in MATLAB/INTLAB. [http://uivtx.cs.cas.cz/˜rohn/matlab](http://uivtx.cs.cas.cz/˜rohn/matlab). [↗](http://uivtx.cs.cas.cz/˜rohn/matlab)
177. J. Rohn and V. Kreinovich (1995), ‘Computing exact componentwise bounds on solutions of linear system is NP-hard’, SIAM J. Matrix Anal. Appl. 16, 415– 420. [find](https://scholar.google.com/scholar?q=J%2E+Rohn+and+V%2E+Kreinovich+%281995%29%2C+%E2%80%98Computing+exact+componentwise+bounds+on+solutions+of+linear+system+is+NP%2Dhard%E2%80%99%2C+SIAM+J%2E+Matrix+Anal%2E+Appl%2E+16%2C+415%E2%80%93+420%2E)
178. S. M. Rump (1980), Kleine Fehlerschranken bei Matrixproblemen. PhD thesis, Universität Karlsruhe. [find](https://scholar.google.com/scholar?q=S%2E+M%2E+Rump+%281980%29%2C+Kleine+Fehlerschranken+bei+Matrixproblemen%2E+PhD+thesis%2C+Universita%CC%88t+Karlsruhe%2E)
179. S. M. Rump (1983), Solving algebraic problems with high accuracy. Habilitationsschrift, published in A New Approach to Scientific Computation (U. W. Kulisch and W. L. Miranker, eds), Academic Press, pp. 51–120. [find](https://scholar.google.com/scholar?q=S%2E+M%2E+Rump+%281983%29%2C+Solving+algebraic+problems+with+high+accuracy%2E+Habilitationsschrift%2C+published+in+A+New+Approach+to+Scientific+Computation+%28U%2E+W%2E+Kulisch+and+W%2E+L%2E+Miranker%2C+eds%29%2C+Academic+Press%2C+p)
180. S. M. Rump (1990), ‘Rigorous sensitivity analysis for systems of linear and non-linear equations’, Math. Comput. 54, 721–736. [find](https://scholar.google.com/scholar?q=S%2E+M%2E+Rump+%281990%29%2C+%E2%80%98Rigorous+sensitivity+analysis+for+systems+of+linear+and+non%2Dlinear+equations%E2%80%99%2C+Math%2E+Comput%2E+54%2C+721%E2%80%93736%2E)
181. S. M. Rump (1994), Verification methods for dense and sparse systems of equations. In Topics in Validated Computations (J. Herzberger, ed.), Elsevier, Studies in Computational Mathematics, Amsterdam, pp. 63–136. [find](https://scholar.google.com/scholar?q=S%2E+M%2E+Rump+%281994%29%2C+Verification+methods+for+dense+and+sparse+systems+of+equations%2E+In+Topics+in+Validated+Computations+%28J%2E+Herzberger%2C+ed%2E%29%2C+Elsevier%2C+Studies+in+Computational+Mathematics%2C+Amsterdam%2C+)
182. S. M. Rump (1999a), [INTLAB: INTerval LABoratory](https://md.research.elamur.ai/free/en/papers/rump-1999-intlab-interval-laboratory/). In Developments in Reliable Computing (T. Csendes, ed.), Kluwer, Dordrecht, pp. 77–104. [http://www.ti3.tu-harburg.de/rump/intlab/index.html](http://www.ti3.tu-harburg.de/rump/intlab/index.html). [↗](http://www.ti3.tu-harburg.de/rump/intlab/index.html)
183. S. M. Rump (1999b), ‘Fast and parallel interval arithmetic’, BIT Numer. Math. 39, 539–560. [find](https://scholar.google.com/scholar?q=S%2E+M%2E+Rump+%281999b%29%2C+%E2%80%98Fast+and+parallel+interval+arithmetic%E2%80%99%2C+BIT+Numer%2E+Math%2E+39%2C+539%E2%80%93560%2E)
184. S. M. Rump (2001a), Rigorous and portable standard functions. BIT Numer. Math. 41, 540–562. [find](https://scholar.google.com/scholar?q=S%2E+M%2E+Rump+%282001a%29%2C+Rigorous+and+portable+standard+functions%2E+BIT+Numer%2E+Math%2E+41%2C+540%E2%80%93562%2E)
185. S. M. Rump (2001b), ‘Computational error bounds for multiple or nearly multiple eigenvalues’, Linear Algebra Appl. 324, 209–226. [find](https://scholar.google.com/scholar?q=S%2E+M%2E+Rump+%282001b%29%2C+%E2%80%98Computational+error+bounds+for+multiple+or+nearly+multiple+eigenvalues%E2%80%99%2C+Linear+Algebra+Appl%2E+324%2C+209%E2%80%93226%2E)
186. S. M. Rump (2003a), ‘Ten methods to bound multiple roots of polynomials’, J. Comput. Appl. Math. 156, 403–432. [find](https://scholar.google.com/scholar?q=S%2E+M%2E+Rump+%282003a%29%2C+%E2%80%98Ten+methods+to+bound+multiple+roots+of+polynomials%E2%80%99%2C+J%2E+Comput%2E+Appl%2E+Math%2E+156%2C+403%E2%80%93432%2E)
187. S. M. Rump (2003b), ‘Structured perturbations I: Normwise distances’, SIAM J. Matrix Anal. Appl. 25, 1–30. [find](https://scholar.google.com/scholar?q=S%2E+M%2E+Rump+%282003b%29%2C+%E2%80%98Structured+perturbations+I%3A+Normwise+distances%E2%80%99%2C+SIAM+J%2E+Matrix+Anal%2E+Appl%2E+25%2C+1%E2%80%9330%2E)
188. S. M. Rump (2003c), ‘Structured perturbations II: Componentwise distances’, SIAM J. Matrix Anal. Appl. 25, 31–56. [find](https://scholar.google.com/scholar?q=S%2E+M%2E+Rump+%282003c%29%2C+%E2%80%98Structured+perturbations+II%3A+Componentwise+distances%E2%80%99%2C+SIAM+J%2E+Matrix+Anal%2E+Appl%2E+25%2C+31%E2%80%9356%2E)
189. S. M. Rump (2006), ‘Eigenvalues, pseudospectrum and structured perturbations’, Linear Algebra Appl. 413, 567–593. [find](https://scholar.google.com/scholar?q=S%2E+M%2E+Rump+%282006%29%2C+%E2%80%98Eigenvalues%2C+pseudospectrum+and+structured+perturbations%E2%80%99%2C+Linear+Algebra+Appl%2E+413%2C+567%E2%80%93593%2E)
190. S. M. Rump (2009), ‘Ultimately fast accurate summation’, SIAM J. Sci. Comput. 31, 3466–3502. [find](https://scholar.google.com/scholar?q=S%2E+M%2E+Rump+%282009%29%2C+%E2%80%98Ultimately+fast+accurate+summation%E2%80%99%2C+SIAM+J%2E+Sci%2E+Comput%2E+31%2C+3466%E2%80%933502%2E)
191. S. M. Rump and S. Graillat (2009), Verified error bounds for multiple roots of systems of nonlinear equations. To appear in Numer. Algorithms; published online at Numer Algor DOI 10.1007/s11075-009-9339-3. — [doi:10.1007/s11075-009-9339-3](https://doi.org/10.1007/s11075-009-9339-3) [find](https://scholar.google.com/scholar?q=S%2E+M%2E+Rump+and+S%2E+Graillat+%282009%29%2C+Verified+error+bounds+for+multiple+roots+of+systems+of+nonlinear+equations%2E+To+appear+in+Numer%2E+Algorithms%3B+published+online+at+Numer+Algor+DOI+10%2E1007%2Fs11075%2D009%2D93)
192. S. M. Rump and S. Oishi (2009), Verified error bounds for multiple roots of non-linear equations. In Proc. International Symposium on Nonlinear Theory and its Applications: NOLTA’09. [find](https://scholar.google.com/scholar?q=S%2E+M%2E+Rump+and+S%2E+Oishi+%282009%29%2C+Verified+error+bounds+for+multiple+roots+of+non%2Dlinear+equations%2E+In+Proc%2E+International+Symposium+on+Nonlinear+Theory+and+its+Applications%3A+NOLTA%E2%80%9909%2E)
193. S. M. Rump and H. Sekigawa (2009), ‘The ratio between the Toeplitz and the unstructured condition number’, Operator Theory: Advances and Applications 199, 397–419. [find](https://scholar.google.com/scholar?q=S%2E+M%2E+Rump+and+H%2E+Sekigawa+%282009%29%2C+%E2%80%98The+ratio+between+the+Toeplitz+and+the+unstructured+condition+number%E2%80%99%2C+Operator+Theory%3A+Advances+and+Applications+199%2C+397%E2%80%93419%2E)
194. S. M. Rump and J. Zemke (2004), ‘On eigenvector bounds’, BIT Numer. Math. 43, 823–837. [find](https://scholar.google.com/scholar?q=S%2E+M%2E+Rump+and+J%2E+Zemke+%282004%29%2C+%E2%80%98On+eigenvector+bounds%E2%80%99%2C+BIT+Numer%2E+Math%2E+43%2C+823%E2%80%93837%2E)
195. S. M. Rump, T. Ogita and S. Oishi (2008), ‘Accurate floating-point summation I: Faithful rounding’, SIAM J. Sci. Comput. 31, 189–224. [find](https://scholar.google.com/scholar?q=S%2E+M%2E+Rump%2C+T%2E+Ogita+and+S%2E+Oishi+%282008%29%2C+%E2%80%98Accurate+floating%2Dpoint+summation+I%3A+Faithful+rounding%E2%80%99%2C+SIAM+J%2E+Sci%2E+Comput%2E+31%2C+189%E2%80%93224%2E)
196. N. V. Sahinidis and M. Tawaralani (2005), ‘A polyhedral branch-and-cut approach to global optimization’, Math. Program. B 103, 225–249. [find](https://scholar.google.com/scholar?q=N%2E+V%2E+Sahinidis+and+M%2E+Tawaralani+%282005%29%2C+%E2%80%98A+polyhedral+branch%2Dand%2Dcut+approach+to+global+optimization%E2%80%99%2C+Math%2E+Program%2E+B+103%2C+225%E2%80%93249%2E)
197. H. Schichl and A. Neumaier (2004), ‘Exclusion regions for systems of equations’, SIAM J. Numer. Anal. 42, 383–408. [find](https://scholar.google.com/scholar?q=H%2E+Schichl+and+A%2E+Neumaier+%282004%29%2C+%E2%80%98Exclusion+regions+for+systems+of+equations%E2%80%99%2C+SIAM+J%2E+Numer%2E+Anal%2E+42%2C+383%E2%80%93408%2E)
198. H. Schichl and A. Neumaier (2005), ‘Interval analysis on directed acyclic graphs for global optimization’, J. Global Optim. 33, 541–562. [find](https://scholar.google.com/scholar?q=H%2E+Schichl+and+A%2E+Neumaier+%282005%29%2C+%E2%80%98Interval+analysis+on+directed+acyclic+graphs+for+global+optimization%E2%80%99%2C+J%2E+Global+Optim%2E+33%2C+541%E2%80%93562%2E)
199. S. P. Shary (2002), ‘A new technique in systems analysis under interval uncertainty and ambiguity’, Reliable Computing 8, 321–419. [find](https://scholar.google.com/scholar?q=S%2E+P%2E+Shary+%282002%29%2C+%E2%80%98A+new+technique+in+systems+analysis+under+interval+uncertainty+and+ambiguity%E2%80%99%2C+Reliable+Computing+8%2C+321%E2%80%93419%2E)
200. G. W. Stewart (1990), ‘Stochastic perturbation theory’, SIAM Rev. 32, 579–610. [find](https://scholar.google.com/scholar?q=G%2E+W%2E+Stewart+%281990%29%2C+%E2%80%98Stochastic+perturbation+theory%E2%80%99%2C+SIAM+Rev%2E+32%2C+579%E2%80%93610%2E)
201. T. Sunaga (1956), Geometry of numerals. Master’s thesis, University of Tokyo. [find](https://scholar.google.com/scholar?q=T%2E+Sunaga+%281956%29%2C+Geometry+of+numerals%2E+Master%E2%80%99s+thesis%2C+University+of+Tokyo%2E)
202. T. Sunaga (1958), ‘Theory of an interval algebra and its application to numerical analysis’, RAAG Memoirs 2, 29–46. [find](https://scholar.google.com/scholar?q=T%2E+Sunaga+%281958%29%2C+%E2%80%98Theory+of+an+interval+algebra+and+its+application+to+numerical+analysis%E2%80%99%2C+RAAG+Memoirs+2%2C+29%E2%80%9346%2E)
203. A. Takayasu, S. Oishi and T. Kubo (2009a), Guaranteed error estimate for solutions to two-point boundary value problem. In Proc. International Symposium on Nonlinear Theory and its Applications: NOLTA’09, pp. 214–217. [find](https://scholar.google.com/scholar?q=A%2E+Takayasu%2C+S%2E+Oishi+and+T%2E+Kubo+%282009a%29%2C+Guaranteed+error+estimate+for+solutions+to+two%2Dpoint+boundary+value+problem%2E+In+Proc%2E+International+Symposium+on+Nonlinear+Theory+and+its+Applications%3A+NOLTA)
204. S. M. Rump Verification methods A. Takayasu, S. Oishi and T. Kubo (2009b), Guaranteed error estimate for solutions to linear two-point boundary value problems with FEM. In Proc. [find](https://scholar.google.com/scholar?q=S%2E+M%2E+Rump+Verification+methods+A%2E+Takayasu%2C+S%2E+Oishi+and+T%2E+Kubo+%282009b%29%2C+Guaranteed+error+estimate+for+solutions+to+linear+two%2Dpoint+boundary+value+problems+with+FEM%2E+In+Proc%2E)
205. Asia Simulation Conference 2009 (JSST 2009 ), Shiga, Japan, pp. 1–8. [find](https://scholar.google.com/scholar?q=Asia+Simulation+Conference+2009+%28JSST+2009+%29%2C+Shiga%2C+Japan%2C+pp%2E+1%E2%80%938%2E)
206. M. Tawaralani and N. V. Sahinidis (2004), ‘Global optimization of mixed-integer nonlinear programs: A theoretical and computational study’, Math. Program. 99, 563–591. [find](https://scholar.google.com/scholar?q=M%2E+Tawaralani+and+N%2E+V%2E+Sahinidis+%282004%29%2C+%E2%80%98Global+optimization+of+mixed%2Dinteger+nonlinear+programs%3A+A+theoretical+and+computational+study%E2%80%99%2C+Math%2E+Program%2E+99%2C+563%E2%80%93591%2E)
207. M. J. Todd (2001), Semidefinite programming. In Acta Numerica, Vol. 10, Cambridge University Press, pp. 515–560. [find](https://scholar.google.com/scholar?q=M%2E+J%2E+Todd+%282001%29%2C+Semidefinite+programming%2E+In+Acta+Numerica%2C+Vol%2E+10%2C+Cambridge+University+Press%2C+pp%2E+515%E2%80%93560%2E)
208. L. N. Trefethen (2002), ‘The SIAM 100-dollar, 100-digit challenge’, SIAM-NEWS 35, 2. [http://www.siam.org/siamnews/06-02/challengedigits.pdf](http://www.siam.org/siamnews/06-02/challengedigits.pdf). [↗](http://www.siam.org/siamnews/06-02/challengedigits.pdf)
209. L. N. Trefethen and R. Schreiber (1990), ‘Average-case stability of Gaussian elimination’, SIAM J. Matrix Anal. Appl. 11, 335–360. [find](https://scholar.google.com/scholar?q=L%2E+N%2E+Trefethen+and+R%2E+Schreiber+%281990%29%2C+%E2%80%98Average%2Dcase+stability+of+Gaussian+elimination%E2%80%99%2C+SIAM+J%2E+Matrix+Anal%2E+Appl%2E+11%2C+335%E2%80%93360%2E)
210. W. Tucker (1999), ‘The Lorenz attractor exists’, CR Acad. Sci., Paris, Sér. I, Math. 328, 1197–1202. [find](https://scholar.google.com/scholar?q=W%2E+Tucker+%281999%29%2C+%E2%80%98The+Lorenz+attractor+exists%E2%80%99%2C+CR+Acad%2E+Sci%2E%2C+Paris%2C+Se%CC%81r%2E+I%2C+Math%2E+328%2C+1197%E2%80%931202%2E)
211. R. H. Tütüncü, K. C. Toh and M. J. Todd (2003), ‘Solving semidefinite-quadratic-linear programs using SDPT3’, Math. Program. B 95, 189–217. [find](https://scholar.google.com/scholar?q=R%2E+H%2E+Tu%CC%88tu%CC%88ncu%CC%88%2C+K%2E+C%2E+Toh+and+M%2E+J%2E+Todd+%282003%29%2C+%E2%80%98Solving+semidefinite%2Dquadratic%2Dlinear+programs+using+SDPT3%E2%80%99%2C+Math%2E+Program%2E+B+95%2C+189%E2%80%93217%2E)
212. L. Vandenberghe and S. Boyd (1996), ‘Semidefinite programming’, SIAM Review 38, 49–95. [find](https://scholar.google.com/scholar?q=L%2E+Vandenberghe+and+S%2E+Boyd+%281996%29%2C+%E2%80%98Semidefinite+programming%E2%80%99%2C+SIAM+Review+38%2C+49%E2%80%9395%2E)
213. J. Vignes (1978), ‘New methods for evaluating the validity of the results of mathematical computations’, Math. Comp. Simul. XX, 227–249. [find](https://scholar.google.com/scholar?q=J%2E+Vignes+%281978%29%2C+%E2%80%98New+methods+for+evaluating+the+validity+of+the+results+of+mathematical+computations%E2%80%99%2C+Math%2E+Comp%2E+Simul%2E+XX%2C+227%E2%80%93249%2E)
214. J. Vignes (1980), Algorithmes Numériques: Analyse et Mise en Oeuvre 2: Equations et Systèmes Non Linéaires, Collection Langages et Algorithmes de l’Informatique, Editions Technip, Paris. [find](https://scholar.google.com/scholar?q=J%2E+Vignes+%281980%29%2C+Algorithmes+Nume%CC%81riques%3A+Analyse+et+Mise+en+Oeuvre+2%3A+Equations+et+Syste%CC%80mes+Non+Line%CC%81aires%2C+Collection+Langages+et+Algorithmes+de+l%E2%80%99Informatique%2C+Editions+Technip%2C+Paris%2E)
215. D. Viswanath (1999) ‘Random Fibonacci sequences and the number 1.13198824 . . .’, Math. Comp. 69, 1131–1155. [find](https://scholar.google.com/scholar?q=D%2E+Viswanath+%281999%29+%E2%80%98Random+Fibonacci+sequences+and+the+number+1%2E13198824+%2E+%2E+%2E%E2%80%99%2C+Math%2E+Comp%2E+69%2C+1131%E2%80%931155%2E)
216. D. Viswanath and L. N. Trefethen (1998), ‘Condition numbers of random triangular matrices’, SIAM J. Matrix Anal. Appl. 19, 564–581. [find](https://scholar.google.com/scholar?q=D%2E+Viswanath+and+L%2E+N%2E+Trefethen+%281998%29%2C+%E2%80%98Condition+numbers+of+random+triangular+matrices%E2%80%99%2C+SIAM+J%2E+Matrix+Anal%2E+Appl%2E+19%2C+564%E2%80%93581%2E)
217. M. Warmus (1956), ‘Calculus of approximations’, Bulletin de l’Academie Polonaise des Sciences 4, 253–259. [find](https://scholar.google.com/scholar?q=M%2E+Warmus+%281956%29%2C+%E2%80%98Calculus+of+approximations%E2%80%99%2C+Bulletin+de+l%E2%80%99Academie+Polonaise+des+Sciences+4%2C+253%E2%80%93259%2E)
218. B. Werner and A. Spence (1984), ‘The computation of symmetry-breaking bifurcation points’, SIAM J. Numer. Anal. 21, 388–399. [find](https://scholar.google.com/scholar?q=B%2E+Werner+and+A%2E+Spence+%281984%29%2C+%E2%80%98The+computation+of+symmetry%2Dbreaking+bifurcation+points%E2%80%99%2C+SIAM+J%2E+Numer%2E+Anal%2E+21%2C+388%E2%80%93399%2E)
219. J. H. Wilkinson (1965), The Algebraic Eigenvalue Problem, Clarendon Press, Oxford. [find](https://scholar.google.com/scholar?q=J%2E+H%2E+Wilkinson+%281965%29%2C+The+Algebraic+Eigenvalue+Problem%2C+Clarendon+Press%2C+Oxford%2E)
220. S. J. Wright (1993), ‘A collection of problems for which Gaussian elimination with partial pivoting is unstable’, SIAM J. Sci. Comput. 14, 231–238. [find](https://scholar.google.com/scholar?q=S%2E+J%2E+Wright+%281993%29%2C+%E2%80%98A+collection+of+problems+for+which+Gaussian+elimination+with+partial+pivoting+is+unstable%E2%80%99%2C+SIAM+J%2E+Sci%2E+Comput%2E+14%2C+231%E2%80%93238%2E)
221. N. Yamanaka, T. Okayama, S. Oishi and T. Ogita (2009), ‘A fast verified automatic integration algorithm using double exponential formula’, RIMS Kokyuroku 1638, 146–158. [find](https://scholar.google.com/scholar?q=N%2E+Yamanaka%2C+T%2E+Okayama%2C+S%2E+Oishi+and+T%2E+Ogita+%282009%29%2C+%E2%80%98A+fast+verified+automatic+integration+algorithm+using+double+exponential+formula%E2%80%99%2C+RIMS+Kokyuroku+1638%2C+146%E2%80%93158%2E)
222. R. C. Young (1931), ‘The algebra of many-valued quantities’, Mathematische Annalen 104, 260–290. [find](https://scholar.google.com/scholar?q=R%2E+C%2E+Young+%281931%29%2C+%E2%80%98The+algebra+of+many%2Dvalued+quantities%E2%80%99%2C+Mathematische+Annalen+104%2C+260%E2%80%93290%2E)
223. Y.-K. Zhu, J.-H. Yong and G.-Q. Zheng (2005), ‘A new distillation algorithm for floating-point summation’, SIAM J. Sci. Comput. 26, 2066–2078. [find](https://scholar.google.com/scholar?q=Y%2E%2DK%2E+Zhu%2C+J%2E%2DH%2E+Yong+and+G%2E%2DQ%2E+Zheng+%282005%29%2C+%E2%80%98A+new+distillation+algorithm+for+floating%2Dpoint+summation%E2%80%99%2C+SIAM+J%2E+Sci%2E+Comput%2E+26%2C+2066%E2%80%932078%2E)
224. G. Zielke and V. Drygalla (2003), ‘Genaue Lösung linearer Gleichungssysteme’, GAMM Mitt. Ges. Angew. Math. Mech. 26, 7–108. [find](https://scholar.google.com/scholar?q=G%2E+Zielke+and+V%2E+Drygalla+%282003%29%2C+%E2%80%98Genaue+Lo%CC%88sung+linearer+Gleichungssysteme%E2%80%99%2C+GAMM+Mitt%2E+Ges%2E+Angew%2E+Math%2E+Mech%2E+26%2C+7%E2%80%93108%2E)
225. S. Zimmermann and U. Mertins (1995), ‘Variational bounds to eigenvalues of self-adjoint eigenvalue problems with arbitrary spectrum’, Z. Anal. Anwendungen 14, 327–345. [find](https://scholar.google.com/scholar?q=S%2E+Zimmermann+and+U%2E+Mertins+%281995%29%2C+%E2%80%98Variational+bounds+to+eigenvalues+of+self%2Dadjoint+eigenvalue+problems+with+arbitrary+spectrum%E2%80%99%2C+Z%2E+Anal%2E+Anwendungen+14%2C+327%E2%80%93345%2E)

*Updated 3 November 2026*