Untangling Complex Systems, page 103
The factor c = ( x f ′( x) ) / f ( x) is called the condition number of the function f at x. If c is not much larger than 1, the numerical problem is well-conditioned. On the other hand, if c >> 1, the problem
is ill-conditioned.2 The output of the computation y is then approximated by the closest machine number y. Therefore, if we compare y with the true value y, we obtain:
y − y
≤ c( MA) + ( MA) [E.8]
y
The ideal algorithm would give y as output. However, the perfect algorithm is usually impossible to
implement. Many numerical algorithms compute “discrete” approximations to some desired “con-
tinuous” quantity. For example, an integral is evaluated numerically by computing a function at a
discrete set of points, rather than at “every” point. The discrepancy between the true answer and the
answer obtained in a practical calculation is called the truncation error, T. Truncation error would
persist even on a hypothetical “perfect” computer that had an infinitely accurate representation and
no roundoff error. The roundoff error is a characteristic of the computer hardware. The truncation
2 For instance, if f ( x) = xα, the condition number is c = x f ′ x f x = x α xα−
( ) / ( )
1 / xα = α . The problem is ill- conditioned
when α >> 1.
534
Appendix E
error, on the other hand, is entirely under the programmer’s control. An algorithm is numerically
stable if it yields in machine arithmetic a result y, such that
y − y
≤ T + c( MA) + ( MA) [E.9]
y
where T is of the order of MA.
e.4 hinTs for furTher reading
Who wants to learn more about errors in numerical computation can consult the books Numerical
Recipes by Press et al. (2007), Rounding Errors in Algebraic Processes by Wilkinson (1994), and
Floating-Point Computation by Sterbenz (1974).
References
Aaronson, S.; 2005, NP-complete problems and physical reality. ACM SIGACT News 36, 30–52.
Abarbanel, H. D. I.; 1996, Analysis of Observed Chaotic Data. Springer-Verlag, New York.
Abelson, H.; Allen, D.; Coore, D.; Hanson, C.; Homsy, G.; Knight, T. F.; Nagpal, R.; Rauch, E.; Sussman, G. J.;
Weiss, R.; 2000, Amorphous computing. Commun. ACM 43, 74–82.
Adamatzky, A.; De Lacy Costello, B.; 2012, Reaction-diffusion computing. pp. 1897–1920 in Handbook of
Natural Computing. Rozenberg, G.; Bäck, T.; Kok, J. N.; (Eds.), Springer-Verlag, Berlin, Germany.
Adleman, L. M.; 1994, Molecular computation of solutions to combinatorial problems. Science 266, 1021–1024.
Adleman, L. M.; 1998, Computing with DNA. Sci. Am. 279, 54–61.
Agladze, K. I.; Krinsky, V. I.; 1982, Multi-armed vortices in an active chemical medium. Nature 296, 424–426.
Aguilar, A.; 2009, What Is Death? A Scientific, Philosophical and Theological Exploration of Life’s End.
Libreria Editrice Vaticana, Vatican City.
Albert, R.; Barabási, A.-L.; 2002, Statistical mechanics of complex networks. Rev. Mod. Phys. 74, 47–97.
Albert, R.; Jeong, H.; Barabási, A.-L.; 2000, Error and attack tolerance of complex networks. Nature 406,
378–382.
Albus, J. S.; Bekey, G. A.; Holland, J. H.; Kanwisher, N. G.; Krichmar, J. L.; Mishkin, M.; Modha, D. S.;
Raichle, M. E.; Shepherd G. M.; Tononi, G.; 2007, A proposal for a decade of the mind initiative.
Science 317, 1321.
Alexander, S.; Orbach, R.; 1982, Density of states on fractals: <
Alon, U.; 2007, An Introduction to Systems Biology: Design Principles of Biological Circuits. Chapman &
Hall/CRC, Boca Raton, FL.
Amdahl, G. M.; 1967, Validity of the single processor approach to achieving large-scale computing capabili-
ties. AFIPS Conference Proc. 30, 483–485.
Amir, Y.; Ben-Ishay, E.; Levner, D.; Ittah, S.; Abu-Horowitz, A.; Bachelet, I.; 2014, Universal computing by
DNA origami robots in a living animal. Nat. Nanotechnol. 9, 353–357.
Amos, S. W.; James, M. R.; 2000, Principles of Transistor Circuits. 9th ed. Newnes Butterworth-Heinemann,
Woburn, MA.
Anderson, J. D. Jr.; 2009, Governing equations of fluid dynamics. Chapter 2, pp. 15–51 in Computational
Fluid Dynamics. Wendt, J. F.; (Ed.), Springer-Verlag, Berlin, Germany.
Andréasson, J.; Pischel, U.; 2015, Molecules with a sense of logic: A progress report. Chem. Soc. Rev. 44,
1053–1069.
Antal, T.; Droz, M.; Magnin, J.; Rácz, Z.; Zrinyi, M.; 1998, Derivation of the Matalon-Packter law for
Liesegang patterns. J. Chem. Phys. 109, 9479–9486.
Arnold, R.; Showalter, K.; Tyson, J. J.; 1987, Propagation of chemical reactions in space. J. Chem. Educ. 64, 740–742.
Arthur, W. B.; 2015, Complexity and the Economy. Oxford University Press, New York.
Ashkenasy, G.; Hermans, T. M.; Otto, S.; Taylor, A. F.; 2017, Systems chemistry. Chem. Soc. Rev. 46,
2543–2554.
Ashkin, A.; 1997, Optical trapping and manipulation of neutral particles using lasers. Proc. Natl. Acad. Sci.
USA 94, 4853–4860.
Atanasov, A. T.; 2014, Length of periods in the nasal cycle during 24-hours registration. Open J. Biophys. 4, 93–96.
Atkins, P.; de Paula, J.; 2010, Atkins’ Physical Chemistry. 9th ed. Oxford University Press, Oxford.
Axelrod, R.; 1976, Structure of Decision: The Cognitive Maps of Political Elites. Princeton University Press, Princeton, NJ.
Axelrod, R.; 1984, The Evolution of Cooperation. Basic Books, New York.
Ayarpadikannan, S.; Lee, H. E.; Han, K.; Kim, H. S.; 2015, Transposable element-driven transcript diversifica-
tion and its relevance to genetic disorders. Gene 558, 187–194.
Bajzer, Ž; Prendergast, F. G.; 1992, Maximum likelihood analysis of fluorescence data. Methods Enzymol.
210, 200–237.
Bak, P.; Paczuski, M.; 1995, Complexity, contingency, and criticality. Proc. Natl. Acad. Sci. USA 92,
6689–6696.
535
536
References
Bak, P.; Tang, C.; Wiesenfeld, K.; 1987, Self-organized criticality: An explanation of 1/f noise. Phys. Rev. Let.
59, 381–384.
Baker, R. E.; Schnell, S.; Maini, P. K.; 2006, A clock and wavefront mechanism for somite formation. Dev.
Biol. 293, 116–126.
Balagaddé, F. K.; You, L.; Hansen, C. L.; Arnold, F. H.; Quake, S. R.; 2005, Long-term monitoring of bacteria
undergoing programmed population control in a microchemostat. Science 309, 137–140.
Ball, P.; 2001, The Self-Made Tapestry: Pattern Formation in Nature. Oxford University Press, New York.
Ball, P.; 2009, Nature’s Patterns. A Tapestry in Three Parts. Branches. Oxford University Press, New York.
Ball, P.; 2012, Computer engineering: Feeling the heat. Nature 492, 174–176.
Bancaud, A.; Huet, S.; Daigle, N.; Mozziconacci, J.; Beaudouin, J.; Ellenberg, J.; 2009, Molecular crowding
affects diffusion and binding of nuclear proteins in heterochromatin and reveals the fractal organization
of chromatin. EMBO J. 28, 3785–3798.
Bancaud, A.; Lavelle, C.; Huet, S.; Ellenberg, J.; 2012, A fractal model for nuclear organization: Current evi-
dence and biological implications. Nucleic Acids Res. 40, 8783–8792.
Bandt, C.; Pompe, B.; 2002, Permutation entropy: A natural complexity measure for time series. Phys. Rev.
Lett. 88, 174102.
Bánsági, T. Jr.; Vanag, V. K.; Epstein, I. R.; 2011, Tomography of reaction-diffusion microemulsions reveals
three-dimensional turing patterns. Science 331, 1309–1312.
Barabási, A.-L.; 2016, Network Science. Cambridge University Press, Cambridge, UK, freely available at
http://barabasi.com/networksciencebook/.
Barabási, A.-L.; Albert, R.; 1999, Emergence of scaling in random networks. Science 286, 509–512.
Barabási, A.-L.; Oltvai, Z. N.; 2004, Network biology: Understanding the cell’s functional organization. Nat.
Rev. Genet. 5, 101–113.
Baranger, M.; 2000, Chaos, Complexity, and Entropy. New England Complex Systems Institute, http://www.
necsi.edu/projects/baranger/cce.html.
Barrio, R. A.; Varea, C.; Aragón, J. L.; Maini, P. K.; 1999, A two-dimensional numerical study of spatial pat-
tern formation in interacting turing systems. Bull. Math. Biol. 61, 483–505.
Bar-Yam, Y.; 2004, A mathematical theory of strong emergence using multiscale variety. Complexity 9, 15–24.
Bavireddi, H.; Bharate, P.; Kikkeri, R.; 2013, Use of Boolean and fuzzy logics in lactose glycocluster research.
Chem. Commun. 49, 9185–9187.
Beasley, D.; 2000, Possible applications of evolutionary computation. pp. 4–19 in Evolutionary Computation
1. Basic Algorithms and Operators. Bäck, T.; Fogel, D. B.; Michalewicz, Z.; (Eds.), Institute of Physics
Publishing, Bristol, UK.
Behrens, T. E. J.; Sporns, O.; 2011, Human connectomics. Curr. Opin. Neurobiol. 22, 1–10.
Beinhocker, E. D.; 2007, The Origin of Wealth: Evolution, Complexity, and the Radical Remaking of
Economics. Harvard Business School Publishing, Boston, MA.
Belousov, B. P.; 1958, A Periodic Reaction and Its Mechanism. Sbornik Referatov po Radiatsionni Meditsine,
Moscow: Medgiz, 145.
Ben-Jacob, E.; Garik, P.; 1990, The formation of patterns in non-equilibrium growth. Nature 343, 523–530.
Ben-Jacob, E.; Levine, H.; 2001, The artistry of nature. Nature 409, 985–986.
Benenson, Y.; Gil, B.; Ben-Dor, U.; Adar, R.; Shapiro, E.; 2004, An autonomous molecular computer for logi-
cal control of gene expression. Nature 429, 423–429.
Bennett, C. H.; 1987, Demons, engines and the second law. Sci. Am. 257, 108–116.
Bennett, C. H.; Bessette, F.; Brassard, G.; Salvail, L.; Smolin, J.; 1992, Experimental quantum cryptography.
J. Cryptol. 5, 3–28.
Bennet, C. H.; DiVincenzo, D. P.; 2000, Quantum information and computation. Nature 404, 247–255.
Berberan-Santos, M. N.; Bodunov, E. N.; Valeur, B.; 2005, Mathematical functions for the analysis of lumi-
nescence decays with underlaying distributions 1. Kohlrausch decay function (stretched exponential).
Chem. Phys. 315, 171–182.
Berger, B.; Leighton, T.; 1998, Protein folding in the hydrophobic-hydrophilic (HP) model is NP-complete.
J. Comput. Biol. 5, 27–40.
Berridge, M. J.; Lipp, P.; Bootman, M. D.; 2000, The versatility and universality of calcium signalling. Nat.
Rev. Mol. Cell Biol. 1, 11–21.
Bérut, A.; Arakelyan, A.; Petrosyan, A.; Ciliberto, S.; Dillenschneider, R.; Lutz, E.; 2012, Experimental veri-
fication of Landauer’s principle linking information and thermodynamics. Nature 483, 187–189.
Bianconi, E.; Piovesan, A.; Facchin, F.; Beraudi, A.; Casadei, R.; Frabetti, F.; Vitale, L.; Pelleri, M. C.; Tassani, S.; Piva, F.; Perez-Amodio, S.; Strippoli, P.; Canaider, S.; 2013, An estimation of the number of cells in the
human body. Ann. Hum. Biol. 40, 463–471.
References
537
Bishop, C. M.; 2006, Pattern Recognition and Machine Learning. Springer, Singapore.
Bissette, A. J.; Fletcher, S. P.; 2013, Mechanisms of autocatalysis. Angew. Chem. Int. Ed. 52, 2–30.
Björn, L. O.; (Ed.) 2015, Photobiology. The Science of Light and Life. 3rd ed. Springer, New York.
Blair, E. P.; Lent, C. S.; 2003, An architecture for molecular computing using quantum-dot cellular automata.
2003 Third IEEE Conference on Nanotechnology, 2003. IEEE-NANO 2003, 2, 402–405.
Blanchard, P.; Devaney, R. L.; Hall, G. R.; 2006, Differential Equations. 3rd ed. Thomson Brooks/Cole,
Belmont, CA.
Blankenship, R. E.; 2002, Molecular Mechanisms of Photosynthesis. Blackwell Science Publishers, Oxford,
UK, Malden, MA.
Boccaletti, S.; Grebogi, C.; Lai, Y.-C.; Mancini, H.; Maza, D.; 2000, The control of chaos: Theory and applica-
tions. Phys. Rep. 329, 103–197.
Boccaletti, S.; Kurths, J.; Osipov, G.; Valladares, D. L.; Zhou, C. S.; 2002, The synchronization of chaotic
systems. Phys. Rep. 366, 1–101.
Boissonade, J.; De Kepper, P.; 1980, Transitions from bistability to limit cycle oscillations. Theoretical analy-
sis and experimental evidence in an open chemical system. J. Phys. Chem. 84, 501–506.
Bolotin, A.; 2014, Computational solution to quantum foundational problems. Phys. Sci. Int. J. 4 (8), 1145–1157.
Bonabeau, E.; Dorigo, M.; Theraulaz, G.; 1999, Swarm Intelligence: From Natural to Artificial System.
Oxford University Press, New York.
Bonan, G. B.; Doney, S. C.; 2018, Climate, ecosystems, and planetary futures: The challenge to predict life in
Earth system models. Science 359, eaam8328.
Boole, G.; 1854, An Investigation of the Laws of Thought on Which Are Founded the Mathematical Theories
of Logic and Probabilities. Macmillan. Reprinted with corrections, Dover Publications, New York,
1958. (reissued by Cambridge University Press, 2009).
Bourzac, K.; 2012, Back to analogue. Nature 483, S34–S36.
Boyle, R. W.; 1987, Gold. History and Genesis of Deposits. Springer, Boston, MA.
Bracewell, R.; 1986, The Fourier Transform and Its Applications. McGraw-Hill, Boston, MA.
Bragard, J.; Velarde, M. G.; 1997, Bénard convection flows. J. Non-Equilib. Thermodyn. 22, 1–19.
Bray, D.; 1995, Protein molecules as computational elements in living beings. Nature 376, 307–312.
Bray, W. C.; 1921, A periodic reaction in homogeneous solution and its relation to catalysis. J. Am. Chem. Soc.
43, 1262–1267.
Bray, W. C.; Liebhafsky, H. A.; 1931, Reactions involving hydrogen peroxide, iodine and iodate ion. I.
Introduction. J. Am. Chem. Soc. 53, 38–48.
Brent, R.; Bruck, J.; 2006, Can computers help to explain biology? Nature 440, 416–417.
Briggs, T. S.; Rauscher, W. C.; 1973, An oscillating iodine clock reaction. J. Chem. Ed. 50, 496.
Brillouin, L.; 1956, Science and Information Theory. Academic Press, New York.
Brochon, J.-C.; 1994, Maximum entropy method of data analysis in time-resolved spectroscopy. Methods
Enzymol. 240, 262–311.
Brostow, W.; Hagg Lobland, H. E.; 2017, Materials: Introduction and Applications. John Wiley & Sons,
Hoboken, NJ.
Brumfiel, G.; 2012, Quantum leaps. Nature 491, 322–324.
Budroni, M. A.; Lemaigre, L.; De Wit, A.; Rossi, F.; 2015, Cross-diffusion-induced convective patterns in
microemulsion systems. Phys. Chem. Chem. Phys. 17, 1593–1600.
Burks, A. W.; Goldstine, H. H.; von Neumann, J.; 1963, Preliminary discussion of the logical design of an
electronic computing instrument. Volume V, pp. 34–79 in John von Neumann Collected works. Taub,
A. H.; (Ed.), The Macmillan, New York.
Bustamante, C.; Keller, D.; Oster, G.; 2001, The physics of molecular motors. Acc. Chem. Res. 34, 412–420.
Bustamante, C.; Liphardt, J.; Ritort, F.; 2005, The nonequilibrium thermodynamics of small systems. Phys.
Today 58, 43–48.
Caldarelli, G.; Catanzaro, M.; 2012, Networks. A Very Short Introduction. Oxford University Press, Oxford.
Cañestro, C.; Yokoi, H.; Postlethwait, J. H.; 2007, Evolutionary developmental biology and genomics. Nat.
Rev. Genet. 8, 932–942.
Cao, L.; 1997, Practical method for determining the minimum embedding dimension of a scalar time series.
Physica D 110, 43–50.
Capra, F.; Luisi, P. L.; 2014, The Systems View of Life. A Unifying Vision. Cambridge University Press,
New York.
Carroll, S. B.; 2005, Endless Forms Most Beautiful: The New Science of Evo Devo. Norton & Company, New York.
Cartwright, J. H. E.; Checa, A. G.; Escribano, B.; Sainz-Díaz, C. I.; 2012, Crystal growth as an excitable
medium. Phil. Trans. R. Soc. A. 370, 2866–2876.
538
References
Castets, V. V.; Dulos, E.; Boissonade, J.; De Kepper, P.; 1990, Experimental evidence of a sustained standing
Turing-type nonequilibrium chemical pattern. Phys. Rev. Lett. 64, 2953–2956.
Castelvecchi, D.; 2016, Can we open the black box of AI. Nature 538, 20–23.
Cavalli-Sforza, L. L.; 2001, Genes, Peoples and Languages. Penguin Books, London.
Cavin, R. K.; Lugli, P.; Zhirnov, V. V.; 2012, Science and engineering beyond Moore’s law. Proc. IEEE 100,
1720–1749.
Carruthers, P.; Chamberlain, A.; 2000, Evolution and the Human Mind: Modularity, Language, and Meta-
Cognition. Cambridge University Press, Cambridge, UK.
Cencini, M.; Cecconi, F.; Vulpiani, A.; 2009, Chaos: From Simple Models to Complex Systems. World
Scientific Publishing, Singapore.
Chandrasekhar, S.; 1961, Hydrodynamic and Hydromagnetic Stability. Clarendon Press, Oxford.
Chaisson, E. J.; 2002, Cosmic Evolution. The Rise of Complexity in Nature. Harvard University Press,
Cambridge, MA.
Chaitin, G.; 2012, Proving Darwin. Making Biology Mathematical. Pantheon Books, New York.
Chance, B.; Pye, E. K.; Ghosh, A. K.; Hess, B.; (Eds.) 1973, Biological and Biochemical Oscillators. Academic Press, New York.
Chase, G. C.; 1980, History of mechanical computing machinery. Ann. Hist. Comput. 2, 198–226.
Chock, P. B.; Stadtman, E. R.; 1977, Superiority of interconvertible enzyme cascades in metabolic regulation:
Analysis of multicyclic systems. Proc. Natl. Acad. Sci. USA 74, 2766–2770.
Christensen, K.; Moloney, N. R.; 2005, Complexity and Criticality. Imperial College Press, London.
Chruściński, D.; 2006, Geometric aspect of quantum mechanics and quantum entanglement. J. Phys. Conf.
