nach oben

Erschienen in:

2018 | OriginalPaper | Buchkapitel

11. Complex Numbers

verfasst von : Jean-Michel Muller, Nicolas Brunie, Florent de Dinechin, Claude-Pierre Jeannerod, Mioara Joldes, Vincent Lefèvre, Guillaume Melquiond, Nathalie Revol, Serge Torres

Erschienen in: Handbook of Floating-Point Arithmetic

Verlag: Springer International Publishing

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

Complex numbers naturally appear in many domains (such as electromagnetism, quantum mechanics, and relativity). It is of course always possible to express the various calculations that use complex numbers in terms of real numbers only. However, this will frequently result in programs that are larger and less clear. A good complex arithmetic would make numerical programs devoted to these problems easier to design, understand, and debug.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Vorheriges Kapitel Evaluating Floating-Point Elementary Functions

Nächstes Kapitel Interval Arithmetic

[32]

M. Baudin. Error bounds of complex arithmetic. Technical report, 2011. Available at http://forge.scilab.org/upload/compdiv/files/complexerrorbounds_v0.2.pdf.

[65]

R. P. Brent, C. Percival, and P. Zimmermann. Error bounds on complex floating-point multiplication. Mathematics of Computation, 76:1469–1481, 2007.MathSciNetCrossRef

[118]

M. Cornea, J. Harrison, and P. T. P. Tang. Scientific Computing on Itanium ^®; -based Systems. Intel Press, Hillsboro, OR, 2002.

[163]

J. W. Demmel and X. Li. Faster numerical algorithms via exception handling. In 11th IEEE Symposium on Computer Arithmetic, pages 234–241, June 1993.

[164]

J. W. Demmel and X. Li. Faster numerical algorithms via exception handling. IEEE Transactions on Computers, 43(8):983–992, 1994.CrossRef

[201]

B. P. Flannery, W. H. Press, S. A. Teukolsky, and W. T. Vetterling. Numerical Recipes in C, 2nd edition. Cambridge University Press, New York, NY, 1992.MATH

[207]

P. Friedland. Algorithm 312: Absolute value and square root of a complex number. Communications of the ACM, 10(10):665, 1967.CrossRef

[258]

N. J. Higham. Accuracy and Stability of Numerical Algorithms. SIAM, Philadelphia, PA, 2nd edition, 2002.CrossRef

[264]

T. E. Hull, T. F. Fairgrieve, and P. T. P. Tang. Implementing complex elementary functions using exception handling. ACM Transactions on Mathematical Software, 20(2):215–244, 1994.CrossRef

[265]

T. E. Hull, T. F. Fairgrieve, and P. T. P. Tang. Implementing the complex arcsine and arccosine functions using exception handling. ACM Transactions on Mathematical Software, 23(3):299–335, 1997.CrossRef

[287]

C.-P. Jeannerod. A radix-independent error analysis of the Cornea-Harrison-Tang method. ACM Transactions on Mathematical Software, 42(3):19:1–19:20, 2016.

[294]

C.-P. Jeannerod, P. Kornerup, N. Louvet, and J.-M. Muller. Error bounds on complex floating-point multiplication with an FMA. Mathematics of Computation, 86(304):881–898, 2017.MathSciNetCrossRef

[295]

C.-P. Jeannerod, N. Louvet, and J.-M. Muller. Further analysis of Kahan’s algorithm for the accurate computation of 2 × 2 determinants. Mathematics of Computation, 82(284):2245–2264, 2013.MathSciNetCrossRef

[296]

C.-P. Jeannerod, N. Louvet, and J.-M. Muller. On the componentwise accuracy of complex floating-point division with an FMA. In 21st IEEE Symposium on Computer Arithmetic (ARITH-21), pages 83–90, April 2013.

[298]

C.-P. Jeannerod, N. Louvet, J.-M. Muller, and A. Plet. Sharp error bounds for complex floating-point inversion. Numerical Algorithms, 73(3):735–760, 2016.MathSciNetCrossRef

[299]

C.-P. Jeannerod, J.-M. Muller, and A. Plet. The classical relative error bounds for computing \(\sqrt{a^{2 } + b^{2}}\) and \(c/\sqrt{a^{2 } + b^{2}}\) in binary floating-point arithmetic are asymptotically optimal. In 24th IEEE Symposium on Computer Arithmetic (ARITH-24), July 2017.

[303]

C.-P. Jeannerod and S. M. Rump. On relative errors of floating-point operations: optimal bounds and applications. Mathematics of Computation, 2016. To appear.

[317]

W. Kahan. Branch cuts for complex elementary functions. In The State of the Art in Numerical Analysis, pages 165–211, 1987.

[319]

W. Kahan. Matlab’s loss is nobody’s gain. Technical report, Computer Science, UC Berkeley, 1998. Available at https://people.eecs.berkeley.edu/~wkahan/MxMulEps.pdf.

[384]

X. Li, J. Demmel, D. H. Bailey, G. Henry, Y. Hida, J. Iskandar, W. Kahan, A. Kapur, M. Martin, T. Tung, and D. J. Yoo. Design, implementation and testing of extended and mixed precision BLAS. Technical Report 45991, Lawrence Berkeley National Laboratory, 2000. https://publications.lbl.gov/islandora/object/ir%3A115848.

[441]

J.-M. Muller. On the error of computing ab + cd using Cornea, Harrison and Tang’s method. ACM Transactions on Mathematical Software, 41(2):7:1–7:8, 2015.MathSciNetCrossRef

[497]

D. M. Priest. Efficient scaling for complex division. ACM Transactions on Mathematical Software, 30(4), 2004.

[521]

S. M. Rump. Ultimately fast accurate summation. SIAM Journal on Scientific Computing, 31(5):3466–3502, 2009.MathSciNetCrossRef

[522]

S. M. Rump. Error estimation of floating-point summation and dot product. BIT Numerical Mathematics, 52(1):201–220, 2012.MathSciNetCrossRef

[531]

S. M. Rump, T. Ogita, and S. Oishi. Accurate floating-point summation part I: Faithful rounding. SIAM Journal on Scientific Computing, 31(1):189–224, 2008.MathSciNetCrossRef

[561]

R. L. Smith. Algorithm 116: Complex division. Communications of the ACM, 5(8):435, 1962.

[573]

G. W. Stewart. A note on complex division. ACM Transactions on Mathematical Software, 11(3):238–241, 1985.CrossRef

Titel: Complex Numbers
verfasst von: Jean-Michel Muller
Nicolas Brunie
Florent de Dinechin
Claude-Pierre Jeannerod
Mioara Joldes
Vincent Lefèvre
Guillaume Melquiond
Nathalie Revol
Serge Torres
Verlag: Springer International Publishing
Buch: Handbook of Floating-Point Arithmetic
Print ISBN: 978-3-319-76525-9

Electronic ISBN: 978-3-319-76526-6

Copyright-Jahr: 2018
DOI: https://doi.org/10.1007/978-3-319-76526-6_11

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Springer Professional "Wirtschaft"