# Complex numerical analysis

What numerical analysis situations become more/less stable, have faster/slower convergence, or are otherwise quite different when dealing with functions of complex variable instead of functions of a real variable?

• Your question is just a little bit vague... Could you suggest a particular "situation" or "algorithm" that you had in mind? It would help us a lot to answer your question. – Paul Jan 15 '12 at 2:28
• The only instance where a complex number appears in numerics I know are Maxwell's equations, but there is no intrinsic difficulty only by some numbers being in $\mathbb C$. Still, if you replace all complex numbers by real vectors or matrices, then you see multiplication by a complex number becomes multiplication by a skew-symmetric matrix. Don't whether this implies anything. – shuhalo Jan 15 '12 at 2:34
• @Martin: The complex field is the natural setting for polynomials due to the fundamental theorem of algebra. Since the eigenvalues of a matrix are the roots of its characteristic polynomial, and are in general complex even for real matrices, linear algebra is most naturally built on top of the complex field. – Jack Poulson Jan 15 '12 at 3:28
• On the other hand, witness for instance the double-shift QR algorithm, which double-shifts precisely to sidestep the use of complex arithmetic. Witness as well the quadratic Jenkins-Traub algorithm, which was designed to find complex roots of polynomials a conjugate pair at a time... – J. M. Jan 15 '12 at 5:17
• I'm somewhat torn on this because to add even more confusion to the mix, there are times where complex numbers are basically just treated as pairs of real numbers for bookkeeping purposes. – Geoff Oxberry Jan 15 '12 at 6:32

Complex numerical differentiation is stable, unlike real numerical differentiation.

See pages 32-33 of "Applied and Computational Complex Analysis" vol 3, Peter Henrici,

"The Complex-Step Derivative Approximation", JOAQUIM R. R. A. MARTINS, PETER STURDZA and JUAN J. ALONSO,

and this Wikipedia article on complex variable methods for numerical differentiation.

• In addition, numerical use of the Cauchy differentiation formula is sometimes a viable algorithm. See also the methods by Lyness and others that hinge on the fast Fourier transform to compute Taylor coefficients of a function (that is, evaluating a sequence of derivatives at a given value). – J. M. Jan 15 '12 at 10:34
• Out of curiosity, besides the Wikipedia article, are there any online resources you could point us to? – Geoff Oxberry Jan 15 '12 at 12:45
• @Geoff: This and this deal with the Lyness approach to differentiation; this article by Squire and Trapp is the original paper detailing the "complex step" approach to numerical differentiation. – J. M. Jan 15 '12 at 14:02

Complex interval arithmetic uses different types of interval, e.g. rectangular or circular, so there's more to consider than when using real intervals.

"Complex interval arithmetic and its applications", Miodrag Petković, Ljiljana Petković

• Why answer your own question three times instead of responding with all three comments at once? – Jack Poulson Jan 15 '12 at 21:31

An article:

"Numerical algorithms based on the theory of complex variable", JN Lyness - Proceedings of the 1967 22nd national conference, 1967