complex arithmetic
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11—20 of 28 matching pages
11: 19.8 Quadratic Transformations
12: 4.45 Methods of Computation
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►For interval-arithmetic algorithms, see Markov (1981).
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§4.45(ii) Complex Variables
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4.45.15
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4.45.16
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13: Bibliography
14: Software Index
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Open Source | With Book | Commercial | |||||||||||||||||||||||
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4.48(ii) Interval Arithmetic | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | a | CoStLy | |||||||||||||||||
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16.27(iii) Complex Arguments | ✓ | a | ✓ | ✓ | ✓ | ✓ | |||||||||||||||||||
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22.22(iii) Complex Argument | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | a | ✓ | |||||||||||||||
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23.24(iii) Complex Argument | ✓ | ✓ | ✓ | ✓ | a | ||||||||||||||||||||
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33.26(iii) Complex arguments | ✓ | a | ✓ | ||||||||||||||||||||||
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15: Bibliography I
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IEEE Standard for Floating-Point Arithmetic.
The Institute of Electrical and Electronics Engineers, Inc..
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IEEE Standard for Interval Arithmetic: IEEE Std 1788-2015.
The Institute of Electrical and Electronics Engineers, Inc..
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IEEE Standard for Interval Arithmetic: IEEE Std 1788.1-2017.
The Institute of Electrical and Electronics Engineers, Inc..
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IEEE International Standard for Information Technology—Microprocessor Systems—Floating-Point arithmetic: IEEE Std 754-2019.
The Institute of Electrical and Electronics Engineers, Inc..
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The eigenvalue problem for infinite compact complex symmetric matrices with application to the numerical computation of complex zeros of and of Bessel functions of any real order
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Linear Algebra Appl. 194, pp. 35–70.
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16: Bibliography C
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Level-Index Arithmetic: An Introductory Survey.
In Numerical Analysis and Parallel Processing (Lancaster, 1987), P. R. Turner (Ed.),
Lecture Notes in Math., Vol. 1397, pp. 95–168.
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Polynomial approximations in the complex plane.
J. Comput. Appl. Math. 18 (2), pp. 193–211.
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An encyclopaedia of cubature formulas.
J. Complexity 19 (3), pp. 445–453.
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The arithmetic-geometric mean of Gauss.
Enseign. Math. (2) 30 (3-4), pp. 275–330.
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Gauss and the arithmetic-geometric mean.
Notices Amer. Math. Soc. 32 (2), pp. 147–151.
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17: 25.15 Dirichlet -functions
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25.15.1
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25.15.2
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►where is the complex conjugate of , and
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►This result plays an important role in the proof of Dirichlet’s theorem on primes in arithmetic progressions (§27.11).
Related results are:
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18: 13.29 Methods of Computation
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13.29.2
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13.29.5
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13.29.6
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13.29.7
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►The accuracy is controlled and validated by a running error analysis coupled with interval arithmetic.
19: 24.17 Mathematical Applications
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24.17.2
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24.17.5
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24.17.7
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24.17.8
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►Bernoulli and Euler numbers and polynomials occur in: number theory via (24.4.7), (24.4.8), and other identities involving sums of powers; the Riemann zeta function and -series (§25.15, Apostol (1976), and Ireland and Rosen (1990)); arithmetic of cyclotomic fields and the classical theory of Fermat’s last theorem (Ribenboim (1979) and Washington (1997)); -adic analysis (Koblitz (1984, Chapter 2)).
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20: 18.39 Applications in the Physical Sciences
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►Kuijlaars and Milson (2015, §1) refer to these, in this case complex zeros, as exceptional, as opposed to regular, zeros of the EOP’s, these latter belonging to the (real) orthogonality integration range.
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18.39.21
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►A major difficulty in such calculations, loss of precision, is addressed in Gautschi (2009) where use of variable precision arithmetic is discussed and employed.
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