arithmetic operations

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1: 1.9 Calculus of a Complex Variable

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Arithmetic Operations

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2: Bibliography K

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C. Kormanyos (2011) Algorithm 910: a portable C++ multiple-precision system for special-function calculations. ACM Trans. Math. Software 37 (4), pp. Art. 45, 27.

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Notes:: This system provides a uniform interface in C++, named e_float, to perform arithmetic operations and to calculate mathematical functions that are implemented in several different MP packages. It is a free, open-source, multiple precision package that allows the computation of many special functions. It supports calculations with 30….300 decimal digits and is interoperable with Microsoft’s CLR, Python and Mathematica.
External Links:: ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: Erratum (V1.0.9) for References, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.
Encodings:: BibTeX

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3: 3.1 Arithmetics and Error Measures

… ►The elementary arithmetical operations on intervals are defined as follows: …

4: Bibliography H

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B. Hayes (2009) The higher arithmetic. American Scientist 97, pp. 364–368.

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External Links:: FullText
Referenced by:: §3.1(iv).
Encodings:: BibTeX

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G. J. Heckman (1991) An elementary approach to the hypergeometric shift operators of Opdam. Invent. Math. 103 (2), pp. 341–350.

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External Links:: ISSN 0020-9910, Document, MathReview (Eric M. Opdam), ZentralBlatt
Referenced by:: §18.37(iii).
Encodings:: BibTeX

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C. Hunter (1981) Two Parametric Eigenvalue Problems of Differential Equations. In Spectral Theory of Differential Operators (Birmingham, AL, 1981), North-Holland Math. Stud., Vol. 55, pp. 233–241.

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External Links:: MathReview (H. Hochstadt), ZentralBlatt
Referenced by:: §28.7.
Encodings:: BibTeX

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5: 18.39 Applications in the Physical Sciences

… ►The fundamental quantum Schrödinger operator, also called the Hamiltonian,

\mathcal{H}

, is a second order differential operator of the form … ►Analogous to (18.39.7) the 3D Schrödinger operator is …where

\mathrm{L}^{2}

is the (squared) angular momentum operator (14.30.12). … ►Here tridiagonal representations of simple Schrödinger operators play a similar role. The radial operator (18.39.28) …

6: 3.6 Linear Difference Equations

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3.6.2

a_{n}\Delta^{2}w_{n-1}+(2a_{n}-b_{n})\Delta w_{n-1}+(a_{n}-b_{n}+c_{n})w_{n-1}% =d_{n},

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Symbols:: $\Delta$ : forward difference operator, $w_{n}$ : sequence, $a_{n}$ : coefficient, $b_{n}$ : coefficient, $c_{n}$ : coefficient and $d_{n}$ : coefficient
Permalink:: http://dlmf.nist.gov/3.6.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §3.6(i), §3.6 and Ch.3

►where

\Delta w_{n-1}=w_{n}-w_{n-1}

\Delta^{2}w_{n-1}=\Delta w_{n}-\Delta w_{n-1}

, and

n\in\mathbb{Z}

. … ►Unless exact arithmetic is being used, however, each step of the calculation introduces rounding errors. …

I. Marquette and C. Quesne (2013) New ladder operators for a rational extension of the harmonic oscillator and superintegrability of some two-dimensional systems. J. Math. Phys. 54 (10), pp. Paper 102102, 12 pp..

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External Links:: MathReview Entry
Referenced by:: §18.36(vi).
Encodings:: BibTeX

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A. Máté, P. Nevai, and W. Van Assche (1991) The supports of measures associated with orthogonal polynomials and the spectra of the related selfadjoint operators. Rocky Mountain J. Math. 21 (1), pp. 501–527.

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External Links:: ISSN 0035-7596, Document, Link, MathReview (T. S. Chihara)
Referenced by:: §18.2(xi).
Encodings:: BibTeX

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D. W. Matula and P. Kornerup (1980) Foundations of Finite Precision Rational Arithmetic. In Fundamentals of Numerical Computation (Computer-oriented Numerical Analysis), G. Alefeld and R. D. Grigorieff (Eds.), Comput. Suppl., Vol. 2, Vienna, pp. 85–111.

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External Links:: MathReview (G. Blanch), ZentralBlatt
Referenced by:: §3.1(iii).
Encodings:: BibTeX

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X. Merrheim (1994) The computation of elementary functions in radix

2^{p}

. Computing 53 (3-4), pp. 219–232.

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Notes:: International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics (Vienna, 1993)
External Links:: ISSN 0010-485X, Document, MathReview, ZentralBlatt
Referenced by:: §4.45(i).
Encodings:: BibTeX

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mpmath (free python library)

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Notes:: Mpmath is a pure-Python library for multiprecision floating-point arithmetic. It provides an extensive set of transcendental functions, unlimited exponent sizes, complex numbers, interval arithmetic, numerical integration and differentiation, root-finding, linear algebra, and much more.
External Links:: mpmath
Referenced by:: § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.
Encodings:: BibTeX

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8: Bibliography C

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C. W. Clenshaw, F. W. J. Olver, and P. R. Turner (1989) 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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External Links:: MathReview Entry, ZentralBlatt
Referenced by:: §3.1(iv).
Encodings:: BibTeX

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D. A. Cox (1984) The arithmetic-geometric mean of Gauss. Enseign. Math. (2) 30 (3-4), pp. 275–330.

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External Links:: ISSN 0013-8584, MathReview (Bruce C. Berndt), ZentralBlatt
Referenced by:: §19.8(i).
Encodings:: BibTeX

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D. A. Cox (1985) Gauss and the arithmetic-geometric mean. Notices Amer. Math. Soc. 32 (2), pp. 147–151.

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External Links:: ISSN 0002-9920, MathReview (Willard Parker)
Referenced by:: §19.8(i).
Encodings:: BibTeX

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H. L. Cycon, R. G. Froese, W. Krisch, and B. Simon (2008) Schrödinger Operators, with Applications to Quantum Mechanics and Global Geometry. Springer Verlag, New York.

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External Links:: ISBN 978-3-540-16758-7
Referenced by:: §1.18(vii), §1.18(x).
Encodings:: BibTeX