interval arithmetic

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21—24 of 24 matching pages

21: 3.8 Nonlinear Equations

… ►If

f(a)f(b)<0

with

a<b

, then the interval

[a,b]

contains one or more zeros of

f

. …All zeros of

f

in the original interval

[a,b]

can be computed to any predetermined accuracy. … ►The convergence is linear, and again more than one zero may occur in the original interval

[x_{0},x_{1}]

. … ►There is no guaranteed convergence: the first approximation

x_{2}

may be outside

[x_{0},x_{1}]

. … ►However, when the coefficients are all real, complex arithmetic can be avoided by the following iterative process. …

22: 24.17 Mathematical Applications

… ►Let

0\leq h\leq 1

and

a, m

, and

n

be integers such that

n>a

m>0

, and

f^{(m)}(x)

is absolutely integrable over

[a,n]

. … ►Let

\mathcal{S}_{n}

denote the class of functions that have

n-1

continuous derivatives on

\mathbb{R}

and are polynomials of degree at most

n

in each interval

(k,k+1)

k\in\mathbb{Z}

. … ►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

L

-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));

p

-adic analysis (Koblitz (1984, Chapter 2)). …

23: Bibliography

… ►

G. Alefeld and J. Herzberger (1983) Introduction to Interval Computations. Computer Science and Applied Mathematics, Academic Press Inc., New York.

ⓘ

Notes:: Translated from the German by Jon Rokne
External Links:: ISBN 0-12-049820-0, MathReview (H. Ratschek), ZentralBlatt
Referenced by:: §3.1(ii).
Encodings:: BibTeX

… ►

G. Almkvist and B. Berndt (1988) Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses,

\pi

, and the Ladies Diary. Amer. Math. Monthly 95 (7), pp. 585–608.

ⓘ

External Links:: ISSN 0002-9890, FullText, Document, MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §19.9(i).
Encodings:: BibTeX

… ►

M. A. Anuta, D. W. Lozier, and P. R. Turner (1996) The MasPar MP-1 as a computer arithmetic laboratory. J. Res. Nat. Inst. Stand. Technol. 101 (2), pp. 165–174.

ⓘ

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

… ►

Arblib (C) Arb: A C Library for Arbitrary Precision Ball Arithmetic.

ⓘ

Notes:: An extended-precision C code for special functions. The library uses arbitrary-precision ball arithmetic. Ball arithmetic is an efficient technique for performing numerical computations with automatic and rigorous tracking of error bounds. Arb is designed with computer algebra and computational number theory in mind, extending the principles behind FLINT to the domain of real and complex numbers.
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.
Encodings:: BibTeX

…

24: Bibliography G

… ►

A. Gil, J. Segura, and N. M. Temme (2006a) Computing the real parabolic cylinder functions

U(a,x)

V(a,x)

. ACM Trans. Math. Software 32 (1), pp. 70–101.

ⓘ

External Links:: ISSN 0098-3500, Document, MathReview (Syamal K. Sen), ZentralBlatt
Referenced by:: §12.18.
Encodings:: BibTeX

►

A. Gil, J. Segura, and N. M. Temme (2006b) Algorithm 850: Real parabolic cylinder functions

U(a,x)

V(a,x)

. ACM Trans. Math. Software 32 (1), pp. 102–112.

ⓘ

External Links:: ISSN 0098-3500, Document, MathReview (Syamal K. Sen), ZentralBlatt
Referenced by:: §12.18, 1st item.
Encodings:: BibTeX

… ►

A. Gil, J. Segura, and N. M. Temme (2011a) Algorithm 914: parabolic cylinder function

W(a,x)

and its derivative. ACM Trans. Math. Software 38 (1), pp. Art. 6, 5.

ⓘ

External Links:: ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §12.18, §12.18, 2nd item, §12.21(ii).
Encodings:: BibTeX

►

A. Gil, J. Segura, and N. M. Temme (2011b) Fast and accurate computation of the Weber parabolic cylinder function

W(a,x)

. IMA J. Numer. Anal. 31 (3), pp. 1194–1216.

ⓘ

External Links:: ISSN 0272-4979, Document, FullText, MathReview (Adam Marlewski), ZentralBlatt
Referenced by:: §12.18, §12.18.
Encodings:: BibTeX

… ►

D. Goldberg (1991) What every computer scientist should know about floating-point arithmetic. ACM Computing Surveys 23 (1), pp. 5–48.

ⓘ

External Links:: ISSN 0360-0300, Document
Referenced by:: §3.1(i).
Encodings:: BibTeX

…