• Corpus ID: 14718329

UNRESTRICTED ALGORITHMS FOR ELEMENTARY AND SPECIAL FUNCTIONS Invited Paper 1

@inproceedings{Brent1980UNRESTRICTEDAF,
  title={UNRESTRICTED ALGORITHMS FOR ELEMENTARY AND SPECIAL FUNCTIONS Invited Paper 1},
  author={Richard P. Brent},
  year={1980}
}
Floating-point computations are usually performed with fixed precision: the machine used may have “single” or “double” precision floating-point hardware, or on small machines fixed-precision floating-point operations may be implemented by software or firmware. Most high-level languages support only a small number of floating-point precisions, and those which support an arbitrary number usually demand that the precision be determinable at compile time. 
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One Citation

Unrestricted Algorithms for Elementary and Special Functions
  • R. Brent
  • Mathematics, Computer Science
    IFIP Congress
  • 1980
TLDR
The topics include: power series methods, use of halving identities, asymptotic expansions, continued fractions, recurrence relations, Newton's method, numerical contour integration, and the arithmetic-geometric mean.

References

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Some New Algorithms for High-Precision Computation of Euler’s Constant
TLDR
Using one of the algorithms, which is based on an identity involving Bessel functions, γ has been computed to 30,100 decimal places and it is shown that, if γ or exp(γ) is of the form P/Q for integers P and Q, then |Q| > 1015000.
Pitfalls in computation, or why a math book isn''t enough
The floating-point number system is contrasted with the real numbers. The author then illustrates the variety of computational pitfalls a person can fall into who merely translates information gained…
A Proposed Standard for Binary Floating-Point Arithmetic
TLDR
This proposed standard facilitates transportation of numerically oriented programs and encourages development of high-quality numerical software.
The Interval Arithmetic Package - Multiple Precision Version.
TLDR
The multiple precision version of the interval arithmetic package documented in MRC Technical Summary Report no. 1755, based on the FORTRAN multiple precision arithemetic package of Brent, is extremely portable.
Computation of π Using Arithmetic-Geometric Mean
A new formula for π is derived. It is a direct consequence of Gauss’ arithmetic-geometric mean, the traditional method for calculating elliptic integrals, and of Legendre’s relation for elliptic…
An Unrestricted Algorithm for the Exponential Function
An algorithm is presented for the computation of the exponential function of real argument. There are no restrictions on the range of the argument or on the precision that may be demanded in the…
An AUGMENT Interface for Brent's Multiple Precision Arithmetic Package
The procedure requuced to interface Brent's multiple premsmn package MP with the AUGMENT precompfler for Fortran is described A method of using the multiple preclsmn arithmetm package m conjunctmn…
Numerical Differentiation of Analytic Functions
TLDR
An algorithm that performs this evaluation for an arbitrary analytic function f(~) is described, and a short FORTRAN program for generating up to 50 leading derivatives is to be found in the algorithm section of this issue.
Big Omicron and big Omega and big Theta
TLDR
I have recently asked several prominent mathematicians if they knew what ~(n 2) meant, and more than half of them had never seen the notation before, so I decided to search more carefully, and to study the history of O-notation and o-notation as well.
On the zeros of the Riemann zeta function in the critical strip
We describe a computation which shows that the Riemann zeta function ζ(s) has exactly 75,000,000 zeros of the form σ+ it in the region 0 < t < 32,585,736.4; all these zeros are simple and lie on the…
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