Extended-Range Arithmetic and Normalized Legendre Polynomials

Smith, John M.; Olver, F. W. J.; Lozier, Daniel W.

doi:10.1145/355934.355940

Cited by 28 publications

(16 citation statements)

References 4 publications

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“…The normalized Legendre polynomialsP m n (x) can be computed by using a recursion technique (Smith et al 1981). The normalization is in accordance with Equation (D.9).…”

Section: G6 Legendre Polynomialsmentioning

confidence: 99%

Numerical Methods

2016

Applied Frequency‐Domain Electromagnetics

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“…The normalized Legendre polynomialsP m n (x) can be computed by using a recursion technique (Smith et al 1981). The normalization is in accordance with Equation (D.9).…”

Section: G6 Legendre Polynomialsmentioning

confidence: 99%

Numerical Methods

2016

Applied Frequency‐Domain Electromagnetics

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“…Smith et al (1981) report that the extended-range arithmetic computation of the ALFs is a factor two slower than the corresponding double-precision computation. Nothing is known about whether this also applies to SHA and SHS; (ii) we want to investigate the numerical stability of publicly available SHA and SHS programs.…”

Section: Introductionmentioning

confidence: 99%

“…The subject of the short note is a special form of computer floating-point arithmetic, which is introduced in Smith et al (1981). The basic idea of this extended-range arithmetic is to allocate a separate storage location to the exponent of each floating-point number.…”

Section: Introductionmentioning

confidence: 99%

Ultra-high degree spherical harmonic analysis and synthesis using extended-range arithmetic

Wittwer

Klees

Seitz

et al. 2007

J Geod

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We present software for spherical harmonic analysis (SHA) and spherical harmonic synthesis (SHS), which can be used for essentially arbitrary degrees and all co-latitudes in the interval (0 • , 180 • ). The routines use extended-range floating-point arithmetic, in particular for the computation of the associated Legendre functions. The price to be paid is an increased computation time; for degree 3, 000, the extended-range arithmetic SHS program takes 49 times longer than its standard arithmetic counterpart. The extended-range SHS and SHA routines allow us to test existing routines for SHA and SHS. A comparison with the publicly available SHS routine GEOGFG18 by Wenzel and HARMONIC SYNTH by Holmes and Pavlis confirms what is known about the stability of these programs. GEOGFG18 gives errors <1 mm for latitudes [−89 • 57.5 , 89 • 57.5 ] and maximum degree 1, 800. Higher degrees significantly limit the range of acceptable latitudes for a given accuracy. HARMONIC SYNTH gives good results up to degree 2, 700

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“…In much the same way, n umbers of extreme size can appear as intermediates in the calculation of practical, more terrestrial quantities. For example : Smith et al 1981 have addressed the problem of computing speci c values of normalized Legendre polynomials," used in the calculation of angular momentum in quantum mechanics and elsewhere. The details of this problem are of little concern here, but a brief summary can be given.…”

Section: Introductionmentioning

confidence: 99%

Handling floating-point exceptions in numeric programs

Hauser

1996

ACM Trans. Program. Lang. Syst.

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There are a number of schemes for handling arithmetic exceptions that can be used to improve the speed (or alternatively the reliability) of numeric code. Overflow and underflow are the most troublesome exceptions, and depending on the context in which the exception can occur, they may be addressed either: (1) through a “brute force” reevaluation with extended range, (2) by reevaluating using a technique known as scaling , (3) by substituting an infinity or zero, or (4) in the case of underflow, with gradual underflow. In the first two of these cases, the offending computation is simply reevaluated using a safer but slower method. The latter two cases are cheaper, more automated schemes that ideally are built in as options within the computer system. Other arithmetic exceptions can be handled with similar methods. These and some other techniques are examined with an eye toward determining the support programming languages and computer systems ought to provide for floating-point exception handling. It is argued that the cheapest short-term solution would be to give full support to most of the required (as opposed to recommended) special features of the IEC/IEEE Standard for Binary Floating-Point Arithmetic. An essential part of this support would include standardized access from high-level languages to the exception flags defined by the standard. Some possibilities outside the IEEE Standard are also considered, and a few thought on possible better-structured support within programming languages are discussed.

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Extended-Range Arithmetic and Normalized Legendre Polynomials

Cited by 28 publications

References 4 publications

Numerical Methods

Numerical Methods

Ultra-high degree spherical harmonic analysis and synthesis using extended-range arithmetic

Handling floating-point exceptions in numeric programs

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