…
►This theorem is employed to increase efficiency in calculating with large numbers by making
use of smaller numbers in most of the calculation.
…
►Even though the lengthy calculation is repeated four times, once for each modulus, most of it only
uses five-digit integers and is accomplished quickly without overwhelming the machine’s memory.
Details of a machine program describing the method together with typical
numerical results can be found in
Newman (1967).
…
…
►Close to the origin
of parameter space, the series in §
36.8 can be
used.
…
►Close to the bifurcation set but far from
, the uniform asymptotic approximations of §
36.12 can be
used.
…
►Direct
numerical evaluation can be carried out along a contour that runs along the segment of the real
-axis containing all real critical points of
and is deformed outside this range so as to reach infinity along the asymptotic valleys of
.
(For the umbilics, representations as one-dimensional integrals (§
36.2) are
used.)
…
►For
numerical solution of partial differential equations satisfied by the canonical integrals see
Connor et al. (1983).
…
►Numerical quadrature is slower than most methods for the standard integrals but can be
useful for elliptic integrals that have complicated representations in terms of standard integrals.
…
…
►are
used in approximating solutions to differential equations with multiple turning points; see §
2.8(v).
…
►In
Olver (1977a, 1978) a different normalization is
used.
…
►In
, the solutions of (
9.13.13)
used in
Olver (1978) are
…The function on the right-hand side is recessive in the sector
, and is therefore an essential member of any
numerically satisfactory pair of solutions in this region.
►Another normalization of (
9.13.17) is
used in
Smirnov (1960), given by
…
…
►In
Colman et al. (2011) an algorithm is described that
uses expansions in continued fractions for high-precision computation of the Gauss hypergeometric function, when the variable and parameters are real and one of the
numerator parameters is a positive integer.
…
…
►Although the series expansions in §§
8.7,
8.19(iv), and
8.21(vi) converge for all finite values of
, they are cumbersome to
use when
is large owing to slowness of convergence and cancellation.
For large
the corresponding asymptotic expansions (generally divergent) are
used instead.
…
►See
Allasia and Besenghi (1987b) for the
numerical computation of
from (
8.6.4) by means of the trapezoidal rule.
…
►A
numerical inversion procedure is also given for calculating the value of
(with 10S accuracy), when
and
are specified, based on Newton’s rule (§
3.8(ii)).
…