…
►As
,
…
►
29.7.5
.
…
►Müller (1966a, b) found three formal asymptotic expansions for a fundamental system of solutions of (
29.2.1) (and (
29.11.1)) as
, one in terms of Jacobian elliptic functions and two in terms of Hermite polynomials.
In
Müller (1966c) it is shown
how these expansions lead
to asymptotic expansions for the Lamé functions
and
.
Weinstein and Keller (1985) give asymptotics for solutions of Hill’s equation (§
28.29(i)) that are applicable
to the Lamé equation.
…
►In other circumstances the power series are prone
to slow convergence and heavy numerical cancellation.
…
►Temme (1997) shows
how to overcome this difficulty by use of the Maclaurin expansions for these coefficients or by use of auxiliary functions.
…
►Similar observations apply
to the computation of modified Bessel functions, spherical Bessel functions, and Kelvin functions.
…
►Similar considerations apply
to the spherical Bessel functions and Kelvin functions.
…
►Newton’s rule (§
3.8(i)) or Halley’s rule (§
3.8(v)) can be used
to compute
to arbitrarily high accuracy the real or complex zeros of all the functions treated in this chapter.
…
…
►We use color
to augment these vizualizations, either
to reinforce the recognition of the height, or
to convey an extra dimension
to represent the phase of complex valued functions.
…
►To provide an easily interpreted encoding of surface heights, a rainbow-like mapping of height
to color is used.
The following figure illustrates the piece-wise linear mapping of the height
to each of the color components red, green and blue, written as
.
…
►In doing this, however, we would like
to place the mathematically significant phase values, specifically the multiples of
correponding
to the real and imaginary axes, at more immediately recognizable colors.
…
►We therefore use a piecewise linear mapping as illustrated below, that takes phase
to red,
to yellow,
to cyan and
to blue.
…
…
►To compute
,
,
to 10D when
,
.
►Four iterations of (
22.20.1) lead
to
.
…
►By application of the transformations given in §§
22.7(i) and
22.7(ii),
or
can always be made sufficently small
to enable the approximations given in §
22.10(ii) to be applied.
The rate of convergence is similar
to that for the arithmetic-geometric mean.
…
►Alternatively,
Sala (1989) shows
how to apply the arithmetic-geometric mean
to compute
.
…
…
►In practice, however, problems of severe instability often arise and in §§
3.6(ii)–
3.6(vii) we show
how these difficulties may be overcome.
…
►It therefore remains
to apply a normalizing factor
.
…
►(This part of the process is equivalent
to forward elimination.)
…
►Similar considerations apply
to the first-order equation
…Thus in the inhomogeneous case it may sometimes be necessary
to recur backwards
to achieve stability.
…
…
►To solve the system
…
►During this reduction process we store the
multipliers
that are used in each column
to eliminate other elements in that column.
…
►The sensitivity of the solution vector
in (
3.2.1)
to small perturbations in the matrix
and the vector
is measured by the
condition number
…
►where
and
are the normalized right and left eigenvectors of
corresponding
to the eigenvalue
.
…
►Lanczos’ method is related
to Gauss quadrature considered in §
3.5(v).
…