NAG Library Routine Document
S21BCF returns a value of the symmetrised elliptic integral of the second kind, via the function name.
|REAL (KIND=nag_wp) S21BCF
||X, Y, Z
S21BCF calculates an approximate value for the integral
, at most one of
is zero, and
The basic algorithm, which is due to Carlson (1979)
and Carlson (1988)
, is to reduce the arguments recursively towards their mean by the rule:
and the function may be approximated adequately by a fifth order power series
The truncation error in this expansion is bounded by
and the recursive process is terminated when this quantity is negligible compared with the machine precision
The routine may fail either because it has been called with arguments outside the domain of definition, or with arguments so extreme that there is an unavoidable danger of setting underflow or overflow.
Note: , so there exists a region of extreme arguments for which the function value is not representable.
Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
Carlson B C (1979) Computing elliptic integrals by duplication Numerische Mathematik 33 1–16
Carlson B C (1988) A table of elliptic integrals of the third kind Math. Comput. 51 267–280
- 1: X – REAL (KIND=nag_wp)Input
- 2: Y – REAL (KIND=nag_wp)Input
- 3: Z – REAL (KIND=nag_wp)Input
On entry: the arguments , and of the function.
and only one of X
may be zero.
- 4: IFAIL – INTEGERInput/Output
must be set to
. If you are unfamiliar with this parameter you should refer to Section 3.3
in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
is recommended. If the output of error messages is undesirable, then the value
is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
. When the value is used it is essential to test the value of IFAIL on exit.
unless the routine detects an error or a warning has been flagged (see Section 6
6 Error Indicators and Warnings
If on entry
, explanatory error messages are output on the current error message unit (as defined by X04AAF
Errors or warnings detected by the routine:
On entry, either X
is negative, or both X
are zero; the function is undefined.
On entry, ; the function is undefined.
On entry, either Z
is too close to zero or both X
are too close to zero: there is a danger of setting overflow. See also the Users' Note
for your implementation.
On entry, at least one of X
is too large: there is a danger of setting underflow. On soft failure the routine returns zero. See also the Users' Note
for your implementation.
In principle the routine is capable of producing full machine precision. However round-off errors in internal arithmetic will result in slight loss of accuracy. This loss should never be excessive as the algorithm does not involve any significant amplification of round-off error. It is reasonable to assume that the result is accurate to within a small multiple of the machine precision.
You should consult the S Chapter Introduction
which shows the relationship of this function to the classical definitions of the elliptic integrals.
This example simply generates a small set of nonextreme arguments which are used with the routine to produce the table of low accuracy results.
9.1 Program Text
Program Text (s21bcfe.f90)
9.2 Program Data
9.3 Program Results
Program Results (s21bcfe.r)