s21bj: FL CL CPP AD PY MB

NAG FL Interface
s21bjf (ellipint_complete_2)

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NAG Library Manual, Mark 30.1

Interfaces: FL CL CPP AD PY MB

NAG FL Interface Introduction

S (Specfun) Chapter Contents

S (Specfun) Chapter Introduction

s21bj: FL CL CPP AD PY MB

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

▸▿ 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1 Purpose

s21bjf returns a value of the classical (Legendre) form of the complete elliptic integral of the second kind, via the function name.

2 Specification

Fortran Interface

Function s21bjf (

dm, ifail)

Real (Kind=nag_wp)	::	s21bjf
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	dm

C Header Interface

#include <nag.h>

double

s21bjf_ (const double *dm, Integer *ifail)

The routine may be called by the names s21bjf or nagf_specfun_ellipint_complete_2.

3 Description

s21bjf calculates an approximation to the integral

E (m) = \int_{0}^{\frac{π}{2}} {(1 - m \sin^{2} θ)}^{\frac{1}{2}} d θ,

where

m \leq 1

The integral is computed using the symmetrised elliptic integrals of Carlson (Carlson (1979) and Carlson (1988)). The relevant identity is

E (m) = R_{F} (0, 1 - m, 1) - \frac{1}{3} m R_{D} (0, 1 - m, 1),

where

R_{F}

is the Carlson symmetrised incomplete elliptic integral of the first kind (see s21bbf) and

R_{D}

is the Carlson symmetrised incomplete elliptic integral of the second kind (see s21bcf).

4 References

NIST Digital Library of Mathematical Functions

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

5 Arguments

1: $dm$ – Real (Kind=nag_wp) Input: On entry: the argument $m$ of the function.

Constraint: $dm \leq 1.0$ .
2: $ifail$ – Integer Input/Output: On entry: ifail must be set to $0$ , $−1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $−1$ means that an error message is printed while a value of $1$ means that it is not.

If halting is not appropriate, the value $−1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $- 1$ or $1$ is used it is essential to test the value of ifail on exit.

On exit: $ifail = 0$ unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

ifail = 0

−1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, $dm = ⟨ value ⟩$ ; the integral is undefined.
Constraint: $dm \leq 1.0$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

In principle s21bjf 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.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

s21bjf is not threaded in any implementation.

9 Further Comments

You should consult the S Chapter Introduction, which shows the relationship between this routine and the Carlson definitions of the elliptic integrals. In particular, the relationship between the argument-constraints for both forms becomes clear.

For more information on the algorithms used to compute