s21bj: FL CL CPP AD PY MB

NAG CL Interface
s21bjc (ellipint_complete_2)

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NAG Library Manual, Mark 30.1

Interfaces: FL CL CPP AD PY MB

NAG CL 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

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Long (impl_details)
Full (nag_info_impl_details)

1 Purpose

s21bjc returns a value of the classical (Legendre) form of the complete elliptic integral of the second kind.

2 Specification

#include <nag.h>

double

s21bjc (double dm, NagError *fail)

The function may be called by the names: s21bjc, nag_specfun_ellipint_complete_2 or nag_elliptic_integral_complete_e.

3 Description

s21bjc 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 s21bbc) and

R_{D}

is the Carlson symmetrised incomplete elliptic integral of the second kind (see s21bcc).

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: $dm$ – double Input: On entry: the argument $m$ of the function.

Constraint: $dm \leq 1.0$ .
2: $fail$ – NagError * Input/Output: The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_REAL: On entry, $dm = ⟨ value ⟩$ ; the integral is undefined.
Constraint: $dm \leq 1.0$ .

7 Accuracy

In principle s21bjc 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.

s21bjc is not threaded in any implementation.

9 Further Comments

You should consult the S Chapter Introduction, which shows the relationship between this function 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