NAG FL Interfaces21bff (ellipint_​legendre_​2)

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1Purpose

s21bff returns a value of the classical (Legendre) form of the incomplete elliptic integral of the second kind, via the function name.

2Specification

Fortran Interface
 Function s21bff ( phi, dm,
 Real (Kind=nag_wp) :: s21bff Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: phi, dm
#include <nag.h>
 double s21bff_ (const double *phi, const double *dm, Integer *ifail)
The routine may be called by the names s21bff or nagf_specfun_ellipint_legendre_2.

3Description

s21bff calculates an approximation to the integral
 $E(ϕ∣m) = ∫0ϕ (1-msin2⁡θ) 12 dθ ,$
where $0\le \varphi \le \frac{\pi }{2}$ and $m{\mathrm{sin}}^{2}\varphi \le 1$.
The integral is computed using the symmetrised elliptic integrals of Carlson (Carlson (1979) and Carlson (1988)). The relevant identity is
 $E(ϕ∣m) = sin⁡ϕ RF (q,r,1) - 13 m sin3⁡ϕ RD (q,r,1) ,$
where $q={\mathrm{cos}}^{2}\varphi$, $r=1-m{\mathrm{sin}}^{2}\varphi$, ${R}_{F}$ 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).

4References

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

5Arguments

1: $\mathbf{phi}$Real (Kind=nag_wp) Input
2: $\mathbf{dm}$Real (Kind=nag_wp) Input
On entry: the arguments $\varphi$ and $m$ of the function.
Constraints:
• $0.0\le {\mathbf{phi}}\le \frac{\pi }{2}$;
• ${\mathbf{dm}}×{\mathrm{sin}}^{2}\left({\mathbf{phi}}\right)\le 1.0$.
3: $\mathbf{ifail}$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 $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{phi}}=⟨\mathit{\text{value}}⟩$.
Constraint: $0\le {\mathbf{phi}}\le \frac{\pi }{2}$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{phi}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{dm}}=⟨\mathit{\text{value}}⟩$; the integral is undefined.
Constraint: ${\mathbf{dm}}×{\mathrm{sin}}^{2}\left({\mathbf{phi}}\right)\le 1.0$.
${\mathbf{ifail}}=-99$
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.
${\mathbf{ifail}}=-399$
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.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7Accuracy

In principle s21bff 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.

8Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
s21bff is not threaded in any implementation.

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 ${R}_{F}$ and ${R}_{D}$, see the routine documents for s21bbf and s21bcf, respectively.
If you wish to input a value of phi outside the range allowed by this routine you should refer to Section 17.4 of Abramowitz and Stegun (1972) for useful identities. For example, $E\left(-\varphi |m\right)=-E\left(\varphi |m\right)$. A parameter $m>1$ can be replaced by one less than unity using $E\left(\varphi |m\right)=\sqrt{m}E\left(\varphi \sqrt{m}|\frac{1}{m}\right)-\left(m-1\right)\varphi$.

10Example

This example simply generates a small set of nonextreme arguments that are used with the routine to produce the table of results.

10.1Program Text

Program Text (s21bffe.f90)

None.

10.3Program Results

Program Results (s21bffe.r)