# NAG CL Interfaces21dac (ellipint_​general_​2)

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## 1Purpose

s21dac returns the value of the general elliptic integral of the second kind $F\left(z,{k}^{\prime },a,b\right)$ for a complex argument $z$.

## 2Specification

 #include
 Complex s21dac (Complex z, double akp, double a, double b, NagError *fail)
The function may be called by the names: s21dac, nag_specfun_ellipint_general_2 or nag_general_elliptic_integral_f.

## 3Description

s21dac evaluates an approximation to the general elliptic integral of the second kind $F\left(z,{k}^{\prime },a,b\right)$ given by
 $F(z,k′,a,b)=∫0za+bζ2 (1+ζ2)(1+ζ2)(1+k′2ζ2) dζ,$
where $a$ and $b$ are real arguments, $z$ is a complex argument whose real part is non-negative and ${k}^{\prime }$ is a real argument (the complementary modulus). The evaluation of $F$ is based on the Gauss transformation. Further details, in particular for the conformal mapping provided by $F$, can be found in Bulirsch (1960).
Special values include
 $F (z, k ′ ,1,1) = ∫ 0 z d ζ (1+ ζ 2 ) (1+k′2 ζ 2 ) ,$
or ${F}_{1}\left(z,{k}^{\prime }\right)$ (the elliptic integral of the first kind) and
 $F(z,k′,1,k′2)=∫0z1+k′2ζ2 (1+ζ2)1+ζ2 dζ,$
or ${F}_{2}\left(z,{k}^{\prime }\right)$ (the elliptic integral of the second kind). Note that the values of ${F}_{1}\left(z,{k}^{\prime }\right)$ and ${F}_{2}\left(z,{k}^{\prime }\right)$ are equal to ${\mathrm{tan}}^{-1}\left(z\right)$ in the trivial case ${k}^{\prime }=1$.
s21dac is derived from an Algol 60 procedure given by Bulirsch (1960). Constraints are placed on the values of $z$ and ${k}^{\prime }$ in order to avoid the possibility of machine overflow.

## 4References

Bulirsch R (1960) Numerical calculation of elliptic integrals and elliptic functions Numer. Math. 7 76–90

## 5Arguments

1: $\mathbf{z}$Complex Input
On entry: the argument $z$ of the function.
Constraints:
• $0.0\le {\mathbf{z}}\mathbf{.}\mathbf{re}\le \lambda$;
• $\mathrm{abs}\left({\mathbf{z}}\mathbf{.}\mathbf{im}\right)\le \lambda$, where ${\lambda }^{6}=1/{\mathbf{nag_real_safe_small_number}}$.
2: $\mathbf{akp}$double Input
On entry: the argument ${k}^{\prime }$ of the function.
Constraint: $\mathrm{abs}\left({\mathbf{akp}}\right)\le \lambda$.
3: $\mathbf{a}$double Input
On entry: the argument $a$ of the function.
4: $\mathbf{b}$double Input
On entry: the argument $b$ of the function.
5: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error 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_COMPLEX
On entry, $|{\mathbf{z}}\mathbf{.}\mathbf{im}|$ is too large: $|{\mathbf{z}}\mathbf{.}\mathbf{im}|=⟨\mathit{\text{value}}⟩$. It must not exceed $⟨\mathit{\text{value}}⟩$.
On entry, ${\mathbf{z}}\mathbf{.}\mathbf{re}<0.0$: ${\mathbf{z}}\mathbf{.}\mathbf{re}=⟨\mathit{\text{value}}⟩$.
On entry, ${\mathbf{z}}\mathbf{.}\mathbf{re}$ is too large: ${\mathbf{z}}\mathbf{.}\mathbf{re}=⟨\mathit{\text{value}}⟩$. It must not exceed $⟨\mathit{\text{value}}⟩$.
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, $|{\mathbf{akp}}|$ is too large: $|{\mathbf{akp}}|=⟨\mathit{\text{value}}⟩$. It must not exceed $⟨\mathit{\text{value}}⟩$.
NE_S21_CONV
The iterative procedure used to evaluate the integral has failed to converge.

## 7Accuracy

In principle the function is capable of achieving full relative precision in the computed values. However, the accuracy obtainable in practice depends on the accuracy of the standard elementary functions such as atan2 and log.

## 8Parallelism and Performance

s21dac is not threaded in any implementation.

None.

## 10Example

This example evaluates the elliptic integral of the first kind ${F}_{1}\left(z,{k}^{\prime }\right)$ given by
 $F1(z,k′)=∫0zdζ (1+ζ2)(1+k′2ζ2) ,$
where $z=1.2+3.7i$ and ${k}^{\prime }=0.5$, and prints the results.

### 10.1Program Text

Program Text (s21dace.c)

### 10.2Program Data

Program Data (s21dace.d)

### 10.3Program Results

Program Results (s21dace.r)