C347 Complete Elliptic Integrals of First, Second, and Third Kind

Function subprograms RELI1C, RELI2C, RELI3C and DELI1C, DELI2C, DELI3C calculate the complete elliptic integrals of the first, second and third kind, respectively.

Function subprograms RELIGC and DELIGC calculate a general complete elliptic integral.

Function subprograms RELIKC, RELIEC and DELIKC, DELIEC calculate the complete elliptic integrals K$(k)$ and E$(k)$ .

On CDC and Cray computers, the double-precision versions DELI1C etc. are not available.

Mainly for reasons of numerical stability, the algorithms are based on the following definitions:

First kind:

$F$1*(k') = ∫0&inf;{dξ(1+ξ2)(1+k'2ξ2)}(k'2> 0).

Second kind:

$F$2*(k',a,b) = ∫0&inf;{a+bξ2(1+ξ2)(1+ξ2)(1+k'2ξ2)} dξ(k'2> 0).

Third kind:

$F$3*(k',p) = ∫0&inf;{1+ξ2(1+pξ2)(1+ξ2)(1+k'2ξ2)} dξ(k'2> 0, p ≠0).

Note that $F$₁^*(k') = F₂^*(k',1,1) =F₃^*(k',1) . For p < 0, the integral $F$₃^* is defined by its principal value.

The general integral:

$G(k\text{'},p,a,b)$	$=$	$\int$0&inf;{a+bξ2(1+pξ2)(1+ξ2)(1+k'2ξ2)} dξ
	$=$	$\int$0π/2{a cos2φ+ b sin2φcos2φ+ p sin2φ}{dφcos2φ+ k'2sin2φ}(k'2> 0).

For p < 0, this integral is defined by its principal value. See Notes for special cases.

The functions K(k) and E(k):

$K(k)$	$=$	$\int$0π/2{dψ1-k2sin2ψ}(|k| < 1),
$E(k)$	$=$	$\int$0π/21-k2sin2ψ dψ(|k| ≤1).

Other common definitions of the complete elliptic integrals and their relations to $F$₁^* , $F$₂^* , $F$₃^* are listed here for convenience ($k2+k\text{'}2=\; 1$ ):
First kind:

$F(k,\pi /2)$	$=$	1*(k') (|k| < 1),
$F(1,k)$	$=$	$\int$01{dη(1-η2)(1-k2η2)}&quad;= &quad;F1*(k') (|k| < 1).

Second kind:

$E(k,\pi /2)$	$=$	2*(k',1,k'2)(|k| ≤1),
$E(1,k)$	$=$	$\int$01{1-k2η21-η2} dη&quad;= &quad;F2*(k',1,k'2) (|k| ≤1).

Third kind:

$\Pi (\pi /2,h,k)$	$=$	$\int$0π/2{dψ(1+hsin2ψ)1-k2sin2ψ}	$=$	$F$3*(k',h+1) (|k| < 1),
$\Pi (1,h,k)$	$=$	$\int$01{dη(1+hη2)(1-η2)(1-k2η2)}	$=$	$F$3*(k',h+1) (|k| < 1).

Structure:

FUNCTION subprograms
User Entry Names:

RELI1C,	RELI2C,	RELI3C,	RELIGC,	RELIKC,	RELIEC
DELI1C,	DELI2C,	DELI3C,	DELIGC,	DELIKC,	DELIEC

Obsolete User Entry Names:

ELLICK$\equiv$ RELIKC,	ELLICE$\equiv$ RELIEC,
DELLIK$\equiv$ DELIKC,	DELLIE$\equiv$ DELIEC

Files Referenced: Unit 6
External References: MTLMTR (N002), ABEND (Z035)

Usage:

In any arithmetic expression, with $AK=k$ and $AKP=k\text{'}$ ,

RELI1C(AKP)	DELI1C(AKP)	$F$1*(k') ,
RELI2C(AKP,A,B)	DELI2C(AKP,A,B)	$F$2*(k',A,B) ,
RELI3C(AKP,AK2,P)	DELI3C(AKP,AK2,P)	$F$3*(k',P) ,
RELIGC(AKP,P,A,B)	DELIGC(AKP,P,A,B)	$G(k\text{'},P,A,B)$ ,
RELIKC(AK)	DELIKC(AK)	K$(k)$ ,
RELIEC(AK)	DELIEC(AK)	E$(k)$ ,

where RELI1C etc are of type REAL, DELI1C etc are of type DOUBLE PRECISION, and AKP, AK, AK2, A, B and P have the same type as the function name.

The redundant parameter AK2 in RELI3C and DELI3C permits improved accuracy when $k2$ is small, i.e. $k\text{'}\; \approx 1$ . In this case, $AK2=\; k2$ should be calculated using higher-precision arithmetic and then truncated before calling the subprogram.

Method:

The evaluation of $F$₁^* , $F$₂^* , $F$₃^*

is based on the Landen transformation, that of $G$ on the Bartky transformation. For details, see Ref. 1--3. For K$(k)$ and E$(k)$

Chebyshev approximations are used (see Ref. 4).

Accuracy:

The REAL functions (except on CDC and Cray computers) have full single-precision accuracy. The REAL functions on CDC and Cray, and the DOUBLE PRECISION functions on all computers have an accuracy approximately two significant digits less than the machine precision.

Restrictions:

1. RELI1C and DELI1C: $AKP\; \ne 0$ .
2. RELI2C and DELI2C: $AKP\; \ne 0$ . or $AKP\; =\; 0$ and $B\; =\; 0$ .
3. RELI3C and DELI3C: AKP*P $\ne$ 0.
4. RELIGC and DELIGC: $AKP\; \ne 0$ .
5. RELIKC and DELIKC: $|AK|\; \le 1$ , RELIEC and DELIEC: .

Error handling:

Error C347.1: Restriction 1 is not satisfied.
Error C347.2: Restriction 2 is not satisfied.
Error C347.3: Restriction 3 is not satisfied.
Error C347.4: Restriction 4 is not satisfied.
Error C347.5: Restriction 5 is not satisfied.
In all cases, the function value is set equal to zero, and a message is written on Unit 6, unless subroutine MTLSET (N002) has been called.

Notes:

Every linear combination of the three special complete elliptic integrals K$(k)$ , E$(k)$ , $\Pi (h,k)$ may be expressed in terms of $G(k\text{'},p,a,b)$ :

$\lambda K(k)\; +\; \mu E(k)$	$=$	$G(k\text{'},1,\lambda +\mu ,\lambda +\mu k\text{'}2)$
$\lambda K(k)\; +\; \mu \Pi (h,k)$	$=$	$G(k\text{'},h+1,\lambda +\mu ,\lambda (h+1)+\mu )$

Special examples are

$K(k)$	$=$	$G(k\text{'},1,1,1),$
$E(k)$	$=$	$G(k\text{'},1,1,k\text{'}2)$
$(K(k)-E(k))/k2$	$=$	$G(k\text{'},1,0,1),$
$(K(k)-k\text{'}2E(k))/k2$	$=$	$G(k\text{'},1,1,0),$
$\Pi (h,k)$	$=$	$G(k\text{'},h+1,1,1),$
$(K(k)-\Pi (h,k))/h$	$=$	$G(k\text{'},h+1,0,1),$

If $ab\; \ge 0$ then $G$ will evaluate any linear combination of K$(k)$ , E$(k)$ , $\Pi (h,k)$ without cancellation (such as would occur, for example, if (K$(k)-$ E$(k))/k2$ were to be computed from values of K$(k)$ and E$(k)$ which had been computed separately.

Other functions which can be represented by $G$ are the Jacobian Zeta function $Z(\Phi ,k)$ and the Heuman Lambda function $\Lambda$₀(Φ,k) (see Ref. 5):

$Z(\Phi ,k)$	$=$	$k2\hspace{0.17em}\{sin\Phi cos\Phi K(k)\}\hspace{0.17em}G(k\text{'},q,0,q)$	$(q\; =\; cos2\Phi +\; k\text{'}2sin2\Phi )$
$\Lambda$0(Φ,k)	$=$	$\{2\pi \}qsin\Phi G(k\text{'},q,1,k\text{'}2)$	$(q\; =\; 1\; +\; k2tan2\Phi ).$

(Quoted from Ref. 3, slightly modified).

The subprograms for $F$₁^* , $F$₂^* are based on the Algol60 procedures cel1, cel2 in Ref. 1, those for $F$₃^* on cel3 in Ref. 2, and those for $G$

on cel in Ref. 3.

References:

1. R. Bulirsch, Numerical calculation of elliptic integrals and elliptic functions, Numer. Math. 7 (1965) 78--90.
2. R. Bulirsch, Numerical calculation of elliptic integrals and elliptic functions II, Numer. Math. 7 (1965) 353--354.
3. R. Bulirsch, Numerical calculation of elliptic integrals and elliptic functions III, Numer. Math. 13 (1969) 305--315.
4. W.J. Cody, Chebyshev approximations for the complete elliptic integrals K and E, Math. Comp. 19 (1965) 105--112.
5. P.F. Byrd and M.D. Friedman, Handbook of elliptic integrals for engineers and scientists, 2nd ed., Springer-Verlag Berlin (1971) 33--37.

C348