|
|
NAG[s21cac] NAG[nag_real_jacobian_elliptic] - Jacobian elliptic functions sn, cn and dn of real argument
|
|
Calling Sequence
s21cac(u, m, sn, cn, dn, 'fail'=fail)
nag_real_jacobian_elliptic(. . .)
Parameters
|
|
u - float;
m - float;
|
|
|
|
On entry: the argument  and the argument  of the functions, respectively.
|
|
|
 , where  .
|
|
|
|
sn - assignable;
cn - assignable;
dn - assignable;
|
|
|
|
Note: On exit the variable sn will have a value of type float, cn will have a value of type float, dn will have a value of type float.
|
|
|
|
'fail'=fail - table; (optional)
|
|
|
|
The NAG error argument, see the documentation for NagError.
|
|
|
|
|
Description
|
|
|
|
Purpose
|
|
nag_real_jacobian_elliptic (s21cac) evaluates the Jacobian elliptic functions sn, cn and dn.
|
|
|
Description
|
|
nag_real_jacobian_elliptic (s21cac) evaluates the Jacobian elliptic functions of argument  and argument  ,
![[[sn(u,|,m),=,sin`ϕ` ],[cn(u,|,m),=,cos`ϕ` ],[dn(u,|,m),=,sqrt(1-msin`ϕ`) ]]](/support/helpjp/helpview.aspx?si=9729/file08691/math84.png)
where  , called the amplitude of  , is defined by the integral
The elliptic functions are sometimes written simply as  ,  and  , avoiding explicit reference to the argument  .
Another nine elliptic functions may be computed via the formulae
![[[cdu,=,cnu/dnu],[sdu,=,snu/dnu],[ndu,=,1/dnu],[dcu,=,dnu/cnu],[ncu,=,1/cnu],[scu,=,snu/cnu],[nsu,=,1/snu],[dsu,=,dnu/snu],[csu,=,cnu/snu]]](/support/helpjp/helpview.aspx?si=9729/file08691/math106.png)
(see Abramowitz and Stegun (1972)).
nag_real_jacobian_elliptic (s21cac) is based on a procedure given by Bulirsch (1960), and uses the process of the arithmetic-geometric mean (16.9 in Abramowitz and Stegun (1972)). Constraints are placed on the values of  and  in order to avoid the possibility of machine overflow.
|
|
|
Error Indicators and Warnings
|
|
"NE_BAD_PARAM"
On entry, argument  had an illegal 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 consult NAG for assistance.
"NE_REAL_2"
On entry,  is too large when used in conjunction with the supplied argument u:  it must be less than  .
On entry,  is too large:  it must be less than  .
|
|
|
Accuracy
|
|
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 SIN and COS.
|
|
|
|
Examples
|
|
|
>
|
u := 0.2:
m := 0.3:
NAG:-s21cac(u, m, sn, cn, dn):
|
|
|
|
See Also
|
|
Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
Bulirsch R (1960) Numerical calculation of elliptic integrals and elliptic functions Numer. Math. 7 76–90
s Chapter Introduction.
NAG Toolbox Overview.
NAG Web Site.
|
|
