NAG CL Interface
s21ccc (jactheta_real)
1
Purpose
s21ccc returns the value of one of the Jacobian theta functions , , , or for a real argument and non-negative .
2
Specification
double |
s21ccc (Integer k,
double x,
double q,
NagError *fail) |
|
The function may be called by the names: s21ccc, nag_specfun_jactheta_real or nag_jacobian_theta.
3
Description
s21ccc evaluates an approximation to the Jacobian theta functions
,
,
,
and
given by
where
and
(the
nome) are real with
.
These functions are important in practice because every one of the Jacobian elliptic functions (see
s21cbc) can be expressed as the ratio of two Jacobian theta functions (see
Whittaker and Watson (1990)). There is also a bewildering variety of notations used in the literature to define them. Some authors (e.g., Section 16.27 of
Abramowitz and Stegun (1972)) define the argument in the trigonometric terms to be
instead of
. This can often lead to confusion, so great care must therefore be exercised when consulting the literature. Further details (including various relations and identities) can be found in the references.
s21ccc is based on a truncated series approach. If
differs from
or
by an integer when
, it follows from the periodicity and symmetry properties of the functions that
and
. In a region for which the approximation is sufficiently accurate,
is set equal to the first term (
) of the transformed series
and
is set equal to the first two terms (i.e.,
) of
where
. Otherwise, the trigonometric series for
and
are used. For all values of
,
and
are computed from the relations
and
.
4
References
Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
Byrd P F and Friedman M D (1971) Handbook of Elliptic Integrals for Engineers and Scientists pp. 315–320 (2nd Edition) Springer–Verlag
Magnus W, Oberhettinger F and Soni R P (1966) Formulas and Theorems for the Special Functions of Mathematical Physics 371–377 Springer–Verlag
Tølke F (1966) Praktische Funktionenlehre (Bd. II) 1–38 Springer–Verlag
Whittaker E T and Watson G N (1990) A Course in Modern Analysis (4th Edition) Cambridge University Press
5
Arguments
-
1:
– Integer
Input
-
On entry: denotes the function to be evaluated. Note that is equivalent to .
Constraint:
.
-
2:
– double
Input
-
On entry: the argument of the function.
-
3:
– double
Input
-
On entry: the argument of the function.
Constraint:
.
-
4:
– 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_INT
-
On entry, .
Constraint: .
On entry, .
Constraint: .
- 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, .
Constraint: .
On entry, .
Constraint: .
7
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.
8
Parallelism and Performance
s21ccc is not threaded in any implementation.
None.
10
Example
This example evaluates at when , and prints the results.
10.1
Program Text
10.2
Program Data
10.3
Program Results