# NAG CL Interfaces21cbc (jacellip_​complex)

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## 1Purpose

s21cbc evaluates the Jacobian elliptic functions $\mathrm{sn}z$, $\mathrm{cn}z$ and $\mathrm{dn}z$ for a complex argument $z$.

## 2Specification

 #include
 void s21cbc (Complex z, double ak2, Complex *sn, Complex *cn, Complex *dn, NagError *fail)
The function may be called by the names: s21cbc, nag_specfun_jacellip_complex or nag_jacobian_elliptic.

## 3Description

s21cbc evaluates the Jacobian elliptic functions $\mathrm{sn}\left(z\mid k\right)$, $\mathrm{cn}\left(z\mid k\right)$ and $\mathrm{dn}\left(z\mid k\right)$ given by
 $sn(z∣k) = sin⁡ϕ cn(z∣k) = cos⁡ϕ dn(z∣k) = 1-k2sin2⁡ϕ,$
where $z$ is a complex argument, $k$ is a real argument (the modulus) with ${k}^{2}\le 1$ and $\varphi$ (the amplitude of $z$) is defined by the integral
 $z=∫0ϕdθ 1-k2sin2⁡θ .$
The above definitions can be extended for values of ${k}^{2}>1$ (see Salzer (1962)) by means of the formulae
 $sn(z∣k) = k1sn(kz∣k1) cn(z∣k) = dn(kz∣k1) dn(z∣k) = cn(kz∣k1),$
where ${k}_{1}=1/k$.
Special values include
 $sn(z∣0) = sin⁡z cn(z∣0) = cos⁡z dn(z∣0) = 1 sn(z∣1) = tanh⁡z cn(z∣1) = sech⁡z dn(z∣1) = sech⁡z.$
These functions are often simply written as $\mathrm{sn}z$, $\mathrm{cn}z$ and $\mathrm{dn}z$, thereby avoiding explicit reference to the argument $k$. They can also be expressed in terms of Jacobian theta functions (see s21ccc).
Another nine elliptic functions may be computed via the formulae
 $cd⁡z = cn⁡z/dn⁡z sd⁡z = sn⁡z/dn⁡z nd⁡z = 1/dn⁡z dc⁡z = dn⁡z/cn⁡z nc⁡z = 1/cn⁡z sc⁡z = sn⁡z/cn⁡z ns⁡z = 1/sn⁡z ds⁡z = dn⁡z/sn⁡z cs⁡z = cn⁡z/sn⁡z$
(see Abramowitz and Stegun (1972)).
The values of $\mathrm{sn}z$, $\mathrm{cn}z$ and $\mathrm{dn}z$ are obtained by calls to s21cac. Further details can be found in Section 9.

## 4References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
Salzer H E (1962) Quick calculation of Jacobian elliptic functions Comm. ACM 5 399

## 5Arguments

1: $\mathbf{z}$Complex Input
On entry: the argument $z$ of the functions.
Constraints:
• $\mathrm{abs}\left({\mathbf{z}}\mathbf{.}\mathbf{re}\right)\le =\sqrt{\lambda }$;
• $\mathrm{abs}\left({\mathbf{z}}\mathbf{.}\mathbf{im}\right)\le \sqrt{\lambda }$, where $\lambda =1/{\mathbf{nag_real_safe_small_number}}$.
2: $\mathbf{ak2}$double Input
On entry: the value of ${k}^{2}$.
Constraint: $0.0\le {\mathbf{ak2}}\le 1.0$.
3: $\mathbf{sn}$Complex * Output
4: $\mathbf{cn}$Complex * Output
5: $\mathbf{dn}$Complex * Output
On exit: the values of the functions $\mathrm{sn}z$, $\mathrm{cn}z$ and $\mathrm{dn}z$, respectively.
6: $\mathbf{fail}$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.
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_COMPLEX
On entry, $|{\mathbf{z}}\mathbf{.}\mathbf{im}|$ is too large: $|{\mathbf{z}}\mathbf{.}\mathbf{im}|=⟨\mathit{\text{value}}⟩$. It must be less than $⟨\mathit{\text{value}}⟩$.
On entry, $|{\mathbf{z}}\mathbf{.}\mathbf{re}|$ is too large: $|{\mathbf{z}}\mathbf{.}\mathbf{re}|=⟨\mathit{\text{value}}⟩$. It must be less than $⟨\mathit{\text{value}}⟩$.
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, ${\mathbf{ak2}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ak2}}\le 1.0$.
On entry, ${\mathbf{ak2}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ak2}}\ge 0.0$.

## 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 SIN and COS.

## 8Parallelism and Performance

s21cbc is not threaded in any implementation.

The values of $\mathrm{sn}z$, $\mathrm{cn}z$ and $\mathrm{dn}z$ are computed via the formulae
 $sn⁡z = sn(u,k)dn(v,k′) 1-dn2(u,k)sn2(v,k′) + i cn(u,k)dn(u,k)sn(v,k′)cn(v,k′) 1-dn2(u,k)sn2(v,k′) cn⁡z = cn(u,k)cn(v,k′) 1-dn2(u,k)sn2(v,k′) - i sn(u,k)dn(u,k)sn(v,k′)dn(v,k′) 1-dn2(u,k)sn2(v,k′) dn⁡z = dn(u,k)cn(v,k′)dn(v,k′) 1-dn2(u,k)sn2(v,k′) - i k2sn(u,k)cn(u,k)sn(v,k′) 1-dn2(u,k)sn2(v,k′) ,$
where $z=u+iv$ and ${k}^{\prime }=\sqrt{1-{k}^{2}}$ (the complementary modulus).

## 10Example

This example evaluates $\mathrm{sn}z$, $\mathrm{cn}z$ and $\mathrm{dn}z$ at $z=-2.0+3.0i$ when $k=0.5$, and prints the results.

### 10.1Program Text

Program Text (s21cbce.c)

### 10.2Program Data

Program Data (s21cbce.d)

### 10.3Program Results

Program Results (s21cbce.r)