emmy.special.elliptic — org.mentat/emmy 0.31.0

This namespace contains function to compute various elliptic integrals in Carlson symmetric form, as well as the Jacobi elliptic functions.

carlson-rc clj/s

(carlson-rc x y)

Computes Carlson’s degenerate elliptic integral, $R_C(x, y)$. x must be nonnegative and y must be nonzero. If y < 0, the Cauchy principal value is returned.

Internal details:

• tiny must be at least 5 times the machine underflow limit
• big at most one fifth the machine maximum overflow limit.

carlson-rd clj/s

(carlson-rd x y z)

Comment from Press, section 6.11, page 257:

'Computes Carlson’s elliptic integral of the second kind, RD(x, y, z). x and y must be nonnegative, and at most one can be zero. z must be positive. TINY must be at least twice the negative 2/3 power of the machine overflow limit. BIG must be at most 0.1 × ERRTOL times the negative 2/3 power of the machine underflow limit.'

This is called Carlson-elliptic-2 in scmutils.

carlson-rf clj/s

(carlson-rf x y z)

From W.H. Press, Numerical Recipes in C++, 2ed. NR::rf from section 6.11

Here's the reference for what's going on here: http://phys.uri.edu/nigh/NumRec/bookfpdf/f6-11.pdf

Comment from Press, page 257:

'Computes Carlson’s elliptic integral of the first kind, RF (x, y, z). x, y, and z must be nonnegative, and at most one can be zero. TINY must be at least 5 times the machine underflow limit, BIG at most one fifth the machine overflow limit.'

A value of 0.08 for the error tolerance parameter is adequate for single precision (7 significant digits). Since the error scales as 6 n, we see that 0.0025 will yield double precision (16 significant digits) and require at most two or three more iterations.'

This is called Carlson-elliptic-1 in scmutils.

carlson-rj clj/s

(carlson-rj x y z p)

Computes Carlson’s elliptic integral of the third kind, RJ(x, y, z, p).

x, y, and z must be nonnegative, and at most one can be zero. p must be nonzero.

If p < 0, the Cauchy principal value is returned. tiny internally must be at least twice the cube root of the machine underflow limit, big at most one fifth the cube root of the machine overflow limit.

elliptic-e clj/s

(elliptic-e k)

(elliptic-e phi k)

Passing k returns the complete elliptic integral of the second kind - see Press, 6.11.20.

The two-arity version returns the Legendre elliptic integral of the second kind E(φ, k). See W.H. Press, Numerical Recipes in C++, 2ed. eq. 6.11.20.

See page 260.

elliptic-f clj/s

(elliptic-f phi k)

Legendre elliptic integral of the first kind F(φ, k). See W.H. Press, Numerical Recipes in C++, 2ed. eq. 6.11.19

See page 260.

elliptic-integrals clj/s

(elliptic-integrals k continue)

Computes the first and second complete elliptic integrals at once, and passes them to the supplied continuation as args K and E.

elliptic-k clj/s

(elliptic-k k)

Complete elliptic integral of the first kind - see Press, 6.11.18.

elliptic-pi clj/s

(elliptic-pi n k)

(elliptic-pi phi n k)

The two-arity call returns the complete elliptic integral of the third kind - see https://en.wikipedia.org/wiki/Carlson_symmetric_form#Complete_elliptic_integrals for reference.

The three-arity call returns the Legendre elliptic integral of the third kind Π(φ, k). See W.H. Press, Numerical Recipes in C++, 2ed. eq. 6.11.21; Note that our sign convention for n is opposite theirs.

See page 260.

jacobi-elliptic-functions clj/s

(jacobi-elliptic-functions u k)

(jacobi-elliptic-functions u k cont)

Direct Clojure translation (via the Scheme translation in scmutils) of W.H. Press, Numerical Recipes, subroutine sncndn.

Calls the supplied continuation cont with sn, cn and dn as defined below.

If no cont is supplied, returns a three-vector of sn, cn and dn.

Comments from Press, page 261:

The Jacobian elliptic function sn is defined as follows: instead of considering the elliptic integral

$$u(y, k) \equiv u=F(\phi, k)$$

Consider the inverse function:

$$y = \sin \phi = \mathrm{sn}(u, k)$$

Equivalently,

$$u=\int_{0}^{\mathrm{sn}} \frac{d y}{\sqrt{\left(1-y^{2}\right)\left(1-k^{2} y^{2}\right)}}$$

When $k = 0$, $sn$ is just $\sin$. The functions $cn$ and $dn$ are defined by the relations

$$\mathrm{sn}^{2}+\mathrm{cn}^{2}=1, \quad k^{2} \mathrm{sn}^{2}+\mathrm{dn}^{2}=1$$

The function calls the continuation with all three functions $sn$, $cn$, and $dn$ since computing all three is no harder than computing any one of them.

k-and-deriv clj/s

(k-and-deriv k)

Returns a pair of:

• the elliptic integral of the first kind, K
• the derivative dK/dk

evaluated at k.