ellipticFunctions
Class Jacobi

java.lang.Object
  ellipticFunctions.Elliptic
      ellipticFunctions.Jacobi

public class Jacobi

extends Elliptic

Implementation for Jacobi's elliptic functions. Attention! This class is not thread save. This is usually no problem, when you do numerics.

Author:

Markus Schmies, Boris Springborn,

Method Summary

static mfc.field.Complex	cn(mfc.field.Complex u, mfc.field.Complex k) Returns cn(u,k).
static void	cn(mfc.field.Complex u, mfc.field.Complex tau, mfc.field.Complex[] thetaConstants, mfc.field.Complex cnOfU) Computes cn(u,k(τ)) for given latiuce paramter τ, and theta constants.
static double	cn(double u, double k) Returns cn(u,k).
static mfc.field.Complex	dn(mfc.field.Complex u, mfc.field.Complex k) Returns dn(u,k).
static void	dn(mfc.field.Complex u, mfc.field.Complex tau, mfc.field.Complex[] thetaConstants, mfc.field.Complex dnOfU) Computes dn(u,k(τ)) for given latiuce paramter τ, and theta constants.
static double	dn(double u, double k) Returns dn(u,k).
static mfc.field.Complex	K_from_k(mfc.field.Complex k) Returns quarter period K as function of the modulus k.
static void	K_from_k(mfc.field.Complex k, mfc.field.Complex K) Computes quarter period K as function of the modulus k.
static mfc.field.Complex	K_from_KPrime_and_tau(mfc.field.Complex KPrime, mfc.field.Complex tau) Returns quarter period K as function of complementary quarter period K' and lattice parameter τ.
static void	K_from_KPrime_and_tau(mfc.field.Complex KPrime, mfc.field.Complex tau, mfc.field.Complex K) Computes quarter period K as function of complementary quarter period K' and lattice parameter τ.
static mfc.field.Complex	k_from_thetaConstants(mfc.field.Complex[] thetaConstants) Returns modulus k as function of theta constants.
static mfc.field.Complex	K_from_thetaConstants(mfc.field.Complex[] thetaConstants) Returns quarter period K as function of theta constants.
static void	k_from_thetaConstants(mfc.field.Complex[] thetaConstants, mfc.field.Complex k) Computes modulus k as function of theta constants
static void	K_from_thetaConstants(mfc.field.Complex[] thetaConstants, mfc.field.Complex K) Computes quarter period K as function of theta constants.
static mfc.field.Complex	KPrime_from_K_and_tau(mfc.field.Complex K, mfc.field.Complex tau) Returns complementary quarter period K' as function of quarter period K and lattice parameter τ.
static void	KPrime_from_K_and_tau(mfc.field.Complex K, mfc.field.Complex tau, mfc.field.Complex KPrime) Computes complementary quarter period K' as function of quarter period K and lattice parameter τ.
static mfc.field.Complex	kPrime_from_k(mfc.field.Complex k) Returns complementary modulus k' as function of the modulus.
static mfc.field.Complex	KPrime_from_k(mfc.field.Complex k) Returns complementary quarter Period K' as function of the modulus k.
static void	kPrime_from_k(mfc.field.Complex k, mfc.field.Complex kPrime) Computes complementary modulus k' as function of the modulus.
static void	KPrime_from_k(mfc.field.Complex k, mfc.field.Complex KPrime) Computes cmplementary quarter period K' as function of the modulus k.
static double	kPrime_from_k(double k) Returns complementary modulus k' as function of the modulus.
static mfc.field.Complex	kPrime_from_thetaConstants(mfc.field.Complex[] thetaConstants) Returns complementary modulus k' as function of theta constants.
static void	kPrime_from_thetaConstants(mfc.field.Complex[] thetaConstants, mfc.field.Complex kPrime) Computes complementary modulus k' as function theta constants.
static mfc.field.Complex	sn(mfc.field.Complex u, mfc.field.Complex k) Returns sn(u,k).
static void	sn(mfc.field.Complex u, mfc.field.Complex tau, mfc.field.Complex[] thetaConstants, mfc.field.Complex snOfU) Computes sn(u,k(τ)) for given latiuce paramter τ, and theta constants.
static double	sn(double u, double k) Returns sn(u,k).
static mfc.field.Complex	tau_from_k(mfc.field.Complex k) Computes lattice parameter τ as function of modulus k.
static void	tau_from_k(mfc.field.Complex k, mfc.field.Complex tau) Computes lattice parameter τ as function of modulus k.
static mfc.field.Complex	tauPrime_from_tau(mfc.field.Complex tau) Returns complementary lattice paramter τ' = -1/τ for lattice parameter τ.
static void	tauPrime_from_tau(mfc.field.Complex tau, mfc.field.Complex tauPrime) Computes complementary lattice paramter τ' = -1/τ for lattice parameter τ.

Methods inherited from class ellipticFunctions.Elliptic

theta, theta, theta0, theta0, theta1, theta1, theta2, theta2, theta3, theta3, theta4, theta4, thetaConstants, thetaConstants

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Method Detail

cn

public static mfc.field.Complex cn(mfc.field.Complex u,
                                   mfc.field.Complex k)

Returns cn(u,k). It computes the lattice paramter τ as function of k, and further the @theta;-constants. Therefore in most cases it is more efficient to hash these values and call cn(Complex,Complex,Complex[],Complex).

Parameters:

u - argument

k - modulus

Returns:

cn(u,k)

See Also:

Elliptic.thetaConstants(Complex), tau_from_k(Complex), cn(Complex,Complex,Complex[],Complex)

cn

public static void cn(mfc.field.Complex u,
                      mfc.field.Complex tau,
                      mfc.field.Complex[] thetaConstants,
                      mfc.field.Complex cnOfU)

Computes cn(u,k(τ)) for given latiuce paramter τ, and theta constants.

Parameters:

u - argument

tau - lattice parameter

thetaConstants - theta constants

cnOfU - sn(u,k(τ)) (on output)

See Also:

Elliptic.thetaConstants(Complex), tau_from_k(Complex), cn(Complex,Complex)

cn

public static double cn(double u,
                        double k)

Returns cn(u,k). It computes the lattice paramter τ as function of k, and further the @theta;-constants. Therefore in most cases it is more efficient to hash these values and call cn(Complex,Complex,Complex[],Complex).

Parameters:

u - argument

k - modulus

Returns:

cn(u,k)

See Also:

Elliptic.thetaConstants(Complex), tau_from_k(Complex), cn(Complex,Complex,Complex[],Complex), cn(Complex,Complex)

dn

public static mfc.field.Complex dn(mfc.field.Complex u,
                                   mfc.field.Complex k)

Returns dn(u,k). It computes the lattice paramter τ as function of k, and further the @theta;-constants. Therefore in most cases it is more efficient to hash these values and call dn(Complex,Complex,Complex[],Complex).

Parameters:

u - argument

k - modulus

Returns:

dn(u,k)

See Also:

Elliptic.thetaConstants(Complex), tau_from_k(Complex), dn(Complex,Complex,Complex[],Complex)

dn

public static void dn(mfc.field.Complex u,
                      mfc.field.Complex tau,
                      mfc.field.Complex[] thetaConstants,
                      mfc.field.Complex dnOfU)

Computes dn(u,k(τ)) for given latiuce paramter τ, and theta constants.

Parameters:

u - argument

tau - lattice parameter

thetaConstants - theta constants

dnOfU - sn(u,k(τ)) (on output)

See Also:

Elliptic.thetaConstants(Complex), tau_from_k(Complex), dn(Complex,Complex)

dn

public static double dn(double u,
                        double k)

Returns dn(u,k). It computes the lattice paramter τ as function of k, and further the @theta;-constants. Therefore in most cases it is more efficient to hash these values and call dn(Complex,Complex,Complex[],Complex).

Parameters:

u - argument

k - modulus

Returns:

dn(u,k)

See Also:

Elliptic.thetaConstants(Complex), tau_from_k(Complex), dn(Complex,Complex,Complex[],Complex), dn(Complex,Complex)

K_from_k

public static mfc.field.Complex K_from_k(mfc.field.Complex k)

Returns quarter period K as function of the modulus k.

Parameters:

k - modulus

Returns:

quarter period #see K_from_k(Complex,Complex)

K_from_k

public static void K_from_k(mfc.field.Complex k,
                            mfc.field.Complex K)

Computes quarter period K as function of the modulus k.

Parameters:

k - modulus

K - quarter period K (on output) #see K_from_k(Complex)

K_from_KPrime_and_tau

public static mfc.field.Complex K_from_KPrime_and_tau(mfc.field.Complex KPrime,
                                                      mfc.field.Complex tau)

Returns quarter period K as function of complementary quarter period K' and lattice parameter τ.

Parameters:

KPrime - complementary quarter period

tau - lattice parameter

Returns:

quarter period (on output)

See Also:

K_from_KPrime_and_tau(Complex,Complex,Complex)

K_from_KPrime_and_tau

public static void K_from_KPrime_and_tau(mfc.field.Complex KPrime,
                                         mfc.field.Complex tau,
                                         mfc.field.Complex K)

Computes quarter period K as function of complementary quarter period K' and lattice parameter τ.

Parameters:

KPrime - complementary quarter period

tau - lattice parameter

K - quarter period (on output)

See Also:

K_from_KPrime_and_tau(Complex,Complex)

k_from_thetaConstants

public static mfc.field.Complex k_from_thetaConstants(mfc.field.Complex[] thetaConstants)

Returns modulus k as function of theta constants.

Parameters:

thetaConstants - theta constants

Returns:

modulus

See Also:

Elliptic.thetaConstants(Complex), k_from_thetaConstants(Complex[],Complex)

K_from_thetaConstants

public static mfc.field.Complex K_from_thetaConstants(mfc.field.Complex[] thetaConstants)

Returns quarter period K as function of theta constants.

Parameters:

thetaConstants - theta constants

Returns:

quarter period

See Also:

Elliptic.thetaConstants(Complex), K_from_thetaConstants(Complex[],Complex)

k_from_thetaConstants

public static void k_from_thetaConstants(mfc.field.Complex[] thetaConstants,
                                         mfc.field.Complex k)

Computes modulus k as function of theta constants

Parameters:

thetaConstants - theta constants

k - modulus (on output)

See Also:

Elliptic.thetaConstants(Complex), k_from_thetaConstants(Complex[])

K_from_thetaConstants

public static void K_from_thetaConstants(mfc.field.Complex[] thetaConstants,
                                         mfc.field.Complex K)

Computes quarter period K as function of theta constants.

Parameters:

thetaConstants - theta constants

K - quarter period (on output)

See Also:

Elliptic.thetaConstants(Complex), K_from_thetaConstants(Complex[])

KPrime_from_K_and_tau

public static mfc.field.Complex KPrime_from_K_and_tau(mfc.field.Complex K,
                                                      mfc.field.Complex tau)

Returns complementary quarter period K' as function of quarter period K and lattice parameter τ.

Parameters:

K - quarter period

tau - lattice parameter

Returns:

complementary quarter period

See Also:

KPrime_from_K_and_tau(Complex,Complex,Complex)

KPrime_from_K_and_tau

public static void KPrime_from_K_and_tau(mfc.field.Complex K,
                                         mfc.field.Complex tau,
                                         mfc.field.Complex KPrime)

Computes complementary quarter period K' as function of quarter period K and lattice parameter τ.

Parameters:

K - quarter period

tau - lattice parameter

KPrime - complementary quarter period (on output)

See Also:

KPrime_from_K_and_tau(Complex,Complex)

kPrime_from_k

public static mfc.field.Complex kPrime_from_k(mfc.field.Complex k)

Returns complementary modulus k' as function of the modulus.

Parameters:

k - modulus

Returns:

complementary modulus k'= sqrt( 1-k*k )

See Also:

kPrime_from_k(Complex,Complex)

KPrime_from_k

public static mfc.field.Complex KPrime_from_k(mfc.field.Complex k)

Returns complementary quarter Period K' as function of the modulus k.

Parameters:

k - modulus

Returns:

complementary quarter Period K'

See Also:

KPrime_from_k(Complex,Complex)

kPrime_from_k

public static void kPrime_from_k(mfc.field.Complex k,
                                 mfc.field.Complex kPrime)

Computes complementary modulus k' as function of the modulus.

Parameters:

k - modulus

kPrime - complementary modulus k'= sqrt( 1-k*k ) (on output)

See Also:

kPrime_from_k(Complex)

KPrime_from_k

public static void KPrime_from_k(mfc.field.Complex k,
                                 mfc.field.Complex KPrime)

Computes cmplementary quarter period K' as function of the modulus k.

Parameters:

k - modulus

KPrime - complementary quarter period K' (on output)

See Also:

KPrime_from_k(Complex)

kPrime_from_k

public static double kPrime_from_k(double k)

Returns complementary modulus k' as function of the modulus.

Parameters:

k - modulus

Returns:

complementary modulus k'= sqrt( 1-k*k )

See Also:

kPrime_from_k(Complex,Complex)

kPrime_from_thetaConstants

public static mfc.field.Complex kPrime_from_thetaConstants(mfc.field.Complex[] thetaConstants)

Returns complementary modulus k' as function of theta constants.

Parameters:

thetaConstants - theta constants

Returns:

complementary modulus

See Also:

Elliptic.thetaConstants(Complex), kPrime_from_thetaConstants(Complex[],Complex)

kPrime_from_thetaConstants

public static void kPrime_from_thetaConstants(mfc.field.Complex[] thetaConstants,
                                              mfc.field.Complex kPrime)

Computes complementary modulus k' as function theta constants.

Parameters:

thetaConstants - theta constants

kPrime - complementary modulus k' (on output)

See Also:

Elliptic.thetaConstants(Complex), kPrime_from_thetaConstants(Complex[])

sn

public static mfc.field.Complex sn(mfc.field.Complex u,
                                   mfc.field.Complex k)

Returns sn(u,k). It computes the lattice paramter τ as function of k, and further the @theta;-constants. Therefore in most cases it is more efficient to hash these values and call sn(Complex,Complex,Complex[],Complex).

Parameters:

u - argument

k - modulus

Returns:

sn(u,k)

See Also:

Elliptic.thetaConstants(Complex), tau_from_k(Complex), sn(Complex,Complex,Complex[],Complex), sn(double,double)

sn

public static void sn(mfc.field.Complex u,
                      mfc.field.Complex tau,
                      mfc.field.Complex[] thetaConstants,
                      mfc.field.Complex snOfU)

Computes sn(u,k(τ)) for given latiuce paramter τ, and theta constants.

Parameters:

u - argument

tau - lattice parameter

thetaConstants - theta constants

snOfU - sn(u,k(τ)) (on output)

See Also:

Elliptic.thetaConstants(Complex), tau_from_k(Complex), sn(Complex,Complex), sn(double,double)

sn

public static double sn(double u,
                        double k)

Returns sn(u,k). It computes the lattice paramter τ as function of k, and further the @theta;-constants. Therefore in most cases it is more efficient to hash these values and call sn(Complex,Complex,Complex[],Complex).

Parameters:

u - argument

k - modulus

Returns:

sn(u,k)

See Also:

Elliptic.thetaConstants(Complex), tau_from_k(Complex), sn(Complex,Complex,Complex[],Complex), sn(Complex,Complex)

tau_from_k

public static mfc.field.Complex tau_from_k(mfc.field.Complex k)

Computes lattice parameter τ as function of modulus k.

Parameters:

k - modulus

Returns:

lattice paramter

See Also:

tau_from_k(Complex,Complex)

tau_from_k

public static void tau_from_k(mfc.field.Complex k,
                              mfc.field.Complex tau)

Computes lattice parameter τ as function of modulus k.

Parameters:

k - modulus

tau - lattice paramter τ

See Also:

tau_from_k(Complex)

tauPrime_from_tau

public static mfc.field.Complex tauPrime_from_tau(mfc.field.Complex tau)

Returns complementary lattice paramter τ' = -1/τ for lattice parameter τ.

Parameters:

tau - lattice paramter τ

Returns:

complementary lattice paramter τ'

See Also:

tauPrime_from_tau(Complex,Complex)

tauPrime_from_tau

public static void tauPrime_from_tau(mfc.field.Complex tau,
                                     mfc.field.Complex tauPrime)

Computes complementary lattice paramter τ' = -1/τ for lattice parameter τ. tau and tauPrime may coinside.

Parameters:

tau - lattice paramter τ

tauPrime - complementary lattice paramter τ'(on output)

See Also:

tauPrime_from_tau(Complex)