public class CarlsonEllipticIntegral extends Object
This utility class computes the various symmetric elliptic integrals defined as: \[ \left\{\begin{align} R_F(x,y,z) &= \frac{1}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{s(t)}\\ R_J(x,y,z,p) &= \frac{3}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{s(t)(t+p)}\\ R_G(x,y,z) &= \frac{1}{4}\int_{0}^{\infty}\frac{1}{s(t)} \left(\frac{x}{t+x}+\frac{y}{t+y}+\frac{z}{t+z}\right)t\mathrm{d}t\\ R_D(x,y,z) &= R_J(x,y,z,z)\\ R_C(x,y) &= R_F(x,y,y) \end{align}\right. \]
where \[ s(t) = \sqrt{t+x}\sqrt{t+y}\sqrt{t+z} \]
The algorithms used are based on the duplication method as described in B. C. Carlson 1995 paper "Numerical computation of real or complex elliptic integrals", with the improvements described in the appendix of B. C. Carlson and James FitzSimons 2000 paper "Reduction theorems for elliptic integrands with the square root of two quadratic factors". They are also described in section 19.36(i) of Digital Library of Mathematical Functions.
| Modifier and Type | Method and Description |
|---|---|
static Complex |
rC(Complex x,
Complex y)
Compute Carlson elliptic integral RC.
|
static double |
rC(double x,
double y)
Compute Carlson elliptic integral RC.
|
static <T extends CalculusFieldElement<T>> |
rC(FieldComplex<T> x,
FieldComplex<T> y)
Compute Carlson elliptic integral RC.
|
static <T extends CalculusFieldElement<T>> |
rC(T x,
T y)
Compute Carlson elliptic integral RC.
|
static Complex |
rD(Complex x,
Complex y,
Complex z)
Compute Carlson elliptic integral RD.
|
static double |
rD(double x,
double y,
double z)
Compute Carlson elliptic integral RD.
|
static <T extends CalculusFieldElement<T>> |
rD(FieldComplex<T> x,
FieldComplex<T> y,
FieldComplex<T> z)
Compute Carlson elliptic integral RD.
|
static <T extends CalculusFieldElement<T>> |
rD(T x,
T y,
T z)
Compute Carlson elliptic integral RD.
|
static Complex |
rF(Complex x,
Complex y,
Complex z)
Compute Carlson elliptic integral RF.
|
static double |
rF(double x,
double y,
double z)
Compute Carlson elliptic integral RF.
|
static <T extends CalculusFieldElement<T>> |
rF(FieldComplex<T> x,
FieldComplex<T> y,
FieldComplex<T> z)
Compute Carlson elliptic integral RF.
|
static <T extends CalculusFieldElement<T>> |
rF(T x,
T y,
T z)
Compute Carlson elliptic integral RF.
|
static Complex |
rG(Complex x,
Complex y,
Complex z)
Compute Carlson elliptic integral RG.
|
static double |
rG(double x,
double y,
double z)
Compute Carlson elliptic integral RG.
|
static <T extends CalculusFieldElement<T>> |
rG(FieldComplex<T> x,
FieldComplex<T> y,
FieldComplex<T> z)
Compute Carlson elliptic integral RG.
|
static <T extends CalculusFieldElement<T>> |
rG(T x,
T y,
T z)
Compute Carlson elliptic integral RG.
|
static Complex |
rJ(Complex x,
Complex y,
Complex z,
Complex p)
Compute Carlson elliptic integral RJ.
|
static Complex |
rJ(Complex x,
Complex y,
Complex z,
Complex p,
Complex delta)
Compute Carlson elliptic integral RJ.
|
static double |
rJ(double x,
double y,
double z,
double p)
Compute Carlson elliptic integral RJ.
|
static double |
rJ(double x,
double y,
double z,
double p,
double delta)
Compute Carlson elliptic integral RJ.
|
static <T extends CalculusFieldElement<T>> |
rJ(FieldComplex<T> x,
FieldComplex<T> y,
FieldComplex<T> z,
FieldComplex<T> p)
Compute Carlson elliptic integral RJ.
|
static <T extends CalculusFieldElement<T>> |
rJ(FieldComplex<T> x,
FieldComplex<T> y,
FieldComplex<T> z,
FieldComplex<T> p,
FieldComplex<T> delta)
Compute Carlson elliptic integral RJ.
|
static <T extends CalculusFieldElement<T>> |
rJ(T x,
T y,
T z,
T p)
Compute Carlson elliptic integral RJ.
|
static <T extends CalculusFieldElement<T>> |
rJ(T x,
T y,
T z,
T p,
T delta)
Compute Carlson elliptic integral RJ.
|
public static double rC(double x,
double y)
The Carlson elliptic integral RCis defined as \[ R_C(x,y,z)=R_F(x,y,y)=\frac{1}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{\sqrt{t+x}(t+y)} \]
x - first symmetric variable of the integraly - second symmetric variable of the integralpublic static <T extends CalculusFieldElement<T>> T rC(T x, T y)
The Carlson elliptic integral RCis defined as \[ R_C(x,y,z)=R_F(x,y,y)=\frac{1}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{\sqrt{t+x}(t+y)} \]
T - type of the field elementsx - first symmetric variable of the integraly - second symmetric variable of the integralpublic static Complex rC(Complex x, Complex y)
The Carlson elliptic integral RCis defined as \[ R_C(x,y,z)=R_F(x,y,y)=\frac{1}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{\sqrt{t+x}(t+y)} \]
x - first symmetric variable of the integraly - second symmetric variable of the integralpublic static <T extends CalculusFieldElement<T>> FieldComplex<T> rC(FieldComplex<T> x, FieldComplex<T> y)
The Carlson elliptic integral RCis defined as \[ R_C(x,y,z)=R_F(x,y,y)=\frac{1}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{\sqrt{t+x}(t+y)} \]
T - type of the field elementsx - first symmetric variable of the integraly - second symmetric variable of the integralpublic static double rF(double x,
double y,
double z)
The Carlson elliptic integral RF is defined as \[ R_F(x,y,z)=\frac{1}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{\sqrt{t+x}\sqrt{t+y}\sqrt{t+z}} \]
x - first symmetric variable of the integraly - second symmetric variable of the integralz - third symmetric variable of the integralpublic static <T extends CalculusFieldElement<T>> T rF(T x, T y, T z)
The Carlson elliptic integral RF is defined as \[ R_F(x,y,z)=\frac{1}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{\sqrt{t+x}\sqrt{t+y}\sqrt{t+z}} \]
T - type of the field elementsx - first symmetric variable of the integraly - second symmetric variable of the integralz - third symmetric variable of the integralpublic static Complex rF(Complex x, Complex y, Complex z)
The Carlson elliptic integral RF is defined as \[ R_F(x,y,z)=\frac{1}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{\sqrt{t+x}\sqrt{t+y}\sqrt{t+z}} \]
x - first symmetric variable of the integraly - second symmetric variable of the integralz - third symmetric variable of the integralpublic static <T extends CalculusFieldElement<T>> FieldComplex<T> rF(FieldComplex<T> x, FieldComplex<T> y, FieldComplex<T> z)
The Carlson elliptic integral RF is defined as \[ R_F(x,y,z)=\frac{1}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{\sqrt{t+x}\sqrt{t+y}\sqrt{t+z}} \]
T - type of the field elementsx - first symmetric variable of the integraly - second symmetric variable of the integralz - third symmetric variable of the integralpublic static double rJ(double x,
double y,
double z,
double p)
The Carlson elliptic integral RJ is defined as \[ R_J(x,y,z,p)=\frac{3}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{\sqrt{t+x}\sqrt{t+y}\sqrt{t+z}(t+p)} \]
x - first symmetric variable of the integraly - second symmetric variable of the integralz - third symmetric variable of the integralp - fourth not symmetric variable of the integralpublic static double rJ(double x,
double y,
double z,
double p,
double delta)
The Carlson elliptic integral RJ is defined as \[ R_J(x,y,z,p)=\frac{3}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{\sqrt{t+x}\sqrt{t+y}\sqrt{t+z}(t+p)} \]
x - first symmetric variable of the integraly - second symmetric variable of the integralz - third symmetric variable of the integralp - fourth not symmetric variable of the integraldelta - precomputed value of (p-x)(p-y)(p-z)public static <T extends CalculusFieldElement<T>> T rJ(T x, T y, T z, T p)
The Carlson elliptic integral RJ is defined as \[ R_J(x,y,z,p)=\frac{3}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{\sqrt{t+x}\sqrt{t+y}\sqrt{t+z}(t+p)} \]
T - type of the field elementsx - first symmetric variable of the integraly - second symmetric variable of the integralz - third symmetric variable of the integralp - fourth not symmetric variable of the integralpublic static <T extends CalculusFieldElement<T>> T rJ(T x, T y, T z, T p, T delta)
The Carlson elliptic integral RJ is defined as \[ R_J(x,y,z,p)=\frac{3}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{\sqrt{t+x}\sqrt{t+y}\sqrt{t+z}(t+p)} \]
T - type of the field elementsx - first symmetric variable of the integraly - second symmetric variable of the integralz - third symmetric variable of the integralp - fourth not symmetric variable of the integraldelta - precomputed value of (p-x)(p-y)(p-z)public static Complex rJ(Complex x, Complex y, Complex z, Complex p)
The Carlson elliptic integral RJ is defined as \[ R_J(x,y,z,p)=\frac{3}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{\sqrt{t+x}\sqrt{t+y}\sqrt{t+z}(t+p)} \]
x - first symmetric variable of the integraly - second symmetric variable of the integralz - third symmetric variable of the integralp - fourth not symmetric variable of the integralpublic static Complex rJ(Complex x, Complex y, Complex z, Complex p, Complex delta)
The Carlson elliptic integral RJ is defined as \[ R_J(x,y,z,p)=\frac{3}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{\sqrt{t+x}\sqrt{t+y}\sqrt{t+z}(t+p)} \]
x - first symmetric variable of the integraly - second symmetric variable of the integralz - third symmetric variable of the integralp - fourth not symmetric variable of the integraldelta - precomputed value of (p-x)(p-y)(p-z)public static <T extends CalculusFieldElement<T>> FieldComplex<T> rJ(FieldComplex<T> x, FieldComplex<T> y, FieldComplex<T> z, FieldComplex<T> p)
The Carlson elliptic integral RJ is defined as \[ R_J(x,y,z,p)=\frac{3}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{\sqrt{t+x}\sqrt{t+y}\sqrt{t+z}(t+p)} \]
T - type of the field elementsx - first symmetric variable of the integraly - second symmetric variable of the integralz - third symmetric variable of the integralp - fourth not symmetric variable of the integralpublic static <T extends CalculusFieldElement<T>> FieldComplex<T> rJ(FieldComplex<T> x, FieldComplex<T> y, FieldComplex<T> z, FieldComplex<T> p, FieldComplex<T> delta)
The Carlson elliptic integral RJ is defined as \[ R_J(x,y,z,p)=\frac{3}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{\sqrt{t+x}\sqrt{t+y}\sqrt{t+z}(t+p)} \]
T - type of the field elementsx - first symmetric variable of the integraly - second symmetric variable of the integralz - third symmetric variable of the integralp - fourth not symmetric variable of the integraldelta - precomputed value of (p-x)(p-y)(p-z)public static double rD(double x,
double y,
double z)
The Carlson elliptic integral RD is defined as \[ R_D(x,y,z)=\frac{3}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{\sqrt{t+x}\sqrt{t+y}\sqrt{t+z}(t+z)} \]
x - first symmetric variable of the integraly - second symmetric variable of the integralz - third symmetric variable of the integralpublic static <T extends CalculusFieldElement<T>> T rD(T x, T y, T z)
The Carlson elliptic integral RD is defined as \[ R_D(x,y,z)=\frac{3}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{\sqrt{t+x}\sqrt{t+y}\sqrt{t+z}(t+z)} \]
T - type of the field elementsx - first symmetric variable of the integraly - second symmetric variable of the integralz - third symmetric variable of the integralpublic static Complex rD(Complex x, Complex y, Complex z)
The Carlson elliptic integral RD is defined as \[ R_D(x,y,z)=\frac{3}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{\sqrt{t+x}\sqrt{t+y}\sqrt{t+z}(t+z)} \]
x - first symmetric variable of the integraly - second symmetric variable of the integralz - third symmetric variable of the integralpublic static <T extends CalculusFieldElement<T>> FieldComplex<T> rD(FieldComplex<T> x, FieldComplex<T> y, FieldComplex<T> z)
The Carlson elliptic integral RD is defined as \[ R_D(x,y,z)=\frac{3}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{\sqrt{t+x}\sqrt{t+y}\sqrt{t+z}(t+z)} \]
T - type of the field elementsx - first symmetric variable of the integraly - second symmetric variable of the integralz - third symmetric variable of the integralpublic static double rG(double x,
double y,
double z)
The Carlson elliptic integral RGis defined as \[ R_{G}(x,y,z)=\frac{1}{4}\int_{0}^{\infty}\frac{1}{s(t)} \left(\frac{x}{t+x}+\frac{y}{t+y}+\frac{z}{t+z}\right)t\mathrm{d}t \]
x - first symmetric variable of the integraly - second symmetric variable of the integralz - second symmetric variable of the integralpublic static <T extends CalculusFieldElement<T>> T rG(T x, T y, T z)
The Carlson elliptic integral RGis defined as \[ R_{G}(x,y,z)=\frac{1}{4}\int_{0}^{\infty}\frac{1}{s(t)} \left(\frac{x}{t+x}+\frac{y}{t+y}+\frac{z}{t+z}\right)t\mathrm{d}t \]
T - type of the field elementsx - first symmetric variable of the integraly - second symmetric variable of the integralz - second symmetric variable of the integralpublic static Complex rG(Complex x, Complex y, Complex z)
The Carlson elliptic integral RGis defined as \[ R_{G}(x,y,z)=\frac{1}{4}\int_{0}^{\infty}\frac{1}{s(t)} \left(\frac{x}{t+x}+\frac{y}{t+y}+\frac{z}{t+z}\right)t\mathrm{d}t \]
x - first symmetric variable of the integraly - second symmetric variable of the integralz - second symmetric variable of the integralpublic static <T extends CalculusFieldElement<T>> FieldComplex<T> rG(FieldComplex<T> x, FieldComplex<T> y, FieldComplex<T> z)
The Carlson elliptic integral RGis defined as \[ R_{G}(x,y,z)=\frac{1}{4}\int_{0}^{\infty}\frac{1}{s(t)} \left(\frac{x}{t+x}+\frac{y}{t+y}+\frac{z}{t+z}\right)t\mathrm{d}t \]
T - type of the field elementsx - first symmetric variable of the integraly - second symmetric variable of the integralz - second symmetric variable of the integralCopyright © 2016-2021 CS GROUP. All rights reserved.