Package rocks.palaiologos.maja

Class Maja

java.lang.Object

rocks.palaiologos.maja.Maja

public class Maja
extends Object

A slick numerics-oriented Mathematical library for Java.

Author:

Palaiologos

Field Summary

Fields

Modifier and Type	Field	Description
static final double	APERY_CONSTANT	Apery's constant - zeta(3).
static final double	ATTO	SI prefix for 10^-18.
static final double	CATALAN_CONSTANT	Catalan's constant.
static final double	CENTI	SI prefix for 10^-2.
static final double	DECA	SI prefix for 10^1.
static final double	DECI	SI prefix for 10^-1.
static final double	E	The value of e.
static final double	EPSILON	The epsilon value for double precision numbers.
static final double	EULER_GAMMA	The Euler-Mascheroni constant.
static final double	EXA	SI prefix for 10^18.
static final double	FEIGENBAUM_CONSTANT	Feigenbaum constant.
static final double	FEMTO	SI prefix for 10^-15.
static final double	GIGA	SI prefix for 10^9.
static final double	GLAISHER_CONSTANT	Glaisher-Kinkelin constant.
static final double	GOLDEN_RATIO	The golden ratio.
static final double	GOLOMB_DICKMAN_CONSTANT	Golomb-Dickman constant.
static final double	HECTO	SI prefix for 10^2.
static final Complex	I	The imaginary unit.
static final double	KHINCHIN_CONSTANT	Khinchin's constant.
static final double	KILO	SI prefix for 10^3.
static final double	LN10	The value of ln(10).
static final double	LN2	The value of ln(2).
static final double	LOG2E	The value of log2(e).
static final double	MEGA	SI prefix for 10^6.
static final double	MICRO	SI prefix for 10^-6.
static final double	MILLI	SI prefix for 10^-3.
static final double	MILLS_CONSTANT	Mills' constant.
static final double	NANO	SI prefix for 10^-9.
static final double	ONE_OVER_E	The value of 1/e.
static final double	ONE_OVER_PI	The value of 1/pi.
static final double	PETA	SI prefix for 10^15.
static final double	PI	The value of pi.
static final double	PI_2	The value of pi/2.
static final double	PI_4	The value of pi/4.
static final double	PICO	SI prefix for 10^-12.
static final double	TERA	SI prefix for 10^12.
static final double	TWO_PI	The value of 2*PI.
static final double	YOCTO	SI prefix for 10^-24.
static final double	YOTTA	SI prefix for 10^24.
static final double	ZEPTO	SI prefix for 10^-21.
static final double	ZETTA	SI prefix for 10^21.

Method Summary

Modifier and Type	Method	Description
static boolean[]	aberth(Complex[] coefficients)	Numerically finds all roots of a polynomial equation P(z) = 0.
static double	abs(double x)	Returns the absolute value of a double precision number.
static long	abs(long x)	Returns the absolute value of a number.
static double	abs(Complex a)	Compute the absolute value of a complex number.
static Dual	abs(Dual a)
static Complex	absparts(Complex x)	Compute the absolute values of parts of a complex number.
static double	acos(double x)	Returns the value of the inverse cosine of a double precision number.
static Complex	acos(Complex a)	Compute the arcus cosine of a complex number.
static Dual	acos(Dual a)
static double	acosh(double x)	Compute the value of the inverse hyperbolic cosine function at x.
static Complex	acosh(Complex a)	Compute the hyperbolic arccosine of a complex number.
static Dual	acosh(Dual a)
static double	acot(double x)	Compute the value of the inverse cotangent (1 / atan(x)) function at x.
static Complex	acot(Complex a)	Compute the arcus cotangent of a complex number.
static Dual	acot(Dual a)
static double	acoth(double x)	Compute the value of the inverse hyperbolic cotangent (1 / atanh(x)) function at x.
static Complex	acoth(Complex a)	Compute the hyperbolic arccotangent of a complex number.
static Dual	acoth(Dual a)
static double	acsc(double x)	Compute the value of the inverse cosecant (1 / asin(x)) function at x.
static Complex	acsc(Complex a)	Compute the arcus cosecant of a complex number.
static Dual	acsc(Dual a)
static double	acsch(double x)	Compute the value of the inverse hyperbolic cosecant (1 / asinh(x)) function at x.
static Complex	acsch(Complex a)	Compute the hyperbolic arccosecant of a complex number.
static Dual	acsch(Dual a)
static double	add(double x, double y)	Adds two double precision numbers together.
static Complex	add(double a, Complex b)	Add a complex number and a real number.
static Dual	add(double a, Dual b)	Perform real-dual addition.
static Complex	add(Complex a, double b)	Add a complex number and a real number.
static Complex	add(Complex a, Complex b)	Add two complex numbers together.
static Dual	add(Dual a, double b)	Perform real-dual addition.
static Dual	add(Dual a, Dual b)	Perform dual addition.
static <T> T	add(AdditiveGroup<T> group, T x, T y)	Subtracts two values through an additive group.
static <T> T	add(AdditiveGroupoid<T> groupoid, T x, T y)	Adds two values through an additive groupoid.
static double	agm(double a, double g)	Computes the Arithmetic-Geometric mean.
static double[]	airy(double x)	Compute the value of the Airy Ai, Ai', Bi and Bi' functions at the specified point.
static Complex[]	airy(Complex x)	Compute the value of the Airy Ai, Ai', Bi and Bi' functions at the specified point.
static double	airyAi(double x)	Compute the value of the Airy Ai function at the specified point.
static Complex	airyAi(Complex x)	Compute the value of the Airy Ai function at the specified point.
static Dual	airyAi(Dual a)
static double	airyAip(double x)	Compute the value of the Airy Ai function's first derivative at the specified point.
static Complex	airyAip(Complex x)	Compute the value of the Airy Ai function's first derivative at the specified point.
static double	airyBi(double x)	Compute the value of the Airy Bi function at the specified point.
static Complex	airyBi(Complex x)	Compute the value of the Airy Bi function at the specified point.
static Dual	airyBi(Dual a)
static double	airyBip(double x)	Compute the value of the Airy Bi function's first derivative at the specified point.
static Complex	airyBip(Complex x)	Compute the value of the Airy Bi function's first derivative at the specified point.
static double	angerJ(double v, double z)
static Complex	angerJ(Complex v, Complex z)
static double	arcLength(Function<Double,Double> df, double a, double b)	Compute the arc length of a curve defined by the function f(x) and its derivative df(x) on the interval [a, b].
static double	arg(Complex z)	Return the argument of a complex number.
static double	asec(double x)	Compute the value of the inverse secant (1 / acos(x)) function at x.
static Complex	asec(Complex a)	Compute the arcus secant of a complex number.
static Dual	asec(Dual a)
static double	asech(double x)	Compute the value of the inverse hyperbolic secant (1 / acosh(x)) function at x.
static Complex	asech(Complex a)	Compute the hyperbolic arcsecant of a complex number.
static Dual	asech(Dual a)
static double	asin(double x)	Returns the value of the inverse sine of a double precision number.
static Complex	asin(Complex a)	Compute the arcus sine of a complex number.
static Dual	asin(Dual a)
static double	asinh(double a)	Compute the value of the inverse hyperbolic sine function at x.
static Complex	asinh(Complex a)	Compute the hyperbolic arcsine of a complex number.
static Dual	asinh(Dual a)
static double	atan(double x)	Returns the value of the inverse tangent of a double precision number.
static Complex	atan(Complex a)	Compute the arcus tangent of a complex number.
static Dual	atan(Dual a)
static double	atan2(double y, double x)	Returns the value of the inverse tangent of two double precision numbers.
static double	atanh(double a)	Compute the value of the inverse hyperbolic tangent function at x.
static Complex	atanh(Complex a)	Compute the hyperbolic arctangent of a complex number.
static Dual	atanh(Dual a)
static int	bernoulliBound(int n)	Returns an integer b such that |B_n| ≤ 2^b for all Bernoulli numbers B_n.
static double	besselI0(double x)	Compute the bessel I0 function at x.
static Dual	besselI0(Dual a)
static double	besselI1(double x)	Compute the bessel I1 function at x.
static double	besselJ0(double x)	Compute the bessel J0 function at x.
static Dual	besselJ0(Dual a)
static double	besselJ1(double x)	Compute the bessel J1 function at x.
static Dual	besselJ1(Dual a)
static double	besselJn(int n, double x)	Compute the bessel Jn function at x.
static Dual	besselJn(int n, Dual a)
static double	besselJv(double n, double x)	Compute fractional order bessel function of n and x.
static Dual	besselJv(double n, Dual a)
static double	besselK0(double x)	Compute the bessel K0 function at x.
static Dual	besselK0(Dual a)
static double	besselK1(double x)	Compute the bessel K1 function at x.
static Dual	besselK1(Dual a)
static double	besselKn(int n, double x)	Compute the bessel Kn function at x.
static Dual	besselKn(int n, Dual a)
static double	besselY0(double x)	Compute the bessel Y0 function at x.
static Dual	besselY0(Dual a)
static double	besselY1(double x)	Compute the bessel Y1 function at x.
static double	besselYn(int n, double x)	Compute the bessel Yn function at x.
static Dual	besselYn(int n, Dual a)
static double	besselYv(double v, double x)	Compute the fractional order bessel Y function of v and x.
static Dual	besselYv(double n, Dual a)
static double	beta(double x, double y)	Compute the beta function of two values.
static Dual	beta(double a, Dual c)	Compute the beta function of a real and dual number.
static Complex	beta(Complex a, Complex b)	Compute the beta function of two complex numbers.
static Dual	beta(Dual a, double c)	Compute the beta function of a dual and real number.
static long	binomial(int n, int k)	Compute the binomial coefficient "n choose k".
static double	cbrt(double x)	Returns the value of the cube root of a double precision number.
static Complex	cbrt(Complex x)	Compute the principal cube root of a complex number.
static Dual	cbrt(Dual a)
static double	ceil(double x)	Returns the value of the smallest integer greater than or equal to a double precision number.
static Complex	ceil(Complex x)	Rounds both the real and imaginary parts of a complex number.
static double	Chi(double x)	Compute the value of the hyperbolic cosine integral function at x.
static Complex	Chi(Complex z)	Compute the value of the complex hyperbolic cosine integral Chi(z).
static Dual	Chi(Dual a)
static double	chiSquaredQuantile(double p, double df)	Computes quantiles for chi-squared probability distribution at probability p .
static Complex	chop(Complex z)	Trim insignificant real/imaginary parts (below machine epsilon), round up numbers where the real/imaginary part is very close to an integer.
static double	Ci(double x)	Compute the value of the cosine integral function at x.
static Complex	Ci(Complex z)	Compute the value of the complex cosine integral Ci(z) at z.
static Dual	Ci(Dual a)
static double	Cin(double x)	Compute the value of the cosine integral function at x.
static Complex	Cin(Complex z)	Compute the value of the complex cosine integral Cin(z) at z.
static Complex	cis(double x)	Compute the value of the cis function of a number.
static double	clamp(double value, double min, double max)	Clamp a value in the range [min, max].
static long	clamp(long value, long min, long max)	Clamp a value in the range [min, max].
static int	compare(double x, double y)	Compare two double precision numbers.
static int	compare(long x, long y)	Compare two integers.
static Complex	conj(Complex a)	Compute the complex conjugate of a complex number.
static double	copySign(double x, double y)	Copy the sign of the second argument to the first argument.
static long	copySign(long x, long y)	Copy the sign of the second argument to the first argument.
static Complex	copySign(Complex x, Complex y)	Copy the sign of the second argument to the first argument.
static double	cos(double x)	Returns the value of the smallest integer greater than or equal to a double precision number.
static Complex	cos(Complex x)	Compute the cosine of a complex number.
static Dual	cos(Dual a)
static double	cosh(double x)	Returns the hyperbolic cosine of a double precision number.
static Complex	cosh(Complex a)	Compute the hyperbolic cosine of a complex number.
static Dual	cosh(Dual a)
static double	cot(double x)	Compute the value of the cotangent (1 / tan(x)) function at x.
static Complex	cot(Complex x)	Compute the cotangent of a complex number.
static Dual	cot(Dual a)
static double	coth(double x)	Compute the value of the hyperbolic cotangent (1 / tanh(x)) function at x.
static Complex	coth(Complex a)	Compute the hyperbolic cotangent of a complex number.
static Dual	coth(Dual a)
static double	csc(double x)	Compute the value of the cosecant (1 / sin(x)) function at x.
static Complex	csc(Complex x)	Compute the cosecant of a complex number.
static Dual	csc(Dual a)
static double	csch(double x)	Compute the value of the hyperbolic cosecant (1 / sinh(x)) function at x.
static Complex	csch(Complex a)	Compute the hyperbolic cosecant of a complex number.
static Dual	csch(Dual a)
static double	dawsonMinus(double x)	Compute the value of the Dawson function (D-) at x.
static Complex	dawsonMinus(Complex z)	Compute the value of the complex Dawson function (D-) at z.
static double	dawsonPlus(double x)	Compute the value of the Dawson function (D+) at x.
static Complex	dawsonPlus(Complex z)	Compute the value of the complex Dawson function (D+) at z.
static double	digamma(double x)	Compute the digamma function of x.
static Complex	digamma(Complex z)	Compute the complex digamma function.
static Dual	digamma(Dual a)
static double	dilog(double n)	Compute the dilogarithm (the value of the Spence function at 1-x) of x.
static Complex	dilog(Complex z)	Compute the complex dilogarithm of z.
static Dual	dilog(Dual a)
static double	div(double x, double y)	Divides two double precision numbers.
static Complex	div(double a, Complex b)	Divide a complex number and a real number.
static Complex	div(Complex a, double b)	Divide a complex number and a real number.
static Complex	div(Complex a, Complex b)	Divide two complex numbers.
static Dual	div(Dual a, Dual b)
static Complex	e1(Complex x)	Compute the value of the exponential integral E1 at the specified point.
static double	Ei(double x)	Compute the value of the exponential integral at x.
static Complex	Ei(Complex x)	Compute the exponential integral using a formula that was revealed to me in a dream:
static Dual	Ei(Dual a)
static Complex	Ein(Complex x)	Compute the value of the complementary exponential integral Ein at the specified point.
static Complex	en(Complex n, Complex z)	Compute the value of the complex generalised exponential integral E_n(z).
static boolean	eq(double x, double y)	Return true if x == y.
static boolean	eq(double x, double y, double tol)	Return true if x is approximately equal to y.
static boolean	eq(double a, Complex b)	Compare two numbers for equality.
static boolean	eq(Complex a, double b)	Compare two numbers for equality.
static boolean	eq(Complex a, Complex b)	Compare two numbers for equality.
static boolean	eq(Complex a, Complex b, double tol)	Compare two complex numbers for equality.
static double	erf(double x)	Compute the value of the error function at x.
static Complex	erf(Complex z)	Compute the complex error function.
static Dual	erf(Dual a)
static double	erfc(double x)	Compute the value of the complementary error function at x.
static Complex	erfc(Complex z)	Compute the complex complementary error function.
static Dual	erfc(Dual a)
static double	erfi(double x)	Compute the value of the imaginary error function at x.
static Complex	erfi(Complex z)	Compute the complex imaginary error function.
static Dual	erfi(Dual a)
static double	exp(double x)	Returns the value of the exponential function of a double precision number.
static Complex	exp(Complex x)	Compute the value of the exponential function of a complex number.
static Dual	exp(Dual a)
static double	expm1(double x)	Returns the value of the exponential function of a double precision number minus one.
static double	fac2(double a)
static double	factorial(long n)	Return the factorial of n as a double-precision.
static float	fastCos(float x)	Return a cached value of cos(x) using Raven's method.
static float	fastSin(float x)	Return a cached value of sin(x) using Raven's method.
static long	fib(int a)	Compute the n-th Fibonacci number using Binet's formula.
static double	floor(double x)	Returns the value of the largest integer less than or equal to a double precision number.
static Complex	floor(Complex x)	Rounds both the real and imaginary parts of a complex number.
static double	fma(double a, double b, double c)	Returns the value of the fused multiply-add operation.
static Complex	fma(Complex x, Complex y, Complex z)	Returns the value of the complex fused multiply-add operation.
static double	fresnelC(double x)	Compute the Fresnel integral C(x).
static Complex	fresnelC(Complex z)	Compute the Fresnel C integral on the complex plane.
static Dual	fresnelC(Dual a)
static double	fresnelS(double x)	Compute the Fresnel integral S(x).
static Complex	fresnelS(Complex z)	Compute the Fresnel S integral on the complex plane.
static Dual	fresnelS(Dual a)
static Pair<Integer,Double>	frexp(double value)	Break floating-point number down into exponent and mantissa
static double	gamma(double x)	Compute the gamma function of x.
static Complex	gamma(Complex a)	Compute the gamma function of a complex number.
static Dual	gamma(Dual a)
static double	gammaP(double a, double x)	Compute the regularized Gamma P function.
static double	gammaQ(double a, double x)	Compute the regularized Gamma Q function.
static double	gcd(double a, double b)	Domain extension of the greatest common division function onto the real line.
static long	gcd(long a, long b)	Compute the greatest common divisor of two integers.
static Complex	gcd(Complex a, Complex b)	Domain extension of the greatest common division function onto the complex plane.
static boolean	ge(double x, double y)	Return true if x >= y.
static int	getExponent(double x)	Returns the exponent of a double precision number.
static boolean	gt(double x, double y)	Return true if x > y.
static double	hurwitzZeta(double x, double a)	Compute the value of the Hurwitz zeta function at x.
static Dual	hurwitzZeta(double s, Dual a)
static Complex	hurwitzZeta(Complex s, Complex a)	Compute the Hurwitz zeta function of complex arguments.
static double	hypergeo1F1(double a, double b, double x)	Computes the Confluent hypergeometric function (1F1) of three arguments.
static double	hypergeo1F2(double a, double b, double c, double x)	Computes the 1F2 case of the generalised hypergeometric function.
static double	hypergeo2F1(double a, double b, double c, double x)	Computes the Gaussian hypergeometric function (2F1) of four arguments.
static double	hypergeo3F0(double a, double b, double c, double x)	Computes the 3F0 case of the generalised hypergeometric function.
static double	hypot(double x, double y)	Returns the length of the hypotenuse of a right triangle with sides of length x and y.
static int	icbrt(int x)	Return the integer cube root of a number.
static long	icbrt(long x)	Return the long integer cube root of a number.
static short	icbrt(short a)	Return the short integer cube root of a number.
static int	ilog10(int x)	Compute the value of the integer logarithm in base 10 of a number
static Complex	integrateGaussLegendreComplex(BiFunction<Complex,Complex,Complex> f, Complex a, Complex b, Complex c, Complex d, int N)	Integrate a dyadic function over a finite interval [a,b] x [c,d] using the Gauss-Legendre quadrature.
static Complex	integrateGaussLegendreComplex(Function<Complex,Complex> f, Complex a, Complex b, int n)	Integrate a C -> C function using the Gauss-Legendre quadrature.
static Complex	integrateGaussLegendreRC(Function<Double,Complex> f, double a, double b, int N)	Integrate a monadic function over a finite interval [a,b] using the Gauss-Legendre quadrature.
static double	integrateGaussLegendreReal(BiFunction<Double,Double,Double> f, double a, double b, double c, double d, int N)	Integrate a dyadic function over a finite interval [a,b] x [c,d] using the Gauss-Legendre quadrature.
static double	integrateGaussLegendreReal(Function<Double,Double> f, double a, double b, int N)	Integrate a monadic function over a finite interval [a,b] using the Gauss-Legendre quadrature.
static Complex	integrateSimpsonRC(Function<Double,Complex> f, double a, double b, int N)	Integrate a monadic function over a finite interval [a,b] using the Simpson rule.
static double	integrateSimpsonReal(Function<Double,Double> f, double a, double b, int N)	Integrate a monadic function over a finite interval [a,b] using the Simpson rule.
static Complex[]	integrateTanhSinhComplex(Function<Complex,Complex> f, Complex a, Complex b, int n, double eps)	Integrate a C -> C function using the Tanh-Sinh quadrature.
static Complex[]	integrateTanhSinhRC(Function<Double,Complex> f, double a, double b, int N, double eps)	Integrate a monadic function over a finite interval [a,b] using the Tanh-Sinh quadrature, especially useful when singularities or infinite derivatives exist at one or both endpoints.
static double[]	integrateTanhSinhReal(Function<Double,Double> f, double a, double b, int N, double eps)	Integrate a monadic function over a finite interval [a,b] using the Tanh-Sinh quadrature, especially useful when singularities or infinite derivatives exist at one or both endpoints.
static int	ipow(int x, int n)	Compute the value of integer x^n.
static long	ipow(long x, long n)	Compute the value of integer x^n.
static short	ipow(short x, short n)	Compute the value of integer x^n.
static boolean	isPerfectSquare(long x)	Determine whether a number is a perfect square.
static boolean	isPowerOfTwo(long value)	Determine if a value is a power of two.
static boolean	isPrime(int n)	Primality test for an integer in range [0, 2^31-1].
static boolean	isPrime(long n)	Primality test for an unsigned integer in range [0, 2^64-1].
static int	isqrt(int x)	Compute the integer square root of a number.
static int	isqrt(long x)	Compute the integer square root of a number.
static int	isqrt(short x)	Compute the integer square root of a number.
static Complex	lambertw(Complex z, long k)	Compute the specified branch of the complex Lambert W function of z.
static double	lambertW0(double x)	Compute Lambert W_0 (x).
static Dual	lambertW0(Dual a)
static double	lambertWm1(double x)	Compute Lambert W_(-1) (x).
static Dual	lambertWm1(Dual a)
static double	landau(double x)	Compute the Landau distribution function.
static double	landau(double x, double mpv, double sigma, boolean norm)	Compute the Landau function with specified most probable value and sigma value.
static double	lcm(double a, double b)	Domain extension of the least common multiple function onto the real line.
static long	lcm(long a, long b)	Compute the least common multiple of two integers.
static Complex	lcm(Complex a, Complex b)	Domain extension of the least common multiple function onto the complex plane.
static boolean	le(double x, double y)	Return true if x <= y.
static double	legendre(double n, double z)
static double	legendreD(double phi, double k)	Compute the Legendre D elliptic integral defined by DLMF §19.2.6
static Complex	legendreD(Complex phi, Complex k)	Compute the Legendre D elliptic integral defined by DLMF §19.2.6
static double	legendreE(double phi, double k)	Compute the Legendre E elliptic integral defined by DLMF §19.2.5
static Complex	legendreE(Complex phi, Complex k)	Compute the Legendre E elliptic integral defined by DLMF §19.2.5
static double	legendreF(double phi, double k)	Compute the Legendre F elliptic integral defined by DLMF §19.2.4
static Complex	legendreF(Complex phi, Complex k)	Compute the Legendre F elliptic integral defined by DLMF §19.2.4
static double	legendrePi(double phi, double alpha, double k)	Compute the Legendre Pi elliptic integral defined by DLMF §19.2.7
static Complex	legendrePi(Complex phi, Complex alpha, Complex k)	Compute the Legendre Pi elliptic integral defined by DLMF §19.2.7
static double	lerchPhi(double z, double s, double a)	Compute the value of the Lerch transcendent function at z, s, a.
static Complex	lerchPhi(Complex z, Complex s, Complex a)	Compute the Lerch transcendent of z, s, and a.
static double	li(double x)	Compute the logarithmic integral of x, defined as li(x) = int(1/log t, t=0..x).
static Complex	li(Complex z)	Compute the complex logarithmic integral of z, defined as li(z) = int(1/log t, t=0..z).
static Dual	li(Dual a)
static double	liGamma(double a, double x)	Compute the value of the lower incomplete gamma function.
static Complex	liGamma(Complex s, Complex z)	Compute the lower incomplete (non-regularised) gamma function of a complex number.
static double	linearInterpolate(double fromValue, double toValue, double progress)	Linearly interpolate between two values.
static double	linearMap(double inRangeStart, double inRangeEnd, double outRangeStart, double outRangeEnd, double value)	Linearly map a value from one range to another.
static double	linearNorm(double rangeStart, double rangeEnd, double value)	Linearly normalise value from a range.
static double	log(double x)	Returns the value of the natural logarithm of a double precision number.
static Complex	log(Complex x)	Compute the natural logarithm of a complex number.
static Dual	log(Dual a)
static double	log10(double x)	Returns the value of the base 10 logarithm of a double precision number.
static double	log1p(double x)	Returns the value of the natural logarithm of a double precision number plus one.
static double	log2(double x)	Returns the value of the base 2 logarithm of a double precision number.
static double[]	logabsgamma(double x)	Computes the logarithm of the absolute value of the gamma function of x.
static double	logbeta(double x, double y)	Compute the logarihtm of the beta function of two values.
static Complex	logbeta(Complex a, Complex b)	Compute the logarithm of the beta function in the complex plane.
static double	loggamma(double x)	Compute the logarithm of the gamma function of x.
static Complex	loggamma(Complex x)	Principal branch of the logarithm of the gamma function.
static double	lommels1(double u, double v, double z)
static boolean	lt(double x, double y)	Return true if x < y.
static double	max(double x, double y)	Returns the larger of two numbers (maximum).
static double	min(double x, double y)	Returns the smaller of two numbers (minimum).
static double	mod(double x, double y)	Returns the modulus of two double precision numbers.
static long	mod(long x, long y)	Returns the modulus of two numbers.
static double	mul(double x, double y)	Multiplies two double precision numbers.
static Complex	mul(double a, Complex b)	Multiply a complex number and a real number.
static Complex	mul(Complex a, double b)	Multiply a complex number and a real number.
static Complex	mul(Complex a, Complex b)	Multiply two complex numbers.
static Dual	mul(Dual a, Dual b)
static <T> T	mul(MultiplicativeGroup<T> group, T x, T y)	Divides two values through a multiplicative group.
static <T> T	mul(MultiplicativeGroupoid<T> groupoid, T x, T y)	Multiplies two values through a multiplicative groupoid.
static boolean	ne(double x, double y)	Return true if x != y.
static boolean	ne(double a, Complex b)	Compare two numbers for inequality.
static boolean	ne(Complex a, double b)	Compare two numbers for inequality.
static boolean	ne(Complex a, Complex b)	Compare two numbers for inequality.
static double	negate(double x)	Negate a real number.
static Complex	negate(Complex x)	Negate a complex number.
static double	newtonRaphson(Function<Double,Double> f, Function<Double,Double> df, double x0, double eps)	Find a root of a monadic function using the Newton-Raphson method.
static double	nextAfter(double x, double y)	Returns the value of the double precision number adjacent to the first argument in the direction of the second argument.
static double	nextDown(double x)	Returns the value of the double precision number adjacent to the first argument in the direction of negative infinity.
static long	nextPowerOfTwo(long value)	Returns the next power of two.
static int	nextPrime(int n)	Find the next prime after n.
static long	nextPrime(long n)	Find the next prime after (unsigned) n.
static double	nextUp(double x)	Returns the value of the double precision number adjacent to the first argument in the direction of positive infinity.
static double	normQuantile(double p)	Computes quantiles for standard normal distribution N(0, 1) at probability p.
static double	pochhammer(double x, double n)	Compute the Pochhammer symbol (x)_n.
static double	pochhammer(double x, int n)	Compute the Pochhammer symbol (x)_n.
static Complex	pochhammer(Complex x, double n)	Compute the Pochhammer symbol (x)_n.
static Complex	pochhammer(Complex x, int n)	Compute the Pochhammer symbol (x)_n.
static Complex	pochhammer(Complex x, Complex n)	Compute the Pochhammer symbol (x)_n.
static double	polygamma(double n, double x)	Compute the value of the n-th polygamma function at x.
static Dual	polygamma(double n, Dual a)
static Complex	polygamma(Complex n, Complex z)	Compute the value of the order-N polygamma function at complex z.
static double	polylog(int n, double x)	Compute the polylogarithm of x.
static Dual	polylog(int n, Dual a)
static Complex	polylog(Complex s, Complex z)	Compute the polylogarithm of s and z.
static double	pow(double x, double y)	Returns the value of the first argument raised to the power of the second argument.
static double	pow(double x, int z)	Compute the value of x^z where x is a double precision floating point number and z is an integer.
static Complex	pow(double a, Complex b)	Compute a to the power of b, where a is a double precision number and b is a complex number.
static Complex	pow(Complex a, long b)	Compute a to the power of b, where a is a complex
static Complex	pow(Complex a, Complex b)	Compute a to the power of b, where a and b are both complex numbers.
static Dual	pow(Dual a, int b)
static double	random()	Returns a random double precision number in range [0, 1).
static double	random(double max)	Returns a random double precision number in range [0, max).
static double	random(double min, double max)	Returns a random double precision number in range [min, max).
static long	random(long max)	Returns a random long integer in range [0, max).
static long	random(long min, long max)	Returns a random long integer in range [min, max).
static long	randomSign()	Return a random sign (-1 or 1).
static double	rem(double x, double y)	Returns the remainder that results from dividing two double precision numbers.
static Complex	rem(Complex a, Complex b)	Domain extension of the remainder function onto the complex plane.
static double	root(double x, int deg)	Compute the n-th root of a complex number.
static Complex[]	root(Complex x, int n)	Find all n-th roots of a complex number.
static long	round(double x)	Rounds a double precision number to the nearest integer.
static Complex	round(Complex x)	Rounds both the real and imaginary parts of a complex number.
static double	scalb(double x, int n)	Returns the correctly rounded value of x*2^n.
static double	sec(double x)	Compute the value of the secant (1 / cos(x)) function at x.
static Complex	sec(Complex x)	Compute the secant of a complex number.
static Dual	sec(Dual a)
static double	sech(double x)	Compute the value of the hyperbolic secant (1 / cosh(x)) function at x.
static Complex	sech(Complex a)	Compute the hyperbolic secant of a complex number.
static Dual	sech(Dual a)
static double	Shi(double x)	Compute the value of the hyperbolic sine integral function at x.
static Complex	Shi(Complex z)	Compute the value of the complex hyperbolic sine integral Shi(z).
static Dual	Shi(Dual a)
static double[]	ShiChi(double x)	Compute both the hyperbolic sine and cosine integral at x.
static Complex[]	ShiChi(Complex z)	Compute the value of the complex hyperbolic sine integral Shi(z) and complex hyperbolic cosine integral Chi(z).
static double	si(double x)	Compute the value of the sine integral function at x.
static Complex	si(Complex z)	Compute the value of the complex sine integral si(z) at z.
static double	Si(double x)	Compute the value of the sine integral function at x.
static Complex	Si(Complex z)	Compute the value of the complex sine integral Si(z) at z.
static Dual	Si(Dual a)
static double	signum(double x)	Returns the sign of a double precision number.
static long	signum(long x)	Returns the sign of a number.
static Complex	signum(Complex x)	Returns the sign of a complex number.
static double	sin(double x)	Returns the value of the sine of a double precision number.
static Complex	sin(Complex x)	Compute the sine of a complex number.
static Dual	sin(Dual a)
static double	sinc(double x)	Returns the sinc function of a double precision number, defined as sin(x) / x, except for x = 0, where sinc(x) = 1.
static Complex	sinc(Complex x)	Returns the sinc function of a complex number, defined as sin(x) / x, except for x = 0, where sinc(x) = 1.
static Dual	sinc(Dual a)
static double	sinh(double x)	Returns the hyperbolic sine of a double precision number.
static Complex	sinh(Complex a)	Compute the hyperbolic sine of a complex number.
static Dual	sinh(Dual a)
static double	solidArea(Function<Double,Double> f, Function<Double,Double> df, double a, double b)	Compute the surface area of a solid of revolution created by rotating the function f(x) about the x-axis.
static double	solidVolume(Function<Double,Double> f, double a, double b)	Compute the volume of a solid of revolution created by rotating the function f(x) about the x-axis.
static double	spence(double n)	Compute the Spence function of x.
static Complex	spence(Complex z)	Compute the Spence function of z.
static Dual	spence(Dual a)
static double	sqrt(double x)	Returns the square root of a double precision number.
static Complex	sqrt(Complex x)	Compute the square root of a complex number.
static Dual	sqrt(Dual a)
static double	squash(double x)	Compute the value of the logistic sigmoid "squash" function at x.
static double	stretch(double x)	Compute the value of the inverse of the logistic sigmoid "squash" function at x.
static double	struve(double v, double x)	Computes the value of the Struve function of order v at x.
static double	struveL0(double x)	Computes the value of the modified Struve function of 0th order.
static double	struveL1(double x)	Computes the value of the modified Struve function of 1st order.
static double	sub(double x, double y)	Subtracts two double precision numbers.
static Complex	sub(double a, Complex b)	Subtract a complex number and a real number.
static Dual	sub(double a, Dual b)
static Complex	sub(Complex a, double b)	Subtract a complex number and a real number.
static Complex	sub(Complex a, Complex b)	Subtract two complex numbers from each other.
static Dual	sub(Dual a, double b)
static Dual	sub(Dual a, Dual b)
static double	tan(double x)	Returns the tangent of a double precision number.
static Complex	tan(Complex x)	Compute the tangent of a complex number.
static Dual	tan(Dual a)
static double	tanh(double x)	Returns the hyperbolic tangent of a double precision number.
static Complex	tanh(Complex a)	Compute the hyperbolic tangent of a complex number.
static Dual	tanh(Dual a)
static double	toDegrees(double x)	Converts the value to degrees from radians.
static double	toRadians(double x)	Converts the value to radians from degrees.
static double	trigamma(double x)	Compute the trigamma function of x.
static Complex	trigamma(Complex z)	Compute the complex trigamma function.
static Dual	trigamma(Dual a)
static double	uiGamma(double a, double x)	Compute the value of the upper incomplete gamma function.
static Complex	uiGamma(Complex s, Complex z)	Compute the upper incomplete (non-regularised) gamma function of a complex number.
static double	ulp(double x)	Returns the value of the unit in last place (ULP) of a double precision number.
static double	weberE(double v, double z)
static Complex	weberE(Complex v, Complex z)
static double	zeta(double x)	Compute the value of the Riemann zeta function at x.
static Complex	zeta(Complex z)	Compute the value of the complex Riemann zeta function at z.

Methods inherited from class java.lang.Object

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

Field Details

DECA

public static final double DECA

SI prefix for 10^1.

See Also:

• Constant Field Values

HECTO

public static final double HECTO

SI prefix for 10^2.

See Also:

• Constant Field Values

KILO

public static final double KILO

SI prefix for 10^3.

See Also:

• Constant Field Values

MEGA

public static final double MEGA

SI prefix for 10^6.

See Also:

• Constant Field Values

GIGA

public static final double GIGA

SI prefix for 10^9.

See Also:

• Constant Field Values

TERA

public static final double TERA

SI prefix for 10^12.

See Also:

• Constant Field Values

PETA

public static final double PETA

SI prefix for 10^15.

See Also:

• Constant Field Values

EXA

public static final double EXA

SI prefix for 10^18.

See Also:

• Constant Field Values

ZETTA

public static final double ZETTA

SI prefix for 10^21.

See Also:

• Constant Field Values

YOTTA

public static final double YOTTA

SI prefix for 10^24.

See Also:

• Constant Field Values

DECI

public static final double DECI

SI prefix for 10^-1.

See Also:

• Constant Field Values

CENTI

public static final double CENTI

SI prefix for 10^-2.

See Also:

• Constant Field Values

MILLI

public static final double MILLI

SI prefix for 10^-3.

See Also:

• Constant Field Values

MICRO

public static final double MICRO

SI prefix for 10^-6.

See Also:

• Constant Field Values

NANO

public static final double NANO

SI prefix for 10^-9.

See Also:

• Constant Field Values

PICO

public static final double PICO

SI prefix for 10^-12.

See Also:

• Constant Field Values

FEMTO

public static final double FEMTO

SI prefix for 10^-15.

See Also:

• Constant Field Values

ATTO

public static final double ATTO

SI prefix for 10^-18.

See Also:

• Constant Field Values

ZEPTO

public static final double ZEPTO

SI prefix for 10^-21.

See Also:

• Constant Field Values

YOCTO

public static final double YOCTO

SI prefix for 10^-24.

See Also:

• Constant Field Values

LN2

public static final double LN2

The value of ln(2).

See Also:

• Constant Field Values

LN10

public static final double LN10

The value of ln(10).

See Also:

• Constant Field Values

LOG2E

public static final double LOG2E

The value of log2(e).

See Also:

• Constant Field Values

PI

public static final double PI

The value of pi.

See Also:

• Constant Field Values

PI_2

public static final double PI_2

The value of pi/2.

See Also:

• Constant Field Values

PI_4

public static final double PI_4

The value of pi/4.

See Also:

• Constant Field Values

TWO_PI

public static final double TWO_PI

The value of 2*PI.

See Also:

• Constant Field Values

ONE_OVER_PI

public static final double ONE_OVER_PI

The value of 1/pi.

See Also:

• Constant Field Values

E

public static final double E

The value of e.

See Also:

• Constant Field Values

ONE_OVER_E

public static final double ONE_OVER_E

The value of 1/e.

See Also:

• Constant Field Values

EPSILON

public static final double EPSILON

The epsilon value for double precision numbers.

EULER_GAMMA

public static final double EULER_GAMMA

The Euler-Mascheroni constant.

See Also:

• Constant Field Values

GOLDEN_RATIO

public static final double GOLDEN_RATIO

The golden ratio.

See Also:

• Constant Field Values

APERY_CONSTANT

public static final double APERY_CONSTANT

Apery's constant - zeta(3).

See Also:

• Constant Field Values

GLAISHER_CONSTANT

public static final double GLAISHER_CONSTANT

Glaisher-Kinkelin constant.

See Also:

• Constant Field Values

CATALAN_CONSTANT

public static final double CATALAN_CONSTANT

Catalan's constant.

See Also:

• Constant Field Values

GOLOMB_DICKMAN_CONSTANT

public static final double GOLOMB_DICKMAN_CONSTANT

Golomb-Dickman constant.

See Also:

• Constant Field Values

MILLS_CONSTANT

public static final double MILLS_CONSTANT

Mills' constant.

See Also:

• Constant Field Values

FEIGENBAUM_CONSTANT

public static final double FEIGENBAUM_CONSTANT

Feigenbaum constant.

See Also:

• Constant Field Values

KHINCHIN_CONSTANT

public static final double KHINCHIN_CONSTANT

Khinchin's constant.

See Also:

• Constant Field Values

I

public static final Complex I

The imaginary unit.

Method Details

add

public static double add(double x,
 double y)

Adds two double precision numbers together.

Parameters:

x -

y -

Returns:

x + y

add

public static <T> T add(AdditiveGroupoid<T> groupoid,
 T x,
 T y)

Adds two values through an additive groupoid.

sub

public static double sub(double x,
 double y)

Subtracts two double precision numbers.

Parameters:

x -

y -

Returns:

x - y

add

public static <T> T add(AdditiveGroup<T> group,
 T x,
 T y)

Subtracts two values through an additive group.

mul

public static double mul(double x,
 double y)

Multiplies two double precision numbers.

Parameters:

x -

y -

Returns:

x * y

mul

public static <T> T mul(MultiplicativeGroupoid<T> groupoid,
 T x,
 T y)

Multiplies two values through a multiplicative groupoid.

div

public static double div(double x,
 double y)

Divides two double precision numbers.

Parameters:

x -

y -

Returns:

x / y

mul

public static <T> T mul(MultiplicativeGroup<T> group,
 T x,
 T y)

Divides two values through a multiplicative group.

mod

public static double mod(double x,
 double y)

Returns the modulus of two double precision numbers.

Parameters:

x -

y -

Returns:

x mod y

mod

public static long mod(long x,
 long y)

Returns the modulus of two numbers.

Parameters:

x -

y -

Returns:

x mod y

rem

public static double rem(double x,
 double y)

Returns the remainder that results from dividing two double precision numbers.

Parameters:

x -

y -

Returns:

x % y

abs

public static double abs(double x)

Returns the absolute value of a double precision number.

Parameters:

x -

Returns:

|x|

See Also:

• Math.abs(double)

abs

public static long abs(long x)

Returns the absolute value of a number.

Parameters:

x -

Returns:

|x|

acos

public static double acos(double x)

Returns the value of the inverse cosine of a double precision number.

Parameters:

x -

Returns:

cos^-1(x)

See Also:

• Math.acos(double)

asin

public static double asin(double x)

Returns the value of the inverse sine of a double precision number.

Parameters:

x -

Returns:

sin^-1(x)

See Also:

• Math.asin(double)

atan

public static double atan(double x)

Returns the value of the inverse tangent of a double precision number.

Parameters:

x -

Returns:

tan^-1(x)

See Also:

• Math.atan(double)

atan2

public static double atan2(double y,
 double x)

Returns the value of the inverse tangent of two double precision numbers.

Parameters:

y -

x -

Returns:

atan2(y, x)

See Also:

• Math.atan2(double, double)

cbrt

public static double cbrt(double x)

Returns the value of the cube root of a double precision number.

Parameters:

x -

Returns:

x^(1/3)

See Also:

• Math.cbrt(double)

ceil

public static double ceil(double x)

Returns the value of the smallest integer greater than or equal to a double precision number.

Parameters:

x -

Returns:

ceil(x)

See Also:

• Math.ceil(double)

cos

public static double cos(double x)

Returns the value of the smallest integer greater than or equal to a double precision number.

Parameters:

x -

Returns:

ceil(x)

See Also:

• Math.ceil(double)

cosh

public static double cosh(double x)

Returns the hyperbolic cosine of a double precision number.

Parameters:

x -

Returns:

cosh(x)

See Also:

• Math.cosh(double)

exp

public static double exp(double x)

Returns the value of the exponential function of a double precision number.

Parameters:

x -

Returns:

e^x

See Also:

• Math.exp(double)

expm1

public static double expm1(double x)

Returns the value of the exponential function of a double precision number minus one.

Parameters:

x -

Returns:

e^x - 1

See Also:

• Math.expm1(double)

floor

public static double floor(double x)

Returns the value of the largest integer less than or equal to a double precision number.

Parameters:

x -

Returns:

floor(x)

See Also:

• Math.floor(double)

hypot

public static double hypot(double x,
 double y)

Returns the length of the hypotenuse of a right triangle with sides of length x and y.

Parameters:

x -

y -

Returns:

sqrt(x ^ 2 + y ^ 2)

See Also:

• Math.hypot(double, double)

log

public static double log(double x)

Returns the value of the natural logarithm of a double precision number.

Parameters:

x -

Returns:

ln(x)

See Also:

• Math.log(double)

log10

public static double log10(double x)

Returns the value of the base 10 logarithm of a double precision number.

Parameters:

x -

Returns:

log10(x)

See Also:

• Math.log10(double)

log1p

public static double log1p(double x)

Returns the value of the natural logarithm of a double precision number plus one.

Parameters:

x -

Returns:

ln(x + 1)

See Also:

• Math.log1p(double)

log2

public static double log2(double x)

Returns the value of the base 2 logarithm of a double precision number.

Parameters:

x -

Returns:

log2(x)

max

public static double max(double x,
 double y)

Returns the larger of two numbers (maximum).

Parameters:

x -

y -

Returns:

max(x, y)

See Also:

• Math.max(double, double)

min

public static double min(double x,
 double y)

Returns the smaller of two numbers (minimum).

Parameters:

x -

y -

Returns:

min(x, y)

See Also:

• Math.min(double, double)

pow

public static double pow(double x,
 double y)

Returns the value of the first argument raised to the power of the second argument.

Parameters:

x -

y -

Returns:

x^y

See Also:

• Math.pow(double, double)

signum

public static double signum(double x)

Returns the sign of a double precision number.

Parameters:

x -

Returns:

-1 if x < 0, 0 if x == 0, 1 if x > 0

See Also:

• Math.signum(double)

signum

public static long signum(long x)

Returns the sign of a number.

Parameters:

x -

Returns:

-1 if x < 0, 1 if x > 0

signum

public static Complex signum(Complex x)

Returns the sign of a complex number.

Parameters:

x -

Returns:

A complex number where the real and imaginary parts correspond to the sign of the real and imaginary parts of x.

sin

public static double sin(double x)

Returns the value of the sine of a double precision number.

Parameters:

x -

Returns:

sin(x)

See Also:

• Math.sin(double)

ceil

public static Complex ceil(Complex x)

Rounds both the real and imaginary parts of a complex number.

Parameters:

x -

Returns:

ceil(x)

floor

public static Complex floor(Complex x)

Rounds both the real and imaginary parts of a complex number.

Parameters:

x -

Returns:

floor(x)

round

public static Complex round(Complex x)

Rounds both the real and imaginary parts of a complex number.

Parameters:

x -

Returns:

round(x)

sinc

public static double sinc(double x)

Returns the sinc function of a double precision number, defined as sin(x) / x, except for x = 0, where sinc(x) = 1.

Parameters:

x -

Returns:

sinc(x)

sinc

public static Complex sinc(Complex x)

Returns the sinc function of a complex number, defined as sin(x) / x, except for x = 0, where sinc(x) = 1.

Parameters:

x -

Returns:

sinc(x)

sinh

public static double sinh(double x)

Returns the hyperbolic sine of a double precision number.

Parameters:

x -

Returns:

sinh(x)

See Also:

• Math.sinh(double)

sqrt

public static double sqrt(double x)

Returns the square root of a double precision number.

Parameters:

x -

Returns:

x^(1/2)

See Also:

• Math.sqrt(double)

tan

public static double tan(double x)

Returns the tangent of a double precision number.

Parameters:

x -

Returns:

tan(x)

See Also:

• Math.tan(double)

tanh

public static double tanh(double x)

Returns the hyperbolic tangent of a double precision number.

Parameters:

x -

Returns:

tanh(x)

See Also:

• Math.tanh(double)

toDegrees

public static double toDegrees(double x)

Converts the value to degrees from radians.

Parameters:

x -

Returns:

x * 180 / PI

See Also:

• Math.toDegrees(double)

toRadians

public static double toRadians(double x)

Converts the value to radians from degrees.

Parameters:

x -

Returns:

x * PI / 180

See Also:

• Math.toRadians(double)

ulp

public static double ulp(double x)

Returns the value of the unit in last place (ULP) of a double precision number.

Parameters:

x -

Returns:

ULP(x)

See Also:

• Math.ulp(double)

fma

public static double fma(double a,
 double b,
 double c)

Returns the value of the fused multiply-add operation.

Parameters:

a -

b -

c -

Returns:

a * b + c

fma

public static Complex fma(Complex x,
 Complex y,
 Complex z)

Returns the value of the complex fused multiply-add operation.

Parameters:

x -

y -

z -

Returns:

x * y + z

nextAfter

public static double nextAfter(double x,
 double y)

Returns the value of the double precision number adjacent to the first argument in the direction of the second argument.

Parameters:

x -

y -

Returns:

nextAfter(x, y)

See Also:

• Math.nextAfter(double, double)

nextUp

public static double nextUp(double x)

Returns the value of the double precision number adjacent to the first argument in the direction of positive infinity.

Parameters:

x -

Returns:

nextUp(x)

See Also:

• Math.nextUp(double)

nextDown

public static double nextDown(double x)

Returns the value of the double precision number adjacent to the first argument in the direction of negative infinity.

Parameters:

x -

Returns:

nextDown(x)

See Also:

• Math.nextDown(double)

scalb

public static double scalb(double x,
 int n)

Returns the correctly rounded value of x*2^n.

Parameters:

x -

n -

Returns:

scalb(x, n)

See Also:

• Math.scalb(double, int)

copySign

public static double copySign(double x,
 double y)

Copy the sign of the second argument to the first argument.

Parameters:

x -

y -

Returns:

copySign(x, y)

See Also:

• Math.copySign(double, double)

copySign

public static long copySign(long x,
 long y)

Copy the sign of the second argument to the first argument.

Parameters:

x -

y -

Returns:

copySign(x, y)

copySign

public static Complex copySign(Complex x,
 Complex y)

Copy the sign of the second argument to the first argument.

Parameters:

x -

y -

Returns:

copySign(x, y)

absparts

public static Complex absparts(Complex x)

Compute the absolute values of parts of a complex number.

Parameters:

x -

Returns:

|re(x)| + i|im(x)|

getExponent

public static int getExponent(double x)

Returns the exponent of a double precision number.

Parameters:

x -

Returns:

exponent(x)

See Also:

• Math.getExponent(double)

round

public static long round(double x)

Rounds a double precision number to the nearest integer.

Parameters:

x -

Returns:

round(x)

See Also:

• Math.round(double)

random

public static double random()

Returns a random double precision number in range [0, 1).

Returns:

random()

random

public static double random(double min,
 double max)

Returns a random double precision number in range [min, max).

Parameters:

min -

max -

Returns:

random() * (max - min) + min

random

public static long random(long min,
 long max)

Returns a random long integer in range [min, max).

Parameters:

min -

max -

Returns:

random() * (max - min) + min

random

public static double random(double max)

Returns a random double precision number in range [0, max).

Parameters:

max -

Returns:

random() * max

random

public static long random(long max)

Returns a random long integer in range [0, max).

Parameters:

max -

Returns:

random() * max

lt

public static boolean lt(double x,
 double y)

Return true if x < y.

Parameters:

x -

y -

Returns:

x < y

le

public static boolean le(double x,
 double y)

Return true if x <= y.

Parameters:

x -

y -

Returns:

x <= y

gt

public static boolean gt(double x,
 double y)

Return true if x > y.

Parameters:

x -

y -

Returns:

x > y

ge

public static boolean ge(double x,
 double y)

Return true if x >= y.

Parameters:

x -

y -

Returns:

x >= y

eq

public static boolean eq(double x,
 double y)

Return true if x == y.

Parameters:

x -

y -

Returns:

x == y

ne

public static boolean ne(double x,
 double y)

Return true if x != y.

Parameters:

x -

y -

Returns:

x != y

compare

public static int compare(double x,
 double y)

Compare two double precision numbers.

Parameters:

x -

y -

Returns:

1 if x > y, 0 if x == y, -1 if x < y

compare

public static int compare(long x,
 long y)

Compare two integers.

Parameters:

x -

y -

Returns:

1 if x > y, 0 if x == y, -1 if x < y

eq

public static boolean eq(double x,
 double y,
 double tol)

Return true if x is approximately equal to y.

Parameters:

x -

y -

tol -

Returns:

abs(x - y) <= tol

isPerfectSquare

public static boolean isPerfectSquare(long x)

Determine whether a number is a perfect square.

Parameters:

x -

Returns:

true if x is a perfect square, false otherwise

linearMap

public static double linearMap(double inRangeStart,
 double inRangeEnd,
 double outRangeStart,
 double outRangeEnd,
 double value)

Linearly map a value from one range to another. Input range must not be empty.

Parameters:

inRangeStart - Input range start

inRangeEnd - Input range end

outRangeStart - Output range start

outRangeEnd - Output range end

value - Value to map

Returns:

Mapped value. Values outside the input range are not clamped to output range

linearNorm

public static double linearNorm(double rangeStart,
 double rangeEnd,
 double value)

Linearly normalise value from a range. Range must not be empty.

Parameters:

rangeStart - Range start normalized to 0

rangeEnd - Range end normalized to 1

value - Value to normalize

Returns:

Normalized value. Values outside the range are not clamped to 0 and 1

linearInterpolate

public static double linearInterpolate(double fromValue,
 double toValue,
 double progress)

Linearly interpolate between two values.

Parameters:

fromValue -

toValue -

progress -

Returns:

fromValue + (toValue - fromValue) * progress

clamp

public static double clamp(double value,
 double min,
 double max)

Clamp a value in the range [min, max].

Parameters:

value -

min -

max -

Returns:

min if value < min, max if value > max, value otherwise

clamp

public static long clamp(long value,
 long min,
 long max)

Clamp a value in the range [min, max].

Parameters:

value -

min -

max -

Returns:

min if value < min, max if value > max, value otherwise

isPowerOfTwo

public static boolean isPowerOfTwo(long value)

Determine if a value is a power of two.

Parameters:

value -

Returns:

true if value is a power of two, false otherwise

nextPowerOfTwo

public static long nextPowerOfTwo(long value)

Returns the next power of two. If the number is already a power of two, this function acts as an identity function.

Parameters:

value -

Returns:

the next power of two

randomSign

public static long randomSign()

Return a random sign (-1 or 1).

Returns:

-1 or 1

fastSin

public static float fastSin(float x)

Return a cached value of sin(x) using Raven's method. The cache is 65KB large.

Parameters:

x - angle in radians, between -2*pi and 2*pi inclusive.

Returns:

sin(x)

fastCos

public static float fastCos(float x)

Return a cached value of cos(x) using Raven's method. The cache is 65KB large.

Parameters:

x - angle in radians, between -2*pi and 2*pi inclusive.

Returns:

cos(x)

icbrt

public static int icbrt(int x)

Return the integer cube root of a number. If x < 0, -cbrt(-x) is returned.

Parameters:

x -

Returns:

floor(cbrt ( x))

icbrt

public static long icbrt(long x)

Return the long integer cube root of a number. If x < 0, -cbrt(-x) is returned.

Parameters:

x -

Returns:

floor(cbrt ( x))

icbrt

public static short icbrt(short a)

Return the short integer cube root of a number. If a < 0, -cbrt(-a) is returned.

Parameters:

a -

Returns:

floor(cbrt ( a))

isqrt

public static int isqrt(int x)

Compute the integer square root of a number. If x < 0, -isqrt(-x) is returned.

Parameters:

x -

Returns:

floor(sqrt ( x))

isqrt

public static int isqrt(long x)

Compute the integer square root of a number. If x < 0, -isqrt(-x) is returned.

Parameters:

x -

Returns:

floor(sqrt ( x))

isqrt

public static int isqrt(short x)

Compute the integer square root of a number. If x < 0, -isqrt(-x) is returned.

Parameters:

x -

Returns:

floor(sqrt ( x))

ilog10

public static int ilog10(int x)

Compute the value of the integer logarithm in base 10 of a number

Parameters:

x -

Returns:

floor(log10 ( x))

ipow

public static int ipow(int x,
 int n)

Compute the value of integer x^n.

Parameters:

x -

n -

Returns:

x^n.

ipow

public static long ipow(long x,
 long n)

Compute the value of integer x^n.

Parameters:

x -

n -

Returns:

x^n.

ipow

public static short ipow(short x,
 short n)

Compute the value of integer x^n.

Parameters:

x -

n -

Returns:

x^n.

frexp

public static Pair<Integer,Double> frexp(double value)

Break floating-point number down into exponent and mantissa

Parameters:

value -

Returns:

A pair of the exponent and the mantissa.

pow

public static double pow(double x,
 int z)

Compute the value of x^z where x is a double precision floating point number and z is an integer.

Parameters:

x -

z -

Returns:

x^z.

Throws:

ArithmeticException - in case of overflow.

airyAi

public static double airyAi(double x)

Compute the value of the Airy Ai function at the specified point.

Parameters:

x -

Returns:

Ai(x)

airyAip

public static double airyAip(double x)

Compute the value of the Airy Ai function's first derivative at the specified point.

Parameters:

x -

Returns:

Ai'(x)

airyBi

public static double airyBi(double x)

Compute the value of the Airy Bi function at the specified point.

Parameters:

x -

Returns:

Bi(x)

airyBip

public static double airyBip(double x)

Compute the value of the Airy Bi function's first derivative at the specified point.

Parameters:

x -

Returns:

Bi'(x)

airy

public static double[] airy(double x)

Compute the value of the Airy Ai, Ai', Bi and Bi' functions at the specified point.

Parameters:

x -

Returns:

a double array of length 4 containing Ai, Ai', Bi and Bi' in that order.

gamma

public static double gamma(double x)

Compute the gamma function of x.

Parameters:

x -

Returns:

gamma(x)

loggamma

public static double loggamma(double x)

Compute the logarithm of the gamma function of x.

Parameters:

x -

Returns:

log(gamma ( x))

digamma

public static double digamma(double x)

Compute the digamma function of x.

Parameters:

x -

Returns:

digamma(x)

trigamma

public static double trigamma(double x)

Compute the trigamma function of x.

Parameters:

x -

Returns:

trigamma(x)

uiGamma

public static double uiGamma(double a,
 double x)

Compute the value of the upper incomplete gamma function.

Parameters:

a -

x -

Returns:

gamma_u(a, x)

liGamma

public static double liGamma(double a,
 double x)

Compute the value of the lower incomplete gamma function.

Parameters:

a -

x -

Returns:

gamma_l(a, x)

pochhammer

public static double pochhammer(double x,
 double n)

Compute the Pochhammer symbol (x)_n.

Parameters:

x -

n -

Returns:

(x)_n

pochhammer

public static double pochhammer(double x,
 int n)

Compute the Pochhammer symbol (x)_n.

Parameters:

x -

n -

Returns:

(x)_n

pochhammer

public static Complex pochhammer(Complex x,
 int n)

Compute the Pochhammer symbol (x)_n.

Parameters:

x -

n -

Returns:

(x)_n

pochhammer

public static Complex pochhammer(Complex x,
 double n)

Compute the Pochhammer symbol (x)_n.

Parameters:

x -

n -

Returns:

(x)_n

pochhammer

public static Complex pochhammer(Complex x,
 Complex n)

Compute the Pochhammer symbol (x)_n.

Parameters:

x -

n -

Returns:

(x)_n

Ei

public static double Ei(double x)

Compute the value of the exponential integral at x.

Parameters:

x -

Returns:

Ei(x)

zeta

public static double zeta(double x)

Compute the value of the Riemann zeta function at x.

Parameters:

x -

Returns:

zeta(x)

zeta

public static Complex zeta(Complex z)

Compute the value of the complex Riemann zeta function at z.

Parameters:

z -

Returns:

zeta(z)

hurwitzZeta

public static double hurwitzZeta(double x,
 double a)

Compute the value of the Hurwitz zeta function at x.

Parameters:

x -

a -

Returns:

zeta(x, a)

polygamma

public static double polygamma(double n,
 double x)

Compute the value of the n-th polygamma function at x.

Parameters:

n -

x -

Returns:

polygamma(n, x)

polygamma

public static Complex polygamma(Complex n,
 Complex z)

Compute the value of the order-N polygamma function at complex z.

Parameters:

n -

z -

Returns:

polygamma(n, z)

beta

public static double beta(double x,
 double y)

Compute the beta function of two values.

Parameters:

x -

y -

Returns:

beta(x, y)

logbeta

public static double logbeta(double x,
 double y)

Compute the logarihtm of the beta function of two values. Uses the identity that log(beta(x, y)) = log(gamma(x)) + log(gamma(y)) - log(gamma(x + y)).

Parameters:

x -

y -

Returns:

log(beta ( x, y))

factorial

public static double factorial(long n)

Return the factorial of n as a double-precision. n must be positive. Faster than using the gamma function.

Parameters:

n -

Returns:

n!

Throws:

ArithmeticException - if n is negative

dilog

public static double dilog(double n)

Compute the dilogarithm (the value of the Spence function at 1-x) of x.

Parameters:

n -

Returns:

dilog(x)

spence

public static double spence(double n)

Compute the Spence function of x.

Parameters:

n -

Returns:

Spence(x)

polylog

public static double polylog(int n,
 double x)

Compute the polylogarithm of x.

Parameters:

n -

x -

Returns:

Li_n(x)

sec

public static double sec(double x)

Compute the value of the secant (1 / cos(x)) function at x.

Parameters:

x -

Returns:

sec(x)

csc

public static double csc(double x)

Compute the value of the cosecant (1 / sin(x)) function at x.

Parameters:

x -

Returns:

csc(x)

cot

public static double cot(double x)

Compute the value of the cotangent (1 / tan(x)) function at x.

Parameters:

x -

Returns:

cot(x)

sech

public static double sech(double x)

Compute the value of the hyperbolic secant (1 / cosh(x)) function at x.

Parameters:

x -

Returns:

sech(x)

csch

public static double csch(double x)

Compute the value of the hyperbolic cosecant (1 / sinh(x)) function at x.

Parameters:

x -

Returns:

csch(x)

coth

public static double coth(double x)

Compute the value of the hyperbolic cotangent (1 / tanh(x)) function at x.

Parameters:

x -

Returns:

coth(x)

asec

public static double asec(double x)

Compute the value of the inverse secant (1 / acos(x)) function at x.

Parameters:

x -

Returns:

asec(x)

acsc

public static double acsc(double x)

Compute the value of the inverse cosecant (1 / asin(x)) function at x.

Parameters:

x -

Returns:

acsc(x)

acot

public static double acot(double x)

Compute the value of the inverse cotangent (1 / atan(x)) function at x.

Parameters:

x -

Returns:

acot(x)

asinh

public static double asinh(double a)

Compute the value of the inverse hyperbolic sine function at x.

Parameters:

a -

Returns:

asinh(a)

acosh

public static double acosh(double x)

Compute the value of the inverse hyperbolic cosine function at x.

Parameters:

x -

Returns:

acosh(x)

atanh

public static double atanh(double a)

Compute the value of the inverse hyperbolic tangent function at x.

Parameters:

a -

Returns:

atanh(a)

asech

public static double asech(double x)

Compute the value of the inverse hyperbolic secant (1 / acosh(x)) function at x.

Parameters:

x -

Returns:

asech(x)

acsch

public static double acsch(double x)

Compute the value of the inverse hyperbolic cosecant (1 / asinh(x)) function at x.

Parameters:

x -

Returns:

acsch(x)

acoth

public static double acoth(double x)

Compute the value of the inverse hyperbolic cotangent (1 / atanh(x)) function at x.

Parameters:

x -

Returns:

acoth(x)

lambertW0

public static double lambertW0(double x)

Compute Lambert W_0 (x).

Parameters:

x -

Returns:

W_0(x)

lambertWm1

public static double lambertWm1(double x)

Compute Lambert W_(-1) (x).

Parameters:

x -

Returns:

W_(- 1) (x)

lerchPhi

public static double lerchPhi(double z,
 double s,
 double a)

Compute the value of the Lerch transcendent function at z, s, a.

Parameters:

z -

s -

a -

Returns:

Lerch(z, s, a)

Throws:

ArithmeticException - if the computation fails unexpectedly due to exceeding the amount of allowed iterations.

dawsonPlus

public static double dawsonPlus(double x)

Compute the value of the Dawson function (D+) at x.

Parameters:

x -

Returns:

D+(x)

dawsonMinus

public static double dawsonMinus(double x)

Compute the value of the Dawson function (D-) at x.

Parameters:

x -

Returns:

D-(x)

erf

public static double erf(double x)

Compute the value of the error function at x.

Parameters:

x -

Returns:

erf(x)

erfc

public static double erfc(double x)

Compute the value of the complementary error function at x.

Parameters:

x -

Returns:

erfc(x)

erfi

public static double erfi(double x)

Compute the value of the imaginary error function at x.

Parameters:

x -

Returns:

erfi(x)

stretch

public static double stretch(double x)

Compute the value of the inverse of the logistic sigmoid "squash" function at x.

Parameters:

x -

Returns:

stretch(x)

squash

public static double squash(double x)

Compute the value of the logistic sigmoid "squash" function at x.

Parameters:

x -

Returns:

squash(x)

Si

public static double Si(double x)

Compute the value of the sine integral function at x.

Parameters:

x -

Returns:

Si(x)

Ci

public static double Ci(double x)

Compute the value of the cosine integral function at x.

Parameters:

x -

Returns:

Ci(x)

Shi

public static double Shi(double x)

Compute the value of the hyperbolic sine integral function at x.

Parameters:

x -

Returns:

Chi

public static double Chi(double x)

Compute the value of the hyperbolic cosine integral function at x.

Parameters:

x -

Returns:

Chi(x)

si

public static double si(double x)

Compute the value of the sine integral function at x.

Parameters:

x -

Returns:

si(x)

Cin

public static double Cin(double x)

Compute the value of the cosine integral function at x.

Parameters:

x -

Returns:

Cin(x)

ShiChi

public static double[] ShiChi(double x)

Compute both the hyperbolic sine and cosine integral at x. Return the results in a two-element double precision array.

Parameters:

x -

Returns:

{Shi(x), Chi(x)}

fresnelC

public static double fresnelC(double x)

Compute the Fresnel integral C(x).

Parameters:

x -

Returns:

C(x)

fresnelS

public static double fresnelS(double x)

Compute the Fresnel integral S(x).

Parameters:

x -

Returns:

S(x)

besselY0

public static double besselY0(double x)

Compute the bessel Y0 function at x.

Parameters:

x -

Returns:

Y0(x)

besselY1

public static double besselY1(double x)

Compute the bessel Y1 function at x.

Parameters:

x -

Returns:

Y1(x)

besselYn

public static double besselYn(int n,
 double x)

Compute the bessel Yn function at x.

Parameters:

n -

x -

Returns:

Yn(x)

besselJ0

public static double besselJ0(double x)

Compute the bessel J0 function at x.

Parameters:

x -

Returns:

J0(x)

besselJ1

public static double besselJ1(double x)

Compute the bessel J1 function at x.

Parameters:

x -

Returns:

J1(x)

besselJn

public static double besselJn(int n,
 double x)

Compute the bessel Jn function at x.

Parameters:

n -

x -

Returns:

Jn(x)

besselI0

public static double besselI0(double x)

Compute the bessel I0 function at x.

Parameters:

x -

Returns:

I0(x)

besselI1

public static double besselI1(double x)

Compute the bessel I1 function at x.

Parameters:

x -

Returns:

I1(x)

besselK0

public static double besselK0(double x)

Compute the bessel K0 function at x.

Parameters:

x -

Returns:

K0(x)

besselK1

public static double besselK1(double x)

Compute the bessel K1 function at x.

Parameters:

x -

Returns:

K1(x)

besselKn

public static double besselKn(int n,
 double x)

Compute the bessel Kn function at x.

Parameters:

n -

x -

Returns:

Kn(x)

gcd

public static long gcd(long a,
 long b)

Compute the greatest common divisor of two integers.

Parameters:

a -

b -

Returns:

gcd(a, b)

gcd

public static double gcd(double a,
 double b)

Domain extension of the greatest common division function onto the real line.

Parameters:

a -

b -

Returns:

gcd(a, b)

gcd

public static Complex gcd(Complex a,
 Complex b)

Domain extension of the greatest common division function onto the complex plane.

Parameters:

a -

b -

Returns:

gcd(a, b)

rem

public static Complex rem(Complex a,
 Complex b)

Domain extension of the remainder function onto the complex plane.

Parameters:

a -

b -

Returns:

a rem b

lcm

public static long lcm(long a,
 long b)

Compute the least common multiple of two integers.

Parameters:

a -

b -

Returns:

lcm(a, b)

lcm

public static double lcm(double a,
 double b)

Domain extension of the least common multiple function onto the real line.

Parameters:

a -

b -

Returns:

lcm(a, b)

lcm

public static Complex lcm(Complex a,
 Complex b)

Domain extension of the least common multiple function onto the complex plane.

Parameters:

a -

b -

Returns:

lcm(a, b)

fib

public static long fib(int a)

Compute the n-th Fibonacci number using Binet's formula. Due to floating point precision issues, this method is only accurate for n ≤= 75.

Parameters:

a -

Returns:

fib(a)

hypergeo2F1

public static double hypergeo2F1(double a,
 double b,
 double c,
 double x)

Computes the Gaussian hypergeometric function (2F1) of four arguments.

Parameters:

a -

b -

c -

x -

Returns:

(2F1)(a, b, c, x)

hypergeo1F1

public static double hypergeo1F1(double a,
 double b,
 double x)

Computes the Confluent hypergeometric function (1F1) of three arguments.

Parameters:

a -

b -

x -

Returns:

(1F1)(a, b, x)

hypergeo1F2

public static double hypergeo1F2(double a,
 double b,
 double c,
 double x)

Computes the 1F2 case of the generalised hypergeometric function.

Parameters:

a -

b -

c -

x -

Returns:

a double array the approximation and the estimated error of (1F2)(a, b, c x)

hypergeo3F0

public static double hypergeo3F0(double a,
 double b,
 double c,
 double x)

Computes the 3F0 case of the generalised hypergeometric function.

Parameters:

a -

b -

c -

x -

Returns:

a double array the approximation and the estimated error of (1F2)(a, b, c x)

struve

public static double struve(double v,
 double x)

Computes the value of the Struve function of order v at x.

Parameters:

v - order of the Struve function

x - argument of the Struve function

Returns:

Struve(v, x)

struveL0

public static double struveL0(double x)

Computes the value of the modified Struve function of 0th order.

Parameters:

x - argument of the Struve function

Returns:

StruveL0(x)

struveL1

public static double struveL1(double x)

Computes the value of the modified Struve function of 1st order.

Parameters:

x - argument of the Struve function

Returns:

StruveL1(x)

logabsgamma

public static double[] logabsgamma(double x)

Computes the logarithm of the absolute value of the gamma function of x.

Parameters:

x -

Returns:

{ log(|Gamma(x)|), sign(Gamma(x)) }

See Also:

• loggamma(double)

besselJv

public static double besselJv(double n,
 double x)

Compute fractional order bessel function of n and x.

Parameters:

n -

x -

Returns:

J_n(x)

besselYv

public static double besselYv(double v,
 double x)

Compute the fractional order bessel Y function of v and x.

Parameters:

v -

x -

Returns:

Y_n(x)

integrateSimpsonReal

public static double integrateSimpsonReal(Function<Double,Double> f,
 double a,
 double b,
 int N)

Integrate a monadic function over a finite interval [a,b] using the Simpson rule. The number of intervals is given by N and the precision of the final result greatly depends on this parameter.

Parameters:

f - function to integrate

a - lower bound

b - upper bound

N - number of intervals, N=10000 tends to give a good approximation in most scenarios.

Returns:

integral of f over [a,b]

Throws:

IllegalArgumentException - if N is not a positive integer

integrateGaussLegendreReal

public static double integrateGaussLegendreReal(Function<Double,Double> f,
 double a,
 double b,
 int N)

Integrate a monadic function over a finite interval [a,b] using the Gauss-Legendre quadrature. The number of intervals is given by N and the precision of the final result greatly depends on this parameter. The computation of an integral using the Gauss-Legendre quadrature involves caching the coefficients required to perform the integration depending on the value of the N parameter. This means that the first call to this method will be slower than subsequent calls with the same value of N. The coefficients are internally cached inside a ConcurrentHashMap.

Parameters:

f - function to integrate

a - lower bound

b - upper bound

N - number of intervals, N=6 tends to give a good approximation in most scenarios. N must be between 1 and 30.

Returns:

integral of f over [a,b]

Throws:

IllegalArgumentException - if N is not a positive integer

integrateTanhSinhReal

public static double[] integrateTanhSinhReal(Function<Double,Double> f,
 double a,
 double b,
 int N,
 double eps)

Integrate a monadic function over a finite interval [a,b] using the Tanh-Sinh quadrature, especially useful when singularities or infinite derivatives exist at one or both endpoints. The Tanh-Sinh quadrature is not as efficient as Gaussian quadrature for smooth integrands.

Parameters:

f - function to integrate

a - lower bound

b - upper bound

N - the degree of the quadrature, usually N=6 is sufficient

eps - desired precision of the result (usually 1.0e-9 is sufficient)

Returns:

an array of double values, first of which is the integral of f over [a,b], while the second is the estimated error.

Throws:

IllegalArgumentException - if N is not a positive integer

integrateSimpsonRC

public static Complex integrateSimpsonRC(Function<Double,Complex> f,
 double a,
 double b,
 int N)

Integrate a monadic function over a finite interval [a,b] using the Simpson rule. The number of intervals is given by N and the precision of the final result greatly depends on this parameter.

Parameters:

f - function to integrate

a - lower bound

b - upper bound

N - number of intervals, N=10000 tends to give a good approximation in most scenarios.

Returns:

integral of f over [a,b]

Throws:

IllegalArgumentException - if N is not a positive integer

integrateGaussLegendreRC

public static Complex integrateGaussLegendreRC(Function<Double,Complex> f,
 double a,
 double b,
 int N)

Integrate a monadic function over a finite interval [a,b] using the Gauss-Legendre quadrature. The number of intervals is given by N and the precision of the final result greatly depends on this parameter. The computation of an integral using the Gauss-Legendre quadrature involves caching the coefficients required to perform the integration depending on the value of the N parameter. This means that the first call to this method will be slower than subsequent calls with the same value of N. The coefficients are internally cached inside a ConcurrentHashMap.

Parameters:

f - function to integrate

a - lower bound

b - upper bound

N - number of intervals, N=6 tends to give a good approximation in most scenarios. N must be between 1 and 30.

Returns:

integral of f over [a,b]

Throws:

IllegalArgumentException - if N is not a positive integer

integrateTanhSinhRC

public static Complex[] integrateTanhSinhRC(Function<Double,Complex> f,
 double a,
 double b,
 int N,
 double eps)

Integrate a monadic function over a finite interval [a,b] using the Tanh-Sinh quadrature, especially useful when singularities or infinite derivatives exist at one or both endpoints. The Tanh-Sinh quadrature is not as efficient as Gaussian quadrature for smooth integrands.

Parameters:

f - function to integrate

a - lower bound

b - upper bound

N - the degree of the quadrature, usually N=6 is sufficient

eps - desired precision of the result (usually 1.0e-9 is sufficient)

Returns:

an array of double values, first of which is the integral of f over [a,b], while the second is the estimated error.

Throws:

IllegalArgumentException - if N is not a positive integer

binomial

public static long binomial(int n,
 int k)

Compute the binomial coefficient "n choose k".

Parameters:

n - the number of elements, n > 0.

k - the number of elements to choose, 0 < k <= n.

Returns:

n! / (k! * (n-k)!)

Throws:

IllegalArgumentException - if n <= 0 or k < 0 or k > n.

newtonRaphson

public static double newtonRaphson(Function<Double,Double> f,
 Function<Double,Double> df,
 double x0,
 double eps)

Find a root of a monadic function using the Newton-Raphson method.

Parameters:

f - the function to find a root for

df - the derivative of the function

x0 - the initial guess

eps - the desired precision of the result

Returns:

a root of the function f within the desired precision unless the iteration limit is exceeded.

aberth

public static boolean[] aberth(Complex[] coefficients)

Numerically finds all roots of a polynomial equation P(z) = 0. There are (n+1) coefficients and n roots. The leading coefficient must not be zero. Errors are returned in a boolean array for each root. Success is indicated by false, i.e. no error. On return each root should be checked against its flag.

Parameters:

coefficients -

Returns:

bernoulliBound

public static int bernoulliBound(int n)

Returns an integer b such that |B_n| ≤ 2^b for all Bernoulli numbers B_n.

Parameters:

n -

Returns:

agm

public static double agm(double a,
 double g)

Computes the Arithmetic-Geometric mean.

Parameters:

a -

g -

Returns:

the Arithmetic-Geometric mean of a and g

fac2

public static double fac2(double a)

angerJ

public static double angerJ(double v,
 double z)

weberE

public static double weberE(double v,
 double z)

angerJ

public static Complex angerJ(Complex v,
 Complex z)

weberE

public static Complex weberE(Complex v,
 Complex z)

lommels1

public static double lommels1(double u,
 double v,
 double z)

legendre

public static double legendre(double n,
 double z)

add

public static Complex add(Complex a,
 Complex b)

Add two complex numbers together.

Parameters:

a -

b -

Returns:

a + b

add

public static Dual add(Dual a,
 Dual b)

Perform dual addition.

Parameters:

a -

b -

Returns:

a + b

add

public static Dual add(Dual a,
 double b)

Perform real-dual addition.

Parameters:

a -

b -

Returns:

a + b

add

public static Dual add(double a,
 Dual b)

Perform real-dual addition.

Parameters:

a -

b -

Returns:

a + b

sub

public static Dual sub(Dual a,
 Dual b)

sub

public static Dual sub(Dual a,
 double b)

sub

public static Dual sub(double a,
 Dual b)

mul

public static Dual mul(Dual a,
 Dual b)

div

public static Dual div(Dual a,
 Dual b)

sin

public static Dual sin(Dual a)

sec

public static Dual sec(Dual a)

cos

public static Dual cos(Dual a)

csc

public static Dual csc(Dual a)

tan

public static Dual tan(Dual a)

cot

public static Dual cot(Dual a)

exp

public static Dual exp(Dual a)

log

public static Dual log(Dual a)

pow

public static Dual pow(Dual a,
 int b)

abs

public static Dual abs(Dual a)

asin

public static Dual asin(Dual a)

acos

public static Dual acos(Dual a)

atan

public static Dual atan(Dual a)

sqrt

public static Dual sqrt(Dual a)

sinh

public static Dual sinh(Dual a)

cosh

public static Dual cosh(Dual a)

tanh

public static Dual tanh(Dual a)

gamma

public static Dual gamma(Dual a)

digamma

public static Dual digamma(Dual a)

besselI0

public static Dual besselI0(Dual a)

besselJ0

public static Dual besselJ0(Dual a)

besselK0

public static Dual besselK0(Dual a)

besselJ1

public static Dual besselJ1(Dual a)

besselY0

public static Dual besselY0(Dual a)

besselK1

public static Dual besselK1(Dual a)

besselJn

public static Dual besselJn(int n,
 Dual a)

besselYn

public static Dual besselYn(int n,
 Dual a)

besselYv

public static Dual besselYv(double n,
 Dual a)

besselKn

public static Dual besselKn(int n,
 Dual a)

besselJv

public static Dual besselJv(double n,
 Dual a)

airyAi

public static Dual airyAi(Dual a)

airyBi

public static Dual airyBi(Dual a)

Chi

public static Dual Chi(Dual a)

Shi

public static Dual Shi(Dual a)

Ci

public static Dual Ci(Dual a)

Si

public static Dual Si(Dual a)

Ei

public static Dual Ei(Dual a)

erf

public static Dual erf(Dual a)

dilog

public static Dual dilog(Dual a)

acot

public static Dual acot(Dual a)

acoth

public static Dual acoth(Dual a)

asec

public static Dual asec(Dual a)

acsc

public static Dual acsc(Dual a)

coth

public static Dual coth(Dual a)

sech

public static Dual sech(Dual a)

csch

public static Dual csch(Dual a)

asinh

public static Dual asinh(Dual a)

acosh

public static Dual acosh(Dual a)

atanh

public static Dual atanh(Dual a)

asech

public static Dual asech(Dual a)

acsch

public static Dual acsch(Dual a)

cbrt

public static Dual cbrt(Dual a)

beta

public static Dual beta(Dual a,
 double c)

Compute the beta function of a dual and real number. The derivative is taken as d/da beta(a,c).

Parameters:

a -

c -

Returns:

beta

public static Dual beta(double a,
 Dual c)

Compute the beta function of a real and dual number. The derivative is taken as d/dc beta(a,c).

Parameters:

a -

c -

Returns:

erfc

public static Dual erfc(Dual a)

erfi

public static Dual erfi(Dual a)

fresnelC

public static Dual fresnelC(Dual a)

fresnelS

public static Dual fresnelS(Dual a)

li

public static Dual li(Dual a)

sinc

public static Dual sinc(Dual a)

spence

public static Dual spence(Dual a)

trigamma

public static Dual trigamma(Dual a)

polygamma

public static Dual polygamma(double n,
 Dual a)

polylog

public static Dual polylog(int n,
 Dual a)

lambertW0

public static Dual lambertW0(Dual a)

lambertWm1

public static Dual lambertWm1(Dual a)

hurwitzZeta

public static Dual hurwitzZeta(double s,
 Dual a)

add

public static Complex add(Complex a,
 double b)

Add a complex number and a real number.

Parameters:

a -

b -

Returns:

a + b

add

public static Complex add(double a,
 Complex b)

Add a complex number and a real number.

Parameters:

a -

b -

Returns:

a + b

sub

public static Complex sub(Complex a,
 Complex b)

Subtract two complex numbers from each other.

Parameters:

a -

b -

Returns:

a - b

sub

public static Complex sub(Complex a,
 double b)

Subtract a complex number and a real number.

Parameters:

a -

b -

Returns:

a - b

sub

public static Complex sub(double a,
 Complex b)

Subtract a complex number and a real number.

Parameters:

a -

b -

Returns:

a - b

mul

public static Complex mul(Complex a,
 Complex b)

Multiply two complex numbers.

Parameters:

a -

b -

Returns:

a * b

mul

public static Complex mul(Complex a,
 double b)

Multiply a complex number and a real number.

Parameters:

a -

b -

Returns:

a * b

mul

public static Complex mul(double a,
 Complex b)

Multiply a complex number and a real number.

Parameters:

a -

b -

Returns:

a * b

div

public static Complex div(Complex a,
 Complex b)

Divide two complex numbers.

Parameters:

a -

b -

Returns:

a / b

div

public static Complex div(Complex a,
 double b)

Divide a complex number and a real number.

Parameters:

a -

b -

Returns:

a / b

div

public static Complex div(double a,
 Complex b)

Divide a complex number and a real number.

Parameters:

a -

b -

Returns:

a / b

conj

public static Complex conj(Complex a)

Compute the complex conjugate of a complex number.

Parameters:

a -

Returns:

conj(a)

abs

public static double abs(Complex a)

Compute the absolute value of a complex number.

Parameters:

a -

Returns:

|a|

sqrt

public static Complex sqrt(Complex x)

Compute the square root of a complex number.

Parameters:

x -

Returns:

sqrt(x)

exp

public static Complex exp(Complex x)

Compute the value of the exponential function of a complex number.

Parameters:

x -

Returns:

exp(x)

log

public static Complex log(Complex x)

Compute the natural logarithm of a complex number.

Parameters:

x -

Returns:

ln(x)

arg

public static double arg(Complex z)

Return the argument of a complex number.

Parameters:

z -

Returns:

Arg(z)

cis

public static Complex cis(double x)

Compute the value of the cis function of a number.

Parameters:

x -

Returns:

cos(x) + i * sin(x)

eq

public static boolean eq(Complex a,
 Complex b)

Compare two numbers for equality.

Parameters:

a -

b -

Returns:

true if a == b, false otherwise

eq

public static boolean eq(double a,
 Complex b)

Compare two numbers for equality.

Parameters:

a -

b -

Returns:

true if a == b, false otherwise

eq

public static boolean eq(Complex a,
 double b)

Compare two numbers for equality.

Parameters:

a -

b -

Returns:

true if a == b, false otherwise

ne

public static boolean ne(Complex a,
 double b)

Compare two numbers for inequality.

Parameters:

a -

b -

Returns:

true if a != b, false otherwise

ne

public static boolean ne(double a,
 Complex b)

Compare two numbers for inequality.

Parameters:

a -

b -

Returns:

true if a != b, false otherwise

ne

public static boolean ne(Complex a,
 Complex b)

Compare two numbers for inequality.

Parameters:

a -

b -

Returns:

true if a != b, false otherwise

eq

public static boolean eq(Complex a,
 Complex b,
 double tol)

Compare two complex numbers for equality.

Parameters:

a -

b -

tol -

Returns:

true if a ~= b, false otherwise

root

public static double root(double x,
 int deg)

Compute the n-th root of a complex number.

Parameters:

x -

deg -

Returns:

x^(1/deg)

root

public static Complex[] root(Complex x,
 int n)

Find all n-th roots of a complex number.

Parameters:

x -

n -

Returns:

x^(1/n)

cbrt

public static Complex cbrt(Complex x)

Compute the principal cube root of a complex number.

Parameters:

x -

Returns:

cbrt(x)

sin

public static Complex sin(Complex x)

Compute the sine of a complex number.

Parameters:

x -

Returns:

sin(x)

cos

public static Complex cos(Complex x)

Compute the cosine of a complex number.

Parameters:

x -

Returns:

cos(x)

tan

public static Complex tan(Complex x)

Compute the tangent of a complex number.

Parameters:

x -

Returns:

tan(x)

cot

public static Complex cot(Complex x)

Compute the cotangent of a complex number.

Parameters:

x -

Returns:

cot(x)

sec

public static Complex sec(Complex x)

Compute the secant of a complex number.

Parameters:

x -

Returns:

csc

public static Complex csc(Complex x)

Compute the cosecant of a complex number.

Parameters:

x -

Returns:

pow

public static Complex pow(Complex a,
 Complex b)

Compute a to the power of b, where a and b are both complex numbers.

Parameters:

a -

b -

Returns:

a^b

pow

public static Complex pow(double a,
 Complex b)

Compute a to the power of b, where a is a double precision number and b is a complex number.

Parameters:

a -

b -

Returns:

a^b

pow

public static Complex pow(Complex a,
 long b)

Compute a to the power of b, where a is a complex

Parameters:

a -

b -

Returns:

a^b

sinh

public static Complex sinh(Complex a)

Compute the hyperbolic sine of a complex number.

Parameters:

a -

Returns:

sinh(a)

cosh

public static Complex cosh(Complex a)

Compute the hyperbolic cosine of a complex number.

Parameters:

a -

Returns:

cosh(a)

tanh

public static Complex tanh(Complex a)

Compute the hyperbolic tangent of a complex number.

Parameters:

a -

Returns:

tanh(a)

coth

public static Complex coth(Complex a)

Compute the hyperbolic cotangent of a complex number.

Parameters:

a -

Returns:

coth(a)

sech

public static Complex sech(Complex a)

Compute the hyperbolic secant of a complex number.

Parameters:

a -

Returns:

sech(a)

csch

public static Complex csch(Complex a)

Compute the hyperbolic cosecant of a complex number.

Parameters:

a -

Returns:

csch(a)

asin

public static Complex asin(Complex a)

Compute the arcus sine of a complex number.

Parameters:

a -

Returns:

asin(a)

acos

public static Complex acos(Complex a)

Compute the arcus cosine of a complex number.

Parameters:

a -

Returns:

acos(a)

atan

public static Complex atan(Complex a)

Compute the arcus tangent of a complex number.

Parameters:

a -

Returns:

atan(a)

acot

public static Complex acot(Complex a)

Compute the arcus cotangent of a complex number.

Parameters:

a -

Returns:

acot(a)

acsc

public static Complex acsc(Complex a)

Compute the arcus cosecant of a complex number.

Parameters:

a -

Returns:

acsc(a)

asec

public static Complex asec(Complex a)

Compute the arcus secant of a complex number.

Parameters:

a -

Returns:

asec(a)

asinh

public static Complex asinh(Complex a)

Compute the hyperbolic arcsine of a complex number.

Parameters:

a -

Returns:

asinh(a)

acosh

public static Complex acosh(Complex a)

Compute the hyperbolic arccosine of a complex number.

Parameters:

a -

Returns:

acosh(a)

atanh

public static Complex atanh(Complex a)

Compute the hyperbolic arctangent of a complex number.

Parameters:

a -

Returns:

atanh(a)

acoth

public static Complex acoth(Complex a)

Compute the hyperbolic arccotangent of a complex number.

Parameters:

a -

Returns:

acoth(a)

asech

public static Complex asech(Complex a)

Compute the hyperbolic arcsecant of a complex number.

Parameters:

a -

Returns:

asech(a)

negate

public static Complex negate(Complex x)

Negate a complex number.

Parameters:

x -

Returns:

-x

negate

public static double negate(double x)

Negate a real number.

Parameters:

x -

Returns:

-x

acsch

public static Complex acsch(Complex a)

Compute the hyperbolic arccosecant of a complex number.

Parameters:

a -

Returns:

acsch(a)

gamma

public static Complex gamma(Complex a)

Compute the gamma function of a complex number.

Parameters:

a -

Returns:

gamma(a)

beta

public static Complex beta(Complex a,
 Complex b)

Compute the beta function of two complex numbers. Defined as beta(a, b) = gamma(a) * gamma(b) / gamma(a + b)

Parameters:

a -

b -

Returns:

beta(a, b)

See Also:

• gamma(Complex)

airyAi

public static Complex airyAi(Complex x)

Compute the value of the Airy Ai function at the specified point.

Parameters:

x -

Returns:

Ai(x)

airyAip

public static Complex airyAip(Complex x)

Compute the value of the Airy Ai function's first derivative at the specified point.

Parameters:

x -

Returns:

Ai'(x)

airyBi

public static Complex airyBi(Complex x)

Compute the value of the Airy Bi function at the specified point.

Parameters:

x -

Returns:

Bi(x)

airyBip

public static Complex airyBip(Complex x)

Compute the value of the Airy Bi function's first derivative at the specified point.

Parameters:

x -

Returns:

Bi'(x)

airy

public static Complex[] airy(Complex x)

Compute the value of the Airy Ai, Ai', Bi and Bi' functions at the specified point.

Parameters:

x -

Returns:

a double array of length 4 containing Ai, Ai', Bi and Bi' in that order.

e1

public static Complex e1(Complex x)

Compute the value of the exponential integral E1 at the specified point. For positive values of real x, E1 and Ei relate as -E1(x) = Ei(-x).

Parameters:

x -

Returns:

E1(x)

Ei

public static Complex Ei(Complex x)

Compute the exponential integral using a formula that was revealed to me in a dream:

Ei(z) = -E1(-z) - log(-z) + 0.5 (log(z) - log(1/z))

Parameters:

x -

Returns:

Ei(x)

Ein

public static Complex Ein(Complex x)

Compute the value of the complementary exponential integral Ein at the specified point. For all complex z, Ein and Ei relate as Ein(z) = E1(z) + EulerGamma + ln z.

Parameters:

x -

Returns:

Ein(x)

See Also:

• e1(Complex)

loggamma

public static Complex loggamma(Complex x)

Principal branch of the logarithm of the gamma function. Defined to be log(gamma(x)) for x > 0. Extended to the complex plane by analytic continuation. The function has a single branch cut on the negative real axis.

It is not generally true that log gamma(x) = log(gamma(x)), though the real parts tend to agree. The benefit of not defining loggamma as is that the latter function has a complicated branch cut structure whereas loggamma is analytic except for on the negative real axis.

On the real line, loggamma is related to logabsgamma.

Parameters:

x -

Returns:

See Also:

• logabsgamma(double)

logbeta

public static Complex logbeta(Complex a,
 Complex b)

Compute the logarithm of the beta function in the complex plane. Defined using complex loggamma as logbeta(a, b) = loggamma(a) - loggamma(b) - loggamma(a + b).

Parameters:

a -

b -

Returns:

logbeta(a, b)

See Also:

• loggamma(Complex)

liGamma

public static Complex liGamma(Complex s,
 Complex z)

Compute the lower incomplete (non-regularised) gamma function of a complex number.

Parameters:

s -

z -

Returns:

ligamma(s, z)

uiGamma

public static Complex uiGamma(Complex s,
 Complex z)

Compute the upper incomplete (non-regularised) gamma function of a complex number.

Parameters:

s -

z -

Returns:

uigamma(s, z)

erf

public static Complex erf(Complex z)

Compute the complex error function.

Parameters:

z -

Returns:

erf(z)

erfc

public static Complex erfc(Complex z)

Compute the complex complementary error function.

Parameters:

z -

Returns:

erfc(z)

erfi

public static Complex erfi(Complex z)

Compute the complex imaginary error function.

Parameters:

z -

Returns:

erfi(z) = -i erf(iz)

dawsonPlus

public static Complex dawsonPlus(Complex z)

Compute the value of the complex Dawson function (D+) at z.

Parameters:

z -

Returns:

D+(z)

dawsonMinus

public static Complex dawsonMinus(Complex z)

Compute the value of the complex Dawson function (D-) at z.

Parameters:

z -

Returns:

D-(z)

fresnelS

public static Complex fresnelS(Complex z)

Compute the Fresnel S integral on the complex plane.

Parameters:

z -

Returns:

FresnelS(z)

fresnelC

public static Complex fresnelC(Complex z)

Compute the Fresnel C integral on the complex plane.

Parameters:

z -

Returns:

FresnelC(z)

digamma

public static Complex digamma(Complex z)

Compute the complex digamma function.

Parameters:

z -

Returns:

digamma(z)

trigamma

public static Complex trigamma(Complex z)

Compute the complex trigamma function.

Parameters:

z -

Returns:

trigamma(z)

li

public static double li(double x)

Compute the logarithmic integral of x, defined as li(x) = int(1/log t, t=0..x).

Parameters:

x -

Returns:

li(x)

li

public static Complex li(Complex z)

Compute the complex logarithmic integral of z, defined as li(z) = int(1/log t, t=0..z).

Parameters:

z -

Returns:

li(z)

Si

public static Complex Si(Complex z)

Compute the value of the complex sine integral Si(z) at z.

Parameters:

z -

Returns:

Si(z)

si

public static Complex si(Complex z)

Compute the value of the complex sine integral si(z) at z.

Parameters:

z -

Returns:

si(z)

Ci

public static Complex Ci(Complex z)

Compute the value of the complex cosine integral Ci(z) at z.

Parameters:

z -

Returns:

Ci(z)

Cin

public static Complex Cin(Complex z)

Compute the value of the complex cosine integral Cin(z) at z.

Parameters:

z -

Returns:

Cin(z)

Shi

public static Complex Shi(Complex z)

Compute the value of the complex hyperbolic sine integral Shi(z).

Parameters:

z -

Returns:

Shi(z)

Chi

public static Complex Chi(Complex z)

Compute the value of the complex hyperbolic cosine integral Chi(z).

Parameters:

z -

Returns:

Chi(z)

ShiChi

public static Complex[] ShiChi(Complex z)

Compute the value of the complex hyperbolic sine integral Shi(z) and complex hyperbolic cosine integral Chi(z).

Parameters:

z -

Returns:

{ Shi(z), Chi(z) }

en

public static Complex en(Complex n,
 Complex z)

Compute the value of the complex generalised exponential integral E_n(z). Uses A&S 5.1.45 E_n(z) = z^(n-1) * uiGamma(1-n, z) to perform computation.

Parameters:

n -

z -

Returns:

E_n(z)

hurwitzZeta

public static Complex hurwitzZeta(Complex s,
 Complex a)

Compute the Hurwitz zeta function of complex arguments.

Parameters:

s -

a -

Returns:

zeta(s, a)

dilog

public static Complex dilog(Complex z)

Compute the complex dilogarithm of z.

Parameters:

z -

Returns:

dilog(z)

spence

public static Complex spence(Complex z)

Compute the Spence function of z.

Parameters:

z -

Returns:

spence(z)

polylog

public static Complex polylog(Complex s,
 Complex z)

Compute the polylogarithm of s and z.

Parameters:

s -

z -

Returns:

polylog(s, z)

Throws:

ArithmeticException - if the amount of numerical algorithm iterations is exceeded

lerchPhi

public static Complex lerchPhi(Complex z,
 Complex s,
 Complex a)

Compute the Lerch transcendent of z, s, and a. A few things to note:

• The implementation may overflow the stack for particularly large, negative values of Re(a). This is being worked on.
• The precision of the output may vary. Worst-case scenario revealed during non-extensive randomised trials is around 0.006% relative error

Parameters:

z -

s -

a -

Returns:

lerch(z, s, a)

lambertw

public static Complex lambertw(Complex z,
 long k)

Compute the specified branch of the complex Lambert W function of z.

Parameters:

z -

k -

Returns:

lambertw(z, k)

chop

public static Complex chop(Complex z)

Trim insignificant real/imaginary parts (below machine epsilon), round up numbers where the real/imaginary part is very close to an integer.

Parameters:

z -

Returns:

iround(z)

solidArea

public static double solidArea(Function<Double,Double> f,
 Function<Double,Double> df,
 double a,
 double b)

Compute the surface area of a solid of revolution created by rotating the function f(x) about the x-axis. The function f(x) must be continuous and differentiable on the interval [a, b]. The area is computed using the Gauss-Legendre quadrature as 2pi * integral of f(x) * sqrt(1 + df(x)^2) from a to b.

Parameters:

f - the function to rotate

df - the derivative of f

a - the lower bound of the interval

b - the upper bound of the interval

Returns:

the area of the solid of revolution

solidVolume

public static double solidVolume(Function<Double,Double> f,
 double a,
 double b)

Compute the volume of a solid of revolution created by rotating the function f(x) about the x-axis. The function f(x) must be continuous and differentiable on the interval [a, b]. The volume is computed using the Gauss-Legendre quadrature as pi * integral of f(x)^2 from a to b (the disk method).

Parameters:

f - the function to rotate

a - the lower bound of the interval

b - the upper bound of the interval

Returns:

the volume of the solid of revolution

integrateTanhSinhComplex

public static Complex[] integrateTanhSinhComplex(Function<Complex,Complex> f,
 Complex a,
 Complex b,
 int n,
 double eps)

Integrate a C -> C function using the Tanh-Sinh quadrature. Performs integration through a straight line contour from a to b.

Parameters:

f - the function to integrate

a - the lower bound of the interval

b - the upper bound of the interval

n - the quadrature degree

eps - the desired accuracy

Returns:

the integral of f from a to b

integrateGaussLegendreComplex

public static Complex integrateGaussLegendreComplex(Function<Complex,Complex> f,
 Complex a,
 Complex b,
 int n)

Integrate a C -> C function using the Gauss-Legendre quadrature. Performs integration through a straight line contour from a to b.

Parameters:

f - the function to integrate

a - the lower bound of the interval

b - the upper bound of the interval

n - the quadrature degree

Returns:

the integral of f from a to b

arcLength

public static double arcLength(Function<Double,Double> df,
 double a,
 double b)

Compute the arc length of a curve defined by the function f(x) and its derivative df(x) on the interval [a, b]. The arc length is computed using the Gauss-Legendre quadrature as the integral of sqrt(1 + df(x)^2) from a to b.

Parameters:

df - the derivative of f

a - the lower bound of the interval

b - the upper bound of the interval

Returns:

the arc length of the curve

legendreF

public static double legendreF(double phi,
 double k)

Compute the Legendre F elliptic integral defined by DLMF §19.2.4

Parameters:

phi -

k -

Returns:

legendreF(phi, k)

legendreE

public static double legendreE(double phi,
 double k)

Compute the Legendre E elliptic integral defined by DLMF §19.2.5

Parameters:

phi -

k -

Returns:

legendreE(phi, k)

legendreD

public static double legendreD(double phi,
 double k)

Compute the Legendre D elliptic integral defined by DLMF §19.2.6

Parameters:

phi -

k -

Returns:

legendreD(phi, k)

legendrePi

public static double legendrePi(double phi,
 double alpha,
 double k)

Compute the Legendre Pi elliptic integral defined by DLMF §19.2.7

Parameters:

phi -

alpha -

k -

Returns:

legendrePi(phi, alpha, k)

legendreF

public static Complex legendreF(Complex phi,
 Complex k)

Compute the Legendre F elliptic integral defined by DLMF §19.2.4

Parameters:

phi -

k -

Returns:

legendreF(phi, k)

legendreE

public static Complex legendreE(Complex phi,
 Complex k)

Compute the Legendre E elliptic integral defined by DLMF §19.2.5

Parameters:

phi -

k -

Returns:

legendreE(phi, k)

legendreD

public static Complex legendreD(Complex phi,
 Complex k)

Compute the Legendre D elliptic integral defined by DLMF §19.2.6

Parameters:

phi -

k -

Returns:

legendreD(phi, k)

legendrePi

public static Complex legendrePi(Complex phi,
 Complex alpha,
 Complex k)

Compute the Legendre Pi elliptic integral defined by DLMF §19.2.7

Parameters:

phi -

alpha -

k -

Returns:

legendrePi(phi, alpha, k)

landau

public static double landau(double x,
 double mpv,
 double sigma,
 boolean norm)

Compute the Landau function with specified most probable value and sigma value.

Parameters:

x - the value to evaluate the function at

mpv - the most probable value

sigma - width of the distribution

norm - whether to normalize the result

Returns:

landau(x, mpv, sigma, norm)

landau

public static double landau(double x)

Compute the Landau distribution function.

Parameters:

x - the value to evaluate the function at

Returns:

landau(x)

gammaP

public static double gammaP(double a,
 double x)

Compute the regularized Gamma P function.

Parameters:

a - order

x - argument

Returns:

gammaP(a, x)

Throws:

ArithmeticException - if arguments are outside of the domain or the iteration count is exceeded.

gammaQ

public static double gammaQ(double a,
 double x)

Compute the regularized Gamma Q function.

Parameters:

a - order

x - argument

Returns:

gammaQ(a, x)

Throws:

ArithmeticException - if arguments are outside of the domain or the iteration count is exceeded.

normQuantile

public static double normQuantile(double p)

Computes quantiles for standard normal distribution N(0, 1) at probability p.

Parameters:

p - probability between 0 and 1.

Returns:

quantile value

Throws:

IllegalArgumentException - if p is not between 0 and 1

chiSquaredQuantile

public static double chiSquaredQuantile(double p,
 double df)

Computes quantiles for chi-squared probability distribution at probability p .

Parameters:

p - probability between 0 and 1.

df - degrees of freedom

Returns:

quantile value

Throws:

IllegalArgumentException - if p is not between 0 and 1

integrateGaussLegendreReal

public static double integrateGaussLegendreReal(BiFunction<Double,Double,Double> f,
 double a,
 double b,
 double c,
 double d,
 int N)

Integrate a dyadic function over a finite interval [a,b] x [c,d] using the Gauss-Legendre quadrature. The number of intervals is given by N and the precision of the final result greatly depends on this parameter. The computation of an integral using the Gauss-Legendre quadrature involves caching the coefficients required to perform the integration depending on the value of the N parameter. This means that the first call to this method will be slower than subsequent calls with the same value of N. The coefficients are internally cached inside a ConcurrentHashMap.

Parameters:

f - function to integrate

a - lower bound

b - upper bound

c - lower bound

d - upper bound

N - number of intervals, N=6 tends to give a good approximation in most scenarios. N must be between 1 and 30.

Returns:

integral of f over [a,b] x [c,d]

Throws:

IllegalArgumentException - if N is not a positive integer

integrateGaussLegendreComplex

public static Complex integrateGaussLegendreComplex(BiFunction<Complex,Complex,Complex> f,
 Complex a,
 Complex b,
 Complex c,
 Complex d,
 int N)

Integrate a dyadic function over a finite interval [a,b] x [c,d] using the Gauss-Legendre quadrature. The number of intervals is given by N and the precision of the final result greatly depends on this parameter. The computation of an integral using the Gauss-Legendre quadrature involves caching the coefficients required to perform the integration depending on the value of the N parameter. This means that the first call to this method will be slower than subsequent calls with the same value of N. The coefficients are internally cached inside a ConcurrentHashMap.

Parameters:

f - function to integrate

a - lower bound

b - upper bound

c - lower bound

d - upper bound

N - number of intervals, N=6 tends to give a good approximation in most scenarios. N must be between 1 and 30.

Returns:

integral of f over [a,b] x [c,d]

Throws:

IllegalArgumentException - if N is not a positive integer

isPrime

public static boolean isPrime(int n)

Primality test for an integer in range [0, 2^31-1].

isPrime

public static boolean isPrime(long n)

Primality test for an unsigned integer in range [0, 2^64-1]. Notice that some values of n that will be interpreted as negative when signed may yield a truthy value, as the number is reinterpreted as unsigned.

nextPrime

public static int nextPrime(int n)

Find the next prime after n.

Throws:

IllegalArgumentException - if n is negative

ArithmeticException - if n is too large

nextPrime

public static long nextPrime(long n)

Find the next prime after (unsigned) n.

Throws:

ArithmeticException - if n is too large