java.lang.Object

org.apfloat.ApfloatMath

public class ApfloatMath extends Object

Various mathematical functions for arbitrary precision floating-point numbers.

Due to different types of round-off errors that can occur in the implementation, no guarantees about e.g. monotonicity are given for any of the methods.

Version:

1.14.0

Author:

Mikko Tommila

See Also:

Method Summary

Modifier and Type

Method

Description

static Apfloat

abs(Apfloat x)

Absolute value.

static Apfloat

acos(Apfloat x)

Inverse cosine.

static Apfloat

acosh(Apfloat x)

Inverse hyperbolic cosine.

static Apfloat

agm(Apfloat a, Apfloat b)

Arithmetic-geometric mean.

static Apfloat

airyAi(Apfloat x)

Airy function Ai.

static Apfloat

airyAiPrime(Apfloat x)

Derivative of the Airy function Ai.

static Apfloat

airyBi(Apfloat x)

Airy function Bi.

static Apfloat

airyBiPrime(Apfloat x)

Derivative of the Airy function Bi.

static Apfloat

asin(Apfloat x)

Inverse sine.

static Apfloat

asinh(Apfloat x)

Inverse hyperbolic sine.

static Apfloat

atan(Apfloat x)

Inverse tangent.

static Apfloat

atan2(Apfloat y, Apfloat x)

Converts cartesian coordinates to polar coordinates.

static Apfloat

atanh(Apfloat x)

Inverse hyperbolic tangent.

static Apfloat

bernoulliB(long n, Apfloat x)

Bernoulli polynomial.

static Apfloat

besselI(Apfloat ν, Apfloat x)

Modified Bessel function of the first kind.

static Apfloat

besselJ(Apfloat ν, Apfloat x)

Bessel function of the first kind.

static Apfloat

besselK(Apfloat ν, Apfloat x)

Modified Bessel function of the second kind.

static Apfloat

besselY(Apfloat ν, Apfloat x)

Bessel function of the second kind.

static Apfloat

beta(Apfloat a, Apfloat b)

Beta function.

static Apfloat

beta(Apfloat x, Apfloat a, Apfloat b)

Incomplete beta function.

static Apfloat

beta(Apfloat x1, Apfloat x2, Apfloat a, Apfloat b)

Generalized incomplete beta function.

static Apfloat

binomial(Apfloat n, Apfloat k)

Binomial coefficient.

static Apfloat

catalan(long precision)

Calculates Catalan's constant, G.

static Apfloat

catalan(long precision, int radix)

Calculates Catalan's constant, G.

static Apfloat

cbrt(Apfloat x)

Cube root.

static Apint

ceil(Apfloat x)

Ceiling function.

static Apfloat

chebyshevT(Apfloat ν, Apfloat x)

Chebyshev function of the first kind.

static Apfloat

chebyshevU(Apfloat ν, Apfloat x)

Chebyshev function of the second kind.

static Apint[]

continuedFraction(Apfloat x, int n)

Generates the first n terms in the continued fraction representation of x.

static Aprational[]

convergents(Apfloat x, int n)

Generates the first n convergents corresponding to the continued fraction of x.

static Apfloat

copySign(Apfloat x, Apfloat y)

Copy sign from one argument to another.

static Apfloat

cos(Apfloat x)

Cosine.

static Apfloat

cosh(Apfloat x)

Hyperbolic cosine.

static Apfloat

coshIntegral(Apfloat x)

Hyperbolic cosine integral.

static Apfloat

cosIntegral(Apfloat x)

Cosine integral.

static Apfloat

digamma(Apfloat x)

Digamma function.

static Apfloat

e(long precision)

Calculates e.

static Apfloat

e(long precision, int radix)

Calculates e.

static Apfloat

ellipticE(Apfloat x)

Complete elliptic integral of the second kind.

static Apfloat

ellipticK(Apfloat x)

Complete elliptic integral of the first kind.

static Apfloat

erf(Apfloat x)

Error function.

static Apfloat

erfc(Apfloat x)

Complementary error function.

static Apfloat

erfi(Apfloat x)

Imaginary error function.

static Apfloat

euler(long precision)

Calculates γ, the Euler-Mascheroni constant.

static Apfloat

euler(long precision, int radix)

Calculates γ, the Euler-Mascheroni constant.

static Apfloat

eulerE(long n, Apfloat x)

Euler polynomial.

static Apfloat

exp(Apfloat x)

Exponent function.

static Apfloat

expIntegralE(Apfloat ν, Apfloat x)

Exponential integral E.

static Apfloat

expIntegralEi(Apfloat x)

Exponential integral Ei.

static Apfloat

fibonacci(Apfloat ν, Apfloat x)

Fibonacci function.

static Apint

floor(Apfloat x)

Floor function.

static Apfloat

fmod(Apfloat x, Apfloat y)

Returns x modulo y.

static Apfloat

frac(Apfloat x)

Extracts fractional part.

static Apfloat

fresnelC(Apfloat x)

Fresnel integral C.

static Apfloat

fresnelS(Apfloat x)

Fresnel integral S.

static Apfloat

gamma(Apfloat x)

Gamma function.

static Apfloat

gamma(Apfloat a, Apfloat x)

Incomplete gamma function.

static Apfloat

gamma(Apfloat a, Apfloat x0, Apfloat x1)

Generalized incomplete gamma function.

static Apfloat

gegenbauerC(Apfloat ν, Apfloat x)

Renormalized Gegenbauer function.

static Apfloat

gegenbauerC(Apfloat ν, Apfloat λ, Apfloat x)

Gegenbauer function.

static Apfloat

glaisher(long precision)

Calculates the Glaisher‐Kinkelin constant, A.

static Apfloat

glaisher(long precision, int radix)

Calculates the Glaisher‐Kinkelin constant, A.

static Apfloat

harmonicNumber(Apfloat x)

Harmonic number.

static Apfloat

harmonicNumber(Apfloat x, Apfloat r)

Generalized harmonic number.

static Apfloat

hermiteH(Apfloat ν, Apfloat x)

Hermite function.

static Apfloat

hypergeometric0F1(Apfloat a, Apfloat x)

Confluent hypergeometric function ₀F₁.

static Apfloat

hypergeometric0F1Regularized(Apfloat a, Apfloat x)

Regularized confluent hypergeometric function ₀F̃₁.

static Apfloat

hypergeometric1F1(Apfloat a, Apfloat b, Apfloat x)

Kummer confluent hypergeometric function ₁F₁.

static Apfloat

hypergeometric1F1Regularized(Apfloat a, Apfloat b, Apfloat x)

Regularized Kummer confluent hypergeometric function ₁F̃₁.

static Apfloat

hypergeometric2F1(Apfloat a, Apfloat b, Apfloat c, Apfloat x)

Hypergeometric function ₂F₁.

static Apfloat

hypergeometric2F1Regularized(Apfloat a, Apfloat b, Apfloat c, Apfloat x)

Regularized hypergeometric function ₂F̃₁.

static Apfloat

hypergeometricU(Apfloat a, Apfloat b, Apfloat x)

Tricomi's confluent hypergeometric function U.

static Apfloat

inverseErf(Apfloat x)

Inverse error function.

static Apfloat

inverseErfc(Apfloat x)

Inverse complementary error function.

static Apfloat

inverseRoot(Apfloat x, long n)

Inverse positive integer root.

static Apfloat

inverseRoot(Apfloat x, long n, long targetPrecision)

Inverse positive integer root.

static Apfloat

inverseRoot(Apfloat x, long n, long targetPrecision, Apfloat initialGuess)

Inverse positive integer root.

static Apfloat

inverseRoot(Apfloat x, long n, long targetPrecision, Apfloat initialGuess, long initialPrecision)

Inverse positive integer root.

static Apfloat

jacobiP(Apfloat ν, Apfloat a, Apfloat b, Apfloat x)

Jacobi function.

static Apfloat

khinchin(long precision)

Calculates Khinchin's constant, K.Uses the default radix.

static Apfloat

khinchin(long precision, int radix)

Calculates Khinchin's constant, K.

static Apfloat

laguerreL(Apfloat ν, Apfloat x)

Laguerre function.

static Apfloat

laguerreL(Apfloat ν, Apfloat λ, Apfloat x)

Generalized Laguerre function.

static Apfloat

legendreP(Apfloat ν, Apfloat x)

Legendre function.

static Apfloat

legendreP(Apfloat ν, Apfloat μ, Apfloat x)

Associated Legendre function of the first kind.

static Apfloat

legendreQ(Apfloat ν, Apfloat x)

Legendre function of the second kind.

static Apfloat

legendreQ(Apfloat ν, Apfloat μ, Apfloat x)

Associated Legendre function of the second kind.

static Apfloat

log(Apfloat x)

Natural logarithm.

static Apfloat

log(Apfloat x, Apfloat b)

Logarithm in arbitrary base.

static Apfloat

logGamma(Apfloat x)

Logarithm of the gamma function.

static Apfloat

logIntegral(Apfloat x)

Logarithmic integral.

static Apfloat

logisticSigmoid(Apfloat x)

Logistic sigmoid.

static Apfloat

logRadix(long precision, int radix)

Gets or calculates logarithm of a radix to required precision.

static Apfloat

max(Apfloat x, Apfloat y)

Returns the greater of the two values.

static Apfloat

min(Apfloat x, Apfloat y)

Returns the smaller of the two values.

static Apfloat[]

modf(Apfloat x)

Split to integer and fractional parts.

static Apfloat

multiplyAdd(Apfloat a, Apfloat b, Apfloat c, Apfloat d)

Fused multiply-add.

static Apfloat

multiplySubtract(Apfloat a, Apfloat b, Apfloat c, Apfloat d)

Fused multiply-subtract.

static Apfloat

negate(Apfloat x)

Deprecated.
Use Apfloat.negate().

static Apfloat

nextAfter(Apfloat start, Apfloat direction)

Returns the number adjacent to the first argument in the direction of the second argument, considering the scale and precision of the first argument.

static Apfloat

nextDown(Apfloat x)

Returns the number adjacent to the argument in the direction of negative infinity, considering the scale and precision of the argument.

static Apfloat

nextUp(Apfloat x)

Returns the number adjacent to the argument in the direction of positive infinity, considering the scale and precision of the argument.

static Apfloat

pi(long precision)

Calculates π.

static Apfloat

pi(long precision, int radix)

Calculates π.

static Apfloat

pochhammer(Apfloat x, Apfloat n)

Pochhammer symbol.

static Apfloat

polygamma(long n, Apfloat x)

Polygamma function.

static Apfloat

polylog(Apfloat ν, Apfloat x)

Polylogarithm.

static Apfloat

pow(Apfloat x, long n)

Integer power.

static Apfloat

pow(Apfloat x, Apfloat y)

Arbitrary power.

static Apfloat

product(Apfloat... x)

Product of numbers.

static Apfloat

random(long digits)

Generates a random number.

static Apfloat

random(long digits, int radix)

Generates a random number.

static Apfloat

randomGaussian(long digits)

Generates a random, Gaussian ("normally") distributed number value with mean 0 and standard deviation 1.

static Apfloat

randomGaussian(long digits, int radix)

Generates a random, Gaussian ("normally") distributed number value with mean 0 and standard deviation 1.

static Apfloat

root(Apfloat x, long n)

Positive integer root.

static Apfloat

round(Apfloat x, long precision, RoundingMode roundingMode)

Deprecated.
Use roundToPrecision(Apfloat,long,RoundingMode).

static Apint

roundToInteger(Apfloat x, RoundingMode roundingMode)

Rounds x to integer using the specified rounding mode.

static Apfloat

roundToMultiple(Apfloat x, Apfloat y, RoundingMode roundingMode)

Rounds x to the nearest multiple of y using the specified rounding mode.

static Apfloat

roundToPlaces(Apfloat x, long places, RoundingMode roundingMode)

Rounds x to the specified number of places using the specified rounding mode.

static Apfloat

roundToPrecision(Apfloat x, long precision, RoundingMode roundingMode)

Rounds the given number to the specified precision with the specified rounding mode.

static Apfloat

scale(Apfloat x, long scale)

Multiply by a power of the radix.

static Apfloat

sin(Apfloat x)

Sine.

static Apfloat

sinc(Apfloat x)

Sinc.

static Apfloat

sinh(Apfloat x)

Hyperbolic sine.

static Apfloat

sinhIntegral(Apfloat x)

Hyperbolic sine integral.

static Apfloat

sinIntegral(Apfloat x)

Sine integral.

static Apfloat

sqrt(Apfloat x)

Square root.

static Apfloat

sum(Apfloat... x)

Sum of numbers.

static Apfloat

tan(Apfloat x)

Tangent.

static Apfloat

tanh(Apfloat x)

Hyperbolic tangent.

static Apfloat

toDegrees(Apfloat x)

Converts an angle measured in radians to degrees.

static Apfloat

toRadians(Apfloat x)

Converts an angle measured in degrees to radians.

static Apint

truncate(Apfloat x)

Truncates fractional part.

static Apfloat

ulp(Apfloat x)

Returns the unit in the last place of the argument, considering the scale and precision.

static Apfloat

w(Apfloat x)

Lambert W function.

static Apfloat

zeta(Apfloat s)

Riemann zeta function.

static Apfloat

zeta(Apfloat s, Apfloat a)

Hurwitz zeta function.

Methods inherited from class java.lang.Object
clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Method Details
- pow
  
  public static Apfloat pow(Apfloat x, long n) throws ArithmeticException, ApfloatRuntimeException
  
  Integer power.
  
  Parameters:
  
  x - Base of the power operator.
  
  n - Exponent of the power operator.
  
  Returns:
  
  x to the n:th power, that is xⁿ.
  
  Throws:
  
  ArithmeticException - If both x and n are zero.
  
  ApfloatRuntimeException
- sqrt
  
  public static Apfloat sqrt(Apfloat x) throws ArithmeticException, ApfloatRuntimeException
  
  Square root.
  
  Parameters:
  
  x - The argument.
  
  Returns:
  
  Square root of x.
  
  Throws:
  
  ArithmeticException - If x is negative.
  
  ApfloatRuntimeException
- cbrt
  
  public static Apfloat cbrt(Apfloat x) throws ApfloatRuntimeException
  
  Cube root.
  
  Parameters:
  
  x - The argument.
  
  Returns:
  
  Cube root of x.
  
  Throws:
  
  ApfloatRuntimeException
- root
  
  public static Apfloat root(Apfloat x, long n) throws ArithmeticException, ApfloatRuntimeException
  
  Positive integer root.
  
  Parameters:
  
  x - The argument.
  
  n - Which root to take.
  
  Returns:
  
  n:th root of x, that is x^1/n.
  
  Throws:
  
  ArithmeticException - If n is zero, or x is negative and n is even.
  
  ApfloatRuntimeException
- inverseRoot
  
  public static Apfloat inverseRoot(Apfloat x, long n) throws ArithmeticException, ApfloatRuntimeException
  
  Inverse positive integer root.
  
  Parameters:
  
  x - The argument.
  
  n - Which inverse root to take.
  
  Returns:
  
  Inverse n:th root of x, that is x^-1/n.
  
  Throws:
  
  ArithmeticException - If x or n is zero, or x is negative and n is even.
  
  ApfloatRuntimeException
- inverseRoot
  
  public static Apfloat inverseRoot(Apfloat x, long n, long targetPrecision) throws IllegalArgumentException, ArithmeticException, ApfloatRuntimeException
  
  Inverse positive integer root.
  
  Parameters:
  
  x - The argument.
  
  n - Which inverse root to take.
  
  targetPrecision - Precision of the desired result.
  
  Returns:
  
  Inverse n:th root of x, that is x^-1/n.
  
  Throws:
  
  IllegalArgumentException - If targetPrecision <= 0.
  
  ArithmeticException - If x or n is zero, or x is negative and n is even.
  
  ApfloatRuntimeException
- inverseRoot
  
  public static Apfloat inverseRoot(Apfloat x, long n, long targetPrecision, Apfloat initialGuess) throws IllegalArgumentException, ArithmeticException, ApfloatRuntimeException
  
  Inverse positive integer root.
  
  Parameters:
  
  x - The argument.
  
  n - Which inverse root to take.
  
  targetPrecision - Precision of the desired result.
  
  initialGuess - Initial guess for the result value, or null if none is available.
  
  Returns:
  
  Inverse n:th root of x, that is x^-1/n.
  
  Throws:
  
  IllegalArgumentException - If targetPrecision <= 0.
  
  ArithmeticException - If x or n is zero, or x is negative and n is even.
  
  ApfloatRuntimeException
- inverseRoot
  
  public static Apfloat inverseRoot(Apfloat x, long n, long targetPrecision, Apfloat initialGuess, long initialPrecision) throws IllegalArgumentException, ArithmeticException, ApfloatRuntimeException
  
  Inverse positive integer root.
  This method is the basis for most of apfloat's non-elementary operations. It is used e.g. in Apfloat.divide(Apfloat), sqrt(Apfloat) and root(Apfloat,long).
  
  Parameters:
  
  x - The argument.
  
  n - Which inverse root to take.
  
  targetPrecision - Precision of the desired result.
  
  initialGuess - Initial guess for the result value, or null if none is available.
  
  initialPrecision - Precision of the initial guess, if available.
  
  Returns:
  
  Inverse n:th root of x, that is x^-1/n.
  
  Throws:
  
  IllegalArgumentException - If targetPrecision <= 0 or initialPrecision <= 0.
  
  ArithmeticException - If x or n is zero, or x is negative and n is even.
  
  ApfloatRuntimeException
- floor
  
  public static Apint floor(Apfloat x) throws ApfloatRuntimeException
  
  Floor function. Returns the largest (closest to positive infinity) value that is not greater than the argument and is equal to a mathematical integer.
  
  Parameters:
  
  x - The argument.
  
  Returns:
  
  x rounded towards negative infinity.
  
  Throws:
  
  ApfloatRuntimeException
- ceil
  
  public static Apint ceil(Apfloat x) throws ApfloatRuntimeException
  
  Ceiling function. Returns the smallest (closest to negative infinity) value that is not less than the argument and is equal to a mathematical integer.
  
  Parameters:
  
  x - The argument.
  
  Returns:
  
  x rounded towards positive infinity.
  
  Throws:
  
  ApfloatRuntimeException
- truncate
  
  public static Apint truncate(Apfloat x) throws ApfloatRuntimeException
  
  Truncates fractional part.
  
  Parameters:
  
  x - The argument.
  
  Returns:
  
  x rounded towards zero.
  
  Throws:
  
  ApfloatRuntimeException
- frac
  
  public static Apfloat frac(Apfloat x) throws ApfloatRuntimeException
  
  Extracts fractional part.
  
  Parameters:
  
  x - The argument.
  
  Returns:
  
  The fractional part of x.
  
  Throws:
  
  ApfloatRuntimeException
  
  Since:
  
  1.7.0
- round
  
  @Deprecated public static Apfloat round(Apfloat x, long precision, RoundingMode roundingMode) throws IllegalArgumentException, ArithmeticException, ApfloatRuntimeException
  
  Deprecated.
  Use roundToPrecision(Apfloat,long,RoundingMode).
  
  Rounds the given number to the specified precision with the specified rounding mode.
  
  Parameters:
  
  x - The number to round.
  
  precision - The precision to round to.
  
  roundingMode - The rounding mode to use.
  
  Returns:
  
  The rounded number.
  
  Throws:
  
  IllegalArgumentException - If precision is less than zero or zero.
  
  ArithmeticException - If rounding is necessary (result is not exact) and rounding mode is RoundingMode.UNNECESSARY.
  
  ApfloatRuntimeException
  
  Since:
  
  1.7.0
- roundToPrecision
  
  public static Apfloat roundToPrecision(Apfloat x, long precision, RoundingMode roundingMode) throws IllegalArgumentException, ArithmeticException, ApfloatRuntimeException
  
  Rounds the given number to the specified precision with the specified rounding mode.
  
  Parameters:
  
  x - The number to round.
  
  precision - The precision to round to.
  
  roundingMode - The rounding mode to use.
  
  Returns:
  
  The rounded number.
  
  Throws:
  
  IllegalArgumentException - If precision is less than zero or zero.
  
  ArithmeticException - If rounding is necessary (result is not exact) and rounding mode is RoundingMode.UNNECESSARY.
  
  ApfloatRuntimeException
  
  Since:
  
  1.11.0
- roundToInteger
  
  public static Apint roundToInteger(Apfloat x, RoundingMode roundingMode) throws IllegalArgumentException, ArithmeticException, ApfloatRuntimeException
  
  Rounds x to integer using the specified rounding mode.
  
  Parameters:
  
  x - The number to round.
  
  roundingMode - The rounding mode to use.
  
  Returns:
  
  The rounded number.
  
  Throws:
  
  ArithmeticException - If rounding is necessary (result is not exact) and rounding mode is RoundingMode.UNNECESSARY.
  
  IllegalArgumentException
  
  ApfloatRuntimeException
  
  Since:
  
  1.11.0
- roundToPlaces
  
  public static Apfloat roundToPlaces(Apfloat x, long places, RoundingMode roundingMode) throws IllegalArgumentException, ArithmeticException, ApfloatRuntimeException
  
  Rounds x to the specified number of places using the specified rounding mode.
  
  Parameters:
  
  x - The number to round.
  
  places - The number of places to round to (in base 10, the number of decimal places).
  
  roundingMode - The rounding mode to use.
  
  Returns:
  
  The rounded number.
  
  Throws:
  
  ArithmeticException - If rounding is necessary (result is not exact) and rounding mode is RoundingMode.UNNECESSARY.
  
  IllegalArgumentException
  
  ApfloatRuntimeException
  
  Since:
  
  1.11.0
- roundToMultiple
  
  public static Apfloat roundToMultiple(Apfloat x, Apfloat y, RoundingMode roundingMode) throws IllegalArgumentException, ArithmeticException, ApfloatRuntimeException
  
  Rounds x to the nearest multiple of y using the specified rounding mode.
  
  Parameters:
  
  x - The number to round.
  
  y - The integer multiple to round to.
  
  roundingMode - The rounding mode to use.
  
  Returns:
  
  The rounded number.
  
  Throws:
  
  ArithmeticException - If rounding is necessary (result is not exact) and rounding mode is RoundingMode.UNNECESSARY.
  
  IllegalArgumentException
  
  ApfloatRuntimeException
  
  Since:
  
  1.11.0
- negate
  
  @Deprecated public static Apfloat negate(Apfloat x) throws ApfloatRuntimeException
  
  Deprecated.
  Use Apfloat.negate().
  
  Returns an apfloat whose value is -x.
  
  Parameters:
  
  x - The argument.
  
  Returns:
  
  -x.
  
  Throws:
  
  ApfloatRuntimeException
- abs
  
  public static Apfloat abs(Apfloat x) throws ApfloatRuntimeException
  
  Absolute value.
  
  Parameters:
  
  x - The argument.
  
  Returns:
  
  Absolute value of x.
  
  Throws:
  
  ApfloatRuntimeException
- copySign
  
  public static Apfloat copySign(Apfloat x, Apfloat y) throws ApfloatRuntimeException
  
  Copy sign from one argument to another.
  
  Parameters:
  
  x - The value whose sign is to be adjusted.
  
  y - The value whose sign is to be used.
  
  Returns:
  
  x with its sign changed to match the sign of y.
  
  Throws:
  
  ApfloatRuntimeException
  
  Since:
  
  1.1
- scale
  
  public static Apfloat scale(Apfloat x, long scale) throws ApfloatRuntimeException
  
  Multiply by a power of the radix.
  
  Parameters:
  
  x - The argument.
  
  scale - The scaling factor.
  
  Returns:
  
  x * x.radix()^scale.
  
  Throws:
  
  ApfloatRuntimeException
- modf
  
  public static Apfloat[] modf(Apfloat x) throws ApfloatRuntimeException
  
  Split to integer and fractional parts. The integer part is simply i = floor(x). For the fractional part f the following is always true:
  0 <= f < 1
  
  Parameters:
  
  x - The argument.
  
  Returns:
  
  An array of two apfloats, [i, f], the first being the integer part and the last being the fractional part.
  
  Throws:
  
  ApfloatRuntimeException
- fmod
  
  public static Apfloat fmod(Apfloat x, Apfloat y) throws ApfloatRuntimeException
  
  Returns x modulo y.
  This function calculates the remainder f of x / y such that x = i * y + f, where i is an integer, f has the same sign as x, and the absolute value of f is less than the absolute value of y.
  If y is zero, then zero is returned.
  
  Parameters:
  
  x - The dividend.
  
  y - The divisor.
  
  Returns:
  
  The remainder when x is divided by y.
  
  Throws:
  
  ApfloatRuntimeException
- multiplyAdd
  
  public static Apfloat multiplyAdd(Apfloat a, Apfloat b, Apfloat c, Apfloat d) throws ApfloatRuntimeException
  
  Fused multiply-add. Calculates a * b + c * d so that the precision used in the multiplications is only what is needed for the end result. Performance can this way be better than by calculating a.multiply(b).add(c.multiply(d)).
  
  Parameters:
  
  a - First argument.
  
  b - Second argument.
  
  c - Third argument.
  
  d - Fourth argument.
  
  Returns:
  
  a * b + c * d.
  
  Throws:
  
  ApfloatRuntimeException
- multiplySubtract
  
  public static Apfloat multiplySubtract(Apfloat a, Apfloat b, Apfloat c, Apfloat d) throws ApfloatRuntimeException
  
  Fused multiply-subtract. Calculates a * b - c * d so that the precision used in the multiplications is only what is needed for the end result. Performance can this way be better than by calculating a.multiply(b).subtract(c.multiply(d)).
  
  Parameters:
  
  a - First argument.
  
  b - Second argument.
  
  c - Third argument.
  
  d - Fourth argument.
  
  Returns:
  
  a * b - c * d.
  
  Throws:
  
  ApfloatRuntimeException
- agm
  
  public static Apfloat agm(Apfloat a, Apfloat b) throws ApfloatRuntimeException
  
  Arithmetic-geometric mean.
  
  Parameters:
  
  a - First argument.
  
  b - Second argument.
  
  Returns:
  
  Arithmetic-geometric mean of a and b.
  
  Throws:
  
  ApfloatRuntimeException
- pi
  
  public static Apfloat pi(long precision) throws IllegalArgumentException, NumberFormatException, ApfloatRuntimeException
  
  Calculates π. Uses default radix.
  
  Parameters:
  
  precision - Number of digits of π to calculate.
  
  Returns:
  
  π accurate to precision digits, in the default radix.
  
  Throws:
  
  NumberFormatException - If the default radix is not valid.
  
  IllegalArgumentException - In case the precision is invalid.
  
  ApfloatRuntimeException
- pi
  
  public static Apfloat pi(long precision, int radix) throws IllegalArgumentException, NumberFormatException, ApfloatRuntimeException
  
  Calculates π.
  
  Parameters:
  
  precision - Number of digits of π to calculate.
  
  radix - The radix in which the number should be presented.
  
  Returns:
  
  π accurate to precision digits, in base radix.
  
  Throws:
  
  NumberFormatException - If the radix is not valid.
  
  IllegalArgumentException - In case the precision is invalid.
  
  ApfloatRuntimeException
- log
  
  public static Apfloat log(Apfloat x) throws ArithmeticException, ApfloatRuntimeException
  
  Natural logarithm.
  The logarithm is calculated using the arithmetic-geometric mean. See the Borweins' book for the formula.
  
  Parameters:
  
  x - The argument.
  
  Returns:
  
  Natural logarithm of x.
  
  Throws:
  
  ArithmeticException - If x <= 0.
  
  ApfloatRuntimeException
- log
  
  public static Apfloat log(Apfloat x, Apfloat b) throws ArithmeticException, ApfloatRuntimeException
  
  Logarithm in arbitrary base.
  The logarithm is calculated using the arithmetic-geometric mean. See the Borweins' book for the formula.
  
  Parameters:
  
  x - The argument.
  
  b - The base.
  
  Returns:
  
  Base-b logarithm of x.
  
  Throws:
  
  ArithmeticException - If x <= 0 or b <= 0.
  
  ApfloatRuntimeException
  
  Since:
  
  1.6
- logRadix
  
  public static Apfloat logRadix(long precision, int radix) throws ApfloatRuntimeException
  
  Gets or calculates logarithm of a radix to required precision. The calculated value is stored in a cache for later usage.
  
  Parameters:
  
  precision - The needed precision.
  
  radix - The radix.
  
  Returns:
  
  Natural logarithm of radix to the specified precision.
  
  Throws:
  
  NumberFormatException - If the radix is invalid.
  
  ApfloatRuntimeException
- exp
  
  public static Apfloat exp(Apfloat x) throws ApfloatRuntimeException
  
  Exponent function. Calculated using Newton's iteration for the inverse of logarithm.
  
  Parameters:
  
  x - The argument.
  
  Returns:
  
  e^x.
  
  Throws:
  
  ApfloatRuntimeException
- pow
  
  public static Apfloat pow(Apfloat x, Apfloat y) throws ArithmeticException, ApfloatRuntimeException
  
  Arbitrary power. Calculated using log() and exp().
  
  Parameters:
  
  x - The base.
  
  y - The exponent.
  
  Returns:
  
  x^y.
  
  Throws:
  
  ArithmeticException - If both x and y are zero, or x is negative and y is not an integer.
  
  ApfloatRuntimeException
- acosh
  
  public static Apfloat acosh(Apfloat x) throws ArithmeticException, ApfloatRuntimeException
  
  Inverse hyperbolic cosine. Calculated using log().
  
  Parameters:
  
  x - The argument.
  
  Returns:
  
  Inverse hyperbolic cosine of x.
  
  Throws:
  
  ArithmeticException - If x < 1.
  
  ApfloatRuntimeException
- asinh
  
  public static Apfloat asinh(Apfloat x) throws ApfloatRuntimeException
  
  Inverse hyperbolic sine. Calculated using log().
  
  Parameters:
  
  x - The argument.
  
  Returns:
  
  Inverse hyperbolic sine of x.
  
  Throws:
  
  ApfloatRuntimeException
- atanh
  
  public static Apfloat atanh(Apfloat x) throws ArithmeticException, ApfloatRuntimeException
  
  Inverse hyperbolic tangent. Calculated using log().
  
  Parameters:
  
  x - The argument.
  
  Returns:
  
  Inverse hyperbolic tangent of x.
  
  Throws:
  
  ArithmeticException - If abs(x) >= 1.
  
  ApfloatRuntimeException
- cosh
  
  public static Apfloat cosh(Apfloat x) throws ApfloatRuntimeException
  
  Hyperbolic cosine. Calculated using exp().
  
  Parameters:
  
  x - The argument.
  
  Returns:
  
  Hyperbolic cosine of x.
  
  Throws:
  
  ApfloatRuntimeException
- sinh
  
  public static Apfloat sinh(Apfloat x) throws ApfloatRuntimeException
  
  Hyperbolic sine. Calculated using exp().
  
  Parameters:
  
  x - The argument.
  
  Returns:
  
  Hyperbolic sine of x.
  
  Throws:
  
  ApfloatRuntimeException
- tanh
  
  public static Apfloat tanh(Apfloat x) throws ApfloatRuntimeException
  
  Hyperbolic tangent. Calculated using exp().
  
  Parameters:
  
  x - The argument.
  
  Returns:
  
  Hyperbolic tangent of x.
  
  Throws:
  
  ApfloatRuntimeException
- acos
  
  public static Apfloat acos(Apfloat x) throws ArithmeticException, ApfloatRuntimeException
  
  Inverse cosine. Calculated using complex functions.
  
  Parameters:
  
  x - The argument.
  
  Returns:
  
  Inverse cosine of x.
  
  Throws:
  
  ArithmeticException - If abs(x) > 1.
  
  ApfloatRuntimeException
- asin
  
  public static Apfloat asin(Apfloat x) throws ArithmeticException, ApfloatRuntimeException
  
  Inverse sine. Calculated using complex functions.
  
  Parameters:
  
  x - The argument.
  
  Returns:
  
  Inverse sine of x.
  
  Throws:
  
  ArithmeticException - If abs(x) > 1.
  
  ApfloatRuntimeException
- atan
  
  public static Apfloat atan(Apfloat x) throws ApfloatRuntimeException
  
  Inverse tangent. Calculated using complex functions.
  
  Parameters:
  
  x - The argument.
  
  Returns:
  
  Inverse tangent of x.
  
  Throws:
  
  ApfloatRuntimeException
- atan2
  
  public static Apfloat atan2(Apfloat y, Apfloat x) throws ArithmeticException, ApfloatRuntimeException
  
  Converts cartesian coordinates to polar coordinates. Calculated using complex functions.
  Computes the phase angle by computing an arc tangent of y/x in the range of -π < angle <= π.
  
  Parameters:
  
  y - The argument.
  
  x - The argument.
  
  Returns:
  
  The angle of the point (x, y) in the plane.
  
  Throws:
  
  ArithmeticException - If y and x are both zero.
  
  ApfloatRuntimeException
- cos
  
  public static Apfloat cos(Apfloat x) throws ApfloatRuntimeException
  
  Cosine. Calculated using complex functions.
  
  Parameters:
  
  x - The argument (in radians).
  
  Returns:
  
  Cosine of x.
  
  Throws:
  
  ApfloatRuntimeException
- sin
  
  public static Apfloat sin(Apfloat x) throws ApfloatRuntimeException
  
  Sine. Calculated using complex functions.
  
  Parameters:
  
  x - The argument (in radians).
  
  Returns:
  
  Sine of x.
  
  Throws:
  
  ApfloatRuntimeException
- tan
  
  public static Apfloat tan(Apfloat x) throws ArithmeticException, ApfloatRuntimeException
  
  Tangent. Calculated using complex functions.
  
  Parameters:
  
  x - The argument (in radians).
  
  Returns:
  
  Tangent of x.
  
  Throws:
  
  ArithmeticException - If x is π/2 + n π where n is an integer.
  
  ApfloatRuntimeException
- sinc
  
  public static Apfloat sinc(Apfloat x) throws ApfloatRuntimeException
  
  Sinc.
  
  Parameters:
  
  x - The argument.
  
  Returns:
  
  sinc(x)
  
  Throws:
  
  ApfloatRuntimeException
  
  Since:
  
  1.14.0
- w
  
  public static Apfloat w(Apfloat x) throws ArithmeticException, ApfloatRuntimeException
  
  Lambert W function. The W function gives the solution to the equation W e^W = x. Also known as the product logarithm.
  This function only gives the solution to the principal branch, W₀. For the real-valued W_-1 branch, use ApcomplexMath.w(Apcomplex,long).
  
  Parameters:
  
  x - The argument.
  
  Returns:
  
  W₀(x).
  
  Throws:
  
  ArithmeticException - If x is less than -1/e.
  
  ApfloatRuntimeException
  
  Since:
  
  1.8.0
- toDegrees
  
  public static Apfloat toDegrees(Apfloat x) throws ApfloatRuntimeException
  
  Converts an angle measured in radians to degrees.
  
  Parameters:
  
  x - The angle, in radians.
  
  Returns:
  
  The angle in degrees.
  
  Throws:
  
  ApfloatRuntimeException
  
  Since:
  
  1.8.0
- toRadians
  
  public static Apfloat toRadians(Apfloat x) throws ApfloatRuntimeException
  
  Converts an angle measured in degrees to radians.
  
  Parameters:
  
  x - The angle, in degrees.
  
  Returns:
  
  The angle in radians.
  
  Throws:
  
  ApfloatRuntimeException
  
  Since:
  
  1.8.0
- product
  
  public static Apfloat product(Apfloat... x) throws ApfloatRuntimeException
  
  Product of numbers. The precision used in the multiplications is only what is needed for the end result. This method may perform significantly better than simply multiplying the numbers sequentially.
  If there are no arguments, the return value is 1.
  
  Parameters:
  
  x - The argument(s).
  
  Returns:
  
  The product of the given numbers.
  
  Throws:
  
  ApfloatRuntimeException
  
  Since:
  
  1.3
- sum
  
  public static Apfloat sum(Apfloat... x) throws ApfloatRuntimeException
  
  Sum of numbers. The precision used in the additions is only what is needed for the end result. This method may perform significantly better than simply adding the numbers sequentially.
  If there are no arguments, the return value is 0.
  
  Parameters:
  
  x - The argument(s).
  
  Returns:
  
  The sum of the given numbers.
  
  Throws:
  
  ApfloatRuntimeException
  
  Since:
  
  1.3
- e
  
  public static Apfloat e(long precision) throws IllegalArgumentException, NumberFormatException, ApfloatRuntimeException
  
  Calculates e. Uses default radix.
  
  Parameters:
  
  precision - Number of digits of e to calculate.
  
  Returns:
  
  e accurate to precision digits, in the default radix.
  
  Throws:
  
  NumberFormatException - If the default radix is not valid.
  
  IllegalArgumentException - In case the precision is invalid.
  
  ApfloatRuntimeException
  
  Since:
  
  1.11.0
- e
  
  public static Apfloat e(long precision, int radix) throws IllegalArgumentException, NumberFormatException, ApfloatRuntimeException
  
  Calculates e.
  
  Parameters:
  
  precision - Number of digits of e to calculate.
  
  radix - The radix in which the number should be presented.
  
  Returns:
  
  e accurate to precision digits, in base radix.
  
  Throws:
  
  NumberFormatException - If the radix is not valid.
  
  IllegalArgumentException - In case the precision is invalid.
  
  ApfloatRuntimeException
  
  Since:
  
  1.11.0
- euler
  
  public static Apfloat euler(long precision) throws IllegalArgumentException, NumberFormatException, ApfloatRuntimeException
  
  Calculates γ, the Euler-Mascheroni constant. Uses default radix.
  
  Parameters:
  
  precision - Number of digits of γ to calculate.
  
  Returns:
  
  γ accurate to precision digits, in the default radix.
  
  Throws:
  
  NumberFormatException - If the default radix is not valid.
  
  IllegalArgumentException - In case the precision is invalid.
  
  ApfloatRuntimeException
  
  Since:
  
  1.10.0
- euler
  
  public static Apfloat euler(long precision, int radix) throws IllegalArgumentException, NumberFormatException, ApfloatRuntimeException
  
  Calculates γ, the Euler-Mascheroni constant.
  
  Parameters:
  
  precision - Number of digits of γ to calculate.
  
  radix - The radix in which the number should be presented.
  
  Returns:
  
  γ accurate to precision digits, in base radix.
  
  Throws:
  
  NumberFormatException - If the radix is not valid.
  
  IllegalArgumentException - In case the precision is invalid.
  
  ApfloatRuntimeException
  
  Since:
  
  1.10.0
- catalan
  
  public static Apfloat catalan(long precision) throws IllegalArgumentException, NumberFormatException, ApfloatRuntimeException
  
  Calculates Catalan's constant, G. Uses the default radix.
  
  Parameters:
  
  precision - Number of digits of G to calculate.
  
  Returns:
  
  G accurate to precision digits, in the default radix.
  
  Throws:
  
  NumberFormatException - If the default radix is not valid.
  
  IllegalArgumentException - In case the precision is invalid.
  
  ApfloatRuntimeException
  
  Since:
  
  1.11.0
  
  Implementation notes:
  
  This implementation is slow.
- catalan
  
  public static Apfloat catalan(long precision, int radix) throws IllegalArgumentException, NumberFormatException, ApfloatRuntimeException
  
  Calculates Catalan's constant, G. Uses the specified radix.
  
  Parameters:
  
  precision - Number of digits of G to calculate.
  
  radix - The radix in which the number should be presented.
  
  Returns:
  
  G accurate to precision digits, in base radix.
  
  Throws:
  
  NumberFormatException - If the radix is not valid.
  
  IllegalArgumentException - In case the precision is invalid.
  
  ApfloatRuntimeException
  
  Since:
  
  1.11.0
  
  Implementation notes:
  
  This implementation is slow.
- glaisher
  
  public static Apfloat glaisher(long precision) throws IllegalArgumentException, NumberFormatException, ApfloatRuntimeException
  
  Calculates the Glaisher‐Kinkelin constant, A. Uses the default radix.
  
  Parameters:
  
  precision - Number of digits of A to calculate.
  
  Returns:
  
  A accurate to precision digits, in the default radix.
  
  Throws:
  
  NumberFormatException - If the default radix is not valid.
  
  IllegalArgumentException - In case the precision is invalid.
  
  ApfloatRuntimeException
  
  Since:
  
  1.11.0
  
  Implementation notes:
  
  This implementation is slow. At the time of implementation no efficient algorithm is known for the Glaisher‐Kinkelin constant.
- glaisher
  
  public static Apfloat glaisher(long precision, int radix) throws IllegalArgumentException, NumberFormatException, ApfloatRuntimeException
  
  Calculates the Glaisher‐Kinkelin constant, A. Uses the specified radix.
  
  Parameters:
  
  precision - Number of digits of A to calculate.
  
  radix - The radix in which the number should be presented.
  
  Returns:
  
  A accurate to precision digits, in base radix.
  
  Throws:
  
  NumberFormatException - If the radix is not valid.
  
  IllegalArgumentException - In case the precision is invalid.
  
  ApfloatRuntimeException
  
  Since:
  
  1.11.0
  
  Implementation notes:
  
  This implementation is slow. At the time of implementation no efficient algorithm is known for the Glaisher‐Kinkelin constant.
- khinchin
  
  public static Apfloat khinchin(long precision) throws IllegalArgumentException, NumberFormatException, ApfloatRuntimeException
  
  Calculates Khinchin's constant, K.Uses the default radix.
  
  Parameters:
  
  precision - Number of digits of K to calculate.
  
  Returns:
  
  K accurate to precision digits, in the default radix.
  
  Throws:
  
  NumberFormatException - If the default radix is not valid.
  
  IllegalArgumentException - In case the precision is invalid.
  
  ApfloatRuntimeException
  
  Since:
  
  1.11.0
  
  Implementation notes:
  
  This implementation is slow. At the time of implementation no efficient algorithm is known for Khinchin's constant.
- khinchin
  
  public static Apfloat khinchin(long precision, int radix) throws IllegalArgumentException, NumberFormatException, ApfloatRuntimeException
  
  Calculates Khinchin's constant, K. Uses the specified radix.
  
  Parameters:
  
  precision - Number of digits of K to calculate.
  
  radix - The radix in which the number should be presented.
  
  Returns:
  
  K accurate to precision digits, in base radix.
  
  Throws:
  
  NumberFormatException - If the radix is not valid.
  
  IllegalArgumentException - In case the precision is invalid.
  
  ApfloatRuntimeException
  
  Since:
  
  1.11.0
  
  Implementation notes:
  
  This implementation is slow. At the time of implementation no efficient algorithm is known for Khinchin's constant.
- gamma
  
  public static Apfloat gamma(Apfloat x) throws ArithmeticException, ApfloatRuntimeException
  
  Gamma function.
  
  Parameters:
  
  x - The argument.
  
  Returns:
  
  Γ(x)
  
  Throws:
  
  ArithmeticException - If x is a nonpositive integer.
  
  ApfloatRuntimeException
  
  Since:
  
  1.9.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. The asymptotic complexity is at least O(n²log n) and it is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the gamma function.
- gamma
  
  public static Apfloat gamma(Apfloat a, Apfloat x) throws ArithmeticException, ApfloatRuntimeException
  
  Incomplete gamma function.
  
  Parameters:
  
  a - The first argument.
  
  x - The second argument.
  
  Returns:
  
  Γ(a, x)
  
  Throws:
  
  ArithmeticException - If a is not a positive integer and x is nonpositive.
  
  ApfloatRuntimeException
  
  Since:
  
  1.10.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. The asymptotic complexity is at least O(n²log n) and it is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the gamma function.
- gamma
  
  public static Apfloat gamma(Apfloat a, Apfloat x0, Apfloat x1) throws ArithmeticException, ApfloatRuntimeException
  
  Generalized incomplete gamma function.
  This function is defined as: Γ(a, x0, x1) = Γ(a, x0) - Γ(a, x1)
  The lower gamma function can be calculated with: γ(a, x) = Γ(a, 0, x)
  
  Parameters:
  
  a - The first argument.
  
  x0 - The second argument.
  
  x1 - The third argument.
  
  Returns:
  
  Γ(a, x0, x1)
  
  Throws:
  
  ArithmeticException - If a is not a positive integer and either x0 or x1 is nonpositive.
  
  ApfloatRuntimeException
  
  Since:
  
  1.10.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. The asymptotic complexity is at least O(n²log n) and it is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the gamma function.
- logGamma
  
  public static Apfloat logGamma(Apfloat x) throws ArithmeticException, ApfloatRuntimeException
  
  Logarithm of the gamma function.
  
  Parameters:
  
  x - The argument.
  
  Returns:
  
  logΓ(x)
  
  Throws:
  
  ArithmeticException - If x is nonpositive.
  
  ApfloatRuntimeException
  
  Since:
  
  1.11.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. The asymptotic complexity is at least O(n²log n) and it is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the gamma function.
- digamma
  
  public static Apfloat digamma(Apfloat x) throws ArithmeticException, ApfloatRuntimeException
  
  Digamma function.
  
  Parameters:
  
  x - The argument.
  
  Returns:
  
  ψ(x)
  
  Throws:
  
  ArithmeticException - If x is a nonpositive integer.
  
  ApfloatRuntimeException
  
  Since:
  
  1.11.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. The asymptotic complexity is at least O(n²log n) and it is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the digamma function.
- polygamma
  
  public static Apfloat polygamma(long n, Apfloat x) throws ArithmeticException, ApfloatRuntimeException
  
  Polygamma function.
  
  Parameters:
  
  n - The order.
  
  x - The argument.
  
  Returns:
  
  ψ⁽ⁿ⁾(x)
  
  Throws:
  
  ArithmeticException - If n is negative or x is a nonpositive integer.
  
  ApfloatRuntimeException
  
  Since:
  
  1.13.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. The asymptotic complexity is at least O(n²log n) and it is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the polygamma function.
- beta
  
  public static Apfloat beta(Apfloat a, Apfloat b) throws ArithmeticException, ApfloatRuntimeException
  
  Beta function.
  
  Parameters:
  
  a - The first argument.
  
  b - The second argument.
  
  Returns:
  
  B(a, b)
  
  Throws:
  
  ArithmeticException - If a or b is a nonpositive integer but a + b is not. Also if both a and b are nonpositive integers.
  
  ApfloatRuntimeException
  
  Since:
  
  1.13.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. The asymptotic complexity is at least O(n²log n) and it is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the beta function.
- beta
  
  public static Apfloat beta(Apfloat x, Apfloat a, Apfloat b) throws ArithmeticException, ApfloatRuntimeException
  
  Incomplete beta function.
  
  Parameters:
  
  x - The first argument.
  
  a - The second argument.
  
  b - The third argument.
  
  Returns:
  
  B_x(a, b)
  
  Throws:
  
  ArithmeticException - If a is a nonpositive integer or x is zero and a is nonpositive or x is negative and a is not an integer. Also if x > 1 and the result is not a polynomial.
  
  ApfloatRuntimeException
  
  Since:
  
  1.13.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. The asymptotic complexity is at least O(n²log n) and it is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- beta
  
  public static Apfloat beta(Apfloat x1, Apfloat x2, Apfloat a, Apfloat b) throws ArithmeticException, ApfloatRuntimeException
  
  Generalized incomplete beta function.
  
  Parameters:
  
  x1 - The first argument.
  
  x2 - The second argument.
  
  a - The third argument.
  
  b - The fourth argument.
  
  Returns:
  
  B_{(x1, x2)}(a, b)
  
  Throws:
  
  ArithmeticException - If a is a nonpositive integer or x1 or x2 is zero and a is nonpositive or x1 or x2 is negative and a is not an integer. Also if x1 > 1 or x2 > 1 and the result is not a polynomial.
  
  ApfloatRuntimeException
  
  Since:
  
  1.13.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. The asymptotic complexity is at least O(n²log n) and it is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- pochhammer
  
  public static Apfloat pochhammer(Apfloat x, Apfloat n) throws ArithmeticException, ApfloatRuntimeException
  
  Pochhammer symbol.
  
  Parameters:
  
  x - The first argument.
  
  n - The second argument.
  
  Returns:
  
  (x)_n
  
  Throws:
  
  ArithmeticException - If x + n is a nonpositive integer but x is not.
  
  ApfloatRuntimeException
  
  Since:
  
  1.13.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. The asymptotic complexity is at least O(n²log n) and it is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the pochhammer symbol.
- binomial
  
  public static Apfloat binomial(Apfloat n, Apfloat k) throws ArithmeticException, ApfloatRuntimeException
  
  Binomial coefficient. Calculated using the gamma(Apfloat) function.
  
  Parameters:
  
  n - The first argument.
  
  k - The second argument.
  
  Returns:
  
  $(\binom{n}{k})$
  
  Throws:
  
  ArithmeticException - If n is a negative integer and k is noninteger.
  
  ApfloatRuntimeException
  
  Since:
  
  1.11.0
- zeta
  
  public static Apfloat zeta(Apfloat s) throws ArithmeticException, ApfloatRuntimeException
  
  Riemann zeta function.
  
  Parameters:
  
  s - The argument.
  
  Returns:
  
  ζ(s)
  
  Throws:
  
  ArithmeticException - If s is 1.
  
  ApfloatRuntimeException
  
  Since:
  
  1.11.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few hundred digits. At the time of implementation no generic fast algorithm is known for the zeta function.
- zeta
  
  public static Apfloat zeta(Apfloat s, Apfloat a) throws ArithmeticException, ApfloatRuntimeException
  
  Hurwitz zeta function.
  
  Parameters:
  
  s - The first argument.
  
  a - The second argument.
  
  Returns:
  
  ζ(s, a)
  
  Throws:
  
  ArithmeticException - If s is 1 or if a is a nonpositive integer or if s is not an integer and a is nonpositive.
  
  ApfloatRuntimeException
  
  Since:
  
  1.11.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few hundred digits. At the time of implementation no generic fast algorithm is known for the zeta function.
- hypergeometric0F1
  
  public static Apfloat hypergeometric0F1(Apfloat a, Apfloat x) throws ArithmeticException, ApfloatRuntimeException
  
  Confluent hypergeometric function ₀F₁.
  
  Parameters:
  
  a - The first argument.
  
  x - The second argument.
  
  Returns:
  
  ₀F₁(; a; x)
  
  Throws:
  
  ArithmeticException - If the function value is not finite.
  
  ApfloatRuntimeException
  
  Since:
  
  1.11.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- hypergeometric0F1Regularized
  
  public static Apfloat hypergeometric0F1Regularized(Apfloat a, Apfloat x) throws ApfloatRuntimeException
  
  Regularized confluent hypergeometric function ₀F̃₁.
  
  Parameters:
  
  a - The first argument.
  
  x - The second argument.
  
  Returns:
  
  ₀F̃₁(; a; x)
  
  Throws:
  
  ApfloatRuntimeException
  
  Since:
  
  1.13.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- hypergeometric1F1
  
  public static Apfloat hypergeometric1F1(Apfloat a, Apfloat b, Apfloat x) throws ArithmeticException, ApfloatRuntimeException
  
  Kummer confluent hypergeometric function ₁F₁. Also known as the confluent hypergeometric function of the first kind.
  
  Parameters:
  
  a - The first argument.
  
  b - The second argument.
  
  x - The third argument.
  
  Returns:
  
  ₁F₁(a; b; x)
  
  Throws:
  
  ArithmeticException - If the function value is not finite.
  
  ApfloatRuntimeException
  
  Since:
  
  1.11.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- hypergeometric1F1Regularized
  
  public static Apfloat hypergeometric1F1Regularized(Apfloat a, Apfloat b, Apfloat x) throws ApfloatRuntimeException
  
  Regularized Kummer confluent hypergeometric function ₁F̃₁. Also known as the regularized confluent hypergeometric function of the first kind.
  
  Parameters:
  
  a - The first argument.
  
  b - The second argument.
  
  x - The third argument.
  
  Returns:
  
  ₁F̃₁(a; b; x)
  
  Throws:
  
  ApfloatRuntimeException
  
  Since:
  
  1.13.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- hypergeometric2F1
  
  public static Apfloat hypergeometric2F1(Apfloat a, Apfloat b, Apfloat c, Apfloat x) throws ArithmeticException, ApfloatRuntimeException
  
  Hypergeometric function ₂F₁. Also known as the Gaussian or ordinary hypergeometric function.
  
  Parameters:
  
  a - The first argument.
  
  b - The second argument.
  
  c - The third argument.
  
  x - The fourth argument.
  
  Returns:
  
  ₂F₁(a, b; c; x)
  
  Throws:
  
  ArithmeticException - If the function value is not finite or real.
  
  ApfloatRuntimeException
  
  Since:
  
  1.11.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- hypergeometric2F1Regularized
  
  public static Apfloat hypergeometric2F1Regularized(Apfloat a, Apfloat b, Apfloat c, Apfloat x) throws ApfloatRuntimeException
  
  Regularized hypergeometric function ₂F̃₁. Also known as the regularized Gaussian or ordinary hypergeometric function.
  
  Parameters:
  
  a - The first argument.
  
  b - The second argument.
  
  c - The third argument.
  
  x - The fourth argument.
  
  Returns:
  
  ₂F̃₁(a, b; c; x)
  
  Throws:
  
  ApfloatRuntimeException
  
  Since:
  
  1.13.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- hypergeometricU
  
  public static Apfloat hypergeometricU(Apfloat a, Apfloat b, Apfloat x) throws ArithmeticException, ApfloatRuntimeException
  
  Tricomi's confluent hypergeometric function U. Also known as the confluent hypergeometric function of the second kind.
  
  Parameters:
  
  a - The first argument.
  
  b - The second argument.
  
  x - The third argument.
  
  Returns:
  
  U(a, b, x)
  
  Throws:
  
  ArithmeticException - If the result would be complex or not finite.
  
  ApfloatRuntimeException
  
  Since:
  
  1.13.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- erf
  
  public static Apfloat erf(Apfloat x) throws ApfloatRuntimeException
  
  Error function.
  
  Parameters:
  
  x - The argument.
  
  Returns:
  
  erf(x)
  
  Throws:
  
  ApfloatRuntimeException
  
  Since:
  
  1.13.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- erfc
  
  public static Apfloat erfc(Apfloat x) throws ApfloatRuntimeException
  
  Complementary error function.
  
  Parameters:
  
  x - The argument.
  
  Returns:
  
  erfc(x)
  
  Throws:
  
  ApfloatRuntimeException
  
  Since:
  
  1.13.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- erfi
  
  public static Apfloat erfi(Apfloat x) throws ApfloatRuntimeException
  
  Imaginary error function.
  
  Parameters:
  
  x - The argument.
  
  Returns:
  
  erfi(x)
  
  Throws:
  
  ApfloatRuntimeException
  
  Since:
  
  1.13.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- inverseErf
  
  public static Apfloat inverseErf(Apfloat x) throws ArithmeticException, ApfloatRuntimeException
  
  Inverse error function.
  
  Parameters:
  
  x - The argument.
  
  Returns:
  
  erf⁻¹(x)
  
  Throws:
  
  ArithmeticException - If |x| is ≥ 1.
  
  ApfloatRuntimeException
  
  Since:
  
  1.13.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- inverseErfc
  
  public static Apfloat inverseErfc(Apfloat x) throws ArithmeticException, ApfloatRuntimeException
  
  Inverse complementary error function.
  
  Parameters:
  
  x - The argument.
  
  Returns:
  
  erfc⁻¹(x)
  
  Throws:
  
  ArithmeticException - If x is ≤ 0 or ≥ 2.
  
  ApfloatRuntimeException
  
  Since:
  
  1.13.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- fresnelS
  
  public static Apfloat fresnelS(Apfloat x) throws ApfloatRuntimeException
  
  Fresnel integral S.
  
  Parameters:
  
  x - The argument.
  
  Returns:
  
  S(x)
  
  Throws:
  
  ApfloatRuntimeException
  
  Since:
  
  1.13.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- fresnelC
  
  public static Apfloat fresnelC(Apfloat x) throws ApfloatRuntimeException
  
  Fresnel integral C.
  
  Parameters:
  
  x - The argument.
  
  Returns:
  
  C(x)
  
  Throws:
  
  ApfloatRuntimeException
  
  Since:
  
  1.13.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- expIntegralE
  
  public static Apfloat expIntegralE(Apfloat ν, Apfloat x) throws ArithmeticException, ApfloatRuntimeException
  
  Exponential integral E.
  
  Parameters:
  
  ν - The first argument.
  
  x - The second argument.
  
  Returns:
  
  E_ν(x)
  
  Throws:
  
  ArithmeticException - If ν is < 0 and x is zero or ν is nonzero and x is negative.
  
  ApfloatRuntimeException
  
  Since:
  
  1.13.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- expIntegralEi
  
  public static Apfloat expIntegralEi(Apfloat x) throws ArithmeticException, ApfloatRuntimeException
  
  Exponential integral Ei.
  
  Parameters:
  
  x - The argument.
  
  Returns:
  
  Ei(x)
  
  Throws:
  
  ArithmeticException - If x is zero.
  
  ApfloatRuntimeException
  
  Since:
  
  1.13.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- logIntegral
  
  public static Apfloat logIntegral(Apfloat x) throws ArithmeticException, ApfloatRuntimeException
  
  Logarithmic integral.
  
  Parameters:
  
  x - The argument.
  
  Returns:
  
  li(x)
  
  Throws:
  
  ArithmeticException - If x is nonpositive or 1.
  
  ApfloatRuntimeException
  
  Since:
  
  1.13.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- sinIntegral
  
  public static Apfloat sinIntegral(Apfloat x) throws ApfloatRuntimeException
  
  Sine integral.
  
  Parameters:
  
  x - The argument.
  
  Returns:
  
  Si(x)
  
  Throws:
  
  ApfloatRuntimeException
  
  Since:
  
  1.13.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- cosIntegral
  
  public static Apfloat cosIntegral(Apfloat x) throws ArithmeticException, ApfloatRuntimeException
  
  Cosine integral.
  
  Parameters:
  
  x - The argument.
  
  Returns:
  
  Ci(x)
  
  Throws:
  
  ArithmeticException - If x is nonpositive.
  
  ApfloatRuntimeException
  
  Since:
  
  1.13.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- sinhIntegral
  
  public static Apfloat sinhIntegral(Apfloat x) throws ApfloatRuntimeException
  
  Hyperbolic sine integral.
  
  Parameters:
  
  x - The argument.
  
  Returns:
  
  Shi(x)
  
  Throws:
  
  ApfloatRuntimeException
  
  Since:
  
  1.13.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- coshIntegral
  
  public static Apfloat coshIntegral(Apfloat x) throws ArithmeticException, ApfloatRuntimeException
  
  Hyperbolic cosine integral.
  
  Parameters:
  
  x - The argument.
  
  Returns:
  
  Chi(x)
  
  Throws:
  
  ArithmeticException - If x is nonpositive.
  
  ApfloatRuntimeException
  
  Since:
  
  1.13.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- airyAi
  
  public static Apfloat airyAi(Apfloat x) throws ApfloatRuntimeException
  
  Airy function Ai.
  
  Parameters:
  
  x - The argument.
  
  Returns:
  
  Ai(x)
  
  Throws:
  
  InfiniteExpansionException - If x is zero.
  
  ApfloatRuntimeException
  
  Since:
  
  1.13.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- airyAiPrime
  
  public static Apfloat airyAiPrime(Apfloat x) throws ApfloatRuntimeException
  
  Derivative of the Airy function Ai.
  
  Parameters:
  
  x - The argument.
  
  Returns:
  
  Ai′(x)
  
  Throws:
  
  InfiniteExpansionException - If x is zero.
  
  ApfloatRuntimeException
  
  Since:
  
  1.13.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- airyBi
  
  public static Apfloat airyBi(Apfloat x) throws ApfloatRuntimeException
  
  Airy function Bi.
  
  Parameters:
  
  x - The argument.
  
  Returns:
  
  Bi(x)
  
  Throws:
  
  InfiniteExpansionException - If x is zero.
  
  ApfloatRuntimeException
  
  Since:
  
  1.13.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- airyBiPrime
  
  public static Apfloat airyBiPrime(Apfloat x) throws ApfloatRuntimeException
  
  Derivative of the Airy function Bi.
  
  Parameters:
  
  x - The argument.
  
  Returns:
  
  Bi′(x)
  
  Throws:
  
  InfiniteExpansionException - If x is zero.
  
  ApfloatRuntimeException
  
  Since:
  
  1.13.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- besselJ
  
  public static Apfloat besselJ(Apfloat ν, Apfloat x) throws ArithmeticException, ApfloatRuntimeException
  
  Bessel function of the first kind.
  
  Parameters:
  
  ν - The order.
  
  x - The argument.
  
  Returns:
  
  J_ν(x)
  
  Throws:
  
  ArithmeticException - If ν is < 0 and ν is not an integer and x is zero. Also if ν is not an integer and x is < 0.
  
  ApfloatRuntimeException
  
  Since:
  
  1.13.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- besselI
  
  public static Apfloat besselI(Apfloat ν, Apfloat x) throws ArithmeticException, ApfloatRuntimeException
  
  Modified Bessel function of the first kind.
  
  Parameters:
  
  ν - The order.
  
  x - The argument.
  
  Returns:
  
  I_ν(x)
  
  Throws:
  
  ArithmeticException - If ν is < 0 and ν is not an integer and x is zero. Also if ν is not an integer and x is < 0.
  
  ApfloatRuntimeException
  
  Since:
  
  1.13.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- besselY
  
  public static Apfloat besselY(Apfloat ν, Apfloat x) throws ArithmeticException, ApfloatRuntimeException
  
  Bessel function of the second kind.
  
  Parameters:
  
  ν - The order.
  
  x - The argument.
  
  Returns:
  
  Y_ν(x)
  
  Throws:
  
  ArithmeticException - If x is ≤ 0.
  
  ApfloatRuntimeException
  
  Since:
  
  1.13.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- besselK
  
  public static Apfloat besselK(Apfloat ν, Apfloat x) throws ArithmeticException, ApfloatRuntimeException
  
  Modified Bessel function of the second kind.
  
  Parameters:
  
  ν - The order.
  
  x - The argument.
  
  Returns:
  
  K_ν(x)
  
  Throws:
  
  ArithmeticException - If x is ≤ 0.
  
  ApfloatRuntimeException
  
  Since:
  
  1.13.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- ellipticK
  
  public static Apfloat ellipticK(Apfloat x) throws ArithmeticException, ApfloatRuntimeException
  
  Complete elliptic integral of the first kind.
  Note that this function uses the definition: $K (x) ⩵ \int_{0}^{\frac{π}{2}} \frac{1}{\sqrt{1 - x \sin^{2} (t)}} ⅆ t$
  
  Parameters:
  
  x - The argument.
  
  Returns:
  
  K(x)
  
  Throws:
  
  InfiniteExpansionException - If x is zero.
  
  ArithmeticException - If x is ≥ 1.
  
  ApfloatRuntimeException
  
  Since:
  
  1.13.0
- ellipticE
  
  public static Apfloat ellipticE(Apfloat x) throws ArithmeticException, ApfloatRuntimeException
  
  Complete elliptic integral of the second kind.
  Note that this function uses the definition: $E (x) ⩵ \int_{0}^{\frac{π}{2}} \sqrt{1 - x \sin^{2} (t)} ⅆ t$
  
  Parameters:
  
  x - The argument.
  
  Returns:
  
  E(x)
  
  Throws:
  
  InfiniteExpansionException - If x is zero.
  
  ArithmeticException - If x is > 1.
  
  ApfloatRuntimeException
  
  Since:
  
  1.13.0
- hermiteH
  
  public static Apfloat hermiteH(Apfloat ν, Apfloat x) throws ApfloatRuntimeException
  
  Hermite function. For integer values of ν gives the Hermite polynomial.
  
  Parameters:
  
  ν - The first argument.
  
  x - The second argument.
  
  Returns:
  
  H_ν(x)
  
  Throws:
  
  ApfloatRuntimeException
  
  Since:
  
  1.14.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- laguerreL
  
  public static Apfloat laguerreL(Apfloat ν, Apfloat x) throws ApfloatRuntimeException
  
  Laguerre function. For integer values of ν gives the Laguerre polynomial.
  
  Parameters:
  
  ν - The first argument.
  
  x - The second argument.
  
  Returns:
  
  L_ν(x)
  
  Throws:
  
  ApfloatRuntimeException
  
  Since:
  
  1.14.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- laguerreL
  
  public static Apfloat laguerreL(Apfloat ν, Apfloat λ, Apfloat x) throws ApfloatRuntimeException
  
  Generalized Laguerre function. For integer values of ν gives the generalized Laguerre polynomial.
  
  Parameters:
  
  ν - The first argument.
  
  λ - The second argument.
  
  x - The third argument.
  
  Returns:
  
  L_ν^λ(x)
  
  Throws:
  
  ApfloatRuntimeException
  
  Since:
  
  1.14.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- legendreP
  
  public static Apfloat legendreP(Apfloat ν, Apfloat x) throws ArithmeticException, ApfloatRuntimeException
  
  Legendre function. For integer values of ν gives the Legendre polynomial.
  
  Parameters:
  
  ν - The first argument.
  
  x - The second argument.
  
  Returns:
  
  P_ν(x)
  
  Throws:
  
  ArithmeticException - If ν is not an integer and x ≤ -1.
  
  ApfloatRuntimeException
  
  Since:
  
  1.14.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- legendreP
  
  public static Apfloat legendreP(Apfloat ν, Apfloat μ, Apfloat x) throws ArithmeticException, ApfloatRuntimeException
  
  Associated Legendre function of the first kind. Gives Legendre functions of type 2.
  
  Parameters:
  
  ν - The first argument.
  
  μ - The second argument.
  
  x - The third argument.
  
  Returns:
  
  P_ν^μ(x)
  
  Throws:
  
  ArithmeticException - If x is ≤ -1 or ≥ 1 and ν or μ is not an integer or μ is not even or μ is not positive and -μ ≤ ν < μ.
  
  ApfloatRuntimeException
  
  Since:
  
  1.14.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- legendreQ
  
  public static Apfloat legendreQ(Apfloat ν, Apfloat x) throws ArithmeticException, ApfloatRuntimeException
  
  Legendre function of the second kind.
  
  Parameters:
  
  ν - The first argument.
  
  x - The second argument.
  
  Returns:
  
  Q_ν(x)
  
  Throws:
  
  ArithmeticException - If x is ≥ 1 or ≤ -1.
  
  ApfloatRuntimeException
  
  Since:
  
  1.14.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- legendreQ
  
  public static Apfloat legendreQ(Apfloat ν, Apfloat μ, Apfloat x) throws ArithmeticException, ApfloatRuntimeException
  
  Associated Legendre function of the second kind. Gives Legendre functions of type 2.
  
  Parameters:
  
  ν - The first argument.
  
  μ - The second argument.
  
  x - The third argument.
  
  Returns:
  
  Q_ν^μ(x)
  
  Throws:
  
  ArithmeticException - If x is ≥ 1 or ≤ -1.
  
  ApfloatRuntimeException
  
  Since:
  
  1.14.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- chebyshevT
  
  public static Apfloat chebyshevT(Apfloat ν, Apfloat x) throws ArithmeticException, ApfloatRuntimeException
  
  Chebyshev function of the first kind. For integer values of ν gives the Chebyshev polynomial of the first kind.
  
  Parameters:
  
  ν - The first argument.
  
  x - The second argument.
  
  Returns:
  
  T_ν(x)
  
  Throws:
  
  ArithmeticException - If x is < -1 and ν is not an integer.
  
  ApfloatRuntimeException
  
  Since:
  
  1.14.0
- chebyshevU
  
  public static Apfloat chebyshevU(Apfloat ν, Apfloat x) throws ArithmeticException, ApfloatRuntimeException
  
  Chebyshev function of the second kind. For integer values of ν gives the Chebyshev polynomial of the second kind.
  
  Parameters:
  
  ν - The first argument.
  
  x - The second argument.
  
  Returns:
  
  U_ν(x)
  
  Throws:
  
  ArithmeticException - If x is ≤ -1 and ν is not an integer.
  
  ApfloatRuntimeException
  
  Since:
  
  1.14.0
- gegenbauerC
  
  public static Apfloat gegenbauerC(Apfloat ν, Apfloat x) throws ArithmeticException, ApfloatRuntimeException
  
  Renormalized Gegenbauer function.
  
  Parameters:
  
  ν - The first argument.
  
  x - The second argument.
  
  Returns:
  
  C_ν⁽⁰⁾(x)
  
  Throws:
  
  ArithmeticException - If ν is zero. Also if x is < -1 and ν is not an integer.
  
  ApfloatRuntimeException
  
  Since:
  
  1.14.0
- gegenbauerC
  
  public static Apfloat gegenbauerC(Apfloat ν, Apfloat λ, Apfloat x) throws ArithmeticException, ApfloatRuntimeException
  
  Gegenbauer function. For nonnegative integer values of ν gives the Gegenbauer polynomial.
  
  Parameters:
  
  ν - The first argument.
  
  λ - The second argument.
  
  x - The third argument.
  
  Returns:
  
  C_ν^λ(x)
  
  Throws:
  
  ArithmeticException - If x is < -1 and ν is not an integer. Also if x is -1 and λ is > 1/2. Also if x is -1 and λ is 1/2 and ν is not an integer.
  
  ApfloatRuntimeException
  
  Since:
  
  1.14.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- jacobiP
  
  public static Apfloat jacobiP(Apfloat ν, Apfloat a, Apfloat b, Apfloat x) throws ArithmeticException, ApfloatRuntimeException
  
  Jacobi function. For nonnegative integer values of ν gives the Jacobi polynomial.
  
  Parameters:
  
  ν - The first argument.
  
  a - The second argument.
  
  b - The third argument.
  
  x - The fourth argument.
  
  Returns:
  
  P_ν^(a,b)(x)
  
  Throws:
  
  ArithmeticException - If ν is not a positive integer and either x is -1 and b is > 0 or x is < -1. Also if ν + a is a negative integer and ν is not an integer.
  
  ApfloatRuntimeException
  
  Since:
  
  1.14.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- fibonacci
  
  public static Apfloat fibonacci(Apfloat ν, Apfloat x) throws ApfloatRuntimeException
  
  Fibonacci function. For nonnegative integer values of ν gives the Fibonacci polynomial.
  
  Parameters:
  
  ν - The first argument.
  
  x - The second argument.
  
  Returns:
  
  F_ν(x)
  
  Throws:
  
  ApfloatRuntimeException
  
  Since:
  
  1.14.0
- eulerE
  
  public static Apfloat eulerE(long n, Apfloat x) throws IllegalArgumentException, ApfloatRuntimeException
  
  Euler polynomial.
  
  Parameters:
  
  n - The first argument.
  
  x - The second argument.
  
  Returns:
  
  E_n(x)
  
  Throws:
  
  IllegalArgumentException - If n < 0.
  
  ApfloatRuntimeException
  
  Since:
  
  1.14.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- bernoulliB
  
  public static Apfloat bernoulliB(long n, Apfloat x) throws IllegalArgumentException, ApfloatRuntimeException
  
  Bernoulli polynomial.
  
  Parameters:
  
  n - The first argument.
  
  x - The second argument.
  
  Returns:
  
  B_n(x)
  
  Throws:
  
  IllegalArgumentException - If n < 0.
  
  ApfloatRuntimeException
  
  Since:
  
  1.14.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- harmonicNumber
  
  public static Apfloat harmonicNumber(Apfloat x) throws ArithmeticException, ApfloatRuntimeException
  
  Harmonic number.
  
  Parameters:
  
  x - The argument.
  
  Returns:
  
  H_x
  
  Throws:
  
  ArithmeticException - If x is a negative integer.
  
  ApfloatRuntimeException
  
  Since:
  
  1.14.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- harmonicNumber
  
  public static Apfloat harmonicNumber(Apfloat x, Apfloat r) throws ArithmeticException, ApfloatRuntimeException
  
  Generalized harmonic number.
  
  Parameters:
  
  x - The first argument.
  
  r - The second argument.
  
  Returns:
  
  H_x^(r)
  
  Throws:
  
  ArithmeticException - If x is a negative integer, unless r is a nonpositive integer. Also if x is < -1 and r is not an integer.
  
  ApfloatRuntimeException
  
  Since:
  
  1.14.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- polylog
  
  public static Apfloat polylog(Apfloat ν, Apfloat x) throws ArithmeticException, ApfloatRuntimeException
  
  Polylogarithm.
  
  Parameters:
  
  ν - The first argument.
  
  x - The second argument.
  
  Returns:
  
  Li_ν(x)
  
  Throws:
  
  ArithmeticException - If ν is ≤ 1 and x is 1 or if x is > 1.
  
  ApfloatRuntimeException
  
  Since:
  
  1.14.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- logisticSigmoid
  
  public static Apfloat logisticSigmoid(Apfloat x) throws ApfloatRuntimeException
  
  Logistic sigmoid.
  
  Parameters:
  
  x - The argument.
  
  Returns:
  
  σ(x)
  
  Throws:
  
  ApfloatRuntimeException
  
  Since:
  
  1.14.0
- random
  
  public static Apfloat random(long digits)
  
  Generates a random number. Uses the default radix. Returned values are chosen pseudorandomly with (approximately) uniform distribution from the range 0 ≤ x < 1. The generated random numbers may have leading zeros and may thus not always have exactly the requested number of significant digits. The precision of the numbers is the requested number of digits minus the number of leading zeros. Trailing zeros do not affect the precision.
  
  Parameters:
  
  digits - Maximum number of digits in the number.
  
  Returns:
  
  A random number, uniformly distributed between 0 ≤ x < 1.
  
  Throws:
  
  NumberFormatException - If the default radix is not valid.
  
  IllegalArgumentException - In case the number of specified digits is invalid.
  
  Since:
  
  1.9.0
- random
  
  public static Apfloat random(long digits, int radix)
  
  Generates a random number. Returned values are chosen pseudorandomly with (approximately) uniform distribution from the range 0 ≤ x < 1. The generated random numbers may have leading zeros and may thus not always have exactly the requested number of significant digits. The precision of the numbers is the requested number of digits minus the number of leading zeros. Trailing zeros do not affect the precision.
  
  Parameters:
  
  digits - Maximum number of digits in the number.
  
  radix - The radix in which the number should be generated.
  
  Returns:
  
  A random number, uniformly distributed between 0 ≤ x < 1, in base radix.
  
  Throws:
  
  NumberFormatException - If the radix is not valid.
  
  IllegalArgumentException - In case the number of specified digits is invalid.
  
  Since:
  
  1.9.0
- randomGaussian
  
  public static Apfloat randomGaussian(long digits)
  
  Generates a random, Gaussian ("normally") distributed number value with mean 0 and standard deviation 1. Uses the default radix.
  
  Parameters:
  
  digits - Maximum number of digits in the number.
  
  Returns:
  
  A random number, Gaussian ("normally") distributed with mean 0 and standard deviation 1.
  
  Throws:
  
  NumberFormatException - If the default radix is not valid.
  
  IllegalArgumentException - In case the number of specified digits is invalid.
  
  Since:
  
  1.9.0
- randomGaussian
  
  public static Apfloat randomGaussian(long digits, int radix)
  
  Generates a random, Gaussian ("normally") distributed number value with mean 0 and standard deviation 1. Uses the default radix.
  
  Parameters:
  
  digits - Maximum number of digits in the number.
  
  radix - The radix in which the number should be generated.
  
  Returns:
  
  A random number, Gaussian ("normally") distributed with mean 0 and standard deviation 1.
  
  Throws:
  
  NumberFormatException - If the radix is not valid.
  
  IllegalArgumentException - In case the number of specified digits is invalid.
  
  Since:
  
  1.9.0
- continuedFraction
  
  public static Apint[] continuedFraction(Apfloat x, int n)
  
  Generates the first n terms in the continued fraction representation of x.
  Note that the result length might be less than n, depending on the input value and precision. The last terms could be incorrect due to accumulating round-off errors.
  
  Parameters:
  
  x - The number whose continued fraction terms should be generated.
  
  n - The maximum number of terms to generate.
  
  Returns:
  
  The continued fraction.
  
  Throws:
  
  IllegalArgumentException - If n is less than one.
  
  Since:
  
  1.12.0
- convergents
  
  public static Aprational[] convergents(Apfloat x, int n)
  
  Generates the first n convergents corresponding to the continued fraction of x.
  Note that the result length might be less than n, depending on the input value and precision. The last convergents could be incorrect due to accumulating round-off errors.
  
  Parameters:
  
  x - The number whose continued fraction convergents should be generated.
  
  n - The maximum number of convergents to generate.
  
  Returns:
  
  The convergents.
  
  Throws:
  
  IllegalArgumentException - If n is less than one.
  
  Since:
  
  1.12.0
- max
  
  public static Apfloat max(Apfloat x, Apfloat y)
  
  Returns the greater of the two values.
  
  Parameters:
  
  x - An argument.
  
  y - Another argument.
  
  Returns:
  
  The greater of the two values.
  
  Since:
  
  1.9.0
- min
  
  public static Apfloat min(Apfloat x, Apfloat y)
  
  Returns the smaller of the two values.
  
  Parameters:
  
  x - An argument.
  
  y - Another argument.
  
  Returns:
  
  The smaller of the two values.
  
  Since:
  
  1.9.0
- nextAfter
  
  public static Apfloat nextAfter(Apfloat start, Apfloat direction)
  
  Returns the number adjacent to the first argument in the direction of the second argument, considering the scale and precision of the first argument. If the precision of the first argument is infinite, the first argument is returned. If both arguments compare as equal then the first argument is returned.
  
  Parameters:
  
  start - The starting value.
  
  direction - Value indicating which of start's neighbors or start should be returned.
  
  Returns:
  
  The number adjacent to start in the direction of direction.
  
  Since:
  
  1.10.0
- nextUp
  
  public static Apfloat nextUp(Apfloat x)
  
  Returns the number adjacent to the argument in the direction of positive infinity, considering the scale and precision of the argument. If the precision of the argument is infinite, the argument is returned.
  
  Parameters:
  
  x - The starting value.
  
  Returns:
  
  The adjacent value closer to positive infinity.
  
  Since:
  
  1.10.0
- nextDown
  
  public static Apfloat nextDown(Apfloat x)
  
  Returns the number adjacent to the argument in the direction of negative infinity, considering the scale and precision of the argument. If the precision of the argument is infinite, the argument is returned.
  
  Parameters:
  
  x - The starting value.
  
  Returns:
  
  The adjacent value closer to negative infinity.
  
  Since:
  
  1.10.0
- ulp
  
  public static Apfloat ulp(Apfloat x)
  
  Returns the unit in the last place of the argument, considering the scale and precision. This is same as the difference between the argument and the value returned from nextUp(Apfloat). If the precision of the argument is infinite, zero is returned.
  For example, ulp of 1. is 1, ulp of 1.1 is 0.1 and ulp of 1.001 is 0.001 (considering significant digits only).
  
  Parameters:
  
  x - The argument.
  
  Returns:
  
  The ulp of the argument.
  
  Since:
  
  1.10.0

Class ApfloatMath

Method Summary

Methods inherited from class java.lang.Object

Method Details

pow

sqrt

cbrt

root

inverseRoot

inverseRoot

inverseRoot

inverseRoot

floor

ceil

truncate

frac

round

roundToPrecision

roundToInteger

roundToPlaces

roundToMultiple

negate

abs

copySign

scale

modf

fmod

multiplyAdd

multiplySubtract

agm

pi

pi

log

log

logRadix

exp

pow

acosh

asinh

atanh

cosh

sinh

tanh

acos

asin

atan

atan2

cos

sin

tan

sinc

w

toDegrees

toRadians

product

sum

e

e

euler

euler

catalan

catalan

glaisher

glaisher

khinchin

khinchin

gamma

gamma

gamma

logGamma

digamma

polygamma

beta

beta

beta

pochhammer

binomial

zeta

zeta

hypergeometric0F1