Package org.apfloat

Class ApcomplexMath

java.lang.Object

org.apfloat.ApcomplexMath

public class ApcomplexMath
extends Object

Various mathematical functions for arbitrary precision complex numbers.

Version:

1.10.0

Author:

Mikko Tommila

See Also:

ApfloatMath

Method Summary

Modifier and Type	Method	Description
static Apfloat	abs(Apcomplex z)	Absolute value.
static Apcomplex	acos(Apcomplex z)	Inverse cosine.
static Apcomplex	acosh(Apcomplex z)	Inverse hyperbolic cosine.
static Apcomplex	agm(Apcomplex a, Apcomplex b)	Arithmetic-geometric mean.
static Apcomplex[]	allRoots(Apcomplex z, int n)	All values of the positive integer root.
static Apfloat	arg(Apcomplex z)	Angle of the complex vector in the complex plane.
static Apcomplex	asin(Apcomplex z)	Inverse sine.
static Apcomplex	asinh(Apcomplex z)	Inverse hyperbolic sine.
static Apcomplex	atan(Apcomplex z)	Inverse tangent.
static Apcomplex	atanh(Apcomplex z)	Inverse hyperbolic tangent.
static Apcomplex	cbrt(Apcomplex z)	Cube root.
static Apcomplex	cos(Apcomplex z)	Cosine.
static Apcomplex	cosh(Apcomplex z)	Hyperbolic cosine.
static Apcomplex	exp(Apcomplex z)	Exponent function.
static Apcomplex	gamma(Apcomplex z)	Gamma function.
static Apcomplex	gamma(Apcomplex a, Apcomplex z)	Incomplete gamma function.
static Apcomplex	gamma(Apcomplex a, Apcomplex z0, Apcomplex z1)	Generalized incomplete gamma function.
static Apcomplex	inverseRoot(Apcomplex z, long n)	Inverse positive integer root.
static Apcomplex	inverseRoot(Apcomplex z, long n, long k)	Inverse positive integer root.
static Apcomplex	log(Apcomplex z)	Natural logarithm.
static Apcomplex	log(Apcomplex z, Apcomplex w)	Logarithm in arbitrary base.
static Apcomplex	negate(Apcomplex z)	Deprecated. Use Apcomplex.negate().
static Apfloat	norm(Apcomplex z)	Norm.
static Apcomplex	pow(Apcomplex z, long n)	Integer power.
static Apcomplex	pow(Apcomplex z, Apcomplex w)	Arbitrary power.
static Apcomplex	product(Apcomplex... z)	Product of numbers.
static Apcomplex	root(Apcomplex z, long n)	Positive integer root.
static Apcomplex	root(Apcomplex z, long n, long k)	Positive integer root.
static Apcomplex	scale(Apcomplex z, long scale)	Multiply by a power of the radix.
static Apcomplex	sin(Apcomplex z)	Sine.
static Apcomplex	sinh(Apcomplex z)	Hyperbolic sine.
static Apcomplex	sqrt(Apcomplex z)	Square root.
static Apcomplex	sum(Apcomplex... z)	Sum of numbers.
static Apcomplex	tan(Apcomplex z)	Tangent.
static Apcomplex	tanh(Apcomplex z)	Hyperbolic tangent.
static Apfloat	ulp(Apcomplex z)	Returns the unit in the last place of the argument, considering the scale and precision.
static Apcomplex	w(Apcomplex z)	Lambert W function.
static Apcomplex	w(Apcomplex z, long k)	Lambert W function for the specified branch.

Methods inherited from class java.lang.Object

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

Method Details

negate

@Deprecated
public static Apcomplex negate(Apcomplex z)
                        throws ApfloatRuntimeException

Deprecated.

Use Apcomplex.negate().

Negative value.

Parameters:

z - The argument.

Returns:

-z.

Throws:

ApfloatRuntimeException

abs

public static Apfloat abs(Apcomplex z)
                   throws ApfloatRuntimeException

Absolute value.

Parameters:

z - The argument.

Returns:

sqrt(x2 + y2), where z = x + i y.

Throws:

ApfloatRuntimeException

norm

public static Apfloat norm(Apcomplex z)
                    throws ApfloatRuntimeException

Norm. Square of the magnitude.

Parameters:

z - The argument.

Returns:

x2 + y2, where z = x + i y.

Throws:

ApfloatRuntimeException

arg

public static Apfloat arg(Apcomplex z)
                   throws ArithmeticException,
ApfloatRuntimeException

Angle of the complex vector in the complex plane.

Parameters:

z - The argument.

Returns:

arctan(y / x) from the appropriate branch, where z = x + i y.

Throws:

ArithmeticException - If z is zero.

ApfloatRuntimeException

scale

public static Apcomplex scale(Apcomplex z,
 long scale)
                       throws ApfloatRuntimeException

Multiply by a power of the radix.

Parameters:

z - The argument.

scale - The scaling factor.

Returns:

z * z.radix()scale.

Throws:

ApfloatRuntimeException

pow

public static Apcomplex pow(Apcomplex z,
 long n)
                     throws ArithmeticException,
ApfloatRuntimeException

Integer power.

Parameters:

z - Base of the power operator.

n - Exponent of the power operator.

Returns:

z to the n:th power, that is zn.

Throws:

ArithmeticException - If both z and n are zero.

ApfloatRuntimeException

sqrt

public static Apcomplex sqrt(Apcomplex z)
                      throws ApfloatRuntimeException

Square root.

Parameters:

z - The argument.

Returns:

Square root of z.

Throws:

ApfloatRuntimeException

cbrt

public static Apcomplex cbrt(Apcomplex z)
                      throws ApfloatRuntimeException

Cube root.

Parameters:

z - The argument.

Returns:

Cube root of z.

Throws:

ApfloatRuntimeException

root

public static Apcomplex root(Apcomplex z,
 long n)
                      throws ArithmeticException,
ApfloatRuntimeException

Positive integer root. The branch that has the smallest angle and same sign of imaginary part as z is always chosen.

Parameters:

z - The argument.

n - Which root to take.

Returns:

n:th root of z, that is z1/n.

Throws:

ArithmeticException - If n is zero.

ApfloatRuntimeException

root

public static Apcomplex root(Apcomplex z,
 long n,
 long k)
                      throws ArithmeticException,
ApfloatRuntimeException

Positive integer root. The specified branch counting from the smallest angle and same sign of imaginary part as z is chosen.

Parameters:

z - The argument.

n - Which root to take.

k - Which branch to take.

Returns:

n:th root of z, that is z1/nei2πsk/n where s is the signum of the imaginary part of z.

Throws:

ArithmeticException - If n is zero.

ApfloatRuntimeException

Since:

1.5

inverseRoot

public static Apcomplex inverseRoot(Apcomplex z,
 long n)
                             throws ArithmeticException,
ApfloatRuntimeException

Inverse positive integer root. The branch that has the smallest angle and different sign of imaginary part than z is always chosen.

Parameters:

z - The argument.

n - Which inverse root to take.

Returns:

Inverse n:th root of z, that is z-1/n.

Throws:

ArithmeticException - If z or n is zero.

ApfloatRuntimeException

inverseRoot

public static Apcomplex inverseRoot(Apcomplex z,
 long n,
 long k)
                             throws ArithmeticException,
ApfloatRuntimeException

Inverse positive integer root. The specified branch counting from the smallest angle and different sign of imaginary part than z is chosen.

Parameters:

z - The argument.

n - Which inverse root to take.

k - Which branch to take.

Returns:

Inverse n:th root of z, that is z-1/ne-i2πk/n.

Throws:

ArithmeticException - If z or n is zero.

ApfloatRuntimeException

allRoots

public static Apcomplex[] allRoots(Apcomplex z,
 int n)
                            throws ArithmeticException,
ApfloatRuntimeException

All values of the positive integer root.

Returns all of the n values of the root, in the order of the angle, starting from the smallest angle and same sign of imaginary part as z.

Parameters:

z - The argument.

n - Which root to take.

Returns:

All values of the n:th root of z, that is z1/n, in the order of the angle.

Throws:

ArithmeticException - If n is zero.

ApfloatRuntimeException

Since:

1.5

agm

public static Apcomplex agm(Apcomplex a,
 Apcomplex b)
                     throws ApfloatRuntimeException

Arithmetic-geometric mean.

Parameters:

a - First argument.

b - Second argument.

Returns:

Arithmetic-geometric mean of a and b.

Throws:

ApfloatRuntimeException

log

public static Apcomplex log(Apcomplex z)
                     throws ArithmeticException,
ApfloatRuntimeException

Natural logarithm.

The logarithm is calculated using the arithmetic-geometric mean. See the Borweins' book for the formula.

Parameters:

z - The argument.

Returns:

Natural logarithm of z.

Throws:

ArithmeticException - If z is zero.

ApfloatRuntimeException

log

public static Apcomplex log(Apcomplex z,
 Apcomplex w)
                     throws ArithmeticException,
ApfloatRuntimeException

Logarithm in arbitrary base.

Parameters:

z - The argument.

w - The base.

Returns:

Base-w logarithm of z.

Throws:

ArithmeticException - If z or w is zero.

ApfloatRuntimeException

Since:

1.6

exp

public static Apcomplex exp(Apcomplex z)
                     throws ApfloatRuntimeException

Exponent function. Calculated using Newton's iteration for the inverse of logarithm.

Parameters:

z - The argument.

Returns:

ez.

Throws:

ApfloatRuntimeException

pow

public static Apcomplex pow(Apcomplex z,
 Apcomplex w)
                     throws ApfloatRuntimeException

Arbitrary power. Calculated using log() and exp().

Parameters:

z - The base.

w - The exponent.

Returns:

zw.

Throws:

ArithmeticException - If both z and w are zero.

ApfloatRuntimeException

acos

public static Apcomplex acos(Apcomplex z)
                      throws ApfloatRuntimeException

Inverse cosine. Calculated using log().

Parameters:

z - The argument.

Returns:

Inverse cosine of z.

Throws:

ApfloatRuntimeException

acosh

public static Apcomplex acosh(Apcomplex z)
                       throws ApfloatRuntimeException

Inverse hyperbolic cosine. Calculated using log().

Parameters:

z - The argument.

Returns:

Inverse hyperbolic cosine of z.

Throws:

ApfloatRuntimeException

asin

public static Apcomplex asin(Apcomplex z)
                      throws ApfloatRuntimeException

Inverse sine. Calculated using log().

Parameters:

z - The argument.

Returns:

Inverse sine of z.

Throws:

ApfloatRuntimeException

asinh

public static Apcomplex asinh(Apcomplex z)
                       throws ApfloatRuntimeException

Inverse hyperbolic sine. Calculated using log().

Parameters:

z - The argument.

Returns:

Inverse hyperbolic sine of z.

Throws:

ApfloatRuntimeException

atan

public static Apcomplex atan(Apcomplex z)
                      throws ArithmeticException,
ApfloatRuntimeException

Inverse tangent. Calculated using log().

Parameters:

z - The argument.

Returns:

Inverse tangent of z.

Throws:

ArithmeticException - If z == i.

ApfloatRuntimeException

atanh

public static Apcomplex atanh(Apcomplex z)
                       throws ArithmeticException,
ApfloatRuntimeException

Inverse hyperbolic tangent. Calculated using log().

Parameters:

z - The argument.

Returns:

Inverse hyperbolic tangent of z.

Throws:

ArithmeticException - If z is 1 or -1.

ApfloatRuntimeException

cos

public static Apcomplex cos(Apcomplex z)
                     throws ApfloatRuntimeException

Cosine. Calculated using exp().

Parameters:

z - The argument.

Returns:

Cosine of z.

Throws:

ApfloatRuntimeException

cosh

public static Apcomplex cosh(Apcomplex z)
                      throws ApfloatRuntimeException

Hyperbolic cosine. Calculated using exp().

Parameters:

z - The argument.

Returns:

Hyperbolic cosine of z.

Throws:

ApfloatRuntimeException

sin

public static Apcomplex sin(Apcomplex z)
                     throws ApfloatRuntimeException

Sine. Calculated using exp().

Parameters:

z - The argument.

Returns:

Sine of z.

Throws:

ApfloatRuntimeException

sinh

public static Apcomplex sinh(Apcomplex z)
                      throws ApfloatRuntimeException

Hyperbolic sine. Calculated using exp().

Parameters:

z - The argument.

Returns:

Hyperbolic sine of z.

Throws:

ApfloatRuntimeException

tan

public static Apcomplex tan(Apcomplex z)
                     throws ArithmeticException,
ApfloatRuntimeException

Tangent. Calculated using exp().

Parameters:

z - The argument.

Returns:

Tangent of z.

Throws:

ArithmeticException - If z is π/2 + n π where n is an integer.

ApfloatRuntimeException

tanh

public static Apcomplex tanh(Apcomplex z)
                      throws ArithmeticException,
ApfloatRuntimeException

Hyperbolic tangent. Calculated using exp().

Parameters:

z - The argument.

Returns:

Hyperbolic tangent of z.

Throws:

ArithmeticException - If z is i (π/2 + n π) where n is an integer.

ApfloatRuntimeException

w

public static Apcomplex w(Apcomplex z)
                   throws ApfloatRuntimeException

Lambert W function. The W function gives the solution to the equation W eW = z. Also known as the product logarithm.

This function gives the solution to the principal branch, W₀.

Parameters:

z - The argument.

Returns:

W0(z).

Throws:

ApfloatRuntimeException

Since:

1.8.0

w

public static Apcomplex w(Apcomplex z,
 long k)
                   throws ArithmeticException,
ApfloatRuntimeException

Lambert W function for the specified branch.

Parameters:

z - The argument.

k - The branch.

Returns:

Wk(z).

Throws:

ArithmeticException - If z is zero and k is not zero.

ApfloatRuntimeException

Since:

1.8.0

See Also:

w(Apcomplex)

product

public static Apcomplex product(Apcomplex... z)
                         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:

z - The argument(s).

Returns:

The product of the given numbers.

Throws:

ApfloatRuntimeException

Since:

1.3

sum

public static Apcomplex sum(Apcomplex... z)
                     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:

z - The argument(s).

Returns:

The sum of the given numbers.

Throws:

ApfloatRuntimeException

Since:

1.3

gamma

public static Apcomplex gamma(Apcomplex z)
                       throws ArithmeticException,
ApfloatRuntimeException

Gamma function.

This implementation is slow, meaning that it isn't a fast algorithm. The asymptotic complexity is something like 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.

Parameters:

z - The argument.

Returns:

Γ(z)

Throws:

ArithmeticException - If z is a nonpositive integer.

ApfloatRuntimeException

Since:

1.9.0

gamma

public static Apcomplex gamma(Apcomplex a,
 Apcomplex z)
                       throws ArithmeticException,
ApfloatRuntimeException

Incomplete gamma function.

This implementation is slow, meaning that it isn't a fast algorithm. The asymptotic complexity is something like 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 incomplete gamma function.

Parameters:

a - The first argument.

z - The second argument.

Returns:

Γ(a, z)

Throws:

ArithmeticException - If the real part of a is nonpositive and z is zero.

ApfloatRuntimeException

Since:

1.10.0

gamma

public static Apcomplex gamma(Apcomplex a,
 Apcomplex z0,
 Apcomplex z1)
                       throws ArithmeticException,
ApfloatRuntimeException

Generalized incomplete gamma function.

This function is defined as: Γ(a, z0, z1) = Γ(a, z0) - Γ(a, z1)

The lower gamma function can be calculated with: γ(a, z) = Γ(a, 0, z)

This implementation is slow, meaning that it isn't a fast algorithm. The asymptotic complexity is something like 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 incomplete gamma function.

Parameters:

a - The first argument.

z0 - The second argument.

z1 - The third argument.

Returns:

Γ(a, z0, z1)

Throws:

ArithmeticException - If the real part of a is nonpositive and either z0 or z1 is zero. For the lower gamma function if a is a nonpositive integer.

ApfloatRuntimeException

Since:

1.10.0

ulp

public static Apfloat ulp(Apcomplex z)

Returns the unit in the last place of the argument, considering the scale and precision. This is maximum of the ulps of the real and imaginary part of the argument. If the precision of the argument is infinite, zero is returned.

Parameters:

z - The argument.

Returns:

The ulp of the argument.

Since:

1.10.0